A ROBOTIC MUSCLE SPINDLE:
NEUROMECHANICS OF INDIVIDUAL AND ENSEMBLE RESPONSE
by
Kristen Nicole Jaax
A dissertation submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
University of Washington
2001
Program Authorized to Offer Degree: Department of Bioengineering
 Copyright 2001
Kristen Nicole Jaax
University of Washington
Graduate School
This is to certify that I have examined this copy of a doctoral dissertation by
Kristen Nicole Jaax
and have found that it is complete and satisfactory in all respects, and that any and all
revisions required by the final examining committee have been made.
Chair of Supervisory Committee:
Blake Hannaford
Reading Committee:
Blake Hannaford
Martin Kushmerick
Francis Spelman
Date:
In presenting this dissertation in partial fulfillment of the requirements for the Doctoral
degree at the University of Washington, I agree that the Library shall make its copies
freely available for inspection. I further agree that the extensive copying of the
dissertation is allowable only for scholarly purposes, consistent with “fair use” as
prescribed in the U.S. Copyright Law. Requests for copying of reproduction of this
dissertation may be referred to Bell and Howell Information and Learning, 300 North
Zeeb Road, Ann Arbor, MI 48106-1346, to whom the author has granted “the right to
reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies
of the manuscript made from microform.”
Signature
Date
University of Washington
Abstract
A Robotic Muscle Spindle:
Neuromechanics of Individual and Ensemble Response
By Kristen Nicole Jaax
Chairperson of Supervisory Committee:
Professor Blake Hannaford
Department of Electrical Engineering and Department of Bioengineering (adjunct)
A mechatronic structural model of the mammalian muscle spindle Ia response was developed and used to
investigate neuromechanical mechanisms contributing to individual spindle dynamics and the information
content of spindle ensemble response. Engineering specifications were derived from displacement,
receptor potential and Ia data in the muscle spindle literature, allowing reproduction of core muscle
spindle behavior directly in hardware. A linear actuator controlled by a software muscle model replicated
intrafusal contractile behavior; a cantilever-based transducer reproduced sensory membrane
depolarization; a voltage-controlled oscillator encoded strain into a frequency signal. Results of
engineering tests met all performance specifications. Data from the biological literature was used first to
tune the model against 5 measures of ramp and hold response, then to validate the fully tuned model
against ramp and hold, sinusoidal and fusimotor response experiments. The response with dynamic or
static fusimotor input was excellent across all studies. The passive spindle response matched well in 5 of
9 measures. Dynamic intramuscular strain data from 28 locations on the surface of a contracting rat
medial gastrocnemius was sent sequentially through the model to reconstruct the Ia ensemble response of
a large population of muscle spindles. Results showed that under dynamic fusimotor stimulation, the
ensemble significantly increased Ia correlation to whole muscle kinematic inputs and that homogeneously
distributed dynamic fusimotor stimulation increased Ia ensemble correlation to muscle velocity in a dosedependent manner. Proposed mechanisms include decorrelation of spindle noise by intramuscular strain
inhomogeneities and fusimotor-dependent noise and nonlinear gains, as well as fusimotor-dependent
velocity selectivity. Potential applications for the robotic model include basic science motor control
research and applied research in prosthetics and robotics.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
TABLE OF CONTENTS
List of Figures................................................................................................................ iii
List of Tables ...................................................................................................................v
Chapter 1: Introduction .................................................................................................1
1.1
Problem Statement ........................................................................................1
1.2
Specific Aims ................................................................................................2
1.3
Dissertation Overview...................................................................................3
Chapter 2: Literature Review........................................................................................5
2.1
The Mammalian Muscle Spindle ..................................................................5
2.1.1
Overview...................................................................................................5
2.1.2
Intrafusal Muscle.......................................................................................7
2.1.3
Neural Transduction and Encoding...........................................................8
2.2
Muscle Spindle Modeling .............................................................................8
2.2.1
Intrafusal Muscle Models..........................................................................8
2.2.2
Transducer and Encoder Models.............................................................14
2.2.3
Biorobotic Models...................................................................................15
2.3
Muscle Spindle Ensemble Response...........................................................16
2.3.1
Ensemble Information Content ...............................................................17
2.3.2
Experimental Data...................................................................................17
2.3.3
Modeling .................................................................................................18
Chapter 3: Mechatronic Design of an Actuated Biomimetic Length and
Velocity Sensor ..........................................................................................20
3.1
Abstract .......................................................................................................20
3.2
Introduction .................................................................................................21
3.2.1
Background .............................................................................................22
3.3
Methods .......................................................................................................23
3.3.1
Design .....................................................................................................23
3.3.2
Implementation .......................................................................................24
3.3.3
Linear Positioning Device.......................................................................27
3.3.4
Modeling .................................................................................................28
3.4
Results .........................................................................................................30
3.4.1
Actuator Performance .............................................................................30
3.4.2
Transducer and Encoder Calibration.......................................................31
3.4.3
Linear Positioning Device Performance .................................................32
3.4.4
Integrated Performance ...........................................................................32
3.5
Discussion ...................................................................................................34
i
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
Chapter 4: A Biorobotic Structural Model of the Mammalian Muscle Spindle
Primary Afferent Response ......................................................................37
4.1
Abstract .......................................................................................................37
4.2
Introduction .................................................................................................37
4.2.1
Prior Literature........................................................................................39
4.2.2
Approach.................................................................................................41
4.3
Methods .......................................................................................................42
4.3.1
Design .....................................................................................................42
4.3.2
Experimental Methods ............................................................................49
4.4
Results .........................................................................................................50
4.4.1
Model Tuning Studies.............................................................................50
4.4.2
Model Validation Studies........................................................................55
4.5
Discussion ...................................................................................................58
4.5.1
Model Tuning..........................................................................................59
4.5.2
Model Validation ....................................................................................65
4.5.3
Summary of Contributions......................................................................69
Chapter 5: Fusimotor Effect on Signal Information Content of Ia Ensemble
Model Reconstructed from Dynamic Intramuscular Strain Data........72
5.1
Summary .....................................................................................................72
5.2
Introduction .................................................................................................73
5.3
Methods .......................................................................................................76
5.3.1
Collecting Local Muscle Fiber Strain Data.............................................76
5.3.2
Calculating Muscle Spindle Ensemble Response ...................................78
5.3.3
Data Analysis: Nonlinearity of Spindle Ensemble Output. ....................79
5.3.4
Data Analysis: Effect of Fixed Fusimotor Stimulation Rate ..................80
5.4
Results .........................................................................................................81
5.4.1
Local Strain Data ....................................................................................81
5.4.2
Ensemble Reconstruction........................................................................82
5.4.3
Nonlinearity of Spindle Ensemble Output. .............................................84
5.4.4
Effect of Fixed Fusimotor Stimulation Rate ...........................................86
5.5
Discussion ...................................................................................................87
5.5.1
Reconstructing the Ensemble Response .................................................88
5.5.2
Effect of Ensemble on Kinematic Information Content .........................91
5.5.3
Dose-Dependent Effect of Homogeneous Fusimotor Stimulation .........91
5.5.4
Conclusions.............................................................................................93
Chapter 6: Conclusions ................................................................................................94
6.1
Summary .....................................................................................................94
6.2
Future Work ................................................................................................96
Bibliography ................................................................................................................100
Appendix A: Technical Drawings..............................................................................112
ii
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
LIST OF FIGURES
Figure 2.1: Mammalian muscle spindle...........................................................................6
Figure 2.2: Artificial Muscle Spindle ............................................................................16
Figure 2.3: Robotic Muscle Spindle ..............................................................................16
Figure 3.1: CAD model of biomimetic sensor...............................................................20
Figure 3.2: Mammalian muscle spindle anatomy ..........................................................22
Figure 3.3: Linear actuator and transducer assembly ....................................................24
Figure 3.4: Transducer platform ....................................................................................25
Figure 3.5: CAD drawing of transducer platform..........................................................26
Figure 3.6: Encoder circuit diagram ..............................................................................27
Figure 3.7: Linear Positioning Device and Robotic Muscle Spindle.............................28
Figure 3.8: Time response of linear actuator ................................................................31
Figure 3.9: Calibration plots for transducer and encoder...............................................31
Figure 3.10: Waveform of frequency modulated square wave ......................................32
Figure 3.11: Time response of LPD...............................................................................32
Figure 3.12: Test of integrated engineering hardware ...................................................33
Figure 3.13: Effect of ramp speed and γ mn input on robotic Ia Response during 6 mm
amplitude ramp and hold.........................................................................................34
Figure 3.14: Comparison of robotic and biological Ia response to sinusoidal stretch
input. .......................................................................................................................35
Figure 4.1: Mammalian muscle spindle.........................................................................39
Figure 4.2: CAD drawing of sensory element design....................................................44
Figure 4.3: CAD drawing of linear actuator design.......................................................45
Figure 4.4: Block diagram of linear actuator controller.................................................47
Figure 4.5: Model parameter tuning study. Ia output metrics during ramp and hold
experiment...............................................................................................................52
Figure 4.6: Model parameter tuning study. Comparison of Ia responses during ramp
and hold input..........................................................................................................53
Figure 4.7: Model parameter tuning study. Sensory region stretch during ramp and
hold stretch applied across whole muscle spindle ..................................................54
Figure 4.8: Completed model validation study. Comparison of Ia response to ramp and
hold position input ..................................................................................................55
Figure 4.9: Completed model validation study. Comparison of depth of modulation of
Ia output in response to varying amplitude of sinusoidal stretch input ..................57
Figure 4.10: Completed model validation study. Comparison of effect of varying γmn
stimulation level on Ia response..............................................................................58
Figure 5.1: Location of 28 markers on surface of rat medial gastrocnemius muscle
fibers used to reconstruct mechanical input to 28 hypothetical muscle spindles....81
Figure 5.2: Dynamic strains recorded at the 28 markers shown in Figure 5.1 ..............82
iii
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
Figure 5.3: Sequence of 28 displacement trajectories laid out in manner in which they
were physically applied to muscle spindle model...................................................82
Figure 5.4: Comparison of ensemble response to kinematic inputs ..............................83
Figure 5.5: Correlation coefficients for multiple regression on whole muscle position
and velocity.............................................................................................................85
Figure 5.6: Correlation between ensemble response and whole muscle velocity under
dynamic fusimotor stimulation ...............................................................................86
Figure 5.7: Correlation between ensemble response and whole muscle position under
static fusimotor stimulation.....................................................................................88
iv
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
LIST OF TABLES
Table 2.1: Anatomical elements included in nonlinear structural models .....................10
Table 2.2: Processes modeled in structural models to generate neural output ..............14
Table 4.1: Parameter values changed during tuning of robotic muscle spindle ............51
v
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
ACKNOWLEDGEMENTS
My thanks go first to my parents, Jim and Suzanne Jaax, whose support and
encouragement of my science education has spanned twenty-three years. Suzanne has
served as an expert proofreader throughout those years, including this dissertation and
the technical papers it has produced.
My love and gratitude go to Ryan Campbell, who served as my sounding board, my
resident computer expert, and in the final months of this dissertation, my right hand.
His help in finding voice recognition software, formatting this dissertation and
performing the many two-handed tasks of daily living were invaluable in completing
this dissertation.
I wish to thank Prof. Blake Hannaford for providing both the guidance and the freedom
we graduate students needed to develop as independent researchers. I also wish to
thank my committee members, Prof. Francis Spelman, Prof. Martin Kushmerick, Prof.
Deirdre Meldrum and Prof. Peter Detwiler for contributing their time and wisdom to the
completion of this dissertation.
My thanks goes out to the members of the BioRobotics Lab for making this journey fun
every step of the way. In particular I wish to thank Glenn Klute, Dan Ferris, Thavida
Maneewarn and Steven Venema. Their informal mentoring was a cornerstone of my
education.
My collaborators have been instrumental in the development of this dissertation.
Pierre-Henry Marbot laid the foundation for this work and has been a source of
knowledge and experience throughout the building of the robotic muscle spindle. C.C.
van Donkelaar and M.R. Drost allowed my vision of the ensemble study to become a
vi
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
reality by making available their intramuscular strain data, collected at a remarkable
seventy sites on the surface of the muscle.
Finally, I am grateful to the Whitaker Graduate Fellowship Program and the University
of Washington Medical Scientist Training Program for their financial support.
vii
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
DEDICATION
To my parents, Jim and Suzanne Jaax, for their unwavering dedication to
& enthusiasm for my education.
viii
1
Chapter 1:
Introduction
1.1
Problem Statement
The biological sensor responsible for measuring muscle length, the muscle spindle, is a
complex neuromuscular organ. The focus of the muscle spindle is a central sensory
region whose strains determine the muscle spindle output, the Ia response. Surrounding
the sensory region, the muscle spindle has an internal muscle, the intrafusal muscle,
whose sole purpose is to mechanically filter the spindle’s position and velocity inputs
thereby shaping the character of the strains that reach the central sensory region. The
muscle spindle also has a dedicated input from the central nervous system (CNS), the
gamma motorneuron, whose sole function is to modulate the intrafusal muscle’s
mechanical properties, creating a way for the organism to actively control the
mechanical filtering of the intrafusal muscle. The complexity of these systems shows
that the organism devotes substantial neurological and muscular resources toward the
goal of controlling the shape of the signal it receives from these sensors. From this
expenditure of resources, one would assume that the spindle’s response is carefully
sculpted to maximize information content. Yet these sensors are known to be noisy and
nonlinear transducers, raising the question of how the central nervous system extracts a
decipherable signal from their response [1].
Recent advances in technology have allowed researchers to implement the mechanisms
of integrated physiological systems, such as the muscle spindle, in robotic hardware.
This field, known as biorobotics, has grown rapidly on the principle that engineers in
the field of robotics are often trying to find solutions to problems that have already been
solved in physiological systems. Biologists, meanwhile, are often working on the
problem of unraveling the mechanisms underlying these same systems. Biorobotics
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
2
brings together the knowledge and experience of both fields to address these shared
problems by using engineering hardware to recreate mechanisms used in biology. In
developing this technology, bioroboticists address several interrelated goals: increasing
understanding of biological systems, discovering novel solutions for engineering
problems and developing components for prosthetic devices. The muscle spindle, with
its complex neuromechanical systems and unusual transducer behavior, is an ideal
candidate for such an approach.
The problem, then, is two-fold. First, to advance the state-of-the-art in biorobotics by
developing the technology needed to accurately reproduce the behavior of the muscle
spindle in precision engineering hardware. Second, to apply the biorobotic muscle
spindle model to the basic science question of whether the CNS could use the ensemble
response of a population of muscle spindles to extract a more decipherable signal of
muscle length and velocity from its muscle spindles.
1.2
Specific Aims
The Specific Aims of this dissertation include:
1) To identify the core neural and mechanical elements of the muscle spindle
necessary to elicit their characteristic Ia response and develop precision
engineering hardware capable of replicating the performance of these core
elements.
2) To integrate the core engineering components into a structural model of the
individual muscle spindle Ia response.
3) To tune and validate the physiological faithfulness of the individual muscle
spindle model against biological Ia data from the literature, proposing
modifications to the underlying biological mechanisms when supported by
evidence from model behavior as well as data from the biological literature.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
3
4) To use the mechatronic model to reconstruct the ensemble response that would
be generated by a population of muscle spindles residing in a single muscle
body.
5) To test whether the ensemble model’s response exhibits increased correlation to
mechanical inputs, length and velocity, as compared to the individual response.
6) To test whether gamma motorneuron stimulation, when homogeneously
distributed across a population of muscle spindles, improves the correlation of
the ensemble model’s response to position or velocity.
1.3
Dissertation Overview
The research content of this dissertation is organized as distinct chapters written in a
manner suitable for independent publication. The chapters collectively describe the
development of a robotic muscle spindle, starting with engineering hardware
development and concluding with a reconstruction of the response of a population of
muscle spindles.
Chapter 2 reviews background and literature pertinent to this dissertation.
Chapter 3, entitled “Mechatronic Design of an Actuated Biomimetic Length and
Velocity Sensor,” describes the engineering aspects of the robotic muscle spindle’s
development. This chapter describes the design and implementation of mechatronic
systems created to capture the behavior of the core elements of the muscle spindle’s
anatomy and physiology. Engineering performance data are presented as well as tests
of the integrated system to demonstrate feasibility.
Chapter 4, entitled “A Biorobotic Structural Model of the Mammalian Muscle Spindle
Primary Afferent Response,” describes the biological modeling aspects of the robotic
muscle spindle’s development. The methods used to integrate the engineering hardware
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
4
into a biological model are described including a modification to current muscle spindle
theory proposed during the tuning process. Results from the tuning studies are
presented as well as the performance of the fully tuned model in a battery of validation
experiments. Biological data from the literature accompany all results to facilitate
comparison.
Chapter 5, entitled “Fusimotor Effect on Signal Information Content of Ia Ensemble
Model Reconstructed from Dynamic Intramuscular Strain Data,” describes the
reconstruction of the ensemble response of a population of muscle spindles.
Collaborators provided data describing the mechanical strains experienced at 28
locations on the surface of a muscle during the course of a muscular contraction. These
data were used to reconstruct the ensemble response of a hypothetical population of 28
muscle spindles. Using this novel methodology, this chapter investigates the
information content of the ensemble response including the influence of the fusimotor
system. The chapter concludes by proposing neuromechanical mechanisms to explain
the observed behavior.
Chapter 6 summarizes the major findings of Chapters 3-5. It also suggests directions for
future work both in muscle spindle physiology and in the development and application
of biorobotic length and velocity sensors.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
5
Chapter 2:
Literature Review
The first section of this chapter describes the anatomy and physiology of the muscle
spindle as well as a review of pertinent literature on the physiology of these
mechanoreceptors. Section two reviews muscle spindle modeling efforts to date, with
emphasis on structural nonlinear models including the prototype artificial muscle
spindle. Section three reviews the topic of muscle spindle ensemble response including
theory, experimental data and modeling efforts.
2.1
The Mammalian Muscle Spindle
2.1.1 Overview
The mammalian muscle spindle, shown in Figure 2.1, resides in the body of its host
muscle. The fusiform-shaped organ consists of long muscle fibers that run the length of
the spindle. Those fibers are called intrafusal muscle fibers, and can be divided both
anatomically and functionally into the sensory region, in the center, and the contractile
region, lying at either end. The sensory region of these fibers is devoid of contractile
tissue, instead behaving in a spring-like manner. The muscle spindle has two types of
sensory nerve endings. Primary endings, or group Ia neurons, wrap around the sensory
region; secondary endings, group II neurons, terminate in endings called flower spray
endings that adhere to the intrafusal fiber. As the cell membranes of these nerve
endings are stretched, strain dependent ion channels in the membrane open. The
resulting flow of ions across the membrane causes local depolarization of the cell,
transducing the strain into an analog receptor potential. This receptor potential is then
encoded at the heminodes of the nerve ending, translating the analog potential into a
train of action potentials, or voltage spikes, whose frequency is proportional to the
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
6
applied strain. This frequency modulated spike train then travels down the Ia or II
nerve axon to the spinal cord.
The contractile regions of the muscle spindle
are essentially muscle fibers. They are filled
with actin and myosin, the force generating
proteins of muscles. These molecules are
aligned to generate tension along the long
axis of the spindle. The γ motorneuron, a
motor nerve fiber specific to muscle
spindles, transmits control signals to the
contractile region from the central nervous
system[2]. These commands govern the
contraction of the intrafusal fiber. The
kinematics of these contractile regions is
further modulated by the unique viscoelastic
properties of muscle tissue.
Figure 2.1: Mammalian muscle spindle
The function of the contractile regions is to
filter incoming displacements, thereby conditioning the nature of the signal reported by
the sensory transducer. An example of this filtering is to keep this central sensory
region taut as the host muscle changes lengths. Hence, if the CNS commands the
biceps to contract by 10%, a γ motorneuron can command the intrafusal fibers of the
bicep’s muscle spindles to also contract by 10%. More sophisticated filtering can be
achieved by driving the host muscle and intrafusal muscle fibers independently. A
simple example of this is to contract the intrafusal fiber by 12% instead of 10% in the
scenario described above, making the nerve endings very taut and thereby increasing
the gain. This is observed during uncertain kinematic situations, such as in a cat being
held by a human[3].
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
7
A further nuance of control in muscle spindles is the presence of three different classes
of fibers within a given muscle spindle, each classified as velocity or position sensitive.
The static nuclear bag and nuclear chain fibers are position sensitive, producing
primarily a response proportional to the magnitude of strain. The dynamic nuclear bag
fibers are velocity sensitive, producing a more complex output approximated by a
weighted sum of strain magnitude and its first derivative. The γ motorneuron system
has separate inputs to the static fibers and the dynamic fibers, allowing the central
nervous system to preferentially amplify the response of just one type of fiber.
Further information on the basic biology of muscle spindles can be found in Kandel et
al.[3], Gladden[4] or the exhaustive review by Hunt[5].
2.1.2 Intrafusal Muscle
Researchers have sought to identify which aspects of the muscle spindle’s behavior
arise from the mechanical properties of the intrafusal muscle since Matthews first
proposed that intrafusal muscles might be responsible for the muscle spindle’s position
and velocity sensitivity[6]. Ottoson and Shepherd opened the door to examining this
question directly by visually recording changes in intrafusal muscle length with
stroboscopic photomicroscopy[7], a technique which has been applied extensively in
the ensuing years[8-11]. This experimental work was driven by the investigation of
several hypotheses about intrafusal muscle mechanisms. Short-range stiffness is one of
the most popular, with the theory that many of the actin-myosin cross-bridges remain
bound during displacements <0.2%[12]. This results in a highly sensitive linear region
of Ia output during small displacements and the initial burst seen during ramp and
holds[13, 14]. Another theory tested with photomicroscopy data is stretch activation,
the theory that stretching a passive dynamic nuclear bag fiber will cause it to contract.
This theory was first proposed by Boyd[15] and has been controversial ever since. The
only definitive evidence for it was obtained with an experimental technique that
damaged the muscle spindle and is thus inconclusive[10, 16].
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
8
Researchers have shown that the distinctive mechanical properties associated with each
of the intrafusal muscle types correlate well to their respective myosin isoforms:
dynamic bag myosin is similar to slow tonic extrafusal fibers, static bag myosin is a
unique isoform similar to extrafusal slow twitch fibers, and nuclear chain fibers are a
fast form similar to developing muscle[4]. This provides support for the assumption
that the differing mechanical properties of the various intrafusal fibers are the result of
mechanisms similar to those of the well studied extrafusal muscle.
2.1.3 Neural Transduction and Encoding
Investigators have sought to understand the mechanisms causing the dependency of Ia
action potential frequency on the receptor potential rate of change. One possible
mechanism, a decrease in AP initiation threshold voltage during dynamic increases in
receptor potential, has been noted in three different studies[17-19]. To date, though, no
investigation has specifically addressed the possibility of unidirectional behavior in the
encoder’s rate dependency.
Researchers have also sought to isolate the source of various spindle behaviors to the
mechanical system vs. the sensory system. Many have concluded that almost all
behaviors, including short-range stiffness, must be mechanical in origin because they
are clearly parallel in the tension record and the receptor potential record[18, 20]. The
exception to this is undershoots which are deemed of chemical origin because they do
not appear in the tension record[14]. Although some work has been done in identifying
the types of ions that are involved in neural transduction, there are no data available on
specific ion channel types or numbers[21].
2.2
Muscle Spindle Modeling
2.2.1 Intrafusal Muscle Models
2.2.1.1 Linear Models
Since Matthew’s observation in the 1930’s that the position and velocity sensitivity of
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
9
the muscle spindle could be caused by mechanical properties of the intrafusal fiber,
virtually all structural (homeomorphic) muscle spindle models have attempted to
account for the mechanics of the intrafusal fiber. Early models ranged from 1st order to
6th order linear models. The linearity limited their applicability to small subsets of the
spindle’s behavior in which the output was known to be linear. One example is
Poppele’s 1970 model[22] which used system identification techniques to empirically
fit transfer functions to Bode plots of deefferented spindles’ responses to sinusoidal
length inputs in the spindle’s linear range. Through his experimental data he concluded
that primary and secondary afferents shared a common mechanical filtering system, but
differed in their transduction and encoding. Hence, he generated one common
mechanical filtering transfer function and two unique transduction/encoding transfer
functions. While the resulting transfer functions matched small amplitude sinusoidal
behavior quite well, the range was limited, applying only to deefferented muscle
spindles in their linear region.
In 1970, Rudjord introduced a structural linear model[23], one in which model elements
corresponded to selected physiological entities in the muscle spindle. The model
consisted of 2 fibers: a nuclear bag fiber and a nuclear chain fiber. Like the Poppele
model, it also generated both primary and secondary output. Modeling of fusimotor
input was omitted from early versions of the model, then included later under the
condition of constant length. Rudjord established an arrangement of springs and
dampers based on spindle anatomy and physiology. He then used experimental data
from the ubiquitous ramp and hold experiment to tune his model parameters. This
model achieved good performance in the small amplitude linear region, but lacked
general applicability.
A few of these early linear models took changes in fusimotor activation, not length, as
their input. Andersson et al.[24] presented such a model in 1968. This model
successfully reproduced spindle response to sinusoidal fusimotor activation of either the
gamma static or gamma dynamic system, but could not account for the nonlinearities
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
10
observed when both systems were stimulated simultaneously.
2.2.1.2 Nonlinear Models
Nonlinear models began appearing in 1981, with the introduction of three non-structural
empirical models with small ranges of applicability. The Houk model[25] defined
power laws capable of describing the spindle’s response during the constant velocity
phase of a ramp and hold after the initial burst has dissipated. Poppele and Quick
introduced a model very similar to his 1970 model[26], except that the transfer
functions were fit to experimental data generated by subjecting the spindle to bandlimited white noise inputs rather than slow sinusoids.
2.2.1.2.1 Hasan Model
In 1983, Hasan published the first structural nonlinear model, responding to the lack of
a comprehensive set of rules describing the dependence of firing output on stretch
input[27]. Table 2.1 gives an overview of the major anatomical features included in this
and the other nonlinear structural models.
Table 2.1: Anatomical elements included in nonlinear structural models
Model
Gamma Gamma
Ia
II
Dynamic Static Nuclear
Static Dynamic Fibers Fibers
Bag 1
Bag 2 Chain
Schaafsma
C
C
X
X
H
H
Winters
C
X
X
Hasan
D
D
X
X
X
Robotic
C
C
X
X
H
H
Where: C=continuous range of levels, D= discrete levels, X=explicitly included in
model, H=included in model as part of hybrid static fiber.
Hasan’s model of the Dynamic Nuclear Bag 1 fiber incorporated a nonlinear
mechanical filter component and a linear transducer/encoder component. Like all
models to date, the Hasan model treated the contractile tissue as an extrafusal muscle.
The transducer/encoder was implemented as a series elastic element. In theory, the
model accounted for both static and dynamic fusimotor input as well as primary and
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
11
secondary output. In practice, however, the model can only represent one fusimotor
state at a time, since the modelers recorded a set of experimental data in each of the 4
desired fusimotor states and retuned their constants to match each of the four states.
One feature that captures some of the complexity of muscle spindles is “resetting” of
short-range stiffness. Hence, if the spindle is stretched slowly enough, the sensitivity
can stay very high. The mathematical expression of the model cannot be solved
analytically and results are obtained through numerical simulation. The model enjoys a
broad applicability to a range of types of motion, successfully reproducing both
sinusoids and ramp and hold.
2.2.1.2.2 Schaafsma Model
Schaafsma et al. developed the most complete mechanical model introduced to
date[28]. Since its introduction in 1991, the same group has introduced three additional
submodels relating the neural aspects of spindle behavior. All of the models are based
on known micro-physiological or micro-anatomical concepts. The complete set of the
four submodels, the Integrated Model of the Mammalian Muscle Spindle, is described
in Otten et al.[29]. The Schaafsma submodel is discussed in this section while the
remaining submodels are presented below in the transducer/encoder section.
The Schaafsma muscle spindle model was the first model to structurally incorporate
fusimotor stimulation during dynamic length changes and widely varying testing
protocols. The model was founded on the belief that complex spindle behavior arises
from the mechanical interaction between the intrafusal muscle tissue and the sensory
region. The Schaafsma mechanical model models only the primary afferent fiber and
consists of two submodels: (a) Bag1: analogous to the dynamic bag 1 fiber, and
responsible for the spindle’s dynamic fusimotor response, and (b) Bag2, a composite of
the bag2 fiber and the nuclear chain fiber, and responsible for the spindle’s static
fusimotor response. Each submodel then consists of a sensory region in series with a
muscular region. The sensory region is represented mechanically as a simple linear
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
12
spring, while the muscular region is represented by an extrafusal muscle model
developed by Otten incorporating sensitivity to length, velocity and activation[30]. The
primary afferent output is computed as a simple linear function of the length of the
sensory region and its first derivative. At any given moment the model’s output is
either entirely due to the Bag1 model or the Bag2 model, whichever is larger. This is an
early implementation of the competitive pacemaking concept elaborated further in one
of the Otten group’s submodels[31]. Schaafsma has incorporated a short-range stiffness
model consisting of 100 fused cross-bridges that make the intrafusal muscle
indistensible until a force exceeding a threshold ruptures one or more of the crossbridges. Parameter values for the Schaafsma model were obtained via a parametric
search using metrics from experimental ramp and hold data from muscle spindles under
a variety of velocities and fusimotor activation levels.
This model has been also been adapted to mimic fusimotor driving of Ia output in
nuclear chain fibers[32]. The model was moderately successful, though it did not
capture some of the subtleties of the biological system such as robustness to length
change.
2.2.1.2.3 Winters Model
Two further models have been introduced since the Schaafsma model, both as parts of
models describing a larger segment of the neuromuscular control system. The Winters
model[33] was developed in the context of providing closed loop feedback for a largescale neuro-musculoskeletal model of the shoulder. Since he was focusing on posture
control studies, Winters chose to model the secondary afferent output of a static nuclear
bag fiber under static gamma motorneuron input. The basic structure of the model
consists of a contractile region in series with a series elastic region. The combined
elements are assumed to span the full length of the host extrafusal muscle. The
contractile element is essentially modeled as a shorter version of the host extrafusal
muscle model with a few basic modifications: (a) no damping, since this is a model of
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
13
static position sensing, (b) the force-length curve of the parallel elastic element has only
a positive slope, required for the stability of the system, (c) constant strain across the
length of the fiber. The muscle model is quite detailed, even including nerve activation
dynamics and calcium dynamics. The sensory element is modeled as a series elastic
element. Its sensitivity is distributed as a gaussian function with strain, with peak
sensitivity at mid strain. The secondary afferent output is computed as a function of the
sensory element’s length plus a very small function of the sensory element’s rate of
change. This model presents an interesting engineering based treatment of many of the
issues addressed in other spindle models.
2.2.1.2.4 Wallace Model
In 1996, Wallace and Kerr developed a model of the ensemble response of ten muscle
spindles, each from a different muscle[34]. He intentionally chose to use a simple
model of individual spindle response rather than a more detailed model such as
Schaafsma’s or Hasan’s.
Wallace’s model predicts primary and secondary afferent
output with no fusimotor input. It is based on Houk’s[27] empirical model which used
a power law to describe the spindle output during the constant velocity region of a ramp
and hold after the initial response has died out. Wallace augmented this model by
introducing a term causing the spindle to fall silent when shortening velocities dropped
below a threshold. He also removed the spindle length dependency of the output,
making it purely a function of velocity. His companion paper to the ensemble model
does a sensitivity analysis and concludes that, in the context of ensemble encoding, the
information transfer is independent of both the fractional power of velocity and absolute
firing levels of the afferents. He did, however, experiment with reintroducing length
sensitivity to the model with three different types of mathematical expressions and
found a small change in observed correlation coefficients resulting from the inclusion of
explicit length-dependent terms.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
14
2.2.2
Transducer and Encoder Models
Classically, the transducer and encoder functions have been lumped together, if they are
distinguished from the mechanical filtering function at all. Major models such as the
Hasan model and the original Schaafsma model depict the generator potential as simply
the stretch across the sensory region and the Ia firing rate as the sum of the generator
potential and its first derivative.
The recent models introduced by the Otten group have placed a new focus on the
transduction and encoding process. The contents of these models, as well as the neural
transduction and encoding aspects of the Schaafsma and Winters models, are shown in
Table 2.2.
Table 2.2: Processes modeled in structural models to generate neural output
Model
Otten Integrated Model
Schaafsma
Winters
Otten, K+ Conductance
Banks
Nerve
Strain
Nerve
Strain
Rate
Membrane
Depolarization
AP
Encoding
Pacemaker
Sites
Nerve
Modeled
X
X
X
X
X
X
X
X
X
X
X
Ia
Ia
II
Ia
Ia
X
The Otten transduction model[35] uses modified Frakenhauser-Huxley equations to
model ion channel dynamics, focusing on the impact of slow potassium conductance
channels. This model is able to account for many of the nonlinear phenomenon
attributed to the mechanical system including the slow decay during hold and the
silence upon release of holds. Although there are little data on ion channel composition
to validate such a model[36], its results suggest an interesting hypothesis worthy of
experimental investigation regarding how much the mechanical vs. neural systems
contribute to the overall dynamics of the muscle spindle.
The encoder models originated with an interesting study correlating histological data
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
15
and neurophysiological recordings. Banks et al.[31] showed that, depending on the
number of nodes of Ranvier separating two sensory endings, the two signals would
either electrotonically couple or exhibit competitive pacemaker interaction. In the first
case, when separated by one node of Ranvier or less, the signals from the separate
nodes would have an averaging effect, modeled as an analog resistive circuit. When
separated by two or more nodes of Ranvier, they instead interacted competitively, with
the faster node sending its action potential antidromically down to the adjacent nodes
thereby inhibiting their output. Otten’s group has implemented a hybrid of the
occlusion (pacemaker) submodel and the electrotonic coupling submodel in their
integrated model, determining the relative contribution of each submodel according to
the number of nodes of Ranvier separating the two sensory endings. Again, there are
few data available regarding pacemaker membrane kinetics and channel composition to
validate the results of such a model. However, it is a useful tool for suggesting new
experiments and as such is a promising step towards a more detailed understanding of
the encoding process.
2.2.3 Biorobotic Models
Biorobotic hardware is a new medium for muscle spindle modeling. In 1993, Marbot
and Hannaford [37, 38] presented the first prototype of a biorobotic muscle spindle
model, the Artificial Muscle Spindle. The device, shown in Figure 2.2, uses a lead
screw actuator for the mechanical filter, a strain gage for the sensory transducer, and an
onboard printed circuit board for encoding the transducer output into a frequency
modulated square wave. This model demonstrated well the feasibility of reproducing
muscle spindle behavior in engineering hardware in tests spanning a wide range of
experimental protocols.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
16
Figure 2.2: Artificial Muscle Spindle, prototype biorobotic muscle spindle.
The mechanical design of the Artificial Muscle Spindle was precarious, though,
resulting in problems with noise, repeatability and mechanical failure. Most notably,
the noise arising from the mechanical design was too great to allow inclusion of the
derivative term in the computation of muscle spindle Ia output. These design
limitations prevented the Artificial Muscle Spindle from fully implementing a structural
model of Ia response, limiting its applicability as a testbed for asking basic science
questions about motor control.
The robotic muscle spindle presented in this dissertation, Figure 2.3, extends this initial
hardware design by implementing a reengineered design in precision hardware and
validating each engineering subsystem against biological performance specifications.
The resulting design has alleviated the previous limitations and provides a robust
platform for modeling all aspects of spindle behavior.
Figure 2.3: Robotic Muscle Spindle
2.3
Muscle Spindle Ensemble Response
The ensemble response of muscle spindles is a relatively new field that has risen to
prominence during the 1990s. The information that can be extracted from a single
spindle’s Ia response is sharply limited by noise and nonlinearities[1]. As a result,
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
17
researchers look to the ensemble response of a population of spindles as a way for the
central nervous system to obtain a decipherable signal of muscle kinematics[39-46].
Unfortunately, the technical difficulty of recording from multiple muscle spindles from
a single muscle has limited the size of simultaneously recorded ensembles to
populations of ten spindles or less[41-44, 47]. This makes ensemble modeling an
attractive alternative, though to date only two models of spindle populations have been
published, both using simple models of individual spindle behavior to examine limb
position encoding by spindle populations spanning multiple muscles [34, 48].
2.3.1 Ensemble Information Content
The question of what parameters might increase ensemble information has received
considerable attention. Ensemble size, simultaneous recording and an intact fusimotor
system all have been shown to increase the ensemble’s ability to discriminate between
sinusoids of varying amplitude[41, 43]. The fusimotor system has been further
implicated as a mechanism by which ketamine application[44] and heteronymous
muscle fatigue[47] degrade ensemble information content. Several investigators have
raised the issue of decorrelating individual muscle spindle responses as a means to
improve spatial filtering of ensemble information content. Proposed mechanisms for
introducing the decorrelation include the fusimotor system behaving as a neural
network [43, 47], random noise introduced by the active fusimotor system [49] and
membrane firing threshold variability [50]. To date, though, such decorrelation
mechanisms have only been tested indirectly [47], theoretically [50] or in small
populations [49].
2.3.2 Experimental Data
Recording from populations of muscle spindles is a relatively new field, with the
majority of the work being done in the 1990s. The amount of experimental data is
limited by the difficulty of recording from a sufficient number of primary muscle
spindle afferents from a single muscle during a reproducible motor task[51]. Within
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
18
this body of work there are two subfields, intra- and intermuscular populations. These
two fields tend to be split into non-human vs. human data, respectively, due to increased
technical difficulties in recording from humans. There are also two further divisions in
the experimental literature: (a) those that record from one spindle at a time, accruing
“ensemble” data over sequential repetitions of the same behavior[51, 52] and (b) those
that record simultaneously from multiple Ia fibers[41, 43].
In terms of specific data, there are several studies by Bergenheim and Johansson
describing recordings of both simultaneous and sequential data from multiple muscle
spindles in anesthetized cats[41-43, 47]. Prochazka et al. set out to compile an
extensive “look-up chart” of data from muscle spindle ensembles during the cat step
cycle[52]. Since then, they have recorded firing profiles of 47 muscle afferents during
the cat step cycle. These data are a sequential recording under a similar scenario, as
opposed to a simultaneous recording, using 34 cats to collect data on the 47 muscle
spindles during free locomotion[53]. In terms of human data, the studies available are
limited to recordings of single muscle spindles from multiple muscles using the
microneurographic technique[54, 55].
2.3.3
Modeling
Two models have been published describing the response of a population of muscle
spindles. Both articles use populations across multiple muscles to examine limb
position encoding, emphasizing distribution of the spindles within the limb. The Scott
and Loeb[48] study focuses on the reasons underlying muscle spindle distribution
across the muscles of the human body. The study models the individual spindle
secondary response using a simple model which includes a sensory element and a noise
source. The variation between spindle outputs is based on the location of the host
muscle and the variation in the injected noise. Wallace and Kerr[34] present a model of
ensemble muscle spindle output using 10 different muscles with a single muscle spindle
per muscle. Spindle output is calculated as a power law of velocity, adapting the model
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
19
from the work of Houk et al.[25]. They are able to show that the ensemble metric,
calculated as the average of the individual spindle outputs, is well correlated with joint
angular velocity, but not joint angular position.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
20
Chapter 3:
Mechatronic Design of an Actuated
Biomimetic Length and Velocity Sensor
3.1
Abstract
Drawing from the rich source of proven and often novel mechanisms in the
biological realm, biomimetic sensors are being successfully developed for
many different transduction tasks. This paper presents such a sensor for
transducing displacements. Our sensor, Figure 3.1, is a robotic analog of
the biological muscle spindle, an actuated sensor which transduces muscle
displacement for kinesthetic awareness.
The mechanical filter exhibits the desired step response with Tr=26 msec,
Ts=54 msec, P.O. = 9.2%, Ess=6.8x10-3mm. The transducer possesses the
desired linear response with a sensitivity of 34nm/Hz. Finally, the encoder
circuitry successfully maps the millivolt output to a pulse frequency range
of 1150Hz to 12.5kHz. Results from integrated system tests show that
with a traditional engineering-based controller the sensor can successfully
detect errors in trajectory tracking introduced by both phase lag and
perturbations. With a physiologically-based controller, it successfully
replicates the major features of muscle spindle response. By physically
realizing the hypothesized core features of a biological muscle spindle in
engineering hardware, we have evoked the type of actuated sensor output
Figure 3.1:
CAD model
of
biomimetic
sensor
seen in the biological muscle spindle, a widely utilized tool of biological
motor control.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
21
3.2
Introduction
In biorobotics research, engineers and biologists come together to implement in steel
and silicon the researcher’s vision of the mechanisms driving a biological process.
Such a project is instructive for all parties involved. Biologists are able to test the
viability and coherency of the proposed mechanisms when challenged with the demands
of the physical world. Meanwhile, engineers are able to draw from this process novel
approaches to age-old tasks such as detecting the properties of the physical
environment.
This paper describes the development of such a biorobotic device, an actuated
biomimetic length and velocity sensor. The design is inspired by length and velocity
sensors found in mammalian muscle tissue called muscle spindles. These organs
contain a spring-like transducer region which lies in series with an internal actuator, the
intrafusal muscle. This tiny actuator receives motor commands from the central
nervous system (CNS), allowing the brain to actively modulate the nature of the output
of the transducer’s sensory region.
The development of an engineering implementation of these sensors poses the
following questions: What elements of the muscle spindle represent core functionality?
How are these functional elements best implemented to form a robust robotic sensor?
Finally, can a non-back-driveable electromechanical system yield the active filtering
and transduction behavior of living muscle and nervous tissue?
We thus address the following hypotheses:
(a) The core functions of a robotic length and velocity sensor based around a structural
model of muscle spindles are mechanical filtering, transduction and encoding. A sensor
which captures these methods can exhibit the type of response seen in muscle spindles.
(b) The electromechanical systems presented here are capable of achieving the
performance specifications necessary to match the physiology of mammalian muscle
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
22
spindles.
3.2.1 Background
The mammalian muscle spindle, shown in
Figure 3.2, consists of long muscle fibers,
called intrafusal fibers, which run the
length of the spindle. Each fiber contains a
sensory region, in the center, and an
actuator region, lying at either end. The
sensory region acts as a passive linear
elastic spring. Ia sensory nerve endings
wrap around these fibers and transduce
stretch of the sensory region into a
depolarization of their membrane.
Heminodes on the Ia axon then encode this
analog depolarization into a frequency
modulated spike train of action potentials
which travel up to the spinal cord.
The actuator region is essentially a normal
muscle fiber, controlled by the input of a
Figure 3.2: Mammalian muscle spindle
anatomy
dedicated signal from the spinal cord, the γ
motor neuron. The function of the actuator
region is to filter incoming displacements, thereby conditioning the nature of the signal
reported by the sensory transducer. The γ motor neuron control of the actuator’s force
production allows the CNS to finely control this process. For instance, it can raise the
sensor’s gain during uncertain kinematic situations by increasing contraction, thereby
increasing the stretch of the sensory region[56].
Several mathematical models of the muscle spindle have been developed[23, 27]. The
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
23
Schaafsma model [28] is one of the most sophisticated models, consisting of a nonlinear
extrafusal muscle in series with the non-contractile sensory element. This model, unlike
the many linear models, is able to approximate muscle spindle behavior for a wide
range of stimuli.
3.3
Methods
3.3.1 Design
Based on an earlier prototype[37, 38], we abstracted the three core elements of muscle
spindle function: mechanical filtering, transduction and encoding. The mechanical
filtering is performed in the biological muscle spindle by the contractile region of the
intrafusal fibers. Our goal is to create an internal actuator with performance
specifications sufficient to mimic intrafusal muscle dynamics. Based on the kinematics
observed in intrafusal muscle response during direct observations[10, 16], this requires
a rise time (Tr) <30 msec, settling time (Ts) <150 msec, percent overshoot (P.O.) =10%,
steady state error (Ess) =0.
The transduction role in the biological muscle spindle is performed by strain-sensitive
ion channels which cause depolarization of the Ia nerve membrane in direct proportion
to the strain applied across the sensory region[3]. The biological transducer exhibits a
resolution of better than 20µm[4] with linear output at small displacements, known as
short-range stiffness. With large displacements, the transducer stiffens, exhibiting
decreased sensitivity[4]. Our design goal is a transducer with similar resolution and a
large linear region, stiffening to lower sensitivity output at the end of its range.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
24
The final role, encoding, is performed at the Ia axon
heminodes in the biological spindle. The analog
depolarization of the transduction sites is translated into a
frequency modulated spike train with a range of
approximately 0 to 400 Hz. Our goal is to create an encoder
that also produces a frequency modulated spike train
proportional to the analog voltage of the transducer.
3.3.2
Implementation
3.3.2.1 Mechanical Filter
The mechanical filtering task is implemented with a low
inertia, direct drive lead screw linear actuator system, Figure
3.3, to achieve the rapid response times of the intrafusal
muscle. A miniature ironless core dc motor (1016-N-006,
MicroMo, Clearwater, FL) is coupled to a cold rolled
stainless steel 2-56 lead screw with a flexible helical
coupling. The system is mounted between a pair of
semicylindrical stainless steel guides with a Delrin AF
bushing aligning the tip of the lead screw and the guides.
The transducer element is mounted on a platform machined
from Delrin AF with a 2-56 thread tapped through its center.
This platform has integral linear bushings which ride in the
track formed between the two semicylindrical housings,
allowing the lead screw rotation to be transformed into linear
motion of the platform. Mounted onto the lead screw system,
the transducer platform forms the end point of the intrafusal
muscle implementation, which lies in series with the transducer.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
Figure 3.3: Linear
actuator and transducer
assembly
25
An encoder (HEM-1016-N-10, MicroMo, Clearwater, FL) is mounted directly to the
motor. Its quadrature signal is read by a dSPACE 1102 controller board (dSPACE
GmbH, Paderborn DE). The encoder data are filtered with a 25Hz 4th order digital
elliptic filter. A PID controller for the complete linear actuator was designed in
MATLAB and Simulink then implemented in C (Real Time Workshop, MathWorks).
3.3.2.2 Transducer
The transduction element, shown in
Figure 3.4 and Figure 3.5, lies in series
between the linear actuator and the distal
end of the position sensor. We
implemented high resolution transduction
between strain and analog voltage with a
pair of strain gaged cantilevers mounted
perpendicular to the axis of sensing. The
cantilevers are machined from stainless
steel shim stock 51 microns thick. An
aluminum stop 0.16 mm above the plane
of the cantilever, machined to a 6.6°
angle, is used to keep the cantilever’s
Figure 3.4: Transducer platform. Consists of
Delrin AF bushing and aluminum stop. Strain
gaged transducer is visible in the gap between
bushing and stop
deflection within its linear elastic range.
One uniaxial polyimide and constantan
alloy self-temperature compensated 120
Ohm strain gage (EA 06 031CF 120, Measurements Group, Raleigh, NC) is mounted to
the bottom surface of each cantilever. The dimensions and materials of the cantilever
were selected such that a maximum of +2000/-0 µstrain would be applied to the foil
matrix of the strain gages during deflection, giving a fatigue life of 108 cycles.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
26
Input displacements are
applied to the cantilevers by
means of nylon coated 3x7
stainless-steel cables.
These cables run through
guide holes both in the
Delrin AF bushing at the
distal end of the position
sensor as well as in the
aluminum stop immediately
adjacent to the cantilevers
to ensure robust and
Figure 3.5: CAD drawing of transducer platform.
repeatable performance,
unmarred by tangling of the cables in the lead screw. The tension of the cables is
transmitted to the cantilever by means of a steel compression sleeve crimped to form a
solid beam. The beam runs the width of the cantilever, thereby minimizing edge effects
on the strain gage film.
3.3.2.3 Encoder
The encoding of the analog voltage into a frequency modulated spike train is
implemented with surface mount integrated circuit (IC) chips on a printed circuit board
mounted directly to the sensor platform. The circuit, Figure 3.6, uses a Wheatstone
bridge configured as a half bridge and is zeroed by a 60 kΩ resistor in parallel with one
of the 120Ω bridge completion resistors. The resulting signal is then immediately
amplified with a gain of 430. The amplified signal is then sent into a 7555 IC, wired in
voltage controlled oscillator mode, resulting in a frequency modulated square wave with
a range of 1150 to 12500 Hz.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
27
R1
120
C5
22pF
R9
390k
VCC
7
R5
1k
LM 308
Strain Gage X
R10
10k
4
Strain Gage Y
6
R4a
150k
Out
5
R6
1k
R4b
100k
7555
2
C1
6.8nF
R2
120
R7a
470k
1
R7b
5100k
C4
1nF
R8a
470k
C3
1nF
R8b
5100k
Figure 3.6: Encoder circuit diagram. Strain sensed by strain gages generates a millivolt potential
across a Wheatstone Bridge. That signal is amplified (LM308 chip) then converted (7555 chip) to a
frequency modulated square wave.
3.3.3
Linear Positioning Device
We designed and built a linear positioning device (LPD), shown in Figure 3.7, to
provide position inputs to the robotic spindle. The actuator was based around a 5.25in
hard drive actuator. The rotary displacement of the precision hard drive motor is
converted to linear displacement by wrapping a metal ribbon 3.17mm wide and 0.10mm
thick around a metal drum rigidly mounted to the motor. A slot machined in the outer
circumference of the drum aligns the metal ribbon with the linear axis. The metal
ribbon is then rigidly mounted to the ball slide of a miniature linear guide. The motion
of this ball slide is defined as the linear position output of the actuator.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
28
5 cm
Figure 3.7: Linear Positioning Device and Robotic Muscle Spindle
To control the actuator’s position, we use an LVDT (LD100-20, Omega, Stamford, CT)
rigidly mounted to the LPD base such that its axis is parallel to the linear slide rail. The
aluminum core of the LVDT is then rigidly fixed to the ball slide. The data are filtered
with a 3rd order 40 Hz Butterworth filter. A separate PID controller was designed for
the LPD and implemented in the same dSPACE system.
To perform experiments on the robotic spindle, the cable from the robotic muscle
spindle is fixed directly to the ball slide of the LPD.
3.3.4
Modeling
3.3.4.1 Mechanical Filtering
The transfer function for position control of the linear actuator is:
X A ( s)
KmP
=
3
V IN ( s ) JLs + ( RJ + LB) s 2 + ( RB + K b K m ) s
( 3.1 )
Where J=inertia, L=motor inductance, R=motor resistance, B=damping, Kb=back EMF
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
29
constant, Km=torque constant, P=thread pitch, XA=linear actuator position and
VIN=motor voltage.
With the parameter values for our system inserted, the transfer function for position
control of the linear actuator is:
X A ( s)
217
=
mm / V
VIN (s) 2.4 *10−6 s 3 + 0.74s 2 + 10s
( 3.2 )
Our desired step response was Tr<30 msec, Ts<150 msec, P.O.=10%, Ess=0. A PID
controller was designed iteratively both in simulation and on the physical hardware to
meet these specifications. The resulting controller gains are KP=100, KI=10, KD=0.5.
This design gives the following theoretical step response: Tr=3.5 msec, Ts=18 msec,
PO=25%, Ess=0.
3.3.4.2 Transducer
We derived the linear relationship between displacement of the transducer, xC, and the
strain of the strain gages, ε, to be:
ε=
3C th ( L − d )
xC
2 L3
( 3.3 )
Where xC = overall input displacement, Cth = cantilever thickness, L = distance from
cantilever base to load, d=distance from cantilever base to center of strain gage.
We designed the transducer with values for L, d, and Cth such that the displacement
range, xC, yielded the desired 2000 µstrain at full-scale deflection. As this transducer is
analog, it has continuous resolution. Hence, the resolution goals were met. Finally, this
model shows the response is linear throughout the transducer’s primary range.
3.3.4.3 Encoder
Accounting for the interaction between the strain gage and the pulse generation circuit,
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
30
we derived the relationship between strain, ε, and frequency, F, as:

 5 R2

 2R
F = .69 R B C + (R B + RC )C ln  1

 5 R2

 R
 1


 ε *G 
 − 5 

 2 + ε *G 


 ε *G 
 − 5 

 2 + ε *G 

−1
( 3.4 )
Where: RB=resistor between 7555 VCC and discharge pins, RC=resistor between 7555
discharge and threshold pins, C=capacitor across 7555 trigger and ground pins, R2/R1 =
amplifier gain, G=gage factor of strain gages.
Values for these parameters were selected to give a frequency range of approximately
1kHz to 14kHz.
3.4
Results
3.4.1 Actuator Performance
Figure 3.8a shows the step response of the linear actuator. The performance metrics for
this step function are: Tr=26msec, Ts=54msec, PO=9.2%, Ess=6.8x10-3mm, which meets
our goal of: Tr<30 msec, Ts<150 msec, P.O.=10%, Ess~0.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
31
On a 30mm/sec ramp trajectory,
Figure 3.8b, the performance metrics
are P.O.=0.30% of absolute position,
overshoot=0.089mm, maximum
error=0.15mm, mean absolute Ess =
0.041mm.
3.4.2 Transducer and Encoder
Calibration
Figure 3.9 shows the combined
Figure 3.8: Time response of linear actuator
implementation of intrafusal muscle (solid line). (a) 1
mm step position input (dotted line), (b) 30 mm/sec
ramp position input (dotted line).
calibration of the transducer and
encoder systems. Calibration is
depicted between Displacement and
Frequency, Figure 3.9a, and Force
and Frequency, Figure 3.9b. In each,
the response is linear at small to
moderate displacements and forces,
followed by a region at the end of the
range exhibiting decreased
sensitivity, reflecting the design
specifications. Figure 3.10
demonstrates the waveform
generated by the encoder circuitry at
both the low and high ends of the
encoder’s working range.
Figure 3.9: Calibration plots for transducer and
encoder, (a) Frequency vs. Displacement, (b)
Frequency vs. Force
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
32
The observed range of strain across
the strain gages is 66µstrain1700µstrain, within the targeted 02000 µstrain range.
3.4.3
Linear Positioning Device
Performance
Figure 3.10: Waveform of frequency modulated
square wave at small sensor displacement (top graph)
and large sensor displacement (bottom graph).
Figure 3.11 shows the ramp response
of the linear positioning device
during a 6 mm/sec ramp and hold.
For PID controller values of P=10,
I=140, D=0.1, the 6mm/sec ramp
performance metrics are
P.O.=0.75%, overshoot=0.017mm,
and mean absolute Ess =0.018mm.
3.4.4 Integrated Performance
Figure 3.11: Time response of LPD (solid line) to
6mm/sec ramp and hold position input (dotted line).
To initially test the performance of
the three core elements as an
integrated system, we programmed our sensor and the LPD testing machine to move
with the same sinusoidal trajectory, separated only by a phase lead. The resulting
performance is shown in Figure 3.12.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
33
As is shown in the Figure 3.12, the
transducer produces a response
proportional to the “error” in the host
muscle’s displacement, as created by
the phase lead of the robotic sensor’s
movement. Like the biological
spindle, it only detects stretching and
not compression forces. Additionally,
the frequency response reflects any
transient perturbations between the
robotic sensor’s motion and the host
muscle’s motion. An example of this
Figure 3.12: Test of integrated engineering hardware.
(a) Trajectory of robotic sensor (solid line) and LPD
(dotted line) for phase lead of 20°°. (b) Frequency
output for phase leads of 8.6°°, 14.3°°, and 20.0°°.
is the local peaks produced at 1.95
sec when the LPD experiences
stiction and briefly deviates from the
sinusoidal trajectory.
Figure 3.13 and Figure 3.14 show the performance of the hardware elements following
full integration and validation with a physiologically based controller. Integration,
tuning and validation details are in Chapter 4. Figure 3.13 shows that the response of
the robotic system to variation in ramp velocities and γ motorneuron (γ mn) activation
levels is well tuned to match the current theory regarding muscle spindle behavior.
Position gain is independent of speed, but dependent on γ mn activation level, which
alters the properties of the linear actuator’s control algorithm. The velocity gain
produces a velocity-dependent offset during the ramps whose magnitude is dependent
on γ mn input rate. The noise exhibited is normally distributed with a standard
deviation of 10.5 Hz, which is typical of active biological muscle spindles which exhibit
normally distributed noise with a standard deviation of ~8 Hz[1]. The model’s time
domain sinusoidal response, Figure 3.14, shows good qualitative correspondence to the
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
34
biological response, including similarities in phase lead and relative amplitudes under
different γ mn levels. In the passive case, response amplitude varies from the biological
data, revealing a limitation of the device. Noise is absent in the biological cases
because these data are the average response of multiple trials.
3.5
Discussion
This paper presents a physically
realized robotic implementation of a
biological length and velocity sensor,
the mammalian muscle spindle. We
set out two hypotheses in this paper.
First, that a sensor that captured the
three core behaviors of mechanical
filtering, transduction and encoding
could exhibit the type of behavior
seen in muscle spindles. Second,
that the electromechanical devices
we selected to implement each of
these core functions could meet the
performance specifications necessary
to express each of these behaviors.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
Figure 3.13: Effect of ramp speed and γ mn input on
robotic Ia Response during 6 mm amplitude ramp and
hold. Left column: No γ mn input (passive); Middle
column: 100 Hz dynamic, 0 Hz static (dynamic); Right
column: 0 Hz dynamic, 100 Hz static (static).
35
Each of the three electromechanical
subsystems met the required
performance specifications to
replicate their biological analogs.
The linear actuator met the desired
time response criteria. The
transducer detected displacements
with the desired resolution and
linearity.
For the encoder subsystem, we met
Figure 3.14: Comparison of robotic and biological (cf.
Hulliger et al. 1977[57]) Ia response to sinusoidal
stretch input. Robotic response (top row) matches
phase lead and shape, but not amplitude, of cat soleus
muscle spindle response (middle row) to sinusoidal
position input (bottom row) under different γ mn
levels: Left column: 0 Hz dynamic, 0 Hz static, Center
column: 87Hz dynamic, 0 Hz static, Right column: 0
Hz dynamic, 100 Hz static. Lengths are reported as
displacements to cat soleus.
our desired frequency range,
although we had intentionally chosen
a range substantially different from
the biological encoder range. First of
all, we desired a frequency range of
several kilohertz, whereas the
biological encoder range is
approximately 0-400Hz. This increase in range is a consequence of needing to increase
sensitivity beyond that of biological muscle spindles. This was necessary because our
sensor will be used in a 1:1 ratio with the host muscle, while biological muscle spindles
are often found in much higher densities. Secondly, our displacement-frequency
relationship is the inverse of the biological spindle’s relationship: in our system,
increasing displacements lead to decreasing frequencies. This choice was made to
minimize the number of integrated circuit chips in the pulse generation circuitry. This,
in turn, allowed the circuit to be mounted directly to the transducer platform. Based on
the fact that all three subsystems met the performance specifications of their biological
analog, the second hypothesis is confirmed.
The results from our test of the integrated system support the first hypothesis as well.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
36
When the three elements are integrated, they produce an output proportional to the
positive displacement discrepancy between the actuator and the LPD, as seen in
biological muscle spindles. Further, they detect this equally well during low frequency
sinusoids and transient perturbations. When integrated using a physiologically-based
controller, the system is able to replicate the major features of the performance of the
full mammalian muscle spindle. Hence, we have shown that the first hypothesis is
correct, these three hardware subsystems are capable of exhibiting the type of sensing
behavior seen in biological muscle spindles.
In conclusion, we have implemented in mechatronic hardware a sensor which replicates
the transducer behavior of a biological length and velocity sensor, the muscle spindle.
Such a device has applications in basic science, as a testbed for studying motor control,
and in prosthetics, as a sensor which communicates in the language of the user’s motor
control system. The question remains, though, as to the suitability of such a device for
engineering applications. An actuated sensor for kinematic measurements such as this
is not commonly employed in engineering applications. We propose that such a system
might be advantageous in situations where the range of the actual transducer is limited,
or for real-time tuning of the sensor’s output to a variety of different kinematic
variables, e.g. length, velocity, or perturbations from a desired length.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
37
Chapter 4:
A Biorobotic Structural Model of the Mammalian Muscle
Spindle Primary Afferent Response
4.1
Abstract
A biorobotic model of the mammalian muscle spindle Ia response was implemented in
precision hardware. We derived engineering specifications from displacement, receptor
potential and Ia data in the muscle spindle literature, allowing reproduction of muscle
spindle behavior directly in the robot’s hardware; a linear actuator replicated intrafusal
contractile behavior, a cantilever-based transducer reproduced sensory membrane
depolarization, and a voltage-controlled oscillator encoded strain into a frequency
signal. Aspects of muscle spindle behavior not intrinsic to the physical design were
added in control software using an adaptation of Schaafsma’s mathematical model. We
tuned the response to biological ramp and hold metrics including peak, mean, dynamic
index, time domain response and sensory region displacement. The model was
validated against biological Ia response to ramp and holds, sinusoids and fusimotor
input. The response with dynamic or static gamma motorneuron input was excellent
across all studies. The passive spindle response matched well in 5 of the 9 measures.
Potential applications include basic science muscle spindle research and applied
research in prosthetics and robotics.
4.2
Introduction
Investigators have been studying the muscle spindle for many years, developing and
testing theories about the physiological origins of its unique transducer properties. One
means of testing these theories has been synthesizing them into a structural model, a set
of mathematical expressions that have direct analogs in the physiological system, to see
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
38
if they exhibit muscle spindle like behavior. While these structural models have offered
substantial insight into the physiology of the muscle spindle, they are limited by their
abstraction from the physical world. This barrier limits their ability to rigorously test
under strict adherence to all physical laws and to apply physically realistic experimental
inputs, e.g. limited bandwidth of stretch inputs. It also deprives them of the opportunity
to gain insights into the muscle spindle through physically implementing their theories
in hardware.
A number of researchers have recognized the potential of building models which span
that gap between idealized mathematical theory and the physical world. The models
these investigators have built implement hypotheses regarding biological mechanisms
on robotic hardware. The primary goal for this breed of biorobotics researchers is to
increase their understanding of biological mechanisms by testing the ability of their
proposed mechanisms to drive real systems replete with physical obstacles such as
friction and inertia. Spin-off applications, though, are inherent to the nature of such a
project. Biorobotic devices are attractive candidates for prosthetics as they are designed
to use the language of the body to replicate its behavior. These devices also offer novel
mechanisms for engineering applications.
The robotic muscle spindle project was thus conceived with the following objectives:
(a) implementing a state-of-the-art structural model in precision robotic hardware and
(b) testing the biological theories which drive the model by rigorously validating the
model’s behavior against biological data from a wide range of experimental protocols.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
39
4.2.1 Prior Literature
4.2.1.1 Biological Muscle Spindles
Static Nuclear
Bag Fiber
Dynamic
Nuclear Bag
Fiber
Nuclear Chain
Fibers
The mammalian muscle spindle, Figure
4.1, is a mechanoreceptor that resides in
the body of extrafusal muscle and
transduces muscle length. Intrafusal
muscle fibers span the length of the
Capsule
spindle and are divided anatomically and
functionally into a sensory region and a
contractile region, which lie in series. The
contractile region is a muscle fiber
aligned to generate tension along the long
axis of the spindle. The sensory region
is a linearly elastic spring devoid of
Ia
Neuron
Output
contractile tissue. Group Ia afferent
neurons wrap around the sensory region,
linearly transducing sensory region strain
into receptor potential. This analog
Gamma Motor
Neuron Input
potential is then encoded into an action
potential train, the Ia response, whose
frequency is thought to be a function of
Figure 4.1: Mammalian muscle spindle. Strain
applied across the organ is transduced into
primary (Ia) afferent output. Input from the γmn
contracts intrafusal fiber tissue at distal ends,
modulating the Ia response.
the receptor potential and its first
derivative[5, 14]. This frequency
modulated spike train then travels down
the Ia axon to the spinal cord. There are
three types of intrafusal fibers: static nuclear bag and nuclear chain fibers transduce
primarily position information while dynamic nuclear bag fibers transduce primarily
velocity information. Commands from the γ motorneuron (γmn) descend from the
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
40
spinal cord and control the contraction of the intrafusal muscle. Two types of γ
motorneurons exist, static and dynamic, which innervate position sensitive fibers and
velocity sensitive fibers, respectively.
B. H. C. Matthews first proposed in the 1930’s that the position and velocity sensitivity
of the muscle spindle could arise from the differing mechanical properties between the
intrafusal muscle and the sensory region[6]. Studies using stroboscopic
photomicroscopy [7, 10, 58] and force transducers [18, 20] to study intrafusal fibers
support this hypothesis, which forms the foundation of the structural model presented in
this paper.
4.2.1.2 Modeling
Researchers have been developing models of the muscle spindle for decades. A large
number of linear models have been developed, but exhibit limited ranges due to the
spindle’s nonlinear behavior[22, 23]. Empirical nonlinear models [25] are in common
use in large neuromuscular models[34] due to their computational simplicity and
broader range. Structural nonlinear models, though computationally intensive, offer a
unique opportunity in that specific model behaviors can be correlated to analogous
physiological mechanisms. A small number of these models have been published
describing all [27-29] or part[31, 35] of the muscle spindle. One such model, the
Schaafsma model[28], was built upon the widely held theory that complex spindle
behavior arises from mechanical interaction between the intrafusal muscle tissue and
the sensory region. It models the primary (Ia) response of a dynamic (bag1) fiber and
static (bag2 and nuclear chain) fiber. Each fiber consists of a linear elastic sensory
region in series with a contractile region. The primary afferent output is computed as a
function of sensory region length and its first derivative. Our robotic muscle spindle
model incorporates parts of the Schaafsma model for aspects of spindle behavior not
intrinsic to the mechatronic design.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
41
4.2.1.3 Robotics
Biorobotic devices are being developed to replicate a variety of aspects of the peripheral
motor control system. Projects include an analog VLSI based model of motorneuron
pools[59], a robotic replica of the upper arm[60, 61] and pneumatic artificial
muscles[60, 62]. The robotic muscle spindle project, initiated by Marbot and
Hannaford[38], presents the first biorobotic model of a muscle spindle. This device
offers the precision engineering and validation required for using it both as a platform
for further spindle research and as a robust peripheral element in higher level biorobotic
models.
4.2.2 Approach
This article describes the design and performance of a biorobotic, structural muscle
spindle model in which the biological behavior is captured through both the
performance characteristics of mechatronic hardware and the modeling algorithms of
the control software. In the Methods section we describe the design and
implementation process by which we integrated three robotic subsystems into a
structural model of the muscle spindle. Technical engineering details of the robotic
subsystem design, implementation and performance are described elsewhere[63].
The tuning and validation process was divided into two independent stages. First we
tuned the model parameters against five data sets obtained from the literature describing
the cat muscle spindle’s response to a ramp and hold position input. The performance
of the robotic muscle spindle in each of these tuning studies is presented in the first half
of the Results section. We then validated the fully tuned robotic muscle spindle against
five additional experiments also obtained from the cat muscle spindle literature. These
validation studies are presented in the second half of the Results section.
In the Discussion section we evaluate the model’s successes and limitations as revealed
by the tuning and validation studies. We also comment on the significance of the model
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
42
including use of the biorobotic modeling technique and potential contributions to
biological theory raised through the modeling process.
4.3
Methods
4.3.1 Design
4.3.1.1 Conceptual Design
4.3.1.1.1 Modeling Approach
In conceptualizing the robotic muscle spindle, we abstracted three core functions from
physiological behaviors intrinsic to the muscle spindle for hardware implementation: (a)
the mechanical filtering produced by intrafusal muscle contractile tissue, (b) the neural
transduction from strain to receptor potential, and (c) the encoding of receptor potential
as an action potential spike train. The medium for implementing each of these
functions was selected from the repertoire of available engineering technology using the
selection criteria that it must (a) meet performance specifications derived from
biological studies on the analogous physiological system, and (b) be miniature enough
to viably mount the full robotic muscle spindle in parallel to a human biceps muscle.
Once the technologies were selected, the specific robotic systems were designed and
implemented to capture as much of the physiological functionality as possible in the
mechanical and electrical behavior of the hardware itself. Aspects of the muscle
spindle’s behavior not intrinsic to the electrical and mechanical design were
implemented in control software using an adaptation of the structural mathematical
model developed by Schaafsma et al.[28].
4.3.1.1.2 Model Framework
The conceptual framework for the model consists of a contractile element in series with
a linear elastic sensory element. External position inputs are applied as a strain across
the whole system. The strain is then unequally distributed between the contractile
element and the linear elastic sensory region. The contractile element’s force
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
43
production is a complex function of its length, velocity and contraction level, while the
sensory element’s force production is a simple linear function of length. The resulting
instantaneous variations in the mechanical properties of the two elements result in the
mechanical filtering behavior of the muscle spindle in which the strain across the
sensory region is different from that applied across the whole muscle spindle.
The model output is then generated as a function of the sensory region strain. The
receptor potential of the muscle spindle model is calculated as a linear function of strain
across the sensory element. This reproduces the neural transduction function of the
muscle spindle. Finally, the model’s output signal, Ia firing frequency, is calculated as
a function of the receptor potential and the receptor potential’s first derivative, thereby
reproducing the muscle spindle encoder function.
The robotic muscle spindle models two fiber types: dynamic and static. These fibers
receive their sole efferent input from the dynamic and static γmn, respectively. Further,
their parameter values model the analogous intrafusal fiber: the dynamic nuclear bag
and a hybrid of the static nuclear bag and nuclear chain fiber, respectively.
4.3.1.2 Design Implementation
4.3.1.2.1 Sensory Element Model
We used published data from the experimental muscle spindle literature to create
performance specifications for the sensory element. These specifications include: (a)
absolute deflection amplitudes greater than 0.24 mm, suitable for a maximum 3:1 scale
model of sensory region deflection, (b) resolution better than 20 µm[4], and (c) a linear
response region at low deflection levels, followed by stiffening and decreasing
sensitivity at increasing amplitudes[4].
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
44
CABLE
The resulting design, Figure 4.2, is a
CANTILEVER
pair of strain-gaged cantilevers. The
base of the cantilevers is rigidly
mounted to a nut that defines the
interface between the contractile
element and the sensory element.
STRAIN GAGE
The cantilevers are connected directly
5 MM
to a pair of cables that provide
external strain inputs across the full
Figure 4.2: CAD drawing of sensory element design.
Displacement of cable with respect to cantilever base
causes bending. Strain gages mounted to cantilevers
transduce bending into millivolt potential change
analogous to Ia receptor potential.
length of the robotic muscle spindle.
Strain between the cable insertion and
the cantilever base is transduced by
electronic circuitry into a millivolt
potential. This millivolt potential, representing the strain across the sensory region, is
then converted into a frequency-modulated spike train and transmitted to the computer
as the output of the sensory element. A description of engineering aspects of this
robotic length sensor is given in Jaax et al.[63]. By successfully meeting all of the
biologically-derived design specifications, this robotic sensor is able to reproduce the
strain-to-millivolt-potential transduction behavior of the sensory element directly in the
mechatronic hardware.
Functionally, the output of the sensory element serves a dual role in the muscle spindle
model. First, it represents the receptor potential that is used to calculate the muscle
spindle output. Secondly, it provides sensory information for the feedback control
algorithm that drives the contractile element. This second role will be addressed in the
Linear Actuator Control Algorithm section below.
4.3.1.2.2 Intrafusal Muscle Model
The contractile element in the model’s conceptual framework is implemented using a
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
45
linear actuator. Muscle-like behavior is produced in the linear actuator by means of the
software algorithm controlling the actuator. Hence, the primary performance
requirement for this device is that it respond to the software controller’s commands
rapidly enough to reproduce the experimentally measured dynamics of intrafusal muscle
tissue. We used published experimental data to identify the following biologicallymotivated performance specifications: (a) a rise time for a 30mm/sec ramp stretch of 22
msec, based on optical measurements of the kinematics of intrafusal motion[4, 16] and
(b) a maximum position error of 0.3 mm during the fastest experimental trajectory, the
30mm/sec ramp and hold. The latter specification arises from the need to keep the
sensory element from exceeding its maximum deflection. We used these two
specifications to identify specific engineering design criteria and design an actuator and
controller that met the required performance specifications. The biologically-motivated
performance specifications were successfully met with the following performance
metrics: (a) the 0-90% rise time on a 30 mm/sec ramp is 21 msec and (b) the maximum
position error is 0.15 mm on a 30 mm/sec ramp. Engineering aspects of the resulting
design, Figure 4.3, are described in detail in Jaax et al.[63].
A software-based control algorithm
MOTOR
NUT
supplies the muscle-like behavior to
the lead screw linear actuator. A
LEAD
SCREW
10 mm
Figure 4.3: CAD drawing of linear actuator design.
Motor rotates threaded rod, driving linear travel of
nut. Muscle model in the control algorithm (see text)
generates muscle-like response to length and γmn
inputs. (top housing removed for visibility)
computational muscle model
calculates the force that should be
present across the contractile element,
Fd, based on its length, velocity and γ
motorneuron firing frequency. The
sensory element measures the actual
force across it, Fa. The difference between these two forces, Fd - Fa is then used as the
error signal, E, to control the linear actuator. The computational muscle model is
described in further detail in the Mathematical Muscle Model section below. Muscle
spindle modeling aspects of the control algorithm are covered in the Linear Actuator
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
46
Control Algorithm section below. Engineering aspects of the position controller, which
ultimately controls the linear actuator, are described elsewhere[63].
4.3.1.2.2.1 Mathematical Muscle Model
The muscle model algorithm is adapted from an extrafusal muscle fiber model
developed by Otten[30]. It calculates force as a function of velocity, length, and γmotorneuron input level. In developing their mathematical muscle spindle model,
Schaafsma et al.[28] retuned the 10 parameters of Otten’s extrafusal fiber to match
intrafusal fiber dynamics by using experimental muscle spindle data as the optimization
target. In implementing this algorithm as our muscle model, we used the structure of
Otten’s muscle fiber model combined with the ten parameter values in the Schaafsma
model. The resulting equation for intrafusal force is:
k F , v > 0
F = ka ,i Fa ,i Fv,i Fq ,i + k p ,i Fp ,i + bi vi +  a ,i e i
 0, vi ≤ 0
( 4.1 )
where i is fiber type (1=dynamic bag1, 2= static bag2 ), ka,i and kp,i are maximum
active and passive isometric force, respectively, Fa,i is active force generated at current
length (normalized), Fv,i is active force generated at current velocity (normalized), Fq,i
is active force generated by gamma stimulation rate (normalized), Fp,i is passive force
generated at current length (normalized), bi is passive damping, vi is velocity of
contractile region, and Fe is force enhancement.
Equations defining Fa, Fv, Fq, Fp are in Otten’s muscle model[30]. Parameters were
freed and tuned when justifiable on either biological grounds or due to subsumption of
the behavior into the mechatronic device. Details regarding the new parameter values
and their justifications are included in the Results and Discussion sections.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
47
4.3.1.2.2.2 Linear Actuator Control Algorithm
Figure 4.4 is a block diagram describing the algorithm used to control the linear
actuator. The force error signal, E, drives the position of the linear actuator. Force
errors arise from three sources: (a) updates to Fd, the desired force, calculated by the
γ-mn input
Mathematical
Muscle
Model
External position
input, C
Nut position, B
Desired
Force,
Fd
+
-
Desired
Force
Position,
Error,
+
1/k
x
E
Convert force to
displacement
Controller
Physical
Plant
Position Controller
Sensory Region
k
Force, Fa
Convert displacement
to force
H
Feedback
Linearization
Nut
Position,
B
-
+
TRANSDUCTION
Compute Ia
Ia output
ENCODING
Sensory element
strain, ε
Figure 4.4: Block diagram of linear actuator controller. Algorithm compares actual force, FA, to
force predicted by muscle model, FD. The difference, E, is used as error signal to drive linear
actuator position. E arises from three sources: updates to FD from muscle model, external position
∆C), and dynamics of control loop.
inputs (∆
muscle model, (b) updates to C, the external position input, and (c) the dynamics of the
control loop. In case (a), the desired muscle force, Fd, calculated by the mathematical
muscle model, serves to maintain continuous strain across the sensory region, adjusting
its magnitude up and down as the mathematical muscle model’s force calculation
varies. In case (b), the external position input, C, maintains a continuous stretch across
the whole spindle equivalent to the input C. As the magnitude of the position input, C,
changes, that instantaneous change, ∆C, is transmitted directly to the position controller
causing the nut to move an identical distance. Finally, in case (c), the dynamics of the
closed loop controller results in transient force errors as the negative feedback loop
works to keep the actual force, Fa, close to the desired force, Fd. A linear scaling factor
was used to tune the magnitude of the muscle model force output, Fd, to the stiffness of
the sensory region to reproduce the sensory region displacements seen in the biological
literature.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
48
4.3.1.2.3 Encoder Model
The function of the encoder, translating the output of sensory transducer into a
biologically accurate Ia action potential frequency, is accomplished in two stages. The
first stage, conversion from millivolt receptor potential to a frequency modulated spike
train, is done onboard the spindle itself to minimize distortion of the signal. The
circuitry design used to accomplish this conversion was designed by Marbot [37]. The
raw frequency signal is transmitted in the range of 1kHz-11kHz to maximize resolution
and then rescaled in the computer. The second stage uses the algorithm adapted from
Schaafsma et al.[28] to convert the raw sensory element output into a Ia signal:
Pi = ltp i × d i
( 4.2 )
Iai = ptr × Pi + h × Pi
( 4.3 )
where Pi is receptor potential, ltpi is the conversion from sensory region length to
potential, di is the displacement of the sensory region beyond the zero firing length, the
length at which there is no mechanical contribution to the receptor potential in the
passive muscle spindle, ptr is the conversion from receptor potential to Ia firing rate, h
is rate sensitivity of encoding from receptor potential to firing rate, and Iai is the firing
rate of muscle spindle Ia afferent.
A 2nd order filter with a cutoff frequency of 20 Hz was implemented on the first
derivative of sensory element strain to minimize propagation of noise extraneous to the
experimental protocol[64]. The 20 Hz cutoff frequency was selected based on Fourier
analysis of the Ia signal that revealed a significant noise source in the motion of the
linear actuator mechanism at frequencies just above 20 Hz. This choice is in agreement
with the opinion stated by PBC Matthews that “frequencies above 20 Hz were not really
relevant for motor control[65].” Given that we are not examining external vibration
protocols, frequencies in excess of 20 Hz are unlikely to be due to the physiology we
are examining.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
49
4.3.2 Experimental Methods
4.3.2.1 Linear Positioning Device
A linear positioning device (LPD) was designed and built to apply position inputs to the
robotic muscle spindle in a manner analogous to that used in experimental muscle
spindle studies[63]. This device has a stroke length of 19 mm, sufficient to allow a
maximum 3:1 scaling of the amplitudes used in the majority of the muscle spindle
literature[10, 58]. The resolution of the LPD’s length sensor is 0.33µm. This is within
0.1µm of the highest resolution length data available in the muscle spindle literature[10,
26, 58]. Using a 2:1 scale in our robotic muscle spindle, the resolution of the LPD
length sensor is greater than the highest resolution length data in the muscle spindle
literature.
4.3.2.2 Experimental Protocols
4.3.2.2.1 Implementing Biological Experimental Protocols
In experiments where we reproduced biological experiments, close attention was paid to
accurately implementing the biological position trajectories. In the case of trajectory
amplitude, physiologists often report stretch amplitudes in terms of the displacement
applied across the entire host muscle body. When this is the case, we assume that this
stretch is proportionally transmitted to the muscle spindle without distortion, and thus
apply the appropriate linear scaling factor to the reported amplitude. The initial spindle
length for an experimental protocol was selected by experimentally identifying the
robotic spindle length at which there was optimal correspondence between the
magnitude of the biological and robotic Ia response across multiple γmn activation
levels. These lengths are reported along with the initial length used in the biological
experiment, if available.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
50
4.3.2.2.2 Scaling
For the purpose of scaling cat soleus and tenuissimus displacements into muscle spindle
strain, we define the optimal spindle length of the biological muscle spindles modeled
here as 11.5 mm, identical to the optimal spindle length in the Schaafsma model[28].
The robotic muscle spindle is a 2:1 scale model of such a biological muscle spindle,
giving it an optimal length of 23 mm. Zero length, the length at which the mechanical
effect on receptor potential is zero in the passive muscle spindle, is set at 10 mm in the
biological muscle spindle[28] and 20 mm in the robotic muscle spindle. The muscle
fiber length of the cat soleus is 42.6mm[66].
4.4
Results
Tuning and validation of the model against data from the muscle spindle literature was
performed in two independent stages. In the first stage we tuned model parameters to
five metrics from the muscle spindle literature describing the muscle spindle’s ramp and
hold response: mean Ia output during ramp, peak Ia output, dynamic index, time domain
response of Ia output, and time domain response of the physical stretching of the
sensory region. The results of this process are presented in the first half of the Results
section. In the second stage we validated the fully tuned model against five additional
experiments from the muscle spindle literature including experimental protocols and
results not used in the tuning studies. The results of these validation studies are
presented in the second half of the Results section.
4.4.1
Model Tuning Studies
This section shows the degree of similarity achieved between robotic and biological
results by tuning the model parameters to replicate these specific sets of biological data
from the muscle spindle literature. The majority of the model parameters retain the
values originally identified by Schaafsma et al.[28]. Changes from these parameter
values, Table 4.1, were justified by one of two reasons: (a) the behavior was subsumed
by the mechatronics of the robotic muscle spindle or (b) there is a biologically-based
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
51
reason for the new value. Further details on specific changes are included in the
Discussion section.
Table 4.1: Parameter values changed during tuning of robotic muscle spindle
Name
New Value
Biologically Motivated
K2
.4
h
15pps(mV/s) -1 , Pi > 0

-1  0pps(mV/s) , Pi ≤ 0
Mechatronically Motivated
Fx
0 FU
0
Fe
-4
b1
8.6x10 FU(mm/s)-1
b2
4.6x10-4 FU(mm/s)-1
Function
static F-v slope
encoder rate sensitivity
cross-bridge rupture
force enhancement
bag1 passive damping
bag2 passive damping
4.4.1.1 Ramp and Hold: Ia Metrics
Optimization of the robotic muscle spindle’s parameters focused primarily on
reproducing three metrics reported by Crowe and Matthews[67] for a biological muscle
spindle given ramp and hold position inputs: mean, peak and dynamic index. Dynamic
index is defined as the change in the Ia output between the end of the ramp and 0.5
seconds after the ramp. Figure 4.5 shows the results of this process overlaid on the
original biological data. The plots present the metrics as a function of ramp velocity as
well as γmn activation level. Figure 4.5a&b depict the mean and peak Ia response
during the ramp, respectively. Figure 4.5c depicts the dynamic index of the Ia response.
The biological metrics from the muscle spindle literature were reported as the
“approximate average for several spikes[67].” In an effort to reproduce this
methodology, we applied a 2nd order 7 Hz low pass filter to the robotic Ia output before
calculating the metrics.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
52
a.)
Mean
b.)
Ia Output (Hz)
300
Peak
300
c.) Dynamic Index
At 5 mm/s, the mean difference
between the robotic and biological
300
metrics is –4.7 Hz with a standard
200
200
200
100
100
100
0
0
0
0 10 20 30
0 10 20 30
Velocity (m m /sec)
deviation of 12.3Hz. Across all ramp
speeds, the mean difference is 1.1 Hz
with a standard deviation of 19.7 Hz.
0 10 20 30
The few notable discrepancies occur
at high velocities. The static robotic
muscle spindle exhibits greater
Figure 4.5: Model parameter tuning study. Ia output
metrics during ramp and hold experiment: (a) mean
response during ramp input, (b) peak response, (c)
dynamic index (see text). Robotic muscle spindle
response (markers with lines) closely matched cat
soleus data (markers without lines, Crowe et al.[67])
for different levels of γmn stimulation (‘+,’ 100 Hz
dynamic, 0 Hz static, “*,” 0 Hz dynamic, 100 Hz static,
“o,” 0 Hz dynamic, 0 Hz static). Displacements refer to
biological host muscle. Final length in biological tissue
(max. physiologic length) similar to robotic muscle
spindle (24.5 mm).
velocity dependency than the
biological muscle spindle,
demonstrated by the increased peak
and dynamic index metrics at high
velocities. Also, the mean response
of the passive robotic muscle spindle
is less than its biological counterpart
at high velocities.
4.4.1.2 Ramp and Hold: Ia Time Domain
Time domain plots of the muscle spindle’s Ia response to a ramp and hold stimulus
allow its characteristic morphology to be observed and tuned. Responses to a 5mm/s
ramp and hold were overlaid in Figure 4.6 to show how closely the fully tuned robotic
model (black) matches Crowe and Matthew’s biological data from an identical
stimulus[67] (grey). Note that in the original biological data the x-sweep rate of the
recording oscilloscope was a linear function of the muscle spindle position input[67].
Accordingly, the time scale of the x-axis only applies to the hold region. We plotted the
robotic muscle spindle data with a similar x-axis distortion during the ramp (solid bar)
to allow direct comparison of the results. These data, as with all time domain Ia
response plots in this article, are filtered with a 2nd order 60 Hz low pass filter.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
53
At all γmn activation levels, the
robotic muscle spindle model
replicates the major elements of the
biological muscle spindle Ia
response. First, the accuracy of the
gain between position and Ia output
is evident in (a) the slope of the Ia
Figure 4.6: Model parameter tuning study.
Comparison of Ia responses (top graph) during ramp
and hold input (bottom graph). Robotic muscle
spindle response (black) closely reproduces cat soleus
muscle spindle response (gray, Crowe et al.[67]) under
varying γmn stimulation levels ((a) 0 Hz dynamic, 0 Hz
static (b) 70 Hz dynamic, 0 Hz static (c) 0 dynamic, 70
Hz static). Solid bar indicates region where x axis is a
function of position input, not time. See text for
details. Lengths refer to displacements of host muscle.
Final length in biological tissue (max physiological
length) similar to robotic muscle spindle (24.5 mm).
response during the ramp and (b) the
magnitude of the Ia response during
the hold. Second, the accuracy of the
gain between velocity and Ia output
is demonstrated by the offset of the
Ia response during the ramp period at
all three γmn activation levels.
4.4.1.3 Ramp and Hold: Sensory
Region Stretching
In building a structural physical model, one of our goals was to accurately reproduce the
mechanical deformations of the two regions of the muscle spindle. The ramp and hold
tuning study in Figure 4.7a depicts the displacement of the sensory region of the robotic
muscle spindle. Figure 4.7c presents for comparison data from Dickson et al.[10]
showing the displacement of a point in a biological muscle spindle 0.3 mm from the
spindle equator, just lateral to the junction between the sensory region and the intrafusal
muscle. Note that since the robotic muscle spindle is a 2x scale model, the actual robot
displacements are 2x the values presented here.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
54
The peak displacement of the robotic
and biological data match with a
value of 21 µm. Further, the ratio
between the peak passive and peak
dynamic response is similar between
the two data sets, with the robotic
muscle spindle exhibiting a slightly
larger peak passive response. Both
cases also exhibit a slow decrease of
the sensory region strain at the end of
ramp, although the time constant for
the robotic spindle is much faster
than the biological spindle showing
we do not fully replicate the slow
decay behavior. Finally, between 0
and 150 msec, the displacement of
both the passive and dynamic sensory
regions show an initial burst spike
typical of short-range stiffness.
These spikes are qualitatively similar,
though the robotic spindle’s initial
burst exhibits a steeper rising slope
than the biological spindle. As the
robotic spindle’s spike behavior is
Figure 4.7: Model parameter tuning study. Sensory
region stretch during ramp and hold stretch applied
across whole muscle spindle. (a) Robotic muscle
spindle sensory region stretch, (b) Input displacement
applied across whole muscle spindle, (c) Displacement
of cat tenuissimus muscle spindle tissue 0.3 mm from
spindle equator, just beyond sensory region (Dickson
et al.[10]). For all graphs, Left column: 0 Hz dynamic,
0 Hz static γmn stimulation (passive), Right column:
100 Hz dynamic, 0 Hz static γmn stimulation
(dynamic). Range and shape of sensory region
displacement closely matches biological data. Lengths
refer to displacements applied directly to biological
muscle spindle. Final length in biological tissue not
available to compare to robotic spindle length (24
mm).
the result of transmitting 100% of the
whole spindle’s displacement to the sensory region, this suggests that in the biological
spindle some displacement does occur across the contractile region during the initial
burst. This mechanism would also explain why the peak occurs later in the biological
spindle. If the applied strain is being absorbed by both the sensory and the contractile
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
55
region, it will take more time for the ramp position input in this experiment to apply
enough strain to achieve the displacement of the sensory region and associated force
necessary to rupture the actin-myosin crossbridges and end the initial burst.
4.4.2
Model Validation Studies
Once the robotic muscle spindle
was completed and tuned, we
validated its performance by
comparing its behavior to a
different set of five experiments
obtained from the muscle spindle
literature. No parameter values in
the robotic muscle spindle were
adjusted while performing this set
of validation studies.
4.4.2.1 Ramp and Hold
The first experiment compares the
ramp and hold response of the
robotic muscle spindle to
biological data from Boyd et
al.[68] (Figure 4.8). In both the
dynamic and static cases, the
morphology of the robotic muscle
spindle response shows a close
correspondence to the biological
data. In the passive case the
Figure 4.8: Completed model validation study (cf.
Boyd et al. 1977[68]). Comparison of Ia response to
ramp and hold position input (bottom row)
Parameters tuned with data from Crowe et al.[67]
(Figure 4.5 and Figure 4.6) and Dickson et al.[10]
(Figure 4.7) applied to data from Boyd et al.
Normalized robotic muscle spindle response (top row)
very closely matches normalized dynamic and static
response of cat tenuissimus muscle spindle (middle
row), although amplitude of passive is small. γmn
stimulation levels: Left column: 0 Hz dynamic, 0 Hz
static (passive), Center column: 100 Hz dynamic, 0 Hz
static (dynamic), Right Column: 0 Hz dynamic, 100
Hz static (static). All Ia responses normalized to
maximum depth of modulation in dynamic response of
respective spindle, robotic or biological. Positions
refer to deformations applied to host muscle. Final
length data for biological muscle spindle not available
to compare to robotic muscle spindle (24.4mm).
morphology is still similar, although an unusually large initial gain in the biological data
results in a large positive 45% offset that is not present in the robotic data. The
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
56
morphological similarities between the robotic and biological data include the position
dependency, velocity dependency and initial burst. The position dependency similarity
can be seen both in the slope of the ramps and the final value of the hold after the
transients have dissipated. The velocity dependency similarity is best seen in the
similarities between the robotic and biological offsets during the ramp. Both the time
course and magnitude of initial burst phenomenon are mimicked nicely in the static and
dynamic robotic muscle spindle data, with the passive robotic data showing an initial
burst with a slightly faster time course. Note that the data are presented with their
scales normalized to the full depth of modulation of that muscle spindle’s dynamic
response, robotic or biological, with zero set as the minimum Ia value in each individual
response. This was done to allow comparison of the morphology despite substantial
differences in the scale of the two responses. Our robotic muscle spindle had a range of
200 Hz in this study while the biological muscle spindle range was only 48 Hz.
The robotic muscle spindle ramp and hold response also matched data from P.B.C.
Matthews[69], but the normalization of Figure 4.8 was not required. The major
discrepancy in the two data sets was a small velocity gain in the robotic muscle
spindle’s passive and dynamic data sets, which results in a smaller offset during the
ramp phase of the robotic passive and dynamic response.
4.4.2.2 Sinusoidal Stretch Experiments
During a 2 mm peak-to-peak amplitude, 1 Hz sinusoidal input, the robotic muscle
spindle’s time domain Ia response closely matched data from Hulliger et al.[57] under
passive, maximal dynamic and maximal static γmn activation. Similarities included a
phase lead of approximately 80° across all γmn activation levels, dynamic γmn input
generating the maximum Ia depth of modulation, and zero Ia output in the passive
muscle spindle at lengths less than the “zero length.” Scaling of the robotic passive
response, though, was notably smaller than the biological response.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
57
The fourth validation study, the effect of the amplitude of a sinusoidal position input on
the depth of modulation of the muscle spindle’s Ia response, is shown in Figure 4.9.
The lines for the static and dynamic
robotic output are very close to the
biological behavior reported by
Hulliger et al.[57]. Further, the
robot’s static and dynamic slopes
exhibit the gain compression
phenomenon. There is a steep linear
relationship between sinusoid
amplitude and Ia response at small
amplitudes, which then abruptly
Figure 4.9: Completed model validation study (cf.
Hulliger et al. 1977[57]). Comparison of depth of
modulation of Ia output in response to varying
amplitude of sinusoidal stretch input. Robotic muscle
spindle data (dashed lines) closely matches cat soleus
muscle spindle data (solid lines) during dynamic γmn
(“+”, 100 Hz dynamic, 0 Hz static) and static γmn
(“o”, 0 Hz dynamic, 100 Hz static) stimulation, while
the passive response (“*”, 0Hz dynamic, 0 Hz static) is
about 25% of experimental amplitude. Amplitudes
refer to displacement of the host muscle. Mean length
of biological spindle (1-2 mm less than physiological
max) similar to robotic spindle (22 mm).
decreases and stabilizes at a
shallower slope at higher sinusoid
amplitudes. In the passive case,
however, the robotic muscle spindle
output is much smaller than its
biological counterpart. To test the
origin of this, a sensitivity analysis
was done on the passive damping
parameter, b1, which had been
reduced from 9.91x10-3 to 8.6x10-4 FU (mm/s)-1 due to the intrinsic damping of the
mechatronics. Restoring this parameter to its original value only increased the
amplitude of the passive response by 5-8 Hz.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
58
4.4.2.3 γ Motorneuron Performance
The final validation experiment, Figure 4.10, shows the effect of varying γmn stimulus
amplitude on the mean Ia output of the robotic and biological muscle spindle. The
robotic Ia response matches the
biological data reported by
Hulliger[70] nicely under both static
and dynamic γmn stimulation.
Indeed, all values for the robotic
muscle spindle response lie within
the standard deviation bars for the
biological experiment, which was
performed on 28 static and 20
dynamic γ mn axons[70]. The slopes
Figure 4.10: Completed model validation study (cf.
Hulliger 1979[70]). Comparison of effect of varying
γmn stimulation level on Ia response. Robotic muscle
spindle data (dotted lines) matches slope and
saturation point of cat soleus muscle spindle response
(solid lines, error bars and shading indicate std. dev.)
under two different types of γmn stimulation (dynamic
“+” and static “*”). Muscle spindle held at constant
length throughout all experiments. Biological muscle
spindle length (2 mm less than physiological max)
similar to robotic muscle spindle (22.5 mm). Note,
robotic data exactly overlap biological data if
inequality allowed between static length (23mm) and
dynamic length (22mm).
in both cases are extremely similar to
their biological counterparts, with
only a 10 Hz offset. Finally, the
saturation point to γmn input
corresponds well at approximately
100Hz. These data were collected
with the robotic muscle spindle held
at the same length for the static and
dynamic tests, 23.5 mm, reproducing
the length constraint from the biological experiment.
4.5
Discussion
This biorobotic model of the muscle spindle tests the spindle mechanism theories which
comprise it by quantitatively assessing their performance in a novel testbed, a physical
model built in robotic hardware. Further, testing and validating the model against
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
59
biological data from numerous experimental protocols across multiple authors has
challenged the universality of its structure. Working to replicate these data from the
literature has given us insight into the sources of the model’s limitations and their
implications. These processes have collectively spawned new hypotheses regarding
spindle function and physiology.
We will first discuss the model tuning, examining which parameters were tuned and
why, as well as its successes and limitations. We will then evaluate the validation
studies for the model’s ability to capture key elements of muscle spindle behavior in a
more general context. Finally, we will conclude by presenting hypotheses about muscle
spindle function generated through the development and validation of this model.
4.5.1
Model Tuning
The initial parameters of the model included six determined by the mechatronics of the
system[63] and twelve intrafusal muscle model parameters, ten of which were identified
by Schaafsma et al.[28] and two of which arise from Otten’s original extrafusal muscle
model[30]. Using this initial parameter set, we compared the model’s performance
against five biological metrics characterizing the ramp and hold response: peak Ia
output, mean Ia output during ramp, dynamic index, Ia response in the time domain, and
sensory region displacement. When discrepancies arose, the responsible parameter was
identified and evaluated according to the following criteria: (a) was there evidence in
the physiology or anatomy of the biological muscle spindle to support changing the
parameter value, and (b) was this parameter duplicated in the mechatronics and the
mathematical muscle model? If either criterion was met, the parameter was freed and
tuned accordingly.
4.5.1.1 Mechatronically Motivated Parameter Changes
The software model of short-range stiffness used in the Schaafsma model was the first
term modified due to subsumption into the mechatronics. Fx, a parameter controlling
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
60
the force threshold above which a single cross-bridge will rupture, was set to zero,
thereby eliminating the short-range stiffness algorithm. We instead modeled it with a
physically analogous mechanism: stiction. In the biological muscle spindle, short-range
stiffness is thought to arise from persistence of bound cross-bridges until a force large
enough to rupture the bonds is placed across the muscle spindle[71]. In our linear
actuator, short-range stiffness arises from the persistence of a surface bond between the
nut and lead screw until a force large enough to rupture the bond is placed across the
robotic muscle spindle. In the active robotic spindle, approximately 33µm of whole
spindle stretch is required to generate a force error signal, E, (Figure 4.4) large enough
to break the surface bond in the linear actuator. This corresponds to 0.15% strain across
the whole spindle, compared to the 0.3% whole spindle strain at which cross bridges are
thought to rupture in the biological muscle spindle[72]. The success of this physical
model in producing an initial burst by transmitting initial displacements directly to the
sensory region is demonstrated by the sensory region displacement, Figure 4.7, and the
Ia response, Figure 4.8.
The second mechatronically motivated change was force enhancement, Fe, which had
been implemented in the muscle model in Eq. 4.1 as a discontinuous force offset term: a
positive constant in lengthening and zero in shortening. Schaafsma et al.[28] added Fe
to the Otten muscle model[30] while tuning the model for intrafusal muscles. We again
removed Fe because the discontinuity introduces significant instability into closed loop
control systems. Further, the effect of the force enhancement term is to increase the
magnitude of the force-velocity term on lengthening, which in the dynamic fiber is
already near maximum. Hence, we omitted this property from our muscle model and
have accounted for its effects elsewhere.
The muscle model’s passive damping term, bi, from Eq. 4.1 was the final change caused
by subsumption by the mechatronics. Since the mechanical plant has intrinsic damping,
the passive damping model is redundant and we reduced its value accordingly.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
61
4.5.1.2 Biologically Motivated Parameter Changes
Biological motivations resulted in two changes: (a) the encoder rate sensitivity, h, was
increased in magnitude and made unidirectional and (b) the slope of the static fiber’s
force-velocity relationship was decreased.
Based on biological data[14], we raised the magnitude of the encoder rate sensitivity
term, h, in Eq. (4.3) to compensate for lack of “Force Enhancement,” Fe. Increasing the
magnitude of h revealed the need for a second change in h: unidirectional rate
sensitivity. In previous models this term has always been symmetrical, driving the Ia
output up or down as the receptor potential rose and fell[27, 28]. On raising the
magnitude of h, though, we observed that falling receptor potentials, e.g. ramp
cessation, led to large sustained non-physiological Ia undershoots. Experimentation
with our model revealed that eliminating h just during falling receptor potentials
allowed the Ia output to maintain its velocity-dependent offset during the ramp, while
eliminating the large non-physiological undershoots.
We found two studies in the biological literature with data to support this theory of
unidirectional rate sensitivity in the transfer function between receptor potential and Ia
frequency. Hunt and Ottoson[14] overlaid on top of an actual Ia response a theoretical
Ia response predicted as a linear function of receptor potential. The actual Ia response
was much greater than predicted during rising receptor potentials, but corresponded
well to the predicted value during falling receptor potentials. Fukami’s data showed
similar results for snake muscle spindles[73]. Hunt and Gladden also observed in their
reviews that Ia output during stretch is proportionally greater than the receptor potential
predicts[4, 5], although neither explicitly addressed Ia output during shortening. Based
on this evidence, we postulate that the encoder transfer function is:
15, Pi > 0
Ia = ptr × Pi + h × Pi , h 
 0, Pi ≤ 0
( 4.4 )
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
62
where ptr = potential to rate conversion factor, Pi = receptor potential, i = fiber type and
h = encoder rate sensitivity. Additional biological experiments could further test this
hypothesis by measuring the relationship between receptor potential and Ia output under
a wider range of experimental protocols than the ramp and hold studied by Hunt and
Ottoson [14]. Such experimentation could also be used to quantify the magnitude of the
encoder rate sensitivity, h.
The second biologically-motivated parameter change was K2, the slope of the static
(bag2) fiber force-velocity curve in Eq. 4.5 below. Due to differences between the bag2
and dynamic (bag1) fiber’s parameter values, removal of the Fe term had a much
smaller effect on the positive stretch sensitivity of the bag2 fiber than the bag1 fiber.
Further, the compensatory increase in the h term was tuned to the bag1 fiber. Thus, to
restore bag2 sensitivity, we needed a parameter to selectively decrease bag2 sensitivity
during stretch. The optimal choice was the bag2 fiber force-velocity relationship from
the original Otten muscle model[30]:
1 − v / V max 2

,v ≥ 0

1 + v /( K 2 × V max 2 )
Fv 
1 + v / V max 2
e2 − (e2 − 1)
,v < 0

1 − 7.56v / (K 2 × V max 2 )
( 4.5 )
where: Fv is the force due to velocity, v is velocity, Vmax2 is the maximum bag2
velocity, e2 is maximum bag2 force due to velocity, and K2 is the slope of the bag2
force-velocity curve. We targeted this relationship for several reasons: (a) it is
biologically accurate to tune F-v of the static fiber independently of the dynamic fiber,
(b) its exact value for intrafusal muscle is still unknown and (c) the available evidence
suggests extremely low viscosity in the static fiber, e.g. fast myosin isoforms[4], driving
in the nuclear chain fiber[74], and extremely small dynamic indices[67]. Based on this
biological support we increased K2 from 0.25 to 0.4, lowering the slope of the static
force-velocity curve.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
63
4.5.1.3 Quality of Fit
The goal of the tuning process was to match the model’s output to five different
measures of the biological muscle spindle’s ramp and hold response. The first three
measures are metrics describing the accuracy of quantitative aspects of the Ia response,
Figure 4.5. The overall closeness of the match is very strong, particularly at 5 mm/s,
the slowest ramp speed. The dynamic and passive responses are quite accurate at all
speeds, reflecting a high quality of fit for the dynamic fiber, which generates both the
dynamic and passive response. At higher velocities the static muscle spindle exhibits
too great a dependence on velocity. This is because the intrinsic damping in the robotic
muscle spindle makes it difficult to replicate the static fiber’s extremely low velocity
gain at high velocities. Sources of damping in the static muscle model, b2 and K2, were
tuned to minimize the damping. A sensitivity analysis on b2, K2, and e2, the static forcevelocity curve’s maximum value, showed that further changes would not appreciably
lower the peak and dynamic index metrics. Hence, the static muscle spindle is slightly
over-dynamic at speeds greater than 15mm/s.
The time domain tuning studies demonstrate how well the qualitative features of the Ia
response were tuned. Figure 4.6 shows that the robotic muscle spindle’s Ia output
echoes the biological Ia output almost exactly at all three γmn input levels, indicating
that it successfully reproduces the qualitative aspects of the biological muscle spindle
response. These aspects include both position and velocity gain. Out of the forcelength relationship of the muscle model comes the dependence of the position gain on
γmn activation. The dependence of the velocity gain on γmn input, exhibited by the Ia
offset during ramps, arises from the muscle model’s force-velocity relationship.
The final tuning measure was physical displacement of the sensory region.
Physiologists have long thought that the mechanical filtering of the intrafusal muscle
generates much of the muscle spindle’s behavior[6]. Optical recordings support this
theory by demonstrating that aspects of the muscle spindle’s nonlinear Ia response are
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
64
present in the dynamic strain of the sensory region[10, 68]. Since this concept forms the
foundation of our structural model, we included this tuning study to (a) ensure that the
major Ia response features are present in the dynamics of the intrafusal muscle model
and (b) tune the range of the sensory region displacement to match the biological data.
Figure 4.7 shows our success in fitting the model to these requirements. The range of
displacements is very similar to the biological range for both the dynamic and passive
case. Further, these graphs show that we have reproduced in our intrafusal mechanics
most of the major features of the Ia response, including both the time course and
magnitude of the initial burst.
4.5.1.4 Muscle Length
In tuning the robotic muscle spindle to match the results of multiple biological
experiments, it quickly became apparent that the initial length at which the study is
performed is an important factor in replicating the Ia response. This phenomenon arises
from several factors. First, the muscle force-length relationship is markedly nonlinear,
meaning both the initial value and the position gain of the Ia response change with
length. Second, the passive muscle spindle’s position sensitivity increases significantly
as a function of length while the active muscle spindle’s position sensitivity exhibits a
slight decrease in the robotic muscle spindle and almost no variation with mean initial
length in the biological spindle. Finally, the passive muscle spindle has zero Ia
response below its zero firing length. To accommodate this, for each experiment we
repeated the experimental protocol at 5 different initial lengths throughout the robotic
muscle spindle’s working range. We then used the relative Ia amplitudes at each of the
γmn activation levels to determine which length best corresponded to the length of the
biological muscle spindle when the data were collected. These initial lengths are
reported in the figure captions along with the approximate initial lengths reported by the
biological investigators, when available.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
65
4.5.2
Model Validation
To validate our model, we obtained Ia response data from five different experiments in
the muscle spindle literature. These experiments differ in several ways from the Crowe
and Matthews studies used for Ia tuning[67]. Four studies come from different authors,
introducing variation in experimental technique. Further, three studies are new types of
experiments: two examine sinusoidal response and one looks at fusimotor response. All
studies used cat spindles: four soleus and one tenuissimus.
The key to this validation was testing the fully tuned model under novel circumstances
to examine its general applicability. Absolutely no modifications to the robotic muscle
spindle were made while performing these studies. The only variable adjusted to get
the best match to specific studies was the initial length at which the experimental
protocol was applied.
4.5.2.1 Ramp and Hold Studies
The validation included two ramp and hold studies to test the robotic muscle spindle’s
response to data from different authors and γmn input levels. In our comparison with
the Boyd et al. study[68], the robot’s static and dynamic responses are similar to the
biological data (Figure 4.8). The passive data are qualitatively similar, but the
biological response exhibits a large positive offset. In our comparison with the
Matthews study[69] there was also qualitative similarity between the biological and
robotic data.
The data in Figure 4.8 were normalized due to range differences which we suspect
result from the fact that our model was tuned to muscle spindles with larger depths of
modulation than the muscle spindles used in Boyd et al.[68]. When we compare the
robotic muscle spindle’s behavior to data[69] from P. B. C. Matthews, the same author
who published the data used for tuning[67], we find that the robotic muscle spindle’s
range is quite accurate.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
66
One aspect of the Boyd et al. passive data is atypical for passive muscle spindles. The
extreme steepness of the initial force-length (F-L) relationship is inconsistent with the
shallow F-L relationship at the end of the ramp, perhaps signaling the presence of
stretch activation. We present this study despite these unusual data, though, because it
nicely illustrates the short range stiffness phenomenon.
The robotic muscle spindle exhibited many characteristic features of biological muscle
spindle Ia response in these validation studies. The static and dynamic data in Figure
4.8 demonstrate nicely the robotic muscle spindle’s ability to mimic the position and
velocity dependency of the Ia response. Further, the initial burst phenomenon is well
illustrated in Figure 4.8. The data show that our mechatronic model of short range
stiffness works well in all three γmn input levels, reproducing both the magnitude and
the time course of the initial burst under static and dynamic γmn stimulation.
4.5.2.2 Sinusoidal Studies
Sinusoidal experiments test whether the robotic muscle spindle model is complete
enough to reproduce a range of muscle spindle behaviors beyond its tuning studies. The
model’s time domain sinusoidal response has good qualitative correspondence to the
biological response. The intrafusal fiber viscoelasticity is evident in the phase lead of
all three responses, as well as in the dynamic bag1 fiber’s large response. Further, the
robotic spindle successfully mimics the passive biological spindle’s zero firing length.
The scale of the robotic passive response, though, is smaller than the biological
response.
The second sinusoidal study (Figure 4.9) was included to test our modeling of the “gain
compression” phenomenon, another manifestation of short range stiffness. Biological
muscle spindles will exhibit a “linear range” with high position-Ia gains at small
amplitude stretches while the cross-bridges are still bound. At larger stretches the
cross-bridges rupture and there is a flattening of the curve to a new lower position-Ia
gain. The robotic muscle spindle data reproduce this behavior very nicely in both the
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
67
dynamic and static cases, not only matching the range of the depth of Ia modulation
extremely well, but also exhibiting a distinct linear range. The passive robotic data,
however, are much smaller than the biological data with an amplitude similar to the
Schaafsma model’s passive response.
In summary, the robotic muscle spindle’s sinusoidal response is good under γmn
activation, exhibiting phase lead, gain compression, and biologically plausible Ia
amplitudes. In the absence of γmn activation, the robotic muscle spindle’s response is
smaller than the physiological response. This behavior will be commented on below.
4.5.2.3 γ Motorneuron Study
The fusimotor validation study was performed to test the response of the robotic muscle
spindle to various frequencies of γmn stimulation (Figure 4.10). The model’s response
matches the slope, magnitude and saturation point of the biological response under both
types of γmn stimulation, static and dynamic. The graph also shows that both the
robotic and biological muscle spindles are more sensitive to variation in static than
dynamic γmn input, reflecting the steeper active force-length relationship of the static
fiber. This figure, combined with the success of the active γ mn cases in each of the
other validation studies, strongly supports the accuracy of the robotic muscle spindle in
replicating the behavior of the biological muscle spindle under active γmn inputs.
4.5.2.4 Limitations
Although in 5 of the 9 measures the robotic muscle spindle’s passive response matched
the biological response quite well, in the remaining four studies its amplitude was much
smaller than the biological response, representing the only major limitation of the
model’s general applicability. We identified three possible sources for this behavior:
failure to correctly identify the initial length, a missing term in the passive model and
stretch activation.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
68
The initial length theory comes from the fact that the passive Ia position sensitivity rises
as a function of spindle length while the static and dynamic spindle’s position
sensitivity decreases (robotic) or increases only slightly (biological). If we performed
these studies at a longer length we would likely be able to replicate the relative
amplitudes of the passive, dynamic and static cases. The absolute magnitude of the Ia
response would then exceed the biological data, but such variability in scaling is
observed in the biological data[75]. This explanation is appealing since only four of the
nine passive experiments exhibited low output amplitudes.
An absent term in the passive muscle spindle model is the second possibility. Careful
examination of the passive sinusoidal time domain response suggests it has insufficient
phase lead, indicative of a missing damping term. We performed a sensitivity analysis
to test the effect such a term might have. Theoretical calculations, confirmed by
experimentation, showed that increasing passive damping by a factor of 10 only
increases the passive Ia depth of modulation by 5-8 Hz during a 1 mm sinusoid. Since
this change is so slight and would have equal effect in the active spindle, we concluded
that the passive damping term was not contributing substantially to the small passive
response.
Stretch activation is the final possibility. If the prediction is true, that the act of
stretching a passive intrafusal fiber can lead to contraction[10], this could account for
the four biological experiments whose passive Ia response amplitude we were unable to
replicate. Unequivocal evidence for this phenomenon has not yet been found. The one
study that reported visual evidence of intrafusal muscle shortening on stretch used a
grip technique that damaged the muscle spindle[16]. Further experiments using
simultaneous recording of intrafusal muscle length, tension and receptor potential
during passive stretching at different velocities may be able to establish whether stretch
activation truly exists and better characterize the kinematic properties of the passive
intrafusal muscle. Perhaps the studies in which the biological Ia amplitudes exceeded
our prediction, e.g. large sinusoidal position inputs, could be used as a guide for the
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
69
types of experimental protocols where unaccounted for behavior such as stretch
activation might occur.
4.5.3
Summary of Contributions
4.5.3.1 First Biorobotic Muscle Spindle Model
Our device and its prototype [37, 38] are the first muscle spindle models to be built
using the biorobotic modeling technique. This technique offers several unique
advantages over traditional software modeling including (a) rigorous adherence to all
physical laws, (b) insights gained through implementing concepts in physical hardware,
(c) the ability to apply realistic inputs directly to the model, (d) educational advantages
of having students physically interact with the model and (e) having a working device
upon completion of the project.
The biorobotic modeling technique significantly enhanced the results of the robotic
muscle spindle project in several respects. First, we realized that a discontinuous force
enhancement term results in an extremely difficult system to control, suggesting that the
biological system exhibits more continuous behavior than that described in the
Schaafsma intrafusal muscle model. Second, we gained insight into the bandwidth of
our model as well as the technology with which the biological tuning data were
collected through building in-house a Linear Positioning Device to apply position
inputs. Third, since our model is physically realized in robust robotic hardware, we can
install it on a robot or prosthetic. This feature is especially significant for researchers
developing biologically accurate biorobotic models of the stretch reflex.
4.5.3.2 Potential Applications to Biological Theory
Ideally, the modeling process is closely coupled with experimentation. We have drawn
extensively upon the work of experimenters to develop and validate this model and in
this final section we hope to offer something in return. While developing this model,
two issues arose from which we wish to postulate two new hypotheses about muscle
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
70
spindle mechanisms. The first issue is force enhancement, implemented in the
Schaafsma model as a discontinuous term that produces a constant positive force offset
during lengthening that is absent during shortening. This type of discontinuity is
extremely difficult for a control system to accommodate and might provide similar
difficulties for the nervous system. We therefore hypothesize that, if force enhancement
does occur in the intrafusal fiber, it has a more continuous form, e.g. sigmoidal.
The second hypothesis we propose is unidirectional rate sensitivity in the encoding
process. Symmetrical rate sensitivity between receptor potential and Ia frequency led to
non-physiological large undershoots on ramp cessation. Investigation of biological data
on the encoding process[14] supports the hypothesis that this rate sensitivity is indeed
only present during increasing receptor potentials, not decreasing. We implemented
this behavior in our model and were able to eliminate the large undershoots on ramp
cessation. Hence, we hypothesize that the true encoding function exhibits only
unidirectional rate sensitivity and encourage further experimentation to test this theory.
The final element we wish to comment on is a functional implication of the relative
length sensitivities of the muscle spindle. In both the robotic muscle spindle model and
biological muscle spindles[76, 77], passive position sensitivity increases substantially as
a function of length while active position sensitivity increases only slightly (dynamic
biological), remains constant (static biological), or decreases slightly (robotic) as a
function of length. These relative effects in which the γmn input stabilized the position
sensitivity[76] made it important to replicate the initial length of the biological muscle
spindle when attempting to match the relative responses of the passive and active
spindles. Such effects may also contribute to biological phenomenon such as the
dependence of ankle joint motion sensitivity on extensor muscle length, observed in the
passive limb[78]. Further, Schafer[79] observed that prestretched passive muscle
spindles replicate the Ia response amplitudes of shorter muscle spindles under dynamic
γmn stimulation and postulated the origin to be prestretch-dependent stretch activation.
Again, the nonlinearity of the passive force-length relationship might contribute to this
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
71
phenomenon, although it could not account for the increased velocity sensitivity.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
72
Chapter 5:
Fusimotor Effect on Signal Information Content of Ia
Ensemble Model Reconstructed from Dynamic Intramuscular
Strain Data
5.1
Summary
1. It is long observed that the transducer characteristics of the muscle spindle Ia
response, e.g. noise and nonlinearity, sharply limit kinematic information. Many
propose the ensemble response as a source of an accurate signal, but technical
difficulties limit experimental population size and fusimotor control.
2. We reconstruct the ensemble response of a hypothetical population of 20-28
muscle spindles from dynamic local strain data from contracting rat medial
gastrocnemius. For 18 contractions in 3 rats, individual Ia responses are
generated by a nonlinear muscle spindle model and then averaged to form
ensemble Ia response.
3. Results under dynamic fusimotor stimulation show significantly improved
correlation to linear function of whole muscle position and velocity in ensemble
vs. individual Ia response.
4. Correlation to whole muscle velocity increased with rate of homogeneously
distributed dynamic fusimotor input and proximity of initial length to optimal
length of extrafusal muscle.
5. The results support our hypotheses that the reconstructed ensemble would
reduce Ia signal nonlinearity and that homogeneously distributed fusimotor
stimulation can suppress ensemble noise and nonlinearities in a dose-dependent
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
73
manner. Proposed mechanisms include decorrelation by intramuscular strain
inhomogeneities, fusimotor-dependent length and velocity selectivity, and
decorrelating effect of fusimotor-dependent noise and nonlinear gains.
5.2
Introduction
The noise and nonlinearity of the individual muscle spindle’s output [1] sharply limits
the kinematic information capacity of the signal produced by a single muscle spindle.
In response to this, many physiologists have looked to the ensemble response of a
population of muscle spindles as the way for the central nervous system (CNS) to get an
accurate signal from these sensors [39, 40, 45, 46]. The population encoding theory is
supported by experiments showing that firing of a single muscle spindle is insufficient
stimulus to elicit perception of motion [40].
The question then arises: What variables might be critical for increasing the ensemble’s
information capacity? Ensemble size, simultaneous recording, and an intact fusimotor
system have been shown to improve information content [41, 43]. In fact, the fusimotor
system has been implicated as the mechanism by which such effects as heteronymous
muscle fatigue [47] and ketamine application [44] can degrade ensemble information
content. Several investigators have raised the issue of decorrelating individual muscle
spindle responses as a means to improve spatial filtering of ensemble information
content. Proposed mechanisms include the fusimotor system behaving as a neural
network [43, 47], random noise introduced by the active fusimotor system [49] and
membrane firing threshold variability [50]. Such decorrelation mechanisms have only
been tested indirectly [47], theoretically [50] or in small populations [49].
The technical difficulties associated with recording the afferent response of a population
of muscle spindles have limited the availability of simultaneously recorded
experimental data to populations of 10 or fewer [41-44, 47]. Sequential recording under
similar experimental conditions has allowed large data sets to be gathered, but the
discontinuities of time, muscle and animal, e.g. 34 cats employed in measuring a total of
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
74
47 muscle spindles [52], limits the ability to study decorrelation of an intramuscular
spindle population.
With the paucity of experimental data, a model becomes an attractive option for
reconstructing the information content of the ensemble response of a large population of
muscle spindles. Further, a model allows one to readily control variables such as
fusimotor stimulation rates across a large population, something not possible in animal
models. Muscle spindle ensemble models in the literature use simple models of
individual spindle behavior to examine limb position encoding by spindle populations
spanning multiple muscles [34, 48]. To date, no ensemble model has been developed
that offers the level of detail necessary to reconstruct the influence of physiologic
variables on suppression of the individual spindle’s noise and nonlinearities in a single
muscle body’s ensemble response.
In this study, we create such a model of the ensemble response of a large population of
muscle spindles residing in a single muscle. Because the noise and nonlinearities in the
spindle’s behavior are the very thing that limit its information content, it is essential that
a model designed to generate physiologically relevant results regarding ensemble
information content be accurate in capturing the nonlinear features of the individual
muscle spindle response. Accordingly, we employ a structural muscle spindle model
that captures the major features of muscle spindle response: position gain, velocity gain,
fusimotor response, gain compression and normally distributed noise [63, 80].
Further, we propose that local strain variation within a muscle is so relevant for
decorrelating individual spindle response that it must also be included in the model to
generate physiologically relevant data. The muscle spindle Ia response has consistently
shown itself to be a function of the local strain directly adjacent to it in studies
comparing the Ia response to maximum strain [81], velocity [82] and contraction of
motor units [83]. Further, recent studies have demonstrated that local strains vary
substantially across the extrafusal muscle’s surface and with respect to muscle origin-
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
75
to-insertion length under a variety of experimental protocols including passive
stretching [84], active contraction [85, 86] and locomotion [87, 88]. Accordingly, to
accurately model the mechanical environments of the members of our muscle spindle
population, we use as mechanical input local surface strain data simultaneously
recorded from multiple locations on contracting muscle tissue using a three-dimensional
determination method.
Once the ensemble model is developed, we then use it to ask two questions about the
ensemble’s information content. The first question is whether the ensemble response
reduces the nonlinearities seen in the individual muscle spindle response, and if so, by
how much. This aim tests whether the sources of variability in the ensemble model
(local strain variability, fusimotor-induced random noise, and fusimotor-induced
nonlinear responses to the strain variability) decorrelate the noise sufficiently to allow it
to be spatially filtered out of the ensemble response. As such, it is an explicit test of the
widely held theory that a large population will reduce the presence of the individual
spindle’s noise and nonlinearities in the ensemble response [39, 40, 45, 46].
The second question goes on to ask whether the rate of homogeneously distributed
fusimotor stimulation improves the correlation between input trajectories and ensemble
response, i.e. if the fusimotor system has a dose-dependent effect on ensemble response.
This objective stems from the observations that an intact fusimotor system improves the
information content of a spindle ensemble. Bergenheim et al.[43] proposed that the
fusimotor system is acting as a neural network to decorrelate the output from each of
the spindles, thereby increasing the ensemble’s discriminative ability for kinematic
variables. While we concur that the neural network mechanism could produce the
observed behavior, we postulate that a simpler mechanism, the differing transducer
properties of the active spindle vs. the passive spindle, could also produce the observed
effect. We put forward the idea that, even at a fixed stimulation level across the
population, the fusimotor system could increase the correlation of the ensemble to
whole muscle position or velocity by (a) increasing the random noise and variability of
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
76
the individual spindle response [49, 50], (b) increasing the decorrelation introduced by
local strain variability by means of the nonlinear mechanical properties ascribed to the
intrafusal fiber [28, 63], and (c) increasing the percentage of the Ia signal that responds
specifically to velocity or length, assuming exclusively dynamic or static fusimotor
input, respectively.
We therefore pose the following hypotheses:
Hypothesis 1: The response of an intramuscular muscle spindle ensemble is a more
linear function of length and velocity than the individual muscle spindle response.
Hypothesis 2a(b): Increasing the rate of homogeneously distributed dynamic (static)
fusimotor stimulation to a muscle spindle population improves the strength of the
correlation between whole muscle velocity (length) and ensemble response.
5.3
Methods
5.3.1 Collecting Local Muscle Fiber Strain Data
The methods for collecting muscle strain data are described in detail elsewhere [86] and
are briefly summarized here. Three male, 12 week old Lewis rats were anaesthetized
with sodium-pentobarbital (Numbutal®, 0.1 ml/kg BW, i.p.) after short-term (<20s)
sedation with CO2. Sodium-pentobarbital was supplemented as necessary. The local
ethical committee approved the experiments. The medial surface of the medial
gastrocnemius was surgically exposed and dissected free of fascia. Approximately 70
fluorescent polystyrene spheres (Bangs Laboratories Inc. Fishers USA) of 0.45 ±
0.05mm diameter were attached to the muscle surface in a uniform distribution with an
interdistance of ~2mm. The calcaneus bone was dissected free of the leg and fixed to a
force transducer. The femur was securely fixed to the lower traverse. Elevation of the
upper traverse allowed control of muscle length with an accuracy of 0.01 mm. Two
electrode wires wrapped around the sciatic nerve supplied pulsed electrical stimulation,
0.7-0.9 Volts at 80-90Hz, to generate maximal muscle contraction force. No increase in
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
77
force was observed with further increases in voltage or frequency. Images of the
markers were captured at 50 fields per second with two synchronized CCD cameras and
digitized for analysis. For each 350 msec contraction, recording began 80 msec prior to
contraction and captured 600 msec of video data. 3D marker tracks were reconstructed
from the pair of digitized images. Strains were calculated for each video frame with
respect to the first image of the data acquisition. Local strains at each marker position
were calculated using the procedure adapted from Peters [89] in which a linear strain
field was assumed within a strain group, defined as a circle (r=4mm) around the marker.
As markers were on the muscle surface, strains calculated were two-dimensional
surface strains. Strain in the third-dimension, normal to the muscle surface, were
assumed to be zero. Strain groups containing less than 5 markers were excluded from
analysis to calculate reliable strain.
Seven trials of isometric contractions were recorded from three rats. The first trial was
performed at just below the muscle’s optimal length, with the remaining trials
performed at –1 mm, +1mm, -2mm, +2mm, -3mm, +3mm, respectively. Only markers
recording muscle fiber motion were included in the study. These markers were
identified by the presence of a negative principal strain aligned with the muscle’s
longitudinal axis during maximal contraction. In addition to exhibiting temporally and
spatially appropriate contractile behavior, microscopic examination of the tissue
confirmed that all markers used in the study lay either directly on the muscle (68
markers) or on the muscle’s proximal border separated from the muscle fibers by a thin
layer of aponeurosis (5 markers). Viable muscle markers were limited to those present
in the data for all seven trials in a given rat. This subset included all muscle fiber
motion markers except those at the distal and lateral edges of the muscle body, with a
total of 25, 20 and 28 markers in rats 1-3, respectively. The trial at –2mm was omitted
from further analysis for all rats due to insufficient numbers of viable markers in Rat 1.
Local muscle fiber orientation was identified as the negative principal strain axis during
peak tetanic contraction. Marker principal strains were rotated to this orientation in
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
78
each frame to generate a timecourse of the local strain that would be experienced by
sensors lying parallel to the muscle fiber. Whole muscle strain was determined by
measuring the displacement between a marker adjacent to the proximal muscle insertion
and a marker at the musculotendonous junction with the Achilles tendon.
5.3.2 Calculating Muscle Spindle Ensemble Response
The timecourse of strain experienced by each marker was run individually through the
model of the mammalian muscle spindle described in Chapter 4. Initial spindle length
was calculated using the assumption of homogeneous strain distribution in the passive
muscle body. In the first frame of each trial the distance was calculated between
markers at the proximal and distal end of the muscle belly whose negative principal
strain during contraction was approximately collinear, giving a reference length along
the muscle fiber axis. Initial strain was then calculated by normalizing this length to the
corresponding length at optimal fiber length. All measurements from the medial
gastrocnemius were reported as strain. The optimal length of the muscle spindle model,
the length at which the intrafusal muscle generates maximal force, was used to
denormalize the strain and calculate physical displacements to apply to the muscle
spindle model.
Fusimotor activation level was constant throughout a given trial. All experiments were
repeated under eight fusimotor stimulation rates: 25, 50, 75 and 100 Hz dynamic, and
25, 50, 75 and 100 Hz static. No experiments were performed with simultaneous
stimulation of the static and dynamic γ motorneurons (γmn).
A linear positioning device applied the strain trajectories to the robotic muscle spindle
one at a time. Ia output was sampled at 1000 Hz and a 5 point moving average was
recorded at 200 Hz using a dSPACE 1102 data acquisition card and the ControlDesk
software interface. Technical details regarding this system are published elsewhere [63].
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
79
The individual muscle spindle Ia responses were compiled into an ensemble metric by
calculating their average response as a function of time:
E (t ) =
1 m
∑ Ian (t )
30 n =1
( 5.1 )
Where: E(t) is the ensemble response, n is the individual trajectory number, m is the
number of spindles in the ensemble, Ian is the afferent output of the muscle spindle for
the nth trajectory, and t is time.
5.3.3 Data Analysis: Nonlinearity of Spindle Ensemble Output.
The goal of this analysis was to test the hypothesis that the ensemble response fits the
model of a linear weighted sum of position and velocity better than the individual
muscle spindle response. To test this, a multiple regression was performed to calculate
the correlation coefficient for the following model:
Ia(t ) = Ax(t ) + B
dx
+C +ε
dt
( 5.2 )
Where: Ia = Ia output (ensemble or individual), x = position input to whole muscle,
dx/dt = velocity input to whole muscle, C = offset in data, ε = residual error. Fisher’s Z
transformation was performed to obtain a normally distributed variable, Z’, describing
the correlation coefficient.
A Student’s paired t-test was calculated between (a) Z’ for ensemble response in a given
trial, rat, and fusimotor activation level, and (b) the average of all individual spindle Z’
values in the population corresponding to that ensemble response. This test was run
separately for data collected under static γmn stimulation and dynamic γmn stimulation.
All statistical computations were performed with the MATLAB statistics toolbox.
For both the nonlinearity and fusimotor studies, we limited our analysis of the static
fusimotor data to the window from 250 msec to the 600 msec to exclude a non 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
80
physiological spike observed on extrafusal contraction initiation (80 msec) in the static
fusimotor stimulated muscle spindle model. An additional non-physiological spike was
discovered later at ~400 msec in all rats and was not excluded. Its source and
implications on the data are described in the discussion. Because of its differing
dynamics, the muscle spindle model under dynamic fusimotor stimulation did not
exhibit these non-physiological behaviors and therefore the dynamic analysis
encompassed the full 600 msec contraction.
5.3.4 Data Analysis: Effect of Fixed Fusimotor Stimulation Rate
The goal of this analysis was to test hypotheses 2a&b: that increasing the rate of
dynamic (static) fusimotor stimulation to a muscle spindle population improves the
strength of the correlation between ensemble response and whole muscle velocity
(length). For each fusimotor rate, the correlation coefficient was calculated between
ensemble response and whole muscle velocity (length). Fisher’s Z transformation was
used to convert the correlation coefficient into a normally distributed variable, Z’.
Sources of variation in the Z’ variable were determined in JMP statistical software
(SAS Institute Inc., Cary, NC) with a repeated measures ANOVA using the following
linear model:
Z’ijk = µ + Gk + Ti + βj + εijk
( 5.3 )
Where: G=Gamma motorneuron treatment level, T= Block for run order (repeated
measure) and initial length, β= Block for rat and number of spindles in the ensemble, ε
= residuals. Significance was determined as p<0.05.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
81
5.4
Results
5.4.1 Local Strain Data
The negative principal strain alignment at each of the markers during peak contraction,
Figure 5.1, was used to assign muscle fiber orientation. The circle denotes muscle
marker location, and the line’s
orientation and length denote the
negative principal strain alignment
and magnitude during peak
contraction, respectively.
Aponeurosis markers are not shown.
Despite the inability to visualize
muscle curvature in this frontal view,
Figure 5.1: Location of 28 markers (o) on surface of
rat medial gastrocnemius muscle fibers used to
reconstruct mechanical input to 28 hypothetical
muscle spindles. Local muscle fiber axes ( | ) are well
aligned across the marker set, suggesting the muscle
fiber axes were correctly assigned as negative
principal strain axis during maximum contraction.
Distal end of muscle at top. All data in Figure 5.1Figure 5.4 are from Trial 6 of Rat 3.
the alignment appears consistent
across all markers, verifying the
muscle fiber orientation assignments.
The magnitude is consistent through
the main muscle body, decreasing
rapidly at the aponeurosis boarder.
Markers at [-.5,-7] and [0,-6] are on the distal border of the aponeurosis. All data in
Figure 5.1-Figure 5.4 are from Rat 3, Trial 6.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
82
The 28 markers attached to the
muscle body show considerable
diversity in the local strain time
course experienced during a typical
isometric contraction, Figure 5.2.
Although strain is similar among the
individual markers during the initial
contraction, during the hold and
relaxation periods, 100-600msec, the
strains experienced by the individual
Figure 5.2: Dynamic strains recorded at the 28
markers shown in Figure 5.1. Strains exhibited
variation in amplitude, velocity, time course and
smoothness during and after a 350 msec contraction
markers vary from one to another in amplitude, velocity, smoothness and final strain.
5.4.2 Ensemble Reconstruction
The middle row of Figure 5.3 shows the 28 marker strain timecourses from Figure 5.2
as they were physically applied to the robotic muscle spindle: 1 trajectory every 1.5
Figure 5.3: Sequence of 28 displacement trajectories (middle row) laid out in manner in which
they were physically applied to muscle spindle model. This protocol generated 28 individual Ia
responses corresponding to 28 hypothetical muscle spindles (top row, 50Hz static γ mn stimulation,
Bottom Row, 50 Hz dynamic γ mn stimulation) which were then pooled to form ensemble response.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
83
seconds. The other two rows depict the robotic muscle spindle’s response to the
individual strains for one replicate of the 50 Hz fusimotor stimulation rate, both static
(top) and dynamic (bottom). The Ia data in Figure 5.3 are low pass filtered at 10Hz to
increase visibility. No filter was applied during data analysis.
The markers in Figure 5.3 exhibit
a wide diversity of strain
trajectories. Amplitude and shape
diversity are particularly evident
in this figure. The muscle
spindle response also exhibits
variation among markers. The
static response is loosely related
to the position input while the
dynamic response closely follows
the velocity input. In the static
response, a large spike occurs at
the peak of contraction as the
input rapidly accelerates from
shortening to lengthening. All
Figure 5.4: Comparison of ensemble response to
kinematic inputs. (a) Under 50 Hz dynamic fusimotor
stimulation ensemble Ia response of muscle spindle
population (solid line, left axis) closely parallels whole
muscle velocity (dashed line, right axis). (b) Under 50 Hz
static fusimotor stimulation baseline ensemble Ia response
(solid line, left axis) loosely follows whole muscle position
(dashed line, right axis), but is dominated by large spike at
onset of relaxation ramp.
responses exhibit the noise and
nonlinearities typical of spindle
Ia response.
Figure 5.4 illustrates the
similarities between the Ia
ensemble response (solid line) and the input trajectory (dotted line). The compliance of
the experimental set-up allowed the muscle body to shorten, resulting in the motion
seen here. The ensemble response under dynamic fusimotor stimulation mimics the
input velocity in both amplitude and phase throughout the 600 msec trial. Under static
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
84
γmn stimulation the ensemble’s correlation to input position is smaller. Although the
baseline amplitude follows the sigmoidally increasing position (175-600 msec), the
response shows a definite velocity offset throughout and is punctuated by large positive
spikes that coincide with large changes in velocity, both during shortening and
lengthening.
5.4.3 Nonlinearity of Spindle Ensemble Output.
Multiple regression analysis of the linearity of the relationship between spindle Ia
output and position and velocity inputs generated a correlation coefficient, Z’, which
quantifies the proportion of the nonlinear Ia response which can be accounted for by a
linear function of position and velocity. The resulting correlation values, Z’, are shown
for a typical case in Figure 5.5a&b. Figure 5.5a shows the response under 100Hz
dynamic γmn input, while Figure 5.5b shows the response under 100Hz static γmn
input. The 6 trials correspond to a single repetition of each of the 6 contractions
performed by a single rat. Under both static and dynamic γmn stimulation, the
ensemble response correlation (solid line) is typically higher than the averaged
correlation of the individual muscle spindle responses (dotted line). The individual
muscle spindle response’s correlation to whole muscle motion (dots) varies greatly,
with an average standard deviation of 0.12 and 0.13 in the dynamic and static trials
shown, respectively. The ensemble’s multiple correlation to position and velocity is,
for all cases, greater than or equal to the ensemble’s single correlation to just position or
velocity under either static or dynamic γmn input, respectively (dash-dot line).
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
85
A paired t-test examined the hypothesis that the ensemble response of a spindle
population has a higher correlation coefficient than the average individual muscle
spindle’s correlation to the whole muscle’s position and velocity. Correlation
coefficients are reported as the
normally distributed coefficient,
Z’. Under dynamic γmn input the
ensemble correlation is
significantly higher than the
average individual response, p <
.0001. The mean difference is
0.22 higher; the mean ensemble
correlation coefficient is 0.62 ±
0.20(std. dev.); the mean
individual correlation coefficient
is 0.40 ± 0.11. Under static γmn
input, the ensemble correlation is
also significantly higher than the
average individual response, p <
Figure 5.5: Correlation coefficients for multiple regression
on whole muscle position and velocity. Correlations are
consistently higher for ensemble response (solid line) than
when the 28 highly variable individual muscle spindle
correlation coefficients (dots) are averaged together
(dotted line). Correlation coefficients for single regression
of ensemble response against single kinematic variable
(dash-dot line) shows strength of fusimotor stimulation in
tuning ensemble selectivity to velocity (5a) or position (5b).
Fusimotor stimulation (5a, 100 Hz dynamic, 5b, 100 Hz
static) constant across all six trials (x-axis) of rat 2.
.0001. The mean difference was
0.08; the mean ensemble
correlation coefficient is 0.40 ±
0.18; the mean individual
correlation coefficient is 0.32
±0.084.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
86
5.4.4
Effect of Fixed Fusimotor Stimulation Rate
Under dynamic fusimotor
stimulation, the effect of
fusimotor stimulation rate on the
correlation between ensemble
response and whole muscle
velocity, Figure 5.6a, is
significant, p < .0001. The mean
correlation values, Z’, increase
with increasing dynamic
fusimotor input with values of
0.469, 0.537, 0.560, 0.583 for the
25, 50, 75 and 100 Hz dynamic γ
fusimotor stimulation rates,
respectively. The effects of run
order and initial length are shown
in Figure 5.6b&c. These plots
indicate strong trends in the data,
though statistical assessment of
the effect is not possible due to
the experimental design. The
plot of run order vs. correlation,
Figure 5.6c reveals oscillations in
Figure 5.6: Correlation between ensemble response and
parallel with the oscillating initial whole muscle velocity under dynamic fusimotor
stimulation (a) increases monotonically with increasing
lengths, as well as decreasing
correlation with repeated muscle
contraction. Plotting muscle
initial length vs. correlation.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
rates of dynamic fusimotor stimulation, (b) peaks at
optimal muscle length then decreases with distance from
optimal muscle length for all rates of dynamic fusimotor
stimulation (o, 25 Hz, x, 50 Hz, +, 75 Hz, ∆, 100 Hz), (c)
decreases with repeated extrafusal contraction for all rates
of fusimotor stimulation.
87
Figure 5.6b, reveals a consistent trend that correlation increases with proximity to
muscle optimal length. Note that in Figure 5.6b 4 of the 21 trials were omitted from
calculating mean correlations because their initial length deviated from the mean initial
length across all rats for that trial by >2%. The data from Rat 2 and Rat 3 from the trial
at –2mm, which was omitted from the statistical analysis due to insufficient markers in
Rat 1, were included in the plot of initial length correlations.
Under static fusimotor stimulation, the effect of fusimotor stimulation rate on the
correlation between whole muscle position and spindle ensemble response, Figure 5.7a,
is significant with p=.0065. The direction of the effect is opposite of what was
predicted. The highest mean correlation, Z’, occurrs in the 25 Hz case, with means of
0.281, 0.255, 0.185, and 0.187 for the 25, 50, 75 and 100 Hz static fusimotor
stimulation rates, respectively. Figure 5.7b&c show the effect of initial length and run
order, respectively. No clear trend of the effect of run order or initial length is evident
in the static fusimotor data.
5.5
Discussion
The aim of this article is to use mechanical data collected from 70 locations on an
actively contracting muscle to reconstruct the ensemble response that would have been
produced by 20-28 muscle spindles scattered throughout that muscle. In doing so, we
asked the question of whether the sources of variability in our ensemble data, strain
inhomogeneity leveraged by the nonlinear transfer function of the active spindle, and
the random noise of the active spindle, increased the information content of the
ensemble response. We further asked whether the variability in the individual Ia output
introduced by fusimotor stimulation could, when applied homogeneously across a
muscle spindle population, produce a dose-dependent improvement in the ensemble
information content.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
88
5.5.1 Reconstructing the
Ensemble Response
5.5.1.1 Local Strain Data
Our thesis presumes variability in
local strain across the muscle
body, and the results (Figure 5.2)
show that there is indeed
substantial variation in amplitude,
time course, velocity and
smoothness between individual
marker strain trajectories.
Further, although the strain
magnitudes seen in the bottom
three markers are small, Figure
5.2 shows that the strain
trajectories experienced by all
markers, including these three, are
typical of the strain trajectories
experienced by contracting
muscle, confirming that they were
correctly identified as muscle, not
aponeurosis, markers.
The experimental protocol
employed to get a sample of
typical local strains experienced
by a population of muscle
spindles was isometric contraction
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
Figure 5.7: Correlation between ensemble response and
whole muscle position under static fusimotor stimulation
(a) decreases monotonically with increasing rates of static
fusimotor stimulation, (b) shows no apparent relation to
initial muscle length for all rates of dynamic fusimotor
stimulation (o, 25 Hz, x, 50 Hz, +, 75 Hz, ∆, 100 Hz), (c)
shows no discernable change with repeated extrafusal
contraction for all rates of fusimotor stimulation. All
correlation values are extremely low, reflecting a
nonphysiological short-range stiffness behavior in the
static muscle spindle model which severely degraded the
ability to reconstruct an accurate correlation value.
89
of the musculotendonous unit, which raises two issues. First, is there sufficient change
in muscle body length to test our hypotheses? Elek et al. [87] showed that in the cat
medial gastrocnemius muscle spindles signal muscle body length, which differs from
changes in origin-to-insertion length [88, 90]. Figure 5.4 supports Elek et al.’s data,
showing there was considerable variation in muscle body length as well as velocity
during our experiments. The second issue is whether fixed fusimotor stimulation, both
static and dynamic, is observed during extrafusal contraction. Studies in which
fusimotor outflow is reconstructed for volitional movements in the cat show that fixed
fusimotor levels provide the best match to experimental data during locomotion (fixed
static fusimotor input) and stretching (fixed dynamic fusimotor input), far
outperforming EMG-linked fusimotor input [56].
Several assumptions are implicit to these data. First, since data are unavailable
regarding the spindle count in the rat medial gastrocnemius, we assumed a population of
20-28 muscle spindles. Spindle counts from related muscles, rat gracilis (13-17) [91]
and cat medial gastrocnemius (46-80) [92], suggest that 20-28 spindles is a reasonable
approximation. The second assumption is that the distribution of those 20-28 spindles
across the muscle’s medial surface (Figure 5.1) is representative of the distribution in a
typical unipennate muscle. The spread across the muscle surface is consistent with the
limited data available[93-95] on the distribution of spindles in rat and cat medial
gastrocnemius. These same data suggest that few muscle spindles in the medial
gastrocnemius lie near to the muscle surface, but other data indicate that spindle output
is closely correlated to the overlying surface strain of the extrafusal muscle[81, 82].
Hence, we conclude that the distribution of our hypothetical spindle population is a
reasonable approximation for the purposes of reconstructing ensemble response.
5.5.1.2 Muscle Spindle Population Response
Our proposal that the variability in the individual response is spatially filtered out of the
ensemble response is supported by the smoothness of the ensemble Ia response (Figure
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
90
5.4) as compared to the noise in the individual Ia responses (Figure 5.3).
Essential to the reliability of our reconstruction of the ensemble response is the
accuracy of the muscle spindle model in generating the pertinent features of the
individual Ia response. Because the questions we ask are sensitive to the nonlinear
aspects of the spindle response, we used a muscle spindle model, the robotic muscle
spindle, which accurately replicates many of the linear and nonlinear features of the Ia
response including position and velocity gain, normally distributed noise, fusimotor
response and gain compression. In tuning and validation studies with protocols similar
to the spindle inputs used in this study, e.g. ramp and hold, sinusoidal and fusimotor
response, the fusimotor-stimulated model reproduced the biological data well in 10 out
of 10 cases. Without fusimotor stimulation, the response matched the biological data in
5 of 9 cases, with non-physiologically small responses in the remaining 4, as described
in Chapter 4. Recognizing this limitation, we restricted the experiments in this study to
active fusimotor stimulation. In using this model, we make the assumption that the use
of two different animal models, the rat for extrafusal motion and the cat for muscle
spindle modeling, does not impair our ability to draw meaningful conclusions from our
reconstruction of the spindle ensemble response.
The large spike observed in the static ensemble response (Figure 5.4b) is a symptom of
a previously unrecognized limitation in the static fusimotor response of the muscle
spindle model: the short-range stiffness model allows the "cross-bridges" to rapidly
reset during fusimotor stimulation. As a result, when the muscle spindle briefly comes
to rest during the contraction plateau, the short-range stiffness model engages, causing a
large spike when the relaxation ramp begins. This type of spike is not observed
physiologically during active fusimotor stimulation [96]. Because of the spike’s large
size and the fact that the sudden cessation of α-motorneuron stimulation caused a
temporal correlation of relaxation ramp initiation across all of the local strains
(~380msec in Figure 5.2), this non-physiological nonlinearity persists in the ensemble
response. This phenomenon is not observed under dynamic fusimotor stimulation
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
91
because the dynamics of the muscle spindle model prevent the short-range stiffness
response from resetting.
5.5.2 Effect of Ensemble on Kinematic Information Content
Under dynamic fusimotor stimulation, the results provide unequivocal support for
hypothesis 1. We find that the ensemble response is far superior to the individual
response in terms of the correlation to position and velocity. Also, the spread in
individual Ia correlations (dots in Figure 5.5a) suggests there is extensive variability
within the spindle population. This evidence supports the proposed mechanism that the
sources of variability in the model are sufficiently decorrelated to allow spatial filtering
of the individual spindle’s noise and nonlinearities.
Under static fusimotor stimulation, the results are unfortunately masked by the large
spike introduced by the muscle spindle model’s non-physiological short-range stiffness
response. All correlations are extremely low with a mean of .30 for the average
individual spindle correlation. The low correlations are exacerbated by the fact that the
very large Ia responses associated with the short-range stiffness occur at the beginning
of the relaxation ramp when length is at a minimum. We did still observe a slight
improvement in correlation in the ensemble response as compared to the individual
spindles. This is likely due to the filtering which occurred in the time windows before
and after the short range stiffness peak.
5.5.3 Dose-Dependent Effect of Homogeneous Fusimotor Stimulation
Under dynamic fusimotor stimulation, the results clearly support our hypothesis,
showing a statistically significant monotonic increase in correlation between velocity
and ensemble response with increasing dynamic fusimotor stimulation rates (Figure
5.6a). This supports our theory that the fusimotor-dependent mechanisms described
above work together to increase ensemble information content by both increasing the
spindle’s selectivity for velocity inputs and increasing the decorrelation effect. The plot
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
92
of correlation vs. length (Figure 5.6b) further supports this theory. It shows that
ensemble information content, defined by the correlation coefficient Z’, peaks when the
contraction is performed at optimal muscle length in a fashion similar to the familiar
active muscle length-tension relationship [30]. This behavior is also consistent with the
theory of fusimotor level increasing information content. In the muscle model used to
drive the intrafusal dynamics, all of the properties that contribute to the active tension
(length-tension, velocity-tension, fusimotor-tension) are multiplied together to calculate
the active component of the muscle’s force. As a result, the length-tension property can
act as a coefficient to modulate the other active properties in a manner similar to
increasing fusimotor stimulation. The possibility also exists that the extrafusal tissue
from which we reconstructed the ensemble response exhibited increasing decorrelation
of its local strain with proximity to optimal muscle length.
Repeated contraction of the extrafusal muscle leads to decreased ensemble information
content (Figure 5.6c). This could be a result of repeated contraction of the extrafusal
muscle increasing correlation between local strains, particularly since the muscle was
not preconditioned. We must, however, temper our conclusions with the following
caveat. Due to confounding of the experimental design, it is impossible to assess the
relative effect of run order vs. initial length (Figure 5.6b&c). The number of
contractions and the deviance of initial length from optimal length could both be having
the same depressive effect on ensemble information content.
Under static fusimotor input, the non-physiological coherent short-range stiffness
nonlinearity overwhelms the correlation between the static ensemble response and
whole muscle length. Through the convergence of two factors: (a) the size of the shortrange stiffness spike is proportional to the stiffness of the intrafusal muscle making it
increase with fusimotor input and (b) the large spike occurs at the onset of the
relaxation ramp where position is at a minimum, we see both very low correlation
coefficients for position and a decreasing correlation with increasing fusimotor input
(Figure 5.7a). Initial length and run order have no appreciable effect (Figure 5.7b&c),
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
93
again likely due to degradation of the correlation by the coherent short-range stiffness.
The degradation observed here is testimony to the power of a coherent noise source to
alter the ensemble’s information content. Perhaps certain types of nonlinearities which
tend to be temporally correlated, such as short-range stiffness, are useful to the CNS and
are handled by behavior-specific decoding mechanisms.
5.5.4
Conclusions
Using physiological data on the local strain distribution during an active contraction, we
reconstruct what the ensemble response might look like from a large population of
muscle spindles. We show that, given the sources of variability in our model, spatial
filtering across a population substantially improves the information content of the signal
sent to the CNS. We also show that in our model the fusimotor stimulation rate
improves ensemble information content even if applied at a fixed rate across the
population. These data support our theory that much of the decorrelation which
suppresses signal distortion in the ensemble is the product of the combined effect of
intramuscular strain inhomogeneities and the nonlinear mechanical properties of the
actively contracting intrafusal muscle.
These studies may reconcile the seemingly disparate views of muscle spindle
nonlinearities between muscle spindle physiologists, who treat nonlinearities as an
important aspect of spindle behavior[4, 5, 72], and physiologists studying higher
organizational levels, who theorize that most nonlinearities will be negligible in the
ensemble response[34]. Our results suggest that, as previously proposed[39, 45, 46, 49,
50], it is in fact the extent of the irregularity of the individual muscle spindle responses,
their decorrelation, that is essential for producing an accurate signal in the ensemble
response. This concept elegantly reconciles what at first glance appear to be
contradictory stances within the physiological community.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
94
Chapter 6:
Conclusions
6.1
Summary
The theme of this dissertation is to develop precision engineering hardware capable of
accurately modeling muscle spindle behavior and to use the process of building and
tuning the model, as well as experimentation with the model, to increase our
understanding of the mechanical and neurological mechanisms by which the body
measures muscle kinematics.
A three element abstraction of muscle spindle behavior was proposed and implemented
in precision engineering hardware. Engineering tests of the individual components
show that their dynamics meet performance metrics derived from the biological
literature; physiologically realistic tests of the integrated robotic muscle spindle show
that the subsystems replicate the physical performance observed in biological muscle
spindles. The transducer hardware matches the displacements observed in the springlike sensory region and uses this physical displacement to replicate the sensory region's
transduction behavior directly in mechatronic hardware. The encoder replicates the
biological encoder's conversion of the analog receptor potential to a frequency
modulated spike train directly in on-board circuitry, using a software-based algorithm to
add positive rate dependency. The contractile element meets the engineering
performance specifications, exhibiting fast, precise and robust linear actuation.
Physiologically realistic tests show that, when driven by a software based muscle
model, the contractile element's physical displacement closely matches the movements
of the biological intrafusal muscle. These tests collectively show that the individual
subsystems of the robotic muscle spindle accurately model the behavior of their
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
95
analogous physiological systems. This lays a solid foundation for investigating the
effect of these subsystem's behaviors on the transduction of muscle kinematics.
The integrated robotic system was tuned against a battery of muscle spindle data and the
physiological faithfulness of the resulting behavior was then validated against a
different set of experimental protocols and results from the biological literature. Under
fusimotor stimulation, the robotic muscle spindle replicates biological behavior well in
all experiments, including ramp and hold and sinusoidal position inputs of varying
speeds and amplitudes as well as a full spectrum of fusimotor stimulation rates, both
static and dynamic. In the passive case, the robotic muscle spindle matches biological
behavior well in 5 of 9 experiments, exhibiting smaller amplitudes than the biological
spindle in the remaining four cases. Thus, under active fusimotor stimulation, the
model enjoys wide applicability to a variety of experimental protocols, with more
limited applicability in the passive case.
During the tuning process, non-physiological undershoots on ramp cessation were
encountered. The bi-directionality of the encoder’s rate dependency was identified as a
likely cause and it was proposed that the encoder rate dependency might instead be
unidirectional. This new hypothesis, supported by data from the biological
literature[14], was implemented and indeed eliminated the non-physiological
undershoots. Hence, the process of building the model led to the proposal of an
alternative hypothesis for spindle encoding which is more consistent with biological
evidence and the systems behavior of the mechatronic model.
Employing a novel methodology, the robotic muscle spindle was then applied to the
task of reconstructing the ensemble response of a population of hypothetical muscle
spindles on the surface of a contracting muscle. Data from collaborators describing
muscle strain time courses at 28 locations on an actively contracting muscle were run
through the robotic spindle to generate the 28 Ia responses that would have been
generated by muscle spindles at each of those locations. The average of those outputs,
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
96
the Ia ensemble response, shows a significantly closer correlation to a linear function of
muscle length and velocity than the individual Ia response. Further, the dynamic
fusimotor input from the central nervous system improves the correlation of the
ensemble response to muscle velocity in a dose-dependent manner.
It is proposed that it is actually the decorrelation of the complexity of the individual
muscle spindle's response, the noise and nonlinearities, that transforms the individual
responses into a much easier to decipher ensemble response. The ensemble's spatial
filtering effect will minimize the influence of noise on the ensemble response if that
noise is decorrelated. The ensemble reconstruction incorporates two major noise
sources, both of which are potentially decorrelated across the population:
inhomogeneous local extrafusal muscle strain and noise whose decorrelation is
dependent on the rate of fusimotor stimulation. These effects could explain the increase
in linearity observed in the ensemble response. Further, the fusimotor system enhances
these decorrelation sources, as well as increasing the individual spindle's selectivity to
specific kinematic variables, thereby providing a mechanism for the observed dosedependent effect on the ensemble response. These neuromechanical hypotheses
elegantly reconcile the noise of the individual Ia response and the nervous system's need
for a decipherable signal of muscle kinematics.
6.2
Future Work
Many different aspects of this dissertation could serve as a starting point for future
work. Candidate areas include experimentating on biological muscle spindles to test
specific hypotheses generated by this robotic modeling research, investigating new
research questions using the ensemble reconstruction technique, expanding the robotic
muscle spindle model, addressing basic science questions in biorobotics, and applying
the robotic muscle spindle to biorobotics applications in prosthetics and engineering.
Experiments to test the biological hypotheses raised by the robotic models in this
dissertation is one area of future work. First, additional biological experiments could
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
97
test the hypothesis of unidirectional encoder rate dependency by recording the
relationship between receptor potential and Ia response under a wider range of
experimental protocols. This method could also be used to quantitatively characterize
the rate dependency. Second, the observation that the passive spindle model matched
biological data well in five cases, but exhibited the same shortcoming in the remaining
four cases, led to the conclusion that a mechanism not included in the model heavily
influences passive behavior. Additional studies could test the proposed candidates
including stretch activation and an omitted passive damping term. Finally, in the
ensemble study three different mechanisms were proposed to explain the improvement
in Ia response with large populations and fusimotor input. Further studies, such as
repeating the experiment with homogeneous local strains, could investigate the specific
effect of each of these mechanisms.
The second area for additional research is applying the ensemble reconstruction
technique to additional research questions. First, anatomical studies have shown that, in
some muscles, muscle spindles are distributed in distinct patterns, such as being
collocated with deep, oxidative fascicles[95]. Reconstruction studies could examine the
types of information coded by different distributions of spindle populations to test the
effect of this selective distribution on ensemble information content. Obtaining strains
from internal locations in the muscle would enhance the power of such an experiment,
as well as provide an interesting comparison to the results presented here. Second, this
technique could be applied to a ramp and hold protocol to examine the ensemble's effect
on specific features such as short-range stiffness. Third, the same questions asked here
could be applied to the secondary spindle response and reapplied to the static Ia
response, pending model modification to omit the non-physiological short-range
stiffness behavior. Finally, running this experiment with and without γ motorneuron
stimulation to the extrafusal fibers would allow one to test the influence of active
contraction on decorrelation of local strains. One could speculate that a passive muscle
would exhibit greater correlation allowing the nonlinear short-range stiffness to persist
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
98
in the ensemble response and act as an early warning against external perturbations in
the resting animal.
The third area for future work is expanding the robotic muscle spindle model.
Candidate areas include: (a) augmenting the onboard circuitry to incorporate
unidirectional rate sensitivity, (b) implementing sarcomere length inhomogeneity in the
muscle model to incorporate phenomena such as stretch activation, local contraction
foci, spread of depolarization across the intrafusal muscle, and a cross-bridge model of
short range stiffness, (c) implementing a detailed model of ion channel transduction,
although the experimental data on which such models are based are limited [35], (d)
building an additional robotic muscle spindle to allow simultaneous stimulation of the
static and dynamic fusimotor fibers and (e) modeling the secondary afferent response.
The fourth area of future work is in applying this research to basic science problems in
biorobotics. This sensor could be used as part of a biorobotic model to study the
behavior of larger neuromuscular systems, as in Chou and Hannaford [61], or as
sensory feedback to train cerebellar learning models. Alternatively, one could
miniaturize the muscle spindle model using MEMs technology and attach a large
population of the devices to a biorobotic muscle to generate a real time ensemble
response.
The final area of future work is the use of the completed robotic muscle spindle as a
sensor for engineering applications. While this model was designed with the aim of
understanding the basic science of muscle spindles, like most biorobotic models it has
obvious applications in prosthetics and engineering as well. The robotic muscle spindle
is a functioning device that can report actuator kinematics in the language of the central
nervous system. As such, it is an attractive candidate for the development of prosthetic
devices. Further, the robotic muscle spindle is an actuated sensor, a type of sensor not
currently in the repertoire of devices used to measure kinematic properties in
engineering. It would be interesting to explore this type of device as a means of
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
99
increasing a transducer’s range in situations where the physical displacement of the
sensing mechanism is limited, or for real-time tuning of the sensor’s output to different
kinematic variables, e.g. absolute length vs. perturbations from a desired length.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
100
Bibliography
[1]
P. B. C. Matthews and R. B. Stein, "The regularity of primary and secondary
muscle spindle afferent discharges," Journal of Physiology, vol. 202, pp. 59-82,
1969.
[2]
A. Prochazka and M. Hulliger, "The continuing debate about CNS control of
proprioception," Journal of Physiology, vol. 513, pp. 315, 1998.
[3]
E. R. Kandel, J. H. Schwarz, and T. M. Jessell, Principles of Neural Science, 3rd
ed. New York: Elsevier, 1991.
[4]
M. H. Gladden, "Mechanical factors affecting the sensitivity of mammalian
muscle spindles," Trends in Neuroscience, vol. 9, pp. 295-297, 1986.
[5]
C. C. Hunt, "Mammalian muscle spindle: Peripheral mechanisms,"
Physiological Reviews, vol. 70, pp. 643-663, 1990.
[6]
B. H. C. Matthews, "Nerve endings in mammalian muscle," Journal of
Physiology, vol. 78, pp. 1-53, 1933.
[7]
D. Ottoson and G. M. Shepherd, "Changes of length within the frog muscle
spindle during stretch as shown by stroboscopic photomicroscopy," Nature, vol.
220, pp. 912-14, 1968.
[8]
Y. Fukami and C. C. Hunt, "Structures in sensory region of snake spindles and
their displacement during stretch," Journal of Neurophysiology, vol. 40, pp.
1121-1131, 1977.
[9]
I. A. Boyd, M. H. Gladden, D. Halliday, and M. Dickson, "Stroboscopic
cinematographic and videorecording of dynamic bag1 fibres during rapid
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
101
stretching of isolated cat muscle spindles," presented at Proceedings of the
International Symposium on Mechanoreceptors, Prague, Czechoslovakia, 1988.
[10]
M. Dickson, M. H. Gladden, D. M. Halliday, and J. Ward, "Fusimotor
mechanisms determining the afferent output of muscle spindles," Progress in
Brain Research, vol. 80, pp. 9-17, 1989.
[11]
K. Diete-Spiff, "Slow contraction of intrafusal muscle fibres of lumbrical muscle
spindles of the dog," Journal of Physiology, vol. 183, pp. 65P-66P, 1966.
[12]
D. K. Hill, "Tension due to interaction between the sliding filaments in resting
striated muscle," Journal of Physiology, vol. 199, pp. 637-684, 1968.
[13]
M. H. Gladden, "Efferent control of human muscle spindles," in The Muscle
Spindle, I. A. Boyd and M. H. Gladden, Eds. New York: Stockton Press, 1985,
pp. 161-166.
[14]
C. Hunt and D. Ottoson, "Impulse activity and receptor potential of primary and
secondary endings of isolated mammalian muscle spindles," Journal of
Physiology, vol. 252, pp. 259-81, 1975.
[15]
I. A. Boyd, "The response of fast and slow nuclear bag fibres and the nuclear
chain fibres in isolated cat muscle spindles to fusimotor stimulation, and the
effect of intrafusal contraction on the sensory endings," Quarterly Journal of
Experimental Physiology, vol. 61, pp. 203-254, 1976.
[16]
R. E. Poppele and D. C. Quick, "Stretch-induced contraction of intrafusal
muscle in the muscle spindle," Journal of Neuroscience, vol. 1, pp. 1069-1074,
1981.
[17]
Y. Fukami, "Receptor potential and impulse initiation in two varieties of
reptilian muscle spindle," Nature, vol. 257, pp. 240-1, 1975.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
102
[18]
C. Hunt and D. Ottoson, "Receptor potential and impulse activity in isolated
mammalian spindles," Journal of Physiology, vol. 230, pp. 49P-50P, 1972.
[19]
D. Ottoson and G. M. Shepherd, "Transducer characteristics of the muscle
spindle as revealed by its receptor potential," Acta Physiologica Scandinavica,
vol. 82, pp. 545-54, 1971.
[20]
C. C. Hunt and R. S. Wilkinson, "An analysis of receptor potential and tension
of isolated cat muscle spindles in response to sinusoidal stretch," Journal of
Physiology, vol. 302, pp. 241-62, 1980.
[21]
C. C. Hunt, R. S. Wilkinson, and Y. Fukami, "Ionic basis of the receptor
potential in primary endings of mammalian muscle spindles," Journal of
General Physiology, vol. 71, pp. 683-98, 1978.
[22]
R. E. Poppele and R. J. Bowman, "Quantitative description of linear behavior of
mammalian muscle spindles," Journal of Neurophysiology, vol. 33, pp. 59-72,
1970.
[23]
T. Rudjord, "A second order mechanical model of muscle spindle primary
endings," Kybernetik, vol. 6, pp. 205-215, 1970.
[24]
B. F. Andersson, G. Lennerstrand, and U. Thoden, "Response characteristics of
muscle spindle endings at constant length to variations in fusimotor activation,"
Acta Physiolologica Scandinavica, vol. 74, pp. 301-18, 1968.
[25]
J. C. Houk, W. Z. Rymer, and P. E. Crago, "Dependence of dynamic response of
spindle receptors on muscle length and velocity," Journal of Neurophysiology,
vol. 46, pp. 143-66, 1981.
[26]
R. E. Poppele and D. C. Quick, "An analysis of muscle spindle behavior using
randomly applied stretches," Neuroscience, vol. 6, pp. 1157-65, 1981.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
103
[27]
Z. Hasan, "A model of spindle afferent response to muscle stretch," Journal of
Neurophysiology, vol. 49, pp. 989-1006, 1983.
[28]
A. Schaafsma, E. Otten, and J. D. van Willigen, "A muscle spindle model for
primary afferent firing based on a simulation of intrafusal mechanical events,"
Journal of Neurophysiology, vol. 65, pp. 1297-1312, 1991.
[29]
E. Otten, K. A. Scheepstra, and M. Hulliger, "An integrated model of the
mammalian muscle spindle," presented at Symposium on Alpha and Gamma
Motor Systems, London, 1995.
[30]
E. Otten, "A myocybernetic model of the jaw system of the rat," Journal of
Neuroscience Methods, vol. 21, pp. 287-302, 1987.
[31]
R. W. Banks, M. Hulliger, K. A. Scheepstra, and E. Otten, "Pacemaker activity
in a sensory ending with multiple encoding sites: The cat muscle spindle
primary ending," Journal of Physiology, vol. 498, pp. 177-99, 1997.
[32]
K. A. Scheepstra, E. Otten, M. Hulliger, and R. W. Banks, "Modelling of
chaotic and regular Ia afferent discharge during fusimotor stimulation,"
presented at Alpha and Gamma Motor Systems, New York, 1995.
[33]
J. M. Winters, "An improved muscle-reflex actuator for use in large-scale neuromusculoskeletal models," Annals of Biomedical Engineering, vol. 23, pp. 35974, 1995.
[34]
K. R. Wallace and G. K. Kerr, "A numerical simulation of muscle spindle
ensemble encoding during planar movement of the human arm," Biological
Cybernetics, vol. 75, pp. 339-350, 1996.
[35]
E. Otten, M. Hulliger, and K. A. Scheepstra, "A model study on the influence of
a slowly activating potassium conductance on repetitive firing patterns of
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
104
muscle spindle primary endings," Journal of Theoretical Biology, vol. 173, pp.
67-78, 1995.
[36]
F. Awiszus, "Critique on the papers of the analysis and modelling section,"
presented at Symposium on Alpha and Gamma Motor Systems, London, 1995.
[37]
P. H. Marbot, "Ia response of a mechanical spindle replica," Ph.D. thesis.
University of Washington, Seattle, WA, 1995.
[38]
P. H. Marbot and B. Hannaford, "The mechanical spindle: A replica of the
mammalian muscle spindle," presented at IEEE Conference on Engineering in
Medicine and Biology, San Diego, CA, 1993.
[39]
S. M. P. Verschueren, P. J. Cordo, and S. P. Swinnen, "Representation of wrist
joint kinematics by the ensemble of muscle spindles from synergistic muscles,"
Journal of Neurophysiology, vol. 79, pp. 2265-2276, 1998.
[40]
S. C. Gandevia, D. I. McCloskey, and D. Burke, "Kinaesthetic signals and
muscle contraction," Trends in Neuroscience, vol. 15, pp. 62-65, 1992.
[41]
H. Johansson, M. Bergenheim, M. Djupsjobacka, and P. Sjolander, "Analysis of
encoding of stimulus separation in ensembles of muscle afferents," presented at
Alpha and Gamma Motor Systems, New York, 1995.
[42]
M. Bergenheim, H. Johansson, J. Pedersen, F. Ohberg, and P. Sjolander,
"Ensemble coding of muscle stretches in afferent populations containing
different types of muscle afferents," Brain Research, vol. 734, pp. 157-166,
1996.
[43]
M. Bergenheim, H. Johansson, and J. Pedersen, "The role of the gamma-system
for improving information transmission in populations of Ia afferents,"
Neuroscience Research, vol. 23, pp. 207-215, 1995.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
105
[44]
M. Bergenheim, A. G. Crenshaw, J. Pederson, and H. Johansson, "The
information transmitted by ensembles of primary spindle afferents is diminished
when ketamine is used as a pre-anaesthetic," Neuroscience Research, vol. 30,
pp. 177-183, 1998.
[45]
U. Windhorst and J. Meyer-Lohmann, "The influence of extrafusal muscle
activity on discharge patterns of primary muscle sindle endings," Pflügers
Archive, vol. 372, pp. 131-138, 1977.
[46]
U. Windhorst, "Origin and nature of correlations in the Ia feedback pathway of
the muscle control system," Biological Cybernetics, vol. 31, pp. 71-79, 1978.
[47]
J. Pedersen, M. Ljubisavljevic, M. Bergenheim, and H. Johansson, "Alterations
in information transmission in ensembles of primary muscle spindle afferents
after muscle fatigue in heteronymous muscle," Neuroscience, vol. 84, pp. 953959, 1998.
[48]
S. H. Scott and G. E. Loeb, "The computation of position sense from spindles in
mono- and multiarticular muscles," Journal of Neuroscience, vol. 14, pp. 75297540, 1994.
[49]
G. Inbar, J. Madrid, and P. Rudomín, "The influence of the gamma system on
cross-correlated activity of Ia muscle spindles and its relation to information
transmission," Neuroscience Letters, vol. 13, pp. 73-78, 1979.
[50]
P. Milgram and G. F. Inbar, "Distortion suppression in neuromuscular
information transmission due to interchannel dispersion in muscle spindle firing
thresholds," IEEE Transactions on Biomedical Engineering, vol. 23, pp. 1-15,
1976.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
106
[51]
A. Prochazka and M. Gorassini, "Ensemble firing of muscle afferents recorded
during normal locomotion in cats," Journal of Physiology, vol. 15, pp. 293-304,
1998.
[52]
A. Prochazka, P. Trend, M. Hulliger, and S. Vincent, "Ensemble proprioceptive
activity in the cat step cycle: Towards a representative look-up chart," in
Progress in brain research, vol. 80, J. H. J. Allum and M. Hulliger, Eds.
Amsterdam: Elsevier, 1989, pp. 61-74.
[53]
A. Prochazka and M. Gorassini, "Models of ensemble firing of muscle spindle
afferents recorded during normal locomotion in cats," Journal of Physiology,
vol. 507, pp. 277-91, 1998.
[54]
A. Valbo, "Activity from skin mechanoreceptors recorded percutaneously in
awake human subjects," Experimental Neurology, vol. 21, pp. 270-89, 1968.
[55]
N. A. Al-Falahe, M. Nagaoka, and A. B. Vallbo, "Response profiles of human
muscle afferents during active finger movements," Brain Research, vol. 113, pp.
325-346, 1990.
[56]
A. Prochazka, M. Hulliger, P. Zangger, and K. Appenteng, ""Fusimotor set":
New evidence for alpha-independent control of gamma-motoneurones during
movement in the awake cat," Brain Research, vol. 339, pp. 136-140, 1985.
[57]
M. Hulliger, P. B. C. Matthews, and J. Noth, "Static and dynamic fusimotor
action on the response of Ia fibers to low frequency sinusoidal stretching of
widely ranging amplitude," Journal of Physiology, vol. 267, pp. 811-838, 1977.
[58]
D. Ottoson and G. M. Shepherd, "Length changes within isolated frog muscle
spindle during and after stretching," Journal of Physiology, vol. 207, pp. 747759, 1970.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
107
[59]
T. H. Zobel, B. C. Harris, and S. P. DeWeerth, "A VLSI biologically-based
motor control circuits," presented at Midwest Symposium on Circuits and
Systems. IEEE, Piscataway, NJ, 1998.
[60]
B. Hannaford, J. M. Winters, C. P. Chou, and P. H. Marbot, "The anthroform
biorobotic arm: A system for the study of spinal circuits," Annals of Biomedical
Engineering, vol. 23, pp. 399-408, 1995.
[61]
C. P. Chou and B. Hannaford, "Study of human forearm posture maintenance
with a physiologically based robotic arm and spinal level neural controller,"
Biological Cybernetics, vol. 7, pp. 285-298, 1997.
[62]
G. K. Klute, J. Czerniecki, and B. Hannaford, "Muscle-like pneumatic actuators
for below-knee prostheses," presented at 7th International Conference on New
Actuators, Bremen, Germany, 2000.
[63]
K. N. Jaax, P. H. Marbot, and B. Hannaford, "Development of a biomimetic
position sensor for robotic kinesthesia," presented at 2000 IEEE/RSJ
International Conference on Intelligent Robots and Systems, Takamatsu, Japan,
2000.
[64]
S. Usui and I. Amidror, "Digital low-pass differentiation for biological signal
processing," IEEE Transactions on Biomedical Engineering, vol. 29, pp. 686693, 1982.
[65]
P. B. C. Matthews, "Review lecture: Evolving views on the internal operation
and functional role of the muscle spindle," Journal of Physiology, vol. 320, pp.
1-30, 1981.
[66]
L. Poliacu-Prose, "De functionele stabiliteit van de knie van de kat," Ph.D.
thesis. Free University of Brussels, PhD Thesis. Brussels, 1985.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
108
[67]
A. Crowe and P. B. C. Matthews, "The effects of stimulation of static and
dynamic fusimotor fibres on the response to stretching of the primary endings of
muscle spindles," Journal of Physiology, vol. 174, pp. 109-131, 1964.
[68]
I. A. Boyd, M. H. Gladden, P. N. McWilliam, and J. Ward, "Control of dynamic
and static nuclear bag fibres and nuclear chain fibres by gamma and beta axons
in isolated cat muscle spindles," Journal of Physiology, vol. 265, pp. 133-162,
1977.
[69]
P. B. C. Matthews, "The differentiation of two types of fusimotor fibre by their
effects on the dynamic response of muscle spindle primary endings," Quarterly
Journal of Experimental Physiology, vol. 47, pp. 324-333, 1962.
[70]
M. Hulliger, "The responses of primary spindle afferents to fusimotor
stimulation at constant and abruptly changing rates," Journal of Physiology, vol.
294, pp. 461-482, 1979.
[71]
C. C. Hunt and D. Ottoson, "Initial burst of primary endings of isolated
mammalian muscle spindles," Journal of Neurophysiology, vol. 39, pp. 324-330,
1976.
[72]
P. B. C. Matthews and R. B. Stein, "The sensitivity of muscle spindle afferents
to small sinusoidal changes of length," Journal of Physiology, vol. 200, pp. 723743, 1969.
[73]
Y. Fukami, "Receptor potential and spike initiation in two varieties of snake
muscle spindles," Journal of Neurophysiology, vol. 41, pp. 1546-56, 1978.
[74]
A. Crowe and P. Matthews, "Further studies of static and dynamic fusimotor
fibres," Journal of Physiology, vol. 174, pp. 132-151, 1964.
[75]
R. B. Botterman and E. Eldred, "Static stretch sensitivity of Ia and II afferents in
the cat's gastrocnemius," Pflügers Archive, vol. 395, pp. 204-211, 1982.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
109
[76]
G. M. Goodwin, M. Hulliger, and P. B. C. Matthews, "The effects of fusimotor
stimulation during small amplitude stretching on the frequency-response of the
primary ending of the mammalian muscle spindle," Journal of Physiology, vol.
253, pp. 175-206, 1975.
[77]
R. E. Poppele, W. R. Kennedy, and D. C. Quick, "A determination of static
mechanical properties of intrafusal muscle in isolated cat muscle spindles,"
Neuroscience, vol. 4, pp. 401-411, 1979.
[78]
K. M. Refshauge and R. C. Fitzpatrick, "Perception of movement at the human
ankle: Effects of leg position," Journal of Physiology, vol. 488, pp. 243-248,
1995.
[79]
S. S. Schafer, "Simulation of dynamic fusimotor effects in the discharge
frequency of Ia afferents by prestretching the muscle spindle," Experimental
Brain Research, vol. 108, pp. 297-304, 1996.
[80]
K. Jaax and B. Hannaford, "A biorobotic model of the mammalian muscle
spindle," Annals of Biomedical Engineering, vol. 28, pp. S-8, 2000.
[81]
J. Meyer-Lohmann, W. Riebold, and D. Robrecht, "Mechanical influence of the
extrafusal muscle on the static behaviour of deefferented primary muscle spindle
endings in cat," Pflügers Archive, vol. 352, pp. 267-278, 1974.
[82]
U. Windhorst, J. Schmidt, and J. Meyer-Lohmann, "Analysis of the dynamic
responses of deefferented primary muscle spindle endings to ramp stretch,"
Pflügers Archive, vol. 366, pp. 233-240, 1976.
[83]
W. Cameron, M. Binder, B. R. Botterman, R. M. Reinking, and D. G. Stuart,
""Sensory partitioning" of cat medial gastrocnemius muscle by its muscle
spindles and tendon organs," Journal of Neurophysiology, vol. 46, pp. 32-47,
1981.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
110
[84]
H. van Bavel, M. R. Drost, D. L. Wielders, J. M. Huyghe, A. Huson, and J. D.
Janssen, "Strain distribution on rat medial gastrocnemius (mg) during passive
stretch," Journal Biomechanics, vol. 29, pp. 1069-1074, 1996.
[85]
H. van Bavel, C. A. J. Holtzer, J. L. van Leeuwen, A. A. M. Muijtjiens, P. J. B.
Willems, and M. R. Drost, "Quantification of local surface strains of cat
gastrocnemius medialis muscle during contraction," Netherlands Journal of
Zoology, vol. 48, pp. 323-334, 1998.
[86]
C. C. van Donkelaar, P. J. B. Willems, A. A. M. Muijtjens, and M. R. Drost,
"Skeletal muscle transverse strain during isometric contraction at different
lengths," Journal of Biomechanics, vol. 32, pp. 755-762, 1999.
[87]
J. Elek, A. Prochazka, M. Hulliger, and S. Vincent, "In-series compliance of
gastrocnemius muscle in cat step cycle: Do spindles signal origin-to-insertion
length?," Journal of Physiology, vol. 429, pp. 237-258, 1990.
[88]
J. A. Hoffer, A. A. Caputi, I. E. Pose, and R. I. Griffiths, "Roles of muscle
activity and load on the relationship between muscle spindle length and whole
muscle length in the freely walking cat," Progress in Brain Research, vol. 80,
pp. 75-85, 1989.
[89]
G. W. M. Peters, "Tools for measurement of stress and strain fields in soft
tissue.," Ph.D. thesis. Maastrict University, PhD Thesis. Maastrict, 1987.
[90]
R. I. Griffiths, "Shortening of muscle fibers during stretch of the active cat
medial gastrocnemius muscle: The role of tendon compliance," Journal of
Physiology, vol. 436, pp. 219-236, 1991.
[91]
L. Puzanová and M. Grim, "The distribution of muscle spindles in some long
extremity muscles of the rat," Sborník Lékarský, vol. 70, pp. 266-272, 1968.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
111
[92]
N. Chin, M. Cope, and M. Pang, "Number and distribution of spindle capsules in
seven hindlimb muscles of the cat.," in Symposium on Muscle Receptors, D.
Barker, Ed. Hong Kong: Hong Kong University Press, 1962, pp. 241-248.
[93]
J. E. Swett and E. Eldred, "Distribution and numbers of stretch receptors in
medial gastrocnemius and soleus muscles of the cat.," Anatomical Record, vol.
137, pp. 453-460, 1960.
[94]
J. E. Swett and E. Eldred, "Relation between spinal level and peripheral location
of afferents in calf muscles of the cat," American Journal of Physiology, vol.
196, pp. 819-823, 1959.
[95]
H. Yellin, "A histochemical study of muscle spindles and their relationship to
extrafusal fiber types in the rat.," American Journal of Anatomy, vol. 125, pp.
31-46, 1969.
[96]
F. E. Emonet-Dénand, "On the subdivision of static and dynamic fusimotor
actions on the primary ending of the cat muscle spindle," Journal of Physiology,
vol. 268, pp. 827-861, 1977.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
112
Appendix A:
Technical Drawings
PCB Board
Circuit Diagram............................................................... 113
PCB Layout..................................................................... 113
CAD Drawings
Cantilever ........................................................................ 114
Spindle Housing.............................................................. 115
Guide............................................................................... 116
Stop ................................................................................. 117
Shim Spacer .................................................................... 118
Nut................................................................................... 119
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
113
R1
120
C5
22pF
R9
390k
VCC
7
R5
1k
R10
10k
LM 308
Strain Gage X
4
Strain Gage Y
6
R4a
150k
Out
5
R6
1k
R4b
100k
7555
2
C1
6.8nF
R2
120
1
R7a
470k
R7b
5100k
C4
1nF
R8a
470k
Wire Connections:
• Ground to 3 pin connector
• Freq. Out to 3 pin connector
• 5v to 3 pin connector
• F to Strain gage Y
• B to C5
• A to C5
• D to dd
• Dd to Strain Gage X
• H to I
• J to Stain Gage X
• C to Strain Gage Y
PCB Traces to Sever
(marked with X)
• Through-hole in R2
• Connection between R2
and LM308
R8b
5100k
C5
dd
j
R1
c
blank
f
R8a&b
& C3
C2
R7a&b
& C4
R5
LM308
a
b
h
5V
Freq Out
7555
Jumper Connections
• Put a jumper between the
2 boards when a line
“comes out” of the board.
C3
1nF
Gnd
R6
R2
d
i
R4a&b
R9
R10
C1
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
114
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
115
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
116
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
117
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
118
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
119
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
120
KRISTEN N. JAAX
(206) 729-8050
[email protected]
4204 NE 95th St. Seattle, WA 98115
EDUCATION
Doctor of Medicine, Anticipated Completion December, 2002
University of Washington, School of Medicine, Seattle, WA.
Doctor of Philosophy in Bioengineering, June, 2001
University of Washington, School of Engineering and School of Medicine, Seattle, WA.
Ph.D. Dissertation: A Robotic Muscle Spindle: Neuromechanics of Individual and Ensemble
Response.
Bachelor of Science in Mechanical Engineering, with distinction, June, 1994
Stanford University, School of Engineering, Palo Alto, CA.
PROFESSIONAL AND ACADEMIC EXPERIENCE
Research Assistant, 7/96-6/01
Biorobotics Laboratory. Department of Electrical Engineering,University of Washington,
Seattle, WA. PI: Blake Hannaford, PhD.
Projects: hardware design and manufacture of miniature displacement sensor & linear
actuator; control algorithm design; printed circuit board design; computational and
mechatronic modeling of muscle mechanics, neural transduction and neural encoding;
mechatronic modeling of individual mammalian muscle spindle; experimental reconstruction &
analysis of multi-sensor integration behavior in muscle spindle population.
Research Assistant, 6/95-8/95
Human Motion Analysis Lab, Department of Physical Medicine and Rehabilitation, University
of Washington, Seattle, WA, PI: Joe Czerniecki
Projects: developed gait analysis software
Biomechanical Engineering Technician, 6/93-9/93
Bone Densitometry Lab, Spinal Cord Injury (SCI) Center, Palo Alto Veterans Affairs, Palo Alto,
CA. PI: B. Jenny Kiratli, PhD.
Projects: analyzed mechanical loading of femur during fracture events to develop a clinical
estimator of fracture risk in Spinal Cord Injury.
Biomechanical Engineering Technician, 6/92-9/92
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
121
Rehabilitation R&D Center. Palo Alto Veterans Affairs, Palo Alto, CA. PI: Eric Sableman,
PhD.
Projects: spine-stabilization mechanisms for Roto-Rest trauma therapy beds; design of
automated quadriplegic transfer device.
Summer Preprofessional, 6/90-9/90, 6/91-9/91
IBM Federal Sector Division, Space Station Data Management System (DMS), Houston, TX.
Supervisor: Bob Brauer.
Projects: translated and updated thermal control software; generated software functionality
document.
Surgical Research Assistant, 6/89-8/89
Dr. Michael DeBakey Summer Surgery Fellowship, Baylor College of Medicine, Houston, TX.
Supervisor: Polk Smith.
Ophthalmic Photography Assistant, 6/87-8/87
Hermann Eye Center, Hermann Hospital, Houston, TX. Supervisor: Sue McCraney.
PUBLICATIONS
Jaax, KN, “A Robotic Muscle Spindle: Neuromechanics of Individual and Ensemble
Response,” Ph.D. Thesis, Department of Bioengineering, University of Washington,
2001.
Jaax, KN, van Donkelaar, C.C., Drost, M.R., Hannaford, B, “Fusimotor Effect on Signal
Information Content of Ia Ensemble Model Reconstructed from Dynamic Intramuscular
Strain Data,” submitted to Journal of Physiology, June, 2001.
Jaax, KN, Hannaford, B, “Mechatronic Design of an Actuated Biomimetic Length and Velocity
Sensor,” submitted to IEEE Transactions on Robotics and Automation, May, 2001.
Jaax, KN, Hannaford, B, “A Biorobotic Structural Model of the Mammalian Muscle Spindle
Primary Afferent Response,” submitted to Annals of Biomedical Engineering,
February, 2001.
Jaax, KN, Marbot, PH, Hannaford B, “Development of a Biomimetic Position Sensor for
Robotic Kinesthesia,” Proceedings of the IEEE/RSJ International Conference on
Intelligent Robots and Systems, Takamatsu, Japan, Nov. 2000. pp. 1255-60.
Jaax, KN, “Biomechanical Analysis of the Role of Wrist Guards in "Split-Top" Forearm
Fractures,” Proceedings of the World Congress 2000 on Medical Physics and
Biomedical Engineering and the 22nd Annual International Conference of the IEEE
Engineering in Medicine and Biology Society, Chicago, IL, July 2000.
Abstracts:
Jaax, KN, “Developing a Robotic Muscle Spindle,” Proceedings of The Whitaker Foundation
Biomedical Engineering Research Grants Conference 2000, La Jolla, CA, August 2000.
Jaax, KN, Hannaford B, “A Biorobotic Model of the Mammalian Muscle Spindle,” Annals of
Biomedical Engineering. 2000. 28(S1). pp. S-8.
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
122
Jaax, KN and BJ Kiratli, “Estimating Risk of Fracture During Activities of Daily Living Within
the Spinal Cord Injured Population,” Journal of Investigative Medicine. 1996. 44(1) pp.
155A.
PRESENTATIONS
Jaax, KN, “A Robotic Muscle Spindle and Other Current Research in the Biorobotics
Laboratory at the University of Washington,” Invited Seminar at ATR Human
Information Processing Research Laboratories, Kyoto, Japan, Nov. 2000.
Jaax, KN, Marbot, PH, Hannaford B, “Development of a Biomimetic Position Sensor for
Robotic Kinesthesia,” IEEE/RSJ International Conference on Intelligent Robots and
Systems, Takamatsu, Japan, Nov. 2000, platform presentation.
Jaax, KN, Hannaford B, “A Biorobotic Model of the Mammalian Muscle Spindle,” Biomedical
Engineering Society Annual Meeting, Seattle, WA, Oct. 2000, platform presentation.
Jaax, KN, “Developing a Robotic Muscle Spindle,” The Whitaker Foundation Biomedical
Engineering Research Grants Conference 2000, La Jolla, CA, August 2000, poster
presentation.
Jaax, KN, “Biomechanical Analysis of the Role of Wrist Guards in "Split-Top" Forearm
Fractures,” World Congress on Medical Physics and Biomedical Engineering, Chicago,
IL, July 2000, platform presentation.
Jaax, KN, and BJ Kiratli, “Estimating Risk of Fracture During Activities of Daily Living
Within the Spinal Cord Injured Population,” Western Medical Student Research
Conference, Carmel, CA, Feb. 1996, platform presentation.
Jaax, KN. “Developing a Mathematical Model of Human Eye Movements to Optimize
Strabismus Surgery,” 2nd Annual Super! Conference, Gainesville, FL, Apr. 1990,
platform presentation.
Jaax, KN. “Developing a Mathematical Model of Human Eye Movements to Optimize
Strabismus Surgery,” Supercomputing ’89, Reno, NV, Nov. 1989, poster presentation.
FELLOWSHIPS AND HONORS
Paul G. Allen Foundation for Medical Research Fellowship, 2001-present
Whitaker Graduate Fellowship in Biomedical Engineering, 1996-present
Medical Scientist Training Program Fellowship, 1994-present
ARCS Fellowship, Seattle Chapter, 1998-2000
Travel Grant, NSF Engineering Education Scholars Workshop at Carnegie Mellon, July 1999
Terman Award, Top 5% of graduating class, Stanford School of Engineering, 1994
President, Tau Beta Pi Engineering Honor Society, Stanford University, 1994
Phi Beta Kappa, 1994
1st Place, SuperQuest National Supercomputing Competition, sponsored by IBM, NSF, and the
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington
123
Cornell Theory Center, 1989
PROFESSIONAL SERVICE & ASSOCIATIONS
Ad Hoc Reviewer, Annals of Biomedical Engineering
Ad Hoc Reviewer, Behavioral & Brain Sciences
Admissions Committee Member, University of Washington School of Medicine, 1995-2000
Member, IEEE, Engineering in Medicine and Biology Society
Member, Biomedical Engineering Society
 2001, K.N. Jaax
Ph.D. Dissertation
University of Washington