Annals of Biomedical Engineering, Vol. 29, pp. 330–339, 2001
Printed in the USA. All rights reserved.
0090-6964/2001/29共4兲/330/10/$15.00
Copyright © 2001 Biomedical Engineering Society
Identification of Static and Dynamic Components of Reflex Sensitivity
in Spastic Elbow Flexors Using a Muscle Activation Model
BRIAN D. SCHMIT and W. ZEV RYMER
Sensory Motor Performance Program, Rehabilitation Institute of Chicago, Department of Physical Medicine and Rehabilitation,
Northwestern University Medical School, Chicago, IL
(Received 5 May 2000; accepted 6 February 2001)
Abstract—Static and dynamic components of the stretch reflex
were studied in elbow flexors of 13 hemiparetic brain-injured
individuals. Constant-velocity joint rotations were applied to
the elbow, and the resulting stretch reflex torque and electromyographic responses were recorded in the biceps brachii
and brachioradialis muscles. Ten elbow extension velocities
between 6 and 150 deg s⫺1 were applied in random order. The
resulting reflex torque response was plotted as a function of
elbow angle and fitted with a mathematical model designed to
depict elbow flexor activation. We found that four of the six
model parameters were essentially independent of test velocity.
Conversely, 73% 共19/26兲 of cases involving the other two
model parameters were dependent on velocity of joint extension 共p⬍0.05兲. We conclude from these results that four of the
model parameters reflect the static reflex response while the
two remaining velocity-dependent parameters reflect the dynamic reflex response. To describe overall velocity dependence
of stretch reflexes in spastic elbow muscles, the two dynamic
reflex parameters were fitted to a fractional exponential function of velocity, similar to a model previously used to describe
spindle firing rate in the cat hindlimb. We found that the mean
velocity exponent of the dynamic reflex parameters was 0.24
⫹ 0.17 共s.d.兲 共N ⫽ 13兲, a value similar to that for muscle
spindle velocity sensitivity in reduced animal preparations. We
conclude that both static and dynamic reflex sensitivities can be
measured by examining different aspects of the torque/angle
relation associated with the reflex response to a large-amplitude
ramp stretch of the elbow. © 2001 Biomedical Engineering
Society. 关DOI: 10.1114/1.1359496兴
able reflex response in uninjured subjects under similar
conditions. In the current study, we hypothesize that the
reflex resistance to stretch of spastic muscle may have
identifiable static 共position兲 and dynamic 共velocity兲 components, similar to well-defined muscle spindle position
and velocity components.18,21 The benefit of studying
these sensitivities is twofold. First, knowledge of the
static and dynamic stretch reflex sensitivities increases
our understanding of the pathophysiology of spasticity
associated with brain injury. Second, relative differences
in position and velocity sensitivities may have functional
consequences in the motor control of spastic muscles,
and may suggest rehabilitation strategies to target position or velocity sensitivity.
The procedure for identification of static and dynamic
reflex sensitivity was based on an earlier model of reflex
torque/joint angle relations in spastic elbow flexors.24
This model accounts for the mechanics of the elbow
flexors and describes the muscle activity as a sigmoid
function of elbow angle. The reflex response is best
expressed as the summation of the response from two
muscle groups, each with a separate activation function.
The model for the torque/angle relation is formulated as
␶ e ⫽ ␴ 1 PCSABid Bi⫹ ␴ 2 PCSABrad Bra⫹ ␴ 2 PCSABRDd BRD ,
Keywords—Spasticity, Stretch reflex, Stroke, Biomechanics.
共1兲
INTRODUCTION
Spasticity, classically defined as a velocity-dependent
reflex resistance to externally imposed movement,13 can
be quantified in brain-injured individuals using mechanical measures of the response to a controlled muscle
stretch.12,23 These types of stretch reflex responses are
only obtained in the spastic individuals, with no measur-
where ␶ e is the measured elbow torque, ␴ is the muscle
stress 共which is proportional to activation兲,
PCSABi,Bra,BRD is the physiological cross sectional area
of the biceps, brachialis, or brachioradialis,1 and
d Bi,Bra,BRD is the moment arm of the biceps, brachialis, or
brachioradialis.19 Note that the moment arm is a function
of elbow angle, and thus the model accounts for changes
in the torque–angle relation resulting from the mechanics
of the musculoskeletal structure. The muscle stresses ␴ 1,2
represent the activation of the muscle groups and are
functions of elbow angle ␪ which fit the relation
Address correspondence to Brian D. Schmit, Department of Biomedical Engineering, Marquette University, P.O. Box 1881, Milwaukee, WI 53201-1881. Electronic mail: [email protected]
W. Zev Rymer is also associated with the Department of Biomedical Engineering at Northwestern University.
330
Stretch Reflexes in Spasticity
再 冉 冊 冎
再 冉 冊 冎
␴ 1 ⫽K 1
␴ 2 ⫽K 2
331
1
␪⫺␮
1
erf
⫹ ,
2
2
冑2 ␤ 1
共2兲
1
␪⫺␮-␣
1
erf
⫹ .
2
2
冑2 ␤ 2
In this formula, erf is the error function, or cumulative
normal distribution. The six model parameters are ␣, ␮,
␤ 1,2 , and K 1,2 . Intuitively, this formulation represents
the muscle stress as the probability of motoneuron firing,
which is a function of joint angle. This description of
muscle activation is appealing because many naturally
occurring phenomena follow a normal probability distribution and good fits were obtained to the data.24 Note
from Eq. 共1兲 that a single muscle stress function ␴ 2 is
used in the model to represent two of the major elbow
flexors, indicating a brachialis/brachioradialis synergy.
This assumed synergy is based upon studies of voluntary
muscle activation patterns3,4 and was consistent with the
bimodal torque patterns observed in these experiments.
Torque responses of the two-muscle synergy were validated using only brachioradialis electromyograms.24
Each parameter of the activation function model has a
unique influence on the description of the stretch reflex
torque/angle response. These effects are summarized in
Fig. 1. The parameters ␮ and ␮ ⫹ ␣ determine the
relative position of the rise in the torque/angle relation.
Thus, an increase in ␮ will shift the response into extension and similarly, a decrease shifts the entire response into flexion as shown in Fig. 1共A兲. In effect, the
parameters ␮ and ␮⫹␣ are the angles at 50% recruitment of the respective muscle group. The ␤ parameters,
akin to the standard deviation of a normal distribution,
directly influence the angular range over which the
torque increases. Thus, a large ␤ results in a large range
of torque rise, or equivalently a lower slope, while a
smaller ␤ has the opposite effect 关Fig. 1共B兲兴. The K
parameters do not affect the angular range of rise, but
scale the torque response as demonstrated in Fig. 1共C兲.
Note that changes in K also increase or decrease the
overall slope of the torque/angle relation. We hypothesized that the parameters ␮, ␣, and ␤ would reflect the
static stretch reflex response due to the fact that the
angular characteristics of the torque response are determined by these parameters. Conversely, we expected the
K parameters to reflect dynamic stretch reflex properties,
since these parameters scale the torque–angle relation,
independent of joint angle.
In the current study, we tested the hypothesis that the
parameters of the activation function model, which are
used to describe the torque/angle relation, distinguish the
motoneuron response to static spindle inputs from the
response to dynamic spindle inputs. Specifically, we pos-
FIGURE 1. The effects of changing activation function model
parameters on the torque–angle response is shown for the
three different types of parameters. Normalized, hypothetical
data are shown. „A… Changing ␮ shifts the entire response to
the left or right, but retains the slope and shape of the response. „B… Changing ␤ changes the slope of the response,
but retains the ‘‘inflection point’’ at its original joint angle.
„C… Increasing K increases the magnitude of the response,
scaling the entire torque–angle relation.
tulated that the model parameters ␣ and ␮ would be
independent of movement speed, indicating that they are
associated with the muscle response to the static spindle
inputs. Conversely, the parameters K 1,2 , were expected
to be correlated with stretch velocity, and therefore associated with dynamic spindle inputs.
Data were obtained from 13 participants who had
suffered a unilateral brain injury more than 1 yr prior to
testing. We found that the parameters ␮, ␣, and ␤ 1,2
were independent of movement speed, suggesting that
they reflect the position sensitivity of the stretch reflex.
In a similar way, the parameters K 1,2 were found to be
correlated with movement speed, suggesting that they
represent the dynamic sensitivity. Both sets of parameters were found to correlate with clinical measures of
elbow spasticity.
METHODS
Subjects
We tested 13 hemiparetic brain-injured subjects,
whose major clinical features are summarized in Table 1.
Subjects had mild to severe levels of spasticity, with
332
SCHMIT and RYMER
TABLE 1. Description of study participants
Type of Affected
Fugl– Time post
Subject injury
arm
Ashwortha Meyerb injury (yr) Age
A
B
C
D
E
F
G
H
I
J
K
L
M
Stroke
Stroke
Stroke
TBIc
Stroke
Stroke
Stroke
Stroke
Stroke
Stroke
TBIc
Stroke
Stroke
Right
Left
Left
Left
Left
Left
Right
Left
Right
Right
Right
Left
Right
2
2
2
3
2
3
3
2
1
4
4
4
2
58
16
21
18
27
12
17
25
40
17
14
15
51
5
3
5
18
3
13
3
3
4
9
8
14
4
41
64
66
48
58
61
53
53
55
58
27
57
61
a
Four point Ashworth scale (see Ref. 2).
Sixty-six point upper extremity motor function score (see Ref. 7).
c
TBI⫽traumatic brain injury.
b
Ashworth scores of 1 共out of 4兲 or greater for the elbow
flexors.2 Motor function was assessed using the 66-point
Fugl–Meyer scale, which uses isolated movements to
assess arm function.7 All experimental procedures were
approved by the Institutional Review Board of Northwestern University and complied with the principles of
the Declaration of Helsinki. Informed consent was obtained prior to each test session.
Procedures
The experimental apparatus and subject preparation
used in this study have been described previously.24
Briefly, displacements of elbow angle were imposed at a
constant angular velocity using a Biodex Rehabilitation/
Testing System 2 共Biodex Medical Systems, Inc., Shirley, NY兲 共hereafter referred to as Biodex兲. The hand and
wrist were cast and affixed to a manipulandum extending
from the Biodex motor. The motor position was adjusted
to achieve a shoulder abduction angle of 80° and shoulder flexion of 3°–10°. With full elbow extension defined
as 180°, minimum elbow angles ranged from 47° to 55°
while maximums were 130°–157°. Each elbow perturbation sequence consisted of a constant velocity stretch of
the elbow flexors, a 10 s pause, and a return to the
starting position. The cycle was repeated every 60 s and
the extension portion of the data was used for further
analysis.
Ten different movement speeds were tested: 6, 15, 30,
45, 60, 75, 90, 105, 120, and 150 deg s⫺1 . Five test trials
were conducted at each speed resulting in a total of 50
elbow stretches. Five test epochs, with each epoch consisting of ten trials 共one trial at each test speed兲 were
applied sequentially. Within each epoch, the movement
speeds were applied in random order, randomized by the
computer. This procedure was implemented to eliminate
any bias associated with the order in which movement
speeds were applied.
Surface electromyograms 共EMGs兲 were made of the
biceps, the brachioradialis, and the lateral head of the
triceps. Electrodes 共ConMed, Model 1700, ConMed
Corp., Utica, NY兲 were placed over each muscle belly on
lightly abraded skin. Electrode leads were connected to
an isolated, differential preamplifier/filter, the signal was
band pass filtered at 10–500 Hz, and preamplified by
1000. Further amplification of 1–100, depending on the
signal amplitude, was performed prior to digitizing the
data.
Torque, position, velocity, and the three EMG signals
were low-pass filtered at 500 Hz and digitized at 1000
Hz using a Macintosh 840AV computer with a National
Instruments NB-MIO data acquisition board 共National
Instruments, Austin, TX兲 and custom LABVIEW software
共National Instruments兲. Data acquisition was coordinated
with applied stretches using computer-generated triggers,
at 1 min intervals, to initiate elbow movements by the
Biodex.
Analysis
Reflex torque of the elbow flexors was estimated for
the extension portion of the perturbation by subtracting
the passive torque, measured at a slow velocity 共6
deg s⫺1 兲, from the torques measured at every other test
velocity, leaving only the stretch reflex torque. First, the
torque data were selected from the extension portion of
each trial, beginning after the inertial artifact that accompanied initiation of the extension movement and terminating immediately prior to the inertial artifact associated
with the deceleration of the movement. Each slow trial
共6 deg s⫺1 ) was examined for muscle activity, based on
the EMGs. A trial with no evidence of muscle activity
was selected for each subject and the torque angle relation was fit with a seventh order polynomial to obtain the
passive torque. The passive torque was then subtracted
from each of the other trials. This manipulation is possible because ‘‘passive’’ torque responses to flexion/
extension are essentially velocity insensitive at the
elbow.8 关Note, however, that this assumption is based
upon data obtained with a smaller angular range 共60°–
120°兲 and a smaller velocity range 共6–90 deg s⫺1 ).] The
resulting elbow torque was low-pass filtered at 5 Hz
using a zero phase delay filter. Sample data for the reflex
torque response to elbow extension are shown in Fig. 2
for five speeds 共30, 60, 90, 120, and 150 deg s⫺1 ).
Reflex torque responses were consistent with previously observed patterns, including a baseline, rise of
torque at angular onset, and leveling of torque at larger
joint angles.24 As a result of these similarities, the previously developed activation function model 关Eqs. 共1兲
Stretch Reflexes in Spasticity
FIGURE 2. The elbow torque–angle relation for subject C is
shown for five of the tested speeds. The median–amplitude
trial is shown for each speed. Increasing the movement
speed increased the magnitude of the response. Movement
speeds include 30, 60, 90, 120, and 150 deg sÀ1 .
and 共2兲兴 was used to parameterize the torque response
for each trial. The model parameters, ␣, ␮, ␤ 1 , ␤ 2 , K 1 ,
and K 2 were estimated using the constr function from the
optimization toolbox of MATLAB 共The Math Works, Inc.,
Natick, MA兲. This nonlinear least-squares optimization
utilizes a sequential quadratic programming method and
enabled us to apply the following constraints to the parameters: ␪ min⬍ ␮ ⫺ ␤ 1 , ␪ min⬍ ␮ ⫹ ␣ ⫺ ␤ 2 , ␪ max⬎ ␮
⫹ ␤ 1 , ␪ max⬎ ␮ ⫹ ␣ ⫹ ␤ 2 , 2⬍ ␤ 1 ⬍50, 2⬍ ␤ 2 ⬍50, K 1
⬎0.2* K 2 , and K 2 ⬎0.2* K 1 . These constraints were
based on previous parameter estimates obtained using an
unconstrained Levenberg–Marquardt optimization procedure with the same model.24 The constrained optimization was utilized because it improved the likelihood of
convergence to reasonable model parameters. Specifically, the onset and plateau angles were constrained
within the stretch range, the rise angles were within
previously convergent values 共2–50兲, and the limits on
the K parameters assured activation of both muscle
groups. A sensitivity analysis of the model parameters,
conducted in a previous study, demonstrated comparable
sensitivity for all six parameters when assessed across
the tested joint range.24 This same study validated the
model using EMGs, recorded simultaneously with the
reflex torque.
The initial estimates of ␮ and ␣ were obtained from
visual inspection of the EMGs. The initial guesses for ␤
were 10°, and for K were 1500 based on previous
results.24 There were at least two possible converging
solutions that fit within the constraints. These two solutions essentially interchanged which muscle group was
activated first. We assumed that by setting the initial
estimates of ␮ and ␣, based on the EMGs of the biceps
and brachioradialis, the solutions would converge correctly.
The dependence of the model parameters on stretch
velocity was tested for each subject. First, model param-
333
eters were obtained for each of the 45 trials 共five trials at
each of nine speeds兲 for all 13 participants in the study.
A linear regression was performed on each parameter,
for each subject, using the velocities at 30 deg s⫺1 and
greater. The 15 deg s⫺1 trials were not used because they
did not reliably produce a reflex response in all participants. The regression slope was tested for significance at
␣⫽0.05, for each parameter, to determine dependence on
velocity. The mean of the velocity-independent parameters was calculated and the velocity-dependent parameters were then modeled to determine the velocity
‘‘threshold’’ and ‘‘gain’’ 共defined below兲.
The velocity-dependent parameters (K 1,2 as shown below兲 were used to identify the characteristics of the velocity response. The velocity response of the cat soleus
muscle spindle has been characterized using the
function10
r⫺r 0 ⫽C 2 共 x⫺x 1 兲v n .
共3兲
In this formulation, r is the spindle firing rate, x is the
muscle length, and v is the speed of lengthening. There
are four parameters that describe the overall relation, r 0 ,
C 2 , x 1 , and n, with the velocity dependence expressed
as an exponential function. We expected that the velocity
dependence could be formulated in a similar way for the
velocity dependent parameters of the model K 1,2 共as
demonstrated below兲. Therefore, we used the following
equation to express velocity dependence:
K 1 ⫹K 2 ⫽C 1 ⫹C 2 v n ,
共4兲
where the sum of the parameters from the activation
function model K 1,2 is now defined as the dependent
variable, v is the speed of stretch and the independent
variable, and C 1,2 and n are the model parameters. In this
formulation, the muscle length has been eliminated because the parameters K 1,2 are independent of length. The
velocity dependence of the torque output, reflected by
the parameters K 1,2 , is assumed to have the same form
as the velocity dependence of spindle firing 关Eq. 共3兲兴.
We applied the constraints n⬍1, C 1 ⬍0. The parameters
to the model were estimated, again using the constr function of MATLAB. Two parameters of interest were singled
out: the velocity threshold was calculated as the intersection of the model with the velocity axis 共the independent
variable兲 and the velocity gain was identified as the exponential parameter, n.
As a final analysis, we determined the correlation of
the activation function model parameters with a clinical
measure of spasticity: the Ashworth score.2 In order to
generalize the response for each parameter type, the sum
of similar mean parameters was identified for each subject. First the mean parameter values were calculated
334
SCHMIT and RYMER
across all eight velocities: 30 deg s⫺1 or greater. Again,
the trials at 15 deg s⫺1 were excluded due to unreliable
reflex responses. Then, the sums, ␮ ⫹( ␮ ⫹ ␣ ), ␤ 1 ⫹ ␤ 2 ,
and K 1 ⫹K 2 were correlated to the Ashworth scores and
the Pearson’s correlation coefficients were examined for
significance 共t test, slope versus 0, ␣⫽0.05兲.
RESULTS
The stretch reflex torque responses of 13 hemiparetic
brain-injured individuals were evaluated using constant
velocity ramp stretches at ten velocities, ranging between
6 and 150 deg s⫺1 . An activation function model was
used to characterize the torque responses and model parameters were analyzed to identify velocity dependence.
Velocity dependent parameters were then modeled as a
function of velocity using a variation of the cat spindle
model.9
Typical torque responses, plotted versus elbow angle,
are shown in Fig. 2 for five speeds 共subject C兲. The trials
shown represent the median trial for each speed, based
on the peak torque response of each trial. Clear increases
in the magnitude of the stretch reflex torque response
were observed with increasing speed of stretch for all 13
subjects tested. However, as evidenced in Fig. 2, a fivefold increase in speed of stretch produced much less than
a fivefold increase in the torque response for speeds
greater than 30 deg s⫺1 . The activation function model
was used to characterize the speed-sensitive component
of the stretch reflex torque response.
The model produced a good fit of the data 共mean
square error ⬍0.1 N m兲 for all subjects, at speeds between 30 and 150 deg s⫺1 . An example is shown in Fig.
3. The activation function model produced estimates of
the activation levels of the biceps and the brachialis/
brachioradialis muscle groups as shown in Fig. 3共A兲.
These activation functions were then used to predict the
elbow torques of each muscle group 关Fig. 3共B兲兴 and
provide an accurate representation of the elbow torque
response to stretch 关Fig. 3共C兲兴. Details of the activation
function model have already been described.24 It is worth
noting that the current application of the model extends
its use to stretch velocities in the range of 30–150
deg s⫺1 . To assess the trial-to-trial variation and reproducibility of the stretch reflex response, the standard deviation of the model parameters were calculated for each
subject, for each speed 共n⫽5 trials兲, and found to have a
mean of 10.2° for ␮, ␮⫹␣, 4.5° for ␤ 1 , ␤ 2 and 809 for
K 1 , K 2 across all subjects.
The effect of stretch velocity on the activation function model parameters was evaluated for each of the 13
subjects. The parameter values for the 40 test trials 共five
trials at each of eight speeds, 30, 45, 60, 75, 90, 105,
120, and 150 deg s⫺1 ) were plotted against test velocity
and a linear regression analysis was done. Data from
FIGURE 3. Activation model fit †Eq. „1…‡ to the torque data
for a single trial „subject J at 45 deg sÀ1 …. Model parameters
were estimated using the constr function in MATLAB. „A… The
calculated muscle stress for the biceps and brachialisÕ
brachioradialis is shown. In this case the biceps preceded
the brachioradialis, but did not come on as strongly. „B… The
resulting elbow torque from each muscle group is shown.
These curves were calculated from the muscle stress, the
physiological cross-sectional area of the muscles „from published values, Ref. 1… and the moment arms „also from published values, Ref. 19…. „C… The model outputs overlaid the
raw data precisely „mean square error Ë0.1 N m….
subject D are shown in Figs. 4共A兲–4共C兲. In this subject,
it appeared as though the model parameters K 1 and K 2
were velocity dependent, while the velocity dependence
of the remaining parameters was not evident. The results
shown in Fig. 4 were consistent with the results from the
remaining 12 subjects.
The slope of the linear regression between each parameter and velocity was calculated for each subject, and
was tested for significance 共t test, ␣ ⫽ 0.05兲. A summary
of these results is shown in Fig. 5. For the ␮ and ␮⫹␣
parameters 关Fig. 5共A兲兴, only eight of the 26 linear regressions 共31%兲 yielded a significant slope 共p⬍0.05兲.
Similarly, only eight of 26 ␤ parameters 共31%兲 关Fig.
5共B兲兴 had a significant dependence on velocity 共p⬍0.05兲.
In addition, when the ␮ and ␮⫹␣ parameters had a
significant effect, it was still small, with R 2 values averaging 0.07 and 0.11 for ␮ and ␮⫹␣, respectively.
Conversely, K 1 and K 2 had average R 2 values of 0.26
and 0.33, respectively.
Stretch Reflexes in Spasticity
FIGURE 4. The model parameters, which were estimated for
each trial, were plotted against movement velocity and a
linear regression was conducted to assess the correlation
between the model parameters and velocity. All parameters
are shown for subject D. „A… The parameters ␮ and „␮¿␣…
showed no obvious trend with velocity. „B… Similarly, parameters ␤ 1 and ␤ 2 demonstrated no obvious trend. „C… However, parameters K 1 and K 2 showed an apparent increase
with movement speed.
We conclude from these data that in general, that ␮,
␣, ␤ 1 , and ␤ 2 are independent of stretch velocity. Thus,
these parameters are likely to reflect the ‘‘static’’ sensitivity of the reflex arc, thereby providing a measure of
the motoneuronal response to changes in joint position
共independent of velocity兲. In considering the parameters
␮, ␣, ␤ 1 , and ␤ 2 as independent of stretch velocity, we
then obtained accurate estimates of the parameter values
by averaging across speeds. The mean parameter values
are shown in Fig. 6 for all study participants.
In contrast, the model parameters K 1 and K 2 had a
significant dependence on velocity in 19 of 26 cases
共73%兲 as shown in Fig. 5共C兲 共p⬍0.05兲. We conclude
from these results that K 1 and K 2 are likely to reflect the
reflex sensitivity to the dynamic spindle afferent response. As a result, we used these parameters to characterize the velocity dependence of the stretch reflex in
spastic elbow flexors. Recall that K 1 and K 2 represent
335
FIGURE 5. The slope of each model parameter vs velocity is
shown for each subject. Slope significance „compared to
zero… is indicated by *„pË0.05…. „A… Eight of 26 ␮ or „␮¿␣…
parameters demonstrated a significant slope. „B… Seven of
26 ␤ parameters had a significant slope. „C… 19 of 26 K
parameters had a significant slope. These data suggest the
generalization that ␮ and ␤ parameters are independent of
velocity while K parameters change with velocity.
FIGURE 6. The mean angular parameters are shown for each
subject. Mean parameters were obtained by averaging
across speeds: „A… The ␮ and ␮¿␣ parameters. „B… The ␤ 1
and ␤ 2 parameters.
336
SCHMIT and RYMER
FIGURE 7. The velocity model
†Eq. „3…‡ is shown for subject G.
The velocity model was fit using
a nonlinear least squares technique „constr function from MATLAB…. The data from trials at velocities of 6 and 15 deg sÀ1 are
not included because these
slower speeds did not reliably
produce reflex responses in all
subjects. The velocity threshold
was identified as the point
where the function crossed zero
„6.5 deg sÀ1 …. The velocity gain
was defined by the exponent of
velocity „0.49….
the attainable level of activity that the muscle reaches in
the stretch reflex, or alternately, K 1 and K 2 can be considered the plateau level of muscle stress at large elbow
angles. Since K 1 and K 2 are velocity dependent, our
results suggest that the potential level of muscle activity
increases with speed of stretch, while the angular dependence 共reflected by ␮, ␣, ␤ 1 , and ␤ 2 ) remains relatively
unchanged.
The velocity dependence of K 1 and K 2 was evaluated
in greater detail using a velocity-dependent reflex model
关Eq. 共4兲兴, which allowed comparisons to the dynamic
response of muscle spindles measured in reduced cat
preparations 关Eq. 共3兲兴. In this analysis we summed K 1
and K 2 to include both muscle groups in the analysis.
The velocity model fit for subject G is shown in Fig. 7.
From this model, two parameters of the dynamic response were calculated. The velocity threshold, or speed
below which a reflex response would not be measured,
was calculated by extrapolation 共identified as the velocity
where K 1 ⫹K 2 ⫽0). This parameter is of clinical interest
because it defines the speed at which the elbow can be
moved without a stretch reflex response. The second
parameter, the velocity gain, was identified as the exponential parameter 共n兲 and was calculated for comparison
with dynamic muscle spindle data derived from reduced
animal preparations.10
The individual velocity thresholds and gains for all
study participants are displayed in Fig. 8. Two subjects,
subjects E and L, had velocity thresholds of zero. A
velocity threshold of zero corresponded with the parameter C 1 at the constraint. The velocity thresholds ranged
from 0.0 to 16.4 deg s⫺1 , with five of the 11 subjects
below the 6 deg s⫺1 level that was used for assessment
of ‘‘passive’’ joint properties. Unfortunately, velocity
thresholds were not measured directly, since the 6 and 15
deg s⫺1 trials did not provide the resolution necessary for
precisely quantifying the velocity threshold. In the
screening of individual trials at 6 deg s⫺1 , we found that
some stretches did elicit reflex activity even at this slow
speed; however, we always found at least one trial in
which reflex activity of the elbow muscles could not be
detected. The velocity exponent (n) ranged from 0.05 to
0.49 as shown in Fig. 8.
As a final analysis, we examined whether the model
parameters correlated with a clinical measure of spasticity 共the Ashworth score兲. In this analysis, we grouped the
parameters by type, using the sums of the mean parameters ␮⫹共␮⫹␣兲, ␤ 1 ⫹ ␤ 2 , and K 1 ⫹K 2 . These sums
were then correlated to the Ashworth score and examined for significance. Even in this small sample, we
FIGURE 8. The estimated velocity threshold and gain is
shown for each subject. „A… The velocity threshold ranged
between 0 and 16.4 deg sÀ1 . Subjects E and L had a threshold of 0 deg sÀ1 corresponding to C 1 Ä0, at the limit of the
constraint. „B… The velocity gain ranged from 0.05 to 0.49
indicating a modest dependence on velocity.
Stretch Reflexes in Spasticity
337
Evidence of an Activation Plateau in Multispeed
Experiments
FIGURE 9. The model parameters K 1 ¿ K 2 and ␮¿„␮… were
both correlated with Ashworth score. „A… K 1 ¿ K 2 was positively correlated with Ashworth. „B… ␮¿„␮¿␣… was negatively
correlated with the Ashworth score. These results indicate
that either parameter is a measure of spasticity, which is
comprised of a static „␮¿„␮¿␣…… and dynamic „ K 1 ¿ K 2 …
component.
Stretch reflex muscle activity demonstrated a clear
tendency to plateau as joint angle became large, supporting previous observations that were made using a single
speed.24 The consistency of the plateau torque response
across test speeds suggests that the leveling of muscle
activity at large elbow angles is not simply a temporal
phenomenon. Note that the parameters ␮, ␣, ␤ 1 , and ␤ 2
did not change with speed of stretch, inexorably linking
the torque plateau with absolute elbow angle, not with
time following torque onset. Thus, the leveling of torque
at large elbow angles is linked to the position, or the
static reflex response.
The static spindle afferent properties have typically
been modeled as a linear firing rate–length response.
This relation is derived from recordings in decerebrate
cat preparations6,14,15,18,16 as well as from microneurographic recordings of the human finger extensor
afferents.5,25 These studies are in striking contrast to our
reflex observations using large amplitude elbow
stretches, suggesting that another mechanism may play a
role in the stretch reflex response of the spastic elbow.
The Dynamic Reflex Response
found that both ␮⫹共␮⫹␣兲 and K 1 ⫹K 2 were significantly correlated with the Ashworth score 共p⬍0.05兲 as
shown in Fig. 9. Note that this correlation is consistent
with the previous observation that the angular threshold
of torque onset of the spastic elbow flexors is correlated
with the Ashworth score.11 If ␮⫹共␮⫹␣兲 decreases, or if
K 1 ⫹K 2 increases, the angular threshold would be expected to decrease.
DISCUSSION
We conclude from this study that spasticity following
hemiparetic brain injury is accompanied by an increase
in both static and dynamic components of the stretch
reflex elicited during ramp and hold angular extension of
the elbow. As a result of these analyses, we believe that
spinal pathways receiving primary spindle afferent input
共group Ia兲 display distinct static and dynamic length sensitivities. Given the association of the ␮ and ␮⫹␣ parameters with the static stretch reflex, and of K 1 and K 2
parameters with the dynamic stretch reflex, it appears
that static and dynamic reflex sensitivity can be readily
quantified using an activation function that includes these
共and other兲 parameters. Model parameters that were associated with static and dynamic reflex properties were
correlated to clinical measures of spasticity indicating
that the clinical perception of spasticity involves both
static and dynamic components.
Unlike the static reflex response, the characteristics of
the dynamic stretch reflex response were similar to those
predicted from recordings of muscle spindle afferents
during imposed ramp stretches in the decerebrate cat
共e.g., Houk et al.10兲. We found that effects of velocity
changes for parameters K 1,2 were steep at low velocities,
leveling as velocity increased. This response for these
parameters is qualitatively identical to the Ia afferent
firing rate in the deefferented, decerebrate cat17 共e.g.,
compare Fig. 7 to Fig. 6 of Matthews17兲.
Houk et al. formulated the spindle firing rate–velocity
response as a fractional exponential, or ‘‘power-law’’
relation,10 which also fits our data for the model parameters, K 1,2 关Eq. 共3兲, Fig. 7兴. Note that for many of the
subjects, the relation could also be fit with a straight line
without a significant loss in the variance accounted for,
so the power law relation was applied based on the
theory of spindle operation, rather than a clear trend in
the data. The velocity exponent 共n兲 of the spindle firing
rate was estimated at 0.3 in the decerebrate cat
hindlimb10 compared to a mean of 0.24 共range 0.05–
0.49兲 for the K 1,2 parameters in this study. However, the
range and estimated standard deviation of the fractional
exponent was larger for our data 共0.17兲 than the data
from the decerebrate cat model 共0.05兲. This difference
may be attributed to greater subject-to-subject variability
in brain injury and the fact that a lower variability in the
response would be expected from a reduced preparation.
SCHMIT and RYMER
338
Others have reported power-law relations for the
muscle spindle with larger exponential fractions. For example, Prochazka and Gorassini reported that an exponent of 0.6 provides a more accurate representation of
muscle spindle firing rate in awake, behaving cats.22 The
difference in spindle firing rate may reflect differences in
fusimotor drive between the imposed movements in decerebrate cats and awake cats producing voluntary leg
movements.
The lower 0.24 power-law relation that was calculated
for stroke reflex data in the current study may be the
result of quantifying the power law using reflex torque
data, rather than spindle firing rate. The reflex torque
measurements include both the spindle dynamic response
as well as the motoneuronal ‘‘processing’’ of the spindle
inputs. For example, a smaller power-law relation was
found for stretch reflex force20 compared to spindle firing
rate10 in decerebrate cats. Therefore, the motoneuron
may be acting to decrease the effects of larger stretch
velocities.
Implications for Quantification of Spasticity
The results of this study have implications for the
quantification of spasticity. Although the activation function model is unlikely to find direct clinical application
due to limited availability of the necessary quantitative
methods in the clinical setting, the findings of this study
are useful for designing and interpreting clinical measures of spasticity.
Previously, it has been demonstrated that clinical
measures of spasticity, such as the Ashworth scale, are
significantly correlated with two parameters from the
torque response to ramp and hold stretches—the angular
threshold of initiation of reflex torque and the absolute
torque measured at a fixed, extended position.11 Conceptually, these two measures correspond with the model
parameters ␮ ⫹共␮⫹␣兲 共angular threshold兲 and K 1 ⫹K 2
共torque at a specified angle兲. Based on the earlier discussion of these model parameters, we conclude that the
angular threshold represents an indirect measure of the
static reflex sensitivity, while the reflex torque near the
end of a ramp stretch reflects the dynamic reflex sensitivity. Thus, our correlations of the model parameters to
Ashworth scores are consistent with these previous results.
The stretch reflex model that was tested in the current
study may be more useful for monitoring spasticity than
previously used measures. As an example, changes in
static and dynamic sensitivity are likely to depend on the
type of therapeutic intervention. Moreover, we envision
that one type of effect 共reduction of either the static or
dynamic reflex component兲 may have significant functional advantages over the other. In summary, the activation function model, which includes static 共velocity
insensitive兲 and dynamic 共velocity sensitive兲 parameters,
provides greater insight into the neurophysiological
mechanisms than currently used spasticity measurement
techniques. As new measurement techniques evolve, the
distinction between static and dynamic parameters are
likely to be useful for spasticity management.
ACKNOWLEDGMENTS
This work was supported by The Ralph and Marion
C. Falk Medical Research Trust, NIDRR Grant No.
H133B30024 and NIH Traineeship Grant No. F32NS10203. We thank Dr. Derek Kamper for his constructive review of the manuscript.
REFERENCES
1
An, K. N., F. C. Hui, B. F. Money, R. L. Linsheid, and E. Y.
Chao. Muscles Across the Elbow Joint: A Biomechanical
Analysis. J. Biomech. 14:659–669, 1981.
2
Ashworth, B. Preliminary trial of carisoprodal in multiple
sclerosis. Practitioner 192:540–542, 1964.
3
Buchanan, T. S., D. P. J. Almdale, J. L. Lewis, and W. Z.
Rymer. Characteristics of synergic relations during isometric
contractions of human elbow muscles. J. Neurophysiol.
56:1225–1241, 1986.
4
Buchanan, T. S., G. P. Rovai, and W. Z. Rymer. Strategies
for muscle activation during isometric torque generation at
the human elbow. J. Neurophysiol. 62:1201–1212, 1989.
5
Edin, B. B., and A. B. Vallbo. Dynamic response of human
muscle spindle afferents to stretch. Neurophysiol. 63:1297–
1306, 1990.
6
Eldred, E., R. Granit, and P. A. Merton. Supraspinal control
of the muscle spindles and its significance. J. Physiol. (London) 122:498–523, 1953.
7
Fugl-Meyer, A. R., L. Jaasko, I. Leyman, S. Olsson, and S.
Steglind. The post-stroke hemiplegic patient. I. A method for
evaluation of physical performance. Scand. J. Rehab. Med.
7:13–31, 1975.
8
Given, J. D., J. P. A. Dewald, and W. Z. Rymer. Joint
dependent passive stiffness in paretic and contralateral limbs
of spastic patients with hemiparetic stroke. J. Neurol. Neurosurg. Psychiatry 59:271–279, 1995.
9
Houk, J. C., and W. Z. Rymer. Neural control of muscle
length and tension. In: The Nervous System, edited by J.M.
Brookhart and V.B. Mountcastle. Bethesda, MD: American
Physiological Society, 1981, pp. 257–324.
10
Houk, J. C., W. Z. Rymer, and P. F. Crago. Dependence of
dynamic response of spindle receptors on muscle length and
velocity. J. Neurophysiol. 46:143–168, 1981.
11
Katz, R. T., G. P. Rovai, C. Brait, and W.Z. Rymer. Objective quantification of spastic hypertonia: Correlation with
clinical findings. Arch. Phys. Med. Rehabil. 73:339–347,
1992.
12
Knutsson, E. Quantification of spasticity. In: Electromyography and Evoked Potentials, edited by A. Struppler and A.
Weindl. Berlin: Springer, 1985, pp. 84–91.
13
Lance, J. W. Pathophysiology of spasticity and clinical experience with baclofen. In: Spasticity: Disordered Motor
Control, edited by R. G. Feldman, R. R. Young, and W. P.
Koella. Chicago, IL: Year Book, 1980, pp. 185–203.
Stretch Reflexes in Spasticity
14
Lennerstrand, G., and U. Thoden. Position and velocity sensitivity of muscle spindles in the cat. II. Dynamic fusimotor
single-fibre activation of primary endings. Acta Physiol.
Scand. 74:16–29, 1968.
15
Matthews, P. B. C. The dependence of tension upon extension in the stretch reflex of the soleus muscle of the decerebrate cat. J. Physiol. (London) 147:521–546, 1959.
16
Matthews, P. B. C. A study of certain factors influencing the
stretch reflex of the decerebrate cat. J. Physiol. (London)
147:547–564, 1959.
17
Matthews, P. B. C. The response of de-efferented muscle
spindle receptors to stretching at different velocities. J.
Physiol. (London) 168:660–678, 1963.
18
Matthews, P. B. C. Muscle spindles: Their messages and
their fusimotor supply. In: The Nervous System, edited by J.
M. Brookhart and V. B. Mountcastle. Bethesda, MD: American Physiological Society, 1981, pp. 189–228.
19
Murray, W. M., S. L. Delp, and T. S. Buchanan. Variation of
muscle moment arms with elbow and forearm positions. J.
Biomech. 28:513–525, 1995.
20
Nichols, T. R., and J. C. Houk, Improvement in linearity and
regulation of stiffness that results from actions of stretch
339
reflex. J. Neurophysiol. 39:119–142, 1976.
Prochazka, A. Proprioceptive feedback and movement regulation. In: Handbook of Physiology, Section 12: Exercise:
Regulation and Integration of Multiple Systems, edited by L.
B. Rowell and J. T. Shepard. New York: American Physiological Society, 1996, pp. 89–127.
22
Prochazka, A., and M. Gorassini. Models of ensemble firing
of muscle spindle afferents recorded during normal locomotion in cats. J. Physiol. (London) 507:277–291, 1998.
23
Rymer, W. Z., and R. T. Katz. Mechanical quantification of
spastic hypertonia. In: Physical Medicine and Rehabilitation:
State of the Art Reviews, edited by R. T. Katz. Philadelphia,
PA: Hanley & Belfus, Inc., 1994, pp. 455–463.
24
Schmit, B. D., Y. Dhaher, J. P. A. Dewald, and W. Z.
Rymer. Reflex torque response to movement of the spastic
elbow: Theoretical analyses and implications for quantification of spasticity. Ann. Biomed. Eng. 27:815–829, 1999.
25
Vallbo, A. B. Afferent discharge from human muscle
spindles in non-contracting muscles: Steady state impulse
frequency as a function of joint angle. Acta Physiol. Scand.
90:303–318, 1974.
21