Articles in PresS. J Neurophysiol (January 21, 2015). doi:10.1152/jn.00421.2014
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The timing of continuous motor imagery – the two-thirds power
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law originates in trajectory planning
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Matan Karklinsky* and Tamar Flash
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Department of Computer Science and Applied Mathematics,
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Weizmann Institute of Science, Rehovot 76100, Israel
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* Corresponding author: [email protected]
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Abbreviated title: The timing of continuous motor imagery
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Number of pages:42
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Number of figures:6
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Number of Tables:2
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Number of words: Abstract-249, Introduction-600, Discussion-2201, Total – 9802
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Acknowledgements:
This study was supported by the EU Seventh Framework Programme (FP7-ICT No 248311
AMARSi). We thank Ronit Fuchs, Elad Schneidman and Edna Schechtman for fruitful
discussions, Naama Kadmon, Yaron Meirovitch and Noam Arkind for their helpful support,
and Moshe Abeles, Yosef Yomdin and Daniel Bennequin for their useful comments. We
thank Jenny Kien for editorial assistance.
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The authors declare no competing financial interests.
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Keywords:
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motor imagery, motor control, trajectory planning, two-thirds power law, equi-affine
geometry
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Copyright © 2015 by the American Physiological Society.
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Abstract
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The two-thirds power law, ‫ ݒ‬ൌ ߛߢ ିଵȀଷ , expresses a robust local relationship between the
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geometrical and temporal aspects of human movement, represented by curvature ߢ and
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speed ‫ݒ‬, with a piecewise constant ߛ. This law is equivalent to moving at a constant equi-
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affine speed and thus constitutes an important example of motor invariance. Whether this
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kinematic regularity reflects central planning or peripheral biomechanical effects has been
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strongly debated.
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Motor imagery, i.e., forming mental images of a motor action, allows unique access to the
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temporal structure of motor planning. Earlier studies have shown that imagined discrete
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movements obey Fitts's law and their durations are well correlated with those of actual
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movements. Hence, it is natural to examine whether the temporal properties of continuous
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imagined movements comply with the two-thirds power law.
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A novel experimental paradigm for recording sparse imagery data from a continuous cyclic
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tracing task was developed. Using the likelihood ratio test we concluded that for most
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subjects the distributions of the marked positions describing the imagined trajectory were
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significantly better explained by the two-thirds power law than by a constant Euclidean
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speed or by two other power law models. Using nonlinear regression, the ߚ parameter
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values in a generalized power law ‫ ݒ‬ൌ ߛߢ ିఉ , were inferred from the marked position
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records. This resulted in highly variable yet mostly positive ߚ values.
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Our results imply that imagined trajectories do follow the two-thirds power law. Our
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findings therefore support the conclusion that the coupling between velocity and curvature
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originates in centrally represented motion planning.
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Introduction
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The theory of invariance in movement task space
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Execution of human movement requires consideration of geometry (movement
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path) and timing (velocity profile along this path). The concept of geometric
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invariance is important in unraveling how the central nervous system selects specific
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paths and timing. The influence of path geometry on timing is well captured by the
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two-thirds power law (Lacquaniti et al. 1983), ߱ ൌ ‫ ߢܥ‬ଶȀଷ , relating angular speed
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߱to curvature ߢ. This law is an example of invariance, because motion satisfying this
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law has constant equi-affine speed (Pollick & Shapiro 1996; Flash & Handzel 1997).
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Other geometries may play a role in movement generation (Bennequin et al. 2010),
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but equi-affine geometry is the most robust example of invariance and is found in
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smooth pursuit eye movements (de'Sparti & Viviani 1997), full body locomotion
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(Hicheur et al. 2005), leg motions (Ivanenko et al. 2002), speech (Tasko et al. 2004)
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and visual perception of motion (Viviani and Stucchi 1992). Here we use an
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equivalent formulation of the two-thirds power law,‫ ݒ‬ൌ ߛߢ ିఉ , relating speed
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‫ݒ‬with curvature ߢ, with exponent valueߚ ൌ ଷ, and ߛ a constant named the velocity
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gain factor.
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Does the two-thirds power law indicate invariance in motor plans?
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The two-thirds power law may derive from biomechanics (the low-pass filtering
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properties of muscles, Gribble and Ostry 1996), it may be a byproduct of oscillatory
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movement generators in joint space (Schaal and Sternad 2001) or of limb dynamics.
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Alternatively, it may arise from the intrinsic encoding of trajectory planning within
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the central nervous system. This latter view is supported by the encoding of
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kinematic features of hand trajectories in the size and direction of the population
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vector reflecting the activities of neural populations within various cortical areas
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(Schwartz and Moran 1999). Additionally, visual perception of the motion of dots
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moving along elliptical trajectories while obeying the two-thirds law evokes larger
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and stronger brain responses than perception of movements complying with other
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laws of motion (Dayan et al. 2007; Meirovitch et al. 2015). These findings support a
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central origin of this law in both motion planning and perception.
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Motor imagery as a tool for deciphering the motor plan
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Motor imagery is the mental simulation of a motion without executing it. We assume
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(following Jeannerod 1995) that motor imagery is functionally equivalent to the
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covert part of movement generation without overt execution. This functional
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equivalence is supported by imaging studies showing similar brain activation
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patterns during motor imagery and execution (Lotze et al. 1999; Hétu et al. 2013;
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Sharma and Baron 2013). It implies that the rules governing the kinematics of
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movement generation should apply to motor imagery (cf. Fitts's law in Decety and
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Jeannerod 1995). While total durations of imagined and generated movements are
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similar (see Guillot et al. 2012 for review but also Rodriguez et al. 2008), little is
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known about the timing profiles and the role of invariances such as the two-thirds
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power law in motor imagery.
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To examine whether imagined movements comply with the two-thirds power law,
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Papaxanthis et al. (2012) recently compared global movement durations of imagined
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and executed movements. The imagined movement durations were correlated with
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the accumulated path curvature but not with the number of curved regions along
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the imagined template. Here we introduce a more direct paradigm for examining the
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kinematics of imagined movements. We show that their speed profiles are similar to
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those of executed movements and are captured by the two-thirds power law. Our
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findings support central origins for the two-thirds power law and our methods may
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open a new frontier in addressing the timing and geometry of continuous
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movements during motor imagery.
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Methods
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Imagine, stop and report - a paradigm for recording imagined motion
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We designed a novel experimental paradigm specifically to extract the properties of
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imagined movements. Our basic method involved stopping the subjects at arbitrary
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times during an imagined drawing motion and asking them to report the imagined
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locations of their hand. By studying the dynamically evolving hand positions
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throughout imagined movements, our paradigm allowed us to examine, not only the
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durations of complete imagined movements, but also their speed profiles.
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During the experiment on imagined motion each subject imagined drawing a given
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cyclical template on a tablet for 1 training session and 4 recording sessions, each
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comprising 30 trials. To prevent influence of prior execution on the imagery task,
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subjects were instructed to never actually draw the shape prior to the completion of
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the imagery part of the experiment.
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Subjects were instructed to imagine a constant pace of 3 to 4 seconds per lap, to
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avoid actually moving their right hand (the one they imagined to be moving), and to
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discard any trial if they were uncertain of its quality, e.g., the fluency of imagery.
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The trials in the main imagery experiment comprised a complex task (Figure 1b) that
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allowed extracting the duration and extent of the full cycles of the imagined
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trajectories, as well as of particular segments within them. The subjects watched the
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template displayed on the tablet with a red dot marking the start of the cycle. An
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auditory cue marked the start of each trial. The subjects closed their eyes and
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imagined a clockwise movement at a constant pace. Each time the imaginary
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movement passed through the red dot, the subjects pressed a key with their left
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hand. Another high-pitched auditory cue sounded at a random interval after the first
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cue. The subjects memorized the imagined hand position at that time and continued
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the imagined movement, again pressing the key with the left hand each time a cycle
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was completed. Two more laps after the second auditory cue a third longer auditory
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cue was played, and the subjects stopped the imagined movement and opened their
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eyes. The subjects marked on the screen the memorized position that their
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imaginary trace had reached at the time of the second auditory cue, unless they
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failed to memorize this position or made some mistake in the imagery process. In
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this case, the subjects marked a box on the upper right corner of the screen to
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discard this record. Each trial was followed by a short rest period.
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In this experiment, the subjects were required to memorize the imagined position at
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the time of the second auditory cue over 2 laps. We conducted another version of
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the experiment to control for the involvement of memory. This version differed in
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that, on hearing the second auditory cue, subjects immediately opened their eyes
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and marked the imagined position of their hand on the tablet, without continuing
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the imagined motion. To indicate the difference between the two experiments we
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denote the main experiment as MEM and the control experiment which lacked the
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long memorization period as NO MEM.
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Normalization and correction of recorded imagery data
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Unlike standard experimental imagery procedures, in which only one duration and
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arc length measurement are recorded for each trial, our MEM experiment gathered
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three durations and their three corresponding arc lengths for each complex trial
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(Figure 1c). The durations measured included the durations of two full cycles of
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imagined movement (marked ܶ௣ and ܶ) and the duration of a specific segment
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contained within the second lap (marked ‫)ݐ‬, which started at the fixed red dot and
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ended at the time of hearing the high-pitched auditory cue. For the full laps the
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matching arc length was simply the constant full lap length (markedܵ) and for the
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specific segment the arc length was measured along the curve between the fixed red
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dot marking the start position and the location which the subject marked on the
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tablet at the end of the trial (marked ‫)ݏ‬.
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In the NO MEM procedure the second lap was not complete as the subject stopped
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the imagery immediately after hearing the auditory cue and did not complete the lap
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containing the specific segment. Therefore, in the NO MEM experiment only ‫ ݐ‬and
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ܶ௣ , but not ܶ, were recorded.
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To overcome the dependency of segment duration on the gain factor ߛ (see
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Discussion for details), we normalized the segment duration with respect to the total
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duration of the same lap, marking it as ‫ݐ‬ǁ ൌ . We also normalized the segment
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duration with respect to the total duration of the prior lap and marked it as ‫ݐ‬ǁ௣ ൌ
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We further normalized the segment arc length with respect to the total lap arc
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length and marked it as ‫ݏ‬ǁ ൌ . These normalizations were performed for each trial
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separately, thus we did not need to assume that the gain factor ߛ had the same
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values in different trials. Measured ‫ݏ‬ǁ and ‫ݐ‬ǁ values for six subjects are shown in Figure
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2.
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Note that‫ݐ‬ǁ,‫ݐ‬ǁ௣ and‫ݏ‬ǁ are all cyclical with period 1, because our analysis considered
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only movements from the last passage through the starting point to the hand
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location imagined by the subject when they heard the second auditory cue. By
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default the values of ‫ݐ‬ǁfell within the range of [0,1]. The‫ݏ‬ǁ values also fell within the
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range of [0,1], with rare exceptions (less than 6% of all trials). When subjects marked
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a location on the tablet in the vicinity of the red dot, i.e., the point at which ‫ݏ‬ǁ ൌ Ͳ,
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which is also the point at which ‫ݏ‬ǁ ൌ ͳ, we had to decide whether to include the
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reported values as located closer to ‫ݏ‬ǁ ൌ Ͳ or to ‫ݏ‬ǁ ൌ ͳ. Our decision was based on
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the value of ‫ݐ‬ǁ. By default we selected the first option and reported ‫ݏ‬ǁ as being closer
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to zero. Only when the value of ‫ݐ‬ǁ was larger than ͲǤ͹ͷdid we choose to report the
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value of ‫ݏ‬ǁ as being closer to 1.This procedure ensured that of the two possible
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interpretations of the data, the one selected had the ‫ݐ‬ǁand‫ݏ‬ǁ values closer to each
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other.
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To avoid ambiguity in interpreting the location of the marked positions around the
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self-intersection points of the limaçon paths, we removed all marked data points
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within a 1 cm radius around these points.
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Recording actual motions and questionnaires
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The second part of the experiments included actual movement recordings. The
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subjects traced trajectories on the tablet, following the template previously used for
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imagery. Two sessions of the drawing task were recorded, each session comprising 6
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trials of 6 laps each (overall 72 repetitions).
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The subjects were instructed to draw as continuously and as smoothly as possible,
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without changing pace and not to pay attention to the accuracy of their drawing.
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They rapidly traced the shapes with their right hand and with their eyes open and
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used their left hand to press a key each time the pen passed through the red dot
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marking the start position.
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Data were taken from the 6 trials of the second session, discarding the first two laps
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and the last lap of each trial (we also manually removed for one lap of a single
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subject due to an extreme deviation from the template). This left 18 laps for each
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subject.
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The subjects were also asked to take the Edinburgh questionnaire, measuring right-
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handedness (Oldfield, 1971), the MIQ-RS questionnaire, measuring movement
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imagery ability (Gregg et al., 2010), and a follow-up questionnaire in which the
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subjects reported whether they actually moved any body part during the imagery
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task. The actual drawing trials, the follow-up questionnaire and the MIQ-RS
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questionnaire were always administered after completion of the imagery task.
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Participants, settings and templates
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Forty-five healthy subjects (19 men and 26 women, ages 18-33) participated in the
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three hour experiment that included imagery recording, actual drawing and
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answering questionnaires. The experimental procedure was approved by the local
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Ethics Committee and written informed consent was obtained from all subjects.
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Subjects sat in front of a WACOM tablet (142 Hz sampling rate), placed on a 100 cm
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high table. The tablet's surface was tilted 65 degrees from the horizontal plane.
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Subjects adjusted the chair height, the distance from the tablet and the keyboard
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location to a comfortable position. They held a pen in their right hand and their left
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hand was placed on the keyboard. Templates displayed on the tablet (see Figure 1a)
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included two limaçons with ratios of 1:3 (L13) and 2:3 (L23) between the arc lengths
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of their inner (smaller) and outer (larger) loops, scaled by a factor of 1.7 compared to
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the limaçons used in Viviani and Flash (1995) and an ellipse (ELL; with a major axis of
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14 cm and minor axis of 3.5 cm with major axis tilted at 45 degrees to the x axis). A
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red mark was displayed on the figural forms (on the top right extremity of ELL and on
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the rightmost extremity of the limaçons) to indicate the starting point of each cycle.
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Data of subjects who failed to comply with one of the following requirements were
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eliminated: subjects who discarded more than 40 data points, leaving less than 80
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valid data points, subjects whose MIQ score was lower than 4, indicating poor
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imagery skill, or whose LQ score was less than or equalͲǤ͹, indicating not being fully
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right-handed. One subject failed to perform the task. After eliminating the data of
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these subjects, the remaining data comprised 7,7,7 MEM subjects and 3,5,5 NO
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MEM subjects, for the L13, L23, ELL templates respectively, and an additional outlier
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subject for the L23 template, whose results were not included in the statistical
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analysis and are reported separately.
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Actual motion during imagery
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A primary concern when examining motion imagery is whether the subject also
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produced some correlated but unreported movement. Subjects performed the
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experiments with eyes closed and with the right hand resting in a fixed position on
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the table. This insured that they did not produce any large hand movements. We
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emphasized to the subjects the need for kinesthetically imagining their right hand
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moving, as opposed to visually imagining it. In a follow-up questionnaire only six of
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the thirty-four subjects reported moving their right hand during the imagery trials
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(always reporting a total movement amplitude smaller than 5 cm). However, all
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subjects reported moving either the head or the closed eyes at least once during
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imagery and one subject reported moving the leg, while another reported moving
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the tongue.
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Behavioral (Papaxanthis et al. 2012) and fMRI studies (Chiew et al. 2012; Raffin et al.
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2012) have shown that imagery activity does not give rise to activity in the arm
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muscles. Eye movements are naturally coupled with imagined hand movements
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(Heremans et al. 2008), but they have no effect on the temporal properties of the
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imagined movements (Gueugneau et al. 2008). Hence, many motor imagery
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experiments have not used EMG or EOG measurements for well performed imagery
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tasks (Rodriguez et al. 2008, 2009). Relying on these observations, we focused on
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natural imagery production, disregarding any secondary muscular effects which may
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have arisen during imagery.
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Statistical analysis of distributions
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Unlike our drawing data, our imagery data were sparse, containing not more than
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120 data samples per subject. The sparseness prevented reconstruction of speed
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profiles as the numeric derivative of the normalized location profile. Instead, we
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examined the distributions of marked locations, represented by the ‫ݏ‬ǁ parameter. In
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the distributions of marked locations along a curve (see Figure 2 for several
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individual subjects' data points and distributions), the number of markings within
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each bin indicated the speed of the imagined movement; the probability of stopping
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within a specific bin was directly related to the time the subject spent imagining the
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movement within this bin. This time is inversely related to the speed of the
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imaginary movement within this bin. Hence, the speed profiles of the imagined
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movements can be inferred from these distributions.
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Qualitatively, for the generalized power law model,‫ ݒ‬ൌ ߛߢ ିఉ , positive ߚ values
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(such as the two-thirds power law, ߚ ൌ ଷ) predict that more markings will occur
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during the more curved segments of a template, where the movement is slower.
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Quantitatively, each value of the exponent ߚpredicts a unique distribution for each
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template. Therefore, to examine the validity of different power laws, we examined
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how well the predicted power law distributions described measured distributions of
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imagery and drawing data.
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Using the measured data we extracted the distribution of both ‫ݐ‬ǁ and ‫ݏ‬ǁ within bins
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covering the range of [0.05,0.95]. For each studied power law we calculated the
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predicted distributions of ‫ݏ‬ǁ based on this law of motion. The measured distribution
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of ‫ݐ‬ǁ established a specific distribution of the predicted values of ‫ݏ‬ǁ ǡ calculated by
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using the normalized speed profile defined by this law of motion. Integrating this
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speed profile gave a prediction of an ‫ݏ‬ǁ value for each value of ‫ݐ‬ǁ and, hence, a
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distribution of ‫ݏ‬ǁ values for each distribution of ‫ݐ‬ǁ values. The observed and the
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modeled distributions of ‫ݏ‬ǁ for six subjects are shown in Figure 2.
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We compared the measured distribution of ‫ݏ‬ǁ for each subject with the
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corresponding distribution of the predicted ‫ݏ‬ǁ (based on the subject’s measured ‫ݐ‬ǁ
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values) using the߯ ଶ goodness of fit test for each law of motion studied here.
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For each law of motion we calculated its log likelihood value,
‫ ܮܮ‬ൌ ෍ ݀௜ Ž ܲ௜
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௜
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where ݀௜ is the number of observed data points in bin ݅, ܲ௜ is the number of data
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points predicted by the model for this bin, and the summation is across all bins.
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To qualitatively compare the extent to which each of two laws of motion matched
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the measured data, we derived the log of the likelihood ratio for the two predicted
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distributions for each subject, calculated as the difference:
Ž ߣ ൌ ‫ܮܮ‬௔௟௧௘௥௡௔௧௜௩௘ െ ‫ܮܮ‬௡௨௟௟
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whereߣ is the likelihood ratio stating how well the alternative model outperforms
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the null model. The likelihood ratio test was performed by comparing ߣto a
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߯ ଶ distribution with 1 degree of freedom.
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We used a similar process for the comparison of an alternative model to several null
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models, where each comparison was made between the alternative model and the
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most likely of the null models.
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We also compared log likelihood values of several models using repeated measures
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ANOVA, which demonstrated which power laws were more likely than others.
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Conformity with normal distributions was tested using Shapiro-Wilk tests, and post
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hoc analysis using ‫ݐ‬-tests was corrected with the Fisher LSD method.
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Statistical analysis of speed profiles as power laws
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In addition to the analysis of the distributions of the marked locations we examined
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the dependency of ‫ݏ‬ǁ on ‫ݐ‬ǁwithin an imagined movement. We considered models of
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normalized speed ‫ݒ‬෤ ൌ
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estimate ߚ, which is the only free parameter of this model (because‫׬‬଴ ‫ݒ‬෤݀‫ݐ‬ǁ ൌ ͳ, so
ௗ௦ǁ
as
ௗ௧ሚ
power laws of the form‫ݒ‬෤ ൌ ߛߢ ିఉ . We wished to
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ߛis uniquely determined). We defined an error term, the cyclic mean squared error
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(CMSE) in ‫ݐ‬ǁto be:
‫ܧܵܯܥ‬ሺ‫ݐ‬ǁǡ ‫ݏ‬ǁ ሻ ൌ ෍ ݀൫݂ ௠௢ௗ௘௟ ൫‫ݏ‬ǁ௝ௗ௔௧௔ ൯ǡ ‫ݐ‬௝ǁ ௗ௔௧௔ ൯
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ʹ
௝
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Where ‫ݐ‬௝ǁ ௗ௔௧௔ ǡ ‫ݏ‬ǁ௝ௗ௔௧௔ are the measured data points, ݂ ௠௢ௗ௘௟ ሺ‫ݏ‬ሻ is a function
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calculating predicted ‫ݐ‬ǁ values for given ‫ݏ‬ǁ values and for a given value ofߚ, and
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݀ሺšǡ ›ሻ ൌ ݉݅݊ሺȁ‫ ݔ‬െ ‫ݕ‬ȁǡ ͳ െ ȁ‫ ݔ‬െ ‫ݕ‬ȁሻis the cyclic distance (used because of the cyclic
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nature of our variables). Nonlinear regression estimated the ߚvalue that minimized
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the CMSE error, along with the confidence intervals for ߚbased onߙ ൌ ͲǤͲͷ.
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This procedure for extracting ߚ values for each whole shape was used for both the
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imagery and the drawing data. We studied the extracted ߚ values using a repeated-
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measures two-way ANOVA, comparing imagery and drawing across the studied
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templates. Conformity with normal distributions was tested using Shapiro-Wilk tests,
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sphericity violations were corrected by the Greenhouse-Geisser method, and post
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hoc analysis using ‫ݐ‬-tests was corrected with the Fisher LSD method.
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For the data from drawing the limaçon shapes we also used the extracted ߚ values
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along with the durations ܶ to calculate the gain factor ߛ. The ߚ values, as well as the
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gain factor values, were compared for pairs of matching small and large loops
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comprising the limacon laps using a Wilcoxon signed-rank test.
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Results
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Distribution of subjects’ marked locations
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A distribution of marked locations corresponds to an imagery speed profile, the
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number of data points in each bin being inversely proportional to the imagery speed
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within this bin (see Methods - Statistical methods for analysis of distributions).
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Qualitatively, the two-thirds power law predicts that more markings will occur during
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the more curved segments of a template. Quantitatively, each value of the exponent
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ߚ in the power law model‫ݒ‬෤ ൌ ߛߢ ିఉ predicts a unique distribution. Therefore, we
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examined how well the predicted power law distributions described measured
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distributions of imagery and drawing data in order to infer evidence of the validity of
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different power laws.
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Figure 2 presents imagery data points and distributions of ‫ݏ‬ǁ for six subjects, while
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Figure 3 presents the grouped resampled distributions of ‫ݏ‬ǁ , for both imagery and
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drawing data of the MEM subjects. Notice that in both figures the number of peaks
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and their locations match those predicted by the two-thirds power law; the subjects'
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tendency to move more slowly in the more curved segments of each shape was
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evident from their tendency to stop more often while passing through the curved
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regions of the shape.
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We tested whether observed distributions matched distributions predicted by model
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power laws, using ߯ ଶ goodness of fit tests. For the grouped data of the MEM
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subjects for each shape, the observed grouped ‫ݐ‬ǁdistribution (and also the observed
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grouped ‫ݏ‬ǁ distribution) were not uniform (߯ ଶ goodness of fit resulted in‫ ݌‬൑ ͲǤͲͲͳ,
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checking for 18 bins of size ͲǤͲͷ covering the range ሾͲǤͲͷǡ ͲǤͻͷሿ). Therefore, for each
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shape we used the values of ‫ݐ‬ǁas inputs to the two power laws withߚ values of 0 and
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ଵ
, corresponding to the constant speed and the two-thirds power law models,
ଷ
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respectively. For neither of the two power law models did the predicted distribution
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of ‫ݏ‬ǁ match the observed distribution of ‫ݏ‬ǁ (for each group of subjects imagining one
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of the three templates, a ߯ ଶ goodness of fit test on the group distribution resulted
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in‫ ݌‬൏ ͲǤͲͲͳfor each of the ߚ values, using the method described above). The
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mismatch between model and observation is visible in Figure 3 and possibly
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indicates the noisy nature of the imagery data. To conclude, the ߯ ଶ goodness of fit
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tests showed that neither of the two power laws perfectly matched our data set.
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Next, we compared different power laws to each other using likelihood ratio tests,
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which do not require a perfect match of one model, but can indicate which of two
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(or more) imperfect models is better for describing the observed distributions. We
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tested whether theߚ ൌ model outperforms the nullߚ ൌ Ͳ model, obtaining
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significantly positive results for the majority of subjects for each of the templates
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(Table 1 summarizes the MEM ‫ݐ‬ǁ data, the MEM ‫ݐ‬ǁ௣ data and the NO MEM ‫ݐ‬ǁ௣ data).
361
This analysis shows that for most subjects, the two-thirds power law described our
362
imagery data better than the constant speed model. This result proved significant for
363
all subject groups based on Fisher's combined probability test, which gave a
364
significant overall effect (‫ ݌‬൏ ͲǤͲͲͳ) for each group of subjects performing a shape,
365
for the MEM‫ݐ‬ǁ, the MEM ‫ݐ‬ǁ௣ and for the NO MEM data separately. Hence, the effect
366
of curvature on the subjects' marked positions, indicating their imagery movement
367
speed, was robust, persisting independently of the choice of a normalization
368
procedure (whether by dividing ‫ݐ‬by ܶ, or byܶ௣ ) and independently of the effect of
369
memory. Using a Pearson correlation test, we examined the correlation between the
370
total imagery score and the ‫ ݌‬values derived from the likelihood ratio test. There was
371
no significant correlation between the two scores. Similar tests applied to the visual
372
and to the kinesthetic imagery scores also showed no significant correlations (‫ ݌‬൐
373
ͲǤͶ for each of the three scores).
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374
We then tested whether this model preference merely reflected a tendency towards
375
positive ߚ values or indicated the use of a more precise value, that of ߚ ൌ . We
376
used a similar likelihood ratio test for distributions to compare how well the MEM‫ݐ‬ǁ
377
data fitted the two-thirds power law of motion (ߚ ൌ ଷ), but now compared to the
378
best of three alternative models (ߚ ൌ ǡ Ͳǡ െ ). For the MEM experiment, for 3 of
379
the 7 subjects of L13, the test gave statistically significant results (‫ ݌‬൏ ͲǤͲͷ),
380
indicating that the ߚ ൌ ଷ law outperformed all other modelsin describing these
381
subjects’ distributions. This was also the case for 3 of the 7subjects of L23 and for 3
382
of the 7 subjects of ELL.
383
An ANOVA test was used to group the individual results comparing the four power
384
law models across multiple subjects. We extracted log likelihood values for each
385
MEM subject's data for each of the four models (ߚ ൌ ଷ ǡ ଷ ǡ Ͳǡ െ ଷ) (see Figure 4).
386
When a specific model yielded a likelihood value of 0 for any particular subject, to
387
avoid having to take the logarithm of such likelihood value, we assigned the minimal
388
likelihood value of the other models to this model. For one subject all four models
389
gave 0 likelihood values, and he was assigned four identical log likelihood values; the
390
mean log likelihood values of the other subjects who imagined moving along the
391
same template, was averaged over all four models. The log likelihood values
392
conformed with a normal distribution (Shapiro-Wilk tests yielded‫ ݌‬൐ ͲǤͶͲ, for each
393
of the four power law models). A one-way repeated-measures ANOVA was applied
394
to the log likelihood values comparing the four power law models. We found a main
395
effect of the ߚ parameter (F(3) = 6.6040, ‫ ݌‬൏ ͲǤͲͲͳ). The mean log likelihood values
396
were -183.8,-177.7,-182.1,-187.4 for ߚ ൌ ǡ Ͳǡ െ respectively. Post hoc analysis
ଵ
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ଷ
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ଶ ଵ
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ଷ ଷ
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17
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397
using paired-sample t-tests (corrected with the Fisher LSD method) resulted in the
398
two-thirds power law model,ߚ ൌ , having the highest log likelihood among all four
399
models (‫ ݌‬ൌ ͲǤͲͲͺ, ‫ ݌‬ൌ ͲǤͲͷͲ which is nearly significant, ‫ ݌‬൏ ͲǤͲͲͳ for
400
comparisons with ߚ ൌ ଷ ǡ Ͳǡ െ ଷ respectively). The ߚ ൌ ଷ model was indistinguishable
401
from the ߚ ൌ Ͳ model (‫ ݌‬ൌ ͲǤͶͶͺ), but almost significantly higher than the ߚ ൌ െ
402
model (‫ ݌‬ൌ ͲǤͳͳʹ). The ߚ ൌ Ͳ model had significantly higher log likelihood values
403
than the ߚ ൌ െ ଷ model (‫ ݌‬ൌ ͲǤͲʹͳ). Thus we concluded that the two-thirds power
404
law explained the imagery distributions better than the other three kinematic
405
models. The worst performing model was the negative ߚ value model.
406
Quantitative analysis of power laws for drawing and imagery
407
So far we only analyzed distributions of the marked location ‫ݏ‬ǁ variable. This analysis
408
did not regard a basic property of our data set, namely the coupling between time, ‫ݐ‬ǁ,
409
and position, ‫ݏ‬ǁ , in the measured data set (see Figure 2). Thus, to complement the
410
analysis in the previous section, we employed a procedure that allowed a more
411
precise extraction of the ߚ values. This used an alternative method of nonlinear
412
regression, which took advantage of this coupling of time and location to reconstruct
413
a speed profile. We performed the nonlinear regression procedure on the drawing
414
and imagery data of each subject, obtaining optimal ߚ values as well as confidence
415
intervals based on ߙ ൌ ͲǤͲͷ(Figure 5 gives individual subject data, Table 2
416
summarizes mean group results). There was no correlation between ߚ values of the
417
MEM drawing and imagery results (Figure 6).
ଵ
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18
418
The optimal ߚ values of the MEM subjects conformed with a normal distribution
419
(Shapiro-Wilk tests yielded‫ ݌‬൐ ͲǤͳ for each set of imagining or drawingߚ values
420
extracted for all subjects executing the same template) and were analyzed using
421
repeated-measures ANOVA. We compared all subjects on the three shapes (shape, a
422
between-subject variable), testing the ߚ values of imagery and drawing for each
423
subject (imagery, a within-subject variable). Sphericity violations were corrected by
424
the Greenhouse-Geisser method. A main effect of shape was found (‫ܨ‬ሺͳǡͳͺሻ ൌ
425
ͻǤͳ͸͹,‫ ݌‬ൌ ͲǤͲͲʹ). No main effect of imagery was found (‫ܨ‬ሺͳǡͳͺሻ ൌ ͲǤͲͳ͸,‫ ݌‬ൌ
426
ͲǤͻͲͲ). There was a shape-imagery interaction effect (‫ܨ‬ሺʹǡͳͺሻ ൌ ͳͲǤ͹Ͷ͸,‫ ݌‬ൌ
427
ͲǤͲͲͳ).
428
To examine the interaction effect, we compared the imagery and drawing results for
429
each shape, using paired-samples t-tests (corrected with the Bonferroni method)
430
giving insignificant differences for the L13 shape (ܶሺ͸ሻ ൌ ͲǤ͸͹ͳ, ‫ ݌‬ൌ ͲǤͷʹ͹) and for
431
the L23 shape (ܶሺ͸ሻ ൌ ͳǤͻͻͻ,‫ ݌‬ൌ ͲǤͲͻʹ), but a significant difference for the ELL
432
shape (ܶሺ͸ሻ ൌ െͷǤ͵ͷͳ,‫ ݌‬ൌ ͲǤͲͲʹ).
433
To understand how ߚ values differ among shapes we performed separate one-way
434
ANOVAs for the imagery and drawing conditions of the MEM subjects. No significant
435
difference among the shapes was found for drawing (‫ܨ‬ሺʹǡͳͺሻ ൌ ͳǤ͸ͷͺ, ‫ ݌‬ൌ ͲǤʹͳͺ),
436
and group optimal ߚ means were 0.38 (SD 0.16) for the L13 subjects, 0.31 (SD 0.08)
437
for the L23 subjects and 0.28 (SD 0.03) for the ELL subjects. For imagery we found a
438
significant difference among the shapes (‫ܨ‬ሺʹǡͳͺሻ ൌ ͳͳǤͲʹͻ,‫ ݌‬ൌ ͲǤͲͲͳ), which a
439
Tukey post hoc analysis attributed to a difference between the ߚvalues of subjects
440
who performed the L23 and ELL shapes (‫ ݌‬ൌ ͲǤͲͲͳ). The group optimal ߚ means
441
were 0.33 (SD 0.15) for the L13 subjects, -0.04 for the L23 subjects (SD 0.45, a high
19
442
between-subject variability which was accompanied by high within-subject
443
variability, as seen in the confidence intervals) and 0.71 (SD 0.20) for the ELL
444
subjects.
445
We examined the control NOMEM subjects to see whether the effects found for the
446
MEM experiment persisted without memorization. The mean optimal ߚ values
447
followed a similar pattern. While for drawing they were 0.33 (SD 0.16) for the L13
448
subjects, 0.39 (SD 0.06) for the L23 subjects and 0.33 (SD 0.03) for the ELL subjects,
449
for imagery they were 0.18 (SD 0.25) for the L13 subject, 0.11 (SD 1.19) for the L23
450
subjects and 0.79 (SD 0.18) for the ELL subjects. Both the noisiness of L23 subjects
451
and the high ߚ values of the ELL shape appear unrelated to the memorization
452
required in the main MEM task.
453
Finally, we report the results of the subject excluded from the statistical analysis. The
454
8th subject performing the L23 template had only 80 valid data points, without any
455
samples of imagery ‫ݐ‬ǁ values in the interval [0.76,0.96]. The likelihood values of all
456
four power laws examined were 0 for her imagery data, rendering the study of this
457
subject's distributions uninformative. The nonlinear regression process yielded an
458
optimal ߚ value of -2.65, which was an extreme outlier, and including it along with
459
the optimal ߚ values of the L23 MEM subjects caused theirߚ values to lose their
460
normal distribution. We found nothing extraordinary in her drawing data, which had
461
an optimal ߚ value of 0.29.
462
Segmented structure of the limaçon drawing data
463
Throughout our imagery study we assumed that each cycle of movement consisted
464
of a single motion segment with a constant unknown velocity gain factor ߛand a
465
constant exponent ߚ. This is a simplified version of a more general model, in which
20
466
different motion segments have different velocity gain factors. We found such a
467
segmented pattern for our drawing data (repeating the results of Viviani and Flash
468
1995). Unfortunately, our current imagery data had insufficient resolution to
469
examine such a segmented model, yet it is reasonable to assume that better suited
470
data can shed light on motion segmentation.
471
We tested whether, in addition to our nonlinear regression process, segmentation
472
described the entire template of the drawing movements better than a single power
473
law. Segmentation of the limaçon curves into two loops (for details see Viviani and
474
Flash 1995), followed by application of the nonlinear regression procedure, yieldedߚ
475
and ߛvalues for each loop of the drawing data. For L13 theߚ values were larger for
476
the short loops than for the long loops (0.369, SD 0.073 for the short loop versus
477
0.274, SD 0.061 for the long loop, ‫ ݌‬ൌ ͲǤͲͲ͸ in a Wilcoxon signed-rank test) and
478
theŽ‘‰ ߛ values were smaller for the short loop than for the long loop (3.977, SD
479
0.501 for the short loop versus 4.654, SD 0.628 for the long loop,‫ ݌‬ൌ ͲǤͲͲʹ in a
480
Wilcoxon signed-rank test). For the L23 template theߚ values did not differ between
481
the short loops and the long loops (0.340, SD 0.064 for the short loop versus 0.275,
482
SD 0.069 for the long loop,‫ ݌‬ൌ ͲǤͳ͹͸ in a Wilcoxon signed-rank test).The
483
Ž‘‰ ߛvalues for the short loops of L23 were almost significantly larger than those of
484
the long loops of L23 (4.771, SD 0.428 for the short loop and 4.936, SD 0.451 for the
485
long loop, ‫ ݌‬ൌ ͲǤͲͷʹ in a Wilcoxon signed-rank test).
486
21
487
Discussion
488
Our results show how the local velocity of continuous imagery of drawing
489
movements depends on the geometry of the imagined shape. The two-thirds power
490
law offered a better description of the observed normalized location distributions
491
than motion with a constant Euclidean speed (Figure 3) and other competing power
492
law models (Figure 4). Furthermore, the estimated ߚ values for the power law
493
strongly tended towards positive values (Figure 5). Our results thus suggest that the
494
source of the two-thirds power law lies in the motion planning stages and that this
495
law is not solely a byproduct of motor execution. Similar conclusions have been
496
reached by e.g., Dayan et al. (2007), Schwartz and Moran (1999), Papaxanthis et al.
497
(2012), and Meirovitch et al. (2015).
498
How ࢼis calculated determines the obtained values
499
As noted by Schaal and Sternad (2001), using nonlinear regression to extract the ߚ
500
values (as we did here) gives results which diverge from ߚ ൌ more than using a
501
linear fit in the log-log space (as in Viviani and Flash 1995). Thus, our method of
502
nonlinear regression is stricter than the standard methods in the motor control
503
literature. These are inapplicable to our sparse imagery data which do not include
504
well sampled velocity profiles. As different methods yield different estimations of
505
the ߚ values, a systematic comparison of these methods is still necessary but is
506
outside the scope of the current work.
507
Imagery results support a central origin for the two-thirds power law
508
The curvature-velocity relation for movement execution can be well quantified as a
509
power law, as shown originally by Lacquaniti et al. (1983). It is crucial to determine to
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22
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ଷ
ଷ
510
what extent the exponent of the power law, ߚ, truly equals , because ߚ ൌ
511
implies a constant equi-affine speed, suggesting that movement is planned in an
512
equi-affine invariant manner. For production data the specific values of ߚ vary with
513
age, coming closer toଷ in adults (Viviani and Schneider 1991). Our nonlinear
514
regression analysis of drawing gave mean group ߚ values ranging from ͲǤʹͺ to ͲǤ͵ͻ.
515
The distributions of the subjects' marked positions here demonstrate that there is a
516
similar curvature-velocity relation for imagery and, indeed, the two-thirds power law
517
(ߚ ൌ ) explained the data better than other power laws (ߚ ൌ ǡ Ͳǡ െ ). Further
518
quantification of this relation using nonlinear regression analysis gave positive
519
meanߚ values for the group, indicating that subjects did imagine slower movements
520
in the more curved regions of each template. However, the resolution of the data did
521
not allow a precise determination of ߚ values giving the best description. The
522
curvature-velocity relation persisted in the control NO MEM experiment, where
523
memory played a less important role, supporting our initial assumption that the
524
memorization in this task was not a major factor, except, perhaps, for its effect on
525
the variance of results.
526
Commonalities and differences between imagery and execution
527
The subjects’ variability ofߚ values for imagery (Results - Quantitative analysis of
528
power laws for drawing and imagery; Figure 5) cannot be explained by their drawing
529
ߚ values, because there was no correlation between them (Figure 6). This lack of
530
correlation may result from increased variability in the imagery data. It is possible
531
that the drawing data were more stereotypical and showed greater invariance due
532
to the subjects’ greater proficiency in drawing versus imagery. On the one hand,
ଵ
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ଵ
ଷ
ଷ
ଷ
23
533
visual and proprioceptive feedback, absent during motor imagery, may give rise to
534
better controlled and less variable behavior (Hoff & Arbib 1993; Todorov & Jordan
535
2002). On the other hand, motor noise may be the cause of a significant portion of
536
movement variability (Van Beers et al. 2004) and is likely to be more influential
537
during motor execution than during imagery.
538
The major mechanism underlying motor imagery may involve a simulation process
539
giving rise to motor predictions (Blakemore and Sirigu 2003). Forward models allow
540
anticipation of the movement outcomes of internal motor commands even in the
541
absence of overt motor execution. Therefore, an internal anticipatory model of a
542
preplanned action may allow prediction of movement progress in space and time
543
(Wolpert and Flanagan 2001) and this planned motion can be made accessible via
544
the reporting of motor imagery. Characteristics common to motor imagery and
545
execution may, therefore, indicate which aspects of the movement are ingrained in
546
the feed forward commands. Differences between imagined and actual movements
547
may indicate what aspects of the observed behavior are likely to be governed by
548
other mechanisms, such as specific mechanical properties of the task and motor
549
system which are not represented by the feedforward models, and unanticipated
550
external perturbations and feedback corrections.
551
Alternative explanations of the two-thirds power law in motor imagery
552
Gribble and Ostry (1996) suggested that the two-thirds power law may reflect
553
smoothing by the low-pass filtering properties of muscles. Such smoothing
554
properties affect actual movement production but are unlikely to affect imagery.
555
Therefore, similar to the increased brain activation during observation of motion
24
556
obeying the two-thirds power law (Dayan et al. 2007; Meirovitch et al. 2015), the
557
timing effects observed for motor imagery in our study probably do not solely reflect
558
these biomechanical muscle properties. These properties, however, may explain
559
some of the differences between our imagery and drawing results.
560
Alternatively, the two thirds-power law could result from actions comprised of
561
oscillatory movements in joint space (Schaal and Sternad 2001). This could account
562
for our findings only if subjects used a mental representation for imagery that fully
563
described the joint angles of the kinematic chain of the arm. Subjects would also
564
have to apply an internal simulation of the forward kinematics transformations from
565
joint to end effector motion along the prescribed path.
566
Movement dynamics may be represented to some extent in motor imagery. The
567
durations of imagined movements, like those of executed movements, depend not
568
only on movement kinematics, but also on the inertial properties of the arm, which
569
vary with arm configuration, and are affected by external loads applied to the arm
570
(Papaxanthis et al. 2002). The inertial properties of the arm appear better
571
represented in motor imagery in adults than in children (Crognier et al. 2013).
572
Therefore, we cannot completely rule out the possibility that a simulated model of
573
movement dynamics yields the observed two-thirds power law behavior for
574
imagined movements. We hope that future studies will determine to what extent
575
motor imagery represents motion dynamics and joint space kinematics in addition to
576
end effector movement kinematics.
25
577
Real-virtual isochrony and the two-thirds power law
578
Durations of motor imagery and production are often positively correlated. This
579
correlation, termed “real-virtual isochrony” in the imagery literature (not to be
580
confused with "isochrony" in the motor control literature) is quite robust, but with
581
some exceptions arising from either lack of proficiency or from an inherent
582
difference between well performed imagery and execution (see Guillot et al. 2012).
583
The cases where real-virtual isochrony fails may provide useful information on the
584
differences between planning and execution. Rodriguez et al. (2009) report an
585
example, where the speed profiles of movement imagery for straight reaching
586
movements did not follow the bell-shaped characteristics of the actual movements.
587
They emphasized that this discrepancy was probably a characteristic of imagined
588
movements of simple, automatic tasks like reaching and not of the more complex,
589
attention-demanding tasks such as used here. Indeed, in our imagery data the basic
590
temporal properties of production did persist, although for the ELL template, which
591
is simpler than the limaçon templates, a mismatch between the ߚ values of imagery
592
versus drawing tasks did appear.
593
The challenges in recording continuous imagery motion
594
Comparisons of the total durations of imagined movements, used extensively in the
595
imagery literature, are limited when it comes to describing the instantaneous
596
kinematics of continuous trajectories. In previous motor imagery studies, during
597
each trial the experimenters recorded only the geometric characteristics of the paths
598
being drawn and the total durations of imagined movement along those paths. The
26
599
speed profiles of the imagined movements cannot be constructed from these data,
600
unless some simplifying assumptions are made.
601
The generalized power law model of the form ‫ ݒ‬ൌ ߛߢ ିఉ is the natural model in the
602
context of power laws. This model has two free parameters. Therefore, for any
603
shape with known curvature profile the measured total duration of the imagined
604
motion must satisfy the equationܶ ൌ ‫ ݐ݀ ׬‬ൌ ‫ ݏ݀ ׬‬ൌ ߛ‫ ߢ ׬‬ఉ ݀‫ݏ‬. For a known value
605
ofܶ this equation allows determination of the value of only one of the two
606
parameters,ߚ and ߛ. Inside a relatively simple movement it can be assumed that ߛis
607
constant, but its value may change between different templates. For instance,
608
Viviani (1986) showed that velocity gain factors, ߛ, for elliptical trajectories, were
609
dependent upon their perimeters and speeds of execution. Hence, when the only
610
available data are the total movement duration of the entire shape, it is not possible
611
to infer the values of the exponent ߚ.
612
The alternative approach used here relies on reconstructing the entire speed profile
613
by using multiple queries on the distance traveled during the imagined movement,
614
and using within-shape normalization. This normalization exploits the fact that the
615
ratios between the durations needed to travel different distances along the same
616
path do not depend on the value of the velocity gain factorߛ but only on the value of
617
the exponentߚ. Therefore, we do not need to assume anything about the velocity
618
gain factor ߛ, except for the weak assumption that it is constant within each cycle of
619
imagining a specific template, and we can extract the values of the ߚ exponent.
620
Analyzing motor imagery, Papaxanthis et al. (2012) recently convincingly reported
621
that the increase in the total duration of imagined movements is correlated with
ଵ
௩
27
622
their cumulative turning angle,‫ ׬‬ȁߢȁ݀‫ݏ‬. Unfortunately, even if one does assume a
623
constant gain factorߛ, the following reasons show that these observations alone do
624
not provide sufficient evidence for a compliance of imagined movements with a
625
generalized power law.
626
First, the generalized power law model predicts an infinite speed in regions of zero
627
curvature. Therefore, straight trajectory segments, included in the analysis of
628
Papaxanthis et al. (2012), should be avoided when examining compliance with the
629
power law. Second, for circular segments, any real value of the ߚ exponent
630
(including zero and negative values) will yield a positive correlation between the
631
duration of the movement and the cumulative turning angle (which is proportional
632
to the arc length of the circular segment). Hence, the value of the exponentߚ can
633
only be determined if the value of the gain factor ߛ is known. Third, movement
634
through cusps and inflection points cannot be accounted for by a simple power law
635
model.
636
In our opinion, these technical yet important limitations have not been sufficiently
637
dealt with in previous studies. Nevertheless, the main novelty of the current work
638
does not lie in any technical matter, but rather in the fact that our paradigm allows
639
examination of the speed profiles of complex continuous motions. We hope that
640
future studies of motor imagery, in addition to reporting comparisons of durations of
641
actual and imagined movements, will reveal in detail the qualitative and quantitative
642
factors governing speed profiles of imagined movements. For example, one direction
643
that may guide future studies of motor imagery is the examination of imagined
644
motions using the framework developed by Bennequin et al. (2009). This suggests
28
645
that movement duration and compositionality arise from cooperation among
646
Euclidian, equi-affine and full affine geometries.
647
Does imagery show the segmentation of movement execution?
648
It is commonly assumed that movement is planned and executed as a composition of
649
elementary units (Morasso and Mussa-Ivaldi 1982; Viviani 1986; Flash and Hochner
650
2005; Polyakov et al. 2009). Given the sparseness of our samples it was difficult to
651
examine segmentation for our imagery data, but we could examine it in our drawing
652
data. One indication of segmentation in the actual drawing of limaçons is that
653
different values of the gain factors ߛ and of the ߚvalues derived for the different
654
loops (Results – Segmented structure of the limaçon drawing data, after Viviani and
655
Flash 1995). Bennequin et al. (2009) described a qualitatively similar segmentation
656
effect, where two distinct linear trends emerged for regions of different curvature in
657
theŽ‘‰ ‫ݒ‬versusŽ‘‰ ߢplot for drawing an ellipse. Such segmentation patterns require
658
more than a single power law for the description of an entire shape. Experimental
659
methods must be improved to investigate how well segmentation models apply to
660
motor imagery. Nonetheless, in the absence of both muscle smoothing effects and
661
the influence of corrections through feedback, motor imagery may allow examining
662
whether trajectory planning is indeed inherently segmented.
663
Further application of motor imagery to the study of continuous motion
664
The consistence of motor imagery observations with Fitts's law (Decety and
665
Jeannerod 1995) and our findings on the two-thirds power law raise the question of
666
the possibility of identifying regularities and invariants of motion planning through
667
motor imagery studies. Motor control theories describing both the task space and
29
668
the joint space have much to benefit from imagery studies examining both timing
669
and geometry of imagined motion. Characterization of motion structure, particularly
670
identification of motion primitives, seems plausible using similar imagery paradigms.
671
Imagery studies of kinematics may also allow distinguishing between movement
672
characteristics arising from trajectory planning (feedforward processes) and
673
properties arising from execution, such as online feedback-based error corrections.
30
674
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675
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34
776
Figure 1: Experimental procedure
777
(a) Each subject performed the experimental procedure using 1 of the 3 templates
778
shown for 4 sessions, each of 30 trials. (b) Each trial was composed of a complex task
779
involving observation, imagery (including key pressing each time the image passed
780
through the red mark, and memorization of imagery location at the beep, marking
781
(the memorized location) on the tablet, and rest (see Methods - Imagine, stop and
782
report - a paradigm for recording imagined motion). Eyes were closed only during
783
the imagery part. (c) Each trial yielded one measurement of each of the five
784
parameters (durations ‫ݐ‬,ܶ,ܶ௣ and matching arc lengths ‫ݏ‬,ܵ,ܵ). Using these
785
parameters we calculated the normalized durations (‫ݐ‬ǁ and ‫ݐ‬ǁ௣ ) matched by
786
normalized arc lengths (‫ݏ‬ǁ ) (see Methods - Normalization and correction of recorded
787
imagery data).
788
35
789
Figure 2: Single subject profiles and distributions
790
Normalized segment length and duration profiles ሺ‫ݐ‬ǁǡ ‫ݏ‬ǁ ሻfor imagery motion (Obs),
791
along with derived distributions of each subject's ‫ݏ‬ǁ data (Obs Density) for six
792
different subjects (S1 … S6), two subjects for each of the three shapes. Also shown
793
are predicted distributions for movement according to the two-thirds power law
794
(model ߚ ൌ ͳȀ͵) and for movement with a constant Euclidean speed (model ߚ ൌ Ͳ),
795
assuming a uniform distribution for ‫ݐ‬ǁ.
796
797
798
799
800
801
802
36
803
Figure 3: Group distributions of ‫ݏ‬ǁ
804
A resampled distribution of ‫ݏ‬ǁ values (Obs Density), such that ‫ݐ‬ǁ distribution was
805
uniform for grouped data for imagery and drawing of MEM subjects. Also shown are
806
predicted distributions for movement according to the two-thirds power law (model
807
ߚ ൌ ͳȀ͵) and for movement with a constant Euclidean speed (model ߚ ൌ Ͳ)
808
assuming a uniform distribution of ‫ݐ‬ǁ.
809
The distributions of ‫ݐ‬ǁ and ‫ݏ‬ǁ were resampled such that the ‫ݐ‬ǁ distribution became
810
uniform. This eliminated the random effect of the ‫ݐ‬ǁ distribution from the presented
811
‫ݏ‬ǁ distribution. These distributions were compared to the predicted distributions of ‫ݏ‬ǁ
812
for the two power laws with the assumption that ‫ݐ‬ǁ was uniformly distributed. This
813
resampling, i.e., the process of randomly selecting one data point (a pair (‫ݐ‬ǁ,‫ݏ‬ǁ ) with ‫ݐ‬ǁ
814
within the bin) was repeated 10000 times for each bin of the ‫ݐ‬ǁ variable and the
815
resulting ‫ݏ‬ǁ values were collected in a histogram for each template.
816
817
818
37
819
Figure 4: Comparison of likelihood values of the four power laws
820
The values of the log likelihoods of the model, mean and standard error for all
821
subjects of the MEM experiment for each of the four power laws ߚ ൌ ଷ ǡ ଷ ǡ Ͳǡ െ ଷ. The
822
two-thirds power law, ߚ ൌ , had higher log likelihood values than the other power
823
laws (‫ ݌‬൑ ͲǤͲͷ, see Results – Distribution of subjects' marked locations).
ଶ ଵ
ଵ
ଷ
824
38
ଵ
825
Figure 5: Nonlinear regression results for ߚ
826
For each of the three templates and for each subject, the results of theߚ values
827
derived by the nonlinear regression process along with confidence intervals for ‫ ݌‬ൌ
828
ͲǤͲͷ, for imagery (top) and drawing data (bottom). MEM subjects, blue; NO MEM
829
subjects, black. Shown for comparison are the constantߚ values of the two-thirds
830
power law (ߚ ൌ ଷ, dotted pink line) and of movement with a constant Euclidean
831
speed (ߚ ൌ Ͳ, red line).
ଵ
832
833
39
834
Figure 6: Comparison of Imagery and drawing ߚ values
835
Comparison of the best ߚ value for each subject for imagery and drawing (see Figure
836
5) in the MEM condition, extracted using the nonlinear regression procedure. No
837
correlation between the imagery and drawing ߚ values is apparent.
838
40
839
Table 1: Single subject results of the likelihood ratio test
840
Summary of single subject results of the likelihood ratio tests on imagery data. For
841
each shape we counted in how many subjects the two-thirds power law model
842
described the distribution of their marked locations better than the constant speed
843
model (‫ ݌‬൏ ͲǤͲͷ), calculated for 14 bins covering the range ሾͲǤͲͷǡͲǤͻͷሿ. Analysis of
844
three data sets is shown in three corresponding rows. The MEM ‫ݐ‬ǁ row shows results
845
for the MEM subjects, with time normalized with respect to the same lap. The MEM,
846
‫ݐ‬ǁ௣ row shows results for the same MEM subjects, with time normalized with respect
847
to the previous lap. The NO MEM, ‫ݐ‬ǁ௣ row shows results for the NO MEM subjects
848
with time normalized with respect to the previous lap (this was the only option for
849
the NO MEM subjects, since they did not complete the lap in which they were
850
stopped by the beep). For most subjects the two-thirds power law was a more
851
suitable description of their movement than a constant speed law.
852
* For two subjects the test failed due to a technical problem arising when bins which
853
were predicted to be empty according to the movement law were not empty in the
854
measured data; the likelihood ratio test forces a zero division.
855
Shape
‫ ݌‬൏ ͲǤͲͷ
Invalid *
Total
MEM,
L13
4
1
7
‫ݐ‬ǁ
L23
5
0
7
ELL
5
0
7
MEM,
L13
6
1
7
‫ݐ‬ǁ௣
L23
3
1
7
ELL
6
0
7
NO
L13
2
0
3
MEM,
L23
4
0
5
‫ݐ‬ǁ௣
ELL
3
0
5
856
41
857
Table 2: Group ߚ values summary
858
Mean and standard deviations of theߚ values extracted using nonlinear regression
859
(see Methods – Statistical analysis of speed profiles as power laws) for drawing and
860
imagery. Each value represents the group value for the corresponding set of
861
subjects.
IMAGERY
DRAWING
L13
L23
ELL
MEM
0.33 (SD 0.15)
-0.04 (SD 0.45)
0.71 (SD 0.20)
NOMEM
0.18 (SD 0.25)
0.11 (SD 1.19)
0.79 (SD 0.18)
MEM
0.38 (SD 0.16)
0.32 (SD 0.08)
0.28 (SD 0.03)
NOMEM
0.33 (SD 0.16)
0.39 (SD 0.06)
0.33 (SD 0.03)
862
42
a
L13
L23
ELL
10 cm
X
b
servatio
Ob
Marking
3.
n
X
2. Imagery
1.
eyes
open/closed
c
4. Rest
beep
sounds
T,S
p
ůĞŌŚĂŶĚ
key press
T,S
t,s
2. Imagery
marking on
tablet
~t = t/T
~s = s/S
~tp= t/Tp
1
L13
1
S1
0.5
0.2
0.1
0
0.5
0.5
1
L23
1
0.2
0.1
0
Normalized segment
ĂƌĐůĞŶŐƚŚ͕Ɛ˜
0.1
0
1
0.2
0.1
0.1
Probablity
0
S4
0
1
S5
0.5
0.2
Obs Density
Model E = 0
Model 1E = 1/3
Obs
0.5
0.5
1
ELL
0.5
1
S3
0.5
0.2
S2
0.5
1
0.5
1
S6
0.5
0.5
Normalized segment
ĚƵƌĂƟŽŶ͕ƚ˜
1
0.2
0.1
0
L13
L23
ELL
0.05
Probablity
Imagery
0.1
0
Obs Density
Model E = 0
Model E = 1/3
Drawing
0.1
0.05
0
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
Normalized segment arc length, ˜s
0
0.25
0.5
0.75
1
log likelihood
оϭϳϬ
оϭϴϬ
оϭϵϬ
оϮϬϬ
2/3
1/3
0
ɴ
оϭͬϯ
Drawing ɴ Imagery
2
2
L13
2
L23
1
0.5
0
1
0.5
0
1
0.5
0
оϭ
оϭ
оϭ
оϮ
оϮ
оϮ
1
0.5
0
1
0.5
0
1
0.5
0
1234567
123
1234567
12345
Subject #
ELL
E=0
E = 1/3
MEM
NO MEM
1234567
12345
ɴ͕ŝŵĂŐĞƌLJ
ϭ
0.5
0
оϬ͘5
>ϭϯ
>Ϯϯ
ELL
оϭ
0
0.2
0.4
ɴ͕ĚƌĂǁŝŶŐ
0.6