Phase Resetting of Human Walking
by
Daniel Klenk
S.B. Mechanical Engineering
Massachusetts Institute of Technology
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanical
Engineering
ARCHIVES
at the
Massachusetts Institute of Technology
MW
s
September 2011
@ 2011 Massachusetts Institute of Technology. All rights reserved.
L iB
The authored hereby grants to MIT permission to reproduce
and to distribute publicly paper and electronics copies of this thesis document in whole or in part
in any medium now know or hereafter created.
A
Signature of Author ................................
Department of mechanical Engineering
August 19, 2011
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Certified by ............................
......
*
.
evilvogan
Sun Jae Professor of Mechanical Engineering and Professm of Brain and Cognitive Sciences
- . .Thesis Sunervisor
Accepted by ................................
.
.......
....-
....... ..
David E. Hardt
Chairman, Depart Committee on Graduate Students
R';
Phase Resetting of Human Walking
by
Daniel Klenk
S.B. Mechanical Engineering
Massachusetts Institute of Technology
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanical
Engineering
Abstract
This thesis is an investigation of the neural control of unimpaired human walking. Specifically,
this work studied the potential for phase resetting of human walking by analyzing results from
treadmill walking experiments. Subjects walked on a treadmill while wearing a robotic device
that attaches to the lower leg, which applied 6 Nm torque perturbations to the ankle that acted to
plantarflex the ankle. The effect of these perturbations on the stride period was then analyzed to
determine the potential for phase resetting of the gait. For the experimental setup used, no phase
resetting was found. This was determined by fitting a Fourier series regression to the data and
finding very low R2 values for all subjects, ranging from 0.04 to 0.10, which implies that no
underlying periodic curve exists in the data. This evidence of zero phase resetting is consistent
with prior work that indicates some type of kinematic controller is present during walking.
Thesis Supervisor: Neville Hogan
Title: Sun Jae Professor of Mechanical Engineering and Professor of Brain and Cognitive
Sciences
Acknowledgments
I must first thank my advisor, Professor Neville Hogan, for devoting significant time and energy
to my work. With his guidance, I have learned much about approaching a challenging problem
and completing the proper analysis to generate a firm, conclusive answer. Professor Hogan's
lucid explanations of various concepts have changed my perspective on engineering.
Next, I would like to thank Dr. Hermano Igo Krebs for raising many valuable questions during
the course of this thesis. Pursuing these questions has improved my understanding of the
problem at hand and the reasoning behind my answer.
I owe Jooeun Ahn many thanks for all of the helpful discussions concerning my work since the
first day I began. I would also like to thank Hyunglae Lee and Panagiotis Artemiadis for
numerous software consultations that greatly expedited my work.
Finally, I thank my parents for their support during these years, the importance of which I could
never fully express. I would not have been able to pursue my desires in life without them.
Table of Contents
Abstract.......................................-------..........
------................................................................
2
Acknowledgments...........................................................................................................................
3
Table of Figures..............................................-................................................................
5
Table of Tables .............................................................................
6
Table of Equations................................--
6
-...........- ....-.....................................................
Chapter 1 Introduction....................................................................................................................7
1.1 Background on the Neural Control of Walking .................................................................
Chapter 2 Phase Resetting Curves.........................................8
2.1 Definition of phase.
.........................................
2.2 Phase Resetting......................................................
7
....................................................
8
.......................................................
9
2.3 Phase Resetting Curves (PRC)..............................................................................................
2.3.1 Phase Resetting Curves: Theory and Applications.....................................................
9
9
2.3.2 Computation of Experimental PRC's.........................................................................
10
Chapter 3 Experimental PRC of Human Walking....................................................................
13
3.1 Subjects ............................------------..........--------............................................................
13
3.2 Experimental Setup............................
13
.......
.......................................................
3.3 Experimental Protocol...................................................................................................
15
3.3.1 Warm-up: Finding the Preferred Walking Speed and Perturbation Demonstration..... 15
3.3.2 Data Collection: Walking with Perturbations.............................................................
16
3.4 Data Analysis......................................................................................................................
17
3.4.1 Knee Brace Angle: Zeroing and Filtering .................................................................
3.4.2 Footswitch Event and Poincard Section Identification..............................................
17
3.4.3 Computing and Classifying the Stride Periods,.........................................................
20
3.4.4 Computing the Intrinsic Stride Period and Verifying it is Constant.........................
3.4.5 Verifying stride settling within two strides ..............................................................
20
17
22
3.4.6 Computing the phase shift............................................................................................
25
3.4.7 Plotting the PRC and a Least Squares Fourier regression
...................
25
3.5 Results ..
gr...............................................
3.5 R esults .......................................--.---.--...---.................................................
26
Chapter 4 Discussion and Conclusions.....................................................................................
4.1 PRC's for all Subjects were Indistinguishable from Zero..............................................
32
32
4.2 Considering the Potential for Artifact in the Zero PRC's ................................................
32
4.3 Interpretation of a Zero PRC for Human Walking..........................................................
33
B ibliography .................................................................................................................................
34
Appendix A: Stride Periods Plots ..............................................................................................
36
Appendix B: Knee plane plots sorted by perturbation phase.....................................................
38
Appendix C: Histograms of perturbation distributions............................................................
56
Appendix D: Regression residuals (h = 10).............................................................................
58
Table of Figures
Figure 2.1 Stable limit cycle with two perturbations applied at different phases. The perturbed
trajectories (dotted red) settle back to the limit cycle with a different phase when compared to the
phase of the respective unperturbed trajectory (green). The perturbation at # delays the phase,
9
while the perturbation at k advances the phase...........................................................................
the
causes
Figure 2.2 Limit cycle oscillator waveform plotted in time. The perturbation
perturbed trajectory to be delayed (negative phase shift), and the phase shifts are computed using
11
the At between the Poincard section crossings. .........................................................................
Figure 3.1 Experimental setup. Subject on treadmill wearing the Anklebot with custom shoe and
15
knee brace. ....................................................................................................................................
Figure 3.2 Example footswitch data with markers of four events: heel strike, heel off, toe strike,
18
and toe off. ....................................................................................................................................
Figure 3.3 Footswitch data showing heel slap event used as a substitute for toe off. A toe off
19
marker is shown at the heel slap events....................................................................................
a
have
periods
stride
and
3
1
the
Stage
that
showing
1,
Figure 3.4 All stride periods for Subject
21
very similar distribution to that of Stage 2................................................................................
Figure 3.5 All stride periods for Subject 8, which indicate that Stages 1 and 3 have different
distributions than Stage 2, and thus Stages 1 and 3 are not usable for computing the intrinsic
21
stride period. .................................................................................................................................
Figure 3.6 Period-3 strides and Stage 1 unperturbed strides plotted in the knee plane for Subject
1. The Period-3 band of trajectories shows only slightly greater variability and was used as a
23
valid representation of the limit cycle behavior.........................................................................
Figure 3.7 Period-3 knee plane plot and Stagel unperturbed knee plane plot for Subject 8........ 23
Figure 3.9 Knee plane plot showing Period-1 trajectories perturbed between phases 0.9 - 1. All
Period-2 strides have settled as all red dots lie inside the settled band. (Subject 1)................. 24
25
Figure 3.10 Perturbation phase distribution for Subject 5 .......................................................
25
.......................................................
Figure 3.11 Perturbation phase distribution for Subject 6
Figure 3.13 Residuals for the regression fit for Subject 1 above. The absence of pattern in the
residuals indicate that increasing the number of Fourier coefficients by increasing h will not
improve the fit...............................................................................................................................
27
2
Figure 3.15 PRC's: Subject 3, 2= 0.06 (left) and Subject 4, R = 0.06 (right)........................ 29
Figure 3.16 PRC's: Subject 5, 2 = 0.10 (left) and Subject 6, 2= 0.07 (right)........................ 30
Figure 3.17 PRC's: Subject 7, 2 = 0.06 (left) and Subject 8, R2 = 0.04 (right)........................ 30
Figure 3.18 PRC: Subject 9, 2 . .................................................................................
31
Table of Tables
Table 3.1 Mean and standard deviatation of physical characteristics and PWS (defined below)
experiment subjects.......................................................................................................................
Table 3.2 The mean and standard deviation for all subjects' stride periods, showing the close
agreement in stride period standard deviation of toe off vs. heel slap.......................................
Table 3.3.......................................................................................................................................
of
13
20
28
Table of Equations
Equation
Equation
Equation
Equation
Equation
2.1....................................................................................................................................
2.2..................................................................................................................................
2.3..................................................................................................................................
3.1..................................................................................................................................
3.2..................................................................................................................................
8
11
11
23
26
Chapter 1 Introduction
1.1 Background on the Neural Control of Walking
All animals have remarkable motor control capabilities thanks to an exceptional neural
controller. In humans, significant progress has been made on understanding the motor control of
the upper limb. However, control of the lower limb has many unanswered questions.
Understanding the control of the lower limb, and more specifically the control of walking, has
potential to improve our understanding of the challenges facing patients with neurological
disease, so that effective rehabilitation therapy can be offered in an attempt to improve their
mobility in everyday life.
Broadly speaking, there are two levels of controllers in the central nervous system that may be
responsible for generating the rhythmic patterns of muscle signals that control walking in
humans: supraspinal and spinal. Currently, the roles these neural controllers play in human
walking is unknown. In animals, it is well established that neural circuits called central pattern
generators (CPG's) lie in the spinal cord of numerous quadrupeds, including mammals and
reptiles, which generate a complete, set of feedforward motor commands that can provide low
level control of muscles for walking. These circuits can do so without any feedback, and require
only a tonic, or "on/off' signal from the brain. This is not to say that the brain plays no role
during walking in animals with CPG's, but rather serves to highlight the existence and role of
such spinal CPG's in several types of animals.
For humans, it is still unknown whether CPG's exist in the spinal cord, and if so, exactly what
role they play in the control of walking. There are many pieces of evidence that they do exist,
but no clear answer is available. There is definitive evidence, however, that a nonlinear
oscillator plays a role at some level in the control of walking. Such evidence is found in a study
by (Ahn & Hogan, 2010), where the stride frequency of humans entrained to an external
perturbation over a finite basin of entrainment.
To further study the role of a nonlinear oscillator in walking control, this thesis uses a tool of
dynamical systems analysis called a phase resetting curve, which is useful for understanding how
a nonlinear oscillator interacts with external perturbations. If a phase resetting curve can be
obtained, it serves as a first step in developing a simple model of the oscillator. The focus of this
thesis has been to determine if a phase resetting curve can be measured for humans, using an
experimental setup where human subjects walked on a treadmill while receiving intermittent
ankle torque perturbations. The final result is that no phase resetting was observed, which is not
the result expected if a nonlinear oscillator plays a prominent role. However, this result is
consistent with the kinematic control of the foot's trajectory, for which there is strong evidence
already in the literature.
Chapter 2 Phase Resetting Curves
This chapter introduces the phase resetting curve, which is a useful tool for studying nonlinear
dynamical systems. First, phase resetting curves are described, and second, their use and
applications are detailed.
2.1 Definition of phase
The phase of an oscillation is simply a unitless variable used to parameterize the oscillation.
Over one period, the phase parameterizes the oscillation by time with respect to the period
(Ermentrout, 2010). The phase of any instant in time during a period of the oscillation is defined
as,
<p(t) =
t-tk
tk+1 tk
Equation 2.1
where the instant in time of interest is t, the time that the current period began is tk, and the time
that the current period ends is tk+I. Thus, the phase increases linearly over the period, where each
period begins with # = 0 and wraps around to zero again at the start of the next period.
While the choice of the location or feature of the oscillation that corresponds to zero phase is
arbitrary in theory, there are some practical considerations in making this choice, discussed in
(Pikovsky, Rosenblum, & Kurths, 2001). The zero phase location in the oscillation is defined
using the idea of a Poincard section. Although this thesis does not look at stability analysis of
any model or system using a Poincard Map, a Poincard section is a useful way to define the zero
phase event of human walking.
For an n-dimensional system, a Poincar6 section is an n-1 dimensional surface that is transverse
to the flow of all trajectories, such that any trajectories that start on the section flow through it,
and not parallel to it (Strogatz, 1994). Also, only crossings in the defined direction are of
interest, thus the direction of the trajectory flow through the Poincard section must be considered.
In this thesis, where a walking human is the system of interest, Poincard sections are
approximated using footswitch data, to be described later.
For an autonomous limit cycle process, (Pikovsky, Rosenblum, & Kurths, 2001) note that the
phase is a marginally stable variable since d$/dt is constant, which is a direct result from the
definition that the phase increases linearly from zero to one over a single period.
Regarding terminology, the terms "phase space" and "phase plane" are not related to the
previously defined "phase" of an oscillation. Also, the terms phase lag and phase lead used to
describe the phase of a linear system relative to the phase of a forcing function differ from phase
"delays" and "advances," which are described in the next section.
2.2 Phase Resetting
A limit cycle's phase may be advanced or delayed by an external perturbation. The following
is a description of this concept using a limit cycle in 2-D phase space on the phase plane, shown
in Figure 2.1. Also, note that the system dynamics in this description are arbitrary. Zero phase is
defined as the Poincard section where the velocity, x, is held equal to zero, and where we are
only interested in Poincar6 section crossings from positive to negative velocity (blue line shows
Poincard section crossing).
At phase 4a, a perturbation is applied which displaces the system off the limit cycle (black line),
resulting in the perturbed trajectory (red dotted line). The perturbed trajectory settles back to the
limit cycle at some further phase indicated by *b'. The unperturbed trajectory (green line) would
have reached phase $b during the settling interval of the perturbed trajectory. The resulting
phase shift for the perturbation applied at *p, A*, is then given by the difference in phases at b
and b': 4b' - 4b. For this perturbation, the system undergoes a phase delay. Using the same
convention, a perturbation applied at 4c is shown to result in a phase advance.
Figure 2.1 Stable limit cycle with two perturbations applied at different phases. The perturbed trajectories (dotted red)
settle back to the limit cycle with a different phase when compared to the phase of the respective unperturbed trajectory
(green). The perturbationat 4 delays the phase, while the perturbation at O advances the phase.
2.3 Phase Resetting Curves (PRC)
2.3.1 Phase Resetting Curves: Theory and Applications
A phase resetting curve (PRC) plots AO vs.
#p,for all
, over the period. In general, a PRC is
found experimentally by applying perturbations to the system, one at a time, subsequently
measuring the resulting phase shifts. The resulting PRC depends on both the oscillator's
dynamics and the shape of the perturbation (Smeal, Ermentrout, & White, 2010).
PRC's are closely related to phase transition curves, or PTC, which were first introduced by
Arthur Winfree for the purpose of studying biological clocks and circadian rhythms (Winfree,
2001). A PRC contains the same information as a PTC, but plotted differently. On a PTC, the
vertical axis shows the final phase after a perturbation, not the phase shift. As with other terms
in nonlinear dynamics that are still in flux, PRC's are sometimes called PTC's and vice versa;
this thesis follows the terminology used in experimental neuroscience.
For limit cycle oscillators that are asymptotically stable, there are several theoretical results that
build upon PRC's that provide highly useful insight into the oscillator's behavior. From a single
state of an asymptotically stable oscillator, a PRC using a weak perturbation can be determined
by numerical or experimental methods. This "infinitesimal" PRC, allows for the construction a
phase model that can be used to predict the oscillator's entrainment behavior in response to an
external periodic perturbation (Galan, Ermentrout, & Urban, 2005). For example, predictions
can be made for the specific phases at which the oscillator will phase-lock to the perturbation.
Further, the phase model can be used to predict synchronization behavior in coupled oscillator
networks, which are a central topic in understanding neural processes (Smeal, Ermentrout, &
White, 2010).
2.3.2 Computation of Experimental PRC's
Except for perhaps the rarest examples, PRC's must be computed numerically or computed from
experimental data. As an alternative illustration of phase resetting, consider the oscillator plotted
in time in Figure 2.2. This description will also detail simple computations for finding PRC's
from data. The dynamics in this 2-D example are arbitrary, and are not intended to match the
previous example in Section 2.2.
Figure 2.2 Limit cycle oscillator waveform plotted in time. The perturbation causes the perturbed trajectory to be
delayed (negative phase shift), and the phase shifts are computed using the At between the Poincard section crossings.
The unperturbed limit cycle oscillator (black line) with intrinsic period To is plotted in time, with
zero phase (black dots) defined by the Poincard section, x = 0, with crossing direction from
negative to positive x (note Poincard sections exist in phase space, not plots in time). At time t,
(phase, #p), a perturbation is applied, resulting in the perturbed trajectory (dotted red line). The
perturbed trajectory reaches the Poincard section (red dot) after period Tp1 . If the assumption can
be made that the oscillator settles back to the limit cycle by the end of Tp1 , then the phase shift is
computed as
A@
At
TO
T= Tp
Equation 2.2
TO
After Tpl, the oscillator continues with period To (Tp2 = To). By repeatedly applying
perturbations at a sufficiently fine distribution of #,'s throughout the limit cycle, and allowing
settling to occur, the phase resetting curve can then be plotted. If the assumption of same-period
settling is violated, where Tp2 # To, then additional periods must be included in computing At. If
the system settles during Tp2, then the phase shift is computed as
At
2T=(Tp+TP 2 )
TO
TO
Equation 2.3
For systems that do not settle in Tpn, the theory referenced above cannot be used without some
modification. In experiments that do require Tr2 for settling, the PRC theory has been
successfully modified to yield useful predictions about synchronization (Oprisan, Prinz, &
Canavier, 2004). Also, Oprisan suggests that beyond 2 periods "the bookkeeping becomes
intractable." While not stated explicitly, this is likely due to variability of the system's period,
which may confound the computation of phase resetting. However, if the magnitude of the
phase shift is large relative to the period variability, then looking beyond period Tn2 may allow
for phase resetting to be estimated (Feldman, Krasovksy, Banina, Lamontagne, & Levin, 2010),
(Nomura, Kobayashi, & Kozuka, 1998).
Chapter 3 Experimental PRC of Human Walking
This chapter describes the experiments conducted to measure a PRC for unimpaired human
walking using a torque perturbation at the ankle. The following sections describe the
experimental setup, protocol, data analysis, and results.
3.1 Subjects
Nine male subjects were recruited on the basis of being acquaintances of the investigator.
Subjects were included on the basis of having no reported history of neural abnormality. All
subjects gave informed consent after receiving an explanation of the experiment. Table 3.1
shows the age, height, weight and chosen walking speed (defined below) for the 9 subjects.
Mean ± std. dev.
Age (yrs.)
Height (ft., in.)
25
5' 11"
2.7
2.8
Weight (lbs.)
165 ±19
PWS (mph)
2.3 ± 0.41
Range
22-31
5'8"- 6'5"
135-195
1.7
-
3.0
Table 3.1 Mean and standard deviatation of physical characteristics and PWS (defined below) of experiment subjects.
3.2 Experimental Setup
Perturbation Device
A robot called the Anklebot was used to apply the plantarflexion perturbations during the PRC
experiments. The Anklebot is a novel, 2 DOF device that can be used to apply torques about two
axes. For a full characterization of the Anklebot, see (Roy, et al., 2009). The Anklebot is
controlled via a Tcl/Tk script running inside a real time Linux kernel. The system computer
collects Anklebot data such as torque commands and ankle position, as well as other sensors
described below, all at a sampling rate of 200 Hz.
The perturbation applied by the Anklebot was a 140ms torque pulse which acted to plantarflex
the ankle, which is the motion undergone when pointing one's toes. The main muscles that
actuate plantarflexion are the soleus and gastrocnemius. The magnitude of the perturbation was
6 Nm when the shank was at a right angle to the sole of the foot. While the magnitude of the
force applied by the Anklebot was constant, the applied torque varied because the moment arm
varied slightly due to the angular travel of the ankle during walking. The minimum moment arm
will occur at maximum deflection, which occurs during maximum ankle plantarflexion, which is
on average about 20' (Perry, 1992). Thus the actual torque applied was in the range of 5.6 - 6
Nm.
The magnitude was chosen to be as large as possible while still remaining within the comfort
range of all subjects. This was determined through several iterations of the experimental
protocol. Subjects reported that the discomfort induced by perturbations occurred during swing
phase, most notably when perturbations occurred just before heel strike. The interruption of heel
strike elicited a surprising sensation and may have evoked a startle response. This effect was not
studied in this thesis, though it may provide interesting information about the feedback used by
the neural controller. The chosen magnitude was relatively light and easy to walk with, and as a
result, any phase shifts were expected to be small in magnitude.
Donning Experimental Hardware
The experimental session began with fitting and attaching the hardware and sensors to the
subject's dominant leg, determined by kicking preference. First, two footswitches used to detect
toe-off and heel strike were taped to the bottom of the bare foot on the big toe and the heel. A
footswitch is a variable resistor that decreases in resistance when pressure is applied. The sock
was then put on, and a custom shoe with a mounting bracket for the Anklebot was put on the foot. Then, an appropriately sized knee brace, also with mounting bracket for the Anklebot, was
attached to knee. The knee brace has a potentiometer for measuring the angle of the knee.
All knee brace straps were tightened firmly to reduce brace sliding but not to an extent to cause
discomfort. In particular, the strap around the upper part of the gastrocnemius was re-tightened
as the final fitting step to ensure support for the weight of the Anklebot. After fitting the knee
brace, the subject was instructed to swing the knee several times to assess proper joint alignment
and general comfort.
Next, a shoulder strap was attached to a loop on the knee brace and run up around the neck as
another means to prevent the brace from sliding. Lastly, the Anklebot was mounted to the knee
brace and shoe mounting brackets.
Figure 3.1 Experimental setup. Subject on treadmill wearing the Anklebot with custom shoe and knee brace.
Zeroing the knee brace
3.3 Experimental Protocol
3.3.1 Warm-up: Finding the Preferred Walking Speed and Perturbation Demonstration
The first task during warm-up was to zero the knee brace. The subject was instructed to stand
with their weight shifted to the non-dominant side, and subsequently asked to rest the dominant
heel in front of their body so that the knee was fully extended against the limit of the knee brace,
if the limit could even be reached. This follows the convention that that a fully extended knee
(straight leg) has an angle equal to zero. A demonstration was provided by the investigator.
Next, the subject was guided through a walking warm up that lasted approximately 10 minutes.
The purpose of this warm up was to allow the subjects to choose a preferred walking speed
(PWS) on the treadmill and to experience a demonstration of the perturbations.
The preferred walking speed was the treadmill speed that the subjects chose to walk for the
duration of the experiment. The purpose of finding a preferred walking speed was twofold.
First, there is some evidence that there is an over ground walking speed that is preferred by the
neural controller. Evidence of such a preference appears as a reduction in hip flexor activity
during swing phase in some subjects studied, to the extent of zero hip flexor activity (Perry,
1992). One hypothesis to account for this observation is that the neural controller takes
advantage of the pendular dynamics of the leg in swing phase. Since settling to the nominal
walking limit cycle is imperative to obtain a PRC, the PWS selection process was done to
accommodate this possible neural preference.
The second purpose, which may be related to the first with regard to efficiency, is to ensure that
the subjects become accustomed to the Anklebot and comfortable with its added mass. Further,
letting the subject choose the speed at which they can comfortably walk for 20 minutes
minimizes fatigue.
The instructions below were verbally provided by the investigator to guide the selection of the
PWS. Subjects were repeatedly reminded that natural, comfortable walking was the primary
objective. To reinforce this objective, the treadmill's speed display was covered. The subjects
alone controlled the speed of the treadmill using buttons on the handrails, which ensured that
they accelerated at their own rate and were not influenced by the investigator.
1. Choose a speed that feels natural and comfortable. [Hold for approximately 1 minute]
2. Increase the speed of the treadmill by several ticks until you feel it is too fast to walk
comfortably for 20 minutes [Hold approximately 30 seconds at "Fast" speed]
3. Decrease the speed back to a comfortable, natural speed [Hold approximately 30 seconds
at "Comfortable" speed]
4. Decrease the speed by several ticks until you feel it is too slow to walk naturally [Hold
approximately 30 seconds at "Slow" speed]
5. Increase the speed back to a comfortable, natural speed [Hold approximately 30 seconds
at "Comfortable" speed]
6. Repeat Steps 1-5
7. Use the final 2 "dround "Comfortable" speed as the PWS
Subjects then stood still on the treadmill for a 2 minute break while the software was started. A
perturbation demonstration was provided, first with the treadmill stopped, and second with
subject walking at the PWS. About 20 perturbations were applied while walking to ensure
subject's familiarity with the setup and also to estimate the subject's stride period, T,,, which
was entered in software to control perturbation timing.
3.3.2 Data Collection: Walking with Perturbations
Next, the data collection portion of the experiment began. Subjects were again instructed to walk
comfortably and naturally. The data collection portion of the experiment was separated into three
stages. Stage 1 consisted of 90 seconds of unperturbed walking, Stage 2 consisted of about 20
minutes of walking with 400 pseudo-randomly timed perturbations, and Stage 3 consisted of
another 90 seconds of unperturbed walking. The perturbation control scheme, detailed below,
was designed to apply 400 perturbations uniformly distributed over the stride period. The phases
at which the perturbations were applied were pseudo-randomly ordered to prevent entrainment,
which was shown to occur for a periodic perturbation with a period within the basin of
entrainment (Ahn & Hogan, 2010). Each subject received the same order of perturbations.
The perturbations during Stage 2 were controlled via the system computer running a Tcl'k
script. The script monitored positive crossings of a threshold knee angle (knee brace
potentiometer voltage) to detect the onset of the large flexion peak during swing phase. After
this event, a perturbation was applied after a time delay equal to some fraction of T,,,. In this
manner, using a uniform distribution of time delays ranging from zero to T,,,, the 400
perturbations were distributed approximately uniformly over the stride period. After each
perturbation, the computer stopped monitoring the knee angle for 1.5T,,,, which ensured that no
perturbations will occur during the next stride period, T 2 . This pause was introduced because
pilot studies indicated that the response to perturbations of the magnitude delivered sometimes
required more than one stride but settled within two strides or less.
The number of perturbations chosen was based on (Galan, Ermentrout, & Urban, 2005), which
showed that for perturbations that produce small phase shifts buried in inevitable noise, a large
number of data points is needed so that regression can recover the true shape of the PRC.
3.4 Data Analysis
This section details the data analysis, which was done using MATLAB v7.12 (The Mathworks,
Natick MA). The data used in the analysis included the knee brace angle, Anklebot torque
commands, and toe and heel switch voltages.
3.4.1 Knee Brace Angle: Zeroing and Filtering
As described in the experimental protocol, the zero knee angle for each subject was found by
asking subjects to fully extend the knee brace. When in this position, the knee angle was
sampled for 2 seconds (2000 ms), and subsequently averaged to provide a zero value for the knee
angle.
The knee angle was used to observe the limit cycle behavior of the subjects when walking. This
entailed finding the angular velocity, thus filtering was needed to remove noise before
differentiating. The knee angle was filtered forwards and backwards using a low pass FIR filter
with cut off frequency of 7.5 Hz to remove noise. No other filtering of data was needed.
3.4.2 Footswitch Event and Poincare Section Identification
Before doing any phase resetting computations, a Poincard section was chosen. In general, any
periodic, identifiable event could be used, however, since the perturbations from the Anklebot
affected the stride kinematics, a Poincard section had to be chosen that could be identified
reliably in the presence of perturbations. For example, the heel switch voltage was used to detect
heel strike, but perturbations just prior to heel strike were found to plantarflex the ankle and raise
the heel, thus delaying the heel strike. Whether this delay resulted in phase resetting was
unknown initially, and alternatives were considered. The final choice, for all subjects, was toe off
because perturbations had no observable effect on this event, likely due to the configuration of
the leg at toe off and the constraint of contact with the ground before toe off.
The method for identifying toe off in the analysis code is described here, and an example result is
shown in Figure 3.2. First, a suitable threshold value for toe off was chosen. Second, the signal
indices of all falling edges that crossed the threshold value were identified. Since the edges have
short times scales, there are relatively few samples per edge. Therefore, when detecting a falling
edge threshold crossing, the identified signal sample may appear below the threshold. Finally,
starting at the first of these indices, all subsequent indices were "scanned" for false events by
ensuring that the n+1* index was separated in time from the n index by a chosen time buffer,
twe, which ranged from 0.5 to 1 second depending on the event being identified. This final
scanning step was critical for proper identification of some subjects' toe off events (and other
events as well), which filtered out many small noise edges that caused false event identifications,
as well as perturbation effects, and spurious toe movement.
The methods for identifying the other events shown in Figure 3.2 were nearly identical except for
the values of the algorithm parameters. These parameters were "tuned" for each subject as the
footswitch signal waveform varied greatly in appearance across the nine subjects. The effects of
the shoe's fit and perturbations also seemed to vary, though this was not studied.
5-
4.--tbe
4
4.5
IV
A
1
off tweshold
Heelstike
Heel oil
2.5-b
714.14.
791i.5
712
712.5
713
713.57174.
ime [sec]
Figure 3.2 Example footswitch data with markers of four events: heel strike, heel off, toe strike, and toe off.
For two of the nine subjects, the toe switch failed during the experiment. Fortunately, the heel switch
signal contained a feature informally called "heel slap," which occurred every stride in these two subjects
(data from several other subjects also had this feature). This event is caused by the heel of the shoe
striking the heel of the foot at some instant near toe off, similar to the behavior of a "flip-flop" sandal.
The validity of this event as a substitute for toe off is evident in Figure 3.2 above, where the sharp edge in
the heel switch data coincides exactly with the falling edge of the toe switch. Figure 3.3 below shows an
example from one of the two subjects in which the toe switch failed. The heel slap event yielded a stride
period standard deviation that was very similar to values obtained from other subjects computed using toe
off, shown in Table 3.2. The algorithm for identifying heel slap was similar to that for toe off, except that
a rising edge threshold was set so that the signal crossed at a time shortly after the onset of the actual
edge. This was necessary to avoid detecting non-heel slap edges that occurred in swing phase. To gain a
better edge estimation in time, the index of heel slap was stepped backward to the previous index to
produce a better measure of the edge in time. This places the marker below the threshold.
5toe
4.5-
heelsp reshd
V Heelsilke
A Heelof
Y Toe stike
4-
A
3.5-
ToeofN
3 -
.5
0.5
791.5
712
712.5
713
timeisee]
713.5
714
714.5
Figure 3.3 Footswitch data showing heel slap event used as a substitute for toe off. A toe off marker is shown at the heel
slap events.
Table 3.2 The mean and standard deviation for all subjects' stride periods, showing the close agreement in stride period
standarddeviation of toe off vs. heel slap.
3.4.3 Computing and Classifying the Stride Periods,
Each stride period is simply the difference in time between adjacent Poincard sections.
All strides during Stage 2, walking with perturbations, were classified into one of three "types,"
which will simplify communication of the analysis performed. The stride during which a
perturbation is applied is called a "Period-I" stride, and the immediately following stride is
called a "Period-2" stride, so that there were 400 of each type of stride. Lastly, the third and final
classification of strides is a Period-3 stride; these strides follow Period-2 strides. They were
located in software by looking for unperturbed strides that occurred after Period-2 strides.
There were approximately 200 Period-3 strides for each subject. The variation of this number
arose because a Period-3 stride only occurred after a perturbation late in a stride, where the
1.5T,,, pause lasted until about halfway into the Period-3 stride. After this pause, if the knee
angle had already passed the rising edge trigger threshold, no perturbation was be triggered for
this Period-3 stride.
3.4.4 Computing the Intrinsic Stride Period and Verifying it is Constant
Before computing phase resetting, the intrinsic stride period was found. To do so, all stride
periods were plotted vs. stride number. In Figure 3.4, the Stage 1 and 3 stride periods show a
similar distribution to the Stage 2 stride periods. However, this is not the case in Figure 3.5,
where there is a visible difference in distributions between Stages. While statistical tests could
be performed to verify that Stages 1 and 3 differ from 2, it was deemed unnecessary as the
Period-3 strides provided a more meaningful measure of the Stage 2 intrinsic period since they
are unperturbed and fully settled; the fact that Period-3 strides have settled will be shown below.
The best measure of the intrinsic period, To, was computed by averaging all Period-3 strides.
Lastly, it is important for the phase resetting computations to verify that the intrinsic period does
not change throughout the perturbed walking in Stage 2. This was verified for all subjects as the
Period-3 strides, and even all strides in Stage 2, show no significant deviation about the dotted To
line. Papers from experimental neuroscience literature report that the intrinsic period of a neuron
may show variability up to 10%, and still allow for a valid PRC to be estimated (Oprisan, Prinz,
& Canavier, 2004). Although it is not stated whether 10% refers to the range (max, min) of the
variability or the standard deviation, in the Results section below it is shown that the standard
deviation of the walking intrinsic period was on the order of 3-4%; well within either possible
interpretation of the limit.
stride period vs. stride number
o
*
*
"
---
*
1.2 -
W
unperturbed walking
Period-1 (perturbed)
Period-2 (unperturbed)
Period-3 (unperturbed)
T ±a
VP
1.53
1-5
0
r
r
200
400
r
600
stride number
r
r
800
1000
Figure 3.4 All stride periods for Subject 1, showing that the Stage 1 and 3 stride periods have a very similar distribution
to that of Stage 2.
stride period vs. stride number
o
.
*
*
1.4-
1.35-
unperturbed walking
Period-1 (perturbed)
Period-2 (unperturbed)
Period-3 (unperturbed)
0
0
0&
S00
8.1.25
0
o
12-
*..
*,.*0e.
%
PS
.
1.151.1 I.r
0
r
200
400
r
600
stride number
r
r
800
1000
Figure 3.5 All stride periods for Subject 8, which indicate that Stages 1 and 3 have different distributions than Stage 2,
and thus Stages 1 and 3 are not usable for computing the intrinsic stride period.
3.4.5 Verifying stride settling within two strides
Another important component of the phase resetting computation was to verify that the perturbed
trajectories settled back to the nominal limit cycle within two strides. To study stride settling,
the knee brace angle was plotted against its angular velocity, which results in a 2-D "knee plane"
plot that shows the limit cycle behavior of human walking. Although human walking dynamics
may have system order far greater than two, this knee plane plot served as a useful way to assess
stride settling.
To verify that a perturbed stride had settled, the limit cycle behavior of the unperturbed gait was
established, along with an appropriate "settled band," within which trajectories were assumed
fully settled. As discussed in the previous section, the Period-3 strides provide the best estimate
of the intrinsic period during perturbed walking. In Figure 3.6 and Figure 3.7, the Period-3
strides are plotted in blue, showing a highly repeatable trajectory with apparently low variability,
or width. To verify that this variability is characteristic of unperturbed, fully settled stride
kinematics, it was compared to the Stage 1 strides also plotted in the knee plane. The Period-3
trajectory showed only slightly greater variability throughout the majority of the stride. Some
subjects, such as Subject 1 in Figure 3.6, showed greater variability and a shift in the smaller
flexion circle which occurs during the stance phase. Nevertheless, the repeatability of the
behavior still indicates the trajectory is not in transient settling or otherwise perturbed. Subject
8's plot shows excellent consistency in the Period-3 trajectory, even when compared to the Stage
1 trajectory. These plots show that the Period-3 strides exhibited repeatable limit-cycle behavior.
They were used to establish a "settled band" to determine whether perturbed strides during stage
2 settled before the end of Period-2.
400
300
200
100
0
0 -100\
-200-
Perod-3 strdes
-300-
strides
Stage 1 unperturbed
-400Cr
0
10
20
30
r
40
knee angle[deg]
r
50
60
70
80
Figure 3.6 Period-3 strides and Stage 1 unperturbed strides plotted in the knee plane for Subject 1. The Period-3 band of
trajectories shows only slightly greater variability and was used as a valid representation of the limit cycle behavior.
400 -
300-
200-
100-
0-100-
-200 -
-300 -
Stage 1eo1 Irpeturbed strides
strides
400
0
10
20
30
40
knee angIe [deg]
50
60
70
s0
Figure 3.7 Period-3 knee plane plot and Stagel unperturbed knee plane plot for Subject 8.
Finally, the perturbed strides were plotted over the Period-3 "settled band" to ensure that at the
conclusion of the Period-2 stride, the trajectory was within this band. To facilitate this visual
inspection, all Period-1 strides and the associated Period-2 strides were sorted by the phase of the
applied perturbation. The phase of the perturbation was computed as
Op =
,
,
Equation 3.1
where t, is the instant of perturbation onset, tpo is the instant of the previous Poincard section
crossing prior to t,, and To is the intrinsic period as computed above. After sorting the Period-I
strides by perturbation phase into ten phase bins with width 0.1, the Period-1 trajectories and the
following Period-2 strides in each bin were plotted. Two example plots are shown in Figure 3.8
and Figure 3.9. As shown in the legend, each Period-1 stride is a red line that begins with a
green marker at the Poincar6 section (toe off) and moves clockwise around the plot. At some
phase during the stride, a perturbation was applied. At the next crossing of the Poincard section,
the trajectory changes color to black for the Period-2 stride, and continues to the Poincar6
section, where it ends at a red dot. The location of these red dots within the settled band is clear
evidence that settling has occurred during the Period-2 strides. If settling did not occur, these red
dots would be located outside of the settling band.
During data analysis, the 10 plots for each subject were inspected for settling.
Period-1 and Period-2 Strides for r, = (0.2 - 0.3)
-
o
*
o
40
Settled band (all Period-3)
Period-1 stride (perturbed)
Period-2 stride (unpert.)
Start of Period-1 stride
End of Period-2 stride
perturbation marker
r
r
80
)70
Knee angle [deg]
Figure 3.8 Knee plane plot showing Period-1 trajectories perturbed at phases between 0.2- 0.3, and the following Period2 trajectories. Note that all red dots lie inside the settled band (blue), which is clear evidence that all Period-2 trajectories
had settled. (Subject 1)
400-
Period-1 and Period-2 Strides for r, = (0.9 - 1)
300-
200-
or100
0Ce
0
o
-1000
-200-
- Settled band (all Period-3)
-
(
-300 H
o
o
0
10
20
30
40
50
60
Period-1 stride (perturbed)
Period-2 stride (unpert.)
Start of Period-1 stride
End of Period-2 stride
perturbation marker
70
80
Knee angle [deg]
Figure 3.9 Knee plane plot showing Period-1 trajectories perturbed between phases 0.9 - 1. All Period-2 strides have
settled as all red dots lie inside the settled band. (Subject 1)
Verifying uniform phase distribution
To ensure the perturbation control software successfully distributed the 400 perturbations
uniformly throughout the stride, histograms were plotted showing the number of perturbations
applied in each of ten phase bins. The data for two subjects below show sufficiently uniform
distributions, with the remaining subjects showing uniformity similar to these, or better.
Anl
PertirbationDisbibtion
50
-40
20
10
0
Figure 3.10 Perturbation phase distribution for Subject 5.
PertubationDistribulion
Figure 3.11 Perturbationphase distribution for Subject 6.
3.4.6 Computing the phase shift
After the intrinsic period was found, verified sufficiently constant and all perturbed trajectories
were shown to settle during Period-2, the phase shifts were computed using Equation 2.3 from
Section 2.3.2.
3.4.7 Plotting the PRC and a Least Squares Fourier regression
With the phase shifts and perturbation phases computed, the PRC was then plotted. However, as
is commonly seen in the literature, simply plotting of the PRC points may not reveal the shape of
the PRC due to unavoidable variability. Since the experimental measurements may have added
noise and variability appears to be an intrinsic property of the underlying behavior, this
experimental method relied on a large number of data points and subsequent regression to
uncover the shape of the PRC. A regression model for the PRC points was chosen, and analysis
was done on the regression to determine if it explained a sufficient amount of variation in the
data.
The chosen regression was a least squared error fit to a Fourier series model. A Fourier series is a
logical and straightforward choice since PRC's are periodic, a fact due the phase wrap-around at
1. The regression equation was
=
ao +
'=1 a cos(n -21rp) + b, sin(n -2r#,)), Equation 3.2
where the h index denotes the number of harmonics used in the fit.
3.5 Results
The top panel in Figure 3.12 shows the PRC data plotted along with Fourier regression with 10
harmonics. Fewer harmonics were tried initially, but the R2, which is a measure of the data
variance accounted for by the regression, was extremely low. The R2 value for h = 10 for this
subject is 0.08, very low, and increasing the number of harmonics further brought little
improvement in R 2. The residuals for h = 10 are plotted in Figure 3.13 and contain no pattern
with approximately zero mean, which indicates further increasing h would do little to improve
the regression fit.
0.2
0.1
0
0.4
0.3
0.6
0.5
0.7
0.8
0.9
0.7
0.8
0.9
'p
Average knee profile
0.1
0.2
0.3
0.4
0.5
0.6
Figure 3.12 The top panel shows the PRC obtained from a least squared error regression to a Fourierseries model with
(h=10). The bottom panel shows an averaged knee profile for the subject, to provide a reference for the configuration of
the leg. (Subject 1)
Regression Residuals (10 harmonics)
0.04
L
L 0
(L
00 00C
0
0
0
0000
0
0
0
0
)o0
0.03
00
0.02
0Q
0.01
0 0
(P
0
0
(30
QCJ)(
0
0
0
0
00
o0
00
0
0000 000
o 00o
aoU
00 00
-0.01,
0
00
0
0
0 0
0000
o
0
0
0
-0.03
0
&
0
0
0
c0
cO
0
CD
000
:
0~
90
%0
O
00
0
0
0
50
100
150
200
0
0
0
o
0
C
0
250
0
COO 0
c
0
00
000
0 0
-0.04
0
0
0
0
000 o
(O00)~@(
0
0
c
000
00
0
0
Q
0
~
0
o
0
d
0
0
0 00e
O 0
?
o
o0
0
-0.02
-0.05
L
00
0
0
o
0
300
9%
000
0 0t
350
400
Figure 3.13 Residuals for the regression fit for Subject 1 above. The absence of pattern in the residuals indicate that
increasing the number of Fourier coefficients by increasing h will not improve the fit.
The PRC results for all 9 subjects are presented here, along with several statistics. All plots used in the
analysis for all subjects are contained in Appendix A: Stride Periods Plots.
There were no adverse events to report aside from the failure of the toe switches, as described in
Section 3.4.2. One subject asked to wear the Anklebot on the non-dominant side because his
dominant knee almost fully recovered from a prior injury. The Anklebot was worn on his nondominant side, and when asked during the experiment about knee pain, the subject reported no
discomfort.
Table 3.3 shows the mean and standard deviation of the Period-3 strides used to compute To, well as all
strides from all three Stages.
Subject
To (Period-3 mean) ± std. dev.
All strides, mean ± std. dev.
1
1.303 ± 0.021
1.308 ± 0.022
2
1.200 ± 0.022
1.201 ± 0.024
3
1.405 ± 0.030
1.404 ± 0.032
4
1.314 ±0.023
1.316 ±0.025
5
1.218 ± 0.017
1.219 ± 0.020
6 (heel slap)
1.270 ± 0.035
1.273 ± 0.036
7
1.308 ±0.030
1.312 ±0.030
8
1.252 ± 0.026
1.257 ±0.028
9 (heel slap)
1.390 ± 0.027
1.392 ± 0.031
Table 3.3 The mean and standard deviation for the intrinsic period To and "All Strides"
The PRC's from each subject are shown in Figure 3.14 through Figure 3.18. Also plotted with
the PRC points are the Fourier regression and the average phase shift in each phase bin. The R2
value for each regression is shown in the figure caption. The averaged knee profile during the
stride is plotted in the bottom panel for referencing the leg configuration.
0
0.0
__
__
1 0~ 1
4;
__1I
10 01
1
F
CmpUedlues
R1
LSiHM
*
Fi
Compuavalues
LSFourer Fit
o
0~
ad_
W
0
0
t
~ ~ ~8t
O
C
qtQ1qTo0
OM~
-
M -_
p
y
-a
'lBin o
b
A
%
c
0
c
CeC
0
.1
43
0.2
4
8
0.5
8
0,7
0L9
Q2, Q3
0010 --%
Q4
0.6
0-)
5
0_
0p
0.7
0.8
Q9
f27
08
Q
1
a_
.p
Average kneeprofile
Average knee profile
P
Q
0.
0.1
M2
4
0.3
0.5
07
as
0.9
0.8
4
Q
05
1
Figure 3.14 PRC's: Subject 1, R2 = 0.07 (left) and Subject 2, R 2 = 0.07 (right)
Q1 -
- -
----
9---
-
-,
-
---
----
d
-
-
-
-
LS Fuir
FI
o
0.1
0.2
03
0.4
i5 n
7
0.9
.
p
0.11
Q2
Avrage kneeprofile
a_7
04
0.5
03
1
0
0o
Q2
03 04
0.5
-
07
Q8
0.9
0.8
Q.7
0.8
OLD9
0.6
(Q7
08
0.9
6
A.--.w
1
Averagekneeprofile
8
j40
0.9
1
_
0.
_
530-
0.1
1
0
0
0-
0
mnkee
LS FoudeFit
o
0
0
oCompuie
n
0
0.2
CO3
0.4
__
05
_
06
0. O2
M.3
0.2
Q3
0.4 0.5
__
L7
0.8
0.9
1
0.1
0.4
O5
4,
Figure 3.15 PRC's: Subject 3, R2 = 0.06 (left) and Subject 4, R2 = 0.06 (right)
1
_
PRC
OL
.1
I
-E- LS FourierFit
0
0
00
9
"1
0
0
0 N ES
i
i
0.1~~~
Bin awage
E)
0.05 ----------
)
00
0
C
0O.
0
01 02 0.3
0
0.4
1.5
0.6
0.7
0.8
__
0_
09
_
0
0
01
0.2
0
0.3
.4
0.5
0
0.6
0
0.7
0.8
09
0.7
0.8
.9
00
Average kneeprolle
__
-
0.1
0
_
1.0
QI I
0 Compuledmalues
LSFourier F
Bin awMge
-
Q2
0.3
___
___
04
. 0.5
0.7
___
0.8
0.9
1
al
0.2
0.3
04
0.5
0.
1
41
Figure 3.16 PRC's: Subject 5,R2 = 0.10 (left) and Subject 6, R2 = 0.07 (right)
r
d
o o - LS FourieFit
C
oc
0
0Co?
00
89
Q1 04
0~~~~
0)0
1
Q
Q B Q0203
6
P
Average kneeprole
2
Q1
1
0
0
Q1
Q2
M3
0.4
OQS
a6
a7
a8
OLD
-0
- - -
0.O2
~
-
0.3
0.4
QS
as
I
Figure 3.17 PRC's: Subject 7, R2 = 0.06 (left) and Subject 8,R2 = 0.04 (right)
a7
as
OLD
1
f
1
0
0
at
o
2
M
.3
04
0.5
wIlue
G
eo8
0p
Compute
LS Fixier Fk
Bi
-
o
n
oP O o-
0
rp1
0
0.6
M7
a
0.9
oln
M 40-__
__
20-
_
0a. M2 0.3
0L4
0.5
0.6
0.7
ao
4?
Figure 3.18 PRC: Subject 9, R2 = 0.05
0.9
__
Chapter 4 Discussion and Conclusions
4.1 PRC's for all Subjects were Indistinguishable from Zero
The Fourier regression for each subject has a very low R2 value, which shows that the regression
barely accounts for the observed variance in the observations of phase resetting. The inescapable
conclusion for this experiment is that the PRC is zero. The Fourier series is an ideal choice for
fitting a PRC due to its periodicity, so there is no reason to pursue other regression models.
Additionally, the plot of residuals for each subject's PRC regression showed no pattern with
approximately zero mean, indicating that the number of coefficients in the regression is
sufficient. Finally, there was no consistent pattern among the subjects.
4.2 Considering the Potential for Artifact in the Zero PRC's
Before interpreting the implications of a zero PRC, the possibility of an inaccurate PRC is
considered here. It is possible that phase resetting may have occurred, but it was not detected by
the analysis method used. However, the method used is very similar to that of other studies that
have successfully measured PRC's for endogenously bursting neurons. Although a neuron is a
drastically different system, both human walking and neurons are capably modeled by limit cycle
oscillators which means the methods of analysis will be necessarily similar.
In other studies, the collection of a large number of PRC points has ensured enough information
is contained in the data to uncover the PRC from noise. The PRC experiments here used exactly
that approach, and still did not find any meaningful regression from the data. Although there is
no reason to suspect excessive noise in this study, one improvement to the analysis that might
reduce noise would be to measure phase resetting using a mixed pair of Poincar6 sections, so that
the total time elapsed is shorter than two full strides. This would require studying the settling
plots and choosing appropriate sections, however there are few options as heel strike was the
only other consistently observed Poincard section, and was found to be corrupted by
perturbations at certain phases.
It is possible that the PRC was found to be zero because the perturbation applied was too weak to
produce a large enough kinematic disturbance. However, this is unlikely as the magnitude used
for the ankle torque perturbation was as large as possible while still remaining tolerable by
subjects, determined by many iterations of the experimental protocol. The plantarflexion torque
pulse seems to delay heel strike when applied just before heel strike, which elicits a sensation of
surprise in subjects. The neural controller may be expecting heel strike feedback at a certain
time, or state of the body, and when this feedback is not present it triggers a startle reflex.
Although this effect was not studied, it provides ideas for experiments to study the neural
controller's feedback requirements.
A final possibility is that no phase resetting was observed due to the speed constraint of the
treadmill. If this were true, then a comparable experiment conducted over ground might show a
non-zero PRC, due to some difference in the neural controller. The study reported here did not
investigate the possible differences between treadmill and over ground walking.
4.3 Interpretation of a Zero PRC for Human Walking
After considering the alternatives above, the clear conclusion from this experiment is that the
PRC is zero. Since the knee angle was clearly perturbed off its nominal trajectory (see Appendix
B: Knee plane plots sorted by perturbation phase), this suggests that some higher level of neural
control may be correcting for the perturbed gait to produce zero phase shift. Which leg is
responsible for the corrective action to maintain the phase cannot be determined, since this
experiment did not collect data for the behavior of the contralateral leg, which undergoes a full
stride during the two periods of settling. This evidence of zero phase resetting is consistent with
other convincing evidence for some form of kinematic control of the lower limb; during walking
under reduced gravity via a body weight support system, the kinematic coordination of the lower
limb was found to be highly repeatable drastic changes in kinetic parameters (Ivankeno, Grasso,
Macellari, & Lacquaniti, 2002). For the upper limb, it is well established that the central nervous
system utilizes kinematic control to generate maximally smooth trajectories in reaching
movements (Flash & Hogan, 1985), thus the ability to control kinematics of the leg, and more
specifically the foot, certainly exists.
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Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization. Cambridge, UK:
Cambridge University Press.
Roy, A., Krebs, H. I., Williams, D. J., Bever, C. T., Forrester, L. W., Macko, R. M., et al. (2009).
Robot-Aided Neurorehabilitation: A Novel Robot for Ankle Rehabilitation. IEEE Trans.
on Robotics, 25(3), 569-582.
Smeal, R. M., Ermentrout, G. B., & White, J. A. (2010). Phase-response Curves and
Synchronized Neural Networks. Phil. Trans. R. Soc. B, 365, 2407-2422.
Strogatz, S. H. (1994). NonlinearDynamics and Chaos. Cambridge, MA: Perseus Books
Publishing.
Winfree, A. T. (2001). The Geometry of Biological Time. New York, NY: Springer.
Appendix A: Stride Periods Plots
stride period vs. stride number (Stbject 1)
stride period vs. stride number (Subject 2)
600
stride number
600
stride number
stride period vs. stride number (Subject 4)
o
*
*
o
-
stride period vs. stride number (Stbject 3)
unperturbed walking
Period-1 (perturbed)
Period-2 (unperturbed)
Period-3 (unperturbed)
T ±
0
00
0
200
400
600
800
stride number
stride period vs. stride number (Subject 5)
0
200
400
600
stride number
800
1000
1000
200
0
200
400
600
800
1000
stride number
stride period vs. stride number (Subject 6)
400
600
stride number
800
1000
stride period vs. stride number (Subject 7)
o
14
stride period vs. stride number (Subject 8)
urperturbed walking
Period-1 (perturbed)
1.4 0
1.4---
Period-2 (unperturbed)
Period-3 (urperturbed)
T o
t
1.35
1.25 121.15
0
200
400
600
stride number
800
1000
stride period vs. stride number (Subject 9)
0
200
400
600
stride number
800
1000
0
200
400
600
stride number
800
1000
Period-iandPedod-2
Stridesfor 4 = (0.1-0.2) (Subject1)
Period-Iand Padod-2
Stides lor 4,= (0 -0.1) (Subject1)
Period-iand Pedod-2Stridesfor4=(0.2-0.3) (Slbjecil)
200
Oe
100
aW
-100
-100
-200
300
0
startof
Pedod-1
vod.
0End
0
-2013
Settled
bard(W4
Peded-3)
(partote~d)
-- Padeod-1stdide
-Peded-a stdide
(urpert)
ofPriod-2 trid.
10
20
s
200
...
30
40
5
an
0
70m
10
!0
30
40
50
60
70
02
0
10
20
3
40
50
B0
70
80
Kneeangle deg]
Period-Iand Period-2Stridesfr 4- (0.5 -0.6) (Subject1)
Kneeangle[deg]
Perinod-1
and Peiod-2Stridesfor 4, (0.4-0.5) (Subject1)
Kneeangle[deg]
Pedod-IandPedod-2
Stridlsor+ =(0.3-0.4) (Subject1)
400-
400-
200
200 -
100 -
100-
--
11-
-100-
-100-
-200-
-300
10
20
30
40
s0
0
70
00
a
10
2D
.ao
41
sM
s0
70
100
Ls0
ao
Kneeangle[deg]
Period-iand Pedsd-2
Strideslor 4 - (0.7- 0.8) (Subject1)
Kneeangle[deg]
PerIdd-iand Period-2Strides
for =(0.6- 0.7) (Subject1)
20
3
3
0
Kneeangle [dog]
Period-I ard Pad d2 Stridesfor 4 - (0.8- 0.9) (Subject1)
400
300
2011
100-
0
10
20
30
40
Kneeangle [deg]
50
02
70
s0
0
10
2
30
40
Knee angle[deg]
50
80
70
so
0
10
an
|0
40
Kneeangle[deg]
50
s0
70
80
al
19~
Kneie angtinrv@L
~Ieg~soc]
I
Stridesfor
Pariod-1
andPeriod-2
----
Settled
Imrd
(all
Perieda
-----
(Pora
Priod-1abide
500 .....
Stridesfor i, - (0.2-0.3) (Subjac 2)
Period-iandPeriod-2
Period-1and Period-2Stdidesfor I4 - (0.1- 0.2) (Subject2)
d
(unpet.)
or
0
400
Per-2 sd
4= (0 - 0.1)(Subject2)
atdde
Stant PededdI
0
o
ErdotPOdod-2strid0
portatie marker-
200-
100I -
-200-
400 -
10
0
00
40
30
20
10
00
70
10
0
10
20
40
30
50
OD
70
0
-10
10
0
40
30
20
50
00
70
00
70
Kneeangle[deg]
Stridesfor #, = (0.5-0.6) (Subjed2)
Period-1and Period-2
Kneeangle[deg]
Stridesfor -(0.4 - 0.5) (Subject2)
Period-iand Pedod-2
Kneeangle[dog]
2)
Period-iandPeriod2 Sridesfor 4 =(0.3- 0.4) (Subject
son -
400 300 -.
200 100 -
-10O
-
-200,1
-10
L
J
t
10
2.
310
4.
50
Br
7
10
0
10
20
40
30
50
00
0
70
10
20
30
40
50
Kneeangle [deg]
Parod-I andPeriod-2Strides 4p=(0.8- 0.9) (Subje
Kneeangle [deg]
Period-1andPedod-2Strideslor 4, (0.7-
Kneeangle[deg[
Period1and Padod-2
Stridesfor 4, = (0.6-0.7) (Subjad2)
ier
0.B)
(Subjec2)
2)
Boo-
400
200
3
10
to-10
00
400
10
0
10
20
30
Kneeargle [deg]
40
00
00
70
.10
-
0
10
20
30
Kneeangle[dog]
40
00
00
7.
10
0
10
20
30
Kneeangle [dog]
40
0
0
7D
Period-iard Pedod-2Suides for
(.9 -
1)
(Stiat 2)
200
so0300200-
00
I-2100a -I
c
1
30
Krm.argleIla
10
Stddes for 4, = (0.1-0.2) (Subject3)
Padd-1 andPedod-2
Podod-1and Pedod-2 Stridesfor 4, - (0.2-
0.3)(Subject3)
400-
200
100
-100
-300
AGO
1
0
10
20
30
40
5o
so
70
0
0
10
20
30
40
Kne angle[dog]
Pedod-IandPriod-2 Strideslor4 (0.4-
Kneeangle[dog]
Peiod-1andPedod-2Stridesfor ,= (0.3-0.4) (Subjed3)
50
an
70
-2a
E0
-10
0
10
30
20
40
50
s0
70
0
Kne angle[deg]
= (0.5-0.6) (Subject3)
Period-1
andPeriod-2Stideasfor4
0.5)(Subject3)
400
200
100
-100
-son
D
0
10
20
30
40
s0
80
70
aa
Kneeangle[dog]
Pedod-1
andPeriod-2
Siddesfor4 =(0.7- 0.8) (Subject3)
Pedod-1 andPedod-2
Stridesfor f, =(0.6- 0.7) (Subject3)
D
0
ao
40
so
so
10
20
Kne angle[degl
Pedod-1andPedod-2
StddeBfor = (0.8- 0.9) (Subject3)
70
ao
4,
.300
400
S20
1
0
-100
.200
.300
10
0
10
20
30
40
Kne angle[deg]
s0
Go
70
so
.10
a
10
20
30
40
Kme angle[dog]
so
60
70
70
-10
0
10
20
30
40
Kneeangle[dog]
so
so
70
en
B
-
8
-L
angulrvel. Ideg/ec]
Knee
Pariod-1ard Period-2Stridasfor 4, = (0 -0. 1) (Subject4)
Period-iand Perod-2
Stridesfor4 =(0.1- 0.2) (Subjed4)
Period-1and Period-2Stridesfor 41,=(0.2-0.3) (Subject4)
400-
200100-
r
-k
100-200-
D
10
20
30
4a5n0
Kneeargle[deg]
Period-Iand Period-2Strides
for4 (0.3-
0
0.4)(Subjed4)
0
10
2D
30
4.
0
Kneeangle[deg]
Period-iand Period-2Stridesfor4 =(0.4- 0.5) (Subjed4)
6
0
S
0
10
20
30
40
5o
so
Kneeangle [deg]
Period-iand Period-2
Stridesfor 4, =(0.5-0.6) (Subject4)
300 -
200 V
1100
0
-200-
-300-
r
400
0
r
i
10
20
i
30
40
5
so
0
70
Kneearge [deg]
Period-iand Period-2
Stridesor =(0.5-0.7) (Subject4)
10
20
30
40
so
so
Kreeangle[deg]
Period-iandPeriod-2Stridesfor 4, = (0.B-0.9) (Subject4)
4
300-
200-
--
100e-
D-
-200-
.300
-
0
10
20
30
Kneeangle [deg]
40
50
a0n
30
40
Kneeangle [deg]
0
02
70
30
40
a[deg]
Knee angl
50
0g7
70
OLOS
~~
M
(~Iot~1s
0*~9pS~p~G~]JSI-p
(is)
00
~
00C
ow
Pedad-.1and Period-2Svides for
4,
(0 -
0.1)(Subject5)
Pad-1 andPeriod-2Stridesfor+,
4
5)
(0.1-0.2) (Subject
Stride for
Parod-1 and Pedod-2
(0.2-0.3) (Subject5)
saa -
400-
o
paab
o
10
0
400
stat orf
10
00
4
To
0
70
a
-10
g
0
10
20
30
40
50
so
70
2O
00
Knee angle
Peariod-1andPeriod-2Srides for, =(0.5- 0.6) (Subject5)
s00-
aoo,
san
4a0-
400 -1
400
S200
[deg]
-10
Knee angle[dg]
Srideafor* = (0.4-0.5) (Subject5)
andParlod-2
Pedod-1
Kneeangle [deg]
Period-1andPeriod-2Stridesfor = (0.3-0.4) (Subject5)
200
4!
--
00
4020
r0
Q10
0
1
20
21
4
00
ao
s
2
0
a
r
0
10
r-
40
20
0
go
ao
10
0
10
20
0a
40
50
s0
71
so
sa
00
21
Kneeangle[dag]
Period-1
and Pariod-2Strideslr4, =(0.8-0.9) (Subject5)
Kneeangle[deg]
Period-1
andPeriod-2
Stridesfor 4= (0.7- 0.8) (Subject5)
Kneeangle [deg]
(0.5-0.7) (Subject5)
Period-1
andPeriod-2Stridealor
400-
-1
0
10
20
21
40
so
Kneeangle[deg]
so
70
2o
90
-10
0
10
20
30
40
s0
[dag]
Kneeangle
so
70
a0
21
-10
0
10
20
21
40
so
Kneeangledesg]
00
70
Padod-1ard Priod-2 Sridlsr
10
0
1
20
Kree
+,.(0.9 - 1) (Stbj s5)
3
angl
Idel
40
*
so
70
80
S
6)
and Period-2Stridesfor,
Period-1
Period-iandPeriod-2Sridesfor 4 - (0-0.1) (Subject
S
0
Sld band(allPd
E-31
Priod-1
abide(Pornu
Piad-2stPde
(p-t.)Stbde
ofPedod-1
Staxt
o
stride
ErdofPedd-2
-
market
prtuaion
0.2)(Subject6)
- (0.1-
Stridesforo= (0.2- 0.3) (Subjd 6)
Pedod-iand Pedod-2
-
1a
-2o
0
10
20
30
40
Kneeangle
50
500-
s0
Idag]
Period-1
and Period-2Stridesfor 4
(0.3-
70
-13
r0
0
10
40
30
20
50
s0
70
0
0
0
Kneeangle Ideg]
forO
Period-Iand Period-2Strides
0.4) (Subject6)
6)
(0.4- 0.5) (Subject
-300
00
70
30
70
30
90
40
50
30
t0
20
Kneeangle[deg]
Stridesfor4, =(0.5-0.6) (Subject6)
Period-1
and Period-2
Sao -
0
300300
2
50
2M200
1
100-
2 -100
0.100-
500
-20 -
400
-30010
0
30
20
10
0
50
4
7
0
D
300
10
20
30
40
50
s0
70
0
Kneeargle [deg]
Stridesfor 4, (0.7-0.8) (Subject6)
Perod-iand Perlod-2
Kneeargle [deg]
6)
Stridesfor 4 (0.6-0.7) (Subject
Period-iand Period-2
10
20
30
40
50
s0
Kneeangle [deg]
Stridesfor 4, = (0.8-0.9) (Subject6)
Period-iandPeriod-2
-1aa
so -3
100-
3
.1o
Sa
-0
-300
.
Me.g
ion
40:0
10
0
30
40
Kneeargie[dag]
So
93
70
o
00
10
20
30
40
Kneeangle [dog]
50
50
70
r
1r
20
r0
00
30
40
50
5
K10
40
Knee angle jdag]
5
so
0
70
0
80
3
90
Period-iard Peulod2Swidus
for =(0.9-1) (Stbjecl6)
I.100-
0m
1073
0
K-
narglo
0 6
a
0
Period-1and Pedcd-2Stides
for #p= (0 -
0.1) (Subject7)
Period-Iard Period-2Stridesfor
4
(0.1-0.2) {S oejct7)
Period-1and Period-2Stridesfor 4, - (0.2 -
0.3)
(Subject7)
400
300
200
-100
0
-200
-300
-400
= (0.3-
0
10
jdag]
Kneeargle
Period-IandPeriod-2Stridesfor
10
30
20
40
50
8D
70
0
.10
Kneeargle [deg]
Period-1andPeriod-2Stridesfor =(0.4- 0.5) (S/b)ect7)
0.4)(Subject7)
0-
400
0
10
30
20
40
ao
50
70
Kme angla[deg]
Period-iandPeriod-2
Sridesor4 = (0.5 -0.6) (Subject7)
400
300-
300
-
200-
1000
-200
-100-
1300
-200-
00
s
-SDD
-E0
-10
0
10
20
30
40
50
Kneeangle [deg]
Period-Iard Period-2Stridelsfor+ (0.6-0.7) (Subject7)
80
70
-10
0
10
.11
--
20
0
40
50
t0
70
Kneeargl [deg
Padlod-land Padiod-2Stidesfiorp -(0.7-0.11)(Stbject7)
400 1
,
0
,
20
,
30
r
40
r
50
r
so
70
Kneeangle[dag]
Period-1
and Period-2
Stridesfor,= (0.8-0.9) (Subject7)
-300
400 -
,
10
10
200 5100-
4200-
.100-30D
D20
-3-100-
-10
0
10
20
30
Kneeangle[deg]
40
50
e0
70
.10
a
10
2.
30.
Kneeangle[deg]
40
50
so
70
-10
0
10
20
30
Kme angla[deg]
40
0
S
70
Pedod-1ard Pedod-2Svrdmsfor #,-(0.9 -1) (Subj- 7)
-10
0
10
20
Krm.angleIdeg]
30
40
50
m
7
Period-1and Period-2Stridesibr
-Satud
band
(all
Pedod-3)
Pedad-1tidde (patisted}0
20D Perod-2
Period-1and Period-2Stridesfor f4,= (0.2-0.3) (Subject 8)
400 -
(urpat)
atide
-S-o
d
Start fPeriod-1
WtidePa
230
4, {0.1 -0.2) (Subject8)
400
400 -
e
0 End
o Pedde
0 pesrtraior n
200-
200
maker
10eIo -
10n
0
-100
1000
-
400
-200 -
-
- -300-
-30-
400-
401
1L
2
o
30
4
Kneeangle[deg]
Period-IandPeiod-2 Stridesfor -
0
(0.3 -
0)
8
(Subject
0
8
0
240
3M0-
(0.4
0.5)
400 -
300-
3000n0
200-
200
Io -
7
so0
1
-100
r
10
40r
r
30
4
-200-
400
0
5
70
r
s0
rL
0
[deg]
4.
Priod-1 and Padiod-2
Stdides for
4M0-
10
Knee angle
20
[dag]
o
r
so
4.
e0
r0
ED
r
50an0
Padond-1
and Padlod-2Stridesfor4.
200 -
100 -
100 -
10n
a -
a
.100-
-10
r0
0
Knee argle
5ro
jdeg]
r8
0
70
r
an
.eorr
a
s
B
6
4
0
[deg]
-
0-
-100
-200--
200
00 Krr-
angle
300 -
200 -
3
0
1
400 -
300-300-
-300-
r
(0.11
- 0.9) (Subjet 8)
200 -
.QU
.
r
Knee
(0.7- C.8) (SubjectB)
400 -
20
7
100
20D-
Perid-1Iand Period-2Stridesfor 4p (0.6- 0.7) (Subject8)
10
88
400r
2o
Knee argle
0
(0.5 -0
(8
0
-a4 -. 300
0
0
2o
--100
-
2
Kne angle [dog)
Period-1 and Period-2 Strides for 4
(Sibject B)
0n
-200-
a
-
400-
20
200-
80
550
Knee angle [deg]
Period-1 and Period-2 Strides for4
B)
40D-
0.
400
-300a
1
r
20
30-
40 d-Knee argia
[deg]
10
20
30
e ng
40 d0 5D
Knee angle Idag]
7[
Ia
a.
Poelq4
Ajo~lus eou~l
Stridesfor o, =(0.1-0.2) (Skj)ect9)
ard Period-2
Period-1
Period-1andPeriod-2Stridesior
4,-
(0.2- 0,3) (Subject 9)
*0'
0
10
40
30
0
S0
70
10
B0
20
30
40
50
so
70
80
-10
0
0
0
10
0
30
40
so
s0
70
0
Kneeangle[deg]
Period-1andPearid-2Stridesfor 4 - (0.5 -0.6) (Subject9)
Kne angle [deg]
Period-1aid Pedod-2Stridesfor 4, - (0.4- 0.5) (Subject9)
Kneeangle[deg]
Period-1ard Period-2Stridesfor 4, =(0.3- 0.4) (Subject9)
.00
00
100
S-10
-am
-so -m
-300
0
10
20
40
30
So.C
so
so
Kneeargle[deg]
Period-IandPeriod-2Stridesor 41,- (0.6-
400-
70
0
I
10
20
30
40
50
so
70
10
Kne angle[dg]
Stridesfor 4 (0.7-0.8) (Subject9)
aid Pedod-2
Period-1
0.7)(Subject9)
so
so
AO
20
30
Kneeangle[deg]
9)
for #, . (0.8-0.9) (Subject
Period-1
andPeriod-2Strides
70
a0
40
70
0
-40
30C
400.
20C
30o
-2a0
~0
10
20
30
40
Knee argle[deg]
5o
so
70
so
0
10
20
30
40
Kneeargle[deg]
sr
ao
70
0
a
10
20
30
Kne angle[deg]
so
so
*\
i
i.
-
4
U-
Appendix C: Histograms of perturbation distributions
Perturbation Distribution (SIbject 2)
Perturbation Distribution (Subject 1)
50
60
4550
40
35
40-
o
30 -
0
-
0
C
CC
-25-
-.
30-
-
20-
20 CL
15 010
10-
5.
0 -0
0.2
Perturbation Distribution (Subject 3)
0.4
0.6
0.
Perturbation Distribution (Subject4)
60C
50400
U
C
0
CO
M
3060 -20
10-
U
U.z
U.4
U.0
0
U.AS
0.6
0.4
0.2
0.8
(P
Perturbation Distribution (Subject 5)
Perturbation Distribution (Subject 6)
60
405040-
40
0
C0
C
0
0
30
30-
-2
t
( 20
'CL 2 0
10
10-
0
0
0
0.2
0.4
0.6
0.8
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Perturbation Distribution (Subject 7)
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Perturbation Distribution (Subject 8)
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Perturbation Distribution (Subject 9)
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30a20
10
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1
Appendix D: Regression residuals (h = 10)
Regression Residuals, h= 10. (Subject 1)
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300
350
4
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50
100
Regression Residuals, h = 10. (Subject 3)
150
200
250
300
350
400
Regression Residuals, h=10. (Subject 4)
0
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Regression Residuals, h- 10. (Subject 7)
Regression Residuals, h- 10. (Subject 8)
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Regression Residuals, h - 10. (Subject 9)
50
100
150
200
250
300
350
400