Finger tapping movements of Parkinson's disease patients automatically rated using nonlinear delay differential equations.
ABSTRACT Parkinson's disease is a degenerative condition whose severity is assessed by clinical observations of motor behaviors. These are performed by a neurological specialist through subjective ratings of a variety of movements including 10-s bouts of repetitive finger-tapping movements. We present here an algorithmic rating of these movements which may be beneficial for uniformly assessing the progression of the disease. Finger-tapping movements were digitally recorded from Parkinson's patients and controls, obtaining one time series for every 10 s bout. A nonlinear delay differential equation, whose structure was selected using a genetic algorithm, was fitted to each time series and its coefficients were used as a six-dimensional numerical descriptor. The algorithm was applied to time-series from two different groups of Parkinson's patients and controls. The algorithmic scores compared favorably with the unified Parkinson's disease rating scale scores, at least when the latter adequately matched with ratings from the Hoehn and Yahr scale. Moreover, when the two sets of mean scores for all patients are compared, there is a strong (r = 0.785) and significant (p<0.0015) correlation between them.
- SourceAvailable from: Luis Antonio Aguirre[show abstract] [hide abstract]
ABSTRACT: In the analysis of a scalar time series, which lies on an m-dimensional object, a great number of techniques will start by embedding such a time series in a d-dimensional space, with d>m. Therefore there is a coordinate transformation phi(s) from the original phase space to the embedded one. The embedding space depends on the observable s(t). In theory, the main results reached are valid regardless of s(t). In a number of practical situations, however, the choice of the observable does influence our ability to extract dynamical information from the embedded attractor. This may arise in problems in nonlinear dynamics such as model building, control and synchronization. To some degree, ease of success will depend on the choice of the observable simply because it is related to the observability of the dynamics. In this paper the observability matrix for nonlinear systems, which uses Lie derivatives, is revisited. It is shown that such a matrix can be interpreted as the Jacobian matrix of phi(s)--the map between the original phase space and the differential embedding induced by the observable--thus establishing a link between observability and embedding theory.Physical Review E 06/2005; 71(6 Pt 2):066213. · 2.31 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: When a global model is attempted from experimental data, some preprocessing might be required. Therefore it is only natural to wonder what kind of effects the preprocessing might have on the modeling procedure. This concern is manifested in the form of recurrent frequently asked questions, such as "how does the preprocessing affect the underlying dynamics?" This paper aims at providing answers to important questions related to (i) data interpolation, (ii) data smoothing, (iii) data-estimated derivatives, (iv) model structure selection, and (v) model validation. The answers provided will hopefully remove some of those doubts and one shall be more confident not only on global modeling but also on various data analyses which may be also dependent on data preprocessing.Chaos (Woodbury, N.Y.) 07/2009; 19(2):023103. · 1.80 Impact Factor
- Neurology 06/1967; 17(5):427-42. · 8.25 Impact Factor
Finger tapping movements of Parkinson’s disease patients automatically
rated using nonlinear delay differential equations
C. Lainscsek,1,2P. Rowat,1L. Schettino,3D. Lee,1D. Song,4C. Letellier,5and H. Poizner1
1Institute for Neural Computation, University of California at San Diego, La Jolla, California 92093-0523,
2Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, La Jolla,
California 92037, USA
3Department of Psychology, Lafayette College Easten, Pennsylvania 18042, USA
4Department of Neurosciences, University of California at San Diego, La Jolla, California 92093-9127, USA
5CORIA UMR 6614, Universite ´ de Rouen, BP 12, F-76801 Saint-Etienne du Rouvray cedex, France
(Received 4 August 2010; accepted 10 January 2012; published online 16 February 2012)
Parkinson’s disease is a degenerative condition whose severity is assessed by clinical observations
of motor behaviors. These are performed by a neurological specialist through subjective ratings of
a variety of movements including 10-s bouts of repetitive finger-tapping movements. We present
here an algorithmic rating of these movements which may be beneficial for uniformly assessing the
progression of the disease. Finger-tapping movements were digitally recorded from Parkinson’s
patients and controls, obtaining one time series for every 10 s bout. A nonlinear delay differential
equation, whose structure was selected using a genetic algorithm, was fitted to each time series and
its coefficients were used as a six-dimensional numerical descriptor. The algorithm was applied to
time-series from two different groups of Parkinson’s patients and controls. The algorithmic scores
compared favorably with the unified Parkinson’s disease rating scale scores, at least when the latter
adequately matched with ratings from the Hoehn and Yahr scale. Moreover, when the two sets of
mean scores for all patients are compared, there is a strong (r¼0.785) and significant (p < 0:0015)
correlation between them. V
C 2012 American Institute of Physics. [doi:10.1063/1.3683444]
Parkinson’s disease (PD) is a common disease affecting
tens of millions of people worldwide. Its cardinal signs
are resting tremor, bradykinesia (slowness clumsiness of
movement), rigidity, and loss of postural reflexes. The
disease evolves slowly and, to adjust medications to the
severity of the disease, there is a need for automatic and
objective evaluation of movements. Such objective move-
ment assessments would supplement subjective clinical
ratings, which are ordinal rather than metric and often
show large inter-rater variability. Rather than using a
spectral based technique, we rated dynamical features of
each individuals’ finger-tapping—one of the items from
the unified Parkinson’s disease rating scale (UPDRS)
used for rating the severity of the disease—by using data
models based on nonlinear delay differential equations
(DDEs). The coefficients of the DDEs are then used to
assess the severity of the disease.
PD is a chronic neurodegenerative disease whose pri-
mary pathophysiology is loss of the dopamine containing
cells in the basal ganglia.1Deprived of their normal dopami-
nergic inputs, nuclei within the basal ganglia become dys-
functional leading to abnormal neural oscillations and
synchronization within multiple basal ganglia-cortical cir-
cuits.2These circuit disturbances lead to the clinical mani-
festations of the disease, which include such motor
impairments as bradykinesia (slow movements), muscle ri-
gidity, resting tremor, and postural instability. The impair-
ment in voluntary movement in PD is characterized by a
number of specific sensorimotor processing deficits, includ-
ing a generalized slowness of movement;3a difficulty in car-
rying out sequential movements;4a reliance on sensory
input, particularly visual input, to guide and correct move-
ment;5,6and difficulties in timing, synchronizing, and coordi-
Since Parkinson’s disease is a degenerative disease, an
effective tool is needed to track changes in its severity and
the effectiveness of remedial treatments. Clinical evaluations
can be costly and difficult to execute consistently over the
long duration of the disease. The first rating scale—the
Hoehn and Yahr (HY) scale10—was designed originally to
be a simple descriptive scale for providing a general estimate
of clinical function, combining functional deficits and objec-
tive signs. 0.5 increments were later introduced.11The origi-
nal HY scale is as follows:
(1) Unilateral involvement only usually with minimal or no
(2) Bilateral or midline involvement without impairment of
(3) Bilateral disease: mild to moderate disability with
impaired postural reflexes; physically independent;
(4) Severely disabling disease; still able to walk or stand
(5) Confinement to bed or wheelchair unless aided.
As in any clinical rating scale, subjective measures are
used in the HY scale and thus may vary among physicians.
C 2012 American Institute of Physics22, 013119-1
CHAOS 22, 013119 (2012)
introduced later12to overcome the inter-rater variability and
to provide a more fine-grained assessment of motor dysfunc-
tion (see Ref. 11 for a review about limitations in the use of
HY scale). The UPDRS is a rating tool to characterize PD se-
verity and to follow the evolution of the disease.13It is com-
prised of four sections: (1) mentation, behavior, and mood, (2)
activity of daily living (ADL), (3) motor function, and (4)
complications of therapy, totaling 45 items. Each item is eval-
uated by interview, or for motor function, by visual scoring of
specific movements and by passive manipulation of the joints.
The most commonly used section of the UPDRS is the motor
section, since motor dysfunction has been the core defining
clinical feature of the disease. The UPDRS motor section has
a maximum score of 108 points representing the worst (total)
disability and 0 corresponding to no disability.
One of the most important characteristics of impaired
movement in PD is bradykinesia, a broad term encompassing
slow, clumsy movements. Bradykinesia is widely tested for in
the motor section of the UPDRS. There are three tests of arm
and hand movements, all involving sequential movements,
which reliably evaluate bradykinesia. These three tests are fin-
ger-to-thumb tapping (item 23), hand opening and closing
(item 24), and hand pronation and supination (item 25). Finger
tapping correlates better with the overall UPDRS clinical
scores than items 24 or 25 (Ref. 27). Moreover, the motor per-
formance of PD patients degrades with task completion far
more easily during sequential opposition of individual finger
movements than during non-individual finger oppositions.28
The dynamics underlying movements of PD patients was,
therefore, investigated by recording 10 s bouts of individual
finger tapping movements. These movements were repetitive
index-thumb oppositions, instructed to be of large constant
amplitude, produced smoothly and rapidly.
While the UPDRS provides an improved instrument for
rating PD motor severity compared to the HY scale and has
overall better clinimetric properties than other scales,14it
does not completely overcome the issues of inter-rater vari-
ability and subjectivity15,16and remains a time consuming
rating technique. Hence, although it is the most studied and
used scale, it cannot be applied as often as desired to track
the progression of the disease.
Consequently, there is a need for automatic, graded
measurements of Parkinsonian motor dynamics that are met-
ric rather than ordinal. For this purpose, commercial acceler-
ometer and gyroscope sensors are often used (see for
instance Ref. 17). The time series these generate have been
analyzed using spectral analysis18or adaptive Fourier
However, spectral analysis has serious limitations when
applied to short time series. The 10 s sessions recommended
for evaluating patients20provide about 30 or 40 oscillations,
which are not enough for an accurate spectral analysis of the
underlying dynamics. To overcome these limitations, we pro-
pose an alternative technique based on non-linear modeling
that is known to be efficient even when applied to short time
series.21–23A global modeling technique extracts from a time
series a set of difference or differential equations whose itera-
tion or integration reproduces the dynamics underlying the
measurements. To enlarge the domain of applicability of this
technique, we do not look for global models, but rather for a
rough approximation of the underlying dynamics by using
delay differential equations, which are known to provide
In the present study, PD motor dysfunction, made appa-
rent by abnormal or dysfunctional finger-tapping dynamics,
was assessed using rough approximations of the underlying
dynamics using delay differential equations (DDE) (Refs.
24–26). DDEs are known to be very flexible, that is, to cap-
ture a wide variety of dynamics with few terms. Moreover,
DDEs are robust against noise contamination. The coeffi-
cient space associated with these DDEs was used to provide
a measure of PD severity. These results then were compared
to those obtained using the UPDRS to rate the quality of the
finger tapping movements.
The remainder of this paper is organized as follows. Sec-
tion II describes the patients and the protocol for recording the
finger tapping movements. Section III discusses dynamical
analysis of the data using a few tools borrowed from nonlinear
dynamical systems theory and shows that these cannot dis-
criminate PD patients from control subjects. Section IV is
devoted to the classification of PD patients and control sub-
jects using optimized coefficients of a selected DDE. We then
compare these results to the UPDRS tapping ratings. Section
V provides some conclusions and perspectives.
II. PATIENTS AND MEASUREMENTS
Thirteen PD patients, together with thirteen controls,
selected so that their ages were in approximate 1-1 correspon-
dence with patients’ ages, were enrolled in our protocol after
signing the informed consent document approved by the
human subjects ethics committee of the University of Califor-
nia at San Diego. We began this study with data from the first
six patients and controls, who we refer to as group i. The next
seven PD patients and controls were enrolled later as part of a
different study and are referred to here as group ii. There is a
slight difference in data acquisition, detailed below.
All patients were studied “off-medication,” as defined in
Ref. 20, using the same operational criteria of having not
taken their anti-parkinsonian medications for at least 12 h
prior to testing; all patients were tested at the same time dur-
ing the day (in the morning); and the same 12 camera 3D
optoelectronic camera system was used (see below). The 3D
movement tracking system was quite precise—each camera
has an optical resolution of 3600 ? 3600 (12 megapixels)
using two linear detectors with 16-bit dynamic range and has
an onboard processor that produces a subpixel resolution of
30000 ? 30000. Given the precision of the movement
recordings and the nature of the human movements being
recorded, a sampling frequency of 120 Hz, as well as that of
480 Hz, was deemed sufficient.
The clinical severity of the 13 PD patients at the time of
testing was rated according to item 23 (finger tapping) of the
UPDRS using the rules as follows:13
1¼Mild slowing and/or reduction in amplitude;
2¼Moderately impaired. Definite and early fatiguing. May
have occasional arrests in movement;
013119-2Lainscsek et al.Chaos 22, 013119 (2012)
3¼Severely impaired. Frequent hesitation in initiating
movements or arrests in ongoing movement; and
4¼Can barely perform the task.
All clinical severity ratings were given by the same
Item 23 counts for 4 points out of a total of 108 for the
entire UPDRS motor score and correlates well with the over-
all motor score. Also, PD patients were given an overall se-
verity rating according to the HY scale (Table I) as
recommended.20They ranged from mild (2) to moderately
All subjects—both PD patients and controls—were
asked to tap the index finger and thumb together making
large, smooth, rapid movements for 10 s, spaced apart by 1
min. Subjects had their eyes open throughout.
To track finger movements, the subjects’ index fingertip
and thumb were fitted with light-weight infrared emitting
diodes (IREDs) approximately 5 mm in diameter. Our active
marker 3D motion capture system (PhaseSpace, Inc., San
Leandro, CA) consisted of twelve cameras placed in a semi-
circle 1-2.5 m from the subject, who was seated at a table.
We calibrated the system prior to each data collection. We
checked that placement of the IRED markers did not perturb
the motion of the digits in the following manner. The
markers were taped to the index fingernail and to the nail of
the thumb, and visually inspected to make sure that no part
of a marker was in contact with the digital interphalangeal
joint of either digit. This left all joints of the hand free to
move. Subjects were then asked to fully flex and extent the
fingers of their hand. The movements were visually
inspected to make sure that there was no restriction in the
motion, and subjects were asked if they felt any restriction.
No restriction of perturbations of the finger movements were
observed or reported. The data may have occlusion artifacts
which occur when fewer than two cameras detect either of
the IRED markers.
The data acquisition protocol for groups i and ii was
almost identical,29but differed in sampling frequency—120
Hz for group i and 480 Hz for group ii—and in camera
placement. The ratio of mild to moderate patient impairment
was roughly the same for both groups.
Three 10 s sessions of individual finger tapping by the
dominant hand were recorded for group i subjects and six
sessions, three for each hand, for group ii subjects. The rates
of recorded occlusions, defined as 100* (number of occluded
data points)/(total number of data points), are reported in
Table II. When occlusions are present in a data set, the
data analysis procedure, described below, causes more data
points to be removed, thus raising the rate of effective
The two groups of patients are characterized by a similar
mean age, (69610) years for group i and (6763) years for
group ii, but the disease duration was significantly different
(p < 0:01) (mean duration was (7:061:3) years for group i
and (11:765:6) years for group ii). Importantly, the mean
UPDRS finger tapping ratings were 24% higher for group i
(2:160:8) than for group ii (1:760:4), reflecting a greater
severity of motor impairment. However, this difference just
failed to reach statistical significance p ? 0:087. These sta-
tistics are summarized in Table III. All p-values reported in
this paper are computed from a Wilcoxon T-test.
Note that group i patients had a larger standard deviation
for their age and UPDRS finger-tapping score, but had a
smaller standard deviation for disease duration. In other
words, group i was less homogeneous than group ii in their
age and UPDRS finger-tapping scores. Since Parkinson’s
disease always progresses with age (i.e., UPDRS scores
increase), one expects to see a correlation between variance
in age and variance in UPDRS score. Table III, columns 4
and 6, is consistent with this observation. Although PD
patients in both groups had typical, idiopathic Parkinson’s
disease, and all were in Hoehn and Yahr stage 2 or 3 of the
disease, patients in group i exhibited greater severity of
motor dysfunction on the UPDRS rating of finger tapping.
Having groups differing in motor severity, even within the
same overall stage of the disease, allow us to examine our
method across a wider range of motor deficits in PD patients
TABLE I. Clinical characteristics of Parkinson’s disease patients for the two groups.
GroupPD patient number Age (years) Sex HandednessDisease duration (years)HY score UPDRS scoreMedicationsa
1.0, 1.0, 1.5, 2.0
3.0, 3.0, 3.0
1.5, 2.0, 2.5
2.0, 2.0, 2.0
1.0, 2.0, 3.0
2.0, 1.5, 2.5
1.0, 1.0, 1.5
1.5, 1.5, 2.0
2.0, 1.5, 1.0
Sel, Lev, Br
Lev, LevR, Sel, Ent, Rot
Lev, Pr, Sel, Am
Lev, Pr, Am
St, Rop, Sel
Lev, Ent, Art
Lev, Pr, Ent
Pr, Sel, Am
LevR, Ent, Pr, Am, Sel
Lev, Pr, St
aMedication codes: Am, Amantadine; Art, Artane (trihexyphenidyl); Br, Bromocriptine; Ent, Entacapone; Lev, Carbidopa/levodopa (regular formulation);
LevR, Carbidopa/levodopa sustained release; Pr, Pramipexole; Ras, Rasagiline; Rop, Ropinirole; Rot, Rotigotine; Sel, Selegiline; and St, Stalevo (Carbidopa/
013119-3 Rating of finger tappings with DDEsChaos 22, 013119 (2012)
who are clinically typical, and without the cognitive and
emotional decline that occurs in more advanced stages of the
We use the data without any preprocessing (not even the
filtering above 10 Hz commonly used when human motions
are investigated), since preprocessing such as filtering or
smoothing could change some dynamical information in the
data. One or more recorded occlusions give rise to several
more effective occlusions after derivatives have been taken
and a particular nonlinear DDE model applied. The rate of
effective occlusions depends on the model used.
So that the same DDEs could be used on the data from
both groups, all the time series of group i were upsampled to
480 Hz using the MATLAB routine “resample.” This upsam-
pling increased the rate of effective occlusions: in each data
set, the first and last 10 data points of each continuous data
segment had to be removed due to the ambiguity of
upsampled data at the beginning or end of a data segment.
Therefore, a time series with one big occlusion has fewer
points removed than the same time series with a lot of small
occlusions since each continuous data segment loses 20 data
points in this procedure. Fig. 1(a) presents the time series of
the raw distances between the thumb and index finger for a
subject in group i, whose movements were recorded at
120Hz. This trial had 7% occluded data points in the origi-
nal, raw, data recording. Fig. 1(b) presents the occlusions af-
ter the data have been upsampled to 480 Hz, and Fig. 1(c)
shows the effective occlusions.
The effective occlusions depend on the computation of
derivatives and on the structure of the DDE model being
used. Depending on the window size used to compute the de-
rivative, data points at both ends of a contiguous segment of
data have to be removed. Finally, consider that the DDE
models used in this paper relate data points at time t to data
points at delayed times t ? sj; with j¼1, 2, 3. The data point
at time t is removed and effectively occluded if the deriva-
tive cannot be computed or the necessary delayed data points
do not exist. If the effective occlusion rate was more than
50% of the time series, the time series was discarded. In
dataset i, 13 out of 34 datafiles had effective occlusion rates
greater than 50% and hence were rejected, and in dataset ii,
no files had effective occlusion rates greater than 50%.
The majority of data files (81%) had no occlusions what-
soever. For those trials in which occlusions did occur, the
small sections of the time series corresponding to the missing
data were simply left blank.
The distance between index finger and thumb was com-
puted at each time step from the raw data files containing the
xyz-coordinates of the finger and thumb IREDs. Typical
time series are shown for a control subject (Fig. 2(a)) and a
PD patient (Fig. 2(b)) from group ii. The cycle time for PD
patients was generally around 200 ms. Both controls and
PDs show variability in the amplitude of finger tapping.
III. DYNAMICAL ANALYSIS
Fig. 2 suggests that finger-tap amplitude might distin-
guish between controls and PD patients. To evaluate whether
there is significant difference in the statistics of the finger-
tapping amplitude An—the difference between the maximum
and the minimum of the distance for the nth tap—we com-
puted the amplitude of each finger tap for all sessions for ev-
ery subject. The standard deviation rAis slightly less for the
control subjects (? rA¼ 0:2260:09) than for the PD patients
(? rA¼ 0:2660:07), but not significantly so (p ¼ 0:1 > 0:05).
Therefore, fluctuations in the finger tapping amplitude cannot
be used to discriminate between control subjects and PD
When the six 10 s sessions are concatenated in the order
of recording, from the first to the last, there is a general
TABLE II. Rate of recorded occlusions for the data of both groups.
GroupControl subjects (%) PD patients (%)
TABLE III. Statistics for age, disease duration, and UPDRS finger-tapping
FIG. 1. (Color online) Distance between thumb and index finger markers
are plotted over time. Example of a time series with 7% occlusions in the
recorded data (a). The dots denote the occluded points. The upsampled data
(b) have an occlusion rate of 16%. In (c) after removing all effective occlu-
sions, 70% of the original data is usable.
013119-4Lainscsek et al.Chaos 22, 013119 (2012)
tendency for a reduction in the finger tapping amplitude (Fig.
3). Could the difference in rates of reduction distinguish
between controls and PD patients? Using the slope of the
regression line (_A) normalized by the mean amplitude (?A),
we find _A=?A ¼ ?0:0360:05 in control subjects and _A=?A
¼ ?0:0860:05 in PD patients. Once again, these differences
are not significant (p ? 0:2 > 0:05). In both groups, small as
well as large amplitude fluctuations can be observed. This
confirms the well known fact that there is significant vari-
ability across different instances of producing a given move-
ment pattern, shown by control subjects as well as by
patients. It is, therefore, necessary to use tools borrowed
from nonlinear dynamical systems theory to investigate the
dynamics underlying finger tapping.
One of the very first steps of a dynamical analysis is to
reconstruct a phase portrait from measurements using delay
or derivative coordinates.30Delay coordinates are here used
with a delay srequal to 25dt where dt ¼
value around a quarter of the pseudo-period of the observed
oscillations.31,32Typical phase portraits are shown for a con-
trol subject (Fig. 4) and a PD patient (Fig. 5), respectively.
Six phase portraits are shown for each subject, one for each
10 s session. In both cases, the tendency of the finger tapping
amplitude to decrease with time is recovered, since the diam-
eter of the phase portrait is larger during the first sessions
(Figs. 4(a) and 5(a)) than during the last ones (Figs. 4(f) and
These phase portraits reveal a highly structured dynam-
ics, that is, in any neighborhood of the phase portraits, trajec-
tories are mainly governed by a unique vector field since
they are locally parallel. This feature suggests a low-
dimensional underlying dynamics. To check this, an embed-
ding dimension was estimated using a false nearest neigh-
bors algorithm.33The rate of true nearest neighbors evolves
in similar ways for control subjects (Fig. 6(a)) and PD
480s, that is, to a
patients (Fig. 6(b)). The curves do not saturate for a dimen-
sion less than 6 (control subjects) or 7 (PD patients). Once
again, it is rather difficult to discriminate control subjects
from PD patients.
The evolution of the rate of true nearest neighbors ver-
sus the dimension of the reconstructed phase portrait in
Fig. 6 is similar to the one observed when the Ro ¨ssler attrac-
tor is reconstructed from the third variable, z, in the Ro ¨ssler
equations (see Fig. 6 in Ref. 34). The rate of true neighbours
decreases after a first maximum when the dimension used
for reconstructing the phase space is equal to three. Such an
oscillation of the rate of true neighbours was also observed
when variable z of the Ro ¨ssler system was used in the recon-
struction procedure. In both cases, the reconstructed phase
portrait presents a zone, known as a lethargy, where each
revolution is not well distinguished from the others.
FIG. 2. Time series of the distance between the thumb and the index finger
during the individual finger tapping for a control subject (a) and a PD patient
(b) from group ii. The sampling rate equals to 480 Hz. Note, that the PD
patient has much reduced movement amplitude. However, there was sub-
stantial overlap in movement amplitude between the control subjects and PD
patients and amplitude alone was not sufficient to discriminate the groups.
FIG. 3. (Color online) Amplitude versus time for individual finger tapping
by control subjects—(a) and (b)—and by PD patients—(c) and (d)—from
group ii. The six 10 s sessions are concatenated to exhibit the tendency. Lin-
ear regression is shown as a red line. A?is the slope of the regression line
and A?is the mean amplitude.
013119-5 Rating of finger tappings with DDEsChaos 22, 013119 (2012)
Consequently, due to limited measurement accuracy, it is no
longer possible to have different pre-images for every pair of
different states: in other words, there is a lack of observabil-
ity. This means that two different states, well distinguished
in the original phase space, cannot be distinguished in the
reconstructed phase portrait.35In the Ro ¨ssler system, this
comes from long lethargies observed in variable z, during
which the system evolves in the x-y plane but not along the
z-axis. As a consequence, there is a domain of the recon-
structed phase space where all revolutions in the attractor
pass through a small cylinder—whose diameter is of the
order of the data resolution—where they cannot be distin-
guished. As in variable z of the Ro ¨ssler system, lethargies
occur in the PD data when the distance between the index
finger and the thumb is near the minimum (Fig. 2). More-
over, all minima are close to the same value, that is, near the
distance between the two IRED markers when the index fin-
ger touches the thumb. Since it is known that variables pro-
viding weak observability of the associated dynamics can
make dynamical analysis more difficult, the weak variable
measured could be at the origin of the difficulties encoun-
tered for discriminating control subjects from PD patients.
IV. CLASSIFICATION USING DYNAMICAL MODELS
A. Delay differential equations
It is known that global modeling is a useful technique
when the time series at disposal are too short to safely perform
a dynamical analysis.23This advantage remains when dynam-
ical models are used for classification. Unfortunately, global
modeling techniques are also known to depend on the choice
of the measured variables,34,36at least when global models
have the form of differential or difference equations without a
strong structure selection.37Such a difficulty is not avoided
when DDEs are used. The main problem arises from the
region of the phase space where trajectories cannot be distin-
guished. Difference or differential equations work with rather
small time steps. Consequently, when investigating time se-
ries with long lethargies, global models cannot capture the
underlying determinism but DDEs can still be applied. This is
why delay differential equations as introduced in Ref. 24 can
be useful. In DDEs, derivatives at time t are related to states at
delayed times t ? s1, t ? s2, etc. where the si’s are larger than
the time steps commonly used in differential or difference
equations. Moreover, the use of different delays leads to a so-
called “non-uniform embedding,” as introduced by Judd and
Mees,38,39which they show to be particularly efficient when
there are widely different timescales in the dynamics.
Typically, a nonlinear delay differential equation has the
_ x ¼ a1xs1þ a2xs2þ a3xs3þ … þ ai?1xsn
… þ alxm
s1þ aiþ1xs1xs2þ aiþ2xs1xs3þ …
s1þ ajþ1xs12xs2þ …
where x¼x(t) and xsj¼ xðt ? sjÞ. In the form (1), a DDE has
n delays, l monomials with coefficients a1;a2;:::;al, and a
degree m of nonlinearity. By a k-term DDE, we mean a DDE
with k monomials selected from the right-hand side of Eq.
(1). Although quite flexible, as for any global modeling tech-
nique, there is a significant gain in accuracy by carefully
selecting the structure of the model.37,40By structure
FIG. 4. Phase portrait reconstructed from the distance between the index
finger and the thumb using the delay coordinates. Case of control subject no.
13 (group ii).
FIG. 5. Phase portrait reconstructed from the distance between the index
finger and the thumb using the delay coordinates. Case of PD patient no. 8
from group ii.
013119-6Lainscsek et al.Chaos 22, 013119 (2012)
selection, we mean retaining only those monomials that
make the most significant contribution to reproducing the
data dynamics. An equally important task is to select the
right time-delays, since they are directly related to the pri-
mary time-scales of the dynamics under study. In the present
work, structure selection is performed using a genetic algo-
rithm (GA). Delays are needed to render the model sensitive
to dynamics on different time-scales, whereas coefficients
are actually the most dependent on dynamical regimes.
Unfortunately, exact relationships between delays and coeffi-
cients are only known for linear systems and cannot be
obtained in this study.
Our aim is to discriminate the dynamics underlying con-
trol subjects from those underlying PD patients. It is not our
purpose to look for a DDE that accurately captures the physi-
ological dynamics underlying the data but to capture major
features of the data sufficiently well to discriminate between
the finger-tapping movements of PD patients and controls
and between Parkinsonian finger-tapping of different degrees
of dysfunction. Even with this restricted task, selecting an
appropriate model structure is crucial. Since the DDE chosen
here plays the role of an approximate global model for repro-
ducing the dynamics underlying the data, it is not possible to
specify which aspects of the data induce changes in a given
coefficient. This can only be done when the model is linear,
which is not the case in the present work.
B. Structure selection using genetic algorithm
A GA is a global problem-solving search algorithm
based on ideas from natural genetics.41,42GAs are widely
used in industrial applications, e.g., Ref. 58, have an exten-
sive body of theory57and often incorporate well-known
Given a problem to be solved, a GA maintains a popula-
tion of individuals, each being a proposed solution to the
problem and drives this population of solutions towards a
collection of solutions at a higher level of fitness. The popu-
lation begins as a collection of random guesses. A method to
measure “fitness” must be provided; typically, for models of
a time-series, this is a mean-square error. A GA uses evolu-
tion operators to generate new individuals from the old ones.
Evolution proceeds iteratively: at each step, new individuals
are born, tested for fitness, and the least fit members of the
population are discarded. New individuals are created by (1)
recombination, (2) mutation, and (3) junk.41,42Recombina-
tion mixes bits of parental solutions to form a new, possibly
better, offspring. Mutation on the other hand modifies a sin-
gle individual, while junk adds new randomly created
Here, given a finger-tapping data set, the problem is to
find the DDE model _ x ¼ Fðxs1;xs2;…Þ that best character-
izes discriminating features of the data. The error at point x
is the difference between the time derivative _ x and the right-
hand side of model (1). The fitness of a model on a data set,
the residual, is the mean of the squared errors at each point
This mean-square error is computed for certain windows of
the data. For the GA part, the window was the length of each
data set and later, for computing the DDE scores, the win-
dows were 200 data points which is about two typical cycles
in the data. Successive data windows overlapped by 20
points. For a model with a selected structure, the coefficients
that minimize the mean-square error are numerically esti-
mated by a singular value decomposition (SVD) algorithm.43
The GA is split in two parts: the first part searches for
optimal time delays and the second part searches for the opti-
mal structure—the most relevant monomials in Eq. (1) based
on the n delays chosen—for a DDE model. Allowing both
the delays and the model structure to vary simultaneously
would significantly complicate the algorithm without signifi-
cant expected gain. Consequently, the GA alternates between
time-delay evolution and model evolution and maintains two
distinct populations: the set of time-delays and a collection
of model structures. For time-delay evolution, the best indi-
vidual in the current population of models is selected and
kept fixed while the population of the remaining time-delays
is evolved, using the fixed model to evaluate the fitness of
each set of time-delays. In the model evolution, the best set
of time-delays is kept fixed while the model population is
evolved, using it in evaluating the fitness of each model.
In the GA used here, the starting population size was
200 individuals for the delay selection part and between 10
and 100 for the model structure selection part, depending on
the order of nonlinearity and the number of delays desired.
The population size changes during the run of the GA: If the
best fitness (lowest mean square error) is constant for around
five generations, the population size is increased in order to
escape possible local minima of the residual. To further
avoid being trapped in local minima, the best five individuals
are taken out for five iterations at this point. During these
five iterations, the mutation rate is increased by 5% and
more randomly created individuals (junk) are added. After
those five generations, the five best individuals that had been
f ? ½_ x ? Fðxs1;xs2;…Þ?2
FIG. 6. Estimating the embedding dimension using a false nearest neighbors
technique. Case of a control subject (a) and a PD patient from group ii. The
fourth finger tapping session was here used.
013119-7 Rating of finger tappings with DDEsChaos 22, 013119 (2012)
removed are added again and are also crossed over with the
existing generation. We found that five generations was an
excellent middle ground between excessive computation
time and the need to avoid entrapment in a local minimum.
This procedure is done twice for each run. The GA is
stopped when the best fitness does not change for 7 new
The GA was run with models having up to six terms and
three delays. Models with fewer delays or fewer terms
yielded a much higher error and were less efficient for our
classification task. More terms and more delays did not lower
the least-square-error significantly and did not improve the
performance. Among those, smaller models with a slightly
higher residual are preferred to models with more terms. If
this did not give satisfying results, the number of terms in
the model and/or the number of delays were increased.
C. Selecting the DDE model structure and delays
We only used data from group ii to run the GA since the
group ii dataset has fewer occlusions. It further was not
upsampled, whereas the upsampling from 120 Hz to 480 Hz
applied to group i could have slightly modified its dynamical
information. Also, group ii is more homogeneous in that the
variance for the age and for the UPDRS score is smaller (Ta-
ble III). As data were available for both hands for group ii,
we used the data for both hands since this provides more
constraint for the GA search and did not change the models
selected by our procedure.
The next question is: when selecting the model form and
delays, does it make any difference whether we use data
from controls and/or PD patients? This question will be
answered in the following two sections: We run the GA on
all the data and then examine the statistics of each of the two
groups, PD and controls, both separately and combined.
It would be preferable to have one DDE model structure
that characterizes the dynamics of finger-tapping, regardless
of the presence of disease. Since delays are closely related to
different dynamical timescales and their interactions, and the
timescale information for PD patients and controls should,
intuitively, be very different, we expect that the delays
selected by the GA will vary according to the presence or ab-
sence of the disease and, when present, on its severity. We
would hope that delays with a low error for controls could
serve as a dynamical “baseline.” Any departure from this
baseline could then be used to assess the severity of the
To conclude: we use data from group ii for selecting the
model structure and test if the selected model (in Sec. I) and
delays (in Sec. II) depend on the presence of the disease in
1. Selecting the model structure from group ii
Group ii consists of seven control subjects and seven PD
patients, and six sessions around 10 s long for each. For each
of the 2 ? 7 ? 6 ¼ 84 recorded time series, we used a GA to
select a model with the best structure from these time series.
The resulting 84 “best” models—with their structures and
associated delays—are independent from each other. Each
model has six ordered terms, chosen from the 19 terms on
the righthand side of the DDE
_ x ¼ a1xs1þ a2xs2þ a3xs3þ a4x2
þ a6xs1xs3þ a7x2
þ a14xs1xs2xs3þ a15xs1x2
s2þ a8xs2xs3þ a9x2
To select the six terms for the model, we proceed as follows.
For j¼1,…,6, we construct a histogram of those terms from
Eq. (3) which occur as the jth term of a selected model.
These are shown in Fig. 7. The most often retained mono-
mials are collected to write the 6-term DDE
_ x ¼ a1xs1þ a2x2
actually used for discriminating control subjects from PD
patients. It is interesting to note that when the selection pro-
cess is run on only the 6 ? 7 ¼ 42 time series recorded from
controls, or alternatively on the 42 time series from PD
patients, the same 6-term model is selected in both cases.
This indicates that the selected model structure captures
finger-tapping dynamics regardless of the presence of the
FIG. 7. (Color online) Histograms for the retained monomials in the 19-
term DDE (3) for the 84 runs of the structure selection algorithm on group ii
data sets. The x-axis is the index of monomials as in Eq. (3).
013119-8Lainscsek et al.Chaos 22, 013119 (2012)
2. Selecting the delays from group ii
In contrast to the model structure, the delays selected by
the GA depend on the data (controls, PDs, or both) used: The
delays selected from the 42 data files of the PD patients are
(22, 60, and 105), the delays selected from the 42 data files
of the controls are (0, 29, and 41), and the delays selected
from the whole data set of 84 files are (0, 23, and 41). Fig. 8
shows the histogram used to select the delays from the 42
data files of the controls.
To discriminate control subjects from PD patients, we
used the coefficient space of model (4), that is, the six-
dimensional space R6ða1;a2;:::;a6Þ. For each session, the
vector of coefficients faig was estimated using a SVD, defin-
ing one point—the center of the cloud of points generated
from the sliding window positions—in the coefficient space.
In order to test the performance of the selected delays, a
K-fold cross-validation44,45was used. The model coefficients
a ¼ ða1;a2;:::;a6Þ were split into K subsamples aK;n (each
subsample is one control subject and one PD subject). We
have K¼6 for group i and K¼7 for group ii according to
the number of patients for each condition in each group.
Each of the K subsamples consists of n outputs an, one a for
each time series.
From the K subsamples, a single subsample an was
retained as the validation data for testing the model and the
remaining K – 1 subsamples were used as training data. For
each validation, the K – 1 PD patients and the K – 1 control
subjects were used to estimate an hyperplane splitting con-
trol subjects from PD patients. The hyperplane was obtained
by running a linear support vector machine (SVM) (Refs.
For each partition, the model was fit to the training data,
and then the model quality assessed on the validation data by
computing the area under the receiver operating characteris-
tic (ROC) curves, A0.49A ROC curve is a plot of the cumula-
tive distribution function P1 of the first class against the
cumulative distribution function P2of the second class (see
Ref. 49, p. 173 for exact definitions). In our case, we plotted
the function PCOof the control subjects against the function
PPDof the PD patients. To compute the area A0under the
ROC curve (following the approach introduced in Ref. 49),
we ranked the outputs of SVM from the largest positive
value to the most negative value. Let ribe the rank of the ith
control subject point. The area under the ROC curve is
A0¼S0? n0ðn0þ 1Þ=2
where S0is the sum of the ranks of the control subject points,
n0the number of control subject, and n1the number of PD
patients. The cross-validation process was repeated K2times
for all possible combinations of training and validation sets.
The average of all K2cross-validations is then the model
quality?A0. This method has the advantage that all observa-
tions are used for both training and validation.
From the average value?A0reported in Table IV, it
clearly appears that classification performance is better for
group i than for group ii. Moreover, the delays estimated
FIG. 8. (Color online) Histograms of the delays for the 42 data files of the
controls. The numbers on the x-axis are the delays and the numbers on the y-
axis are the number of occurrences. The dominant three peaks are
s ¼ ð0;29;41Þ.
TABLE IV. Delays and model quality.?A0for the different sets of delays
obtained from group ii data sets.?A0is the mean value of the estimates A0of
the area under the ROC curves49,50according to Eq. (5) from a K-fold cross-
Group iGroup ii
Patients and controls
TABLE V. Statistics on the predictive accuracy obtained with model (4)
and delays selected using the control subjects from group ii.
Group i Group ii
FIG. 9. Comparing the UPDRS finger-tapping ratings and HY ratings for
Parkinson’s disease. Difference plotted versus mean, where S1¼ SUPDRS
and S2¼ SHY. The solid horizontal lines indicate the 95% confidence limits.
The dashed lines indicate the mean difference: positive for group i and nega-
tive for group ii.
013119-9 Rating of finger tappings with DDEsChaos 22, 013119 (2012)
with control subjects provide better results than using those
estimated with PD patients. This means that the delays esti-
mated from the control group can serve as a dynamical
“baseline” that captures dynamical information about the
nonlinear dynamics of healthy subjects. Any departure from
this baseline can then be used to assess the severity of the
We performed a second K-fold, leave-one-out cross-val-
idation,50as follows. The subsamples are made of one PD
patient or one control subject. Results are reported in Table
V, where the control subjects from group ii were used for
selecting the optimal delays. The sensitivity value 0.57 in
Table V means, surprisingly, that the model—its structure
and delays—selected from group ii data sets was not able to
accurately discriminate patients and control subjects of the
same group. However, we have seen from Table III that for
group ii, the standard deviation of the UPDRS finger-tapping
scores is 0.4 whereas the corresponding figure for group i is
double that, at 0.8. A smaller standard deviation suggests
that it should be harder to discriminate amongst group ii
patients than amongst group i patients. This is consistent
with the low sensitivity score in Table V.
D. Comparison of UPDRS, HY, and DDE scores
Although the UPDRS scale is known to be more reliable
than the HY-scale,14the latter is still widely used. In the case
of group i, the two scales are significantly correlated
(r¼0.80 and p < 0:0002), whereas for group ii, the correla-
tion is very poor (r¼0.330 and p¼0.2). In addition, a
Bland-Altman test51reveals that for both groups, there is a
general correspondence between the two clinical measures
with, in group i, a single point slightly outside the 95% confi-
dence limit (Fig. 9(a)).
For every subject, there are up to 3 time-series from the
dominant hand but due to windowing, each time-series gen-
erates a cloud of 6-dimensional points in coefficient space.
Each cloud was replaced by its center-point, giving up to 3
points in coefficient space for every subject, both controls
and PD. To define a central point for the 13 controls, we
took the center of these center-points and used this as the
central point, or base, for controls. The 1-dimensional DDE
scores for the PD patients are then defined to be the distance
D from the PD center-points for each time series to this base
point for controls.
For group i, distances D are significantly correlated
(r ¼ 0:80 and p < 0:0002) with the UPDRS finger tapping
scores (Fig. 10(a)). To be able to directly compare the dis-
tance D to the UPDRS scale using a Bland-Altman test, we
inverted the linear regression
D ¼ ?99 þ 141SUPDRS
to obtain a rescale distance
~S ¼D þ 99
We thus obtained an assessment of the Parkinson’s disease
finger-tapping dysfunction which correlates better with the
UPDRS finger tapping scale than HY-scale (Fig. 11). Such a
comparison is not reasonable with results for group ii since
(Fig. 10(b)) shows that the correlation between the UPDRS
finger tapping scale and distance D is rather low (r¼0.44
and p < 0:05). It is rather difficult to validate our scorings
using a DDE because there is no obvious “gold standard” to
use as a reference. The quite different results obtained with
the two groups of patients may result from the difference in
the UPDRS finger-tapping scores: compared to the HY-
scale, these scores overestimate the severity of Parkinson’s
FIG. 10. (Color online) Comparing the UPDRS and DDE ratings for Parkin-
son’s disease finger-tapping.
FIG. 11. Comparing the UPDRS and DDE ratings for Parkinson’s disease
finger-tapping: Difference versus mean. S1¼ SUPDRSand S2¼ S~. Data sets
from group i.
013119-10Lainscsek et al.Chaos 22, 013119 (2012)
disease for group i and underestimate it for group ii (Figs.
9(a) and 9(b)). Moreover the 95% confidence limits are 61
for group i and 61:5 for group ii. Our technique provides a
measure which we have shown is related to Parkinson’s dis-
ease severity, but a larger group of patients will be needed to
obtain a better estimation.
E. Comparison of UPDRS and DDE scores for groups i
and ii combined
Clinically, the mean scores for each subject are the main
item of interest, so we plotted, for all subjects from groups i
and ii together, the mean DDE distance?D vs. the mean
UPDRS finger-tapping score. This plot is not shown but was
similar to the plots in Fig. 10. The linear regression was
obtained, as in Eq. (6), and then inverted, as in Eq. (7), to
give the DDE-rescaled distance
~SG¼ 1 þ?D=150:
Fig. 12 shows~SGplotted against the mean UPDRS finger-
tapping score. The regression line is the diagonal, due to the
rescaling, and the two scales are significantly correlated
(r¼0.78 and p < 0:0015). (Note that r is invariant under
rescaling.) The difference vs. means plot (Fig. 13) has a 95%
confidence limit¼1.05, slightly less than the confidence
limit 1.2 for the group i ratings (Fig. 11). This 95% confi-
dence limit is just over 25% of the range [0, 4], which, being
less than 30%, is considered an acceptable percentage value
introducinga newmeasurement technique in
A. Basic findings
An objective assessment of the severity of Parkinson’s
disease is needed to augment clinical evaluations which are
time consuming, expensive, and present inter-rater variabili-
ty. We proposed a novel algorithmic means of assessing PD
motor dysfunction through use of dynamical models based
on delay differential equations. In this method, we selected
the best structure of a nonlinear delay differential equation
using a genetic algorithm fitted to a given movement time se-
ries. The coefficients of the nonlinear delay differential equa-
tion then were used as a six-dimensional numerical
descriptor of the movement. This method was applied to re-
petitive finger-to-thumb tapping movements of PD patients
and matched control subjects. We showed that under certain
conditions—when there was substantial variation in disease
severity across PD patients within a group and a small rate
of marker occlusions—it was possible to obtain a structure
for the model which, when associated with appropriate time
delays, adequately discriminated PD patients from control
subjects. When we compared the algorithmic scores derived
with our method with clinical rating scales of PD motor se-
verity, we found that the algorithmic scores compared favor-
ably with the unified Parkinson’s disease rating scale.
Finally, when we took a subject-centered view of our results
and compared each subject’s mean rescaled DDE score with
each subject’s mean UPDRS score, we obtained a correlation
of 0.785 (p < 0:0015), which shows that our DDE-based
algorithmic scoring technique could be considered clinically
The method for non-linear dynamical analysis of move-
ment presented here can be used to characterize and assess
not only the degree of dysfunction of movements of Parkin-
son’s patients but also the nature of the movement structure
of a wide variety of motor disorders, including the move-
ment dysfunction due to stroke, or that due to developmental
motor dysfunction in childhood. Such quantitative and objec-
tive non-linear dynamical analysis can provide a new tool
for clinicians to use in tracking recovery of function, pro-
gression of dysfunction, or the effectiveness of medical, sur-
gical, or physical treatments in reversing dysfunction.
B. Study limitations
The total number of PD patients tested was small, six
patients in group i, and seven in group ii. A much larger sam-
ple size will be needed to more fully determine the reliability
of our classification technique. Moreover, the data collection
protocols we used differed slightly for the two groups of
patients studied: group i data were recorded at 120 Hz and
group ii at 480 Hz. This difference in sampling rate might
have led to the greater number of marker occlusions
observed in the recordings of group i, an effect accentuated
after upsampling to 480 Hz. Alternatively, the greater num-
ber of marker occlusions in group i may have resulted from
FIG. 12. (Color online) Comparing the mean rescaled DDE and mean
UPDRS, finger-tapping ratings for all subjects from groups i and ii com-
bined: r ¼ 0:785ðp < 0:0015Þ.
FIG. 13. Comparing the mean rescaled DDE and mean UPDRS finger-
tapping ratings for all subjects together: Difference versus mean. S1¼~SG
and S2¼ SUPDRS. 95% confidence limit¼1.05.
013119-11 Rating of finger tappings with DDEsChaos 22, 013119 (2012)
group i patients having greater PD severity than those of
group ii. More severely affected patients often rotate their
hands or otherwise occlude view of one of the markers at
certain points during the movement. Having groups differing
in motor severity, even within the same overall stage of the
disease, allowed us to examine our method across a wider
range of motor deficits in PD patients who are clinically typi-
cal, and without the cognitive and emotional decline that
occurs in more advanced stages of the disease. We also note
that most aspects of the data collection protocols for the two
groups were virtually identical: The same 12 camera opto-
electronic 3D movement tracking system was used for both
groups; IREDs were affixed in the same locations on the
index finger and thumb and in the same manner, for both
groups; data were collected in the same experimental room
in both groups; the same duration of the finger tapping move-
ment was recorded for each group; the same patient selection
criteria were used for each group, namely, diagnosis of typi-
cal, idiopathic Parkinson’s disease of Hoehn and Yahr stage
II-III in severity (mild to moderate); patients in both groups
were tested at least 12 h off of their anti-parkinsonian medi-
cations; testing sessions occurred in the mornings; and con-
trol subjects were age-matched to the PD patients in the
same manner for both groups.
A further limitation of our study is that the clinical rat-
ing scales used, the UPDRS and the Hoehn and Yahr scales,
provide only somewhat global measures of motor severity.
The UPDRS is the most widely used scale for rating PD se-
verity, but even its functional divisions of deficits are quite
broad (e.g., mild, moderate, or severe). The UPDRS has
recently undergone a movement disorders society-sponsored
revision to correct identified shortcomings.52This revised
scale has been undergoing clinimetric testing and is begin-
ning to be used clinically. Use of such enhanced clinical rat-
ing scales could provide a more precise and objective
reference for comparison with our algorithmically derived
scores, and thus, to more adequately assess the reliability of
our classification technique.
C. Future studies
Future investigations planned in our laboratory will test
not only finger tapping movements but also a variety of other
repetitive movements that are also impaired in PD, such as
rapid, alternating forearm supination-pronation. Currently, it
still is an open question whether individual finger tapping is
sufficient to evaluate the severity of Parkinson’s disease. The
non-linear dynamical analysis presented here could also be
used to help determine the effects of aging on loss of sensori-
motor function. It is known that sensorimotor function
degrades with aging.53Moreover, in normal aging, we lose
approximately 4.7% of our dopamine-containing cells per
decade,54and, indeed, a key element of Parkinson’s disease
is the age-dependent reduction of dopamine coupled to a dis-
ease process that produces an exponential loss of dopamine
cells.54Thus, elderly subjects provide a naturally occurring
condition of mild dopamine depletion that is associated with
loss of sensorimotor function. However, the degree of motor
dysfunction at various age spans has not been systematically
characterized. Use of the non-linear, dynamical analysis pre-
sented here could help provide such a systematic characteri-
zation. Likewise, the techniques presented here could be
used to characterize and assess the degree of motor dysfunc-
tion in a wide range of motor disorders, varying from motor
disorders due to stroke to those due to developmental child-
D. DDE data analysis technique used in this paper
The technique described here, using GAs to find a DDE
model that can be used as classifier, has been used
previously24–26and is currently being improved and applied
to various medical data sets. Technical and mathematical
details of this technique and further applications are under
investigation and are planned to be published separately.
This work was supported in part by ONR MURI Grant
No. N00014-10-1-0072 (HP), NIH Grant No. 2 R01 NS036449
(HP), by NSF Grant No. SBE-0542013 to the Temporal
Dynamics of Learning Center, an NSF Science of Learning
Center, and by NSF Grant ENG-1137279 (EFRI M3C).
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