1 Understanding Rhythmic Dynamics and Synchronization in Human Gait through Dimensionality Reduction Jie Zhang 1,∗ , Kai Zhang 2 , Jianfeng Feng 3,4 , Michael Small 1 1 Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China 2 Life Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA, 94720 3 Center for Computational Systems Biology, Fudan University, Shanghai, People’s Republic of China 4 Department of Computer Science and Mathematics, Warwick University, Coventry CV4 7AL, UK ∗ E-mail: [email protected]
Transcript
1
Understanding Rhythmic Dynamics and Synchronization inHuman Gait through Dimensionality ReductionJie Zhang1,∗, Kai Zhang2, Jianfeng Feng3,4, Michael Small1
1 Department of Electronic and Information Engineering, Hong Kong PolytechnicUniversity, Hung Hom, Kowloon, Hong Kong, China2 Life Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road,Berkeley, CA, 947203 Center for Computational Systems Biology, Fudan University, Shanghai, People’sRepublic of China4 Department of Computer Science and Mathematics, Warwick University, Coventry CV47AL, UK∗ E-mail: [email protected]
2
Reliable characterization of locomotor dynamics of walking is vi-tal to understanding the neuromuscular control of human loco-motion, and bares practical values in diagnosis of neurologicaldisorders and quantitative assessment of therapeutic interven-tions. However, the inherent oscillation and noise ubiquitous inbiological systems pose great challenges to current methodolo-gies such as linear and nonlinear time series analysis techniques.To this end, we exploit the state of the art technology in patternrecognition and specifically, dimensionality reduction techniques,and propose to reconstruct and characterize the dynamics accu-rately on scale of the gait cycle. This is achieved by deriving alow-dimensional representation of the cycles through global op-timization, which effectively preserves the topology of the cyclesthat are embedded in a high dimensional Euclidian space. Ourapproach demonstrates a clear advantage in capturing the in-trinsic dynamics and the subtle synchronization patterns fromuni/bivariate oscillatory signals over traditional methods. Ap-plication to human gait data for healthy subjects and diabet-ics reveals significant difference in the fractal dynamics of anklemovements and ankle-knee coordination, but not in knee move-ments. These results indicate that the impaired sensory feedbackfrom the feet due to diabetes does not influence the knee move-ment in general, and that normal human walking is not criticallydependent on the feedback from peripheral nervous system.
Author Summary
Complex physiological rhythms arise from a large variety of biological systems, from heartbeat to therhythmic movement during walking. Accurately extracting and characterizing the fluctuations behind thebiological rhythms is a fundamental problem, which holds the key to understanding the mechanisms thatgovern the dynamics of biological system. Usually biological signals demonstrate nonlinear oscillatorypattern superimposed by irregular fluctuations or noise, which renders traditional spectral method andnonlinear techniques less effective. We propose a novel approach to highlight the intrinsic fluctuationsmasked by the periodic component and noise through advanced dimension-reduction techniques. We useour method to analyze the knee and ankle locomotion of healthy subjects and those with diabetics, andfind it is capable of extracting the intrinsic dynamics and identifying the subtle synchronization patternbetween knee and ankle. We find that although the two groups demonstrate remarkable difference in thedynamics of ankle movement and ankle-knee synchronization, the knee movement of both groups show
3
similar fractal dynamics. These results suggest that sensory feedback from peripheral nerve system (likethe feet) does not play an important role in regulating the motor control of human walking.
Introduction
Complex physiological rhythms and synchronization processes are ubiquitous in biological systems andare fundamental to life [1]. The human heartbeat [2, 3], walking [4] and vocal cords vibration [5], bloodpressure and respiration [6], white blood-cell count and tremor in patients [7, 8], epidemic dynamics[9] all demonstrate stable oscillatory patterns but highly irregular fluctuations over time, and thesesignals are also known as approximately periodic or pseudoperiodic data. The fluctuation overlying theoscillatory patterns, or specifically, the cycle-to-cycle variability, arise from the combined effects from thechanging environment, nonlinear nature inherent in biological systems, and noise. It contains a wealth ofinformation regarding the health or disease status of an individual subject. Usually, little or no a prioriknowledge or models that governs the underlying system are available. Therefore reliably characterizingand quantifying such biological rhythms through data-driven approaches have important applications indiagnosis of disease and the understanding of dynamics of biologic control systems [10].
Traditionally, rhythmic signals are fruitfully analyzed by linear methods like Fourier transform andpower spectrum analysis. However, physiological signals as outputs of complex biological systems aretypically nonlinear and non-stationary, which can not be properly characterized by linear methods. Anumber of new techniques based on nonlinear system theory [11,12] have also been intensively applied, likecorrelation dimension [13] and Lyapunov exponents [14]. Although the chaotic measures may provide newinsights into the nonlinear nature of the system, they are severely hampered by the the cyclic trend andnoise from various sources [15,16]. Recent attempts include producing pseudoperiodic surrogate data [17],or performing a transformation from time domain to dual complex network domain [18, 19]. Generally,there are still no systematic and robust approaches to handle such oscillatory time series. The Fourieranalysis decompose the signal into harmonics that span over the entire time-line, thus information abouthow the each cycle change over time are averaged out. Similarly, nonlinear measures are also based onaveraged properties of phase space attractor. This lack of discrimination among the individual cycles callsfor more advanced signal processing techniques. Inspired by the advances in dimension reduction [20], wepropose a novel and robust approach that can effectively capture the dynamics of cycle-to-cycle variation,which is highly preferred in analyzing oscillatory data like human gait.
Human walking is a highly complex, rhythmic process which was found to exhibit long-range cor-relation and self-similarity, and has attracted sustained interest over the past decades [21–23]. Thefluctuations overlying the cyclic trend in human walking may reflect valuable information about the neu-romuscular processes responsible for normal and pathological locomotor patterns. In particular, the strideinterval (SI) (the duration of each gait cycle) has been intensively studied to quantify the physiological orpathological state associated with walking. It has been found that the fractal dynamics (i.e., long rangecorrelation) are altered with aging and disease [10,22].
The stride interval reflects the duration of each cycle, and a wealth of information contained in thewaveform of the gait cycle lost. Our approach utilizes the full waveform of gait cycle, and is expected toextract more information relevant to motor control of walking. We apply our method to the locomotiondata collected from two groups of people — the healthy subjects and neuropathy patients suffered fromdiabetics, aiming at finding whether the extracted dynamical fluctuation of the knee and ankle movementsas well as their synchronization patterns can vary between the healthy and diabetics group. Specifically,we want to find out whether the impaired sensory feedback due to diabetes can lead to different dynamicsof the knee and ankle locomotion compared with the healthy subject. Our results have revealed markeddifference between the two groups in terms of ankle movement and ankle-knee synchronization, whichshed lights to the understanding of the neuromuscular control involved in human walking.
The general problem of dimension reduction has a long history. With the advances in data collection,dimension reduction revives as a prominent tool to unravel the high dimensional structure emerged invarious disciplines. For example, it has been widely applied to the gene and protein expression profilefor disease prognostication [20]. Generally, the large number of dimension reduction approaches can becategorized into linear methods, including the principle component analysis, multidimensional scaling[24], and nonlinear ones such as state-of-the-art Isomap [24], laplacian eigenmaps [25] and local linearembedding [24]. Usually biomedical data process nonlinear structures and that nonlinear dimensionalityreduction methods might be more appropriate [20]. Here we use Laplacian Eigenmaps [25], which isbased on spectral graph theory and projects the high-dimensional data into low dimension so that twopoints nearby on the manifold are kept near to each other. We first illustrate with a benchmark datafrom the chaotic Rossler system described by:
⎧⎨⎩
x′ = −(y + z)y′ = x + 0.398yz′ = 2 + z(x − 4)
(1)
The time series from the X-component (see Figure 1A), demonstrates a strong periodic componentbut irregular fluctuations, therefore serves as an ideal example of approximately periodic signal withnontrivial dynamics. Motivated by the fact that such data usually exhibit a highly redundant patternsin the form of repeated cycles, we can partition the time series into individual cycles Ci’s (i = 1, ..., k)at the peaks or troughs in the time series, see Figure 1A [26]. Each individual cycle can then be takenas a high dimensional vector xi (i = 1, ..., k), whose dimension equals the number of points in thatcycle. Our goal is to map these multiple, high-dimensional cycles to a set of new, low-dimensional(preferably one dimensional) representation, or embedding yi’s, such that the proximity relations amongxi’s are maximally preserved in their low-dimensional counterparts yi’s. In the case that the cycles xi
are reduced to 1D (yi being a scalar), the derived yi (i = 1, ..., k) constitute a new time series whichrepresent the dynamics on the cycle scale of the original data.
To achieve this, a weighted graph ξ is constructed, where each node corresponds to a cycle xi andedges are assigned between all pairs of nodes. We use Wij to denote the similarity between cycle xi andxj , which can be chosen conveniently as the correlation coefficient ρij = Cov(xi, xj)/(σxiσxj ) (in casexi and xj differ in length, shift the shorter vector on the longer one until ρij maximizes). Then, thelow-dimensional representation yi’s can be cast as the solution of the following optimization problem,min
∑i,j∈ξ Wij ||yi − yj ||2, which penalizes those mappings where nearby points xi’s are relocated far
apart in the space of yi’s. In case of univariate yi’s, the objective can be written as yT Ly, wherey = [y1, y2, ..., yk]�, L = D − W is the graph Laplacian, and D is the diagonal degree matrix such thatDii =
∑j Wji.
The above constrained minimization is solved by the generalized eigenvalue problem Lyi = λiDyi,where λi’s (i = 1, 2, ..., k) are eigenvalues sorted in an ascending order, and yi’s are the correspondingeigenvectors. The minimum eigenvalue λ1 is zero, corresponding to an eigenvector (y1) of all 1’s. There-fore it is degenerate and the optimal solution is actually provided by y2, the eigenvector of the secondsmallest eigenvalue [25]. As is shown in Figure 1B, the eigenvector y2(i) provides a unique, cycle-scalerepresentation of the original time series by reducing each full cycle to a single point. We use a generalnotation c(i) (c means “cycle”, and i indicates the ith cycle) for such simplified representation of thedata. Note that other dimension reduction schemes can also be adopted. For example, we can computethe distance among cycles [26], and multidimensional scaling can be readily applied in this case to reduce
5
the cycles to scalars. The c(i) derived from multidimensional scaling and the Laplacian Eigenmap in factyield quite similar results, see Figure 2.
The Advantage of Characterizing Dynamics on Cycle Scale
A popular method in nonlinear time series analysis is to reduce a continues flow to a series of discretepoints, called Poincare section method. The Poincare section is the intersection of flow data in the statespace with a hyperplane transversal to the flow. Thus each cycle in the data are simplified into a singlepoint on Poincare section, which preserves many properties of periodic or pseudoperiodic orbits. Now wecompare c(i) obtained by dimension reduction and the Poincare section points P (i) obtained by collectingthe local minimum points in the data. As can be seen Figure 3, the return plots of c(i) and P (i) showsimilar quadratic form. Further calculation of the chaotic measures such as Lyapunov exponent andcorrelation dimension indicates that they have the same dynamical origin.
One problem with P (i) series is that it is highly susceptible to noise that is inevitable in biologicaldata. To see this, we plot return map (i.e., plot of x(i+1) versus its previous values x(i)) for c(i) and P (i)series obtained from noisy X component, see Figure 3. We find that although both return maps display aclear quadratic form intrinsic to the chaotic Rossler system, the return map of P (i) are more vulnerableto noise as the points are more dispersed than that of c(i). We use the variance σ2 of the least-squire-fitto the quadratic function (y = ax2 + bx + c) to quantify the influenced by noise, with σ2
p(i) = 4.570,almost 5 times larger than that of c(i) (σ2
c(i) = 0.9360). The extracted c(i) is less influenced by noisebecause we take each “cycle” rather than a point as a basic unit. Furthermore, the acquisition of c(i) isbased on an optimization process that preserves the proximity relation among all cycles simultaneously,while P (i) is obtained by treating each cycle independent of one another. It is reasonable to expect theformer to excavate more useful, global patterns than the latter. Finally, the applicability of dimensionreduction techniques is generally justifiable, considering the low correlation dimension of most real worldpseudoperiodic data. The trajectories in phase space have similar orientations for nearby cycles (see, e.g.,Figure 5). Such redundancy can be effectively removed through dimension reduction, leaving only theuseful degree of freedom.
Detecting Degree of Synchronization from Bivariate Oscillatory Data
Another interesting phenomena associated with rhythmic process is synchronization between self-sustainedoscillators, which plays an important role in understanding the coordination or cooperation in biolog-ical systems [27, 28]. Several different types of synchronization have been observed, such as completesynchronization [29] generalized synchronization [30], and phase synchronization [31].
The evaluation of degree of synchronization from the outputs of coupled systems is of particularinterest to study the interactions biological systems. For example, the consistency of mutual nearestneighbors and the peakness of the phase difference distribution are used to characterize the dynamicalinterdependence [32] and phase synchronization [31], respectively. Here we are interested in the case wheretwo processes are phase synchronized, but the synchronization strength is hard to estimate due to noiseand non-phase-coherence, or is not sensitive enough due to the mask of strong phase synchronization.For example, the knee and ankle moves perfectly in phase during human walking. In this case the phasesynchronization index will be high for both healthy person and diabetics and it cannot reveal the subtledifference in the degree of synchronization masked by the strong phase synchronization and noise.
Unlike the traditional methods, we propose to quantify the degree of synchronization between twonoisy, phase-synchronized oscillatory processes on the cycle scale through the reduced representation c(i)of the original data, therefore avoid the influence of phase synchronization. We first illustrate this usingthe X and Y components of the noisy Rossler system, which are perfectly in phase and suitable as abenchmark data. The two time series are first segmented into cycles by the segmentation scheme of eitherX or Y , then we apply dimension reduction to both and obtain reduced representation, i.e., cX and cY
6
for X ad Y time series. We find that the extracted cX and cY can successfully reveal the synchronizationpattern in presence of noise, which is demonstrated by an uprising trend in the corresponding scatter plot(Figure 4B). The Poincare section points, however, are hardly informative of the degree of synchronizationdue to the presence of noise, as PX and PY show no correlation (Figure 4A) at all.
Results
Data Description
Now we apply our approaches proposed in the method section to human walking data from two groups:the healthy controls (CO) and neuropathy patients (NP, with significant diabetic neuropathy), each with10 subjects [4]. The kinematic data were collected from a portable data-logger equipped on the subjectsduring continuous overground walking for 10 minutes (sampled at 66.7 Hz). Three electrogoniometerswere placed on the approximate joint centers of the hip, knee, and ankle joints of the right leg to measuretheir sagittal plane movements. Here we consider the signals measured from knee and ankle jointsmovement.
Characterizing Human Locomotion Dynamics
Human locomotion is a highly complex, rhythmic process that involves multilevel control of central ner-vous system and feedback from various peripheral sensors. Typically, the human gait time series (seeFigure 5 for knee and ankle movement) exhibit a stable frequency while irregular stride-to-stride fluc-tuation. For biological signals with highly repetitive patterns, vital information regarding the philologyand pathophysiology of the subject is hidden in the cycle-to-cycle variation. Accurately extracting thisfluctuation and characterizing its dynamics are expected to provide important insights into the underly-ing neuromuscular control of walking and yield greater diagnostic information. For example, the strideinterval (SI), defined as the time duration of each gait cycle, has been widely used to study the humangait [21,22,33–35]. It was shown that SI series displays a long-range correlations intrinsic to the healthylocomotor system.
The SI series contains the information of the duration of gait cycles. Another source of informationconsists in the waveforms of the gait cycles, which is lost in SI. As can be seen in Figure 8, the strideinterval from the gait data seems to contain insufficient information to reflect the intrinsic dynamicsdue to digitization. Therefore it is natural to expect that c(i), which is obtained by comparing thewaveforms of each cycle (therefore preserves the full dynamical pattern within each cycle) to keep morevaluable information about the locomotion dynamics. Meanwhile, c(i) successfully removes the periodictrend that obscures the underlying dynamics. Therefore it has the same advantage as SI series becauseit characterizes the dynamics on the scale of gait cycle. In the following we will demonstrate that c(i)obtained by Laplacian Eigenmap can extract the dynamical fluctuation overlying the cyclic trend moreeffectively than the SI series, so that we can distinguish clearly between the healthy and pathologicalgroups and make inference on neuromuscular control of human walking, especially on the role of sensoryfeedback from the feet in regulating fractal dynamics of walking.
First we check the ankle movement (see Figure 5B), and use Laplacian eigenmap to extract thefluctuation c(i) on cycle scale for the two groups. To further characterize the dynamics we compute thepower spectrum density (PSD) of c(i), see the top row in Figure 6. We find that most CO subjectsdemonstrate broad band spectrums (i.e., 1/f noise) that scale as 1/fβ , with β = 0.76(mean)± 0.23(std)(see Figure 7A), indicating the presence of long range correlation (i.e., the strides separated by a largetime span are still statistically correlated). In comparison, the power spectrum of the diabetic patients aremostly flat resembling white noise processes (β = 0.39 ± 0.16), which means that the strides at differenttimes are mostly uncorrelated. The values of β in the two groups are statistically different (P = 0.001).
7
This difference, however, has not been found with either the stride interval (SI) series (see the middlerow in Figure 6) or the raw data (see the bottom row in Figure 6).
The absence of long range correlation in the ankle kinematics of the NP group suggests the alterationof the locomotor pattern in the lower limbs of neuropathy patients. This is due to the loss of peripheralsensation in the lower limbs, which arises from the gradual dying back of nerves from the fingers and toestypical of the diabetics. Despite the deterioration in peripheral nerve system, the knee movement of theNP group are still found to demonstrate a stable long range correlation indistinguishable from the COgroup (βCO = 0.63(mean)±0.21(std), βNP = 0.62(mean)±0.19(std), which are statitcally identical withP = 0.965), see Figure 7B. These results suggest that the impaired peripheral feedback caused by thedying nerves in the feet does not influence the upper-limb dynamics, which led us to another fundamentalproblem in human walking, i.e., what is the role of sensory feedback in adjusting the locomotor dynamics?To understand this, we furthermore examine the degree of synchronization between the knee and anklemovement using still dimension reduction.
Assessing Synchronization between Knee and Ankle Movement
Human walking involves the coordination of two major joints, i.e., the knee and the ankle, whose move-ments during continuous walking are obviously in phase due to the physical connection between them.However, we find that correlation between knee and ankle movement for the two groups can hardly bedistinguished by the phase index of the signal due to presence of strong phase synchronization. Also,noise tends to destroy the local structure in phase space and thus hampers the dynamical dependencymeasures [36]. To circumvent these difficulties, we propose to compare the dynamics of the two timeseries by using their Laplacian Embedding c(i)’s. Note that each time series can be segmented by eitherits own local maximums, or those of its partner series (shown in Figure 5). Therefore we will segmenteach time series twice and compute the averaged correlation coefficients ρij ’s between cankle and cknee
for these two segmentation schemes.We examine the interrelation between the extracted c(i) from the ankle and knee data. For healthy
subjects, the scatter plots demonstrate a significant uprising trend (Figure 9A), indicating that the kneeand ankle movements are highly synchronized. The correlation coefficient ρ between cankle(i) and cknee(i)for the healthy group takes on a high value: ρ = 0.68(mean) ± 0.19(var), see Figure 7C. For diabetics,however, there is little correlation between cankle(i) and cknee(i), as is manifested in randomly distributedpoints in Figure 9B. The correlation coefficient in this case is also low: ρ = 0.26(mean)± 0.18(var), andthe values of ρ is statistically different for the two groups (P = 0.001). Again, the discrimination betweenCO and NP groups cannot be achieved by SI series, which always exhibits a strong correlation between thetwo joints (Fig . 9 (C)-(D)), corresponding to high degree of phase synchronization. Finally, we point outthat a more comprehensive description of synchronization can be achieved by examining more Laplacianeigenvectors. In the current case the single eigenvector y2 already encodes the primary variability and isthus sufficient for the discriminative task.
The lack of significant synchronization between ankle and knee movements observed in diabetic pa-tients suggests the “incoordination” between the ankle and knee movements. This is due to the gradualdying back of the nerves in foot and toes of the diabetics, which is unable to produce sufficient neuralfeedback for the lower limb to be coordinated with the upper limb. In section “Characterizing Humanlocomotion Dynamics” we have found that both the ankle and knee movements for healthy subjectsdemonstrate long range correlation, while for patients, only the knee movement show long range corre-lation. This finding is consistent with the result here in this section, i.e., the ankle and knee movementare more correlated for healthy people than for the diabetics.
8
Discussion
Central Nerve control over Peripheral Nerve system
A fundamental question concerning human walking is the origin of the long range correlation (or 1/fnoise) found in the gait data, the mechanism of which are not exactly clear [22]. Generally, the locomotorsystem incorporates inputs from both the central nervous system such as the motor cortex and basalganglia, and peripheral inputs and sensory feedbacks. Both these two kinds of inputs are suggested tobe possible reasons for the presence of the long range correlation [4] in normal human walking.
In section “Characterizing Human locomotion Dynamics”, we found that although the locomotiondynamics of the ankle shows significant difference between the normal persons and the patients in termsof long range correlation, their knee movements demonstrate similar scaling properties. These resultssupport the belief that the impaired peripheral feedback from the sensors in feet of diabetics influencesonly the lower limb locomotion while not that of the knees. We therefore conclude that the humanwalking is not critically dependent on the feedback from peripheral feedback of the lower-limb, and thatthe central nervous system is playing a vital role in regulating the locomotor dynamics. In fact it hasbeen found that pathology in central nervous system, such as Huntington’s disease, can result in a loss oflong range correlation in the gait dynamics [22]. For diabetics patients, although the peripheral sensoryfeedback is weakened, their central nervous system is not damaged and still plays an important role inadjusting the locomotion dynamics. This is why their knee locomotion still demonstrate fractal dynamics.Finally, it was pointed out [33] that the diabetics may still retained proximal somatosensory inputs, andvisual or vestibular feedback information. Further study need to be do done to clarify the role of thesefactors in regulating the fractal dynamics of human walking.
Acknowledgments
We thank Dr. J. B. Dingwell for providing the human gait data.
References
1. Glass L (2001) Synchronization and rhythmic processes in physiology. Nature 410: 277–284.
2. Bezerianos A, Bountis T, Papaioannou G, Polydoropoulos P (1995) Nonlinear time series analysisof electrocardiograms. Chaos: An Interdisciplinary Journal of Nonlinear Science 5: 95.
3. Babloyantz A, Destexhe A (1988) Is the normal heart a periodic oscillator? Biological cybernetics58: 203–211.
4. Dingwell J, Cusumano J (2000) Nonlinear time series analysis of normal and pathological humanwalking. Chaos: An Interdisciplinary Journal of Nonlinear Science 10: 848.
5. Little M, McSharry P, Moroz I, Roberts S (2006) Testing the assumptions of linear predictionanalysis in normal vowels. The Journal of the Acoustical Society of America 119: 549.
6. Small M, Judd K, Lowe M, Stick S (1999) Is breathing in infants chaotic? Dimension estimatesfor respiratory patterns during quiet sleep. Journal of Applied Physiology 86: 359.
7. Haurie C, Dale D, Mackey M (1998) Cyclical neutropenia and other periodic hematological disor-ders: a review of mechanisms and mathematical models. Blood 92: 2629.
8. Edwards R, Beuter A (2000) Using time domain characteristics to discriminate physiologic andparkinsonian tremors. Journal of Clinical Neurophysiology 17: 87.
9
9. Olsen L, Schaffer W (1990) Chaos versus noisy periodicity: alternative hypotheses for childhoodepidemics. Science 249: 499.
10. Goldberger A, Amaral L, Hausdorff J, Ivanov P, Peng C, et al. (2002) Fractal dynamics in phys-iology: alterations with disease and aging. Proceedings of the National Academy of Sciences 99:2466.
11. Kantz H, Schreiber T (2004) Nonlinear time series analysis. Cambridge Univ Pr.
12. Small M (2005) Applied nonlinear time series analysis: applications in physics, physiology andfinance. World Scientific Pub Co Inc.
13. Grassberger P, Procaccia I (1983) Characterization of strange attractors. Physical review letters50: 346–349.
14. Wolf A, Swift J, Swinney H, Vastano J (1985) Determining Lyapunov exponents from a time series.Physica D: Nonlinear Phenomena 16: 285–317.
15. Cazelles B (1992) How predictable is chaos? Nature 355: 25–26.
16. Abarbanel H, Brown R, Sidorowich J, Tsimring L (1993) The analysis of observed chaotic data inphysical systems. Reviews of Modern Physics 65: 1331–1392.
17. Small M, Yu D, Harrison R (2001) Surrogate test for pseudoperiodic time series data. PhysicalReview Letters 87: 188101.
18. Zhang J, Small M (2006) Complex network from pseudoperiodic time series: Topology versusdynamics. Physical review letters 96: 238701.
19. Zhang J, Sun J, Luo X, Zhang K, Nakamura T, et al. (2008) Characterizing pseudoperiodic timeseries through the complex network approach. Physica D: Nonlinear Phenomena 237: 2856–2865.
20. Lee G, Rodriguez C, Madabhushi A (2008) Investigating the efficacy of nonlinear dimensionalityreduction schemes in classifying gene and protein expression studies. IEEE/ACM Transactions onComputational Biology and Bioinformatics : 368–384.
21. Hausdorff J, Peng C, Ladin Z, Wei J, Goldberger A (1995) Is walking a random walk? Evidencefor long-range correlations in stride interval of human gait. Journal of Applied Physiology 78: 349.
22. Hausdorff J, Mitchell S, Firtion R, Peng C, Cudkowicz M, et al. (1997) Altered fractal dynamics ofgait: reduced stride-interval correlations with aging and Huntington’s disease. Journal of AppliedPhysiology 82: 262.
23. Bartsch R, Plotnik M, Kantelhardt J, Havlin S, Giladi N, et al. (2007) Fluctuation and synchro-nization of gait intervals and gait force profiles distinguish stages of Parkinson’s disease. PhysicaA: Statistical Mechanics and its Applications 383: 455–465.
24. Roweis S, Saul L (2000) Nonlinear dimensionality reduction by locally linear embedding. Science290: 2323.
25. Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representa-tion. Neural computation 15: 1373–1396.
26. Zhang J, Luo X, Small M (2006) Detecting chaos in pseudoperiodic time series without embedding.Physical Review E 73: 16216.
10
27. Pikovsky A, Rosenblum M, Kurths J (2001) Synchronization A Universal Concept in NonlinearSciences. Cambridge University Press.
28. Arenas A, Dıaz-Guilera A, Kurths J, Moreno Y, Zhou C (2008) Synchronization in complex net-works. Physics Reports 469: 93–153.
29. Pecora L, Carroll T (1990) Synchronization in chaotic systems. Physical review letters 64: 821–824.
30. Kocarev L, Parlitz U (1996) Generalized synchronization, predictability, and equivalence of unidi-rectionally coupled dynamical systems. Physical review letters 76: 1816–1819.
31. Rosenblum M, Pikovsky A, Kurths J (1996) Phase synchronization of chaotic oscillators. PhysicalReview Letters 76: 1804–1807.
32. Schiff S, So P, Chang T, Burke R, Sauer T (1996) Detecting dynamical interdependence andgeneralized synchrony through mutual prediction in a neural ensemble. Physical Review E 54:6708–6724.
33. Gates D, Dingwell J (2007) Peripheral neuropathy does not alter the fractal dynamics of strideintervals of gait. Journal of applied physiology 102: 965.
34. Dingwell J, Kang H, Marin L (2007) The effects of sensory loss and walking speed on the orbitaldynamic stability of human walking. Journal of biomechanics 40: 1723–1730.
35. Kang H, Dingwell J (2008) Separating the effects of age and walking speed on gait variability. Gait& posture 27: 572–577.
36. Quiroga R, Arnhold J, Grassberger P (2000) Learning driver-response relationships from synchro-nization patterns. Physical Review E 61: 5142–5148.
1 Figure Legends
Figure 1. Illustration of transforming a pseudo-periodic time series X(t) into a new seriesc(i) by reducing each cycle in X(t) to a point. (A) Time series form X-component of the chaoticRossler system, which demonstrates obvious oscillatory pattern. It can be divided into consecutivecycles at local minimum points (denoted by triangles). (B) A new representation of the oscillatory timeseries X(t) on the cycle scale, with each point in c(i) corresponding to a cycle in X(t).
Figure 2. Return plots of c(i) extracted from oscillatory data X(t) and Poincare sectionpoints P (i). The time series c(i) is extracted from the original data using two dimension reductiontechniques, i.e., the Laplacian Eigenmaps and Multidimentional scaling.
11
Figure 3. Return plots of Poincare section points P (i) and c(i). (A) Return plot for P (i) thatis obtained by collecting the local minimums Xmin and (B) Return plot for c(i) extracted from X(t) bydimension reduction. The Rossler system here is corrupted by 5% dynamical noise and 30%measurement noise. Obviously c(i) is less influenced by noise than P (i).
Figure 4. Correlation between X and Y component of noisy Rossler system as is revealedby Poincare section points P (i) and c(i). (A) Correlation between PX(i) and PY (i). (B)Correlation between cX(i) and cY (i). The Rossler system is corrupted by 5% dynamical noise and 30%measurement noise.
Figure 5. Time series (upper panel) and the corresponding phase space reconstructions(lower panel) of knee and ankle locomotion from a healthy subject. (A) Knee locomotiondata and (B) Ankle locomotion data. The time series are typically non-phase-coherent, demonstratingmulti-oscillation within each cycle. This is also evident from the multi-center rotations of the attractorin phase space (lower panel). The two time series are divided into consecutive cycles by their respectivelocal maximum points.
Figure 6. Power spectrum density (PSD) for a typical healthy subject (the left panel) anda diabetes patient (the right panel). (A) A healthy subject and (B) a diabetes patient. The top,middle and bottom rows are PSDs for the extracted c(i), stride interval series, and the original ankledata, respectively. It is obvious that the PSDs for the stride interval series and the original data showno significant difference between the healthy subject and the diabetic patient. The sampling rate of c(i)can be taken as the mean stride interval, and the log-log PSDs are then fitted with a linear functionusing least squire regression.
Figure 7. Mean and standard deviation of the derived statistics for control (CO) andneuropathy (NP) groups. (A) The slope of the least-squire-fit to the power spectrum density for theankle movements. (B) The slope of the least-squire-fit to the power spectrum density for the kneemovements. (C) The correlation coefficient ρ between cankle(i) and cknee(i).
Figure 8. Time series of the extracted c(i) and the stride interval (SI) from anklemovement data of a healthy subject. (A) c(i) time series, which demonstrates significantfluctuation that is not observed in original ankle movement data. (B) Stride interval series extractedfrom the ankle movement data, which, due to digitization in data collection, loses much informationabout the original dynamics.
12
Figure 9. Synchronization pattern between knee and ankle locomotion revealed by cknee(i)and cankle(i)( upper panel) and stride interval series SIknee(i) and SIankle(i) (lowerpanel).(A) A healthy subject and (B) a diabetes patient. As can be seen in the lower panel, the strideinterval series SI(i) cannot distinguish the healthy from the diabetics, as the scatter plots betweenSIknee(i) and SIankle(i) show similar uprising trends for both subjects.
