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Conceived and designed the experiments: JZ KZ JF MS. Analyzed the data: JZ KZ. Contributed reagents/materials/analysis tools: JZ KZ JF MS. Wrote the paper: JZ KZ JF MS.

The authors have declared that no competing interests exist.

Reliable characterization of locomotor dynamics of human walking is vital to understanding the neuromuscular control of human locomotion and disease diagnosis. However, the inherent oscillation and ubiquity of noise in such non-strictly periodic signals pose great challenges to current methodologies. To this end, we exploit the state-of-the-art technology in pattern recognition and, specifically, dimensionality reduction techniques, and propose to reconstruct and characterize the dynamics accurately on the cycle scale of the signal. This is achieved by deriving a low-dimensional representation of the cycles through global optimization, which effectively preserves the topology of the cycles that are embedded in a high-dimensional Euclidian space. Our approach demonstrates a clear advantage in capturing the intrinsic dynamics and probing the subtle synchronization patterns from uni/bivariate oscillatory signals over traditional methods. Application to human gait data for healthy subjects and diabetics reveals a significant difference in the dynamics of ankle movements and ankle-knee coordination, but not in knee movements. These results indicate that the impaired sensory feedback from the feet due to diabetes does not influence the knee movement in general, and that normal human walking is not critically dependent on the feedback from the peripheral nervous system.

Complex physiological rhythms arise from a large variety of biological systems that include natural pacemakers as well as feedback mechanisms, from the heartbeat to the rhythmic movement of human walking. Accurately extracting and characterizing the fluctuations underlying the biological rhythms is a fundamental problem which holds the key to understanding the mechanisms that govern the dynamics of biological systems. Usually such signals demonstrate certain oscillatory patterns, with each period displaying irregular fluctuation, or nontrivial dynamics, over time. This renders traditional spectral methods and nonlinear techniques less effective. We propose a novel approach to highlight the intrinsic fluctuations masked by the periodic component and noise through advanced dimension-reduction techniques, and apply it to human gait data from healthy subjects and diabetics. We find that this approach is capable of extracting the intrinsic dynamics and identifying the subtle synchronization pattern between knee and ankle. We find that although the two groups of individuals demonstrate remarkable differences in the dynamics of ankle movement and ankle-knee synchronization, the knee movement of both groups show similar dynamics. These results suggest that sensory feedback from a peripheral nerve system (like the feet) does not play an important role in regulating the motor control of human walking.

Complex physiological rhythms and synchronization processes are ubiquitous in biological systems and are fundamental to life

Traditionally, rhythmic signals are fruitfully analyzed by linear methods like the Fourier transform and power spectrum analysis. However, physiological signals as outputs of complex biological systems are typically nonlinear and non-stationary, and can not be properly characterized by linear methods. A number of new techniques based on nonlinear dynamical system theory

Human walking is a highly complex, rhythmic process which was found to exhibit long-range correlation and self-similarity, and has attracted sustained interest over the past decades

The general problem of dimension reduction has a long history. With advances in data collection, dimension reduction has reemerged as a prominent tool to unravel the high dimensional structure emerging in various disciplines. For example, it has been widely applied to gene and protein expression profiling for disease classification and prognostication

The time series from the

(A) Time series form

To achieve this, a weighted matrix

The above constrained minimization is solved by the generalized eigenvalue problem

The time series

The applicability of dimension reduction techniques is generally justifiable, considering the low correlation dimension of most real world pseudoperiodic data. For this kind of data, the trajectories of nearby cycles in phase space usually have similar orientations. Such redundancy can be effectively removed through dimension reduction, leaving only the useful degree of freedom. Finally, it is worthwhile to mention that for long time series with large number of cycles, the nyström method can be adopted to solve large scale spectral clustering problem

A popular method in nonlinear time series analysis is to reduce a continues flow to a series of discrete points, called a Poincaré section. The Poincaré section is the intersection of flow data in the state space with a hyperplane transversal to the flow. Thus each cycle in the data is simplified into a single point on the Poincaré section, which preserves many properties of periodic or pseudoperiodic orbits. Now we compare

One problem with the

(A) Return plot for

Another interesting phenomena associated with rhythmic process is synchronization between self-sustained oscillators, which plays an important role in understanding coordination or cooperation in biological systems

The evaluation of degree of synchronization from the outputs of coupled systems is of particular interest to study the interactions in biological systems. For example, the consistency of mutual nearest neighbors and the peakness of the phase difference distribution are used to characterize the dynamical interdependence

To solve this problem, we propose to quantify the degree of synchronization between two noisy, phase-synchronized oscillatory processes

(A) Correlation between

Now we apply the method proposed in previous section to human gait data collected from two groups: the healthy controls (CO) and neuropathic patients (NP, with significant diabetic neuropathy), each with 10 subjects

Human locomotion is a highly complex, rhythmic process that involves control from subcortical locomotor brain regions and feedback from various peripheral sensors. Typically, the human gait time series (see

(A) Knee locomotion data. (B) Ankle locomotion data. The time series are typically non-phase-coherent, demonstrating multi-oscillation within each cycle. This is also evident from the multi-center rotations of the attractor in phase space (lower panel). The two time series are divided into consecutive cycles by their respective local maximum points.

The SI series contains the information of the duration of each gait cycle. Another source of information consists in the waveforms of the gait cycles, which is not reflected in SI series

(A)

First we check the ankle movement (see

(A) A healthy subject. (B) A diabetes patient. The top, middle and bottom rows are PSDs for the extracted

(A) The slope

The absence of long range correlation in the ankle kinematics of the NP group suggests the alteration of the locomotor pattern in the lower limbs of neuropathy patients. This is due to the loss of peripheral sensation in the lower limbs, which arises from the gradual dying back of nerves from the fingers and toes typical of diabetes. Despite the deterioration in peripheral nervous system, the

Human walking involves the coordination of two major joints, i.e., the knee and the ankle, whose movements 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 be distinguished by the phase index of the signal due to the presence of strong phase synchronization. Also, noise tends to destroy the local structure in phase space and thus hampers the dynamical dependence measures

(A) A healthy subject. (B) A diabetes patient. As can be seen in the lower panel, the stride interval series

The lack of significant synchronization between ankle and knee movements observed in diabetic patients suggests the “incoordination” between the ankle and knee movements. This may arise from the gradual deterioration of the nerves in foot and toes of the diabetics, which is unable to produce sufficient neural feedback for the lower limb to be coordinated with the upper limb. In section “Characterizing Human Locomotion Dynamics” we have found that both the ankle and knee movements for healthy subjects demonstrate long range correlation, while for patients, only the knee movement show long range correlation. This finding is consistent with the result obtained here, i.e., the ankle and knee movement are more synchronized for healthy people than for the diabetics. Finally it should be noted that our study is limited by the relatively small sample size (each group has 10 subjects). Therefore significance tests are performed to verify the differences observed between the two groups of individuals. The current conclusion will be further validated on a larger data base available in the future.

A fundamental question concerning human walking is the origin of the long range correlation (or

In section “Characterizing Human Locomotion Dynamics”, we found that although the locomotion dynamics of the ankle shows significant difference between the normal persons and the patients in terms of long range correlation, their knee movements demonstrate similar scaling properties. These results support the belief that the impaired peripheral feedback from the sensors in the feet of diabetics influences only the lower limb locomotion while not that of the knees. We therefore conclude that human walking is not critically dependent on the feedback from peripheral feedback of the lower-limb, and that the central nervous system is playing a major role in regulating locomotor dynamics. In fact it has been found that pathology in central nervous system, such as Huntington's disease, can result in a loss of long range correlation in the gait dynamics

Our approach may be of great relevance, and is expected to provide more accurate and robust characterization and diagnostics to the complex oscillatory data observed in general biological and engineering fields. Reconstructing the dynamics on the cycle scale also brings new vitality to a number of other approaches which are otherwise not suitable for analyzing rhythmic data directly, such as detrended fluctuation analysis

Another striking example of physiological rhythms and their interaction is the complex, human cardiovascular system (CVS)

Finally, it is worthwhile to note that our approach can also be applied to time-course microarray data

β and ρ values for Control (CO) and Neuropathic (NP) groups.

(0.03 MB PDF)

We thank Dr. J. B. Dingwell for providing the human gait data.

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