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Research Article

Basic Medicine

Neuroscience

Gait variability refers to natural stride-to-stride fluctuations that are present during locomotion. Evidence shows that an optimal state of inherent variability in time domain is associated with health and that this variability is characterized by a nonlinear deterministic structure that enables a healthy person to navigate through an environment in a flexible, but stable manner. Many ailments may be associated with a loss of this flexibility. Nonlinear tools which characterize the rhythmicity, irregularity, or complexity of the system may be used to quantify gait variability in the context of a deterministic dynamical system. Since the source of variability in gait time series is nonlinear deterministic in nature, we employed surrogate data analysis to ascertain this determinism. It has been observed that neurodegenerative disorder patients (in particular, amyotrophic lateral sclerosis, Parkinson’s disease, and Huntington disease) suffer from altered gait rhythms, with an increase in both the magnitude of stride-to-stride fluctuations and perturbations in fluctuation dynamics. To test for this locomotor instability, we employed spectral entropy, spectral centroid, sample entropy, and multi-level Lempel-Ziv complexity to explore the optimal range of healthy gait variability for a specific task and the corresponding ranges for neurodegenerative disorders in the respective nonlinear domain. It is found that the nonlinear measure shows a specific range corresponding to optimum variability of the healthy gait time series in time domain for a specific task. The values in the nonlinear domain either below or above the optimal range signify pathology.

Keywords:gait variability, irregularity, multi-level Lempel-Ziv complexity, nonlinear tools, optimum variability, rhythmicity, sample entropy, spectral centroid, spectral entropy

RecievedJun 1 2015 Accepted: Aug 10 2015 Published: Sep 30 2015

CitationKamath C (2015) A novel perspective to assessment of neurodegenerative disorder gait. Science Postprint **1**(2): e00051. doi: 10.14340/spp.2015.09A0001.

Copyright©2014 The Authors. Science Postprint published by General Healthcare Inc. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs 2.1 Japan (CC BY-NC-ND 2.1 JP) License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

Funding No funding is accepted from any source.

Competing interest The author declares that he has no competing interests.

Ethics The data is taken from a public domain made available to the public by physionet.org. PhysioNet is an online forum for dissemination and exchange of recorded biomedical signals and open-source software for analyzing them, by providing facilities for cooperative analysisof data and evaluation of proposed new algorithms. Hence, the quaestion of ethical approval does not arise.

Corresponding author: Chandrakar Kamath

Address: ShanthaNilaya, 107, Ananthnagar, Manipal-576104, India

E-mail: chandrakar.kamath@gmail.com

Data availabilityThe data was taken from a public domain made available to the public by physionet.org. PhysioNet.

http://www.physionet.org/physiobank/database/gaitndd/

All biological systems possess a natural property, namely inherent variability which can be described as the normal variations that occur in motor performance across multiple repetitions of a specific task 1, 2. Gait variability refers to natural stride-to-stride fluctuations that are present in motor performance during locomotion 3 and are observed across multiple repetitions of a task 4. Goldberger et al. found that a healthy system has a certain amount of inherent variability and that many pathologies exhibit increasingly periodic behavior and loss of such variability 5. Pool proposed that deterministic variability may provide a healthy flexibility of the heart, brain, and other parts of the body and that a loss of this flexibility may indicate different ailments 6. Besides cardiac and neural systems, such deterministic variability is also known to exist in human locomotion. The proposal that nonlinear determinism is healthy has motivated studies to develop clinical measures of health based on methods of nonlinear dynamics 7, 8. Evidence shows that an optimal state of inherent variability in time domain is associated with health and that this variability is characterized by a nonlinear deterministic structure that enables a healthy person to navigate through an environment in a flexible, but stable manner 4. This healthy variability can be associated with physiological factors such as neural control, muscle function, and posture4. Increased variability in a movement pattern from this optimal state renders the locomotor system noisier, random, unfocused, unpredictable, and unstable, leading to drunken-sailor-like walking. On the other hand, decreased variability from this optimal state makes the locomotor system more rigid, highly predictable, and less adaptable to perturbations, leading to robot-like walking 7. Both these situations render the system less adaptable to perturbations and represent unhealthy states. It is discovered that normal gait is a complex cyclic phenomenon which is not random but contains order and can be characterized through nonlinear mathematical descriptors 8, 9. It has also been observed that depending upon the nature and severity of the disorder/disease the variability can be either too periodic and predictable or too random and unpredictable 9, 10. Thus, we propose that optimal gait variability lies between too much variability and complete repeatability. This implies that gait variability either below or above the optimum variability range can be a marker of the gait impairment.

Also, interesting similarities have been observed in the rhythmic variability of normal cardiac, brain, and locomotion systems 11 and alterations in these physiological rhythms have been associated with disease states 12. This means that nonlinear methods suitable for cardiac and brain signals can also tried on gait signals. Recent studies have shown that it is possible to explore gait analysis under three broad independent factors, referred to as rhythm, variability, and pace 13, 14. In this study, the focus is on the first two categories. Gait variability can be assessed by using linear and nonlinear tools, which are complementary to each other. Linear tools measure the magnitude or amount of variation of gait variability. But they provide no information about the evolution of the gait signal with respect to time. Linear tools include the statistics of range, standard deviation, and coefficient of variation. Nonlinear tools, on the other hand, give us additional information about the structure of the variability, which describes the evolution of movement over time. Variation in how a motor behavior emerges in time can be best captured by nonlinear measures where the temporal organization of variability is quantified by the degree to which values emerge in an orderly manner. From this perspective, nonlinear tools which characterize the rhythmicity, irregularity, or complexity of the system may be used to quantify gait variability in the context of a deterministic dynamical system. We hypothesized that for a specific task while mapping from temporal gait variability domain to the nonlinear domain, the optimum variability range always gets mapped to an intermediate region which corresponds to an underlying potential to adapt to changing task demands that allows an effective cooperation among the various participating subsystems. The values in this domain either below or above that corresponding to optimum region, may serve to identify gait impairment/pathology. We demonstrate this through some examples. It has been observed that neurodegenerative disorder patients suffer from altered gait rhythms, with an increase in both the magnitude of stride-to-stride fluctuations and perturbations in fluctuation dynamics. To test for this locomotor instability, we employed spectral entropy (SpecEn), spectral centroid (SpecCen), sample entropy (SampEn) and multi-level Lempel-Ziv complexity (MLZC) to explore the optimal range of healthy gait variability for a specific task and the corresponding ranges for neurodegenerative disorders in the respective nonlinear domain. These variability measures represent different aspects of locomotor behavior, the first indicates rhythmicity, the second is a measure of spectral shape, the third indicates irregularity, and the fourth indicates complexity. Since the source of variability in gait time series is nonlinear deterministic in nature, we employed surrogate data analysis to ascertain this determinism. There have been numerous reports of gait variability 15–29, but only very few researchers have studied and reported the relationship between the aforesaid measures and stride-to-stride variability in healthy control (HC) subjects and patients with neurodegenerative disorders, namely amyotrophic lateral sclerosis (ALS), Parkinson’s disease (PD), and Huntington disease (HD). SpecEn is a convenient way of quantifying the distribution of spectral power. We hypothesized that the power spectra of stride-to-stride variability of healthy subjects will be different from those of neurodegenerative diseases and these differences get reflected in SpecEn and SpecCen of the corresponding individuals. The SpecCen is an estimate of the centre of gravity of the spectrum 30, 31. SpecEn has been widely used, mostly in speech recognition 32, 33. It is also employed as electroencephalographic measure of anaesthetic drug effect 34, 35. SpecCen has been seldom used in speech analysis 36. However, both the SpecEn and SpecCen, are not yet sufficiently experimented in gait analysis. The purpose of this study is to compare SpecEn and SpecCen of stride-to-stride variability of healthy subjects with those of patients with neurodegenerative disorders from the perspective of optimal variability state. We employed SampEn and MLZC to study the irregularity and complexity of the gait signals from the point of optimal variability. The study also shows that methods from nonlinear dynamics may be beneficial in understanding and describing variability and thereby identify health status.

The paper is organized as follows: The immediately following section describes the database supplied by Hausdorff et al., which is widely used in stride analysis. The next section discusses the desired pre-processing of the gait data used in this work. The subsequent section deals with the coefficient of variation (CV), a linear measure commonly used to evaluate fluctuation magnitude of stride-to-stride variability. The next four sections explain in detail the nonlinear SpecEn, SpecCen, SampEn, and MLZC methods. Statistical section discusses the statistical tests used for decision making and classification and its significance on the diagnosis.

The database used in this study is contributed by Hausdorff et al. 15, 37–40 and can be downloaded from the http://www.physionet.org/physiobank/database/gaitndd/. The database includes stride time series from 13 ALS patients (10 males and 3 females, age mean ± standard deviation (SD): 55.6 ± 12.8 years), 15 PD patients (10 males and 5 females, age mean ± SD: 66.80 ± 10.85 years), 20 HD patients (6 males and 14 females, age mean ± SD: 46.65 ± 12.60 years), and 16 healthy control (HC) subjects (2 males and 14 females, age mean ± SD: 39.3 ± 18.5 years). The separate file containing clinical information for each subject such as gender, age, height, weight, walking speed, and a measure of disease severity or duration is also provided. For PD patients the disease severity measure is the Hohn and Yahr score (a higher score indicating more advanced disease), for HD patients this measure is the total functional capacity measure (a lower score indicates more advanced functional impairment), for ALS patients the measure is expressed in months since the diagnosis of the disease, and for healthy control subjects, an arbitrary zero is specified. It was confirmed that the patients free from other pathologies which might lead to lower extremity weakness only participated. It is to be noted that the heights and weights in the four groups were not significantly different. Over the duration of treatment the medication usage was not changed. It was also confirmed that the healthy subjects were free from visual, respiratory, cardiovascular, or other neurological diseases.

The subjects from the four groups (healthy and three neurodegenerative disorder groups) were asked to walk at their normal pace up and down a 77 m long hallway for 5 min. To measure the gait rhythm and the timing of the gait cycle, force sensitive insoles were place inside or under subject’s shoes. These sensors generate a measure proportional to the force applied to the ground during movement. The output from the footswitches which corresponds to force signal is sampled at 300 Hz and digitized using an analog-to-digital converter and then stored in a recorder. The data recorded is then analyzed using a validated software that determined initial and end contact times (and also, stride and swing times) of each stride.

Before the application of the method of analysis it is necessary to pre-process the gait data. To minimize the start-up effects the samples in the first 20 seconds of the recordings are removed 37. Over the monitoring interval of 5 minutes, each time the subject reached the end of the hall-way the subject had to turn around and continue walking. It is found that the strides associated with these turning events constituted mostly outliers and must be removed from the rest of the time series. A median filter is applied to remove those data points that are 3 SDs greater than or less than the median value 37, 40, where SD is the standard deviation of the time series. This implies that those samples which lie outside the range (median ± 3.SD) are outliers and hence, can be removed. In the removal process, median value and not mean value of the time series has been used because some outliers possessed large values and will affect the computation of the mean.

Representative examples of stride time series, over first 54 strides, for the healthy control and three neurodegenerative disease groups are depicted in Figure 1. It is very clear that unlike healthy control group which shows smaller fluctuations in the stride intervals, all the three disease groups exhibit larger fluctuations.

Currently, there is no gold standard to quantify the stride variability and the most commonly used linear analysis techniques include standard deviation and coefficient of variation (CV). It is not possible to use the usual standard deviation to compare measurements from different populations with different units. To circumvent this problem, statisticians defined a linear measure of variation, CV, which permits us to compare the scatter of variables expressed in different units 41. While SD is a measure of absolute variability, CV is a measure of relative variability, relative to the population mean. Two of the other CV strengths are that it is a simple measure to calculate and interpret. It is important to note that this measure is not sensitive to changes in the ordering of the stride intervals or stride dynamics. That is to say, randomly ordering the time series will not affect this measure. In this work, CV of the original stride time series is used to evaluate the magnitude of stride-to-stride variability and gait unsteadiness.

CV is defined as the ratio of the standard deviation (SD) σ to mean µ expressed as a percentage, CV = σ * 100 µ^{-1}. Also, most of the times the SD of the time series generally increases or decreases as the mean increases or decreases, so that division by mean removes it as a factor in the variability. Thus CV is a standardization of the SD which permits comparison of variability estimates irrespective of magnitude of the time series. It shows the extent of variability in relation to mean of the population. It is a distributional metric and provides a measure of only relative variability.

An important point to be noted is that a linear measure, like CV focuses on the magnitude of variation in a distribution without any concern about the ordering of data points, while nonlinear measures, like SpecEn, SpecCen, SampEn, and MLZC of the gait time series, are explicitly concerned with the temporal structure of the data under examination.

Because spectral methods have a better developed theory, are quicker to calculate, and appear to perform on almost on par with phase-space methods 42, we selected SpecEn. Powell et al. were the first to introduce SpecEn based on the peaks of the Fourier transform, as a measure of regularity 43. Jantti et al. have shown that it is better to treat SpecEn as a measure of how much sinusoidal the signal is 44. Hence, we hypothesized that SpecEn can be used as a measure of rhythmicity. SpecEn estimates changes in the amplitude components of the power spectrum of the time series, with the amplitude components at each frequency of the normalized power spectrum as the probabilities in the entropy evaluation 45. SpecEn is a convenient way of quantifying the distribution of spectral power. In order to obtain SpecEn feature, first the Fourier spectrum and then the spectral energy is computed for each segment 46. Next the probability density function for the spectrum is estimated by normalization over all the frequency components as below:

where pi is the probability density corresponding to frequency component fi, S(fi) is the spectral energy for the same frequency component, and N is the total number of frequency components (bins) in the fast Fourier transform. The corresponding SpecEn is defined as

The actual spectral range is arbitrary, and has a considerable influence on the computation of H. Hence, SpecEn is usually normalized on a scale of 0 to 1 by dividing the expression by log(N). A perfect sine wave has only one nonzero spectral component centered at its fundamental frequency, which is normalized to 1 in the probability density function, after the normalization process. This gives the minimal value for the SpecEn of zero. Other similar frequency profiles with the spectral energy at specific frequencies, will lead to correspondingly lower values for SpecEn. By contrast, true white noise will have spectral energy distributed over the entire range of frequencies, with a flat spectrum. This gives the maximal value for the SpecEn of one. We use SpecEn as a measure of rhythmicity. A larger value of H implies lower rhythmicity and a smaller value implies a higher rhythmicity. In this study, we used N = 20.

If the dynamics that generated the time series is not known or if the time series is noisy, in that case it is essential to investigate whether the amount of nonlinear deterministic dependencies is worth analyzing further or to treat the time series as stochastic. Hence, one of the first steps before applying the nonlinear technique to the data is to investigate if the application of such technique is justified. The main reason behind this rationale is that linear stochastic processes can generate very complicated looking signals and that not all the structures that we observe in the data are likely to be due to nonlinear dynamics of the system. The method of surrogate data test, introduced by Theiler et al. 47, has been a popular validating test to address this issue. This test facilitates to find out if the regularity of the data is most likely due to nonlinear deterministic structure or due to variations in system parameters or due to random inputs to the system.

This section presents a brief sketch of the idea in that connection. The starting point is to create an ensemble of random nondeterministic surrogate data sets that have the same mean, variance, and power spectrum as the original time series. The measured topological properties of the surrogate data sets are compared with those of the original time series. If, in case, the surrogate data sets and original data yield the same values for the topological properties (within the standard deviation of the surrogate data sets) then the null hypothesis that the original data is random noise cannot be ruled out. On the other hand, if the data under test is generated by a nonlinear process, the value for the topological property would be different from that of the surrogate data, and the null hypothesis that a linear method characterizes the data can be rejected.

The method of computing surrogate data sets with the same mean, variance, and power spectrum as the original time series, but otherwise random is as follows: First find the Fourier transform of the original time series, then randomize the phases, and find the inverse Fourier transform. The resulting time series is that of the surrogate data. More details can be found in 47.

Spectral centroid frequency or simply, spectral centroid (SC) is the balancing point of the spectrum or more appropriately, an estimate of the centre of gravity of the spectrum 30, 31. It is also a measure of spectral shape. SC behaves as a formant-like feature, as it provides approximate location of the formant frequencies 30. It is defined as the average frequency weighted by the values of the normalized energy of each frequency component in the spectrum.

where, SC represents spectral centroid and F[k] is the amplitude corresponding to bin k in the DFT spectrum. In this study, we used N = 20.

Unlike SpecEn, embedded entropy provides information about how the signal under investigation changes with time by comparing a delayed version with itself 43. Both approximate entropy and sample entropy fall under the category of embedded entropy. Approximate entropy algorithm counts each sequence as matching itself to avoid the occurrence of ln(0) in the calculations 48. This leads to bias of approximate entropy and as an implication approximate entropy lacks two important properties. First, approximate entropy depends more on record length and will lead to lower values for short records. Second, it lacks relative consistency. This means, if approximate entropy for one data set is higher than the other, it should, but does not remain higher for all conditions tested. This drawback is important and it is to be noted that approximate entropy is a relative measure for comparing data sets 48. These problems associated with approximate entropy are resolved by using SampEn which reduces the above mentioned bias. The sample entropy, on the other hand, “is largely independent of record length and shows relative consistency where approximate entropy does not” 48. However, despite its advantages over approximate entropy, the use of SampEn is not widespread. Occasionally, SampEn has been used as a nonlinear regularity measure in gait stability and variability analysis under different brain related disorders 49–51. The SampEn represents the negative natural logarithm of the conditional probability that two sequences similar for m points remain similar at the next point 48. Given a time series with N data points, x_{1}, x_{2}, …, xN. To compute SampEn m-dimensional vector sequences, yi(m) = {xi, xi_{+1}, …, xi + m _{- 1}}, where 1 ≤ i ≤ N - m _{+ 1}. Then

where θ is the Heaviside function, ||.||_{∞} is the maximum norm defined by ||yj(m) – yi(m)||_{∞} = max_{0 }≤ k ≤ m _{- 1}|xj _{+} k – xi _{+} k|. The sum in the above equation represents the number of vectors yj(m) that are within a circular distance r from yi(m) in the reconstructed phase space. However, the cases of self matches indicated by j = i are avoided from the count. This brings down the bias in the estimation of SampEn 48. In the next step the density is computed as

Computations similar to above are then performed on a (m + 1) - dimensional reconstructed space to arrive at the equations below.

We thus have,

B(r) and A(r) respectively represent the total number of template matches in an m-dimensional and (m + 1) - dimensional phase space within a tolerance r.

The sample entropy being defined as the negative of natural logarithm of the conditional probability that a dataset of length N, having repeated itself for m samples within a specified tolerance r, will also repeat itself for m + 1 samples without allowing self matches, we compute the same as

In this work, we chose m = 3, d = 0.15*SD and N = 400. Basically the SampEn increases with the irregularity of the time series.

The Lempel-Ziv complexity (LZC) algorithm was proposed by Lempel and Ziv to evaluate the randomness of finite sequences. It is rather a simple-to-compute nonparametric measure of complexity suitable for finite length one-dimensional signals related to the number of distinct substrings and the rate of their recurrence. Larger values of LZC imply higher complexity data. LZC has been used rarely as a nonlinear complexity measure in comparing the complexity of gait patterns in patients with Parkinson’s disease with those of normal subjects 52, 53. Since LZC analyzes finite symbol-sequences it is essential that the given signal must first be coarse-grained. Mostly, a transformation into two symbols using a single partition/threshold is employed. In this study, however, hypothesizing that conversion with more symbols lead to more information from the signal, a transformation into three symbols using two partitions/thresholds is employed 54. We designate this LZC by multi-level LZC (MLZC). We find that the latter approach to coarse-graining yields a better diagnosis performance.

The two thresholds T_{1} and T_{2}, in the above equation are defined as follows: T_{1} = xm - |xmin|/16 and T_{1} = xm + |xmax|/16. xm, xmin, and xmax are respectively, the median, minimum, and maximum values of the gait segment under consideration. This symbol sequence is scanned from left to right and a complexity counter c(N) is incremented by one unit every time a new subsequence pattern is encountered in the scanning process, and the immediate next symbol is regarded as the beginning of the next subsequence pattern. The MLZC can be estimated using the following algorithm 54.

**1. **Let P denote the original string sequence i.e. P = {s_{1}, s_{2}, s_{3},…}, with si defined as in Eq. (11). Let S and Q denote two subsequences of P and SQ be concatenation of S and Q. Also, let SQπ be a sequence derived from SQ after its last character is deleted (π implying deletion of last character in the sequence) and υ(SQπ) denote the vocabulary of all different subsequences of SQπ.

**2. **At the beginning, the complexity counter c(N) = 1, S = s_{1}, Q = s_{2}, SQ = s_{1}, s_{2}, and therefore, SQπ = s_{1}.

**3. **In general, with S = s_{1}, s_{2}, s_{3},…,sr and Q = sr _{+ 1}, SQπ = s_{1}, s_{2}, s_{3},…,sr. If Q belongs to υ(SQπ) then Q is subsequence of SQπ and not a new sequence.

**4. **With S intact, change Q to sr _{+ 1}, sr _{+ 2} and check if Q belongs to υ(SQπ) or not.

**5. **Keep repeating previous steps until Q does not belong to υ(SQπ). Now Q = sr_{ + 1}, sr_{ + 2},…,sr + i is not a subsequence of SQπ = s_{1}, s_{2},..,sr_{ + }I _{- 1}. So increase c(N) by 1.

**6. ** Thereafter, S is renewed to S = s_{1}, s_{2},…, s_{r + i} and Q to Q = s_{s + }I_{+ 1.}

**7. ** Repeat the previous steps until Q is the last character. At this point in time, the number of subsequences in P is c(N), which corresponds to measure of complexity.

To arrive at a measure of complexity independent of sequence length, c(N) must be normalized. If the length of the sequence is n and the number of different symbols is α, it has been shown that the upper bound of c(N) is 55

where εN is a small quantity and εN → 0 (N → ∞). In general, N/log_{α}(N) is the upper limit of c(N), i.e.,

For the present conversion α = 3, b(N)=N/log_{3}(N) and c(N) can be normalized by b(N) as

C(N), the normalized MLZC, reflects the arising rate of new patterns along with the sequence and thus captures the temporal structure of the sequence. A larger value of MLZC means that the chance of generating a new pattern is greater, so the sequence is more complex, and vice versa. In this work we evaluate the evolution of new patterns in gait time series of healthy subjects and patients with neurodegenerative disorders.

Kruskal-Wallis tests are used to evaluate the statistical differences among the measures used, namely SpecEn, SpecCen, SampEn, and MLZC of the gait of the four groups (HC and neurodegenerative disorder classes). To strengthen the comparison a p-value <0.01 is considered statistically significant. If significant differences between classes are found, then the diagnostic ability of the nonlinear analysis method to discriminate gait of HC from a specific neurodegenerative disorder is evaluated using Mann-Whitney rank sum test. Again, the statistical significance is fixed at p <0.01. These non-parametric tests are used because they make no assumption about the underlying distribution of the data.

After pre-processing the gait data as explained above, it is necessary to normalize the data by subtracting from each sample the mean of the time series and dividing the result by the standard deviation of the time series. Normalization procedure removes most of the very large within and across-subject variability in the signal under consideration. Normalization also tends to produce values that are more exchangeable across different laboratories and research studies.

In this study, first we investigate the linear statistics of stride time series of healthy controls and neurodegenerative diseases. Each gait record (left and right), in each group, is divided into segments, with 20 strides per segment. A thumb rule to select segment length is that it must be long enough to reliably estimate the measure of interest, while it must be short enough to accurately capture local activities. For each segment the variability measures are computed and the results of a particular group are averaged. Table 1 shows the linear measures of fluctuation magnitude of stride-to-stride variability characteristic of HC and neurodegenerative disorder groups (right and left) time series. All the variability measures are expressed as mean ± SD. The table shows differences among the three patient groups, possibly indicating differences in neuropathology. It is found that the average stride time is longer in all the neurodegenerative disorder groups compared to that of control group. The SD showed a significant increase in the diseased groups (both left and right). The CV is also considerably increased in neurodegenerative disorder groups compared to those of the control group. The CV of patients with ALS, was nearly twice while with PD (right-foot) and HD (right-foot), was nearly thrice as that observed in control subjects. These results indicate that the magnitude of stride-to-stride variability in ALS, PD, and HD patients is significantly increased by neurodegenerative diseases. From the results it is found that both left-foot and right-foot time series in the control group show only very small differences in fluctuation magnitude of stride variability which reflect functional differences in the contribution of each limb to propulsion and control during walking. This is because of the fact that the data was collected through hall-way usual straight walking (without any obstacles). The same inference, however, is not true in regard to patients from neurodegenerative disorder groups for the same task. This implies that the degenerative effect of the neurodegenerative disorder on the fluctuation magnitude variability of left-foot and right-foot strides is not the same. This conclusion is in agreement with those of the earlier studies 56. Kruskal-Wallis tests are performed to evaluate the statistical differences between the different measures of four groups. The test detected significant group differences. In the case of CV, p ~ 0 and chi-square >197.29 for left-foot stride analysis while p ~ 0 and chi-square >211.39 for right-foot stride analysis.

In PD and HD groups right foot is significantly affected by the disease while in ALS group left foot is more affected.

Unlike a linear measure which focuses on the magnitude of variation in a distribution irrespective of the order in which data points accumulate, a nonlinear measure is explicitly concerned with the temporal evolution of structure of the data variability and hence, may unravel more meaningful information. Now, we investigate dynamic features of the stride-to-stride intervals of the four groups (HC and neurodegenerative disorder classes) by using SpecEn, SpecCen, SampEn, and MLZC. Since, right-foot and left-foot stride dynamics analyses yield almost similar results, the nonlinear analyses have been carried out only on right-foot strides. We find that for SpecEn, SpecCen, and MLZC, short stride-to-stride sequences in the order of 20 consecutive stride intervals is reasonable to obtain optimum results. However, for SampEn stride-to-stride sequences in the order of 400 consecutive stride intervals is necessary for optimum results. This is because SampEn is extremely sensitive to parameter choices, especially for very short data sets with length ≤200 samples and hence, for relative consistency and for better separation between groups more samples (larger than 200) are necessary 57. For this, each gait record (right-foot) is divided into segments each with 20 or 400 samples, depending upon the measure, in the case of HC as well as neurodegenerative disorder groups. We carefully avoided nonstationary segments of the gait records, since usually stationarity is a requirement for the above nonlinear measures. The results of comparison of SpecEn, SpecCen, SampEn, and MLZC in the four groups (HC, ALS, PD, and HD) using Kruskal-Wallis test is shown in Table 2. All the values are expressed as mean ± SD. It is found that all the nonlinear measures are statistically significant. It can be observed that HC group shows intermediate values for SpecEn (0.449 ± 0.006) in the normalized range of 0 to 1. Compared to HC group the neurodegenerative disorder groups (ALS: 0.469 ± 0.016, PD: 0.474 ± 0.016, HD: 0.520 ± 0.036) show an increase. This means HC group exhibits higher rhythmicity than the disorder groups. The lowest rhythmicity is found in the HD group. Decreased rhythmicity is an indication of loss of gait control mechanism. SpecCen, which represents centre of gravity of the spectrum, also shows a similar behavior, being moderate in the case of HC group and highest in the case of HD group (HC: 0.118 ± 0.001, ALS: 0.126 ± 0.005, PD: 0.127 ± 0.004, HD: 0.142 ± 0.013). In the SpecEn and SpecCen domains, 0.449 ± 0.006 and 0.118 ± 0.001, correspond to their respective optimum state and any deviation from this would imply gait impairment. Thus, the power spectral analysis (through SpecEn and SpecCen) revealed that there is a change in the frequency content (rhythmicity) when moving from healthy to disease state.

All the values are expressed as mean ± SD. The symbol ~ means approximately. The p-value is so small that it can be treated as zero.

Now we adopt an approach suggested by Hausdorff 40 and show (1) the robustness of our approach through the estimates of mean SpecEn and SpecCen and (2) the effectiveness of the median filter in preserving the underlying dynamics while removing the extraneous data points associated with the changes in gait direction. For this, we repeat the procedure of computation of SpecEn and SpecCen for all the four groups, but this time without removing the outliers due to the turning events. The results of testing robustness of SpecEn, and SpecCen in the four groups (HC, ALS, PD, and HD) using Kruskal-Wallis test, when outliers are retained is shown in Table 3. All the values are expressed as mean ± SD. It is found that both the nonlinear measures are statistically significant. Much more, comparison of the rows corresponding to SpecEn and SpecCen in Tables 2 and 3 reveals that there is not much difference in the gait measure values, in their respective groups, before and after the application of median filter. This implies that the median filter applied was apparently effective in minimizing any effects due to the walking protocol (turning around), at the same time preserving the intrinsic variability. The Kruskal-Wallis test before and after application of median filter showed that SpecEn and SpecCen remain independently associated with the intrinsic walking patterns. This indicates the robustness of our approach through the estimates of mean SpecEn and SpecCen.

All the values are expressed as mean ± SD. The symbol ~ means approximately. The p-value is so small that it can be treated as zero.

To test the presence of deterministic structures in the gait time series and thereby ascertain appropriateness of the application of our nonlinear approach, we carried out surrogate data analysis using SpecEn. Fifteen surrogate series for each of the original gait series are constructed as explained in the above section. The mean of surrogate SpecEn values for the fifteen surrogate series are computed and compared with that of the original series. Table 4 shows results of surrogate data analysis of HC, ALS, PD, and HD groups. The values are expressed as mean ± SD. From the Table 4 it is found that in the HC group SpecEn of the original series is 0.449 ± 0.006 and that of the surrogate series is 0.459 ± 0.004. In the ALS group, SpecEn of the original series is 0.469 ± 0.016 and that of the surrogate series is 0.587 ± 0.019. In the PD and HD groups, SpecEn of the original series are 0.474 ± 0.016 and 0.520 ± 0.036, respectively, and those of the surrogate series are 0.511 ± 0.011 and 0.719 ± 0.063, respectively. The statistical significance of the differences between the nonlinear measures of the original and surrogate series, during straight walking, in the four groups investigated using Mann-Whitney rank sum tests is also specified in the Table 4. Interestingly, comparison between the respective SpecEn of the original and surrogate gait series reveals highly significant differences (p-value <0.0001) implying that the relevant patterns in the original time series cannot be considered present by chance. This indicates that the fluctuations observed in the original gait time series are not randomly derived, but they are deterministic in nature. This justifies the application of our nonlinear approach.

All the values are expressed as mean ± SD.

It can be noticed from Table 2 that the optimum variability range for SampEn in the case of HC group is 1.714 ± 0.129. Compared to HC group the neurodegenerative disorder groups exhibit decreased SampEn values (ALS: 0.658 ± 0.077, PD: 1.266 ± 0.117, HD: 1.210 ± 0.219). This means HC group exhibits higher irregularity, which is decreased in the neurodegenerative disorder groups. The lowest irregularity can be seen in ALS group. It is also found from Table 2 that HC group exhibits optimum variability range for MLZC as 0.436 ± 0.054. Compared to HC group the neurodegenerative disorder groups show an increased range of MLZC values in the normalized range of 0 to 1 (ALS: 0.656 ± 0.138, PD: 0.656 ± 0.137, HD: 0.842 ± 0.136). This implies HC group exhibits lower complexity, which is increased in the neurodegenerative disorder groups. The highest complexity can be noticed HD group. Increased complexity shows increased randomness in gait patterns and indication of gait balance impairments.

Next, we evaluated the statistical significance of SpecEn and SpecCen between HC and neurodegenerative disease groups using Mann-Whitney rank sum test. The results are depicted in Table 5. The p-values of the tests show that SpecEn and SpecCen have diagnostic potential to separate HC group from neurodegenerative disease groups. The diagnostic ability of SpecEn and SpecCen in separating the four groups (HC, ALS, PD, and HD groups) is also demonstrated through the scatter plots made for SpecEn vs. SpecCen shown in Figure 2. The results of quadratic regression analysis for best fit are depicted in Table 6. The corresponding coefficient of determination, R^{2} and root mean squared error (RMSE) for the best quadratic fit are also shown in the Table. The three coefficients for the four cases are distinctly different indicating distinct separation into classes.

The red colored curve shows best quadratic fit.

Coeft_{2}, Coeft_{1}, and Coeft_{0} represent respectively, quadratic, linear, and constant coefficients. R^{2} represents Coefficient of determination and RMSE represents root mean squared error.

We also evaluated the statistical significance of SampEn and MLZC between HC group and neurodegenerative disease groups using Mann-Whitney rank sum test. The results are tabulated in Table 7. The tests illustrate that while MLZC has diagnostic ability to separate the HC group from all the neurodegenerative disease groups, SampEn can readily separate HC from ALS and PD. Compared to other measures SampEn can only barely distinguish the HC group from HD group.

The purpose of this study was to (1) test the hypothesis that for a specific walking task while mapping from temporal gait variability domain to the nonlinear domain, the healthy optimum variability range gets mapped to an intermediate region and that values in this domain either below or above that corresponding to optimum region, may serve to identify gait impairment/pathology; (2) show that methods from nonlinear dynamics may be beneficial in understanding and describing variability and thereby identify health status. To the best of our knowledge this is the first study to investigate gait variability in neurodegenerative patients from the point of view of optimal variability. We assessed the gait variability of patients with neurodegenerative disorders (ALS, PD, and HD) relative to healthy control subjects using four nonlinear tools (SpecEn, SpecCen, SampEn, and MLZC). It is essential to note that though the two linear tools (SD and CV of stride time) can serve as accurate measures of only magnitude of motor variability within the system, they do not explain the underlying neural processes of human locomotion.

From the Results section above, it is found that the control group shows hardly any difference between the right and left mean stride times. This difference is moderate in ALS and PD groups, while quite significant in the case of HD. It is also found that the mean stride time is longer in all the neurodegenerative disorder groups compared to that of control group. The control group shows minimum magnitude variability (SD) while all the disorder groups exhibit considerably larger magnitude variability. The maximum magnitude variability is shown by PD and HD groups for right side. The above differences among the three patient groups possibly indicate differences in neuropathology for the same walking conditions. The more the gait variability the greater is the degree of neurological impairment.

From the Results section above, it is also observed that the control and ALS groups show hardly any difference between the right and left stride time CVs. On the other hand, PD and HD groups exhibit significant differences between right and left CVs. In both PD and HD groups the right foot is more affected by the disease. It is also found that the stride time CV is also considerably increased in neurodegenerative disorder groups compared to those of the control group. The CV of patients with ALS and PD (right-foot) was nearly twice, while CV with HD (right-foot) was nearly thrice as that seen in control subjects. These results indicate that the magnitude of stride-to-stride variability in ALS, PD, and HD patients is significantly increased by neurodegenerative diseases.

From the results of SD and CV, it is found that both left-foot and right-foot time series in the control group show only very small differences in fluctuation magnitude of stride variability which reflect functional differences in the contribution of each limb to propulsion and control during walking. This is because of the fact that the data was collected through hall-way usual straight walking (without any obstacles). The same inference, however, is not true in regard to patients from neurodegenerative disorder groups for the same task. This implies that the degenerative effect of the neurodegenerative disorder on the fluctuation magnitude variability of left-foot and right-foot strides is not the same. In the present study with the data base we have used, among 13 patients with ALS in approximately 7 patients the degenerative effect of the neurodegenerative disorder was more pronounced on the left-foot, among 15 patients with PD in approximately 8 patients the degenerative effect of the disorder was more pronounced on the right-foot, and among 20 HD patients in approximately 11 patients the degenerative effect of the neurodegenerative disorder was more prominent on the right-foot. Similar conclusions are in agreement with those of the earlier studies 56. Kruskal-Wallis tests performed to evaluate the statistical differences between the linear measures of four groups detected significant group differences. In the literature increased magnitude of variability (SD and CV) has also been used as a predictor of falling 53.

Surrogate data test was employed to test the presence of deterministic structures in the gait time series. The test showed significant difference between the original gait time series and their surrogate counterparts. This implies that the fluctuations observed in the time series may reflect nonlinear deterministic processes by the neuromuscular system. This also ascertains the appropriateness of the application of our nonlinear approach. Statistical tests revealed that all the nonlinear measures are statistically significant.

The diagnostic ability of SpecEn and SpecCen in separating the four groups (HC, ALS, PD, and HD groups) demonstrated through the scatter plots made for SpecEn vs. SpecCen further ascertains the robustness of the approach. The quadratic regression analysis shows that the three coefficients of the four groups are different from each other indicating distinct separation into classes. In the case of SpecEn, SpecCen, and MLZC measures, the optimum gait variability of HC group mapped to 0.449 ± 0.006, 0.118 ± 0.001, and 0.436 ± 0.054, respectively. The increased gait variability in all the three neurodegenerative disorder groups mapped to almost non-overlapping ranges above the respective optimum. In the case of SampEn, however, the increased gait variability in all the three neurodegenerative disorder groups mapped to almost non-overlapping ranges below the optimum (0.436 ± 0.054). Since each group maps to a different range in the nonlinear domain it implies that the stride fluctuation dynamics is different for each group. It is possible to represent motor variability by a continuum. The two ends of this continuum correspond to total regularity and total randomness. A healthy optimum motor variability is somewhere between the two ends.

The above discussion provides a strong ground for our hypothesis that each nonlinear measure shows a specific range corresponding to optimum variability of the healthy gait time series in time domain for a specified task. This range is characterized by a nonlinear deterministic structure which enables a healthy person to navigate through an environment in a flexible, but stable manner. The values in the nonlinear domain either below or above the optimal range signify pathology. The more the deviation from the optimal range the severe is the neurological impairment. However, further research is needed to more fully characterize and to understand the clinical impact of gait variability in individuals with neurological impairment.

A limitation of this study is the small sample size. Factors like high variance, age differences, and differing male-to-female ratios between groups will have an impact on the results when statistical analyses are carried out on small sample sizes. However, it has been shown that the effect of gender on usual gait patterns is considerably small 58. Though the effect of age on gait is complex, the effect of neurodegenerative disorders considerably predominates over the aging effects 37, 40, 58. This implies that the discrimination using this method stands irrespective of the above limitations.

Evidence shows that an optimal state of inherent variability in time domain is associated with health and that this variability is characterized by a nonlinear deterministic structure that enables a healthy person to navigate through an environment in a flexible, but stable manner. It is possible to represent motor variability by a continuum. The two ends of this continuum correspond to total regularity and total randomness. A healthy optimum motor variability is somewhere between the two ends. Each of the nonlinear measures, SpecEn, SpecCen, SampEn, or MLZC, shows a specific range corresponding to optimum variability of the HC gait time series. The values in the nonlinear domain either below or above the optimal range signify pathology and separate ALS, PD, and HD from HC. The more the deviation from the optimal range the severe is the disease.

All works was made by Chandrakar K, except for taking raw data from subjects.

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