A new measure for quantifying the bilateral coordination of human gait: effects of aging and Parkinson s disease

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1 DOI 1.17/s RESEARCH ARTICLE A new measure for quantifying the bilateral coordination of human gait: effects of aging and Parkinson s disease Meir Plotnik Æ Nir Giladi Æ Jeffrey M. Hausdorff Received: 3 January 27 / Accepted: 31 March 27 Ó Springer-Verlag 27 Abstract The bilateral coordination of locomotion has been described in detail in animal studies and to some degree in man; however, the mechanisms that contribute to the bilateral coordination of gait in humans are not fully understood. The objective of the present study was to develop a measure for quantifying the bilateral coordination of gait and to evaluate the effects of aging and Parkinson s disease (PD) on this new metric. To this end, we compared the gait of healthy older adults to that of healthy young adults and patients with PD. Specifically, we defined the stride duration of one foot as a gait cycle or 36, determined the relative timing of contra-lateral heel-strikes, and defined this as the phase, u (ideally, u = 18 for every step). The sum of the coefficient of variation of u and the mean absolute difference between u and 18 was defined as the phase coordination index (PCI), representing variability and inaccuracy, respectively, in phase generation. PCI values were higher (poorer bilateral coordination) in patients with PD in M. Plotnik (&) N. Giladi J. M. Hausdorff Movement Disorders Unit, Tel-Aviv Sourasky Medical Center, 6 Weizmann St, Tel-Aviv 64239, Israel meirp@tasmc.health.gov.il N. Giladi J. M. Hausdorff Department of Physical Therapy, Sackler Faculty of Medicine, Tel-Aviv University, Tel Aviv, Israel N. Giladi Department of Neurology, Sackler Faculty of Medicine, Tel-Aviv University, Tel Aviv, Israel J. M. Hausdorff Division on Aging, Harvard Medical School, Boston, USA comparison to the healthy older adults (P <.6). Although gait speed and stride time variability were similar in the healthy young and older adults, PCI values were significantly higher among the healthy elderly subjects compared to the young adults (P <.1). Regression analysis suggests that only about 4% of the variance in the values of PCI can be explained by the combination of gait asymmetry (as defined by the differences in each leg s swing times), gait speed and stride time variability, pointing to the independent nature of this new metric. This study demonstrates that bilateral coordination of gait deteriorates with aging, further deteriorates in PD, and is not strongly associated with other spatio-temporal features of gait. Keywords Blateral coordination Aging Parkinson s disease Gait Introduction Bipedal human gait is generally considered to be symmetric (Sadeghi et al. 2) and to be governed by antiphase activation pattern of the two legs (Prokop et al. 1995). The quantification of bilateral coordination of human locomotion has, to date, generally been studied by asking subjects to perform locomotion tasks under non-natural conditions. For example, the left right phase relationship of the lower limbs was investigated during pedaling movements on a stationary bicycle (Abe et al. 23). Dietz and colleagues preformed a series of interesting experiments using split-belt treadmills that impose different gait speeds for each leg to examine the degree of coordination between the legs in these conditions (for review: Dietz 22). These studies found that

2 healthy subjects are able to control each leg separately to a remarkable degree, while this ability declines with disease. Traditionally, the neuronal mechanisms responsible for intralimb coordination of muscle activation during walking have been associated with spinal locomotor centers, which consist of central pattern generators (CPGs), i.e., local neuronal circuits that generate rhythmic stepping movements by alternating activity between groups of flexor and extensor muscles of a limb (for review: Dietz 23). The close coordination of muscle activation between the two legs, a requirement for the regulation of successful human walking, is achieved through reciprocal operation of CPGs on both sides of the spinal cord. This reciprocity is mediated in the rat by commissural fibers in the ventral part of the spinal cord, resulting in strict alternating activity of motor neurons in segmental pairs of ventral roots (Kjaerulff and Kiehn 1996). In humans too, the basic interleg coordination is based on bilateral antiphasing, i.e., initiation of the swing period of one leg is synchronized with the stance period of the contra-lateral leg. Interestingly, this pattern develops in the early stages of infant walking (e.g., Pang and Yang 21). The general effects of aging on gait have been previously described [for a review see (Ashtom-Miller 25)]. The most apparent change is reduction in gait speed at ages above 65 years of age (e.g., Bohannon 1997). Many other changes, such as shorter step and stride lengths and decreased ankle extension have also been reported (e.g., Winter et al. 199). Gait changes with aging can be attributed to age-associated decline in musculo-skeletal function (Morley 23) as well as changes in the central nervous system (Hausdorff et al. 1997). In Parkinson s disease (PD), basal ganglia dysfunction contributes to more significant gait disturbances such as abnormal scaling of stride length (Morris et al. 1996; Sofuwa et al. 25) and impaired pacing of gait rhythmicity (Hausdorff et al. 1998). Split-belt treadmills studies suggest that PD may affect bilateral coordination abilities (Dietz et al. 1995); however, its impact on left right lower extremity coordination during normal walking is largely unknown. In the present study, we develop a new measure for quantifying the bilateral coordination during usual groundlevel walking, based on the degree of consistency and accuracy in the left right stepping coordination. We apply this approach to test the following hypotheses: (1) that aging impairs the bilateral coordination during gait, and, (2) that PD is further associated with a decrease in this ability. To test these hypotheses, we quantified the gait of three subject groups: healthy young adults, healthy elderly subjects, and elderly patients with PD. Methods Subjects We studied 14 healthy elderly subjects, 15 healthy young adults and 21 patients with idiopathic PD, as defined by the UK Brain Bank criteria (Gelb et al. 1999; Hughes et al. 21). Patients with PD were recruited from the outpatient clinic of the Movement Disorders Unit at the Tel-Aviv Sourasky Medical Center. Elderly subjects were recruited from several sources in the community: patients spouses, local senior centers, and volunteers from the community. Young adults were volunteers from the community. PD patients were included if their disease stage was 2 3 on the Hoehn and Yahr clinical staging (Hoehn and Yahr 1967; Goetz et al. 24) and if they were treated with anti-parkinsonian medications. We excluded PD patients with motor response fluctuations and patients who experienced freezing of gait (FOG). None of the PD patients experienced (FOG) during the experiments or reported, in response to explicit questioning, experiencing freezing in the past. Thus, PD patients can be considered to have a mild to moderate disease severity. Subsets of the data used described in the current study were previously described (Yogev et al. 25, 26). All subjects were invited to participate if they were able to ambulate independently, they did not have dementia [as determined by DSM IV criteria and scores >24 on the Mini Mental State Exam (MMSE) (Folstein et al. 1975)], and they were between 6 and 85 years of age in the case of the PD patients and elderly subjects, or between 2 and 3-years old in the case of the young adults. Subjects were excluded if they had clinically significant musculo-skeletal, cardiovascular or respiratory diseases, history of stroke, head trauma or any other neurological disease, major depression or uncorrected visual disturbances. Except for the existence of Parkinson s disease itself, similar criteria were applied to the healthy elderly and the PD group in an effort to exclude subjects with underlying disease likely to affect gait. While subtle degenerative changes cannot be completely ruled out, the application of identical criteria to all groups should minimize such influences. This study was approved by the Human Studies Committee of Tel-Aviv Sourasky Medical Center and all subjects gave their written informed consent according to the declaration of Helsinki, prior to entering the study. Experimental procedure: clinical evaluation and walking protocol After providing informed written consent, the PD patients and the elderly subjects were first cognitively assessed

3 using the MMSE test. The Unified Parkinson s Disease Rating Scale (UPDRS) was used to quantify disease severity and extra-pyramidal sign among the PD patients (Fahn et al. 1987). In order to quantitatively assess gait, subjects were instructed to walk at their normal pace on level ground for 2 min in a 25-m long, 2-m wide, well-lit corridor. Forcesensitive insoles were used in order to obtain a time series of the vertical ground reaction force generated by each of the feet while walking. Measurements from both feet were synchronized. Offline analysis of gait Temporal gait parameters, i.e., stride time, the time between two consecutives heel strikes of the same leg, and swing times, were determined using off-line computerized analysis of the force signal, which identified initial contact of the foot with the ground ( heel strike ) and then, the departure of the foot from the ground ( toe off ), as previously described (Hausdorff et al. 1998; Yogev et al. 25). In order to focus on the assessment of undisturbed continuous normal walking, turns were automatically excluded from the analysis. Thus, the gait segments that were analyzed were composed from straight line, steady state walking. Average gait speed was determined using a stopwatch by measuring the average time the subject walked the middle 8 m of the walking path. Average stride length was determined from gait speed and stride time. Because of this indirect approach, stride length measures may have been less accurate than other gait variables. Gait asymmetry was assessed by comparing the swing times performed by one leg with respect to those performed by the other using the following formula: Gait Asymmetry ¼ 1 j lnðsswt/lswtþj where SSWT and LSWT stand for mean value of swing time for the leg with the short and long mean swing time, respectively (Plotnik et al. 25). Interlimb coordination was also determined off-line, as described in the next section. Assessment of left right coordination of gait Bilateral coordination of gait was assessed by quantifying the phase relationship between the step timing of the left and right legs. To evaluate the phase relationship between right and left legs stepping during gait, we modified the methodological approach introduced by Abe et al. (23), in their study of coordination during bipedal bicycling and defined the phase, u, and the phase coordination index (PCI). Each stride (from one heel strike to the consecutive heel strike of the same leg) defines one gait cycle. The step time, i.e., the time between the start of a gait cycle and the time point when the other leg s heel strike occurs, is used to determine the phase. Normalizing the step time with respect to the stride time, defines the phase of the ith stride (u i; denoted in degrees). In order to preserve uniformity across all subjects, we first calculated the mean values of the swing times for both legs and used the leg with the higher value of the average swing time as the reference for the gait cycles and calculated phase values for the other leg. Hence, u i is defined as: u i ¼ 36 t Si t Li ð1þ t Liþ1 ð Þ t Li where t Li and t Si denote the time of the ith heel strike of the legs with the long and short swing times, respectively, and t L(I +1) > t Si > t Si (see Fig. 1). The denominator in Eq. (1) is the stride time of the leg with the long swing time and the pre-factor 36 is used to transform the variable into degrees (or relative cycle timing). The PCI incorporates the assessment of the accuracy and consistency of phase generation and is the primary outcome. From symmetry considerations, in order to maintain successful walking, u should have the value of approxi- Fig. 1 A cartoon illustrating the phase determination of one leg s stepping with respect to the gait cycle determined by the other leg. A small floating ball is depicted adjacent to the leg, which determines the full gait cycle. The time difference between consecutive heel strikes of the two feet, normalized to the gait cycle duration, is defined as the phase

4 mately 18 (i.e., step time should be equal to the half the gait cycle for each step). Therefore, the level of accuracy in phase generation, i.e., how close are the series of generated phases to the value 18, was assessed by the mean value of the series of absolute differences between the phase at each stride and 18 and was denoted as u_abs: u ABS ¼ ju i 18 j: In order to assess the level of consistency in phase generation across all of each subject s strides, we also calculated the coefficient of variation of the mean of u for each subject. This variable is denoted as u_cv [%]. Although u_abs and u_cv describe different aspects of left right coupling, there is an algebraic relation between these two measures. Appendix A analytically describes the relationship between u_cv and u_abs. Since these variables may be associated with each other with dependence on, for example, the standard deviation of u as a covariate, we defined one combined summary measure, the phase coordination index: PCI = u_cv + Pu_ABS, where Pu_ABS = 1 (u_abs/18). Thus, PCI is the sum of two percentile values, reflecting both accuracy and consistency of phase generation. Statistical analysis To test the two hypotheses, we first applied analysis of variance (ANOVA, SPSS software) to the PCI parameter. If a significant group effect was observed, the young adults and PD group were compared to the reference healthy elderly subjects to evaluate any effects of aging and/or PD using Student t tests (two sided). The other outcome measures, u, u_cv and u_abs, were evaluated in separate ANOVA models similarly to examine the role of accuracy and consistency of phase generation separately. A P value less than or equal to.5 was considered statistically significant. Additional study questions were tested using correlation analyses and multiple regression analyses. Results Demographic and basic gait characteristics of the study groups are summarized in Table 1. The mean age of the PD patients was similar to that of the healthy elderly subjects. Male/female ratios were not significantly different in any of the groups (P >.19). Gait speed and stride length were significantly lower among the PD patients as compared to the healthy elderly subjects (P <.1), but were similar in the healthy young and elderly subjects. Similarly, stride time variability was Table 1 Characteristics of the three study groups Young adults (n = 15) Elderly subjects (n = 14) PD (n = 21) Age (years) 26.3 ±.5 # 69.1 ± ± 1.5 Gender (m/f) 7/8 7/7 16/5 Gait speed (m/s) 1.35 ± ± ±.5* Stride length (m) 1.45 ± ± ±.5* Stride time 1.86 ± ± ±.16* variability (%) Gait asymmetry.84 ±.19 # 1.52 ± ± 1.64* Hoehn & Yahr 2.3 ±.1 stage UPDRS score 35.8 ± 2.6 * and # denote statistically significant difference (P <.5) between the older adults and the patients with PD and between the older adults and the young adults, respectively, based on two tailed t-test comparisons except for Chi square comparison for gender. Entries are mean ± SE, except for gender. UPDRS: Unified Parkinson s disease rating scale increased in the patients with PD, compared to the healthy elderly subjects (P =.3), but was similar in the young and elderly groups. Gait asymmetry values were significantly higher in the PD patients as compared to the elderly subjects (P <.4) and higher in the healthy elderly subjects, compared to the young adults (p <.5). The effects of aging and Parkinson s disease on the PCI In all three groups, mean values of the phase (u) were close to the gold standard, ideal value of 18. Group differences were not statistically significant (see Table 2, last row). Figure 2 depicts the u values generated during the walking of one representative subject from each group. For the healthy young adult, u values barely fluctuated and were relatively close to the ideal line of 18. For the elderly subject, u values were more variable and more distant, in general, from the line of 18. Both of these features were further aggravated in the PD patient. These differences are reflected in the PCI values, which were smallest in the young adult and highest in the PD patient (see the legend of Fig. 2). At the group level, PCI values were lowest in the healthy young subjects, higher in the healthy elderly subjects, and higher still in the subjects with PD (see Fig. 3; Table 2). Compared to the healthy elderly subjects, PCI values were significantly lower in the healthy young adults (P =.1) and were significantly higher in the PD group (P =.6). The two components of the PCI, u_cv and u_abs, followed similar trends.

5 Table 2 Effects of aging and PD on the bilateral coordination parameters Young adults Older adults PD ANOVA F P value Phase Coordination Index (%) 2.47 ±.15 (.1)* 3.3 ± ±.61 (.6) 1.4 <.1 u_cv (%) 1.29 ±.1 (.8) 1.6 ± ±.19 (<.1) 2.2 <.1 u_abs (deg) 2.12 ±.18 (.6) 3.6 ± ±.84 (.48) u (deg) 18.1 ±.4 (.75) ± ± 1.1 (.17) Enteries are means ± SE * Numbers in the parentheses are the P values compared to the healthy older adults Stepping phase (deg) Young PCI=1.8% Elderly PCI=4.2% PD PCI=8.2% Stride # Fig. 2 Stepping phase values are plotted for a healthy young adult (left), a healthy older adult (middle) and one PD patient (right). For the young adult, stepping phases are scattered close to the 18 line. The mean value of u was Values of u_abs and u_cv were 1.4 and 1.%, respectively. For the healthy elderly subject (middle), stepping phase values are more dispersed with a mean value of u of Values of u_abs and u_cv were 3.6 and 2.2%, respectively. Stepping phase values are dispersed further and farther from the 18 line in the case of the PD patient (right panel). The mean u value was Values of u_abs and u_cv were 9. and 3.2%, respectively The relationship between PCI and other gait measures Correlation analysis (all three groups pooled) indicates that the PCI was moderately correlated with gait asymmetry (Spearman s q =.53, P <.1) and inversely correlated with gait speed (Spearman s q <.53, P <.1), but was not strongly related to stride time variability (Spearman s q =.25, P =.8). Multiple regression analysis estimates that only 42% of the variance in PCI can be explained by gait speed, gait asymmetry and stride time variability (all groups combined; see Table 3). Discussion In this study, a new measure, the PCI, was introduced to evaluate the bilateral coordination of gait. This metric was used to quantify the ability of young, elderly and parkinsonian subjects to coordinate left right stepping on level ground. Mean values of phase were close to 18 in all three groups. This finding confirms earlier results from animal (e.g., Kjaerulff and Kiehn 1996) and from human (e.g., Berger et al. 1984) studies, that anti-phasing between one leg s stepping and that of the other is a relatively robust feature of gait. Nonetheless, aging and Parkinson s disease influenced left right coordination, as discussed further below. Bilateral coordination of rhythmic movements Fig. 3 Bar chart demonstrating the PCI differences between the three study groups as reflected by the mean values of PCI in each group (see also Table 2). Error bars SEM Human walking can be considered as a motor task that requires bilateral coordination between two unilateral

6 Table 3 Association between the phase coordination index (PCI) and other measures of gait Correlation of PCI with: All subjects Healthy subjects only PD only Gait Speed*.53 (<.1).8 (.667).55 (.18) Stride Time Variability.25 (.8).7 (.77).32 (.446) Gait Asymmetry.53 (<.1).37 (.48).41 (.77) Multiple Regression Total Variance R 2 =.42 R 2 =.13 R 2 =.2 Enteries are Spearman s q (P values), except for the last row where the variance in the PCI explained by gait speed, stride time CV, and gait asymmetry is shown based on multiple regression (R-Pearson s coeficient). *Almost identical results were obtained if stride length were substituted for gait speed rhythmic movements, where smooth coordination takes place between the motor output of the hands and legs. The nature of this coordination is still not fully understood (for review: Zehr and Duysens 24). Nonetheless, it has been shown that bilateral coordinated rhythmic voluntary hand movements are normally performed either in-phase (synchronous) or anti-phase (e.g., Lee et al. 1996). In the latter case, brain activity involves extensive fronto-parieto-temporal cortical activations, including the supplementary motor area (SMA, Ullen et al. 23). Thus, the SMA has been considered as the brain locus where this coordination mainly occurs (Obhi et al. 22; Debaere et al. 24). Human locomotion mostly involves anti-phase activation patterns (exceptions include the butterfly-stroke and the breast-stroke swimming styles). The new metric, PCI, examines the accuracy of the anti-phase coordination and the consistency of the gait pattern. Increased values of PCI among PD patients emphasize the potential role of the basal ganglia, presumably through their projections to the SMA, in the coordination of the anti-phase pattern of gait. The effect of age and PD on PCI Phase coordination index values were increased in the healthy elderly group as compared to the young adults. It may, therefore, be suggested that the aging brain has a limited capacity to accurately and consistently time the gait cycle of one leg with respect to the other. At the same time, the generation of symmetric gait is also compromised, but perhaps to a lower extent (compare the P values in Tables 1, 2). These results and the similarities between the young adults and the healthy elderly with respect to other gait features (Table 1) suggest that the bilateral motor control of gait may be particularly sensitive to the effects of aging. In the PD group, PCI values were higher as compared to the age-matched healthy subjects. This finding complements the previous works which observed alterations in gait asymmetry both in patients with de novo (Baltadjieva et al. 26), mild PD (Yogev et al. 26) and more advanced PD (Plotnik et al. 25). While head to head comparisons were not made, it is interesting to note that the degree of gait asymmetry is apparently larger in patients with motor response fluctuations (Plotnik et al. 25), compared to those without (the present study). Taken together, these observations suggest that both left right phase coordination of stepping and gait asymmetry are impaired in PD, and one can speculate further, that initial deterioration of bilateral motor control of gait is related to the early stages of the disease, and further deterioration would emerge as the disease progresses. The present findings are generally in agreement with and compliment previous studies in PD. Abe and colleagues (Abe et al. 23) studied left right leg phase relationship during pedaling movements performed on stationary bicycles in which the left pedal was mechanically uncoupled from the right pedal. Among PD patients, unsteady phase patterns were observed, while the control group generally maintained a 18 phase difference between the legs. In split-belt treadmills experiments by Dietz et al. (1995) in which each belt was running at a different speed, healthy subjects readily tolerated differences in the speed of each belt, while PD patients quickly reached the limits of their walking capabilities as belt speed differences increased. The phase coordination index depicts a distinct feature of gait From a functional point of view, our results suggest that bilateral coordination of gait is a relatively distinct feature of walking (recall Table 3). In healthy subjects, gait asymmetry, gait speed and gait variability are not strongly correlated with the PCI. Even in PD, where a reduced stride length is a dominant feature of the impaired locomotion (Morris et al. 1996), the PCI was only weakly associated with stride length and gait speed. The relationship between left right swing times (i.e., gait asymmetry) might reflect the degree of similarity in the motor function (i.e., programming and activation) related to leg propulsion on both sides of the body. On the other hand, left right step phase coordination (i.e., PCI) may represent the degree to which the rhythmic process of

7 stepping with one leg is coordinated with the rhythmic process of stepping with the other one. Theoretically, therefore, GA and PCI are distinctive from each other. To illustrate the unique properties of PCI and GA, consider the hypothetical shortening or elongation of the duration of one foot s swing times while keeping stride times constant (recall, the stance period is complementary to swing within the stride cycle). This will increase gait asymmetry, but will have no effect on phase coordination. Patterns as such can be seen in the gait of amputees (Zmitrewicz et al. 26). The moderate correlation between gait asymmetry and PCI in the PD group, on the other hand, may be explained by general impaired locomotor function in these patients (recall Tables 1, 2). Thus, the correlation between gait asymmetry and PCI might simply be a by-product of poor gait control in these patients, rather then any cause and effect relationship between gait asymmetry and left right phasing. Future directions Several aspects of the present study call for further investigation. The first is a more comprehensive understanding of the step-to-step control of the left right phase coordination and its relationship to other gait and motor control features. For example, does the bilateral coordination affect the overall gait rhythmicity or is it vice versa; i.e., does the basic rhythm of stepping govern the capacity to generate symmetric and anti-phased stepping? Alternatively, is there a stride-to-stride adjustment mechanism that corrects stepping phasing in order to maintain bilaterally coordinated walking? Another important question, which is not presently addressed, is whether impairments in bilateral coordination of gait are significant to the appearance of episodic gait failures. For example, it would be important to study the PCI properties in patients with PD who experience freezing of gait (FOG). In the present study, PD patients with motor response fluctuations were not included and patients who experience FOG were also excluded. It might be interesting to observe the levels of bilateral coordination of gait (e.g., PCI) just prior to the occurrence of FOG. The association between PCI, disease duration, severity and age should be examined in a larger population, in particular in light of the present finding that PCI, unlike other gait disturbances (c.f. Table 1) points to reduced bilateral coordination capacity with aging. The PCI is larger in PD compared to the healthy elderly group and larger in the healthy elderly compared to the young. This suggests that for this feature of gait, PD may be considered an exaggerated form of aging; however, we do not wish to imply that this is the case in general. In the case of elderly idiopathic fallers, gait asymmetry was increased when the subjects had to cope with an additional cognitive task during their walking (Yogev et al. 26). This observation suggests that mental distraction may lead to impaired control of bilateral function, which then causes instability and falls. The affects of mental distraction and the relationship between cognitive function, attention and bilateral coordination also require additional studies. The development of this new metric of the bilateral coordination and the present findings of changes in the PCI with aging and PD enable future studies that can directly address these questions and the factors that contribute to the bilateral coordination of gait. Acknowledgments We thank the subjects for their participation, time and effort. We thank Ms. Galit Yogev, Ms. Michal Leshem and Mr. Ronny Bartsch for their invaluable assistance and Dr. Eli Plotnik for mathematical advice. This work was supported in part by the Inheritance Fund of the Israeli Ministry of Health, NIH grants AG- 141, RR-13622, HD and AG-8812, by the US-Israel Bi- National Science Foundation, by the Parkinson s Disease Foundation (PDF), New York and the National Parkinson Foundation (NPF), Miami USA, and by the European Union Sixth Framework Program, FET contract no , Dynamic Analysis of Physiological Networks (DAPHNet). Appendix: Analytical and numerical analyses of the relationship between the parameters u_cv and u_abs In this appendix, we present analytical and numerical analyses that describe the relationship between the two components of the PCI defined to incorporate the estimates by which a subject is constantly and accurately generating left right stepping phases while walking. Analytical analysis Denote by u = u 1,u 2,, u N, a set of N measurements of the phase u. Let u be the mean value of u, i.e. u ¼ 1 N u i ; and d the standard deviation of u given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 d ¼ t ðu u N i Þ 2 The coefficient of variation of u is defined by u CV ¼ d u Define: ð2þ ð3þ ð4þ

8 u ABS ¼ 1 N From (4) we obtain: ju i 18 j ð5þ P N ðu CVÞ 2 ¼ ðu i uþ 2 N u 2 ; ð6þ and (5) yields: ðu ABSÞ 2 ¼ 1 N 2 þ2 XN j¼iþ1 ju i 18 j jðu i 18 Þðu j 18 Þj! 2 " ¼ 1 # N 2 ðu i 18 Þ 2 ð7þ But the left term in the square parentacese of Eq. 7 can be written as: Equation 11 suggests that quadratic quadratic relation describes in part the analytic dependency between u_cv and u_abs. At the same time this dependency may vary substantially, given a series of u i as expressed by the middle and the left terms of the right part of the equation. In the following paragraphs, a numerical analysis describes how the relation between u_cv and u_abs is altered with accordance to different series of u i. Numerical analysis To explore a priori relations between u_cv and u_abs, three separate artificial data sets were established. For each data set 31 left right stepping phase series of 267 strides each 2 15 A SD=1.1 1 N 2 ðu i 18 Þ 2 ¼ 1N 2 ¼ 1 N 2 ½jðu i uþþðu 18 ÞjŠ 2 ðu i uþ 2 þ 2 N 2 ðu i uþ½dušþ 1 N 2 N½DuŠ2 where [Du] is defined as follows: ½DuŠ ¼u 18 Note that the middle term in Eq. 8 can be written as: ð8þ ϕ_abs ( ) B SD= N 2 ðu i uþ½duš ¼ 2 ½DuŠ ðnu NuÞ ¼ N2 ð9þ Returning to Eq. 7 and using the relation described in Eq. 6, ABS 2 (u) can be now written as: 25 2 C SD=17.6 ðu ABSÞ 2 ¼ u2 N ðu CVÞ2 þ ½DuŠ2 N þ 2 N u 2 1 j¼iþ1 or alternatively: jðu i 18 Þðu j 18 Þj ðu CVÞ 2 ¼ N u 2 ðu ABSÞ2 ½DuŠ2 u 2 2 N u 2 1 j¼iþ1 jðu i 18 Þðu j 18 Þj ð1þ ð11þ ϕ_cv (%) Fig. 4 Numerical analyses depicting the relation between u_abs and u_cv. The trace in each panel shows calculated values of u_abs and u_cv when the standard deviation of the mean value of u is kept constant while the mean value of u varies between 165 and 195 (left most and right most pairs of values, respectively, in all panels)

9 were established. Each of the 31 series represents data obtained from a different walking human subject. Across all series in each data set, the mean values of u ranged from 165 to 195 by increments of 1 (total of 31 mean values). The distribution of u values around the mean value was identical for all 31 series, i.e. the standard deviation of the mean (SD) was identical for all 31 series. SD values were assigned to be 1.1, 4.4, 17.6 in the first (A), second (B) and thirds (C), data sets, respectively. Thus for each data set, 31 pairs of u_cv and u_abs could be generated. Figure 4, depicts the relationships between u_abs and u_cv for each of the data sets. On the abscissa the increase in the values of u_cv corresponds to the decrease in the mean value of u from 195 to 165 (a trivial outcome from the fact that SD was maintained constant). It can be seen that the corresponding u_abs values decrease as the mean value of u approaches 18 and increase again as the mean value of u continues to decrease towards 165. It may be concluded that in a given value of SD there would be a monotonous change in the values of u_abs as a function of u_cv, in half of the range of u_cv, and opposite monotonous change in the second range of u_cv. As can be seen by the different range of variation which is covered by ordinate axes in all three panels, we may conclude that actual experimental data collected may in part be more influenced by the direct dependency between u_cv and u_abs, while for other terms this direct dependency is less obvious. Therefore, a priori for critical statistical tests based on this analysis it was determined to treat u_cv and u_abs as inter-correlated parameters. References Abe K, Asai Y, Matsuo Y, Nomura T, Sato S, Inoue S, Mizukura I, Sakoda S (23) Classifying lower limb dynamics in Parkinson s disease. 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