Lyapunov exponent and surrogation analysis of patterns of variability: profiles in new walkers with and without down syndrome

Abstract

In previous studies we found that preadolescents with Down syndrome (DS) produce higher amounts of variability (Smith et al., 2007) and larger Lyapunov exponent (LyE) values (indicating more instability) during walking than their peers with typical development (TD) (Buzzi & Ulrich, 2004). Here we use nonlinear methods to examine the patterns that characterize gait variability as it emerges, in toddlers with TD and with DS, rather than after years of practice. We calculated Lyapunov exponent (LyE) values to assess stability of leg trajectories. We also tested the use of 3 algorithms for surrogation analysis to investigate mathematical periodicity of toddlers' strides. Results show that toddlers' LyE values were not different between groups or with practice and strides of both groups become more periodic with practice. The underlying control strategies are not different between groups at this point in developmental time, although control strategies do diverge between the groups by preadolescence.

Figure 1

Visual analogy of what Lyapunov Exponent (LyE) calculates, using the variability across cycles of the knee marker of a toddler with Down syndrome (DS). 1a is the knee marker vertical position time series, 1b shows 3 strides extracted from the time series (1a) and overlaid as position vs. corresponding velocity and 1c demonstrates a magnified version of an isolated segment of the state space to show the divergence between neighboring trajectories. One important point to note is that LyE values for data in this manuscript were calculated in an 8-dimensional space, not the 2-dimensional space pictured here.

Figure 2

Exemplar time series of knee marker data in the vertical axis across months of walking experience. Although the behavior is clearly periodic by observation, there is enough variability across strides in time, amplitude, and shape that it is difficult to identify the mathematical rules for defining each period (TD = typical development, DS = Down syndrome).

Figure 3

Exemplar vertical direction time series from treadmill walking for one toddler with typical development with 4 months walking experience (W.E.). For the hip marker, the cyclic motion of the leg is not clear. The toe displays cyclic motion; but noise in the time data, particularly in the troughs, would decrease accuracy of results. The knee marker displays cyclic motion with the least amount of noise.

Figure 4

Success rates of surrogation increase across time using Theiler’s Algorithm 0 to create surrogate data. Success is defined as identifying more periodicity in knee vertical direction time series than in their randomly-generated surrogate counterparts (TD = typical development, DS = Down syndrome).

Figure 5

Mean LyE values for the left knee vertical direction by group across time. Error bars represent 1 standard deviation (TD = typical development, DS = Down syndrome).