Walking at stability's edge

During self-paced human walking, the variability in inter- stride intervals exhibit fractal dynamics characterized by long--range correlations having a power-law decay with exponent $\alpha$. We used diffusion fluctuation analysis (DFA) to estimate $\alpha$ as a function of the roughness of the walking surface for eight (8) healthy subjects (1200-1400 inter- stride intervals for each walking surface). For each subject the highest $\alpha$ (mean 0.96, range 0.88- 1.10) occured for walking on a running track and $\alpha$ was $15-20\%$ lower for walking on either a relatively smoother (tennis hard court) or a rougher (dirt path) surface. These observations are captured by a stochastic discrete time cubic map: $I_{i+1}=a(\xi_i)I_i - bI^3_i + \eta_i$, where $I_i$ is the $i$--th inter--stride time, $a(\xi_i)=a_o (\xi) + \xi_i$ describes parametric, colored noise where $a_0(\xi)$ is a constant that depends on surface roughness and $\xi_i$ is colored noise with mean zero, $\eta_i$ is low--intensity additive white noise, and $b$ is a constant. As the roughness, and hence $a_0(\xi)$, of the walking surface increases, the fluctuations in the inter--stride interval are predicted to obey a power law whose exponent changes non-monotonically: the highest values of $\alpha$ determined with DFA occur when $a_0(\xi)$ is close to the deterministic stability boundary $a=1$. Thus the neural control of walking appears to involve a dynamical system tuned close to the edge of stability subjected to the effects of parametric noise.