Maximum Lyapunov exponents as predictors of global gait stability: a modelling approach

To examine the stability of human walking, methods such as local dynamic stability have been adopted from dynamical systems theory. Local dynamic stability is calculated by estimating maximal finite time Lyapunov exponents (λ(S) and λ(L)), which quantify how a system responds continuously to very small (i.e. "local") perturbations. However, it is unknown if, and to what extent, these measures are correlated to global stability, defined operationally as the probability of falling. We studied whether changes in probability of falling of a simple model of human walking (a so-called dynamic walker) could be predicted from maximum finite time Lyapunov exponents. We used an extended version of the simplest walking model with arced feet and a hip spring. This allowed us to change the probability of falling of the model by changing either the foot radius, the slope at which the model walks, the stiffness of the hip spring, or a combination of these factors. Results showed that λ(S) correlated fairly well with global stability, although this relationship was dependent upon differences in the distance between initial nearest neighbours on the divergence curve. A measure independent of such changes (the log(distance between initially nearest neighbours after 50 samples)) correlated better with global stability, and, more importantly, showed a more consistent relationship across conditions. In contrast, λ(L) showed either weak correlations, or correlations opposite to expected, thus casting doubt on the use of this measure as a predictor of global gait stability. Our findings support the use of λ(S), but not of λ(L), as measure of human gait stability.