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PubMed2004

Random Boolean network model exhibiting deterministic chaos.

This paper considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes, which may vary from one node to another. This is an extension of a model studied by Andrecut and Ali [Int. J. Mod. Phys. B 15, 17 (2001)]], who consider the same number of parents for all nodes. We make use of the same Boolean rule as Andrecut and Ali, provide a generalization of the formula for the probability of finding a node in state 1 at a time t, and use simulation methods to generate consecutive states of the network for both the real system and the model. The results match well. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show that the route to chaos is due to a cascade of period-doubling bifurcations which turn into reversed (period-halving) bifurcations for certain combinations of parameter values.

Biophysics↗

PubMed2004

Scalar equations for synchronous Boolean networks with biological applications.

One way of coping with the complexity of biological systems is to use the simplest possible models which are able to reproduce at least some nontrivial features of reality. Although two value Boolean models have a long history in technology, it is perhaps a little bit surprising that they can also represent important features of living organizms. In this paper, the scalar equation approach to Boolean network models is further developed and then applied to two interesting biological models. In particular, a linear reduced scalar equation is derived from a more rudimentary nonlinear scalar equation. This simpler, but higher order, two term equation gives immediate information about both cycle and transient structure of the network.

Computational Biology↗

PubMed2003

Nonlinear dynamics indicates aging affects variability during gait.

OBJECTIVE: To investigate the nature of variability present in time series generated from gait parameters of two different age groups via a nonlinear analysis. DESIGN: Measures of nonlinear dynamics were used to compare kinematic parameters between elderly and young females. BACKGROUND: Aging may lead to changes in motor variability during walking, which may explain the large incidence of falls in the elderly. METHODS: Twenty females, 10 younger (20-37 yr) and 10 older (71-79 yr) walked on a treadmill for 30 consecutive gait cycles. Time series from selected kinematic parameters of the right lower extremity were analyzed using nonlinear dynamics. The largest Lyapunov exponent and the correlation dimension of all time series, and the largest Lyapunov exponent of the original time series surrogated were calculated. Standard deviations and coefficient of variations were also calculated for selected discrete points from each gait cycle. Independent t-tests were used for statistical comparisons. RESULTS: The Lyapunov exponents were found to be significantly different from their surrogate counterparts. This indicates that the fluctuations observed in the time series may reflect deterministic processes by the neuromuscular system. The elderly exhibited significantly larger Lyapunov exponents and correlation dimensions for all parameters evaluated indicating local instability. The linear measures indicated that the elderly demonstrated significantly higher variability. CONCLUSIONS: The nonlinear analysis revealed that fluctuations in the time series of certain gait parameters are not random but display a deterministic behavior. This behavior may degrade with physiologic aging resulting in local instability. RELEVANCE: Elderly show increased local instability or inability to compensate to the natural stride-to-stride variations present during locomotion. We hypothesized that this may be the one of the reasons for the increases in falling due to aging. Future efforts should attempt to evaluate this hypothesis by making comparisons to pathological subjects (i.e. elderly fallers), and examine the sensitivity and specificity of the nonlinear methods used in this study to aid clinical assessment.

Adult↗