Steady States, Limit Cycles, and Chaos in Models of Complex Biological Networks

Theoretical models of neural and genetic control networks consisting of N interacting elements are cast in the form of an N-dimensional system of piecewise linear (PL) ordinary differential equations. The state transition diagram of this system, which represents transitions between distinct volumes in the N-dimensional phase space, is given as a directed graph on an N-dimensional hypercube in which each edge has only one orientation. Analytical results establish steady-states and limit cycles in these systems, and numerical results have identified chaotic dynamics for N ≥ 6.