CHAOS IN HIGH-DIMENSIONAL NEURAL AND GENE NETWORKS

Neural and gene networks are often modeled by differential equations. If the continuous threshold functions in the differential equations ar e replaced by step functions, the equations become piecewise linear (P L equations). The flow through the state space is represented schemati cally by paths and directed graphs on an n-dimensional hypercube. Clos ed pathways, called cycles, may reflect periodic orbits with associate d fixed points in a chosen Poincare section. A return map in the Poinc are section can be constructed by the composition of fractional linear maps. The stable and unstable manifolds of the fixed points can be de termined analytically. These methods allow us to analyze the dynamics in higher-dimensional networks as exemplified by a four-dimensional ne twork that displays chaotic behavior. The three-dimensional Poincare m ap is projected to a two-dimensional plane. This much simpler piecewis e linear two-dimensional map conserves the important qualitative featu res of the flow.