Chaos in high-dimensional neural and gene networks

Abstract

Neural and gene networks are often modeled by differential equations. If the continuous threshold functions in the differential equations are replaced by step functions, the equations become piecewise linear (PL equations). The flow through the state space is represented schematically by paths and directed graphs on am n-dimensional hypercube. Closed pathways, called cycles, may reflect periodic orbits with associated fixed points in a chosen Poincaré section. A return map in the Poincaré section can be constructed by the composition of fractional linear maps. The stable and unstable manifolds of the fixed points can be determined analytically. These methods allow us to analyze the dynamics in higher-dimensional networks as exemplified by a four-dimensional network that displays chaotic behavior. The three-dimensional Poincaré map is projected to a two-dimensional plane. This much simpler piecewise linear two-dimensional map conserves the important qualitative features of the flow.