On the Phase-Space Dynamics of Systems of Spiking Neurons. II: Formal Analysis

Abstract

We begin with a brief review of the abstract dynamical system that models systems of biological neurons, introduced in the original article. We then analyze the dynamics of the system. Formal analysis of local properties of flows reveals contraction, expansion, and folding in different sections of the phase-space. The criterion for the system, set up to model a typical neocortical column, to be sensitive to initial conditions is identified. Based on physiological parameters, we then deduce that periodic orbits in the region of the phase-space corresponding to normal operational conditions in the neocortex are almost surely (with probability 1) unstable, those in the region corresponding to seizure-like conditions are almost surely stable, and trajectories in the region corresponding to normal operational conditions are almost surely sensitive to initial conditions. Next, we present a procedure that isolates all basic sets, complex sets, and attractors incrementally. Based on the two sets of results, we conclude that chaotic attractors that are potentially anisotropic play a central role in the dynamics of such systems. Finally, we examine the impact of this result on the computational nature of neocortical neuronal systems.