Brain Dynamics, Chaos and Bessel Functions

Abstract

By resorting to Freeman's observations showing that the distribution functions of impulse responses of cortex to sensory stimuli resemble Bessel functions, we study brain dynamics by considering the equivalence of spherical Bessel equation, in a given parametrization, to two oscillator equations, one damped and one amplified oscillator. The study of such a couple of equations, which are at the basis of the formulation of the dissipative many-body model, reveals the structure of the root loci of poles and zeros of solutions of Bessel equations, which are consistent with results obtained using ordinary differential equation techniques. We analyze stable and unstable limit cycles and consider thermodynamic features of brain functioning, which in this way may be described in terms of transitions between chaotic gas-like and ordered liquid-like behaviors. Nonlinearity dominates the dynamical critical transition regimes. Linear behavior, on the other hand, characterizes superpositions within self-organized neuronal domains in each dynamical phase. The formalism is consistent with the observed coexistence in circular causality of pulse density fields and wave density fields.