Dynamics of Neuronal Populations: Eigenfunction Theory, Part 1, Some Solvable Cases

A novel approach to cortical modeling was introduced by Knight et al. (1996). In their presentation cortical dynamics is formulated in terms of interacting populations of neurons, a perspective that is in part motivated by modern cortical imaging (For a review see Sirovich and Kaplan (2002)). The approach may be regarded as the application of statistical mechanics to neuronal populations, and the simplest exemplar bears a kinship to the Boltzmann equation of kinetic theory. The disarming simplicity of this linear equation hides the deeply complex behavior it produces. A purpose of this paper is to investigate and reveal its intricacies by treating a series of solvable special cases. In particular we will focus on issues that relate to the spectral analysis of the underlying operators. A fairly thorough treatment is presented for a simple, but not trivial example, which has important consequences for more general situations.