A Thermodynamic Model of Mesoscale Neural Field Dynamics: Derivation and Linear Analysis

ABSTRACT Motivated by previous research suggesting that mesoscopic collective activity has the defining characteristics of a turbulent system, we postulate a thermodynamic model based on the fundamental assumption that the activity of a neuron is characterized by two distinct stages: a sub-threshold stage, described by the value of mean membrane potential, and a transitional stage, corresponding to the firing event. We therefore distinguish between two types of energy: the potential energy released during a spike, and the internal kinetic energy that triggers a spike. Formalizing these assumptions produces a system of integro-differential equations that generalizes existing models [Wilson and Cowan, 1973, Amari, 1977], with the advantage of providing explicit equations for the evolution of state variables. The linear analysis of the system shows that it supports single- or triple-point equilibria, with the refractoriness property playing a crucial role in the generation of oscillatory behavior. In single-type (excitatory) systems this derives from the natural refractory state of a neuron, producing “refractory oscillations” with periods on the order of the neuron refractory period. In dual-type systems, the inhibitory component can provide this functionality even if neuron refractory period is ignored, supporting mesoscopic-scale oscillations at much lower activity levels. Assuming that the model has any relevance for the interpretation of LFP measurements, it provides insight into mesocale dynamics. As an external forcing, theta may play a major role in modulating key parameters of the system: internal energy and excitability (refractoriness) levels, and thus in maintaining equilibrium states, and providing the increased activity necessary to sustain mesoscopic collective action. Linear analysis suggest that gamma oscillations are associated with the theta trough because it corresponds to higher levels of forced activity that decreases the stability of the equilibrium state, facilitating mesoscopic oscillations.