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

Abstract

<jats:title>ABSTRACT</jats:title><jats:p>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.</jats:p>

An investigation into the nonlinear coupling between CA1 layers and the dentate gyrus.

“…As a global oscillation engaging multiple brain regions simultaneously, the theta rhythmicity in the hippocampus is likely expressed as afferent synaptic forcing distributed across the hippocampus. As discussed in Qin et al (2019), and consistent with Berg, Willumsen, and Lindén (2019) and Gutierrez, O’Leary, and Marder (2013), the activity increase owing to the increase of theta power provides a higher background firing rate, increases the local internal energy of the neural network, and facilitates the collective action oscillations and waves in the gamma frequency range. The transfer of energy across scales from the macroscale theta forcing through the mesoscale gamma range to microscale dissipative processes governing individual neuron physics is consistent with a “turbulent energy cascade” (Sheremet et al, 2019).…”

“…Sorting hippocampal LFP spectra by rat speed (Mc-Naughton, Barnes, & O'Keefe, 1983) produces a well ordered evolution of power distribution across scales, which may be interpreted as a process of internal redistribution of energy in a forced multiscale system under increased power input (Sheremet et al, 2016(Sheremet et al, , 2018(Sheremet et al, , 2019. Moreover, theoretical modeling of mesoscopic collective activity (Qin et al, 2019, extending the work of Amari, 1975, 1977Cowan, Neuman, & van Drongelen, 2016;Wilson & Cowan, 1972, supports the observations that collective activity is weakly dissipative (i.e., persistent enough to exhibit "traveling wave" properties; Lubenov & Siapas, 2009;Muller, This document is copyrighted by the American Psychological Association or one of its allied publishers.…”

“…Here, we examine the nonlinear coupling between gamma and theta both within and across hippocampal layers. The opportunity for this investigation is brought about by the recently proposed turbulence hypothesis for mesoscopic (gamma) brain activity (Qin, Maurer, & Sheremet, 2019; Sheremet, Qin, Kennedy, Zhou, & Maurer, 2019), and its implications about the relationship between mesoscopic and global oscillations in the hippocampus.…”

“…Spectral analysis of mesoscopic activity (Qin et al, 2019; Sheremet, Burke, & Maurer, 2016; Sheremet et al, 2019; Zhou et al, 2019) shows that gamma processes are weakly nonlinear and stochastic. Sorting hippocampal LFP spectra by rat speed (McNaughton, Barnes, & O’Keefe, 1983) produces a well ordered evolution of power distribution across scales, which may be interpreted as a process of internal redistribution of energy in a forced multiscale system under increased power input (Sheremet et al, 2016, 2018, 2019).…”

“…The spectral flux decreases significantly in the neighborhood of ( f ≃ 120 Hz), where the physical laws governing neural activity transition from mesoscopic collective activity (gamma) to individual neuron activity (microscopic). The latter is dominated by dissipation, hence the response to increase in forcing is significantly lower (e.g., Qin et al, 2019). The accumulation of power in the gamma band could be explained by a lower efficiency of cross-scale energy fluxes at the crossover point in the vicinity of 120 Hz, which produces a spectral flux bottleneck (Meyers & Meneveau, 2008; Nazarenko, 2011; Proment, Nazarenko, & Onorato, 2009).…”