Kinetic theory for neuronal network dynamics

Abstract

Abstract. We present a detailed theoretical framework for statistical descriptions of neuronal networks and derive (1 + 1)-dimensional kinetic equations, without introducing any new parameters, directly from conductance-based integrate-and-fire neuronal networks. We describe the details of derivation of our kinetic equation, proceeding from the simplest case of one excitatory neuron, to coupled networks of purely excitatory neurons, to coupled networks consisting of both excitatory and inhibitory neurons. The dimension reduction in our theory is achieved via novel moment closures. We also describe the limiting forms of our kinetic theory in various limits, such as the limit of mean-driven dynamics and the limit of infinitely fast conductances. We establish accuracy of our kinetic theory by comparing its prediction with the full simulations of the original point-neuron networks. We emphasize that our kinetic theory is dynamically accurate, i.e., it captures very well the instantaneous statistical properties of neuronal networks under time-inhomogeneous inputs.Key words. Visual cortex, coarse-grain, fluctuation, diffusion, correlation.AMS subject classifications. 82C31, 92B99, 92C20 IntroductionMathematically, the vast hierarchy of the multiple spatial and temporal scales in the cortical dynamics presents a significant challenge to computational neuroscience. While we may devise ever more efficient numerical methods for simulations of dynamics of large-scale neuronal networks [1, 2, 3, 4, 5], basic computational constraints will eventually limit our simulation power. In order to "scale-up" computational models to sufficiently large regions of the cortex to capture interesting cortical processing (such as optical illusions related to "contour completion"), and to gain qualitative understanding of the cortical mechanisms underlying this level of cortical processing, a major theoretical issue is how to derive effective dynamics under a reduced representation of large-scale neuronal networks. Therefore, a theoretical challenge is to develop efficient and effective representations for simulating and understanding the dynamics of larger, multi-layered networks. As suggested, for example, by the laminar structure of cat's or monkey's primary visual cortex, in which many cellular properties such as orientation preference are arranged in regular patterns or maps across the cortex [6,7,8,9,10], some neuronal sub-populations may be effectively represented by coarse-grained substitutes. Thus, we may partition the two-dimensional cortical layers into coarse-grained patches, each sufficiently large to contain many neurons, yet sufficiently small that these regular response properties of the individual neurons within each patch are approximately the same for each neuron in the patch. These regular response properties are then treated as constants throughout each coarse-grained patch.

The role of fluctuations in coarse-grained descriptions of neuronal networks

“…Note also that the Boltzmann Equation (2.3) is not exact, due to the summed input from all the neurons in the network being only approximately Poisson, and thus the term (2.3c) is valid only in an asymptotic sense. A more mathematically rigorous derivation of the Boltzmann Equation (2.3) is given in the appendix of [31].…”

“…A typical coarse-graining procedure shares much in common with the derivation of the hydrodynamic equations from the Boltzmann kinetic theory of molecular dynamics [14,15]. It replaces groups of neurons, so small that the neurons contained in them share similar properties and are uniformly connected yet sufficiently large that a statistical description is applicable, by neuronal tissue patches whose dynamics reflects the average neuronal response in the patch [31,32,95]. This type of coarse graining also naturally follows from the laminar structure of the brain and the plethora of feature-preference maps [11,12,16,39,44,60,72,103,116,129,138,142].…”

“…For example, a simple Fokker-Planck-type coarsegrained model captures a bifurcation between stable, fluctuation-driven, and bistable, mean-driven, network operating regimes, whose understanding has been suggested to be of critical importance in properly tuning a network modeling orientation selectivity in the primary visual cortex [31,65]. A yet more basic model has been able to account for neuronal activity patterns in the primary visual cortex during drug-induced visual hallucinations [17,18].…”

“…Fluctuations of synaptic inputs, neuronal conductances, and voltages play an important role in the functioning of neuronal networks, from determining individual neurons' spike times and firing rates to possibly being involved in the neural code [6,29,31,32,65,70,86,97,99,108,117,127,130,139,140]. Although sufficiently strong stimuli drive a neuronal network into the mean-driven operating regime, in which its dynamics are determined primarily by the mean drive arising from the stimulus and the network response, while the fluctuations in the spike trains arriving at the individual neurons can be neglected, many networks in the brain are believed to function in the fluctuation-driven regime, in which synaptic-input fluctuations are the primary cause of neuronal firings [6,70,86,117,127,139,140].…”

Synchrony and Asynchrony for Neuronal Dynamics Defined on Complex Networks

“…This hybrid system is thus a mean-field model for the neuronal network. Mean-field models for the dynamics of populations of neurons have been studied extensively (e.g., [31,[86][87][88][89][90][91][92]) and typically lead to deterministic equations for an idealized "infinite number of neurons" limit. The fact that K is kept finite in the large N limit, i.e.…”