How well do reduced models capture the dynamics in models of interacting neurons?

Abstract

This paper introduces a class of stochastic models of interacting neurons with emergent dynamics similar to those seen in local cortical populations. Rigorous results on existence and uniqueness of nonequilibrium steady states are proved. These network models are then compared to very simple reduced models driven by the same mean excitatory and inhibitory currents. Discrepancies in firing rates between network and reduced models are investigated and explained by correlations in spiking, or partial synchronization, working in concert with "nonlinearities" in the time evolution of membrane potentials. The use of simple random walks and their first passage times to simulate fluctuations in neuronal membrane potentials and interspike times is also considered.

Model Reduction Captures Stochastic Gamma Oscillations on Low-Dimensional Manifolds

“…We start with the Markovian integrate-and-fire model (MIF) first proposed in Li et al ( 2019 ), and hereafter referred to as the “full model.” As an analogy of the conventional IF model (see Methods), the MIF brings us two additional conveniences: First, the discretized states of Markovian dynamics make theoretical analysis easier as the probability flow from one state to another is now straightforward; Second, the Markov properties of the MIF enable the computation of the invariant measure of gamma oscillations directly from the probability transition matrix.…”

“…Recently, Young and collaborators have examined the dynamical properties of gamma oscillations in a large-scale neuronal network model of monkey V1 (Chariker et al, 2018 ). To further theoretical understanding, in Li et al ( 2019 ),introduced a relatively tractable stochastic model of interacting neuronal populations designed to capture the essential network features underlying gamma dynamics. Through numerical simulations and analysis of three dynamical regimes (“homogeneous,” “regular,” and “synchronized”), they identified how conductance properties (essentially, how long after each spike the synaptic interactions are fully felt) can regulate the emergence of gamma frequency synchronization.…”

“…Here, we present a sequence of model reductions of the spiking network models based on Li et al ( 2019 ). We first present in detail our methods on a small, homogeneous network of 100 neurons (75 excitatory and 25 inhibitory), exhibiting gamma frequency oscillations.…”

“…We first present in detail our methods on a small, homogeneous network of 100 neurons (75 excitatory and 25 inhibitory), exhibiting gamma frequency oscillations. Inspired by Li et al ( 2019 ) and Li and Xu ( 2019 ), to achieve dimensional reduction, we assume that the spiking activities during gamma oscillations and their temporally organization are mainly governed by one simple variable, namely, which sub-threshold neurons are only a few postsynaptic spikes from firing. Thus, in terms of dynamical dimensions, the number of network states is drastically reduced (although still too large for any meaningful analytical work, since 2 n is astronomical even for n = 100).…”

A strategy for mapping biophysical to abstract neuronal network models applied to primary visual cortex

“…Refractority . The consideration of refractority is one the most significant factors that lead to discrepancies between discrete neuron networks and most mean-field approaches [ 50 ]. Discrete network models explicitly describe each neuron in the network, and thus all their states over time.…”