3 research outputs found
Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis
We study coarse pattern formation in a cellular automaton modelling a
spatially-extended stochastic neural network. The model, originally proposed by
Gong and Robinson [36], is known to support stationary and travelling bumps of
localised activity. We pose the model on a ring and study the existence and
stability of these patterns in various limits using a combination of analytical
and numerical techniques. In a purely deterministic version of the model, posed
on a continuum, we construct bumps and travelling waves analytically using
standard interface methods from neural fields theory. In a stochastic version
with Heaviside firing rate, we construct approximate analytical probability
mass functions associated with bumps and travelling waves. In the full
stochastic model posed on a discrete lattice, where a coarse analytic
description is unavailable, we compute patterns and their linear stability
using equation-free methods. The lifting procedure used in the coarse
time-stepper is informed by the analysis in the deterministic and stochastic
limits. In all settings, we identify the synaptic profile as a mesoscopic
variable, and the width of the corresponding activity set as a macroscopic
variable. Stationary and travelling bumps have similar meso- and macroscopic
profiles, but different microscopic structure, hence we propose lifting
operators which use microscopic motifs to disambiguate between them. We provide
numerical evidence that waves are supported by a combination of high synaptic
gain and long refractory times, while meandering bumps are elicited by short
refractory times
Firing rate and spatial correlation in a stochastic neural field model
This paper studies a stochastic neural field model that is extended from our
previous paper [14]. The neural field model consists of many heterogeneous
local populations of neurons. Rigorous results on the stochastic stability are
proved, which further imply the well-definedness of quantities including mean
firing rate and spike count correlation. Then we devote to address two main
topics: the comparison with mean-field approximations and the spatial
correlation of spike count. We showed that partial synchronization of spiking
activities is a main cause for discrepancies of mean-field approximations.
Furthermore, the spike count correlation between local populations are studied.
We find that the spike count correlation decays quickly with the distance
between corresponding local populations. Some mathematical justifications of
the mechanism of this phenomenon is also provided.Comment: second draf
Autonomous learning of nonlocal stochastic neuron dynamics
Neuronal dynamics is driven by externally imposed or internally generated
random excitations/noise, and is often described by systems of stochastic
ordinary differential equations. A solution to these equations is the joint
probability density function (PDF) of neuron states. It can be used to
calculate such information-theoretic quantities as the mutual information
between the stochastic stimulus and various internal states of the neuron
(e.g., membrane potential), as well as various spiking statistics. When random
excitations are modeled as Gaussian white noise, the joint PDF of neuron states
satisfies exactly a Fokker-Planck equation. However, most biologically
plausible noise sources are correlated (colored). In this case, the resulting
PDF equations require a closure approximation. We propose two methods for
closing such equations: a modified nonlocal large-eddy-diffusivity closure and
a data-driven closure relying on sparse regression to learn relevant features.
The closures are tested for stochastic leaky integrate-and-fire (LIF) and
FitzHugh-Nagumo (FHN) neurons driven by sine-Wiener noise. Mutual information
and total correlation between the random stimulus and the internal states of
the neuron are calculated for the FHN neuron.Comment: 26 pages, 12 figures, First author: Tyler E. Maltba, Corresponding
author: Daniel M. Tartakovsk