Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis

Authors

Avitabile, D. and Wedgwood, K.C.

Abstract

We study coarse pattern formation in a cellular automaton modelling a spatially-extended stochastic neural network. The model, originally proposed by Gong and Robinson (Phys Rev E 85(5):055,101(R), 2012), 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 field 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 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.

DOI arXiv PDF

BibTeX

@article{avitabile2017macroscopic,
  author  = {Avitabile, Daniele and Wedgwood, Kyle CA},
  title   = {Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis},
  journal = {Journal of Mathematical Biology},
  year    = {2017},
  volume  = {75},
  number  = {4},
  pages   = {885--928},
  doi     = {10.1007/s00285-016-1070-9}
}