1,081 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 (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
Multiscale motion and deformation of bumps in stochastic neural fields with dynamic connectivity
The distinct timescales of synaptic plasticity and neural activity dynamics
play an important role in the brain's learning and memory systems.
Activity-dependent plasticity reshapes neural circuit architecture, determining
spontaneous and stimulus-encoding spatiotemporal patterns of neural activity.
Neural activity bumps maintain short term memories of continuous parameter
values, emerging in spatially-organized models with short term excitation and
long-range inhibition. Previously, we demonstrated nonlinear Langevin equations
derived using an interface method accurately describe the dynamics of bumps in
continuum neural fields with separate excitatory/inhibitory populations. Here
we extend this analysis to incorporate effects of slow short term plasticity
that modifies connectivity described by an integral kernel. Linear stability
analysis adapted to these piecewise smooth models with Heaviside firing rates
further indicate how plasticity shapes bumps' local dynamics. Facilitation
(depression), which strengthens (weakens) synaptic connectivity originating
from active neurons, tends to increase (decrease) stability of bumps when
acting on excitatory synapses. The relationship is inverted when plasticity
acts on inhibitory synapses. Multiscale approximations of the stochastic
dynamics of bumps perturbed by weak noise reveal the plasticity variables
evolve to slowly diffusing and blurred versions of that arising in the
stationary solution. Nonlinear Langevin equations associated with bump
positions or interfaces coupled to slowly evolving projections of plasticity
variables accurately describe the wandering of bumps underpinned by these
smoothed synaptic efficacy profiles.Comment: 19 pages, 11 figure
Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise
We consider reaction-diffusion equations that are stochastically forced by a
small multiplicative noise term. We show that spectrally stable travelling wave
solutions to the deterministic system retain their orbital stability if the
amplitude of the noise is sufficiently small.
By applying a stochastic phase-shift together with a time-transform, we
obtain a semilinear sPDE that describes the fluctuations from the primary wave.
We subsequently develop a semigroup approach to handle the nonlinear stability
question in a fashion that is closely related to modern deterministic methods
- …