arXiv subjects

Lawrence M. Ward

Publications and source records attributed to Lawrence M. Ward.

4 recordsLinked to original sources

arXiv2020

Plastic systemic inhibition controls amplitude while allowing phase pattern in a stochastic neural field model

Oscillatory phase pattern formation and amplitude control for a linearized stochastic neuron field model was investigated by simulating coupled stochastic processes defined by stochastic differential equations. It was found, for several choices of parameters, that pattern formation in the phases of these processes occurred if and only if the amplitudes were allowed to grow large. Stimulated by recent work on homeostatic inhibitory plasticity, we introduced static and plastic (adaptive) systemic inhibitory mechanisms to keep the amplitudes stochastically bounded in subsequent simulations. The systems with static systemic inhibition exhibited bounded amplitudes but no sustained phase patterns, whereas the systems with plastic systemic inhibition exhibited both bounded amplitudes and sustained phase patterns. These results demonstrate that plastic inhibitory mechanisms in neural field models can stochastically control amplitudes while allowing patterns of phase synchronization to develop. Similar mechanisms of plastic systemic inhibition could play a role in regulating oscillatory functioning in the brain.

q-bio.NC↗

arXiv2019

Rapidly forming, slowly evolving, spatial patterns from quasi-cycle Mexican Hat coupling

A lattice-indexed family of stochastic processes has quasi-cycle oscillations if its otherwise-damped oscillations are sustained by noise. Such a family performs the reaction part of a discrete stochastic reaction-diffusion system when we insert a local Mexican Hat-type, difference of Gaussians, coupling on a one-dimensional and on a two-dimensional lattice. Quasi-cycles are a proposed mechanism for the production of neural oscillations, and Mexican Hat coupling is ubiquitous in the brain. Thus this combination might provide insight into the function of neural oscillations in the brain. Importantly, we study this system only in the transient case, on time intervals before saturation occurs. In one dimension, for weak coupling, we find that the phases of the coupled quasi-cycles synchronize (establish a relatively constant relationship, or phase lock) rapidly at coupling strengths lower than those required to produce spatial patterns of their amplitudes. In two dimensions the amplitude patterns form more quickly, but there remain parameter regimes in which phase synchronization patterns form without being accompanied by clear amplitude patterns. At higher coupling strengths we find patterns both of phase synchronization and of amplitude (resembling Turing patterns) corresponding to the patterns of phase synchronization. Specific properties of these patterns are controlled by the parameters of the reaction and of the Mexican Hat coupling.

q-bio.NC↗

arXiv2019

Noise Sharing and Mexican Hat Coupling in a Stochastic Neural Field

A diffusion-type coupling operator biologically significant in neuroscience is a difference of Gaussian functions (Mexican Hat operator) used as a spatial-convolution kernel. We are interested in pattern formation by \emph{stochastic} neural field equations, a class of space-time stochastic differential-integral equations using the Mexican Hat kernel. We explore, quantitatively, how the parameters that control the shape of the coupling kernel, coupling strength, and aspects of spatially-smoothed space-time noise, influence the pattern in the resulting evolving random field. We confirm that a spatial pattern that is damped in time in a deterministic system may be sustained and amplified by stochasticity. We find that spatially-smoothed noise alone causes pattern formation even without direct spatial coupling. Our analysis of the interaction between coupling and noise sharing allows us to determine parameter combinations that are optimal for the formation of spatial pattern.

q-bio.NC↗

arXiv2015

A Kuramoto coupling of quasi-cycle oscillators

A family of stochastic processes has quasi-cycle oscillations if the oscillations are sustained by noise. For such a family we define a Kuramoto-type coupling of both phase and amplitude processes. We find that synchronization, as measured by the phase-locking index, increases with coupling strength, and appears, for larger network sizes, to have a critical value, at which the network moves relatively abruptly from incoherence to complete synchonization as in Kuramoto couplings of fixed amplitude oscillators. We compare several aspects of the dynamics of unsynchronized and highly synchronized networks. Our motivation comes from synchronization in neural networks.

math.DS↗