Spatially coherent oscillations in neural fields with inhibition and adaptation. I. One-dimensional domains.

Abstract

We study Hopf bifurcation of stationary activity bumps to localized, spatially coherent oscillations in a family of elementary neural field models involving nonlocal synaptic excitation and inhibition with Heaviside firing rate nonlinearity and local linear adaption, both with and without a localized input inhomogeneity, on the one-dimensional spatial domain (-∞,∞), including cases of interacting pairs of neural fields. (We treat the same neural fields on two-dimensional spatial domain R^{2} separately.) A main focus is to categorize how the underlying symmetries of the nonlinear operators in this family of equations give rise to a related set of spatially coherent time-periodic solutions that bifurcate via Hopf bifurcation with respect to different spatial eigenmodes, each with different spatial structures being selected as a result of the relative balance of synaptic inhibition to excitation. A general framework is constructed to analyze stationary bump solutions in a neural field model containing N neural fields with M linear gating variables that modulate different neural fields. Under a basic set of symmetry assumptions on the synaptic weight functions and the input homogeneity, we show that all such neural fields have two broad classes of eigenmodes with either even or odd spatial symmetry. When these eigenmodes destabilize via Hopf bifurcation, it leads to various types of spatially coherent, time-periodic oscillations that can take the form of breathing bumps or breathers that expand and contract and sloshing bumps or sloshers which move side-to-side. Analytical treatments combined with numerical simulations provide a more complete picture of the emergence of these periodic activity patterns and novel secondary bifurcations are found to occur, including torus, period-doubled, and Rossler band-like dynamics. Interacting pairs of neural fields that support bumps, breathers, and sloshers can lead to novel spatially coherent oscillations with different patterns of synchrony and spatial positioning depending on the type of synaptic interactions between the neural fields including novel in-phase and antiphase breathers and sloshers. A novel transition from a slosher to a spatially localized, traveling periodic wave is also found. The approach is extended to the case of multibump solutions and bifurcations leading to various multibump breathers and sloshers are observed.