Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities

Abstract

Abstract. In this paper, we consider instabilities of localized solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilize spatially localized solutions. For a scalar model with Heaviside firing rate function, we calculate symmetric one-bump and ring solutions explicitly and use an Evans function approach to predict the point of instability and the shapes of the dominant growing modes. Our predictions are shown to be in excellent agreement with direct numerical simulations. Moreover, beyond the instability our simulations demonstrate the emergence of multi-bump and labyrinthine patterns. With the addition of spike-frequency adaptation, numerical simulations of the resulting vector model show that it is possible for structures without rotational symmetry, and in particular multi-bumps, to undergo an instability to a rotating wave. We use a general argument, valid for smooth firing rate functions, to establish the conditions necessary to generate such a rotational instability. Numerical continuation of the rotating wave is used to quantify the emergent angular velocity as a bifurcation parameter is varied. Wave stability is found via the numerical evaluation of an associated eigenvalue problem.

Two-Dimensional Bumps in Piecewise Smooth Neural Fields with Synaptic Depression

“…There have been few studies regarding the existence and stability of standing bumps in 2D neural fields [9,15,25,29]. Laing and Troy [25] were the first to introduce partial differential equation (PDE) methods to study symmetry-breaking of rotationally symmetric bumps.…”

“…Laing and Troy [25] were the first to introduce partial differential equation (PDE) methods to study symmetry-breaking of rotationally symmetric bumps. Since then, PDE methods have been used to study the formation of multiple bump solutions, Turing patterns, traveling spots, and labyrinthine patterns in 2D neural fields with local negative feedback [9,29]. In addition, standard stability analysis of stimulus-driven 2D neural fields with local linear negative feedback revealed symmetry-breaking breathers [15].…”

“…Substituting (3.3) back into (3.2) yields We can calculate the double integral in (3.5) using the Hankel transform and Bessel function identities, as in [14,29]. Thus, we find that…”

“…For the sake of illustration consider a Mexican hat weight distribution given by a combination of modified Bessel functions of the second kind [14,26,29]:…”

“…Moreover, following previous studies of 2D neural field models [20,25,26,29,35], we can transform system (3.1) into a fourthorder PDE, which is computationally less expensive to evaluate. Using the fact that the corresponding Hankel transform of…”