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

In this paper we consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised 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

Target Patterns in a 2-D Array of Oscillators with Nonlocal Coupling

We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction-diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when $\varepsilon$, the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in $\varepsilon$. The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in $\varepsilon$ does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be "correct" to all orders in $\varepsilon$. We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution, which obtained using matched asymptotics.Comment: 39 pages, 1 figur

Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression

We study the spatiotemporal dynamics of a two-dimensional excitatory neuronal network with synaptic depression. Coupling between populations of neurons is taken to be nonlocal, while depression is taken to be local and presynaptic. We show that the network supports a wide range of spatially structured oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. The particular form of the oscillations depends on initial conditions and the level of background noise. Given an initial, spatially localized stimulus, activity evolves to a spatially localized oscillating core that periodically emits target waves. Low levels of noise can spontaneously generate several pockets of oscillatory activity that interact via their target patterns. Periodic activity in space can also organize into spiral waves, provided that there is some source of rotational symmetry breaking due to external stimuli or noise. In the high gain limit, no oscillatory behavior exists, but a transient stimulus can lead to a single, outward propagating target wave

Reproducing Kernel Space Method for the Solution of Linear Fredholm Integro-Differential Equations and Analysis of Stability

We present a numerical method to solve the linear Fredholm integro-differential equation in reproducing kernel space. A simple algorithm is given to obtain the approximate solutions of the equation. Through the comparison of approximate and true solution, we can find that the method can effectively solve the linear Fredholm integro-differential equation. At the same time the numerical solution of the equation is stable