Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integrodifferential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. In this paper we present a review of such techniques and show how recent advances have opened the way for future studies of neural fields in both one and two dimensions that can incorporate realistic forms of axo-dendritic interactions and the slow intrinsic currents that underlie bursting behaviour in single neurons.

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

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Abstract. In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model, and a limiting case is shown to recover recent results of Zhang [Differential Integral Equations, 16 (2003), pp. 513-536]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speeds. Such fronts may be connected, and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally, we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses. Key words. traveling waves, neural networks, integral equations, Evans functionsAMS subject classification. 92C20 DOI. 10.1137/0406059531. Introduction. Traveling waves in neurobiology are receiving increased attention from experimentalists, in part due to their ability to visualize them with multielectrode recordings and imaging methods. In particular, it is possible to electrically stimulate slices of pharmacologically treated tissue taken from the cortex [19], hippocampus [31], and thalamus [28]. For cortical circuits such in vitro experiments have shown that, when stimulated appropriately, waves of excitation may occur [12,40]. Such waves are a consequence of synaptic interactions and the intrinsic behavior of local neuronal circuitry. The propagation speed of these waves is of the order cm s −1 , an order of magnitude slower than that of action potential propagation along axons. The class of computational models that are believed to support synaptic waves differ radically from classic models of waves in excitable systems. Most importantly, synaptic interactions are nonlocal (in space) and involve communication (space-dependent) delays (arising from the finite propagation velocity of an action potential) and distributed delays (arising from neurotransmitter release and dendritic processing). In many continuum models

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