Abstract
In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These nonlocal models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynamic (oscillatory) Turing instability of the homogeneous steady state. In this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability. The appropriate amplitude equations are found to be the coupled mean-field Ginzburg–Landau equations describing a Turing–Hopf bifurcation with modulation group velocity of O ( 1 ) . Importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio-temporal kernels defining the neural field equation of interest. Indeed our results cover not only models with axonal or dendritic delays but also those which are described by a more general distribution of delayed spatio-temporal interactions. We illustrate the predictive power of this form of analysis with comparison against direct numerical simulations, paying particular attention to the competition between standing and travelling waves and the onset of Benjamin–Feir instabilities.
Highlights
The ability of neural field models to exhibit complex spatio-temporal dynamics has been studied intensively since their introduction by Wilson and Cowan [1]
The latter observation is consistent with the failure of models without axonal delay to demonstrate dynamic pattern formation when the connectivity is of inverted wizard hat type
In this paper we have studied pattern formation in a broad class of neural field models
Summary
The ability of neural field models to exhibit complex spatio-temporal dynamics has been studied intensively since their introduction by Wilson and Cowan [1]. Delays arising from the processing of incoming synaptic inputs by passive dendritic trees may be incorporated into neural field models, as in the work of Bressloff [17] In both cases it is known that these space-dependent delays can lead to a dynamic Turing instability of a homogeneous steady state. These were first found in neural field models by Bressloff [17] for dendritic delays and more recently by Hutt et al [18] for axonal delays.
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