Abstract
In this paper, we study the effect of two distinct discrete delays on the dynamics of a Wilson-Cowan neural network. This activity-based model describes the dynamics of synaptically interacting excitatory and inhibitory neuronal populations. We discuss the interpretation of the delays in the language of neurobiology and show how they can contribute to the generation of network rhythms. First, we focus on the use of linear stability theory to show how to destabilize a fixed point, leading to the onset of oscillatory behaviour. Next, we show for the choice of a Heaviside nonlinearity for the firing rate that such emergent oscillations can be either synchronous or anti-synchronous, depending on whether inhibition or excitation dominates the network architecture. To probe the behaviour of smooth (sigmoidal) nonlinear firing rates, we use a mixture of numerical bifurcation analysis and direct simulations, and uncover parameter windows that support chaotic behaviour. Finally, we comment on the role of delays in the generation of bursting oscillations, and discuss natural extensions of the work in this paper.
Highlights
Delays arise naturally in models of neurobiological systems
We focus on the dynamics of two-population neural models with the incorporation of two discrete delays
The periodic and chaotic behaviours of the type seen above are of great interest in neural systems, as are ‘bursting’ oscillations (Coombes & Bressloff 2005)
Summary
Delays arise naturally in models of neurobiological systems. For example, the finite speed of an action potential (AP) propagating along an axon means that spike signalling between neurons depends upon how far apart they are. For an excellent review of the role of time delays in neural systems, we refer the reader to the paper by Campbell (2007). The effects of such delays can be quite. We will work with the well-known Wilson & Cowan (1972) model Such activity-based models are expected to provide a caricature of the behaviour of more realistic spiking networks when the time scale of synaptic processing is much longer than the membrane time constant of a typical cell (Ermentrout 1998). This paper will show how to analyse a delayed neural network with a hybrid approach, combining linear stability theory, the construction of periodic orbits (for piecewise constant nonlinear firing rate functions) and numerical techniques
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