Coombes Stephen, Laing Carlo
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK.
Philos Trans A Math Phys Eng Sci. 2009 Mar 28;367(1891):1117-29. doi: 10.1098/rsta.2008.0256.
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.
在本文中，我们研究了两种不同的离散延迟对威尔逊 - 考恩神经网络动力学的影响。这种基于活动的模型描述了突触相互作用的兴奋性和抑制性神经元群体的动力学。我们讨论了从神经生物学角度对延迟的解释，并展示了它们如何有助于网络节律的产生。首先，我们专注于使用线性稳定性理论来展示如何使一个固定点失稳，从而导致振荡行为的出现。接下来，对于激发率选择海维赛德非线性函数的情况，我们表明这种涌现的振荡可以是同步的或反同步的，这取决于抑制或兴奋在网络结构中占主导地位。为了探究平滑（S 型）非线性激发率的行为，我们结合使用数值分岔分析和直接模拟，并发现了支持混沌行为的参数窗口。最后，我们评论了延迟在爆发性振荡产生中的作用，并讨论了本文工作的自然扩展。
