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Towards a neuronal field theory

Wenlian Lu1, Xuejuan Zhang1, Enrico Rossoni2 and Jianfeng Feng2
1Centre for Computational Systems Biology, Fudan University, Shanghai, P.R. China
2Department of Computer Science and Mathematics, Warwick University, Coventry CV4 7AL, UK

Abstract: Can we understand the dynamical behaviour of leaky integrate-and-fire (LIF) networks, which is the major, and possibly the only analytically tractable tool we employ in computational neuroscience? To answer the question, here we present a novel theoretical framework on LIF networks by including the first order moment (mean firing rate) and the second order moment (variance and correlation). The activity of LIF network is equivalent to a Gaussian random field and the classical Amari-Cowan-Wilson neural field is a special, but unrealistic case. Our analyses reveal many interesting phenomena of LIF networks. With a small clamped correlation, the network will finally become silent when the inhibitory input increases; but the network could enter into a chaotic region when the correlation is stronger and in this case the input-output firing rate is no longer a sigmoidal function. For a feed-forward spiking network, our setup allows us to prove that all neuronal activities rapidly synchronize, a well-known fact in the literature observed in numerical simulations. Finally, we test our MNN with the associative memory setting and, surprisingly, it is found that in the region where the inhibitory input is stronger than the excitatory input, the network optimally retrieves the stored memories.