On a Gaussian neuronal field model

Can we understand the dynamic behaviour of leaky integrate-and-fire (LIF) networks, which present the major, and possibly the only, analytically tractable tool we employ in computational neuroscience? To answer this question, here we present a theoretical framework on the spike activities of LIF networks by including the first order moment (mean firing rate) and the second order moment statistics (variance and correlation), based on a moment neuronal network (MNN) approach. The spike activity of a LIF network is approximated as a Gaussian random field and can reduce to the classical Wilson-Cowan-Amari (WCA) neural field if the variances vanish. Our analyses reveal several interesting phenomena of LIF networks. With a small clamped correlation and strong inhibition, the firing rate response function could be non-monotonic (not sigmoidal type), which can lead to interesting dynamics. For a feedforward and recurrent neuronal network, our setup allows us to prove that all neuronal spike activities rapidly synchronize, a well-known fact observed in both experiments and numerical simulations. We also present several examples of wave propagations in this field model. Finally, we test our MNN with the content-dependent working memory setting. The potential application of this random neuronal field idea to account for many experimental data is also discussed.