Integro-differential equations and the stability of neural networks with dendritic structure.

We analyse the effects of dendritic structure on the stability of a recurrent neural network in terms of a set of coupled, non-linear Volterra integro-differential equations. These, which describe the dynamics of the somatic membrane potentials, are obtained by eliminating the dendritic potentials from the underlying compartmental model or cable equations. We then derive conditions for Turing-like instability as a precursor for pattern formation in a spatially organized network. These conditions depend on the spatial distribution of axo-dendritic connections across the network.