Dynamics of compartmental model recurrent neural networks

We show how the dynamics of a recurrent network of compartmental model neurons can be formulated in terms of a set of coupled nonlinear Volterra integrodifferential equations in which the state of each neuron is represented by a single scalar variable. The associated convolution kernels are determined by the single neuron membrane potential response function, which can be calculated explicitly for arbitrary dendritic tree topologies. Our integral equation approach provides a compact and analytically tractable method for studying the effects of dendritic structure on network dynamics. We illustrate this by deriving conditions for the onset of oscillations in a compartmental model network that depend both on the interneuron connection weights and on the internal dendritic structure of the individual neurons. It is shown that it is possible for a symmetrically connected compartmental model network to oscillate.