Neural Activity Measures and Their Dynamics

We provide an asymptotically justified derivation of activity measure evolution equations (AMEE) for a finite size neural network. The approach takes into account the dynamics for each isolated neuron in the network being modeled by a biophysical model, i.e. Hodgkin-Huxley equations or their reductions. By representing the interacting network as self and pairwise interactions, we propose a general definition of spatial projections of the network, called activity measures, that quantify the activity of a network. We show that the evolution equations that govern the dynamics of the activity measure shadow the activity measure of the network (i.e. the two quantities stay close to each other for all times) for general interactions and various asymptotic dynamics. The AMEE effectively serve as a dimensionality reduction technique for the complex network when spatial synchrony and coherence are present and allow to a priori predict network dynamics that would not be guessed from individual neuron behavior. To demonstrate an explicit derivation of such a reduction, we consider the mean measure for a network of interacting FitzHugh-Nagumo neurons. Computational results comparing the full network dynamics with the mean AMEE model of identical and nonidentical FitzHugh-Nagumo and FitzHugh-Rinzel neurons validate the shadowing theorems and expose the various resulting AMEE models that allow to describe the mean of the network.