Neural Activity Measures and Their Dynamics

We provide an asymptotically justified derivation of activity measure evolution equations (AMEEs) 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 AMEEs effectively serve as a dimensionality reduction technique for the complex network when spatial synchrony and coherence are present and allow us to a priori predict network dynamics that would not be guessed from individual neuron behav...