A Minimal Network of Brain Dynamics: Hierarchy of Approximations to Quasi-critical Neural Network Dynamics

We present an interacting branching model of neural network dynamics, incorporating key biological features such as inhibition with several types of inhibitory interactions. We establish a hierarchy of analytical mean-field approximations to the model, which characterizes nonequilibrium phase transitions between disorder and ordered phases, and perform a stability analysis. Generically, inhibitory neurons increase the stability of the model dynamics. The model is consistent with the quasi-criticality hypothesis in that it displays regions of maximal dynamical susceptibility and maximal mutual information predicated on the strength of the external stimuli. Directed percolation emerges as the universality class of the critical transition of the model, consistent with some previous experimental data and models. In the unstable phase, chaotic dynamics emerge, which may be linked to the occurrence of epileptic seizures.