Coherent oscillations in balanced neural networks driven by endogenous fluctuations

We present a detailed analysis of the dynamical regimes observed in a balanced network of identical quadratic integrate-and-fire neurons with sparse connectivity for homogeneous and heterogeneous in-degree distributions. Depending on the parameter values, either an asynchronous regime or periodic oscillations spontaneously emerge. Numerical simulations are compared with a mean-field model based on a self-consistent Fokker–Planck equation (FPE). The FPE reproduces quite well the asynchronous dynamics in the homogeneous case by either assuming a Poissonian or renewal distribution for the incoming spike trains. An exact self-consistent solution for the mean firing rate obtained in the limit of infinite in-degree allows identifying balanced regimes that can be either mean- or fluctuation-driven. A low-dimensional reduction of the FPE in terms of circular cumulants is also considered. Two cumulants suffice to reproduce the transition scenario observed in the network. The emergence of periodic collective oscillations is well captured both in the homogeneous and heterogeneous setups by the mean-field models upon tuning either the connectivity or the input DC current. In the heterogeneous situation, we analyze also the role of structural heterogeneity.
The balance of excitation and inhibition represents a crucial aspect of brain dynamics explaining the highly irregular fluctuations observed in several parts of the brain. The identification of macroscopic phases emerging spontaneously in balanced neural networks is particularly relevant in neuroscience since classifying them and establishing their robustness (generality) can help to understand and control brain functions. Focusing on pulse-coupled quadratic integrate-and-fire neurons, we illustrate and describe in a quantitative way the asynchronous dynamics and the emergence of collective oscillations. Our main assumption is that the spontaneous current fluctuations emerging in the network due to the sparseness of the connections can be assimilated to (white) noise whose amplitude is determined self-consistently. In this way, the dimensionality of the collective dynamics is “reduced” to that of a nonlinear Fokker–Planck equation, and quite an effective reduction to a few degrees of freedom is also implemented.