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 (QIF) neurons with a sparse connectivity for homogeneous and heterogeneous in-degree distribution. 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.

Splay states and two-cluster states in ensembles of excitable units

Focusing on systems of sinusoidally coupled active rotators, we study the emergence and stability of periodic collective oscillations for systems of identical excitable units with repulsive all-to-all interaction. Special attention is put on splay states and two-cluster states. Recently, it has been shown that one-parameter families of such systems, containing the parameter values at which the Watanabe–Strogatz integrability takes place, feature an instantaneous non-local exchange of stability between splay and two-cluster states. Here, we illustrate how in the extended families that circumvent the Watanabe–Strogatz dynamics, this abrupt transition is replaced by the “gradual transfer” of stability between the 2-cluster and the splay states, mediated by mixed-type solutions. We conclude our work by recovering the same kind of dynamics and transfer of stability in an ensemble of voltage-coupled Morris–Lecar neurons.