Population Dynamics in Networks of Izhikevich Neurons with Global Delayed Coupling

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2293-2322, September 2024.
Abstract.We investigate the collective dynamics of a network of heterogeneous Izhikevich neurons with global constant-delay coupling using a mean-field approximation, valid in the thermodynamic limit. The introduction of a biologically motivated synaptic current expression and a spike frequency adaptation mechanism give rise to significantly different bifurcation structures. Our study emphasizes the impact of heterogeneity in the quenched current, adaptation intensity, and synaptic delay on the emergence of collective oscillations. The effects of heterogeneity and adaptation vary across different scenarios but essentially result from the balance of excitatory drives, including input currents that cause neurons to spike, adaptation currents that terminate spiking, and synaptic currents that predominantly favor spiking in excitatory networks but hinder it in inhibitory cases. Our perturbation and bifurcation analysis reveal interesting transitions in the behavior in both limits of extremely weak heterogeneity and coupling strength. Finally, our analysis indicates that synaptic delays exhibit little impact on the generation of collective oscillations in weakly coupled heterogeneous networks. This effect becomes more pronounced with increasing heterogeneity. Moreover, a larger delay does not necessarily enhance the likelihood of oscillations, especially in weakly adapting neural networks. Beyond that, delays primarily function as an excitatory drive, promoting the emergence of oscillations and even inducing new macroscopic dynamics. Specifically, torus bifurcations may occur in a single population of neurons without an external drive, serving as a crucial mechanism for the emergence of population bursting with two nested frequencies.