How effective delays shape oscillatory dynamics in neuronal networks

Synaptic, dendritic and single-cell kinetics generate significant time delays that shape the dynamics of large networks of spiking neurons. Previous work has shown that such effective delays can be taken into account with a rate model through the addition of an explicit, fixed delay (Roxin et al. (2005,2006) [29,30]). Here we extend this work to account for arbitrary symmetric patterns of synaptic connectivity and generic nonlinear transfer functions. Specifically, we conduct a weakly nonlinear analysis of the dynamical states arising via primary instabilities of the asynchronous state. In this way we determine analytically how the nature and stability of these states depend on the choice of transfer function and connectivity. We arrive at two general observations of physiological relevance that could not be explained in previous work. These are: 1 — fast oscillations are always supercritical for realistic transfer functions and 2 — traveling waves are preferred over standing waves given plausible patterns of local connectivity. We finally demonstrate that these results show good agreement with those obtained performing numerical simulations of a network of Hodgkin–Huxley neurons.