The number of synaptic inputs and the synchrony of large, sparse neuronal networks

The prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchroni ze large populations of neurons. We study synchronization in models of larg e networks of spiking neurons with random sparse connectivity. Synchrony oc curs only when the average number of synapses, M, that a cell receives is l arger than a critical value, M-c. Below M-c, the system is in an asynchrono us state. In the limit of weak coupling, assuming identical, neurons, we re duce the model to a system of phase oscillators that are coupled via an eff ective interaction, Gamma. In this framework, we develop an approximate the ory for sparse networks of identical neurons to estimate M-c analytically f rom the Fourier coefficients of Gamma. Our approach relies on the assumptio n that the dynamics of a neuron depend mainly on the number of cells that a re presynaptic to it. We apply this theory to compute M-c for a model of in hibitory networks of integrate-and-fire (I&F) neurons as a function of the intrinsic neuronal properties (e.g., the refractory period T-r), the synapt ic time constants, and the strength of the external stimulus, I-ext. The nu mber M-c is found to be nonmonotonous with the strength of I-ext. For T-r = 0, we estimate the minimum value of M-c over all the parameters of the mod el to be 363.8. Above M-c, the neurons tend to fire in smeared one-cluster states at high firing rates and smeared two-or-more-cluster states at low f iring rates. Refractoriness decreases M-c at intermediate and high firing r ates. These results are compared to numerical simulations. We show numerica lly that systems with different sizes, N, behave in the same way provided t he connectivity, M, is such that 1/M-eff = 1/M - 1/N remains constant when N varies. This allows extrapolating the large N behavior of a network from numerical simulations of networks of relatively small sizes (N = 800 in our case). We find that our theory predicts with remarkable accuracy the value of M-c and the patterns of synchrony above M-c, provided the synaptic coup ling is not too large. We also study the strong coupling regime of inhibito ry sparse networks. All of our simulations demonstrate that increasing the coupling strength reduces the level of synchrony of the neuronal activity. Above a critical coupling strength, the network activity is asynchronous. W e point out a fundamental limitation for the mechanisms of synchrony relyin g on inhibition alone, if heterogeneities in the intrinsic properties of th e neurons and spatial fluctuations in the external input are also taken int o account.