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

Abstract

The prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons. We study synchronization in models of large networks of spiking neurons with random sparse connectivity. Synchrony occurs only when the average number of synapses, M that a cell receives is larger than a critical value, $M_c$. Below $M_c$, the system is in an asynchronous state. In the limit of weak coupling, assuming identical neurons, we reduce the model to a system of phase oscillators which are coupled via an effective interaction, $\Gamma$. In this framework, we develop an approximate theory for sparse networks of identical neurons to estimate $M_c$ analytically from the Fourier coefficients of $\Gamma$. Our approach relies on the assumption that the dynamics of a neuron depend mainly on the number of cells that are presynaptic to it. We apply this theory to compute $M_c$ for the integrate-and-fire (\IF) model as a function of the intrinsic neuronal properties (\eg the refractory period $T_r$), the synaptic time constants and the strength of the external stimulus, $\Iapp$. When the neurons are inhibitory, $M_c$ is found to be non-monotonous with the strength of $\Iapp$. For $T_r=0$, we estimate the minimum value of $M_c$ over all the parameters of the model to be $363.8$. Above $M_c$, the neurons tend to fire in: 1) smeared one cluster states at high firing rates and 2) smeared two or more cluster states at low firing rates. For excitatory interactions synchrony can be achieved only if the firing rate is not too high. However, our estimates of $M_c$ are, in general, much smaller than for inhibitory networks for similar level of activity. Above $M_c$ excitatory networks settle into smeared 1-cluster states.Refractoriness decreases $M_c$ at intermediate and high firing rates. These results are compared against numerical simulations. We show numerically that systems with different sizes, $N$, behave in the same way provided the connectivity, $M$, is such a way that $1/ \Meff = 1 / M - 1 / N$ remains constant when $N$ varies. This allows one to extrapolate 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 coupling is not too large. We also study the strong coupling regime of inhibitory 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. We point out that there is a fundamental limitation for the mechanisms of synchrony relying on inhibition alone, if heterogeneities in the intrinsic properties of the neurons and spatial fluctuations in the external input are also taken into account.