Abstract
Neuronal network models often assume a fixed probability of connection between neurons. This assumption leads to random networks with binomial in-degree and out-degree distributions which are relatively narrow. Here I study the effect of broad degree distributions on network dynamics by interpolating between a binomial and a truncated power-law distribution for the in-degree and out-degree independently. This is done both for an inhibitory network (I network) as well as for the recurrent excitatory connections in a network of excitatory and inhibitory neurons (EI network). In both cases increasing the width of the in-degree distribution affects the global state of the network by driving transitions between asynchronous behavior and oscillations. This effect is reproduced in a simplified rate model which includes the heterogeneity in neuronal input due to the in-degree of cells. On the other hand, broadening the out-degree distribution is shown to increase the fraction of common inputs to pairs of neurons. This leads to increases in the amplitude of the cross-correlation (CC) of synaptic currents. In the case of the I network, despite strong oscillatory CCs in the currents, CCs of the membrane potential are low due to filtering and reset effects, leading to very weak CCs of the spike-count. In the asynchronous regime of the EI network, broadening the out-degree increases the amplitude of CCs in the recurrent excitatory currents, while CC of the total current is essentially unaffected as are pairwise spiking correlations. This is due to a dynamic balance between excitatory and inhibitory synaptic currents. In the oscillatory regime, changes in the out-degree can have a large effect on spiking correlations and even on the qualitative dynamical state of the network.
Highlights
Network models of randomly connected spiking neurons have provided insight into the dynamics of real neuronal circuits
I have conducted numerical simulations of two canonical networks as a function of the in-degree and out-degree distributions of the network connectivity. For both the purely inhibitory (I), as well as the excitatory– inhibitory (EI) networks, it was the in-degree which most strongly affected the global, dynamical state of the network. In both cases, increasing the variance of the in-degree drove a transition in the dynamical state: in the I network oscillations were abolished while in the EI network, oscillations were generated when the E-to-E in-degree was broadened
In a standard random network with identical neurons, the gain of the network in the spontaneous state can be expressed as the slope of the nonlinear transfer function which converts the total input to neurons into an output, e.g., a firing rate
Summary
Network models of randomly connected spiking neurons have provided insight into the dynamics of real neuronal circuits. Networks operating in a balanced state in which large excitatory and inhibitory inputs cancel in the mean, can self-consistently and robustly account for the low, irregular discharge of neurons seen in vivo (van Vreeswijk and Sompolinsky, 1996, 1998; Amit and Brunel, 1997b; Brunel, 2000) Such network models can explain the skewed, long-tailed firing rate distributions observed in vivo (Amit and Brunel, 1997a) as well as the elevated, irregular spiking activity seen during the delay period in a working memory task in monkeys (Barbieri and Brunel, 2007). The modulation of ongoing oscillations in these networks to time-varying external stimuli has been shown to agree well with local field potential recordings in monkey visual cortex (Mazzoni et al, 2008) Despite their relative simplicity, network models of randomly connected spiking neurons can reproduce an array of non-trivial, experimentally observed measures of neuronal dynamics. How robust are these results to changes in the network connectivity?
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