GABAergic interneurons play a major role in the emergence of various types of synchronous oscillatory patterns of activity in the central nervous system. Motivated by these experimental facts, modeling studies have investigated mechanisms for the emergence of coherent activity in networks of inhibitory neurons. However, most of these studies have focused either when the noise in the network is absent or weak or in the opposite situation when it is strong. Hence, a full picture of how noise affects the dynamics of such systems is still lacking. The aim of this letter is to provide a more comprehensive understanding of the mechanisms by which the asynchronous states in large, fully connected networks of inhibitory neurons are destabilized as a function of the noise level. Three types of single neuron models are considered: the leaky integrate-and-fire (LIF) model, the exponential integrate-and-fire (EIF), model and conductance-based models involving sodium and potassium Hodgkin-Huxley (HH) currents. We show that in all models, the instabilities of the asynchronous state can be classified in two classes. The first one consists of clustering instabilities, which exist in a restricted range of noise. These instabilities lead to synchronous patterns in which the population of neurons is broken into clusters of synchronously firing neurons. The irregularity of the firing patterns of the neurons is weak. The second class of instabilities, termed oscillatory firing rate instabilities, exists at any value of noise. They lead to cluster state at low noise. As the noise is increased, the instability occurs at larger coupling, and the pattern of firing that emerges becomes more irregular. In the regime of high noise and strong coupling, these instabilities lead to stochastic oscillations in which neurons fire in an approximately Poisson way with a common instantaneous probability of firing that oscillates in time.

Recent experimental results have shown that GABAergic interneurons in the central nervous system are frequently connected via electrical synapses. Hence, depending on the area or the subpopulation, interneurons interact via inhibitory synapses or electrical synapses alone or via both types of interactions. The theoretical work presented here addresses the significance of these different modes of interactions for the interneuron networks dynamics. We consider the simplest system in which this issue can be investigated in models or in experiments: a pair of neurons, interacting via electrical synapses, inhibitory synapses, or both, and activated by the injection of a noisy external current. Assuming that the couplings and the noise are weak, we derive an analytical expression relating the cross-correlation (CC) of the activity of the two neurons to the phase response function of the neurons. When electrical and inhibitory interactions are not too strong, they combine their effect in a linear manner. In this regime, the effect of electrical and inhibitory interactions when combined can be deduced knowing the effects of each of the interactions separately. As a consequence, depending on intrinsic neuronal proper-ties, electrical and inhibitory synapses may cooperate, both promoting synchrony, or may compete, with one promoting synchrony while the other impedes it. In contrast, for sufficiently strong couplings, the two types of synapses combine in a nonlinear fashion. Remarkably, we find that in this regime, combining electrical synapses with inhibition ampli-fies synchrony, whereas electrical synapses alone would desynchronize the activity of the neurons. We apply our theory to predict how the shape of the CC of two neurons changes as a function of ionic channel conduc-tances, focusing on the effect of persistent sodium conductance, of the firing rate of the neurons and the nature and the strength of their interac-tions. These predictions may be tested using dynamic clamp techniques.

Population rate models provide powerful tools for investigating the principles that underlie the cooperative function of large neuronal systems. However, biophysical interpretations of these models have been ambiguous. Hence, their applicability to real neuronal systems and their experimental validation have been severely limited. In this work, we show that conductance-based models of large cortical neuronal networks can be described by simplified rate models, provided that the network state does not possess a high degree of synchrony. We first derive a precise mapping between the parameters of the rate equations and those of the conductance-based network models for time-independent inputs. This mapping is based on the assumption that the effect of increasing the cell's input conductance on its f-I curve is mainly subtractive. This assumption is confirmed by a single compartment Hodgkin-Huxley type model with a transient potassium A-current. This approach is applied to the study of a network model of a hypercolumn in primary visual cortex. We also explore extensions of the rate model to the dynamic domain by studying the firing-rate response of our conductance-based neuron to time-dependent noisy inputs. We show that the dynamics of this response can be approximated by a time-dependent second-order differential equation. This phenomenological single-cell rate model is used to calculate the response of a conductance-based network to time-dependent inputs.