Abstract
It is hypothesized that cortical neuronal circuits operate in a global balanced state, i.e., the majority of neurons fire irregularly by receiving balanced inputs of excitation and inhibition. Meanwhile, it has been observed in experiments that sensory information is often sparsely encoded by only a small set of firing neurons, while neurons in the rest of the network are silent. The phenomenon of sparse coding challenges the hypothesis of a global balanced state in the brain. To reconcile this, here we address the issue of whether a balanced state can exist in a small number of firing neurons by taking account of the heterogeneity of network structure such as scale-free and small-world networks. We propose necessary conditions and show that, under these conditions, for sparsely but strongly connected heterogeneous networks with various types of single-neuron dynamics, despite the fact that the whole network receives external inputs, there is a small active subnetwork (active core) inherently embedded within it. The neurons in this active core have relatively high firing rates while the neurons in the rest of the network are quiescent. Surprisingly, although the whole network is heterogeneous and unbalanced, the active core possesses a balanced state and its connectivity structure is close to a homogeneous Erdös-Rényi network. The dynamics of the active core can be well-predicted using the Fokker-Planck equation. Our results suggest that the balanced state may be maintained by a small group of spiking neurons embedded in a large heterogeneous network in the brain. The existence of the small active core reconciles the balanced state and the sparse coding, and also provides a potential dynamical scenario underlying sparse coding in neuronal networks.