The Complexity of Dynamics in Small Neural Circuits

Mean-field approximations are a powerful tool for studying large neural networks. However, they do not describe well the behavior of networks composed of a small number of neurons. In this case, major differences between the mean-field approximation and the real behavior of the network can arise. Yet, many interesting problems in neuroscience involve the study of mesoscopic networks composed of a few tens of neurons. Nonetheless, mathematical methods that correctly describe networks of small size are still rare, and this prevents us to make progress in understanding neural dynamics at these intermediate scales. Here we develop a novel systematic analysis of the dynamics of arbitrarily small networks composed of homogeneous populations of excitatory and inhibitory firing-rate neurons. We study the local bifurcations of their neural activity with an approach that is largely analytically tractable, and we numerically determine the global bifurcations. We find that for strong inhibition these networks give rise to very complex dynamics, caused by the formation of multiple branching solutions of the neural dynamics equations that emerge through spontaneous symmetry-breaking. This qualitative change of the neural dynamics is a finite-size effect of the network, that reveals qualitative and previously unexplored differences between mesoscopic cortical circuits and their mean-field approximation. The most important consequence of spontaneous symmetry-breaking is the ability of mesoscopic networks to regulate their degree of functional heterogeneity, which is thought to help reducing the detrimental effect of noise correlations on cortical information processing.Author Summary: The mesoscopic level of brain organization, describing the organization and dynamics of small circuits of neurons including from few tens to few thousands, has recently received considerable experimental attention. It is useful for describing small neural systems of invertebrates, and in mammalian neural systems it is often seen as a middle ground that is fundamental to link single neuron activity to complex functions and behavior. However, and somewhat counter-intuitively, the behavior of neural networks of small and intermediate size can be much more difficult to study mathematically than that of large networks, and appropriate mathematical methods to study the dynamics of such networks have not been developed yet. Here we consider a model of a network of firing-rate neurons with arbitrary finite size, and we study its local bifurcations using an analytical approach. This analysis, complemented by numerical studies for both the local and global bifurcations, shows the emergence of strong and previously unexplored finite-size effects that are particularly hard to detect in large networks. This study advances the tools available for the comprehension of finite-size neural circuits, going beyond the insights provided by the mean-field approximation and the current techniques for the quantification of finite-size effects.