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.
Fasoli D, Cattani A, Panzeri S (2016) The Complexity of Dynamics in Small Neural Circuits. PLoS Comput Biol 12(8): e1004992. doi:10.1371/journal.pcbi.1004992