From Local Chaos to Critical Slowing Down: A Theory of the Functional   Connectivity of Small Neural Circuits
  Functional connectivity is a fundamental property of neural networks that quantifies the segregation and integration of information between cortical areas. Due to mathematical complexity, a theory that could explain how the parameters of mesoscopic networks composed of a few tens of neurons affect the functional connectivity is still to be formulated. Yet, many interesting problems in neuroscience involve the study of networks composed of a small number of neurons. Based on a recent study of the dynamics of small neural circuits, we combine the analysis of local bifurcations of multi-population neural networks of arbitrary size with the analytical calculation of the functional connectivity. We study the functional connectivity in different regimes, showing that external stimuli cause the network to switch from asynchronous states characterized by weak correlation and low variability (local chaos), to synchronous states characterized by strong correlations and wide temporal fluctuations (critical slowing down). Local chaos typically occurs in large networks, but here we show that it can also be generated by strong stimuli in small neural circuits. On the other side, critical slowing down is expected to occur when the stimulus moves the network close to a local bifurcation. In particular, strongly positive correlations occur at the saddle-node and Andronov-Hopf bifurcations of the network, while strongly negative correlations occur when the network undergoes a spontaneous symmetry-breaking at the branching-point bifurcations. These results prove that the functional connectivity of firing-rate network models is strongly affected by the external stimuli even if the anatomical connections are fixed, and suggest an effective mechanism through which biological networks can dynamically modulate the encoding and integration of sensory information. 
Diego Fasoli, Anna Cattani, Stefano Panzeri
q-bio.NC math.DS