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Functional connectivity is a fundamental property of neural networks thatquantifies the segregation and integration of information between corticalareas. Due to mathematical complexity, a theory that could explain how theparameters of mesoscopic networks composed of a few tens of neurons affect thefunctional connectivity is still to be formulated. Yet, many interestingproblems in neuroscience involve the study of networks composed of a smallnumber of neurons. Based on a recent study of the dynamics of small neuralcircuits, we combine the analysis of local bifurcations of multi-populationneural networks of arbitrary size with the analytical calculation of thefunctional connectivity. We study the functional connectivity in differentregimes, showing that external stimuli cause the network to switch fromasynchronous states characterized by weak correlation and low variability(local chaos), to synchronous states characterized by strong correlations andwide temporal fluctuations (critical slowing down). Local chaos typicallyoccurs in large networks, but here we show that it can also be generated bystrong stimuli in small neural circuits. On the other side, critical slowingdown is expected to occur when the stimulus moves the network close to a localbifurcation. In particular, strongly positive correlations occur at thesaddle-node and Andronov-Hopf bifurcations of the network, while stronglynegative correlations occur when the network undergoes a spontaneoussymmetry-breaking at the branching-point bifurcations. These results prove thatthe functional connectivity of firing-rate network models is strongly affectedby the external stimuli even if the anatomical connections are fixed, andsuggest an effective mechanism through which biological networks candynamically modulate the encoding and integration of sensory information.

From Local Chaos to Critical Slowing Down: A Theory of the Functional Connectivity of Small Neural Circuits