  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.
