Towards a mathematical definition of functional connectivity

Functional connectivity is a neurobiological notion, informally stating that there would be a strong dependence between neurons and that this dependence might be useful in understanding the way the brain encodes stimuli, programs actions, etc. However, in practice such strong dependencies are often reconstructed via Hawkes processes based on an amazingly small number of neurons, because of the very scarce observation of this very complex and huge network. We derive new simple equations, which explain how the ideal Hawkes reconstruction is linked to the covariance between the observed neurons. These equations help us in particular to understand what the Hawkes reconstruction does in two settings, synchronization and classical point process asymptotics. Moreover they might help us to also understand what is qualitatively happening at the scale of the huge unobserved network, paving the path for a possible mathematical definition of functional connectivity.