Linear Response of General Observables in Spiking Neuronal Network Models

The activity of a neuronal network, characterized by action potentials (spikes), is constrained by the intrinsic properties of neurons and their interactions. When a neuronal network is submitted to external stimuli, the statistics of spikes changes, and it is difficult to disentangle the influence of the stimuli from the intrinsic dynamics. Using the formalism of Gibbs distributions, which are a generalization of Maximum Entropy distributions to non-stationary distributions, and generalization of Markov chains to infinite memory, we analyze this problem in a specific model (Conductance-based Integrate-and-Fire), where the neuronal dynamics depends on the history of spikes of the network. We derive a linear response formula allowing to quantify the influence of a weak amplitude external stimuli on the average value of arbitrary observables. This formula clearly disentangles the effect of the stimuli, intrinsic neuronal dynamics, and network connectivity. Upon some approximations, it reduces to a convolution, allowing to recover a standard formulation in computational neuroscience.