Study of perturbations of an oscillating neuronal network via phase-amplitude response functions

Phase reduction is a powerful tool for understanding the behavior of perturbed oscillators. It allows for the description of high-dimensional oscillatory systems in terms of a single variable, the phase. Alternatively, mean-field models are a viable option to make the analysis of large systems more tractable. In this work, we apply a phase-amplitude technique on a mean-field model for a network of quadratic integrate-and-fire neurons, which is exact in the thermodynamic limit. This methodology allows us to compute the global isochrons and isostables of the system, and a generalization of the phase response curve beyond the limit cycle constraint: the phase and amplitude response functions. We compare the perturbed dynamics of the oscillating mean-field system with its N-dimensional counterpart, which also exhibits synchronized spiking, and observe how the response functions are able to predict accurately the evolution of the network. Moreover, since the model exhibits slow-fast dynamics, the method yields a dimensionality reduction restricted to the slow stable manifold of the system