Analysis of networks where discontinuities and nonsmooth dynamics collide: understanding synchrony

Integrate-and-fire networks have proven remarkably useful in modelling the dynamics of real world phenomena ranging from earthquakes, to synchrony in neural networks, to cascading activity in social networks. The reset process means that such models are inherently discontinuous. Moreover, for jump interactions, which are a common choice for many physical systems, the models are also nonsmooth. For synchronous network states these processes can occur simultaneously, and care must be taken with the mathematical analysis of solution stability. This leads to an ordering problem, that has no counterpart in smoothly coupled limit cycle systems. Here we develop a set of network saltation matrices that can be used with an appropriate ordering to determine the instability of a synchronous network state. Moreover, we show that smoothed versions of jump interactions do not capture the behaviour of the nonsmooth model. Synchrony in the smoothed model with reset is analysed using a generalised master stability function (MSF), and the eigenspectra for smooth and nonsmooth interactions are compared. We find that the one determined by the MSF organises that found from the analysis of the nonsmooth model, though the latter has further eigenvalues that can destabilise the synchronous state.