Abstract
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.
Highlights
The European Physical Journal Special Topics outside of neuroscience, and typically to model systems with a threshold such as stickslip models of earthquakes [5], and social network models where a new behaviour is enacted by an individual once threshold is reached [6]
A set of techniques developed for the study of impact oscillators, such as those described by Muller to calculate Lyapunov exponents [7], allow for a natural extension of many of the techniques from smooth dynamical systems theory with the aid of saltation operators
There is no analogous master stability function (MSF) style approach for this case, here we show that an appropriate linear stability analysis can be developed by applying network saltation operations according to the order in which perturbed state variables at the node level cross threshold
Summary
The European Physical Journal Special Topics outside of neuroscience, and typically to model systems with a threshold such as stickslip models of earthquakes [5], and social network models where a new behaviour is enacted by an individual once threshold is reached [6]. We first consider smooth synapses and show the master stability function (MSF) approach for smooth systems can be augmented to generate a spectral problem for the stability of the synchronous state This spectrum is defined by the zeros of a complex function, and this function is succinctly expressed in terms of the IF node parameters and the Fourier transform of the synaptic filter. A smooth synaptic filter with a steep rise time might be thought to give qualitatively similar network behaviour as for one with a jump and decay, that this is not the case In this limit we find that the spectra from the smooth analysis coincides with that from the nonsmooth one, yet the latter has further eigenvalues which can lead to different stability properties.
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