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

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.

A master stability function approach to cardiac alternans

“…To further unravel the complexity of microscopic Ca 2+ alternans, we here compute the master stability function (MSF) for the network (Pecora and Carroll 1998). This approach has been instrumental in understanding instabilities of synchronous network states and has recently been generalised to more structured network dynamics such as cluster states and to the case of nearly-identical oscillators (Lai et al 2018;Coombes and Thul 2016;Ladenbauer et al 2013;Sun et al 2009;Pecora et al 2000). A key input for the computation of the MSF is the synchronous network state and its Jacobian.…”

The Salted Kalman Filter: Kalman filtering on hybrid dynamical systems

“…The saltation matrix is a valuable tool for analysis and control in a wide variety of fields such as general bifurcations theory [26][27][28][29][30], power circuits [22,23,[31][32][33][34][35][36][37][38][39], rigid body systems [40][41][42][43][44][45], chemical processing [46], and hybrid neuron models [24,25,[47][48][49][50]. Fig.…”

“…Fig. 2: The saltation matrix has been used in many different fields, including the control of legged robots in tasks such as (a) a quadrupedal backflip [14] and (b) robust bipdeal walking [21]; the analysis and control of power circuits such as (c) supervising control of a buck converter [22] and (d) the bifurcation behavior of DC drives [23]; and the modelling of neural activity in the brain such as (e) the stability analysis of a Wilson-Cowan neural mass model [24] and (f) the modelling of synaptic filter behavior [25].…”

Advances in nonlinear dynamics of complex networks: adaptivity, stochasticity, and delays

“…A method to recover the network connectivity from the observed dynamics was suggested [19]. New tools for the analysis of the synchronous states in networks with nonsmoothness and discontinuity were presented [20]. The conditions for the asymptotic periodicity in networks of degrade-and-fire oscillators were obtained [21].…”