Clusters in nonsmooth oscillator networks

Abstract

For coupled oscillator networks with Laplacian coupling, the master stability function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory, this approach has recently been extended to treat more general cluster states. However, the MSF and its generalizations require the determination of a set of Floquet multipliers from variational equations obtained by linearization around a periodic orbit. Since closed form solutions for periodic orbits are invariably hard to come by, the framework is often explored using numerical techniques. Here, we show that further insight into network dynamics can be obtained by focusing on piecewise linear (PWL) oscillator models. Not only do these allow for the explicit construction of periodic orbits, their variational analysis can also be explicitly performed. The price for adopting such nonsmooth systems is that many of the notions from smooth dynamical systems, and in particular linear stability, need to be modified to take into account possible jumps in the components of Jacobians. This is naturally accommodated with the use of saltation matrices. By augmenting the variational approach for studying smooth dynamical systems with such matrices we show that, for a wide variety of networks that have been used as models of biological systems, cluster states can be explicitly investigated. By way of illustration, we analyze an integrate-and-fire network model with event-driven synaptic coupling as well as a diffusively coupled network built from planar PWL nodes, including a reduction of the popular Morris-Lecar neuron model. We use these examples to emphasize that the stability of network cluster states can depend as much on the choice of single node dynamics as it does on the form of network structural connectivity. Importantly, the procedure that we present here, for understanding cluster synchronization in networks, is valid for a wide variety of systems in biology, physics, and engineering that can be described by PWL oscillators.

Recent advances in coupled oscillator theory

“…While the synchronous solution (resp., antiphase) solutions are always stable (resp., unstable) bistability emerges through a saddle node bifurcation as ρ is increased. This bifurcation cannot be observed in the more well-established O( ) accurate reduction strategy (27). Panel C shows H 4 (Υ) which determines the order accurate behavior of the isostable coordinates in (26).…”

“…After numerically computing the required functions z(θ), i(θ), b(θ), and c(θ) as well as terms related to the synaptic coupling from (22) using methods described in [39], we compute each h i and subsequent H i function. In panels A and B of Figure 5, the thick black line shows H 1 (Υ) − H 1 (−Υ), the first order accurate coupling function from (27). Note that the shape of this function has no dependence on the coupling strength, ρ, and changing ρ will not alter stable fixed points of (27).…”

“…In panels A and B of Figure 5, the thick black line shows H 1 (Υ) − H 1 (−Υ), the first order accurate coupling function from (27). Note that the shape of this function has no dependence on the coupling strength, ρ, and changing ρ will not alter stable fixed points of (27). Panel C shows H 4 (Ψ), which influences the order dynamics of the isostable coordinates.…”

“…They were able to compute the adjoint and iPRC of the planar IF model due to its explicitly solvable nature. Coombes et al extended the master stability function to the case of coupled piecewise linear oscillators in 2016 [9], a result which as been further extended to include integrate-and-fire models with state and time-dependent interactions [27].…”

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

“…, N . Equations (27) and (28) capture the jump dependence of the perturbed system, and also allow us to construct the vector z ( T m pi ) that appears in (25). Using the above we may now write (25) in the succinct form…”