Self-sustaining oscillations in complex networks of excitable elements

Abstract

Random networks of symmetrically coupled, excitable elements can self-organize into coherently oscillating states if the networks contain loops (indeed loops are abundant in random networks) and if the initial conditions are sufficiently random. In the oscillating state, signals propagate in a single direction and one or a few network loops are selected as driving loops in which the excitation circulates periodically. We analyze the mechanism, describe the oscillating states, identify the pacemaker loops and explain key features of their distribution. This mechanism may play a role in epileptic seizures.PACS numbers: 89.75. Hc, 05.65.+b, 87.19.lj, 87.19.xm The coherent oscillation (CO) of a collection of units that are non-oscillatory on their own is relevant to biological and physical sciences: CO has been identified and analyzed in populations of excitable biological cells ( [7]. In contrast with well-studied synchronization phenomena of self-oscillating units [8], in these cases the ability to oscillate derives from the interactions of the elements. CO can occur also on complex networks if the nodes are excitable [9][10] or even monostable [9], provided that the network contains loops, and that the directional symmetry of couplings is somehow broken to allow a signal to propagate in a single direction around a loop [10].Networks of these types include some neural [11][12] and genetic regulatory networks.[9] Some studies of excitable networks have been inspired by target and spiral waves in continuous media. [13][7] [10] In complex networks, loops are both generic and abundant. [15][14] While short loops of length L ≪ N (where N is the network size) are rare in large random networks, ones with L ln N occur generically in numbers growing exponentially with N . Their number also grows roughly exponentially with L up to a maximum at a length L m ∼ N.[15] [14] In this letter we show that random excitable networks readily self-organize into a CO state following a transient phase during which one or a few loops are dynamically selected as driving (or pacemaker) loops. We describe the mechanism of signal propagation and gain an understanding of the resulting distributions of the oscillating states and associated driving loops.We consider networks of diffusively coupled, excitable elements with dynamics described by the Bär model [16] N is the number of nodes, u i and v i are dynamical variables, D is the coupling strength, and A ij is the adjacency matrix. a, b, and ε are parameters, for which we adopt the values a = 0.84, b = 0.07, and ε = 0.04. In the absence of coupling, the individual nodes display excitable dynamics, with a stable equilibrium at (u, v) = (0, 0) and an excitation threshold u th ≈ 0.1. The dynamical equations were integrated numerically using a fourth-order Runge-Kutta algorithm with time step ∆t = 0.1. For the topology, we chose the undirected random regular network of degree 3 (RRN3), where nodes are randomly and symmetrically connected with the constraint that all have the same degr...

Waves and Patterns on Regular Graphs

“…As long as the graph contained a cycle of 5 or more nodes without shortcuts, there seemed to be patterns possible. In a recent paper [14], McGraw and Menzinger looked at the existence of sustained activity in a random network with 200 nodes and exactly 3 connections per node. For modest strength coupling, they found that the probability of a sustained oscillation was nearly 1, but that this dramatically fell off as the coupling strength increased; in this case, initial data went to synchrony.…”

Sustained dynamics of a weakly excitable system with nonlocal interactions

“…Network connections of nonoscillatory elements, for instance, give rise to such dynamics and have been studied in various contexts, including gene networks [1], epidemic spreading dynamics [2][3][4][5], and generic excitable units [6][7][8][9][10], Excitable units undergo oscillations in many ways: Simple two excitable systems can exhibit sustained dynamics when delay-coupled [11]; spatially extended excitable media can produce sustained spiral waves by introducing a perturbation leading to the formation of a spiral core [12]; one can even consider interactions at a distance through nonlocal links embedded in spatially extended systems [13][14][15][16], which eventually forms network structures composed of wave propagation and nonlocal interactions.…”

Memory-induced mechanism for self-sustaining activity in networks

“…There is a large body of work on collective activity patterns in neuronal systems [30][31][32][33][34][35]. This activity relates to proactive functions of the brain, e.g., attention and memory [36,37].…”

“…The main mechanism for self-sustaining collective activity in such systems is based on the existence of cycles in networks, which effectively serve as feedback loops [15,38]. Such cycles, also called reverberators, are abundant even in random networks [31][32][33]. Recent research has focused on understanding how the loops contribute to sustained activity and how their embedding in a network affects the activity patterns [31][32][33].…”

“…Such cycles, also called reverberators, are abundant even in random networks [31][32][33]. Recent research has focused on understanding how the loops contribute to sustained activity and how their embedding in a network affects the activity patterns [31][32][33]. Besides neuronal networks, self-sustained collective oscillations are observed in other systems [39].…”