Long-range interaction effects on coupled excitable nodes: traveling waves and chimera state

Abstract

In this paper, analytical and numerical studies of the influence of the long-range interaction parameter on the excitability threshold in a ring of FitzHugh-Nagumo (FHN) system are investigated. The long-range interaction is introduced to the network to model regulation of the Gap junctions or hemichannels activity at the connexins level, which provides links between pre-synaptic and post-synaptic neurons. Results show that the long-range coupling enhances the range of the threshold parameter. We also investigate the long-range effects on the network dynamics, which induces enlargement of the oscillatory zone before the excitable regime. When considering bidirectional coupling, the long-range interaction induces traveling patterns such as traveling waves, while when considering unidirectional coupling, the long-range interaction induces multi-chimera states. We also studied the difference between the dynamics of coupled oscillators and coupled excitable neurons. We found that, for the coupled system, the oscillation period decreases with the increasing of the coupling parameter. For the same values of the coupling parameter, the oscillation period of the Oscillatory dynamics is greater than the oscillation period of the excitable dynamics. The analytical approximation shows good agreement with the numerical results.

Keywords: Electrical synapse; Excitable behavior; Long-range interaction; Multi-chimera state; Traveling waves.

Figure 1

Influence of the distance between neurons |k − j| on the long-range coupling σ for σ₀ = 1. The long-range coupling decreases with the distance between neurons.

Figure 2

Variation of the excitability threshold a, when σ₀ = 5, (a) with r, for R = 10 (black dashed line), R = 15 (blue dashed line), R = 20 (red dashed line), R = 65 (green dashed line), (b) with R for r = 0.8.

Figure 3

Variation of the excitability threshold a, (a) with r for R = 10 and σ₀ = 5, (b) with σ₀ for R = 10 and r = 0.8.

Figure 4

Influence of the long-range coupling on the appearing of oscillations at the excitable state when a = 1.27, (a) for r = 0.1 or for r = 0.98, (b) for r = 0.72, (c) for r = 0.8, (d) for r = 0.85.

Figure 5

(a) Phase portrait (trajectories in red line and nullclines in blue dashed line) of a single FitzHugh-Nagumo system and (b) the corresponding time series of activator x(t) for a = 0.1.

Figure 6

(a) Phase portrait (trajectories in red line and nullclines in blue dashed line) of coupled oscillatory FitzHugh-Nagumo neurons and (b) the corresponding time series of activator x_j(t) for a = .35.

Figure 7

(a) Phase portrait (trajectories in red line and nullclines in blue dashed line) of coupled excitable FitzHugh-Nagumo neurons and (b) the corresponding time series of activator x_j(t) for a = 1.35.

Figure 8

Snapshot of coupled excitable FitzHugh-Nagumo neurons for a = 1.35.

Figure 9

Analytical approximations of the period for an FHN system. (a) Oscillatory dynamics for a = 0.35 (blue line), (b) excitable dynamics for a = 1.35 (red line).

Figure 10

Analytical approximations of the period of excitable coupled FHN in term of the long range parameter r for a = 1.35.

Figure 11

Oscillation period of excitable coupled FHN in term of the coupling parameter for a = 1.35. Period obtained from numerical simulations (black circles), period obtained from analytical approximation of Eq. (16) (solid red line), period obtained from analytical approximation of Eq. (18) with the parameter e = 0.22 (solid black line).

Figure 12

Traveling wave by increasing the long-range r. (a) For r = 0.53 and (b) for r = 0.54.

Figure 13

Traveling patterns generated by increasing the long-range r. Panels present the space-time plot of the node variables x_j and their corresponding snapshot. (a) For r = 0.55, (b) for r = 0.56.

Figure 14

Traveling action potential for r = 0.7. (a) The space-time plot of the node variables x_j. (b) The times series of membrane potential x_j.

Figure 15

Multi-chimera states. (a) For r = 0.53. (b) for r = 0.54.