Synchronization of coupled different chaotic FitzHugh-Nagumo neurons with unknown parameters under communication-direction-dependent coupling

Abstract

This paper investigates the chaotic behavior and synchronization of two different coupled chaotic FitzHugh-Nagumo (FHN) neurons with unknown parameters under external electrical stimulation (EES). The coupled FHN neurons of different parameters admit unidirectional and bidirectional gap junctions in the medium between them. Dynamical properties, such as the increase in synchronization error as a consequence of the deviation of neuronal parameters for unlike neurons, the effect of difference in coupling strengths caused by the unidirectional gap junctions, and the impact of large time-delay due to separation of neurons, are studied in exploring the behavior of the coupled system. A novel integral-based nonlinear adaptive control scheme, to cope with the infeasibility of the recovery variable, for synchronization of two coupled delayed chaotic FHN neurons of different and unknown parameters under uncertain EES is derived. Further, to guarantee robust synchronization of different neurons against disturbances, the proposed control methodology is modified to achieve the uniformly ultimately bounded synchronization. The parametric estimation errors can be reduced by selecting suitable control parameters. The effectiveness of the proposed control scheme is illustrated via numerical simulations.

Figure 1

Behavior of identical FHN neurons under EES: (a) bifurcation diagram of the first neuron; (b) bifurcation diagram of the second neuron; (c) largest Lyapunov exponent for the first neuron; (d) largest Lyapunov exponent for the second neuron; (e) bifurcation diagram of the synchronization error between the coupled neurons.

Figure 2

Behavior of different FHN neurons under EES: (a) bifurcation diagram of the first neuron; (b) bifurcation diagram of the second neuron; (c) largest Lyapunov exponent for the first neuron; (d) largest Lyapunov exponent for the second neuron; (e) bifurcation diagram of the synchronization error between the coupled neurons.

Figure 3

Nonsynchronous behavior of the two different FHN neurons under EES: (a) phase portrait of the first neuron; (b) phase portrait of the second neuron; (c) phase portrait of the activation potentials for nonsynchronous behavior.

Figure 4

Effects of time-delay due to separation between the two neurons under different gap junction strengths: (a) bifurcation diagram of the synchronization error for τ = 0.001, (b) bifurcation diagram of the synchronization error for τ = 15; (c) bifurcation diagram of the synchronization error for τ = 30; (d) bifurcation diagram of the synchronization error for τ = 40.

Figure 5

Effects of the unidirectional gap junctions in a medium between the two neurons: (a) bifurcation diagram of the synchronization error for g ₂ = 1; (b) bifurcation diagram of the synchronization error for g ₂ = 0.8; (c) bifurcation diagram of the synchronization error for g ₂ = 0.5; (d) bifurcation diagram of the synchronization error for g ₂ = 0.01.

Figure 6

Synchronization of the two different coupled chaotic FHN neurons under EES by application of the proposed control scheme. The controller was applied for the time t ≥ 185: (a) phase portrait of the first neuron; (b) phase portrait of the second neuron; (c) synchronization error plot.