Asynchronous Chaos and Bifurcations in a Model of Two Coupled Identical Hindmarsh – Rose Neurons

Abstract

We study a minimal network of two coupled neurons described by the Hindmarsh – Rose model with a linear coupling. We suppose that individual neurons are identical and study whether the dynamical regimes of a single neuron would be stable synchronous regimes in the model of two coupled neurons. We find that among synchronous regimes only regular periodic spiking and quiescence are stable in a certain range of parameters, while no bursting synchronous regimes are stable. Moreover, we show that there are no stable synchronous chaotic regimes in the parameter range considered. On the other hand, we find a wide range of parameters in which a stable asynchronous chaotic regime exists. Furthermore, we identify narrow regions of bistability, when synchronous and asynchronous regimes coexist. However, the asynchronous attractor is monostable in a wide range of parameters. We demonstrate that the onset of the asynchronous chaotic attractor occurs according to the Afraimovich – Shilnikov scenario. We have observed various asynchronous firing patterns: irregular quasi-periodic and chaotic spiking, both regular and chaotic bursting regimes, in which the number of spikes per burst varied greatly depending on the control parameter.

Excitation of a Group of Two Hindmarsh – Rose Neurons with a Neuron-Generated Signal

“…2b). In this entire interval of currents the slave system is bistable: the stable equilibrium coexists with a quasi-periodic attractor [22]. The mean interspike intervals of the quasi-periodic regime vary with I from T q 1 ≈ 9.05 at I = 5.40 to T q 2 ≈ 6.85 at I = 6.20.…”

“…The other parameters have the following fixed values: a = 1, b = 3, c = 1, d = 5, s = 4 and x 0 = − 8 5 [14]. In the slave system we have chosen the coupling strengths between the neurons to be equal: D 23 = D 32 = 0.1, the same as in [22]. The strength of the unidirectional connection D 12 is one of the control parameters.…”

“…However, since only the second neuron is connected to the first one, this symmetry is broken when D 12 = 0. Now we briefly describe possible types of dynamics in this subsystem, following [22]. In the area of low external currents, I < 1.2895, there exists a stable equilibrium point for which x 2 = x 3 , y 2 = y 3 , z 2 = z 3 .…”

“…Notice that for I > 1.2760 there also exists an asynchronous attractor (with windows of periodicity), corresponding to chaotic bursting modes [22], making the slave system bistable in the interval 1.2760 < I < I AH . In this bistability region it is possible to force the slave system to transit from the stable equilibrium over the separatrix into the basin of another attractor with an input signal of a finite length.…”

“…There have been a number of works dedicated to studying models of biological neurons, neuronal networks and the Hindmarsh -Rose model in particular (see, e. g., [12][13][14][15][16][17][18][19][20][21] and references therein). Dynamics and firing patterns of two coupled Hindmarsh -Rose neurons can be significantly different from those of one neuron [22]. We are interested in excitation of the group of two neurons in a stable quiescent state by different firing patterns typical for biological neurons.…”