Transient dynamics and multistability in two electrically interacting FitzHugh–Nagumo neurons

Abstract

We analyze the existence of chaotic and regular dynamics, transient chaos phenomenon, and multistability in the parameter space of two electrically interacting FitzHugh–Nagumo (FHN) neurons. By using extensive numerical experiments to investigate the particular organization between periodic and chaotic domains in the parameter space, we obtained three important findings: (i) there are self-organized generic stable periodic structures along specific directions immersed in a chaotic portion of the parameter space; (ii) the existence of transient chaos phenomenon is responsible for long chaotic temporal evolution preceding the asymptotic (periodic) dynamics for particular parametric combinations in the parameter space; and (iii) the existence of various multistable domains in the parameter space with an arbitrary number of attractors. Additionally, we also prove through numerical simulations that chaos, transient chaos, and multistability prevail even for different coupling strengths between identical FHN neurons. It is possible to find multistable attractors in the phase and parameter spaces and to steer them apart by increasing the asymmetry in the coupling force between neurons. Such a strategy can be essential to experimental matters, as setting the right parameter ranges. As the FHN model shares the crucial properties presented by the more realistic Hodgkin–Huxley-like neurons, our results can be extended to high-dimensional coupled neuron models.

Multiple solutions and transient chaos in a nonlinear flexible coupling model

“…The particular solutions are characterized by the period in the response expressed as the multiplicity of the excitation period. Similar coexistence can be obtained from multistable systems; however, in the case period one solutions are multiplied naturally [30,31]. In the case when the system is affected by external excitation with a frequency of ω = 0.677, there are three coexisting solutions, one 1 T-periodic and two 2 T-periodic ones.…”

“…This phenomenon was observed in electric circuits [26], standard nonlinear dynamical models such as the logistic mapping [27], Duffing equation [28] and during experimental tests of the pendulum model [29]. In general, the term "transient chaos" is used when describing a chaotic response, which after some time becomes a periodic or quasi-periodic response [30,31]. Intervals of transient chaos are observed in systems with two or more solutions characterized by basins of attraction with a fractal border (the so-called Mielnikow's chaos [32][33][34]).…”

The Impedance of Spiking Neurons Coupled by Time‐Delayed Interaction

“…The analysis of coupled delayed differential equations has been amply studied for the analysis of the stability of neural networks. [45][46][47] An approach to a system of mutually restrained neurons can be established by using a set of elementary neuron dynamical equations as (1-2) and introducing a coupling function C, as follows [17,19,23,[48][49][50][51][52]…”

“…The analysis of coupled delayed differential equations has been amply studied for the analysis of the stability of neural networks. [ 45–47 ] An approach to a system of mutually restrained neurons can be established by using a set of elementary neuron dynamical equations as (1–2) and introducing a coupling function C , as follows [ 17,19,23,48–52 ] whereis a coupling function with a delay time . This system of equations allows us to study not only coupled neurons but also the coupling of neuronal subensembles that operate synchronically.…”

“…The dynamical features resulting from the interaction of the paradigmatic FitzHugh-Nagumo (FHN) [13,14] neurons, which is the topic of this paper, have been extensively studied. [12,[15][16][17][18][19] With the introduction of the time delay interaction the coupled FHN neurons display a transition from the original disordered motions to periodic ones, which is accompanied by complex bifurcation properties. [20][21][22][23][24][25] Impedance spectroscopy (IS) is a multipurpose characterization technique widely used in materials science and electrochemistry.…”