The role of resonator neuron in the dynamics of two coupled integrator and resonator neurons of different types of excitability

Abstract

In this manuscript, a silent resonator neuron is coupled with a spiking integrator neuron through the gap junction, when the coupled neurons are of different types of excitability and none of the coupled neurons exhibit mixed mode oscillations and bursting oscillations intrinsically. By using dynamical systems theory (e.g. the bifurcation theory), all the observed oscillation patterns and the transition mechanisms between them are investigated, when one of the coupling strengths is fixed and the other is varied. It is noticeable that, there is an interval in the parameter space, for the parameter values within which the coupled system is multi-stable. This multistability corresponds to the coexistence of mixed mode oscillations, bursting oscillations and subthreshold oscillations of the resonator neuron. In addition, some interval in the parameter space is introduced such that, for the values of the coupling strength within which the resonator neuron is in tonic spiking mode, while for the values of the coupling strength outside which the resonator neuron exhibits subthreshold oscillations. It is also verified that the final synchronization of the coupled neurons actually corresponds to the synchronization of tonic spiking oscillations of the integrator neuron and one-bursting oscillations of the resonator neuron.

Figures (23)

• Figure 1: The fixed parameters values of the system (\ref{['1']}).
• Figure 2: $(1),(2)$. Bifurcation diagram of the system (\ref{['1']}) for two different values $m'_{1/2}$ when $I$ is the bifurcation parameter. Here, the color red indicates the stable equilibrium, the color green indicates the stable limit cycle and the blue color indicates the unstable limit cycle of the system (\ref{['1']}). (1) For $m'_{1/2}=-30$, by increasing I, stable equilibrium of the system undergoes Saddle-Node bifurcation on Invariant Circle ($SNIC$ bifurcation), then a stable limit cycle, $W_1$, appears. (2) For $m'_{1/2}=-45$, stable equilibrium of the system undergoes Subcritical Hopf bifurcation, hence it becomes unstable. Then the state of the system tends to the stable limit cycle $W_2$. (3),(4). The frequency-current relation. (3) For $m'_{1/2}=-30$, spiking frequency can be arbitrarily low depending on the strength of the applied current, i.e. the system shows type $I$ excitability. (4) For $m'_{1/2}=-45$, the spiking frequency is in a certain positive band depending on the strength of the applied current, i.e. the system shows type $II$ excitability. The fixed parameters values of the system (\ref{['1']}) have been listed in the Figure .
• Figure 3: When $q_1 = q_2 = 0$ the coupled system \ref{['2']} is bistable, a stable limit cycle and a stable two-dimensional torus. (1),(2). Three-dimensional image, $(V_2,V_1,n_2)$, and the corresponding voltage time series of $"I"$ and $"II"$ for $(1)$ the stable limit cycle and $(2)$ the stable two-dimensional torus.
• Figure 4: The fixed parameters values of the system \ref{['2']}.
• Figure 5: A. Three-dimensional image, $(V_1,V_2,n_2)$, of the bistability, i.e. co-existance of a stable limit cycle and a stable two-dimensional torus for $q_{1} =0.05$ and $q_{2}=0.04$. B. The dynamics on the smooth torus. (1) periodic dynamics for $q_2=0.082$ and (2) quasi-periodic dynamics for $q_2=0.03$.

Theorems & Definitions (30)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 3.1
• Proposition 3.2
• proof
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Proposition 3.6