Controlling switching between birhythmic states in a new conductance-based bursting neuronal model

Abstract

A birhythmic conductance-based neuronal model with fast and slow variables is proposed to generate and control the coexistence of two different attracting modes in amplitudes and frequencies. However, periodic bursting, chaotic spiking and bursting have not been clearly observed there. The control of bistability is investigated in a three-dimensional birhythmic conductance-based neuronal model. We consider slow processes in neuron materialized by an adaptation variable coupled to system in the presence of an external sinusoidal current applied. By using the harmonic balance method, we obtain the frequency–response curve in which membrane potential resonance with his corresponding frequency is controlled by varying a specific parameter. At the resonance frequency, bifurcation and Lyapunov exponent diagrams versus a control parameter are obtained. They reveal a coexistence of two different complex attractors, namely periodic and chaotic spiking, periodic and chaotic bursting. By using the control parameter as the slow variable, the system can switch from bistable to monostable behavior. This is done by destroying subthreshold (small) oscillation of the neuron. The role of adaptation variable in neuron system is then to filter many existing electrical processes and permit to adapt the system by the multiple transitions states in the chosen electrical mode. A fairly good agreement is observed between analytical and numerical results.

Appendix A

$$\begin{aligned} a= & {} -g_L-0.5g_{Na}+0.5g_{Na}\\&\dfrac{\beta _m+E_{Na}}{\gamma _m}-g_{Na}\dfrac{\beta _m^3}{6\gamma _m^3}\\&- 0.5g_{Na}E_{Na}\dfrac{\beta _m^2}{\gamma _m^3}+g_{Na}\\&\dfrac{\beta _m^5}{15\gamma _m^5} \\&+g_{Na}E_{a}\dfrac{\beta _m^4}{3\gamma _m^5}-8.5g_{Na}\\&\dfrac{\beta _m^7}{315\gamma _m^7}-59.5g_{Na}E_{Na}\dfrac{\beta _m^6}{315\gamma _m^7}\\ b= & {} 0.5g_{Na}\dfrac{1}{\gamma _m^3}+0.5g_{Na}E_{Na}\dfrac{1}{3\gamma _m^3}-2g_{Na}\\&\dfrac{\beta _m^3}{3\gamma _m^5}- 2E_{Na}g_{Na}\dfrac{\beta _m^2}{3\gamma _m^5}\\&+59.5g_{Na}\dfrac{\beta _m^5}{105\gamma _m^7}\\&+59.5g_{Na}E_{Na}\dfrac{\beta _m^4}{63\gamma _m^7}\\ \theta _1= & {} -g_{Na}\dfrac{\beta _m}{3\gamma _m^3}+g_{Na}E_{Na}\\&\dfrac{1}{15\gamma _m^5}-59.5g_{Na}\dfrac{\beta _m^3}{63\gamma _m^7}- 59.5g_{Na}E_{Na}\dfrac{\beta _m^2}{63\gamma _m^7},\\ \theta _2= & {} 59.5g_{Na}\dfrac{\beta _m}{315\gamma _m^7}\\&+8.5g_{Na}E_{Na}\dfrac{1}{315\gamma _m^7},\\ B= & {} -g_kE_k,\\ \alpha= & {} -g_aE_a ,\\ \gamma _1= & {} \dfrac{1}{6\gamma _w}-\dfrac{\beta _w}{24\gamma _w^2}\\&-\dfrac{5\beta _w^2}{16\gamma _w^3}+\dfrac{11\beta _w^4}{144\gamma _w^5}- \dfrac{47\beta _w^6}{1080\gamma _w^7}-\dfrac{17\beta _w^8}{1680\gamma _w^9},\\ \gamma _2= & {} \dfrac{1}{48\gamma _w^2} +\dfrac{5\beta _w}{16\gamma _w^3}-\dfrac{11\beta _w^3}{72\gamma _w^5}+\dfrac{329\beta _w^5}{2520\gamma _w^7}\\&+\dfrac{204\beta _w^7}{5040\gamma _w^9},\\ \gamma _{3}= & {} -\dfrac{\beta _w^2}{24\gamma _w^2}-\dfrac{1}{3},\\ \gamma _{4}= & {} \frac{\beta _z^2}{48\gamma _z^2}+\frac{5\beta _z^3}{48\gamma _z^3}\\&-\frac{11\beta _z^5}{720\gamma _z^5}+\frac{47\beta _z^7}{7560\gamma _z^7} +\frac{17\beta _z^9}{15120\gamma _z^9},\\ c= & {} \frac{\beta _w^2}{48\gamma _w^2}\!+\!\frac{5\beta _w^3}{48\gamma _w^3}-\frac{11\beta _w^5}{720\gamma _w^5}\!+\!\frac{47\beta _w^7}{7560\gamma _w^7} \!+\!\frac{17\beta _w^9}{15120\gamma _w^9},\\ r= & {} \dfrac{\beta _z^2}{24\gamma _z^2}+\dfrac{1}{3},\\ s= & {} \dfrac{1}{r}\left[ \dfrac{1}{6\gamma _w}-\dfrac{\beta _w}{24\gamma _w^2}\right. \\&\left. -\dfrac{5\beta _w^2}{16\gamma _w^3}+\dfrac{11\beta _w^4}{144\gamma _w^5}- \dfrac{47\beta _w^6}{1080\gamma _w^7}-\dfrac{17\beta _w^8}{1680\gamma _w^9}\right] \end{aligned}$$

The above constants can be estimated if we refer to the parameters of the improved ML neuronal model defined in [12], for example with $C=1$,$g_L=2.0$,$E_L=-70.0$,$g_{Na}=20.0$,$E_{Na}=50.0$,$g_K=20.0$,$E_K=-100.0$, $g_a=2.0$,$E_a=-100.0$, $\beta _m=-1.2$,$\gamma _m=18.0$,$\beta _w=-20.0$,$\gamma _w=15.0$, $\beta _z=-21.0$,$\gamma _z=10.0$ and $\phi _w=\phi _z=0.15,$ we can say that these constants will be of the order: $0<a\le 2.85$; $0<b\le 236$; $0<\theta _1\le 989$; $0<\theta _2 \le 1.76\times 10^4$; $0<B\le 189.19$; $0<\gamma _1\le 106.5$; $0<\alpha \le 200.0$ $0<\gamma _2\le -0.3$; $0<s\le -213$; $0<r\le 0.45$; $0\le \gamma _4\le \cdots$ and $0\le c\le \cdots$.