Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris–Lecar neuronal model

Abstract

Necessary and sufficient conditions are given for the asymptotic stability and instability of a two-dimensional incommensurate order autonomous linear system, which consists of a differential equation with a Caputo-type fractional-order derivative and a classical first-order differential equation. These conditions are expressed in terms of the elements of the system’s matrix, as well as of the fractional order of the Caputo derivative. In this setting, we obtain a generalization of the well-known Routh–Hurwitz conditions. These theoretical results are then applied to the analysis of a two-dimensional fractional-order Morris–Lecar neuronal model, focusing on stability and instability properties. This fractional-order model is built up taking into account the dimensional consistency of the resulting system of differential equations. The occurrence of Hopf bifurcations is also discussed. Numerical simulations exemplify the theoretical results, revealing rich spiking behavior. The obtained results are also compared to similar ones obtained for the classical integer-order Morris–Lecar neuronal model.

Appendices

A Proof of Proposition 1.

We compute:

$$\begin{aligned} ^cD^qg(x)&=\frac{1}{\Gamma (-q)}\int \limits _{0}^{x}(x-t)^{-q-1}[g(t)-g(0)]\mathrm{d}t\\&=\frac{1}{\Gamma (-q)}\int \limits _{0}^{x}(x-t)^{-q-1}[f(at)-f(0)]\mathrm{d}t\\&=\frac{1}{a\Gamma (-q)}\int \limits _{0}^{ax}\Big (x-\frac{s}{a}\Big )^{-q-1}[f(s)-f(0)]\mathrm{d}t\\&=\frac{1}{a\Gamma (-q)}\int \limits _{0}^{ax}a^{q+1}(ax-s)^{-q-1}[f(s)-f(0)]\mathrm{d}t\\&=\frac{a^{q}}{\Gamma (-q)}\int \limits _{0}^{ax}(ax-s)^{-q-1}[f(s)-f(0)]\mathrm{d}t\\&=a^q (^cD^qf)(ax) \end{aligned}$$

It follows that:

$$\begin{aligned} ^cD^q g(x)=a^q\cdot ^cD^qf(ax),\quad \text {for any }a\ne 0, \end{aligned}$$

which completes the proof. $\square$

B Deduction of the nondimensional system (11)

Starting from system (9), we consider the substitutions

$$\begin{aligned} v(t)=kV(\alpha t)\quad ,\quad n(t)=N(\alpha t), \end{aligned}$$

where $\alpha$ and k will be deduced in the following.

Applying Proposition 2, we have:

$$\begin{aligned} ^cD^qv(t)&=k\cdot ^cD^q[V(\alpha t)]\\&=k\alpha ^q(^cD^qV)(\alpha t)\\&=k\alpha ^q\frac{1}{C_\mathrm{m}(q)}\Big [g_{Ca}M_\infty (V(\alpha t))(V_{Ca}-V(\alpha t))+\\&\quad + \, g_KN(\alpha t)(V_K-V(\alpha t))\\&\quad + \, g_L(V_L-V(\alpha t))+I\Big ]\\&=k\alpha ^q\frac{1}{C_\mathrm{m}(q)}\Big [g_{Ca}M_\infty \Big (\frac{v(t)}{k}\Big )\Big (V_{Ca}-\frac{v(t)}{k}\Big )+\\&\quad + \, g_K n(t)\Big (V_K-\frac{v(t)}{k}\Big )\\&\quad + \, g_L\Big (V_L-\frac{v(t)}{k}\Big )+I\Big ]\\&=\frac{\alpha ^q}{C_\mathrm{m}(q)}\Big [g_{Ca}m_\infty (v)(kV_{Ca}-v)+\\&\quad + \, g_Kn(kV_K-v)+g_L(kV_L-v)+kI\Big ] \end{aligned}$$

and therefore, it makes sense to choose $k=\frac{1}{V_{Ca}}$.

Furthermore, with the notations from Sect. 3, we obtain:

$$\begin{aligned} ^cD^qv(t)&=R_\mathrm{m}\Big (\frac{\alpha }{\tau }\Big )^q\Big [g_{Ca}m_\infty (v)(1-v)\\&\qquad \,+g_K n(v_K-v)+ g_L(v_L-v)+\tilde{I}\Big ]. \end{aligned}$$

At this step, it is easy to see that it makes sense to consider $\alpha =\tau$, which leads to:

$$\begin{aligned} ^cD^qv(t)&=R_\mathrm{m}\Big [g_{Ca}m_\infty (v)(1-v)+g_K n(v_K-v)\\&\quad +g_L(v_L-v)+\frac{I}{V_{Ca}}\Big ]\\&=\,\gamma _{Ca}m_\infty (v)(1-v)\\&\quad +\gamma _K n(v_K-v)+\gamma _L(v_L-v)+\tilde{I}. \end{aligned}$$

As for the second equation, applying Proposition 2, and taking into account that $\alpha =\tau$, we obtain:

$$\begin{aligned} ^cD^pn(t)&=\alpha ^p(^cD^p)(\alpha t)\\&=\,(\alpha \overline{\lambda _N})^p\lambda (V(\alpha t))[N_\infty (V(\alpha t))-N(\alpha t)]\\&=\,(\tau \overline{\lambda _N})^p\lambda \Big (\frac{v(t)}{k}\Big )\Big [N_\infty \Big (\frac{v(t)}{k}\Big )-n(t)\Big ]\\&=\,(\tau \overline{\lambda _N})^p\cdot \ell (v)[n_\infty (v)-n], \end{aligned}$$

Therefore, the nondimensional system (11) is found.