Fractional order induced bifurcations in Caputo-type denatured Morris-Lecar neurons

TL;DR

This work extends the denatured Morris-Lecar neuron model to Caputo-type fractional-order dynamics to incorporate memory effects, analyzing both a single cell and a symmetric two-cell dML dimer with two coupling schemes. The authors derive explicit stability conditions and Hopf-bifurcation thresholds in terms of the memory index $\beta$ and coupling parameters, supported by rigorous numerics using FdeSolver.jl. They show that fractional-order memory drives rich oscillatory behavior, including tonic spiking, bursting, and mixed-mode oscillations, with Hopf bifurcations giving rise to stable limit cycles as $\beta$ increases, and saddle-node bifurcations marking equilibrium multiplicity changes in the 2D and 4D settings. The results reveal how coupling strength shapes the stability regions in the $(I,\beta)$ plane and demonstrate the interplay between memory and synaptic coupling in shaping neuronal dynamics, with practical implications for reduced-order modeling of memory-reliant excitable tissue. The study provides analytical formulas for critical $\beta$ and robust numerical demonstrations, and it points to future work on network extensions, stochastic perturbations, and incommensurate fractional orders.

Abstract

We set up a system of Caputo-type fractional differential equations for a reduced-order model known as the {\em denatured} Morris-Lecar (dML) neurons. This neuron model has a structural similarity to a FitzHugh-Nagumo type system. We explore both a single-cell isolated neuron and a two-coupled dimer that can have two different coupling strategies. The main purpose of this study is to report various oscillatory phenomena (tonic spiking, mixed-mode oscillation) and bifurcations (saddle-node and Hopf) that arise with variation of the order of the fractional operator and the magnitude of the coupling strength for the coupled system. Various closed-form solutions as functions of the system parameters are established that act as the necessary and sufficient conditions for the stability of the equilibrium point. The theoretical analysis are supported by rigorous numerical simulations.

Figures (14)

• Figure 1: $I_\infty(x)$ as a function of $x$ with parameters \ref{['eq:param']}. The two extrema are denoted by $I_{\rm max}$ and $I_{\rm min}$.
• Figure 2: Three different branches of the equilibrium points with $I$ set as (a) $0.0001< I_{\min}$, (b) $0.019 > I_{\max}$, (c) $I_{\min}$, (d) $I_{\max}$, (e) $0.11 \in (I_{\min}, I_{\max})$. Other parameters are set as \ref{['eq:param']}. Panels (a) and (b) have a unique equilibrium, panels (c) and (d) have two equilibriums, and panel (e) has three equilibriums.
• Figure 3: Phase portraits and time series with varying $\beta$ for \ref{['eq:DML_2D']}. We have set $I = 0.019>I_{\max}$. The other parameters are set as \ref{['eq:param']}. This gives $\beta^* \approx 0.98233$. We notice that \ref{['eq:DML_2D']} converges to a stable equilibrium for $\beta <\beta^*$. As $\beta \ge \beta^*$, the equilibrium loses stability through a Hopf bifurcation and a stable limit cycle appears (showing tonic spiking in the dynamical variables).
• Figure 4: Bifurcation diagrams with varying $\beta$ for \ref{['eq:DML_2D']} at $I > I_{\rm max}$. Specifically we have (a) $I=0.019$ whose corresponding phase portraits are displayed in Fig. \ref{['fig:pp']} and (b) $I = 0.022$. Other parameters are set following \ref{['eq:param']}. The vertical blue lines indicate the $\beta^*$ for each of the cases where a Hopf bifurcation occurs. The $\beta^*$ values for these cases are approximately (a) $0.98233$ and (b) $0.98772$.
• Figure 5: Hopf bifurcation curve $\beta^*$ for \ref{['eq:DML_2D']} separating the stable and the unstable regions of the equilibrium point $(x^*, y^*)$ on the $(I, \beta)$-plane. The vertical broken lines correspond to $I = 0.019$ and $I=0.022$. The system undergoes a Hopf bifurcation when $\beta$ crosses the curve from the bottom for a particular $I$ value. Other parameter values are set as \ref{['eq:param']}.

Theorems & Definitions (16)

• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Theorem 4.4
• proof
• Theorem 5.1
• proof