Mixed Mode Oscillations in a Three-Timescale Coupled Morris-Lecar System

Abstract

Mixed mode oscillations (MMOs) are complex oscillatory behaviors of multiple-timescale dynamical systems in which there is an alternation of large-amplitude and small-amplitude oscillations. It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov-Hopf bifurcation (DHB) of the fast subsystem. While MMOs in two-timescale systems have been extensively studied, less is known regarding MMOs emerging in three-timescale systems. In this work, we examine the mechanisms of MMOs in coupled Morris-Lecar neurons with three distinct timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work yields important insights into conditions under which the two separate mechanisms in two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, particularly against timescale variations.

Figures (26)

• Figure 1: Time traces of the model \ref{['eq:main']} for different values of $g_{\rm syn}$. MMOs are observed for $g_{\rm syn}=4.3$ and $g_{\rm syn}=4.4$.
• Figure 2: Bifurcation curves of DHB (red) and CDH singularities (blue) for \ref{['eq:main']} with respect to $g_{\rm syn}$. Specifically, these curves represent the upper DHB and upper CDH, corresponding to larger $V_1$ and $V_2$ values. The lower CDH and DHB, associated with smaller $V_i$ values, are not presented here. The two vertical asymptotes are given by $g_{\rm syn} \approx 4.2628$ and $g_{\rm syn} \approx 4.3213$.
• Figure 3: Regions of MMOs (yellow) and non-MMOs (blue) of the full system \ref{['eq:main']} in $(\phi_2, C_1)$-space for (A) $g_{\rm syn}=4.3$ and (B) $g_{\rm syn}=4.4$. Increasing $C_1$ slows down the fast variable $V_1$, whereas increasing $\phi_2$ speeds up the superslow variable $w_2$. The timescales of $w_1$ and $V_2$ remain unaffected. The red star marks the default parameter values of $C_1$ and $\phi_2$ as given in Table \ref{['tab:par']}. (A) $g_{\rm syn}=4.3$. While MMOs are robust to increasing $C_1$ and decreasing $\phi_2$, decreasing $C_1$ or increasing $\phi_2$ leads to multiple transitions between MMOs and non-MMOs (crossings between the dashed lines with the yellow/blue boundaries). (B) $g_{\rm syn}=4.4$. MMOs are robust to changes of both $C_1$ and $\phi_2$ over the ranges of $0.1\leq C_1\leq 20$ and $0.1\leq \phi_2\leq 8e-3$. Note that the MMOs at $g_{\rm syn}=4.4$ will eventually vanish for $C_1$ and $\phi_2$ large enough at which there is no more timescale separation (data not shown).
• Figure 4: Projections to $(V_1,V_2,w_2)$-space of the critical manifold fold surfaces $L_s$ (blue surface) for (A, C) $\rm g_{syn}=4.3$ and (B, D) $\rm g_{syn}=4.4$. Also shown are the curves of folded singularities ${\color{black}{\mathcal{M}}}$ including folded node (solid green), folded saddle (dashed green), and two types of folded saddle-nodes $\mathrm{FSN}$ (blue star: $\mathrm{FSN}^1$; cyan star: $\mathrm{FSN}^2$). The yellow curve consists of mostly folded foci points and small segments of other singularities (e.g., folded node, folded saddle) that are barely visible and hence are not displayed here. In the top two panels (A, B) when $\delta\neq 0$, an $\mathrm{FSN}^1$ point (blue star) is $O(\delta)$ close to a CDH singularity (blue diamond), whereas an $\mathrm{FSN}^2$ (cyan star) is far away from any CDH. In the lower two panels (C, D) at the singular limit $\delta=0$, the $\mathrm{FSN}^1$ point becomes a CDH singularity (blue star overlapping with blue diamond). The center subspace of an $\mathrm{FSN}^1$ (resp., an $\mathrm{FSN}^2$) is denoted by a pink plane (resp., a yellow plane). It follows that the center manifolds of both $\mathrm{FSN}^1$ and $\mathrm{FSN}^2$ are transverse to $L_s$.
• Figure 5: Projection of the singular orbit (green) and the solution trajectory $\Gamma_{(\varepsilon,\delta)}$ (black) of the full system \ref{['eq:main']} onto the phase plane of ($V_2,w_2$) system with parameters given in Table \ref{['tab:par']}. The red curve is the $V_2$-nullcline, which is the projection of the superslow manifold $M_{SS}$. The dark and light green symbols mark the key transitions between the slow and superslow sections of the singular orbit and the perturbed oscillation, respectively: the star and square indicate the transitions from the slow to the superslow motions, and the circle and triangle mark the transitions from superslow to slow sections at the fold of the $V_2$-nullcline. The circled numbers indicate four phases of the oscillations: superslow excursions along $M_{SS}$ during ① and ③ and slow jumps at the fold of $M_{SS}$ during ② and ④.

Theorems & Definitions (3)

• Remark 2.1
• Remark 2.2
• Remark 2.3