^{1}, Tasso J. Kaper

^{2}and Martin Krupa

^{1}

### Abstract

This article concerns the phenomenon of Mixed-Mode Bursting Oscillations (MMBOs). These are solutions of fast-slow systems of ordinary differential equations that exhibit both small-amplitude oscillations (SAOs) and bursts consisting of one or multiple large-amplitude oscillations (LAOs). The name MMBO is given in analogy to Mixed-Mode Oscillations, which consist of alternating SAOs and LAOs, without the LAOs being organized into burst events. In this article, we show how MMBOs are created naturally in systems that have a spike-adding bifurcation or spike-adding mechanism, and in which the dynamics of one (or more) of the slow variables causes the system to pass slowly through that bifurcation. Canards are central to the dynamics of MMBOs, and their role in shaping the MMBOs is two-fold: saddle-type canards are involved in the spike-adding mechanism of the underlying burster and permit one to understand the number of LAOs in each burst event, and folded-node canards arise due to the slow passage effect and control the number of SAOs. The analysis is carried out for a prototypical fourth-order system of this type, which consists of the third-order Hindmarsh-Rose system, known to have the spike-adding mechanism, and in which one of the key bifurcation parameters also varies slowly. We also include a discussion of the MMBO phenomenon for the Morris-Lecar-Terman system. Finally, we discuss the role of the MMBOs to a biological modeling of secreting neurons.

This study introduces a new mechanism for generating complex oscillations in systems of differential equations with fast and slow time scales. At the heart of the model, there is a third-order system of equations, which exhibits spike-adding transitions. Then, when the bifurcation parameter that controls the spike-adding transition is allowed to evolve slowly in time, a new mechanism for generating oscillations arises. The resulting solutions possess alternating segments of small-amplitude slow oscillations (SAOs) and bursts, which consist of large-amplitude fast oscillations (LAOs). Hence, the solutions are termed Mixed-Mode Bursting Oscillations (MMBOs), in analogy with the known Mixed-Mode-Oscillations (MMO) that consist of alternating SAOs and LAOs without the LAOs being organized in bursts. Moreover, MMBOs may be periodic or aperiodic. The large-amplitude oscillations in the bursts are created by the slow passage through the spike-adding bifurcation, and the number of small-amplitude oscillations is controlled by the system parameters and by the presence of a certain type of singularity in the slow subsystem. While the results are presented for the Hindmarsh-Rose (H-R) model, they apply to a broad class of fast-slow neuronal oscillators, including the Morris-Lecar-Terman model and others that exhibit the spike-adding transition. The analysis also raises timely mathematical questions related to understanding the new dynamics created by combining bursting and MMO dynamics.

M.D. and M.K. acknowledge the support of the INRIA Project-team SISYPHE and Large-Scale Action REGATE. M.D. thanks the Department of Mathematics and Statistics at Boston University, where part of this work was completed. The research of T.K. was supported in part by NSF DMS 1109587.

I. INTRODUCTION

II. BRIEF REVIEW OF CLASSICAL CANARD EXPLOSION

III. SPIKE-ADDING CANARD EXPLOSION IN THE HINDMARSH-ROSE SYSTEM

IV. MMBOs AS A SLOW PASSAGE THROUGH A SPIKE-ADDING CANARD EXPLOSION

A. Understanding MMBOs as a slow passage through a canard explosion

B. Controlling the number of SAOs in MMBOs using folded node theory

C. Slow manifolds and sectors of rotation

V. DISCUSSION AND FUTURE WORK

### Key Topics

- Manifolds
- 44.0
- Explosions
- 28.0
- Bifurcations
- 19.0
- Time series analysis
- 8.0
- Number theory
- 7.0

## Figures

Simulation of system (1)–(4) for , and for i = x, y, I. The orbit shown here is clearly of MMBO type, that is, a succession of small-amplitude slow oscillations and large-amplitude fast oscillations. The observed MMBO pattern is irregular and combines transitions of the type 4^{1}, 4^{2}, and 4^{3}.

Simulation of system (1)–(4) for , and for i = x, y, I. The orbit shown here is clearly of MMBO type, that is, a succession of small-amplitude slow oscillations and large-amplitude fast oscillations. The observed MMBO pattern is irregular and combines transitions of the type 4^{1}, 4^{2}, and 4^{3}.

Canard explosion at the lower fold in the van der Pol system: the slow and the fast dynamics must be as shown and the slow nullcline (shown in black), must cross the fold transversely. Four cycles are presented (in blue): headless canard in panel (a), maximal canard in panel (b), canard with head in panel (c), and relaxation oscillation in panel (d). In each panel, the left plot corresponds to a phase plane representation of the cycle together with the fast cubic nullcline S 0 and the slow linear nullcline (both in black); the right panel shows the time series of the x variable during the cycle.

Canard explosion at the lower fold in the van der Pol system: the slow and the fast dynamics must be as shown and the slow nullcline (shown in black), must cross the fold transversely. Four cycles are presented (in blue): headless canard in panel (a), maximal canard in panel (b), canard with head in panel (c), and relaxation oscillation in panel (d). In each panel, the left plot corresponds to a phase plane representation of the cycle together with the fast cubic nullcline S 0 and the slow linear nullcline (both in black); the right panel shows the time series of the x variable during the cycle.

Panel (a) shows the bifurcation diagram of the fast subsystem of the Hindmarsh-Rose burster, i.e., Eqs. (1) and (2) , with z acting as a parameter. All parameter values are taken to be the classical ones (Sec. I ) and I is fixed at the value corresponding to the canard explosion taking place in the full system and displayed in Fig. 4 , that is, I = 1.3278138. Panels (b1)-(b3) show snapshots of the phase portrait of the fast system for three different values of z, namely, z = 1.449 (lower homoclinic), z = 1.75 and z = 2.180 (upper homoclinic). For each value of z in between the lower and upper homoclinics, the unstable manifold of the saddle returns to a neighbourhood of the saddle and gives the leading-order location of the spike in the full system.

Panel (a) shows the bifurcation diagram of the fast subsystem of the Hindmarsh-Rose burster, i.e., Eqs. (1) and (2) , with z acting as a parameter. All parameter values are taken to be the classical ones (Sec. I ) and I is fixed at the value corresponding to the canard explosion taking place in the full system and displayed in Fig. 4 , that is, I = 1.3278138. Panels (b1)-(b3) show snapshots of the phase portrait of the fast system for three different values of z, namely, z = 1.449 (lower homoclinic), z = 1.75 and z = 2.180 (upper homoclinic). For each value of z in between the lower and upper homoclinics, the unstable manifold of the saddle returns to a neighbourhood of the saddle and gives the leading-order location of the spike in the full system.

Spike-adding phenomenon: time series and phase portraits for the sequence of limit cycle canards observed in the transition from 0 to 1 spike in the Hindmarsh-Rose system (1)–(3) . Parameter values are those given below Eqs. (1)–(3) , with and I varying by an exponentially small amount around the value of 1.3278138026. The top plot in each panel (a) to (i) shows the time series of the x-variable, with the period normalised to 1. The bottom plot shows the limit cycle canard in the phase plane (x, z). In each frame, the behaviour of the trajectory at the end of the canard segment can be understood by looking at the phase portrait of the fast system (1) and (2) shown in Fig. 3 .

Spike-adding phenomenon: time series and phase portraits for the sequence of limit cycle canards observed in the transition from 0 to 1 spike in the Hindmarsh-Rose system (1)–(3) . Parameter values are those given below Eqs. (1)–(3) , with and I varying by an exponentially small amount around the value of 1.3278138026. The top plot in each panel (a) to (i) shows the time series of the x-variable, with the period normalised to 1. The bottom plot shows the limit cycle canard in the phase plane (x, z). In each frame, the behaviour of the trajectory at the end of the canard segment can be understood by looking at the phase portrait of the fast system (1) and (2) shown in Fig. 3 .

Spike-adding phenomenon: time series and phase plane projections for limit cycles along the transition from 1 spike to 2 spikes. The first spike, created during the previous spike-adding transition, remains in essentially the same place during the transition in which the second spike is generated. Along the explosion that leads to the addition of a second spike, the parameter I varies by an exponentially amount around the value of 1.3317831217.

Spike-adding phenomenon: time series and phase plane projections for limit cycles along the transition from 1 spike to 2 spikes. The first spike, created during the previous spike-adding transition, remains in essentially the same place during the transition in which the second spike is generated. Along the explosion that leads to the addition of a second spike, the parameter I varies by an exponentially amount around the value of 1.3317831217.

Periodic MMBOs of system (1)–(4) , for (panel (a) for the x variable and panel (c) for the I variable) and (panels (b) and (d) for the x variable and the I variable, respectively); in all panels, hx = 5 and . In panels (c) and (d), the dashed red curve indicates the value of I corresponding to the folded node. The unlabelled panels to the side of panel (a) and panel (b) show zooms into one burst of the corresponding MMBO, with 8 LAOs (left panel) and 2 LAOs (right panel).

Periodic MMBOs of system (1)–(4) , for (panel (a) for the x variable and panel (c) for the I variable) and (panels (b) and (d) for the x variable and the I variable, respectively); in all panels, hx = 5 and . In panels (c) and (d), the dashed red curve indicates the value of I corresponding to the folded node. The unlabelled panels to the side of panel (a) and panel (b) show zooms into one burst of the corresponding MMBO, with 8 LAOs (left panel) and 2 LAOs (right panel).

Periodic MMBOs of system (1)–(4) , for k = 0.45 (panel (a)) and k = 0.35 (panel (b)) and for ; in both panels hx = 5 and . Decreasing allows to find MMBOs with the correct number of SAOs as predicted by folded node theory via formula (16). In both cases, the number of LAOs is very large, about 200.

Periodic MMBOs of system (1)–(4) , for k = 0.45 (panel (a)) and k = 0.35 (panel (b)) and for ; in both panels hx = 5 and . Decreasing allows to find MMBOs with the correct number of SAOs as predicted by folded node theory via formula (16). In both cases, the number of LAOs is very large, about 200.

Periodic 192^{3}-MMBOs of system (1)–(4) , obtained for k = 0.45, that is, 1/μ ≈ 6.74, and , and projected onto the (x, z, I)-space; also shown are the critical manifold S 0 and the lower fold curve F ^{–}. Panel (b) is an enlargement of panel (a) near the folded node .

Periodic 192^{3}-MMBOs of system (1)–(4) , obtained for k = 0.45, that is, 1/μ ≈ 6.74, and , and projected onto the (x, z, I)-space; also shown are the critical manifold S 0 and the lower fold curve F ^{–}. Panel (b) is an enlargement of panel (a) near the folded node .

Periodic 1^{2}-MMBOs Γ of system (1)–(4) , obtained for k = 0.45, that is, 1/μ ≈ 6.74, and , and projected onto the (x, z, I)-space; also shown are a numerical computation of an attracting slow manifold . We highlight three orbits on : the strong canard γ s as well as two (unlabelled) secondary canards; these define three rotation sectors named si , i = 1, 2, 3. The number of SAO in a given solution depends on which sector it lies in; in particular, in sector si solutions have i SAOs.

Periodic 1^{2}-MMBOs Γ of system (1)–(4) , obtained for k = 0.45, that is, 1/μ ≈ 6.74, and , and projected onto the (x, z, I)-space; also shown are a numerical computation of an attracting slow manifold . We highlight three orbits on : the strong canard γ s as well as two (unlabelled) secondary canards; these define three rotation sectors named si , i = 1, 2, 3. The number of SAO in a given solution depends on which sector it lies in; in particular, in sector si solutions have i SAOs.

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