# American Institute of Mathematical Sciences

December  2017, 22(10): 4003-4039. doi: 10.3934/dcdsb.2017205

## Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations

 1 Department of Mathematics, University of Pittsburgh, Pittsburgh, USA 2 The Mathematical Neuroscience Team, CIRB-Collége de France, (CNRS UMR 7241, INSERM U1050, UPMC ED 158, MEMOLIFE PSL), Paris, France, Inria Paris, Mycenae Team, Paris, France 3 Math., Gdańsk University of Technology, Gdańsk, Poland 4 Department of Mathematics, Brandeis University, Waltham MA 02454, USA 5 Laboratoire de Mathématiques et Modélisation d'Évry (LaMME), CNRS UMR 8071, Université d'Évry-Val-d'Essonne, France

* Corresponding author: justyna.signerska@pg.edu.pl

Received  November 2016 Revised  June 2017 Published  August 2017

Fund Project: J. E. Rubin was partly supported by US National Science Foundation awards DMS 1312508 and 1612913. J. Signerska-Rynkowska was supported by Polish National Science Centre grant 2014/15/B/ST1/01710.

This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left-and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes.

Citation: Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 4003-4039. doi: 10.3934/dcdsb.2017205
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The geometry of MMOs: (Upper row) Phase plane with $v$ and $w$ nullclines (dashed black) and stable (red) and unstable (blue) manifolds of the saddle; the stable manifold winds around the repulsive singular point. The reset line $\{v=v_R\}$ (solid vertical line) intersects the stable manifold, separating out regions such that trajectories emanating from each undergo a specific number of small oscillations (colored segments, here from 0 to 3 below the $w$-nullcline and from $3.5$ to $0.5$ above). (Lower rows) The solution for one given initial condition in each segment. Note that the time interval varies in the different plots (indicated on the $x$-axis). Simulations had initial conditions $v=v_R=0.012$ and $w$ chosen within the different intervals on the reset line.
Geometry of the phase plane with indication of the points relevant in the characterization of the adaptation map $\Phi$. In this example, there are only $p=2$ intersections of $\{ v=v_R\}$ with $\mathcal{W}^s$ (thus $p_1=1$).
Typical topology of manifolds and sections in Lemma 3.2: we consider the correspondence map between sections $S_s$ (red) and $S_u$ (orange) transverse, respectively, to the stable and unstable manifolds (black lines) of the saddle (orange circle). Typical trajectories are plotted in blue. The key arguments are the characterization of correspondence maps associated with the linearized system (upper left inset) between two transverse sections $S_s'$ and $S_u'$, and the smooth conjugacy between the nonlinear flow and its linearization.
Partitions of $(d, \gamma)$ parameter space (for fixed values of the other parameters) according to geometric properties of the map $\Phi$ for the quartic model ($F=v^4+2av$, $a=\varepsilon =0.1$, $b=1$, $I=0.1175$ and $v_R=0.1158$) assuming only two intersections of the reset line with the stable manifold (see text for further information).
The orientation-preserving maps $\Psi_l$ (green) and $\Psi_r$ (blue) enveloping the lift $\Psi$ (red line), which is non-monotonic and admits negative jumps, for the adaptation map $\Phi$ (blue dashed curve) in the overlapping case.
Phase plane structure, $v$ signal generated along attractive periodic orbits and sequence of $w$ reset values for two sets of parameter values for which the map $\Phi$ is in the non-overlapping case (C4). In both cases, $v_R=0.1$ and $\gamma=0.05$. The top case ($d=0.08$) illustrates the regular spiking behavior corresponding to the rotation number $\varrho = 0$. The bottom case ($d=0.08657$) displays a complex MMBO periodic orbit with associated rational rotation number.
Phase plane (inset) and adaptation map (top) fulfilling condition (C4) and the additional condition $\Phi(\alpha)<w_1<\Phi(\beta)$, along with the associated MMBO orbit of system (1) (bottom). The rotation number is equal to $0.5$, hence the $v$ signal along the orbit is a periodic alternation of a pair of spikes and one small oscillation. The parameter values of the system corresponding to this simulation are $v_R=0.1$, $\gamma=0.05$ and $d=0.087$.
Rotation number as a function of $d$. The parameter values $v_R=0.1$ and $\gamma=0.05$ have been chosen such that the adaptive map $\Phi$ fulfills condition (C4) for any value of $d \in [0.08, 0.092]$. Theorem 4.3 applies here, and the rotation number varies as a devil's staircase, as shown in the bottom plot. The top panels show the adaptation map and corresponding attractive periodic orbit at the $d$ values labelled correspondingly in the rotation number plot; note that the rotation number for case (b) is a rational number between 1/3 and 1/2
Rotation intervals for the lifts $\Psi_{d, l}, \Psi_{d, r}$ of the adaptation maps $\Phi_d$ for a range of $d$. The parameter value $\gamma=0.05$ has been chosen so that $\Phi_d$ remains in the overlapping case for all $d \in [0.0745, 0.0825]$.
Rotation numbers according to $(d, \gamma)$. Left panel: rotation number of the point $w=0$ together with the boundaries of the regions A to E corresponding to the different subcases when $w_1$ is the unique discontinuity of the adaptation map lying in the interval $[\beta, \alpha]$ (see text for more details). Right panel: rotation numbers of the left and right lifts $\Psi_l$ and $\Psi_r$ associated with $\Phi$ for $(d, \gamma)$ varying along the blue segment drawn in the inset
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