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Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations
December  2017, 22(10): 3967-4002. doi: 10.3934/dcdsb.2017204

## Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos

 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 Faculty of Appl. Phys. and 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

In a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents mathematical analysis showing that the system can support bursts of any period as a function of model parameters, and that these are organized in a period-incrementing structure. In continuous dynamical systems with resets, such structures are complex to analyze. In the present context, we use the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we show that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We show that the period-incrementing structure persists for the full system with non-constant adaptation, but the transitions are more complex. We investigate the presence of chaos at the transitions.

Citation: Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3967-4002. doi: 10.3934/dcdsb.2017204
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Bifurcations of the adaptive exponential model and its saddle-node (brown), Hopf (green), saddle homoclinic (purple) and Bogdanov-Takens (BT) bifurcations in the $(I, b)$ parameter plane. The analytical curve separating regions of unstable focus and unstable node is added in dashed blue. Typical phase planes in the different regions of interest are depicted as smaller insets. They feature the nullclines (dashed black) and the stable manifold (red).
The period-incrementing structure of $\Phi$ as $v_{R}$ is varied for the quartic model (system (1) with parameters (5)). Blue points: 100 iterates of the map $\Phi$ after a transient of $100$ spikes (most points are overlapping and we only see the periodic points appearing, except in the chaotic region) as a function of $v_{R}$
An example of the adaptation map for small $\varepsilon$, annotated with notation used throughout the paper. Left: attractive (green, $\mathcal{C_{\varepsilon}^-}$) and repulsive (red, $\mathcal{C_{\varepsilon}^+}$) slow manifolds of the subthreshold system as perturbations of the critical manifold $\mathcal{C}$ (black curve). As stated in the main text (in order from largest to smallest $w$): $w_{\varepsilon}^+$ denotes the $w$-coordinate of the upper intersection point of $\mathcal{C_{\varepsilon}^+}$ with $\{v = v_R\}$, $w^*$ is the $w$-coordinate of $\mathcal{C} \cap \{ v=v_R \}$, $(v_F, w_F)$ denotes the minimum point of $\mathcal{C}$, $w_{\varepsilon}^-$ is the $w$-coordinate of $\mathcal{C_{\varepsilon}^-} \cap \{ v = v_F \}$, $w^{**}$ denotes the $w$-coordinate of the intersection of the $w$-nullcline (black line) with $\{ v = v_R \}$. Right: Associated adaption map $\Phi_{\varepsilon}$. The grey region corresponds to the interval $[w^*, \xi]$, where $\xi$ is defined as the largest value such that $\Phi'(\xi) = -1$ (see section 3), while $p_{\varepsilon} := \lim_{w\to\infty} \Phi_{\varepsilon}(w)$.
Bifurcation structure and periodic solutions of $\Phi_0$. Left: bifurcation diagram of $\Phi_{0}$ as a function of $w^{*}$ illustrating the period-incrementing structure as described in Proposition 2. Blue lines represent the stable periodic orbits of the system. Middle: bifurcation diagram of $\Phi_{0}$ as a function of $v_{R}$ in the case of the quartic model with standard parameters (5). The red curve represents the value of $w^{*}$ for each $v_R$. Right: plot of $\Phi_{0}$ and the associated period-4 orbit for $v_{R}=1.3$ (green line in the middle plot).
Illustration for the proof of Proposition 3 with $k=4$, together with the orbits of $w^*$ (blue) and $\tilde{w}$ (green) (see the statement of Proposition 3).
Maps $\Phi_{\varepsilon}$ and bifurcation sequences for the standard quartic model (5) and various values of $\varepsilon$. For $\varepsilon=0.05$, the diagram is very close to the bifurcation diagram of $\Phi_{0}$ depicted in Fig. 4 (we superimposed the plot of the map $F(v)+I$ in orange to emphasize this similarity). As $\varepsilon$ is increased, the map $\Phi$ slowly deviates from $\Phi_{0}$; the associated bifurcation diagrams conserve the overall period-incrementing structure, but with larger transition regimes characterized by the presence of chaos. We also note that as $v_{R}$ increases, the bifurcation structure becomes more similar to the singular limit.
Orbits of $\Phi$ for the standard quartic model (5) with $\varepsilon=0.4$, for values of $v_{R}$ spanning the period-incrementing transition between bursts with 2 and 3 spikes. A period-2 orbit undergoes a period-doubling giving rise to a period-4 orbit, which itself loses stability, yielding chaotic spiking, before the system stabilizes on a period-3 orbit.
Metric chaos in the quartic integrate-and-fire neuron with standard parameters (5) and $\varepsilon=0.4$. (A) Adaptation map in the case where $\Phi^{5}(w^{*})=w^{f}$, a fixed point, found by fine-tuning the value of the reset voltage (here, $v_{R}=1.2226$). The sequence of iterates is chaotic (strictly positive Lyapunov exponent (30) with for instance $\kappa>6$ for all $w$ tested). (B) Iterates of $\Phi$ in the same setting yield a complex chaotic sequence. The $15\, 000$ final iterates are depicted, together with the obtained distribution of points.
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