Spike-Adding Mechanisms in a Three-Timescale System: Insights from the FitzHugh-Nagumo Model with Periodic Forcing

TL;DR

This work analyzes spike-adding in a three-timescale neuronal system driven by low-frequency periodic forcing, using geometric singular perturbation theory to identify how the number of spikes per burst changes with input amplitude and frequency. By reformulating the forced FitzHugh–Nagumo model into a three-dimensional autonomous system and computing its critical and super-critical manifolds, the authors reveal spike-adding mechanisms tied to folded saddle and folded node canards, and they translate these findings into boundaries in parameter space. The results are corroborated in a more realistic Morris–Lecar model, showing a robust spike-adding structure across models and linking the dynamics to biologically relevant brain rhythms. Overall, the paper advances understanding of how slow-fast canard dynamics organize bursting patterns in multi-timescale neuronal systems and offers quantitative tools to predict spike-count transitions.

Abstract

In this work, we investigate the spike-adding mechanism in a class of three-dimensional fast-slow systems with three distinct timescales, inspired by the FitzHugh-Nagumo (FHN) model driven by periodic input. First, we numerically generate a spike-adding diagram for the FHN model by varying the frequency and amplitude of the input, revealing that as the frequency decreases and the amplitude increases, the number of spikes within each burst grows. We demonstrate that a similar spike-adding structure occurs in the more realistic, periodically forced Morris-Lecar neuronal model. Next, we apply methods from geometric singular perturbation theory to compute critical and super-critical manifolds of the fast-slow system. We use them to characterize the emergence of new burst-spikes in the FHN model, when the periodic forcing resembles a low frequency-band brain rhythm. We then describe how the uncovered spike-adding mechanism defines the boundaries that separate regions with different spike counts in the parameter space.

Figures (12)

• Figure 1: Left panel: The spike-adding diagram of FHN system \ref{['E:FHN-1']} is shown as the frequency $\omega$ and amplitude $E$ of the periodic input vary. Four regions (from left to right) indicate four different behaviors of the system in terms of its number of spikes ($m$) versus the number of input periods ($n$). A The system exhibits bursting, i.e., $m>n$. This region is plotted in more detail in Figure \ref{['fig:bifurcation_E_vs_w_region_I']}; B The system exhibits tonic spikes, i.e., $m=n$; C The system exhibits mixed mode oscillations, i.e., $m<n$; D No spikes occur ($m=0$). Right panel: A trajectory of each region from the left panel is shown over two periods of the input ($n=2$) and $m=6, 2, 1$, and $0$, respectively. The ticks at the top of the figure depict possible values $\omega=\varepsilon=0.08$, $\omega = \sqrt\varepsilon\approx 0.3$, $\omega=1$, given that $\varepsilon=0.08$ in system \ref{['E:FHN-1']}.
• Figure 2: Left panel: An enhancement of the blue region from Figure \ref{['fig:bifurcation_E_vs_w']}, which marks the spike-adding diagram of FHN system \ref{['E:FHN-1']} as the frequency $\omega$ changes between 0 and $\varepsilon$ and amplitude $E$ varies between 0.3 and 0.7. For each input period, the system can exhibit no spikes ($m=0$, light-gray), one spike ($m=1$, gray), or a burst with multiple spikes ($m>1$, diverse shades of blue). The squares indicate parameter values for exemplary trajectories. Right panel: Solutions of \ref{['E:FHN-1']}, with $(\omega,E)$ indicated by a--d on the left panel, plotted over one period of the input ($n=1$). Shown here is an increasing sequence of $E$ values with $\omega$ fixed that yields increasing spike counts. Note that $\varepsilon=0.08$ in system \ref{['E:FHN-1']}.
• Figure 3: Bifurcation diagram for the three-dimensional FHN. Parameter regions according to folded equilibria: The blue regions (I, II, and III) show the primary parameter region considered in this paper for spike-adding. In regions I, II, III the FHN model admits folded equilibria only on $\mathcal{L}_{-}$. The lines separating regions I from II and II from III are $E=E^{*}_{\ell,\delta}$ and $E=E^{**}_{\ell,\delta}$, with $E^{*}_{\ell,\delta}$, $E^{**}_{\ell,\delta}$ given by \ref{['E:saddle-node-left']}, \ref{['E:Enode']} respectively. The orange regions (IV, V, and VI) admit additional folded equilibria that lie on $\mathcal{L}_{+}$. The lines separating regions III from V, and V from VI, are $E=E^{*}_{r,\delta}$ and $E=E^{**}_{r,\delta}$ with $E^{*}_{r,\delta}$, $E^{**}_{r,\delta}$ given by Eqs. \ref{['E:saddle-node-right']} and \ref{['E:Enode-right']}. Note that the horizontal axis closely resembles that of Figure \ref{['fig:bifurcation_E_vs_w_region_I']} where $\omega=\varepsilon\delta\in[0,0.08]$ and $\varepsilon=0.08$.
• Figure 4: Bursting solution with critical and supercritical manifold.A The critical manifold, $\mathcal{M}$, is separated into attracting ($\mathcal{M}_{a-}$, $\mathcal{M}_{a+}$; blue) and repelling ($\mathcal{M}_{r}$; red) regions. The attracting and repelling regions intersect along fold lines, $\mathcal{L}_{\pm}$. The folded node ($N\ell$) and folded saddle ($S\ell$) are shown as a square and circle, respectively, along the fold lines. The supercritical manifold, $\mathcal{Z}$ (magenta), is a curve along the critical manifold. A bursting trajectory of system \ref{['E:FHN-1']} with three spikes (forcing parameters $E=0.55$, $\omega=0.0149354$) is shown with $\theta =\omega t$ over one input period. This trajectory contains canard segments, following passage near the folded node and folded saddle, that remain near $\mathcal{M}_{a+}$ for non-negligible time on the fast timescale. B The same trajectory projected onto the $\theta$--$x$ axis. Colors and symbols follow the same conventions as in panel A.
• Figure 5: Parameter region of interest for bursting. Cusp-like boundaries for changing spike counts are located where a folded node and folded saddle exist on $\mathcal{L}_{-}$ (region II in Figure \ref{['F:ROI']}; see also Figure \ref{['fig:bifurcation_E_vs_w_region_I']}). The regions below the saddle-node (SN) bifurcation on $\mathcal{L}_{-}$ ($E=E^{*}_{\ell,\delta}$) and above the curve marking the transition from folded node to folded focus ($E=E^{**}_{\ell,\delta}$) are grayed-out. The parameter region above the SN on $\mathcal{L}_{+}$ ($E=E^{*}_{r,\delta}$) is shown in white. In our analysis we constrained input forcing parameters to $E^{*}_{\ell,\delta} < E < E^{*}_{r,\delta}$. Note that the horizontal axis closely resembles that of Figure \ref{['fig:bifurcation_E_vs_w_region_I']} where $\omega=\varepsilon\delta\in[0,0.08]$ and $\varepsilon=0.08$.

Theorems & Definitions (11)

• Remark 1
• Proposition 1: Existence of folded equilibria
• Proof 1
• Remark 2
• Proposition 2: Type and stability of folded equilibria
• Proof 2
• Remark 3
• Proposition 3
• Proof 3
• Remark 4