Diffusive synchronization of phase waves in the FitzHugh-Nagumo system

TL;DR

The paper analyzes diffusive synchronization of phase waves in the FitzHugh–Nagumo system by treating synchronization as convergence to frequency-synchronized traveling wave trains. It develops a slow–fast geometric framework, employs blow-up desingularization around nonhyperbolic folds, and uses Lin’s method together with Riccati-transform decouplings to derive an explicit main formula connecting spectral parameters to Bloch frequencies. The main results show that the phase-wave trains are spectrally stable with a diffusive mode characterized by $d_ ext{eff}\sim\frac{2\kappa c^3}{L_0}\varepsilon^{2/3}$ and a group-velocity contribution $i\lambda'_\ ext{ε}(0)=c_g-c$, while also identifying finite-wavelength instabilities in system variants. This work provides a rigorous mechanism for diffusion-driven phase synchronization in multi-scale reaction–diffusion systems and delivers a robust analytical toolkit (Lin’s method, blow-up, exponential trichotomies, Riccati) applicable to other folded slow–fast patterns and pattern-forming fronts.

Abstract

We analyze synchronization of relaxation oscillations in multiple-timescale reaction-diffusion systems. Interpreting synchronization as convergence to frequency-synchronized wave-train solutions, we resolve for the first time the case of phase waves. These waves are nearly phase-synchronized relaxation oscillations, featuring quasistationary plateaus of length $\varepsilon^{-1}$ separated by fast transition layers, where $\varepsilon\ll1$ is the timescale separation parameter. Tracking the decay of modulations via a Bloch-wave eigenfunction analysis, we find a remarkably weak interaction strength of order $\varepsilon^{8/3}$. This weak layer interaction and many of the technical difficulties arise from repeated scattering of eigenfunctions through fold points at the ends of the quasistationary plateaus. We capture this by combining a novel geometric desingularization approach with Lin's method, exponential trichotomies, and the Riccati transform. While our spectral stability analysis yields diffusive synchronization of all phase waves in the FitzHugh-Nagumo system, it also identifies potential finite-wavelength instabilities, which we realize in a system variant.

Figures (16)

• Figure 1: Spacetime plot of profiles $u(x,t)$ corresponding to spatially homogeneous oscillations (left) and traveling wave trains (right) obtained numerically in \ref{['eq:FHN_pde']} for $a=0.2, \gamma=1, \varepsilon = 0.001$.
• Figure 2: Illustration of the fast decay of perturbations of trigger waves (left) compared to the weak relaxation of random perturbations of phase waves (right); see Appendix \ref{['s:dns']} for details on implementation.
• Figure 3: Localized perturbation of trigger waves (left column) and phase waves (right column) in the first component by $10^{-2}\exp{(-x^2/100)}$. Top: space-time plot of $u$-component (only partial domain on left) shows quick relaxation in the trigger-wave case, with perturbation only visible up to $t=200$, and a persistent defect in the phase wave case visible up to $t=15000$ (note the different spatial and temporal plot ranges). Middle: Snapshots of perturbation profiles, subtracting a closest perfect periodic wave train from the solution. Perturbations are modulated and large near interfaces. They slowly travel to the left with the group velocity and decay in amplitude as their width grows (again fast in the trigger- and slow in the phase-wave case). Bottom: The decay is diffusive, which is illustrated here by plotting the square of the width of the region where the perturbation exceeds $10^{-5}$ versus time. Data and linear fit with slope $4d_\mathrm{eff}$ shows, as $\varepsilon$ decreases, the increase $d_\mathrm{eff}\sim 1/\varepsilon$ in the trigger case, and the decrease $d_\mathrm{eff}\sim \varepsilon^{2/3}$ in the phase wave case; see Appendix \ref{['s:dns']} for details on implementation.
• Figure 4: The left schematic diagram depicts the left, middle, and right branches $\mathcal{M}^\mathrm{l,m,r}_0$ of the critical manifold $\mathcal{M}_0$, the locations of the fold points $(u,w)=(u_1,f(u_1))$ and $(u,w)=(\bar{u}_1,f(\bar{u}_1))$, and the nullcline $u-\gamma w-a=0$, under the conditions $0<a<1/2$ and $0<\gamma<\gamma_*(a)$. These conditions ensure that \ref{['eq:FHN_kinetics']} is in the oscillatory regime and exhibits relaxation oscillations. In particular, they exclude configurations in which equilibria lie on the outer branches $\mathcal{M}^\mathrm{l,r}_0$ of the critical manifold, such as the excitable and bistable regimes depicted in the top right and bottom right insets, respectively.
• Figure 5: Left column: (Top panel) Speed-period relation from numerical continuation of \ref{['eq:FHN_twode']} for fixed $a=0.2, \gamma=1, \varepsilon=0.001$. (Bottom panel) Plot of the temporal frequency $\omega(\ell)\coloneqq \ell c(\ell)$, where $\ell\coloneqq \tfrac{2\pi}{L_\varepsilon}$ is the spatial wavenumber. The value $c_*(a)$ denoting the transition from trigger to phase waves in the singular limit is marked by a red circle in both plots. Middle column: Profiles $u$ (solid) and $w$ (dashed) for a phase wave at $c=2$ (top panel) and a trigger wave at $c=0.4$ (bottom panel), marked by a green diamond and a black square, respectively, in the left panels. Note that the spatial period scales with the speed $c$, so that the slower trigger waves are much more narrowly spaced compared with the phase waves when plotted on the same spatial scale. Right column: phase space plots of the phase- and trigger-wave trains (solid green) from middle panels along with the cubic nullcline $w=f(u)$ (dashed red).

Theorems & Definitions (66)

• Theorem 1.1
• Theorem 1.2: Main result
• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• proof : Proof of Theorem \ref{['thm:spectral_stability']}
• Proposition 3.1