Wave-driven phase wave patterns in a ring of FitzHugh-Nagumo oscillators

TL;DR

This work introduces a two-field FitzHugh-Nagumo model comprising a cytoplasmic disc driving a cortical ring via unidirectional coupling to study wave-driven phase patterns. By varying the coupling strength, diffusion, and driving pulses, it identifies three external-regime outcomes—oscillatory, excitable, and non-excitable—each producing distinct phase-wave phenomena; a reduced linear-phase oscillator analysis links these patterns to driving pulse width and speed, yielding a phase-wavelength relation $\lambda = vT$. The findings illuminate how diffusion and pacemaker-driven waves shape cortical-like dynamics, offering a framework applicable to cellular signaling and other excitable-media contexts, and suggesting extensions to 3D or heterogeneous systems where spiral-turbulence may arise. Overall, the study demonstrates that phase patterns, traveling pulses, and connective-pulse phenomena emerge from simple unidirectional coupling and diffusion under a pacemaker-driven regime, with broad implications for interpreting wave-like activity in biology and beyond.

Abstract

We explore a biomimetic model that simulates a cell, with the internal cytoplasm represented by a two-dimensional circular domain and the external cortex by a surrounding ring, both modeled using FitzHugh-Nagumo systems. The external ring is dynamically influenced by a pacemaker-driven wave originating from the internal domain, leading to the emergence of three distinct dynamical states based on the varying strengths of coupling. The range of dynamics observed includes phase patterning, the propagation of phase waves, and interactions between traveling and phase waves. A simplified linear model effectively explains the mechanisms behind the variety of phase patterns observed, providing insights into the complex interplay between a cell's internal and external environments.

Figures (14)

• Figure 1: Model consisting of two coupled FHN Systems A. The depicted system consists of a disc (internal system) encircled by a ring (external system), each governed by the FHN equation within a circular area of radius $R=50$. The internal system is divided into two distinct zones: an oscillatory Region 1 centered within a radius $r=10$ at $\boldsymbol{x}=(-25,0)$; and an adjacent excitable Region 2 (see model parameters at Tab. \ref{['tab:modelParameters']}). The external system mirrors the parameters of Region 1 but incorporates a coupling factor $c$ that modulates its parameter $a(u_i)$ based on the internal field $u_i$, as illustrated by the white dashed line and the black arrow in panel B. B. Representation in the parameter space ($a,b$) that illustrates all the areas with distinct dynamical behaviors that appear in our model. Panels C-E further elaborate on these dynamics by presenting time series and phase space trajectories (including nullclines) for oscillatory, excitable, and non-excitable states, respectively.
• Figure 2: Pacemaker-induced traveling pulses. A. Pulses are initiated at the pacemaker region and propagate across the domain as traveling waves. B. Kymograph showing the pulses as they approach and reach the domain's boundary. A new pulse is initiated at intervals of $t=t_0+T$, where $t_0$ marks the initiation of the preceding pulse, ensuring a continuous generation and propagation of pulses to the boundary. See the associated dynamics in Supplementary Movie 1. The simulations are performed with the model's standard parameter set (see Tab. \ref{['tab:modelParameters']}).
• Figure 3: External ring dynamics driven by a constant wave pulse. A. In regimes with small coupling, phase patterns emerge (Supp.Movie 2). B. At intermediate coupling levels, a coexistence of slow traveling pulses alongside both slow and fast phase waves is observed (Supp.Movie 3). C. With large coupling, phase waves exhibiting phase shifts become prominent, characterized by a noticeable contraction in wave profiles (Supp.Movie 4). The simulations are performed with the model's standard parameter set (see Tab. \ref{['tab:modelParameters']}).
• Figure 4: From a limit cycle oscillator to a phase oscillator. A. The depicted limit cycle is transformed into a phase oscillator by employing the arctangent function to define the phase variable. B. This transformation yields a new variable $\theta$ that exhibits periodic oscillations, maintaining the original cycle's two fast and two slow segments. The parameters used here are consistent with those in Fig. \ref{['fig:Fig1']}C.
• Figure 5: Changes in phase velocity when changing system parameters A. Conceptual Framework: Illustration of the setup within phase space, highlighting how to quantify changes in phase velocity and accumulated phase changes over time, using the standard parameter set (see Tab. \ref{['tab:modelParameters']}) while changing the parameter $a$). B. Dynamics of accumulated phase changes: Visual representation of the phase differences that arise for different initial conditions and durations for three distinct changes of the parameter $a$ ($\Delta a$). C. Regions in phase space with different velocities, highlighting the "fast" regions as depicted in panel B. Close to $\theta_{\text{FP}}$ there is a slow "bottleneck" region. D. Accumulated phase difference after one complete cycle, in function of various initial phases, and for different $\Delta a$. E. The maximum (indicative of faster regions) and minimum (indicative of slower regions) phase differences identified in panel B, across all parameter variations considered in panel D. The maximum phase differences align linearly with modulation intensity ($\text{max}(\Delta\theta)=(10.538\pm0.001)\Delta a+1.000$), while the minimum phase differences exhibit an exponential decay relationship ($\text{min}\Delta\theta=0.001(\Delta a)^{-0.913\pm 0.001}$).