Bifurcation Analysis of the Driven FitzHugh-Nagumo Oscillator: Prediction and Experiment

TL;DR

This study develops a comprehensive 2D regime map for the sinusoidally driven FitzHugh-Nagumo oscillator in the ($f$,$c$) plane and validates it against an analog circuit. By transforming the non-autonomous system to an autonomous form and applying bifurcation analysis with XPPAUT, the authors identify periodic, subharmonic, and chaotic dynamics, including an island where the drive frequency is unstable. The work reveals two dominant families of subharmonic limit cycles and confirms the numerical regime map with circuit measurements, strengthening confidence in driven excitable-system modeling. The findings illuminate fundamental resonance phenomena in simple excitable systems and provide a dense resource for planning experiments involving more complex FHN configurations.

Abstract

Bifurcation analysis is applied to the FitzHugh-Nagumo oscillator driven by a sinusoidal source. A numerically generated 2d regime map showing the variety of oscillatory dynamics in the parameter space of source frequency and amplitude agrees well with a map created from analog circuit measurements. Application of the sinusoidal source to the fast variable's first-order differential equation produces an island in the map in which oscillations at the source frequency are unstable and the behavior is dominated by two distinct families of subharmonic limit cycles and by chaos. Previously published maps are portions of the map shown here and are shown to be consistent with it. The more detailed and comprehensive regime map presented here should facilitate the understanding of this foundational system and thereby aid the ongoing research involving more complicated implementations of the Fitzhugh-Nagumo system.

Figures (27)

• Figure 1: Numerical $i$-continuation bifurcation plot showing the maximum value of $u$ for no sinusoidal source. Red is fixed point, green is limit cycle. Solid (dashed) line indicates stable (unstable). Inset shows stable limit cycle time-series of $u$ (solid) and $v$ (dashed) at indicated value $i=0.25$. Value $i=0.1$ is used for this paper. $\epsilon=0.01$, $a=0.25$, and $b=1$.
• Figure 2: Numerical $f$-continuation bifurcation plot showing the maximum value of $u$ for the P1, P2, P3, P4, and P6 LCs. $\epsilon=0.01$, $a=0.25$, $b=1$, $i=0.1$, and $c=0.05$. Solid (dashed) line indicates stable (unstable) LC. PD and LP bifurcation points are indicated. Insets show stable LC time-series of $u$ and $v$ (blue) for P1 at $f=0.012$, P2 at $f=0.025$ and P4 at $f=0.038$.
• Figure 3: Analog circuit for the FitzHugh Nagumo system with sinusoidal source $V(t)$. $R=10^4 \Omega$, $C=10^{-9}$ f.
• Figure 4: Numerical $f$-$c$ regime map for $\epsilon=0.01$, $a=0.25$, $b=1$, and $i=0.1$. P1-P4 and P6 LC regions are indicated. Frequency $f$ and source amplitude $c$ are dimensionless. LP lines are dashed, PD lines are solid. Chaos indicated by diagonal lines.
• Figure 5: Numerical time series of the P3 LC for $f=0.033, c=0.05$. $u$ solid line, $v$ dashed line. Sinusoidal source is shown in red.