FitzHugh-Nagumo Neuronal Units

Updated 4 February 2026

FitzHugh–Nagumo neuronal units are two-variable models that simplify neuron excitability by coupling a fast membrane potential with a slow recovery process.
The model facilitates analytical and numerical investigations of key phenomena such as bifurcations, synchronization, and multistability in both isolated and coupled networks.
Coupled FHN units reveal complex dynamics including phase locking, Arnold tongues, and mixed-mode oscillations, providing insights into coordinated neural behavior.

The FitzHugh–Nagumo (FHN) neuronal unit is a canonical two-variable dynamical system that captures the essential mechanisms of excitability and spiking in neurons. Abstracted from the full Hodgkin–Huxley model, FHN units combine a fast "membrane potential" variable with a slow "^{^{^{^{1^{^{^{^"}}}}}}} variable to reproduce the core features of neuronal action potentials, robustness to noise, and the emergence of rich collective dynamics in neuronal networks. The model's tractability enables systematic analytical and numerical investigation of bifurcations, synchronization, phase locking, and multistability, both in isolation and when coupled in small or large networks.

1. Mathematical Formulation and Single-Unit Dynamics

The canonical FHN oscillator consists of two nonlinear ODEs for the activator ( $u(t)$ ) and inhibitor/recovery ( $w(t)$ ) variables, with parameters controlling threshold, timescale separation, and recovery rate. The Campbell–Waite form is

$\dot u = c(w + u - \tfrac13 u^3 + J(t)), \quad \dot w = -\frac{1}{c}(u - a + b w)$

where $u$ is the membrane potential, $w$ is the recovery variable, $a$ sets threshold, $b$ the slope of recovery, $c$ is (inverse) timescale separation, and $J(t)$ is potentially an external drive (usually set to zero for analysis of the autonomous system). In absence of external input, the dynamics of a single FHN unit are restricted to fixed-point equilibrium or periodic oscillation; chaotic dynamics are not attainable in two dimensions (Hoff et al., 2015).

Phase plane analysis reveals that the system is excitable—small perturbations decay, but suprathreshold perturbations yield a large excursion ("spike"). Bifurcations occur as parameters (notably $a$ or input $J$ ) are varied. For typical neuronal parameter ranges, supercritical Hopf bifurcations appear at critical thresholds, giving rise to limit cycles (tonic spiking) and transitions between rest and sustained activity (Cebrián-Lacasa et al., 2024, Gürbüz et al., 29 Jan 2025).

2. Coupling Schemes and Network Extensions

Two FHN units, when coupled, form a minimal network supporting substantially richer dynamics than the single-unit case. The principal coupling configurations are:

Unidirectional (master–slave):

$\begin{cases} \dot x_1 = c(y_1 + x_1 - \tfrac13 x_1^3) + \gamma(x_1 - x_2) \ \dot y_1 = -\frac{1}{c}(x_1 - a + b y_1) \ \dot x_2 = c(y_2 + x_2 - \tfrac13 x_2^3) \ \dot y_2 = -\frac{1}{c}(x_2 - a + b y_2) \end{cases}$

The second unit (driver) influences the first, but not vice versa (Hoff et al., 2015).

Bidirectional (diffusive):

$\begin{cases} \dot x_1 = c(y_1 + x_1 - \tfrac13 x_1^3) + \gamma(x_1 - x_2) \ \dot y_1 = -\frac{1}{c}(x_1 - a + b y_1) \ \dot x_2 = c(y_2 + x_2 - \tfrac13 x_2^3) + \gamma(x_2 - x_1) \ \dot y_2 = -\frac{1}{c}(x_2 - a + b y_2) \end{cases}$

Both units couple symmetrically.

Phase-locking and tree/topology: Directed chains and more general trees can be considered, with sufficient coupling strength for complete phase locking throughout the network. A sufficient condition is $g_{j\to i} > 1 - b\varepsilon$ for every directed edge $j\to i$ (Davison et al., 2018).
Generalized networks: For networks consisting of $N$ FHN neurons coupled by electrical synapses (gap-junctions), the dynamics are given by

$\dot v_i = f(v_i,w_i) + D\sum_{j\in\mathcal{N}(i)}(v_j - v_i), \quad \dot w_i = g(v_i,w_i)$

where $D$ is the gap junction conductance (Canuto et al., 2013).

3. Bifurcation and Dynamical Phenomena in Coupled Units

Even in a two-unit network, the system exhibits dynamics unobtainable in a single FHN oscillator. Key phenomena include:

Arnold tongues: In unidirectional forcing at sufficiently strong coupling, wedge-shaped tongues of robust periodic (frequency-locked) oscillation appear amidst quasiperiodicity or chaos. These tongues are organized by the period-adding (Stern–Brocot/Farey) rule: between tongues of periods $P$ and $Q$ , a tongue of period $P+Q$ appears. Saddle-node bifurcation curves bound these tongues; Neimark–Sacker curves mark torus birth (Hoff et al., 2015).
Multistability: For the same parameters but different initial conditions, distinct attractors (regular periodic orbits, tori, chaotic attractors) may coexist, with basins of attraction showing complex, often fractal structures (Hoff et al., 2015).
Mixed-mode oscillations (MMOs): When timescale separation is leveraged ( $\varepsilon\to 0$ limit), canard-induced MMOs (alternating small- and large-amplitude oscillations) are found, with folded singularities and folded saddle-node bifurcations organizing parameter regions. Canards and MMOs require crossing of specific bifurcation loci in parameter space, typically appearing for intermediate coupling strengths (Davison et al., 2018).
Phase-locking: Sufficiently strong unidirectional or diffusive coupling results in all downstream units entrained to the upstream oscillator, with a robust phase relationship determined by the root oscillator (Davison et al., 2018).

4. Numerical and Analytical Techniques

Quantitative study of FHN networks relies on multiple concurrent approaches:

Lyapunov exponent spectra: Calculation of the full Lyapunov spectrum determines regions of periodic, quasiperiodic, and chaotic behavior. The largest Lyapunov exponent, $\lambda_1$ , distinguishes between stable, marginal, and chaotic regimes (Hoff et al., 2015).
Isoperiodic diagrams: Automated period-counting captures the structure of periodicity windows, including tongues and their period-adding sequences (Hoff et al., 2015).
Numerical continuation: Curves of saddle-node, Hopf, Neimark–Sacker, and period-doubling bifurcations are computed and superimposed to link observed dynamical transitions to their organizing bifurcations (Hoff et al., 2015).
Singular perturbation analysis and geometric methods: For small $\varepsilon$ , critical manifolds, folds, and canard structures can be analyzed using Fenichel theory and canard explosion mechanisms. Folded node/saddle transitions determine emergence of MMOs (Davison et al., 2018).
Discriminant variety (algebraic geometry): For small networks, discriminant varieties and cylindrical algebraic decomposition rigorously partition parameter space into regions of distinct stability and number of equilibria (Hanan et al., 2010).

5. Influence of Noise and Extreme Fluctuations

Stochastic forcing, whether in parameter drift or explicit noise sources, can qualitatively alter the behavior of FHN units and networks:

Coherence resonance: In excitable regimes, noise can optimize spike train regularity at an intermediate noise amplitude. Inter-spike interval variability (coefficient of variation) versus noise amplitude demonstrates a non-monotonic minimum (Medeiros et al., 2011).
Noise-induced extreme events: Even a single FHN oscillator under weak white noise can experience rare, large-amplitude excursions not predicted by deterministic bifurcation theory (extreme events). These are analyzable via large fluctuation theory and Wentzell–Freidlin path minimization; mean escape times scale as $T(D) \sim D^{3/2} \exp(S^*/D)$ with calculated action $S^*$ (Hariharan et al., 29 Jan 2025).
Poisson statistics: At low noise, spike generation is sufficiently rare that interspike intervals are exponentially distributed, as in spontaneous firing of biological neurons (Medeiros et al., 2011).
Activation in small networks: In coupled units, the activation paths and firing order can become noise-dependent, and collective noise-driven bifurcations emerge, including a universal three-regime structure in mean activation times (long, plateau, short) tied to stochastic Hopf bifurcations in cumulant-closed moment systems (Franović et al., 2015).

6. Excitability Classification and Physiological Implications

The FHN model and its extensions organize neuron excitability into several canonical regimes, depending on parameters and model variants:

Type II excitability (classical FHN): Oscillation onset at a Hopf bifurcation with finite-frequency oscillations and a finite amplitude at onset (Cebrián-Lacasa et al., 2024).
Type I excitability: In parameter regimes supporting a homoclinic (SNIC) bifurcation, arbitrarily low frequency oscillations emerge at threshold.
Novel types and bistabilities: Extensions that include additional gating (e.g., calcium conductances) as cooperative variables yield both pitchfork and transcritical bifurcations, generating bistability (multiple fixed points), slow passage effects (spike latency), plateau oscillations, and complex bursting (Franci et al., 2011, Franci et al., 2012).
Multistability and synchronization: Coupled FHN networks generically support multiple attractors; synchronization and complete phase-locking occur at strong coupling, and intricate dynamics (chimera states, cluster synchronization) arise in suitable network architectures.

7. Spatially Extended and Geometric Generalizations

FHN units also form the basis for spatially continuous models:

Reaction–diffusion equations: Adding diffusive coupling leads to traveling pulse, wave train, spiral wave, and Turing pattern dynamics, with precise instabilities and bifurcations determined by the interplay of diffusion, local kinetics, and spatial heterogeneity (Cebrián-Lacasa et al., 2024, Ambrosio, 2019).
Nontrivial geometry: On non-flat domains such as undulated cylinders (modeling axons), geometric effects can be rigorously reduced to effective 1+1D FHN systems with modified diffusion coefficients, justified via exponential contraction ("spontaneous symmetrization") of nonradial modes and accurate tracking of the radial mean (Karali et al., 9 Apr 2025).
Continuum network limits: Scaling limits of discrete gap-junction networks recover reaction–diffusion–convection PDEs, with effective diffusion and drift coefficients determined by the local topology and coupling scaling (Canuto et al., 2013).
Families of exact solutions: Nonclassical symmetries yield uncountably infinite families of explicit closed-form solutions in the PDE, including exponential traveling waves, heteroclinic fronts, and periodic Jacobi-elliptic patterns (Ramji et al., 2024).

The FitzHugh–Nagumo paradigm, through its balance of biophysical realism and analytical accessibility, remains foundational in the study of neuronal excitability, oscillations, synchronization, and pattern formation. When assembled into coupled units or spatial networks, FHN models reveal mathematically rigorous mechanisms underlying the diversity of neural responses, robust multi-attractor dynamics, and the emergence of complex temporal and spatial structures characteristic of excitable biological tissues (Hoff et al., 2015, Davison et al., 2018, Cebrián-Lacasa et al., 2024).