Supersymmetric Neuromorphic FitzHugh--Nagumo model

Technical Summary: Supersymmetric Spectral Description of Stochastic FitzHugh–Nagumo Dynamics

Problem Statement
The paper addresses the challenge of mapping the stochastic dynamics of a neuromorphic antiferromagnetic (AFM) system onto a quantum-like formalism without asserting the existence of microscopic quantum coherence. Specifically, the authors seek to describe the stochastic FitzHugh–Nagumo (FHN) model, derived from the overdamped dynamics of a Néel vector in an AFM/heavy-metal bilayer, using a supersymmetric (SUSY) spectral framework. The core difficulty lies in the fact that the full two-dimensional FHN drift is non-gradient, preventing a global scalar-potential representation. The authors aim to bridge classical stochastic relaxation spectra with an "auxiliary" two-state amplitude representation to diagnose coherence in neuromorphic spiking, while rigorously defining the limits of this approximation.

Methodology
The study proceeds through a multi-stage reduction and transformation process:

1. Neuromorphic Derivation: Starting from the Euler-Lagrange-Rayleigh equations for a constrained Néel vector on a unit sphere, the authors perform a local normal-form reduction. By introducing a north-pole ansatz and expanding around a local operating point, they transform the tangential Néel-vector dynamics into the canonical FHN normal form. This establishes a direct link between magnetic control parameters (spin torque, anisotropy) and the effective FHN parameters ($v, w$).
2. Stochastic Formulation: The deterministic FHN model is promoted to an Itô stochastic differential equation to include thermal and electronic fluctuations. This leads to a two-dimensional Fokker-Planck equation for the probability density $P(V, W, \theta)$.
3. Adiabatic and SUSY Reduction: Utilizing the time-scale separation ($\epsilon \ll 1$), the slow recovery variable $W$ is frozen. This reduces the problem to a one-dimensional Fokker-Planck equation for the fast activation variable $V$ with a quartic potential.
4. Supersymmetric Factorization: The one-dimensional Fokker-Planck generator undergoes a similarity transformation, mapping it to an imaginary-time Schrödinger equation with a Hermitian, positive-semidefinite Hamiltonian $\hat{H} = A^\dagger A$. This allows for a supersymmetric factorization where the bosonic Hamiltonian $\hat{H}_B$ and its fermionic partner $\hat{H}_F$ share a spectrum (except for the ground state).
5. Coherence Diagnostic: The authors construct a normalized auxiliary two-state amplitude by projecting the similarity-transformed probability density onto the two lowest eigenmodes. They derive a localized-basis coherence metric, $C^{(\chi)}_{l2,LR}$, expressed in terms of time-dependent basin populations ($p_L, p_R$) and their equilibrium weights ($\pi_L, \pi_R$).

Key Contributions and Results

• Spectral Characterization of Relaxation: The authors demonstrate that the first nonzero relaxation eigenvalue of the Fokker-Planck operator is exactly the ground-state energy of the supersymmetric partner Hamiltonian. In the weak-noise bistable regime, this eigenvalue is approximated by the sum of the two interwell Kramers rates ($k_{L\to R} + k_{R\to L}$), providing a spectral characterization of interwell equilibration.
• Auxiliary Coherence vs. Classical Populations: A central result is the derivation of the relationship between the auxiliary amplitude weights and classical basin populations. The maximum coherence ($1/\sqrt{2}$) occurs when the normalized amplitude weights are equal, which generally corresponds to unequal basin populations ($p_L \neq p_R$) unless the potential is symmetric. This distinguishes the auxiliary coherence diagnostic from simple population mixing.
• Validity Limits in Spiking Dynamics: The paper rigorously tests the localized two-state approximation against deterministic spiking trajectories. It finds that while fixed-recovery relaxation exhibits a coherence pulse without a spike, deterministic spiking trajectories frequently exit the bistable domain (crossing the folds where $|W - I_0| = 2/3$) and traverse regions with insufficient barrier separation. Consequently, the localized two-state approximation is not uniformly valid throughout a full spike cycle.
• Distinction from Quantum Coherence: The authors explicitly state that the derived "coherence" is a property of the chosen normalized amplitude representation of classical stochastic dynamics. It does not, by itself, establish microscopic quantum coherence in the magnetic device.

Significance and Claims
The paper claims to provide a "controlled supersymmetric construction" that connects stochastic relaxation spectra to an occupation-based coherence diagnostic. Its significance lies in:

1. Mapping Neuromorphic Systems: Establishing a rigorous mathematical bridge between antiferromagnetic spin-Hall nano-oscillators and the FHN model, and further mapping this stochastic system into a Schrödinger-like form via the Nelson map.
2. Diagnostic Tool: Offering a specific diagnostic ($C^{(\chi)}_{l2,LR}$) to analyze the dynamics of neuromorphic spiking, while clarifying that this diagnostic is basis-dependent and distinct from physical quantum coherence.
3. Defining Regimes of Validity: Delimiting the application of the two-state approximation. The authors emphasize that the approximation holds for slow recovery variables and weak noise but breaks down during rapid spiking excursions where the system leaves the bistable domain or encounters low barriers.

The work concludes that while the framework successfully links spectral relaxation rates to auxiliary amplitude coherence, applying this to full neuromorphic spiking cycles requires direct propagation of the stochastic distribution rather than reliance on deterministic waveforms alone. The results characterize an auxiliary representation of classical dynamics, not the emergence of quantum effects in the hardware.