Self-induced stochastic resonance: A physics-informed machine learning approach

TL;DR

The paper tackles predicting self-induced stochastic resonance (SISR) in a slow–fast neuronal model by marrying stochastic dynamical systems theory with physics-informed machine learning. It introduces NASP-PINN, a noise-aware neural integrator built on an MLP backbone that learns the stochastic FitzHugh–Nagumo dynamics while enforcing data fidelity, dynamical residuals, and a barrier-based timescale-matching constraint derived from Kramers’ escape theory. The composite loss improves data efficiency, convergence, and generalization, enabling accurate long-horizon rollouts and correct replication of how spike-coherence (CV) depends on noise $\sigma$, excitability $a$, and timescale separation $\varepsilon$. The results establish NASP-PINN as an effective surrogate for complex stochastic multiscale dynamics, offering interpretable insights and a framework extendable to other noise-driven systems with similar slow–fast structure.

Abstract

Self-induced stochastic resonance (SISR) is the emergence of coherent oscillations in slow-fast excitable systems driven solely by noise, without external periodic forcing or proximity to a bifurcation. This work presents a physics-informed machine learning framework for modeling and predicting SISR in the stochastic FitzHugh-Nagumo neuron. We embed the governing stochastic differential equations and SISR-asymptotic timescale-matching constraints directly into a Physics-Informed Neural Network (PINN) based on a Noise-Augmented State Predictor architecture. The composite loss integrates data fidelity, dynamical residuals, and barrier-based physical constraints derived from Kramers' escape theory. The trained PINN accurately predicts the dependence of spike-train coherence on noise intensity, excitability, and timescale separation, matching results from direct stochastic simulations with substantial improvements in accuracy and generalization compared with purely data-driven methods, while requiring significantly less computation. The framework provides a data-efficient and interpretable surrogate model for simulating and analyzing noise-induced coherence in multiscale stochastic systems.

Figures (10)

• Figure 1: Excitable regime of the FHN model. (a) Phase portrait with two trajectories from different initial conditions converging to the fixed point $(v,w)=(0,0)$. (b) Time series of $v(t)$ (blue) and $w(t)$ (green) for the black trajectory in (a). Parameters: $a=0.1$, $b=1.0$, $c=2.0$.
• Figure 2: Potential landscape $U(v,w,a)$ for different $(a,w)$. The asymmetry of the double-well potential varies with $a$ and $w$: (a)--(c) right asymmetry for small $a$, (e) symmetry at intermediate $a$ for $w=0$, (d),(f) asymmetry at intermediate $a$ for $w\neq0$, and (g)--(i) left asymmetry for large $a$. $\Delta U_{\ell}(w,a)$ and $\Delta U_{r}(w,a)$ denote the left and right potential barriers.
• Figure 3: Monotonic dependence of the potential barriers $\Delta U_{\ell}(w,a)$ (blue) and $\Delta U_{r}(w,a)$ (red) on $w \in [w_{\min}, w_{\max}]=[-0.04775, 0.04775]$ at $a=0.5$.
• Figure 4: Effect of the excitability parameter $a$ on SISR. (a1)--(a3) Time series of $v(t)$ (blue) and $w(t)$ (red) at $\sigma=0.05$. (b1)--(b3) Corresponding phase portraits. (c) $\mathrm{CV}$ vs. noise intensity $\sigma$ for different $a$. Smaller $a$ values yield stronger SISR at weaker noise intensities. Parameters: $\varepsilon=0.00025$, $b=1.0$, $c=2.0$.
• Figure 5: Effect of the timescale parameter $\varepsilon$ on SISR. (a1)--(a3) Time series of $v(t)$ (blue) and $w(t)$ (red) at $\sigma=0.05$. (b1)--(b3) Corresponding phase portraits. (c) $\mathrm{CV}$ vs. $\sigma$ for different $\varepsilon$. Smaller $\varepsilon$ enhances SISR coherence at lower noise intensities. Parameters: $a=0.05$, $b=1.0$, $c=2.0$.