Mean-square spectral stability of stochastic FitzHugh–Nagumo networks with noise

Bing Zhou; Manrui Zhou; Xiangrong Liu; Qianqian Zheng; Bing Zhou; Manrui Zhou; Xiangrong Liu; Qianqian Zheng

doi:10.3934/math.20261213

AIMS Mathematics

2026, Volume 11, Issue 9: 30619-30642. doi: 10.3934/math.20261213

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Research article

Mean-square spectral stability of stochastic FitzHugh–Nagumo networks with noise

1.
School of Information Engineering, Zhongyuan Institute of Science and Technology, Xuchang 461000, China
2.
School of Science, Xuchang University, Xuchang 461000, China

Received: 05 July 2026 Revised: 02 September 2026 Accepted: 08 September 2026 Published: 20 September 2026
MSC : 34D20, 60H10, 92B20

This study investigates mean-square spectral instability in network-organized FitzHugh-Nagumo systems under multiplicative noise. By linearizing the stochastic dynamics and analyzing the spectrum of the second-moment operator, we state the exact mean-square stability condition and derive an explicit, generally conservative, spectral condition obtained with the Lyapunov choice $ P = I $. The framework distinguishes direct noise acting on the membrane-potential variable, recovery-channel noise, and their combined influence. Numerical simulations illustrate how these noise channels and network connectivity change finite-network variability and synchronization diagnostics. Direct noise more strongly shifts the determinant component of the conservative spectral boundary in the parameter regimes studied, whereas recovery-channel noise produces a more gradual change. The results provide a mathematical and numerical framework for studying noise-dependent second-moment stability in excitable networks; they are not intended as evidence for clinical seizure regulation or as a demonstrated neuromodulation strategy.
- FitzHugh–Nagumo model,
- multiplicative noise,
- network,
- mean-square stability,
- synchronization
Citation: Bing Zhou, Manrui Zhou, Xiangrong Liu, Qianqian Zheng. Mean-square spectral stability of stochastic FitzHugh–Nagumo networks with noise[J]. AIMS Mathematics, 2026, 11(9): 30619-30642. doi: 10.3934/math.20261213

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Abstract

This study investigates mean-square spectral instability in network-organized FitzHugh-Nagumo systems under multiplicative noise. By linearizing the stochastic dynamics and analyzing the spectrum of the second-moment operator, we state the exact mean-square stability condition and derive an explicit, generally conservative, spectral condition obtained with the Lyapunov choice $ P = I $. The framework distinguishes direct noise acting on the membrane-potential variable, recovery-channel noise, and their combined influence. Numerical simulations illustrate how these noise channels and network connectivity change finite-network variability and synchronization diagnostics. Direct noise more strongly shifts the determinant component of the conservative spectral boundary in the parameter regimes studied, whereas recovery-channel noise produces a more gradual change. The results provide a mathematical and numerical framework for studying noise-dependent second-moment stability in excitable networks; they are not intended as evidence for clinical seizure regulation or as a demonstrated neuromodulation strategy.

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