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Synchronization by noise for traveling pulses

Christian Kuehn, Joris van Winden

TL;DR

The paper provides the first rigorous proof of synchronization by noise for traveling pulses in SPDEs, using a carefully constructed isochronal phase reduction to reduce the dynamics to a one-dimensional phase SDE for the pulse position. By establishing ergodicity, negative Lyapunov exponent, and irreducibility properties for the reduced system, the authors show that two initially separated pulses driven by the same noise synchronize in probability on the time window $σ^{-2} \ll t \ll \exp(σ^{-2})$, with a lift from the reduced model back to the full SPDE under mild conditions. The approach hinges on a precise phase-orientated description, with a rigorous analysis of the phase map derivatives and the noise-induced drift, and yields insights into when and how noise can enforce spatial synchronization of propagating pulses. The results have potential implications for neuroscience and cardiac dynamics, and lay groundwork for extending phase reduction and synchronization results to higher-dimensional patterns and alternative noise structures.

Abstract

We consider synchronization by noise for stochastic partial differential equations which support traveling pulse solutions, such as the FitzHugh-Nagumo equation. We show that any two pulse-like solutions which start from different positions but are forced by the same realization of a multiplicative noise, converge to each other in probability on a time scale $σ^{-2} \ll t \ll \exp(σ^{-2})$, where $σ$ is the noise amplitude. The noise is assumed to be Gaussian, white in time, colored and periodic in space, and non-degenerate only in the lowest Fourier mode. The proof uses the method of phase reduction, which allows one to describe the dynamics of the stochastic pulse only in terms of its position. The position is shown to synchronize building upon existing results, and the validity of the phase reduction allows us to transfer the synchronization back to the full solution.

Synchronization by noise for traveling pulses

TL;DR

The paper provides the first rigorous proof of synchronization by noise for traveling pulses in SPDEs, using a carefully constructed isochronal phase reduction to reduce the dynamics to a one-dimensional phase SDE for the pulse position. By establishing ergodicity, negative Lyapunov exponent, and irreducibility properties for the reduced system, the authors show that two initially separated pulses driven by the same noise synchronize in probability on the time window , with a lift from the reduced model back to the full SPDE under mild conditions. The approach hinges on a precise phase-orientated description, with a rigorous analysis of the phase map derivatives and the noise-induced drift, and yields insights into when and how noise can enforce spatial synchronization of propagating pulses. The results have potential implications for neuroscience and cardiac dynamics, and lay groundwork for extending phase reduction and synchronization results to higher-dimensional patterns and alternative noise structures.

Abstract

We consider synchronization by noise for stochastic partial differential equations which support traveling pulse solutions, such as the FitzHugh-Nagumo equation. We show that any two pulse-like solutions which start from different positions but are forced by the same realization of a multiplicative noise, converge to each other in probability on a time scale , where is the noise amplitude. The noise is assumed to be Gaussian, white in time, colored and periodic in space, and non-degenerate only in the lowest Fourier mode. The proof uses the method of phase reduction, which allows one to describe the dynamics of the stochastic pulse only in terms of its position. The position is shown to synchronize building upon existing results, and the validity of the phase reduction allows us to transfer the synchronization back to the full solution.
Paper Structure (30 sections, 24 theorems, 104 equations, 1 figure)

This paper contains 30 sections, 24 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Synchronization by noise
  4. Phase reduction
  5. Proof strategy
  6. Outline
  7. Setting, assumptions and preliminaries
  8. Notation
  9. Analytic setting and linear stability
  10. Noise
  11. Probability, RDS, synchronization
  12. Phase reduction
  13. Isochronal phase
  14. Isochron map derivatives
  15. Reduced phase SDE
  16. ...and 15 more sections

Key Result

Theorem 1

Suppose that the times $(t_{\sigma})_{\sigma > 0}$ satisfy for some $q \in (0,2)$. Then for any $x,y \in {\mathbb R}$, we have as $\sigma \to 0$.

Figures (1)

  • Figure 1: Trajectories starting at $z_2$ and $z_3$ 'squeeze' near $z_1$ and afterwards return to their initial position.

Theorems & Definitions (59)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • Remark 3.2
  • ...and 49 more