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Contraction and concentration of measures with applications to theoretical neuroscience

Simone Betteti, Francesco Bullo

TL;DR

The paper extends contraction theory to stochastic systems with spatially inhomogeneous diffusion and non-gradient drift terms, deriving conditions for existence and uniqueness of stationary measures and providing explicit $W_2$-convergence rates. It introduces a $c$-strong $B_r$-contractivity framework to analyze mass concentration around stable equilibria and deepest-energy minima in multistable settings, including a drift of the form $f(x)=-P(x)\nabla E(x)$. Theoretical results are illustrated via Hopfield networks, showing how memory retrieval dynamics under noise align with the predicted concentration patterns around dominant equilibria. These insights illuminate the interplay between contraction, diffusion inhomogeneity, and potential landscapes in noisy neural and dynamical systems, with practical implications for designing reliable memory-like retrieval under environmental fluctuations.

Abstract

We investigate the asymptotic behavior of probability measures associated with stochastic dynamical systems featuring either globally contracting or $B_{r}$-contracting drift terms. While classical results often assume constant diffusion and gradient-based drifts, we extend the analysis to spatially inhomogeneous diffusion and non-integrable vector fields. We establish sufficient conditions for the existence and uniqueness of stationary measures under global contraction, showing that convergence is preserved when the contraction rate dominates diffusion inhomogeneity. For systems contracting only outside of a compact set and with constant diffusion, we demonstrate mass concentration near the minima of an associated non-convex potential, like in multistable regimes. The theoretical findings are illustrated through Hopfield networks, highlighting implications for memory retrieval dynamics in noisy environments.

Contraction and concentration of measures with applications to theoretical neuroscience

TL;DR

The paper extends contraction theory to stochastic systems with spatially inhomogeneous diffusion and non-gradient drift terms, deriving conditions for existence and uniqueness of stationary measures and providing explicit -convergence rates. It introduces a -strong -contractivity framework to analyze mass concentration around stable equilibria and deepest-energy minima in multistable settings, including a drift of the form . Theoretical results are illustrated via Hopfield networks, showing how memory retrieval dynamics under noise align with the predicted concentration patterns around dominant equilibria. These insights illuminate the interplay between contraction, diffusion inhomogeneity, and potential landscapes in noisy neural and dynamical systems, with practical implications for designing reliable memory-like retrieval under environmental fluctuations.

Abstract

We investigate the asymptotic behavior of probability measures associated with stochastic dynamical systems featuring either globally contracting or -contracting drift terms. While classical results often assume constant diffusion and gradient-based drifts, we extend the analysis to spatially inhomogeneous diffusion and non-integrable vector fields. We establish sufficient conditions for the existence and uniqueness of stationary measures under global contraction, showing that convergence is preserved when the contraction rate dominates diffusion inhomogeneity. For systems contracting only outside of a compact set and with constant diffusion, we demonstrate mass concentration near the minima of an associated non-convex potential, like in multistable regimes. The theoretical findings are illustrated through Hopfield networks, highlighting implications for memory retrieval dynamics in noisy environments.

Paper Structure

This paper contains 6 sections, 5 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Contracting drifts and convergence to stationary measures
  4. $B_{r}$-contracting drifts and concentration of stationary measures
  5. Input-driven Hopfield networks and stochastic memory retrieval
  6. Conclusions

Key Result

Theorem 5

Let $\{\mathbf{X}_{t}\}_{t\geq 0}$ be the unique strong solution to eq. eq: SDE-ref with drift $f$ and diffusion $G$ satisfying Assumption as: LL. Let $f$ be c-strongly contracting w.r.t. the $2$-norm. If $c>L_{G}/2$, then the solution $\upmu_{t}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ to eq. eq: KFE-ref

Figures (3)

  • Figure 1: Visual example of potentials $\mathrm{E}(x)$ associated to globally contracting and $B_{r}$-contracting vector fields $f$. (a) Globally contracting vector field associated to a convex potential. (b) $B_{r}$-contractive vector field associated to a mostly convex potential, with a small concave region.
  • Figure 2: A stochastic Hopfield model with a globally contracting drift term. (a) Energy function associated to the Hopfield model, with a unique global minimum in the origin. (b) The stationary measure is a gaussian centered at the origin - the globally asymptotically stable equilibrium point of the drift term. The black arrows beneath panels (a-b) are the streamlines associated to the drift $f$. (c) Exponential trend of convergence towards stationarity, conformably with the results of Theorem \ref{['thm: stat']}.
  • Figure 3: A stochastic Hopfield model with a $B_{r}$-contracting drift term. (a) Energy associated to the multistable Hopfield model, with antisymmetric minima w.r.t. to the origin. As observable, the minima $(3,-3)$ and $(-3,3)$ associated with the input $u_{2}$ are far deeper than the minima $(1,1)$ and $(-1,-1)$ associated with the input $u_{1}$. (b) The stationary measure concentrates its mass around the stable equilibrium points associated with the deepest Energy $\mathrm{E}(x)$ minima, as predicted by Theorem \ref{['thm: mass']}. (c) Exponential trend of convergence towards stationarity, conformably with the results in PM:23.

Theorems & Definitions (15)

  • Definition 2: Infinitesimal generator
  • Definition 3: Wasserstein metric
  • Definition 4: c-strong contractivity
  • Theorem 5: Contraction of measures
  • proof
  • Proposition 7: Contraction for different diffusions
  • proof
  • Definition 8: $c$-strong $B_{r}$-contractivity
  • Proposition 10: Mass sinks
  • proof
  • ...and 5 more