Learning non-stationary Langevin dynamics from stochastic observations of latent trajectories

Mikhail Genkin,Tatiana A Engel,Owen Hughes

doi:10.1038/s41467-021-26202-1

Mikhail Genkin, Tatiana A Engel + Show 1 more

Open Access

https://doi.org/10.1038/s41467-021-26202-1

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Abstract

Many complex systems operating far from the equilibrium exhibit stochastic dynamics that can be described by a Langevin equation. Inferring Langevin equations from data can reveal how transient dynamics of such systems give rise to their function. However, dynamics are often inaccessible directly and can be only gleaned through a stochastic observation process, which makes the inference challenging. Here we present a non-parametric framework for inferring the Langevin equation, which explicitly models the stochastic observation process and non-stationary latent dynamics. The framework accounts for the non-equilibrium initial and final states of the observed system and for the possibility that the system’s dynamics define the duration of observations. Omitting any of these non-stationary components results in incorrect inference, in which erroneous features arise in the dynamics due to non-stationary data distribution. We illustrate the framework using models of neural dynamics underlying decision making in the brain.

Highlights

Many complex systems operating far from the equilibrium exhibit stochastic dynamics that can be described by a Langevin equation
We focus on onedimensional (1D) Langevin dynamics representing a decisionmaking process on the domain x ∈ [−1; 1]
The equilibrium dynamics in the groundtruth potential predict lower probability density at the domain center than in the data, the likelihood is lower for the ground-truth potential than for the inferred shallow potential when incorrect initial distribution is used in the likelihood calculation. These results demonstrate that all three nonstationary components are critical for the accurate inference of non-stationary Langevin dynamics, and omitting any of them results in incorrect inference that accounts for the non-stationary data statistics by artifacts in the potential shape

Summary

Introduction

Many complex systems operating far from the equilibrium exhibit stochastic dynamics that can be described by a Langevin equation. The framework accounts for the non-equilibrium initial and final states of the observed system and for the possibility that the system’s dynamics define the duration of observations Omitting any of these non-stationary components results in incorrect inference, in which erroneous features arise in the dynamics due to nonstationary data distribution. Langevin equations are used to model stochastic evolution of complex systems such as neural networks[2,3,4,5], motile cells[6], swarming animals[7], carbon nanotubes[8], financial markets[9], or climate dynamics[10]. The inference of Langevin equations from data is crucial to enable efficient analysis, prediction, and optimization of complex systems

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