Dynamical Effects of Multiplicative Feedback on a Noisy System

TL;DR

This paper investigates how the interpretation of stochastic differential equations (SDEs) for noisy systems with feedback can change under different operational conditions, affecting system stability and behavior.

Contribution

It experimentally demonstrates that the calculus convention (Itô or Stratonovich) for a physical system can switch depending on conditions, impacting long-term stability predictions.

Findings

A noisy electric circuit shifts from Stratonovich to Itô calculus under certain conditions.

The transition depends on the ratio of noise correlation time to feedback delay.

Such transitions can significantly alter the predicted stability of the system.

Abstract

Intrinsically noisy mechanisms drive most physical, biological and economic phenomena, from stock pricing to phenotypic variability. Frequently, the system's state influences the driving noise intensity, as, for example, the actual value of a commodity may alter its volatility or the concentration of gene products may regulate their expression. All these phenomena are often modeled using stochastic differential equations (SDEs). However, an SDE is not sufficient to fully describe a noisy system with a multiplicative feedback, because it can be interpreted according to various conventions -- in particular, It\^{o} calculus and Stratonovitch calculus --, each of which leads to a qualitatively different solution. Which convention to adopt must be determined case by case on the basis of the available experimental data; for example, the SDE describing electrical circuits driven by a noise are known to obey Stratonovich calculus. Once such an SDE-convention pair is determined, it c an be employed to predict the system's behavior under new conditions. Here, we experimentally demonstrate that the convention for a given physical system may actually vary under varying operational conditions. We show that, under certain conditions, a noisy electric circuit shifts to obey It\^o calculus, which may dramatically alter the system's long term stability. We track such Stratonovich-to-It\^o transition to the underlying dynamics of the system and, in particular, to the ratio between the driving noise correlation time and the feedback delay time. We briefly discuss ramifications of our conclusions for biology and economics: the possibility of similar transitions and their dramatic consequences should be recognized and accounted for where SDEs are employed to predict the evolution of complex phenomena.