Singularity of information flow at the Hopf bifurcation point

In information thermodynamics, information flow is quantified by the learning rate. In the second law of information thermodynamics, the entropy production rate is bounded from below by the learning rate, revealing the inseparability of energy and information flows.  

In biological systems with regulatory mechanisms, oscillations often emerge. When the system exhibits oscillatory dynamics, variables tend to follow each other, implying information flow exists. Therefore, elucidating the relationship between oscillations and information flow is essential for understanding biochemical reactions.

In this presentation, we investigate the relationship between oscillatory behavior and information flow. Specifically, we focus on a chemical reaction system exhibiting a Hopf bifurcation, and describe the dynamics of chemical concentrations by the Langevin equation. We analyze the singular behavior of the learning rate at the Hopf bifurcation point by extending a singular perturbation method to the Langevin equation. Moreover, we demonstrate that the learning rate obtained from numerical simulations is quantitatively reproduced by this perturbative analysis. Finally, we show that, in the deterministic limit, the learning rate exhibits a non-smooth change at the Hopf bifurcation point.