Thermodynamics of collective enhancement of precision

The circadian oscillator exhibits remarkably high temporal precision, despite its exposure to several fluctuations. The central mechanism that protects the oscillator from fluctuations is a collective enhancement of precision where a population of coupled oscillators displays higher temporal precision than that achieved without coupling. Since coupling is essentially information exchange between oscillators, we herein investigate the relation between the temporal precision and the information flow between oscillators in the linearized Kuramoto model by using stochastic thermodynamics. For general coupling, we find that the temporal precision is bounded from below by the information flow. We generalize the model to incorporate a time-delayed coupling and demonstrate that the same relation also holds for the time-delayed case. Furthermore, the temporal precision is demonstrated to be improved in the presence of the time delay, and we show that the increased information flow is responsible for the time-delay-induced precision improvement.

Figure 1

Analogy between (a) the Maxwell demon and (b) the CEP. (a) When a system $\mathcal{X}$ is coupled to another system $\mathcal{Y},\mathcal{X}$'s medium entropy $\mathrm{\Delta }{S}_{\mathcal{X}}^m$, which is the heat from $\mathcal{X}$ to the medium (divided by the temperature), can be negative. (b) When an oscillator $\mathcal{X}$ is coupled to another oscillator $\mathcal{Y},{\mathcal{V}}_{\mathcal{X}}$ (the temporal variance with coupling) can be smaller than ${\mathcal{V}}_{\mathcal{X}}^{\mathrm{w}/\mathrm{o}}$(the temporal variance without coupling).

Figure 2

Coupled oscillators model. (a) Adopted coupled oscillator model for $N=7(V=\{1,2,...,7\left\}\right)$. Each oscillator is entrained by a periodic signal with the angular frequency $\mathrm{\Omega }$ and is subject to noise ${\xi }_i\left(t\right)$. (b) Observable set $V_O$ and latent set $V_L$, where $V_O=\{1,2,5\}$ and $V_L=\{3,4,6,7\}$ in this example.

Figure 3

Variance ${\mathcal{V}}_X$ and its lower bound ${\mathcal{V}}_X^{\mathrm{LB}}$ as a function of $R=N_L/N_O$ for (a) $D=0.001$ and (b) $D=0.1$. The solid, long-dashed, and dashed lines denote ${\mathcal{V}}_X^{\mathrm{LB}},{\mathcal{V}}_X$ with $K=1$, and ${\mathcal{V}}_X$ with $K=\infty$, respectively, obtained analytically. The squares, crosses, circles, and triangles denote ${\mathcal{V}}_X^{\mathrm{LB}}$ with $K=1,{\mathcal{V}}_X^{\mathrm{LB}}$ with $K=10,{\mathcal{V}}_X$ with $K=1$, and ${\mathcal{V}}_X$ with $K=10$, respectively, obtained by Monte Carlo simulation. The other parameters are $N_O=10,L=1,\omega =1$, and $\mathrm{\Omega }=1$.

Figure 4

Variance ${\mathcal{V}}_X$ and its lower bound ${\mathcal{V}}_X^{\mathrm{LB}}$ for a time-delayed case. (a) ${\mathcal{V}}_X$ and ${\mathcal{V}}_X^{\mathrm{LB}}$ as a function of $R=N_L/N_O$ for the time-delayed case ($\tau =0.02$) and $K=1$. The dotted and dot-dashed lines denote ${\mathcal{V}}_X$ and ${\mathcal{V}}_X^{\mathrm{LB}}$ for $\tau =0.02$, respectively, obtained analytically. The solid and long-dashed lines denote ${\mathcal{V}}_X^{\mathrm{LB}}$ and ${\mathcal{V}}_X$, respectively, for the nondelayed case ($\tau =0$) and are identical to those in Fig. 3. The circles and triangles denote ${\mathcal{V}}_X$ and ${\mathcal{V}}_X^{\mathrm{LB}}$, respectively, for $\tau =0.02$ obtained by Monte Carlo simulations. (b) Variance ${\mathcal{V}}_X$ and its lower bound ${\mathcal{V}}_X^{\mathrm{LB}}$ as a function of the time-delay $\tau$ for $K=1$ and $N=N_O=10$. The solid and dashed lines denote ${\mathcal{V}}_X$ and ${\mathcal{V}}_X^{\mathrm{LB}}$, respectively, obtained analytically. The circles and triangles denote ${\mathcal{V}}_X$ and ${\mathcal{V}}_X^{\mathrm{LB}}$, respectively, obtained by Monte Carlo simulations. In (a) and (b), the other unspecified parameters are the same as those in Fig. 3.