Information-theoretic versus thermodynamic entropy production in autonomous sensory networks

For sensory networks, we determine the rate with which they acquire information about the changing external conditions. Comparing this rate with the thermodynamic entropy production that quantifies the cost of maintaining the network, we find that there is no universal bound restricting the rate of obtaining information to be less than this thermodynamic cost. These results are obtained within a general bipartite model consisting of a stochastically changing environment that affects the instantaneous transition rates within the system. Moreover, they are illustrated with a simple four-states model motivated by cellular sensing. On the technical level, we obtain an upper bound on the rate of mutual information analytically and calculate this rate with a numerical method that estimates the entropy of a time series generated with a simulation.

Figure 1

Network and transition rates of a simple model of an internal process $Y$ coupled to an external one $X$.

Figure 2

The rate of mutual information $\mathcal{I}$, estimated with the numerical method explained in the text, the upper bound ${\mathcal{I}}^{\left(u\right)}$, and the thermodynamic entropy production $\sigma$ as a function of $k_{-}$ for $k_{+}=25$ and $\gamma =1$. Here and in the following figures the common unit of time is arbitrary and the error in the numerics is less than the size of the symbols.

Figure 3

The lower and upper bounds on the mutual information rate and the numerical simulation result as a function of $\tau$ for $k_{-}=5$, $k_{+}=25$, and $\gamma =1$. The simulations were done for a time series with $N={10}^{10}$.

Figure 4

The entropy production rate $\sigma$, the upper bound ${\mathcal{I}}^{\left(u\right)}$, and the numerically determined rate of mutual information $\mathcal{I}$ as a function of $a$ for the thermodynamically consistent model shown in the inset. The other parameters are set to $k_{\mathrm{on}}=k_{\mathrm{off}}=1$, ${\kappa }_{-}=1/10$, ${\kappa }_{+}=4/10$, ${\omega }_{-}=1/10$, and ${\omega }_{+}=6/10$.