# Information Thermodynamics for Time Series of Signal-Response Models

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## Abstract

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## 1. Introduction

## 2. Introduction to Continuous Information Thermodynamics

#### 2.1. Entropy Production in Heat Baths

#### 2.2. Feedbacks and Information

#### 2.3. Irreversible Entropy Production

## 3. Bivariate Time Series Information Thermodynamics

#### 3.1. Setting and Definition of Causal Representations

#### 3.2. Definition of Mapping Irreversibility and the Standard Integral Fluctuation Theorem

#### 3.3. Ito Inequality for Time Series

#### 3.4. The Mapping Irreversibility Density

## 4. The Fluctuation Theorem for Time Series of Signal-Response Models

## 5. Applications

#### 5.1. The Basic Linear Response Model

#### 5.2. Receptor-Ligand Systems

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

OU | Ornstein-Uhlenbeck process |

BLRM | basic linear signal-response model |

## Appendix A. Mapping Irreversibility in the BLRM

**Figure A1.**Numerical verification of the analytical solution for the entropy production ${\Phi}_{\tau}^{xy}$ with observational time $\tau $ in the BLRM. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. The slight down-deviation for small $\tau $ is due to the finite box length in the discretized space, while the up-deviation for $\tau \to \infty $ is due to the finite number of samples.

## Appendix B. Backward Transfer Entropy in the BLRM

## Appendix C. The Causal Influence Rate Converges to the Horowitz-Esposito Information Flow in the BLRM

**Figure A2.**Probability currents $\overrightarrow{J}$ in the BLRM, estimated with $\tau =0.1$. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. The space coordinates are in units of the standard deviation.

## Appendix D. Numerical Convergence of the Mapping Irreversibility to the Entropy Production in the Feedback Cooling Model

## Appendix E. Numerical Estimation of the Entropy Production in the Bivariate Gaussian Approximation

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**Figure 1.**Complete causal representation. The arrows represent the way we factorize the joint probability density. In the complete case the causal representation is fully degenerate: $p\left({\zeta}_{\tau}^{xy}\right)=p({x}_{t},{y}_{t})\xb7p({x}_{t+\tau},{y}_{t+\tau}|{x}_{t},{y}_{t})=p({x}_{t},{y}_{t})\xb7p\left({x}_{t+\tau}\right|{x}_{t},{y}_{t})\xb7p\left({y}_{t+\tau}\right|{x}_{t},{y}_{t},{x}_{t+\tau})=p({x}_{t},{y}_{t})\xb7p\left({y}_{t+\tau}\right|{x}_{t},{y}_{t})\xb7p\left({x}_{t+\tau}\right|{x}_{t},{y}_{t},{y}_{t+\tau})$.

**Figure 2.**Causal representation of signal-response models. The joint probability density is factorized into $p\left({\zeta}_{\tau}^{xy}\right)=p({x}_{t},{y}_{t})\xb7p\left({x}_{t+\tau}\right|{x}_{t})\xb7p\left({y}_{t+\tau}\right|{x}_{t},{y}_{t},{x}_{t+\tau})$.

**Figure 3.**Mapping irreversibility ${\Phi}_{\tau}^{xy}$, backward transfer entropy ${T}_{y\to x}(-\tau )$ and causal influence ${C}_{x\to y}\left(\tau \right)$ in the BLRM as a function of the observational time interval $\tau $. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. All graphs are produced using R [41].

**Figure 4.**Mapping irreversibility density $\psi ({x}_{t},{y}_{t})$ for the BLRM at $\tau =0.5<\frac{1}{\beta}<{t}_{rel}$. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. Both $\psi ({x}_{t},{y}_{t})$ and the space $(x,y)$ are expressed in units of standard deviations.

**Figure 5.**Mapping irreversibility density $\psi ({x}_{t},{y}_{t})$ for the BLRM at $\tau =25>{t}_{rel}>\frac{1}{\beta}$. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. Both $\psi ({x}_{t},{y}_{t})$ and the space $(x,y)$ are expressed in units of standard deviations.

**Figure 6.**Mapping irreversibility and backward transfer entropy in our model of receptor-ligand systems (Equation (22)). The parameters are ${k}_{on}=5$, ${k}_{off}=1$, and $h=2$.

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Auconi, A.; Giansanti, A.; Klipp, E.
Information Thermodynamics for Time Series of Signal-Response Models. *Entropy* **2019**, *21*, 177.
https://doi.org/10.3390/e21020177

**AMA Style**

Auconi A, Giansanti A, Klipp E.
Information Thermodynamics for Time Series of Signal-Response Models. *Entropy*. 2019; 21(2):177.
https://doi.org/10.3390/e21020177

**Chicago/Turabian Style**

Auconi, Andrea, Andrea Giansanti, and Edda Klipp.
2019. "Information Thermodynamics for Time Series of Signal-Response Models" *Entropy* 21, no. 2: 177.
https://doi.org/10.3390/e21020177