# Applications of Information Theory Methods for Evolutionary Optimization of Chemical Computers

## Abstract

**:**

## 1. Introduction

_{3}) is used as the catalyst, then the BZ reaction is photosensitive and illumination with the blue light produces $B{r}^{-}$ ions that inhibit the reaction [17,18,19]. After illumination of such an oscillatory medium, excitations are rapidly damped and the system reaches a stable, steady state. On the other hand, the oscillatory behavior re-appears immediately after the illumination is switched off [20]. The existence of external control is very important for information processing applications because it allows inputting information into the computing medium [10,11,12]. For the analysis presented below, it is sufficient to assume that the controlling factor has an inhibiting effect. In the following part of the paper, following the analogy with the photosensitive BZ-reaction, I use the word illumination to describe the control factor.

## 2. Results

#### 2.1. The Time Evolution Model of an Oscillator Network

#### 2.2. The Computing Medium Made of Interacting Oscillators and Its Evolutionary Optimization

#### 2.3. Chemical Algorithms for Verification of Which of the Two Numbers Is Larger

#### 2.4. Shadows on Optimization Towards the Maximum Mutual Information

## 3. Discussion: Why can a Small Network of Chemical Oscillators So Accurately Determine Which of the Two Numbers Is Larger?

## 4. Conclusions and Perspectives

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

BZ | Belousov–Zhabotinsky |

## References

- Feynman, R.P. The Feynman Lectures on Computation; Hey, A.J.G., Allen, R.W., Eds.; Addison-Wesley: Boston, MA, USA, 1996. [Google Scholar]
- Haken, H. Brain Dynamics, Synchronization and Activity Patterns in Pulse-Coupled Neural Nets with Delays and Noise; Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Toth, A.; Showalter, K. Logic gates in excitable media. J. Chem. Phys.
**1995**, 103, 2058–2066. [Google Scholar] [CrossRef] [Green Version] - Steinbock, O.; Kettunen, P.; Showalter, K. Chemical wave logic gates. J. Phys. Chem.
**1996**, 100, 18970–18975. [Google Scholar] [CrossRef] - Motoike, I.N.; Yoshikawa, K. Information operations with an excitable field. Phys. Rev. E
**1999**, 59, 5354–5360. [Google Scholar] [CrossRef] - Sielewiesiuk, J.; Gorecki, J. Logical functions of a cross junction of excitable chemical media. J. Phys. Chem. A
**2001**, 105, 8189–8195. [Google Scholar] [CrossRef] - Adamatzky, A.; De Lacy Costello, B. Experimental logical gates in a reaction–diffusion medium: the XOR gate and beyond. Phys. Rev. E
**2002**, 66, 046112. [Google Scholar] [CrossRef] - Steinbock, O.; Toth, A.; Showalter, K. Navigating complex labyrinths—Optimal paths from chemical waves. Science
**1995**, 267, 868–871. [Google Scholar] [CrossRef] [Green Version] - Agladze, K.; Magome, N.; Aliev, R.; Yamaguchi, T.; Yoshikawa, K. Finding the optimal path with the aid of chemical wave. Physica D
**1997**, 106, 247–254. [Google Scholar] [CrossRef] - Kuhnert, L. A new optical photochemical memory device in a light-sensitive chemical active medium. Nature
**1986**, 319, 393–394. [Google Scholar] [CrossRef] - Kuhnert, L.; Agladze, K.I.; Krinsky, V.I. Image processing using light-sensitive chemical waves. Nature
**1989**, 337, 244–247. [Google Scholar] [CrossRef] - Rambidi, N.G.; Maximychev, A.V. Towards a biomolecular computer. Information processing capabilities of biomolecular nonlinear dynamic media. BioSystems
**1997**, 41, 195–211. [Google Scholar] [CrossRef] - Tyson, J.J. What everyone should know about the Belousov–Zhabotinsky reaction. In Frontiers in Mathematical Biology; Levin, S.A., Ed.; Springer: Berlin/Heidelberg, Germany, 1994; pp. 569–587. [Google Scholar]
- Epstein, I.R.; Pojman, J.A. An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos; Oxford University Press: New York, NY, USA, 1998. [Google Scholar]
- Sielewiesiuk, J.; Gorecki, J. Passive Barrier as a Transformer of Chemical Signal Frequency. J. Phys. Chem. A
**2002**, 106, 4068–4076. [Google Scholar] [CrossRef] - Adamatzky, A.; De Lacy Costello, B.; Asai, T. Reaction–Diffusion Computers; Elsevier: New York, NY, USA, 2005. [Google Scholar]
- Burger, M.; Field, R.J. Oscillations and Traveling Waves in Chemical Systems; Wiley: New York, NY, USA, 1985. [Google Scholar]
- Krug, H.-J.; Pohlmann, L.; Kuhnert, L. Analysis of the modified complete oregonator accounting for oxygen sensitivity and photosensitivity of Belousov-Zhabotinsky systems. J. Phys. Chem.
**1990**, 94, 4862–4866. [Google Scholar] [CrossRef] - Kádár, S.; Amemiya, T.; Showalter, K. Reaction mechanism for light sensitivity of the Ru(bpy)
_{3}^{2+}-catalyzed Belousov-Zhabotinsky reaction. J. Phys. Chem. A**1997**, 101, 8200–8206. [Google Scholar] [CrossRef] - Gizynski, K.; Gorecki, J. Chemical memory with states coded in light controlled oscillations of interacting Belousov–Zhabotinsky droplets. Phys. Chem. Chem. Phys.
**2017**, 19, 6519–6531. [Google Scholar] [CrossRef] - Gruenert, G.; Gizynski, K.; Escuela, G.; Ibrahim, B.; Gorecki, J.; Dittrich, P. Understanding Computing Droplet Networks by Following Information Flow. Int. J. Neural Syst.
**2015**, 25, 1450032. [Google Scholar] [CrossRef] [Green Version] - Gizynski, K.; Gruenert, G.; Dittrich, P.; Gorecki, J. Evolutionary design of classifiers made of droplets containing a nonlinear chemical medium. MIT Evol. Comput.
**2017**, 25, 643–671. [Google Scholar] [CrossRef] - Gizynski, K.; Gorecki, J. Cancer classification with a network of chemical oscillators. Phys.Chem.
**2017**, 19, 28808–28819. [Google Scholar] [CrossRef] [Green Version] - Fogel, D.B. An introduction to simulated evolutionary optimization. IEEE Trans. Neural Netw.
**1994**, 5, 3–14. [Google Scholar] [CrossRef] [Green Version] - Weicker, K. Evolutionare Algorithmen; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Gorecki, J.; Gorecka, J.N.; Adamatzky, A. Information coding with frequency of oscillations in Belousov– Zhabotinsky encapsulated disks. Phys. Rev. E
**2014**, 89, 042910. [Google Scholar] [CrossRef] [Green Version] - Gorecki, J.; Yoshikawa, K.; Igarashi, Y. On chemical reactors that can count. J. Phys. Chem. A
**2003**, 107, 1664–1669. [Google Scholar] [CrossRef] - Gizynski, K.; Gorecki, J. A Chemical System that Recognizes the Shape of a Sphere. Comput. Methods Sci. Technol.
**2016**, 22, 167–177. [Google Scholar] [CrossRef] [Green Version] - Muzika, F.; Schreiberova, L.; Schreiber, I. Chemical computing based on Turing patterns in two coupled cells with equal transport coefficients. RSC Adv.
**2014**, 4, 56165–56173. [Google Scholar] [CrossRef] - Muzika, F.; Schreiberova, L.; Schreiber, I. Discrete Turing patterns in coupled reaction cells in a cyclic array. React. Kinet. Mech. Catal.
**2016**, 118, 99–114. [Google Scholar] [CrossRef] - Szymanski, J.; Gorecka, J.N.; Igarashi, Y.; Gizynski, K.; Gorecki, J.; Zauner, K.P.; de Planque, M. Droplets with information processing ability. Int. J. Unconv. Comput.
**2011**, 7, 185–200. [Google Scholar] - Guzowski, J.; Gizynski, K.; Gorecki, J.; Garstecki, P. Microfluidic platform for reproducible self-assembly of chemically communicating droplet networks with predesigned number and type of the communicating compartments. Lab Chip
**2016**, 16, 764–772. [Google Scholar] [CrossRef] - Mallphanov, I.L.; Vanag, V.K. Fabrication of New Belousov-Zhabotinsky Micro-Oscillators on the Basis of Silica Gel Beads. J. Phys. Chem. A
**2020**, 124, 272–282. [Google Scholar] [CrossRef] - Kuze, M.; Horisaka, M.; Suematsu, N.J.; Amemiya, T.; Steinbock, O.; Nakata, S. Chemical Wave Propagation in the Belousov-Zhabotinsky Reaction Controlled by Electrical Potential. J. Phys. Chem. A
**2019**, 123, 4853–4857. [Google Scholar] [CrossRef] - Field, R.J.; Noyes, R.M. Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys.
**1974**, 60, 1877–1884. [Google Scholar] [CrossRef] [Green Version] - Rovinsky, A.B.; Zhabotinsky, A.M.; Irving, R. Mechanism and mathematical model of the oscillating bromate–ferroin–bromomalonic acid reaction. J. Phys. Chem.
**1984**, 88, 6081–6084. [Google Scholar] [CrossRef] - Cover, T.M.; Thomas, J.A. Elements of Information Theory; Wiley-Interscience: New York, NY, USA, 2006. [Google Scholar]
- Holley, J.; Jahan, I.; De Lacy Costello, B.; Bull, L.; Adamatzky, A. Logical and arithmetic circuits in Belousov-Zhabotinsky encapsulated disks. Phys. Rev. E
**2011**, 84, 056110. [Google Scholar] [CrossRef] [Green Version] - Grzybowski, B.A. Chemistry in Motion: Reaction-Diffusion Systems for Micro- and Nanotechnology; Wiley-Interscience: New York, NY, USA, 2009. [Google Scholar]

**Figure 1.**(

**a**) The geometrical interpretation of the problem of which of the two numbers is larger. The areas $y>x$ and $y<x$ are colored red and green respectively, (

**b**) I postulate that the problem of which of the two numbers is larger can be solved with the illustrated network of three coupled oscillators. Having in mind the symmetry of the problem, I assume that oscillators #1 and #2 are inputs of x and y. The role of oscillator #3, the network parameters and location of the output oscillators are determined by the evolutionary optimization.

**Figure 2.**Graphical illustration of the event-based model used to describe the time evolution of an oscillator. Three phases: excited (green), refractory (brown) and responsive (red) follow one after another. The arrow marks the direction of time. Numbers give lengths of corresponding phases.

**Figure 3.**(

**a**) The mutual information as a function of an evolutionary step for the classifier in Figure 1b if the number of excitations observed on a single oscillator is used as the output, (

**b**) the numbers of cases corresponding to different numbers of excitations and for different relationships between x and y. The records from ${F}_{>}(L=100,000)$ were used. The red and green bars correspond to $y>x$ and $y<x$, respectively. The number of cases recorded on different oscillators are shown.

**Figure 4.**The location of correctly and incorrectly classified pairs $(x,y)\in [0,1]\times [0,1]$ if the number of excitations observed on a single oscillator is used as the output. Correctly classified pairs in which $y>x$ and $y<x$ are marked by red and green points, respectively. Incorrectly classified pairs in which $y>x$ and $y<x$ are marked by blue and black points, respectively.

**Figure 5.**(

**a**) The mutual information as the function of an evolutionary step for the classifier in Figure 1b if the pair of excitation numbers observed on two selected oscillators is used as the output. (

**b**) The table that translates the numbers of excitations observed on oscillators #1 and #2 into the classifier answer. The red and green points correspond to $y>x$ and $y<x$, respectively. (

**c**) The increase in classifier accuracy as a function of ${t}_{max}$ observed during the optimization.

**Figure 6.**The location of correctly and incorrectly classified pairs $(x,y)\in [0,1]\times [0,1]$ if the pair of excitation numbers observed on oscillators #1 and #2 is used as the output. Correctly classified pairs in which $y>x$ and $y<x$ are marked by red and green points, respectively. Incorrectly classified pairs in which $y>x$ and $y<x$ are marked by blue and black points, respectively.

**Figure 7.**(

**a**) The countour plot of $Accuracy(k,l)$ and (

**b**) the mutual information $MI(k,l)$ for the classifier discussed in Section 2.4 as functions of k and l. The contour lines in (a) are equally distributed between $0.5$ and $1.0$.

**Figure 8.**(

**a**) The countour plot of the mutual information illustrated in Figure 7b with a hypothetical optimization path shown in red. (

**b**) The accuracy (blue curve) and the double of mutual information (red curve) along the path marked in (

**a**).The value $s=0$ describes the point $(0,0.5)$.

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gorecki, J.
Applications of Information Theory Methods for Evolutionary Optimization of Chemical Computers. *Entropy* **2020**, *22*, 313.
https://doi.org/10.3390/e22030313

**AMA Style**

Gorecki J.
Applications of Information Theory Methods for Evolutionary Optimization of Chemical Computers. *Entropy*. 2020; 22(3):313.
https://doi.org/10.3390/e22030313

**Chicago/Turabian Style**

Gorecki, Jerzy.
2020. "Applications of Information Theory Methods for Evolutionary Optimization of Chemical Computers" *Entropy* 22, no. 3: 313.
https://doi.org/10.3390/e22030313