# Information Processing Using Networks of Chemical Oscillators

## Abstract

**:**

## 1. Introduction

## 2. Information Processing with Oscillator Networks

#### 2.1. Classification Type Problems

#### 2.2. The Node Model

#### 2.3. The Model of a Network

#### 2.4. Top-Down Design of Computing Networks

- -
- The observation time ${t}_{max}$;
- -
- All parameters for a model of chemical oscillations inside a node; for the Oregonator model, they are $\epsilon $, q and f;
- -
- Parameters ${t}_{start}$ and ${t}_{end}$ that translate an input value into the illumination of an input oscillator (cf. Equation (5));
- -
- The rates for reactions responsible for interactions between oscillators (${\alpha}_{j}$, ${\beta}_{j,i}$);
- -
- Location of input and normal oscillators;
- -
- Finally, the illumination times for all normal oscillators ${t}_{illum}\left(i\right)$.

- (1)
- My attention is restricted to classifiers formed by $m=3$ oscillators;
- (2)
- There have to be input oscillators for each coordinate in the network and a normal oscillator. Keeping in mind the symmetry of the considered network, we can assume that node #1 is the normal oscillator and nodes #2 and #3 are the inputs of x- and y-coordinates, respectively;
- (3)
- The system symmetry reduces the number of parameters in the networks because: ${\alpha}_{2}={\alpha}_{3}$, ${\beta}_{1,2}={\beta}_{1,3}$, ${\beta}_{2,1}={\beta}_{3,1}$ and ${\beta}_{2,3}={\beta}_{3,2}$.

## 3. What Is the Color of a Point on the Japanese Flag? (As Seen by the Networks)

- -
- If one or two maxima are observed, then the record represents a point in the sun area of the training dataset;
- -
- If three or more maxima are observed, then the record represents a point outside the sun area.

- -
- If a single maximum of the activator is observed, then the record represents a point in the sun area of the training dataset;
- -
- If we record no maxima, then the processed data represent a point outside the sun area.

- -
- If the value of ${J}_{u}\ge 0.1$, then the record represents a point in the sun area of the training dataset;
- -
- If the value of ${J}_{u}<0.1$, then the record represents a point outside the sun area.

- -
- If the value of ${J}_{v}\ge 0.2$, then the record represents a point in the sun area of the training dataset;
- -
- If the value of ${J}_{v}<0.2$, then the record represents a point outside the sun area.

## 4. Discussion and Conclusions

- -
- If one or two maxima are observed, then the point is within the sun area;
- -
- If three or more maxima are observed, then the point is outside the sun area.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviation

BZ | Belousov–Zhabotinsky |

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**Figure 1.**The geometrically inspired problem of determining the color of a randomly selected point located on the Japanese flag formed by the central red disk and the surrounding white area. The flag is represented by the Cartesian product $[-0.5,0.5]\times [-0.5,0.5]$), and the disk radius is $r=1/\sqrt{(}2\pi )$; thus, the areas of the sun and the white region are equal.

**Figure 2.**(

**a**) Time-dependent illumination $\varphi \left(t\right)=(1.001+tanh(-10\ast (t-{t}_{illum})))/10$ for ${t}_{illum}=5$. (

**b**,

**c**) The character of oscillations for the 2-variable Oregonator models used in simulations: (

**b**) Model II: $f=1.1$, $q=0.002$, $\u03f5=0.3$; (

**c**) Model I: $f=1.1$, $q=0.0002$, $\u03f5=0.2$. Red and blue curves represent concentrations of activator (u) and inhibitor (v), respectively. The values of $\alpha $ are $0.5$ (

**b**) and $0.7$ (

**c**).

**Figure 3.**The idea of a computing oscillator network. Circles represent network nodes that are chemical oscillators. The nodes have different characteristics. The upper one (#1) is a normal one, and its illumination function is fixed. The bottom nodes (#2) and (#3) are inputs of x and y coordinates, respectively. The arrows interlinking oscillators represent reactions that exchange the activators between nodes. The arrows directed away mark activator decay (reaction 3).

**Figure 4.**The time evolution of activator (the red curves) and inhibitor (the blue curves) observed on all nodes of the network defined by the parameters listed in the first line of Table 1. The coordinates of the input point are: $(-0.25,0.25)$. The green line marks the threshold for the activator maximum. There are 2 maxima at all nodes in the network.

**Figure 5.**The time evolution of activator (the red curves) and inhibitor (the blue curves) observed on all nodes of the network defined by the parameters listed in the first line of Table 1. The coordinates of the input point are: $(-0.29,0.29)$. The green line marks the threshold for the activator maximum. There are 3 maxima of $u\left(t\right)$ on nodes #1 and #3 and two maxima of $u\left(t\right)$ on node #2 within the observation time $[0,{t}_{max}]$. The green line marks the threshold for the activator maximum.

**Figure 6.**The time evolution of the activator (the red curve) and the inhibitor (the blue curve) on node #1 of the network defined by the parameters listed in the second line of Table 1: (

**a**) ${u}_{1}\left(t\right)$ and ${v}_{1}\left(t\right)$ for the point inside the red area $(-0.25,0.28)$; (

**b**) ${u}_{1}\left(t\right)$ and ${v}_{1}\left(t\right)$ for the point outside the red area $(-0.39,-0.43)$.

**Figure 7.**The time evolution of the activator at node #1 of the network defined by the parameters listed in the third line of Table 1: (

**a**) ${u}_{1}\left(t\right)$ for the point $(-0.25,0.28)$ located inside the red area; (

**b**) ${u}_{1}\left(t\right)$ for the point $(-0.39,-0.43)$ located outside the red area. The red shaded area below the function represents the integral of ${J}_{u}={\int}_{0}^{{t}_{max}}{u}_{1}\left(t\right)dt$, considered as the network output.

**Figure 8.**The time evolution of the inhibitor at node #3 of the network defined by the parameters listed in the fourth line of Table 1: (

**a**) ${v}_{3}\left(t\right)$ for the point $(-0.25,0.28)$ located inside the red area; (

**b**) ${v}_{3}\left(t\right)$ for the point $(-0.39,-0.43)$ located outside the red area. The blue shaded area below the function represents the integral ${J}_{v}={\int}_{0}^{{t}_{max}}{v}_{3}\left(t\right)dt$, considered as the network output.

**Figure 9.**The answer of the network defined by the parameters listed in the first line of Table 1 to the records of the training dataset. Subfigures (

**a**,

**c**,

**d**) are probability distributions of obtaining a given number of activator maxima on nodes #1, #2 and #3, respectively. The red bars correspond to points inside the red area; the blue bars refer to points outside the red area. Subfigure (

**b**) illustrates correctly (yellow and red) and incorrectly (green and blue) classified points of the training dataset when node #1 is used as the output.

**Figure 10.**The answer of the network defined by the parameters listed in the second line of Table 1 to the records of the training dataset. Subfigures (

**a**,

**c**,

**d**) are probability distributions of obtaining a given number of activator maxima on nodes #1, #2 and #3, respectively. The red bars correspond to points inside the red area; the blue bars refer to points outside the red area. Subfigure (

**b**) illustrates correctly (yellow and red) and incorrectly (green and blue) classified points of the training dataset when node #1 is used as the output.

**Figure 11.**The answer of the network defined by the parameters listed in the third line of Table 1 to the records of training dataset. (

**a**) The probability distribution of obtaining the value of ${J}_{u}={\int}_{0}^{{t}_{max}}{u}_{1}\left(t\right)dt$ in the intervals $[k\ast 0.025,(k+1)\ast 0.025)$ for $k\in \{1,2,3,4\}$. The red bars correspond to points inside the red area; the blue bars refer to points outside the red area. Subfigure (

**b**) illustrates correctly (yellow and red) and incorrectly (green and blue) classified points of the training dataset.

**Figure 12.**The answer of the network defined by the parameters listed in the fourth line of Table 1 to the records of training dataset. (

**a**) The probability distribution of obtaining the value of ${J}_{v}={\int}_{0}^{{t}_{max}}{v}_{1}\left(t\right)dt$ in the intervals $[k\ast 0.025,(k+1)\ast 0.025)$ for $k\to \{1,2,3,4\}$. The red bars correspond to points inside the red area; the blue bars refer to points outside the red area. Subfigure (

**b**) illustrates correctly (yellow and red) and incorrectly (green) classified points of the training dataset. As in (

**a**,

**b**) but for node #3: (

**c**) the probability distribution of obtaining the value of ${J}_{u}={\int}_{0}^{{t}_{max}}{v}_{3}\left(t\right)dt$ in the intervals $[k\ast 0.025,(k+1)\ast 0.025)$ for $3\le k\le 10$. Subfigure (

**d**) illustrates correctly (yellow and red) and incorrectly (green and blue) classified points of the training dataset.

**Figure 13.**The answer of networks defined by the parameters listed in Table 1 to a testing dataset of 100,000 records. Yellow and red points are classified correctly. The network gives a wrong answer on points marked green (they belong to the sun but are classified as located outside it) and points marked blue (they are located outside the sun but are classified as belonging to the red area). Subfigures (

**a**–

**d**) correspond to networks with parameters listed in lines 1–4 of Table 1, respectively. The accuracy of these networks is (

**a**) 0.957, (

**b**) 0.984, (

**c**) 0.976 and (

**d**) 0.979, respectively.

**Table 1.**The parameters of networks that give the best correlations between the time evolution of the output oscillator and the point color.

Oregonator | Method | ${\mathit{t}}_{\mathbf{max}}$ | ${\mathit{t}}_{\mathbf{start}}$ | ${\mathit{t}}_{\mathbf{end}}$ | ${\mathit{t}}_{\mathbf{illum}}\left(1\right)$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}={\mathit{\alpha}}_{3}$ | ${\mathit{\beta}}_{1,2}={\mathit{\beta}}_{1,3}$ | ${\mathit{\beta}}_{2,1}={\mathit{\beta}}_{3,1}$ | ${\mathit{\beta}}_{2,3}={\mathit{\beta}}_{3,2}$ |
---|---|---|---|---|---|---|---|---|---|---|

Model I | activator maxima | 34.4 | 11.6 | 19.9 | 10.1 | 0.87 | 0.72 | 0.16 | 0.43 | 0.29 |

Model II | activator maxima | 8.45 | 3.77 | 8.03 | 5.42 | 0.96 | 0.46 | 0.53 | 0.38 | 0.42 |

Model II | u-integral | 8.43 | 3.77 | 7.41 | 5.00 | 0.65 | 0.50 | 0.83 | 0.26 | 0.29 |

Model II | v-integral | 9.06 | 3.77 | 7.56 | 5.71 | 0.75 | 0.44 | 0.60 | 0.29 | 0.33 |

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Gorecki, J.
Information Processing Using Networks of Chemical Oscillators. *Entropy* **2022**, *24*, 1054.
https://doi.org/10.3390/e24081054

**AMA Style**

Gorecki J.
Information Processing Using Networks of Chemical Oscillators. *Entropy*. 2022; 24(8):1054.
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**Chicago/Turabian Style**

Gorecki, Jerzy.
2022. "Information Processing Using Networks of Chemical Oscillators" *Entropy* 24, no. 8: 1054.
https://doi.org/10.3390/e24081054