**1. Introduction**

Chemical computers are everywhere because all living organisms use them for acquiring and transmitting information and for decision-making. Animals and humans, using chemical computers represented by their nervous systems and brains [1], are able to control complex life processes such as orientation in space, navigation in crowded environments, creating models of the environment they live in, developing self-awareness and even predicting the future. This demonstrates that Nature-made chemical computers can perform very complex computational tasks with low energy consumption.

The information-processing industry is based on semiconductor technology. The unprecedented success of this technology in machine information processing [2] is possible because semiconductor logic gates are highly reliable. They are characterized by a long time of error-free operation and can be downsized to the nanoscale [3]. The gates can be concatenated within a single integrated circuit and perform more complex information processing functions. These properties of semiconductor information-processing devices perfectly match the bottom-up design strategy, according to which more complex operations are represented by combinations of simpler tasks for which constructions of corresponding circuits have already been developed [4].

The concept of logic gates and binary information coding, so successfully used for information processing with semiconductor devices, has strongly influenced other fields of unconventional computation, including the use of chemicals [5]. There are many reports on chemical realization of logic gates or binary operations [6–14]. The fact that a chemical medium allows for implementation of all basic logic gates proves that a universal chemical

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

Academic Editor: Adam Gadomski

Received: 7 July 2022 Accepted: 26 July 2022 Published: 31 July 2022

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**Copyright:** © 2022 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 (https:// creativecommons.org/licenses/by/ 4.0/).

computer can be constructed. In exceptional cases, molecular logic gates used as molecular probes offer an interesting alternative to standard techniques [14]. However, most chemical logic gates, especially those constructed with a reaction-diffusion medium [15], are neither small nor fast. For the Belousov–Zhabotinsky (BZ) reaction, the output signal of a gate appears a few seconds after the input is introduced [7,8,10]. For other chemical media, this time can be much longer. In the case of information coded in DNA molecules, it may take a few hours before the gate answer is obtained [16]. In typical experimental conditions, the time of reliable chemical gate operation is measured in hours, not years as for semiconductors. In my opinion, it would be difficult to make a chemical device in which many chemical gates are concatenated, communicate and demonstrate stable functionality for a long time.

I think the BZ reaction [17,18] is the most frequently used medium in experimental studies on chemical computation. The BZ reaction is a complex catalytic oxidation of an organic substrate (usually malonic acid) in an acidic environment [19,20]. Two stages of the BZ reaction can be visually identified. One is fast oxidation of the catalyst, and the other is slow reduction of the catalyst by an organic substrate. The solution color reflects concentrations of catalyst in the oxidized and reduced forms. Therefore, many types of nonlinear evolution of the medium as oscillations or spatio-temporal patterns can be easily observed. If the BZ reaction proceeds in a spatially distributed medium, then local excitation corresponding to a high concentration of the reaction activator can propagate in space in the form of a concentration pulse. This type of behavior resembles the propagation of nerve impulses in living organisms. As a result, the BZ reaction has attracted attention as a medium for experiments with neuron-like chemical computing [21–23].

Within the most popular approach to computing with BZ medium, it is assumed that information is transmitted by propagating pulses of the oxidized form of catalyst. For binary coding, the presence of a pulse represents the logic TRUE state, and the state with a low concentration of the catalyst in the oxidized form is the logic FALSE state [13,21]. If the ruthenium complex (*Ru*(*bpy*)2<sup>+</sup> <sup>3</sup> ) is used as the reaction catalyst, then the BZ reaction becomes photosensitive [24,25] and can be externally controlled by illumination. Oscillations can be inhibited by light, which means that for the same initial concentrations of reagents, the medium oscillates in the dark, is excitable at a low light intensity, and shows a strongly attracting stable steady-state when illumination is strong. In a medium with the photosensitive BZ reaction, excitable channels through which signals can propagate can be formed by specific illumination of a spatially distributed medium. Using a suitable geometry of excitable and non-excitable channels, one can control the propagation of excitations and, for example, make a signal diode [26], a memory cell [27,28] or logic gates [13,15]. However, in typical applications, such gates are big (with an area of about 1 cm2), and a single operation takes more than 10 s [20]. Therefore, the bottom-up approach from gates to complex information processing tasks does not look promising if binary information coding is used with BZ medium.

Literature reports demonstrate that a chemical medium can be efficiently applied for specific computing tasks. Parallel processing of information by the medium as a whole is a common feature of efficient implementations. The classic example is the Adleman experiment proving that the Hamiltonian path problem can be solved with DNA molecules [29,30]. Another example is the so-called prairie-fire algorithm for verifying if there is a path linking two randomly selected points in a labyrinth. This problem can be solved by a labyrinth formed from an excitable medium where stable pulses of excitation can propagate [31,32]. If there is a path linking two points, an excitation generated at one of the points will then appear at the other, and the time difference between excitation and detection can be used to estimate the shortest path linking these points. Yet another famous computing application of a chemical medium working in parallel is the image processing of black and white photos performed using a photosensitive variant of the BZ reaction proceeding in a uniform, spatially distributed system [33,34]. In such a medium, image processing is the consequence of a non-homogeneous initial state generated by initial illumination with intensity

proportional to the grayscale of pixels of the processed image. In all methods mentioned above, the output information is coded in the time evolution of the computing medium.

However, the number of man-written parallel algorithms that can be efficiently executed using a chemical medium of parallel algorithms for chemical computers is very limited. A top-down design strategy offers a promising approach for finding new ones. The strategy can be summarized as follows. In the beginning, we select a problem we want to solve and the computing medium that is supposed to do it. Next, we define how the input information is introduced and how the output is extracted from observing medium evolution. The top-down approach can be applied if the properties of the medium—and thus of the medium evolution—can be controlled by a number of adjustable parameters. Within this strategy, we are supposed to find the values of parameters for which the medium answer (the output) gives the most accurate solution to the considered problem. To perform such optimization, we need a number of examples (the training dataset) that can be used to verify the accuracy of computation performed by the medium.

Networks of interacting chemical oscillators seem to be an interesting candidate for a chemical computing medium. Networks of BZ oscillators can be assembled using droplets containing reagents stabilized by lipids dissolved in the surrounding oil phase [35–37]. The idea of information processing with networks of interacting chemical reactors was formulated in [38]. In such an approach, a node is defined by a set of reactions sharing the specific reagents. Interaction between nodes consists of reactions providing the exchange of reagents between nodes. In [38–41], the authors focused on nodes that show excitable or bistable behavior; thus, the concentration of reagents in a single node can evolve towards one of two values that can be interpreted as corresponding to binary logical values [6]. It has been demonstrated that such nodes can perform logic gate operations. The idea of computing with oscillator networks generalizes the approach described above. First, the dynamics of a node are more complex. Second, the node answer is not a stationary state but reflects the character of evolution observed within a finite time interval. As the system is continuously evolving, the time of observation is an important parameter.

Several theoretical studies demonstrate the computing potential of oscillator networks [42–44]. It has been shown that oscillator networks could be optimized to perform classification tasks [45,46] and process information with the best possible use of the chemical medium. In this paper, I am concerned with the previously reported determination of color for a randomly selected point on the Japanese flag [47]. I introduce a few new elements to computing oscillator networks, including the comparison between two Oregonator models for node evolution that exhibit the different character of oscillations and generalization of the node–node interaction model with coupling parameters individually adjusted for pairs of nodes. Moreover, a new concept of extracting the output information from the network is proposed. In all previous papers on the subject, the number of activator maxima observed on a selected network node was regarded as the network answer. Here the total amount [48] of activator or inhibitor observed on the output oscillator is regarded as the network output. The results presented below demonstrate that such an approach is equally useful and leads to similar accuracy in determining the color of a point on the flag based on its coordinates as the number of activator maxima.

The paper is organized as follows. The information on the computational problem I am concerned with, the mathematical model of the time evolution of a node and of the network, and the optimization procedure are described in Section 2. Section 3 contains obtained results and their discussion. The final section presents verification of obtained results and presents suggestions for future studies.

#### **2. Information Processing with Oscillator Networks**

In this section, I present general information on the types of problems that can be solved with oscillator networks. Moreover, I discuss the network structure and a chemical model used to simulate its evolution.
