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Edited by: Annette F. Taylor, The University of Sheffield, United Kingdom

Reviewed by: Vladimir Karl Vanag, Immanuel Kant Baltic Federal University, Russia; Pier Luigi Gentili, University of Perugia, Italy

This article was submitted to Physical Chemistry and Chemical Physics, a section of the journal Frontiers in Chemistry

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Chemical computing is something we use every day (e.g., in the brain), but we can still not explore and master its potential in human-made experiments. It is expected that the maximum computational efficiency of a chemical medium can be achieved if information is processed in parallel by different parts of the medium. In this paper, we use computer simulations to explore the efficiency of chemical computing performed by a small network of three coupled chemical oscillators. We optimize the network to recognize the white and red regions of the Japanese flag. The input information is introduced as the inhibition times of individual oscillators, and the output information is coded in the number of activator maxima observed on a selected oscillator. We have used the Oregonator model to simulate the network time evolution and the evolutionary optimization to find the best network for the considered task. We have found that even a network of three interacting oscillators can recognize the color of a randomly selected point with 95% accuracy.

The success of semiconductor technology in machine information processing is the consequence of a highly efficient realization of logic gates characterized by a long time of error-free operation. The gates can be downsized to the nanoscale and concatenated to make more complex information processing devices. The semiconductor technology perfectly matches the bottom-up design scenario of information processing systems according to which more complex operations are represented by the combination of simpler tasks for which constructions of corresponding circuits have already been developed (Feynman et al.,

The usefulness of logic gates and binary information coding demonstrated by semiconductor devices has strongly influenced other fields of unconventional computation including the chemical one (Adamatzky,

On the other hand, living organisms use chemistry for information processing and do so with significant efficiency for various classes of algorithms, including sound and image recognition, orientation in space, or navigation in crowded environments. This observation demonstrates that a chemical medium can be efficiently applied for specific computing tasks and presumably solve them using a highly parallel approach. Applications of parallel chemical computation have been reported in the literature. The classic example is the Adleman demonstration that the Hamiltonian path problem can be solved with DNA molecules (Adleman,

The number of examples where a chemical medium can be efficiently used for computing is, however, limited. A top-down design strategy offers a promising method for the identification of the 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 the observation of 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 of the considered problem. To do this, we need a number of examples (the training dataset) that can be used to verify the accuracy of computation performed by the medium.

In this paper, we concentrate on the geometrically oriented problem illustrated in

_{illum} = 4).

We postulated that the chemical medium composed of three coupled oscillators, illustrated in

In a number of recent papers, we considered flow of information (Grüunert et al.,

The paper is organized as follows. The description of the mathematical model of the network time evolution and the optimization procedure are described in the next section. Section 3 contains obtained results and their discussion. The conclusions summarize obtained results and present suggestions for the future studies.

We postulate that the problem of attributing color to a point on the Japanese flag defined by its coordinates can be approximately solved by a network of interacting chemical oscillators (cf. _{j} and _{j} represent concentrations of the activator (_{j}) and the inhibitor (_{j}) for reactions proceeding in the oscillator #j. The equations describing the time evolution of _{j} and _{j} read:
_{j} and _{j}, and

The last term in Equation (1) describes the additional decay of activator with the reaction rate α, which can be represented by the following reaction:
_{j}(_{j}(_{ilum}(_{j}(_{j} = 0.0002 and _{j} = 0.0002. For long durations, ϕ_{j}(_{illum} = 4.

We assumed that oscillators in the network are coupled via the transport of the activator. This type of coupling was observed in our experiment on BZ-droplets stabilized by a solution of lipids in decane (Szymanski et al.,

The interactions between oscillations in the considered network are indicated by the linking lines shown in _{k} and _{j} of these oscillators:

Within our model the time evolution of the network is described by the following set of kinetic equations:
_{j} in the last two terms in Equation (7) cancel out reduce to

Mathematically, the terms resulting from processes (5) and (6) have a similar form to those describing the coupling between CSTRs resulting from the exchange of equal volumes of reagents (and assuming that the is no transport of inhibitor). For the considered parameters of the Oregonator model (as well as a few other we tried), we found the interactions between oscillators described by reactions (5,6) were difficult to control without the process (3) because the model, depending on the value of β, gave too weak or too strong coupling between oscillators. The introduction of reaction (3) allowed us to control the value of activator concentration around its maximum and to moderate these interactions without the need to optimize all parameters of the Oregonator model.

Following our previous studies, we assume that an oscillator in the network can perform one of two functions (Gizynski and Gorecki, _{end} and _{start} in Equations (9) and (10) are identical. The values of _{end} and _{start} were the subject of network optimization procedure. The relationship between _{ilum} and the input value, obtained for the Japanese flag problem, is illustrated in

The network can also include so-called normal oscillators inhibited for a fixed time that is not related to the values of

We also assumed that the output information is coded in the number of activator maxima observed on one of the network oscillators within the time interval [0, _{max}]. As we show later, the choice of the output oscillator results directly from the network optimization. The full definition of a computing network therefore includes the number of oscillators in the network, their types, the method of inputting

The parameters that were modified during the network optimization procedure were:

- The type of oscillator #3 and, in the case it was a normal oscillator, its illumination time _{ilum}(3),

- The length of time interval _{max}, within which the network evolution was observed,

- The times _{start} and _{end},

- The reaction rates α and β.

The values of network parameters and the parameters describing the relationship between the input values and the time-dependent illuminations of input oscillators determine the medium evolution. For each set of parameters, we solved the set of Equations (7, 8) numerically and studied the time evolution of the network for any values of ^{−4} time step. The output information corresponding to a specific input was extracted from the numerical solution as the number of activator maxima observed at the selected oscillator within the time interval [0, _{max}].

_{ilum}(3) = 6.37, _{max} = 20.23, _{start} = 3.78, _{end} = 12.10, α = 0.849, β = 0.251.

The method of the top-down optimization of a computing network has been described in details in our previous papers (Gizynski and Gorecki, _{S} = {(_{n}, _{n}, _{n}), _{S} of _{n},

_{S}. The red points are in the sun area, the blue ones outside it. _{ilum} and the input value, obtained for the optimized network solving the Japanese flag problem.

We postulate that information about the point color can be extracted from the number of activator maxima recorded on a selected oscillator of the network during the time interval [0, _{max}]. The quality of an oscillator network for solving a specific problem can be estimated in the following way. Let us consider a record (_{n}, _{n}, _{n}) ∈ _{S} and study the network evolution for the input (_{n}, _{n}). Assume that _{1}(_{2}(_{3}(_{j} = {_{j}(_{j} is defined as (Cover and Thomas, _{j} is defined as _{j} = {(_{n}, _{j}(_{j}) is maximal was selected as the network output. The maximum _{j}) was used as the measure of network fitness in our optimization program.

The use of mutual information gives the quantitative measure of the network usefulness without the need to specify how to translate the number of activator maxima into the output. On the other hand, it has been shown that there is no monotonic relationship between the accuracy and mutual information (Gorecki,

The network parameters, such as the type of oscillator #3, its inhibition time, the method for inputting the predictor values, or the type of interactions between oscillators, undergo optimization to achieve the highest mutual information on a representative dataset of cases. Both systematic methods of optimization and random trial and error ones can be applied. We have found (Gizynski and Gorecki, _{ilum}(3), _{max}, _{start}, _{end}, α and β] were randomly generated. The fitness of each network was calculated using the whole training dataset. The next generation comprised of 20% of most fit networks of the previous population and of 80% of offspring generated by recombination and mutation operations applied to oscillators from top 50% networks of the previous population. We randomly selected two parents from 50% of the fittest networks and next recombined randomly their parameters to obtain an offspring. After recombination, we applied mutations to randomly selected parameters. The probability of this operation was selected, such that on average, a single parameter of the network was mutated. The maximum change in the chosen parameter value was restricted to 10% of the original one. Next, the fit of networks belonging to the new generation were calculated, and the procedure was repeated. The optimization procedure was continued for 1,000 generations.

The optimization procedure returned the network illustrated in _{ilum}(3) = 6.37. The other parameters of the network are the following: _{max} = 20.23, _{start} = 3.78, _{end} = 12.10, α = 0.849, and β = 0.251. The oscillator #3 is also the output one. For each input, we observed one or two activator maxima at the output oscillator. _{S}, such a majority rule leads to (369 + 393)/800 ~ 0.95 accuracy. It is interesting that the distribution of incorrectly attributed cases is not rotationally symmetric.

For more objective evaluation of the accuracy of the optimized network we considered a large test dataset _{T} of 100,000 random, uniformly distributed points in the square [0, 1] × [0, 1]. _{T} and the number of activator maxima observed on the output oscillator within the time interval [0, _{max}]. The red light points are located inside the sun disk and produced a single maximum (48,028 cases). The light blue points are located in the surrounding area, and they forced two maxima of the output oscillator (47,117 cases). Using the majority rule introduced for the training dataset _{S} we can say that the total number of correctly located points was 95,145 thus the classifier accuracy is ~95%. The dark colors mark points that are incorrectly attributed. The dark red points are located outside the sun, but they force a single activator maximum (2,967 cases). The dark blue points produced two activator maxima, but they were located in the sun area (1,888 cases).

The figure shows how the optimized network sees the Japanese flag. Each dot corresponds to a record from the testing dataset that contains 100,000 records. The red points produce a single activator maximum and are thus considered to be sun. For the records represented by the blue points, two maxima are observed, and they are thus classified as the surrounding white area. The light red and light blue points are those that are classified correctly. The dark red points belong to the white region in

The red points in _{D}(_{U}(

In this paper, we investigated whether a chemical computer can solve the color determination problem for a point on the Japanese flag. We considered the flag of Japan because it represents more complex geometrical structure than the striped pattern common for many flags like the Polish or the French ones. For example the flag of France can be represented by a unit square with the blue points for (_{start} = 0. The inhibition time of input oscillator [_{illum}(1)] is proportional to the value of _{illum}(1) = _{end}], and we can thus select _{end} such that a small number of oscillations appear in the red area, more of them are seen for the white points, and the largest number of oscillations is observed if the input represents coordinates of a blue point. For example, if _{end} = 32.4, then the oscillator with parameters given in this paper observed for _{max} = 47 shows 4, 3, and 2 maxima of activator for blue, white and red regions respectively.

The idea of a neural network has inspired the considered medium structure, composed as it is of interacting, individual units. Classical image recognition methods are based on multilayered neural networks in which the output of an artificial neuron is a single number (MacKay, _{max} it was observed. It seems that to determine the color of a point on the Japanese flag with a given accuracy with a standard neural network, one needs more nodes than the number of oscillators used by our medium (Zammataro,

We have demonstrated that a simple network of just three coupled chemical oscillators can predict the color of a randomly selected point on the Japanese flag with 95% accuracy. Another interesting result is that the network delivers a fast answer. The output information if a point belongs to the white or the red region on the flag appears just within two oscillation periods. Our simulations were based on the Oregonator model that is more realistic than the event-based model previously used in papers on chemical classifiers (Gizynski and Gorecki,

It is worth noticing that the points classified as belonging to the sun area group into an interesting, horned shape illustrated in

Results presented in the paper were obtained based on computer simulations, but the Oregonator model can qualitatively describe BZ-reaction and therefore brings information for potential experiments on the chemical computation. Systems of interacting oscillators have been studied experimentally using a few techniques (Vanag and Epstein,

We believe the maximization of information processing functionality based on optimization of the mutual information can be combined with other types of complex chemical dynamics, such as excitability or multistability. However, in typical experiments, oscillations are robust, whereas other types of non-linear behavior are more difficult to control, stabilize, and repeat. Additionally, our results illustrate that even a small network of oscillators can have significant information processing potential.

All datasets presented in this study are included in the article.

JG was responsible for the idea of the presented study, network optimization, and presentation of results. AB helped to develop the model for interactions between oscillators and verified the obtained results using her own simulation code. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.