Information Processing Using Networks of Chemical Oscillators

Abstract

I believe the computing potential of systems with chemical reactions has not yet been fully explored. The most common approach to chemical computing is based on implementation of logic gates. However, it does not seem practical because the lifetime of such gates is short, and communication between gates requires precise adjustment. The maximum computational efficiency of a chemical medium is achieved if the information is processed in parallel by different parts of it. In this paper, I review the idea of computing with coupled chemical oscillators and give arguments for the efficiency of such an approach. I discuss how to input information and how to read out the result of network computation. I describe the idea of top-down optimization of computing networks. As an example, I consider a small network of three coupled chemical oscillators designed to differentiate the white from the red points of the Japanese flag. My results are based on computer simulations with the standard two-variable Oregonator model of the oscillatory Belousov−Zhabotinsky reaction. An optimized network of three interacting oscillators can recognize the color of a randomly selected point with >98% accuracy. The presented ideas can be helpful for the experimental realization of fully functional chemical computing networks.

Keywords: Japanese flag problem; Oregonator model; chemical computing; network; oscillators; top-down design.

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: $f=1.1$, $q=0.002$, $ϵ=0.3$; (c) Model I: $f=1.1$, $q=0.0002$, $ϵ=0.2$. 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.