Time-dependent wave selection for information processing in excitable media

We demonstrate an improved technique for implementing logic circuits in light-sensitive chemical excitable media. The technique makes use of the constant-speed propagation of waves along defined channels in an excitable medium based on the Belousov-Zhabotinsky reaction, along with the mutual annihilation of colliding waves. What distinguishes this work from previous work in this area is that regions where channels meet at a junction can periodically alternate between permitting the propagation of waves and blocking them. These valvelike areas are used to select waves based on the length of time that it takes waves to propagate from one valve to another. In an experimental implementation, the channels that make up the circuit layout are projected by a digital projector connected to a computer. Excitable channels are projected as dark areas and unexcitable regions as light areas. Valves alternate between dark and light: Every valve has the same period and phase, with a 50$\%$ duty cycle. This scheme can be used to make logic gates based on combinations of or and and-not operations, with few geometrical constraints. Because there are few geometrical constraints, compact circuits can be implemented. Experimental results from an implementation of a four-bit input, two-bit output integer square root circuit are given.

Figure 1

The $z$ or ($x$ and not $y$) gate from Ref. [3]. For the case $x=0$, $y=0$, and $z=1$ (a) shows waves that originated at the $z$ input and the constant 1 input and (b) shows these waves at a later time. For the case $x=1$, $y=0$, and $z=0$ (c) shows waves that originated at the $x$ input and the constant 1 input and (d) shows these waves at a later time. For both cases, waves arriving at the output channels are synchronized, but a comparison of (b) and (d) illustrates that for the latter case it takes longer for waves to reach the output than in the former case.

Figure 2

Domino fork. Letters denote events. Input events ($a$, $c$, and $e$) correspond to a domino toppling into the fork and output events ($b$, $d$, and $f$) correspond to a domino toppling out of the fork.

Figure 3

Simulation of a fork structure in BZ excitable media. (a) Splitting at a fork junction in a simulation of subexcitable BZ-reaction media. (b) Directional propagation at a fork junction in a simulation of subexcitable BZ-reaction media.

Figure 4

Experimental implementation of a fork structure in BZ excitable media. The scale bar is 3 mm. The level of illumination in dark regions was estimated at 0.5 mW cm${}^{-2}$. The level of illumination in light regions was estimated at 6.5 mW cm${}^{-2}$. (a) Splitting at a fork junction in subexcitable BZ-reaction media [compare with Figure 3a]. (b) Directional propagation at a fork junction in subexcitable BZ-reaction media [compare with Fig. 3b].

Figure 5

A $\mathsf{T}$-shaped region of excitable medium. The region is fully excitable, so a wave entering one terminal will propagate throughout the region. A unit of length is shown. A unit of time is defined as the time that it takes a wave to propagate one unit of length.

Figure 6

A $\mathsf{T}$-shaped region of excitable medium with valves at its terminals. This region implements a two-input or gate.

Figure 7

A wave propagating from $b$ to the $a$ or $b$ output. Squares with crosses inside represent closed valves. The left-hand case in this figure shows the wave shortly after $t_0$. Subsequent cases show the progress of the wave every 0.5 time units until shortly after $t_0+1.5$. The wave never reaches $a$ because it is blocked by a closed valve.

Figure 8

A $\mathsf{T}$-shaped region of excitable medium with an additional input. This structure implements the function ($a$ or $b$) and not $c$.

Figure 9

Region of excitable medium that implements the function $a$ and not $b$. The $a$ and not $b$ output emerges from two different terminals and the $b$ input is also made available at an output.

Figure 10

Four-bit integer square root circuit. Here $x_n$ is the $n\text{th}$ binary digit of $x$ and $y_n$ is the $n\text{th}$ binary digit of $y$. The logic gate symbols and excitable media channels are shaded to show the correspondence between the two. The key refers to the previous figures in which the excitable media structures that are used here were introduced.

Figure 11

Pattern for the four-bit square root circuit. Gray areas are areas of constant excitability. Dark areas are valves that alternate between excitable and unexcitable. White areas are unexcitable.

Figure 12

Block diagram of the experimental setup.

Figure 13

Experiment and simulation for cases $x=3$, 6, and 15 (note that for the $x=6$ case no wave emerges from $y_0$ because the valve at $y_0$ is closed when a wave reaches it): (a) $x=3$ experiment, halfway through; (b) $x=3$ simulation, halfway through; (c) $x=3$ experiment, complete; (d) $x=3$ simulation, complete; (e) $x=6$ experiment, halfway through; (f) $x=6$ simulation, halfway through; (g) $x=6$ experiment, complete; (h) $x=6$ simulation, complete; (i) $x=15$ experiment, halfway through; (j) $x=15$ simulation, halfway through; (k) $x=15$ experiment, complete; and (l) $x=15$ simulation, complete.

Figure 14

A not gate that turns a leading wave into a lagging wave and a lagging wave into a leading wave.

Figure 15

Structures for crossing over $x$ and $y$: (a) crossover structure for $x$ and $y$, in which $x$ will propagate to $x^{\prime }$ and alternatively $y$ will propagate to $y^{\prime }$, but the crossover fails if both $x$ and $y$ occur at the same time, and (b) crossover structure for $x$ and $y$ that will work even if $x$ and $y$ occur at the same time.

Figure 16

Arbitrarily shaped region of excitable media with three terminals with valves. Distances between pairs of valves are shown.

Figure 17

Mechanism that will synchronize waves with valve timing. All points along valve $v$ are the same distance from $b$ so that a wave originating from $a$ will reach $b$ at a constant time after $v$ opens, regardless of the exact time at which it entered $a$.