Logical and arithmetic circuits in Belousov-Zhabotinsky encapsulated disks

Excitation waves on a subexcitable Belousov-Zhabotinsky (BZ) substrate can be manipulated by chemical variations in the substrate and by interactions with other waves. Symbolic assignment and interpretation of wave dynamics can be used to perform logical and arithmetic computations. We present chemical analogs of elementary logic and arithmetic circuits created entirely from interconnected arrangements of individual BZ encapsulated cell-like disk. Interdisk wave migration is confined in carefully positioned connecting pores. This connection limits wave expansion and unifies the input-output characteristic of the disks. Circuit designs derived from numeric simulations are optically encoded onto a homogeneous photosensitive BZ substrate.

Figure 1

Example of time-lapse image creation. The image shown in (f) is the accumulation of successive images shown from central wave initiation in (a) to extinction in (e). Time lapses periods are 50 time steps and the refractory tail of the inhibitor ($u$) shown in blue [gray inner shadow of propagating ring in (a)–(e)] is not shown in the time-lapse image (f) to improve clarity.

Figure 2

(a) Disks as reaction vesicles in the left-hand-side network or communication channels in the right-hand-side network. (b) The effect of connection angle; the two input signals combine in the left network to produce an output. Adjusting the angle of the lower input in the right network alters the result of the collision and no output is produced. (c) The effect of connection efficiency. The large-aperture (6 SP) connection in the left network results in a broadly spreading beam. Conversely, a smaller (4 SP) aperture connection in the right network creates a narrow beam wave.

Figure 3

Experimental setup: A light sensitive catalyst, ${\mathrm{Ru}\left(\mathrm{bpy}\right)}_3{}^{3+}$-loaded silica gel, is immersed in catalyst-free BZ reaction solution in a thermostat-controlled (G) Petri dish (E). (F) is a peristaltic pump. A reactor provides thermostat-controlled reaction solution and removes effluent (J). The reaction solution reservoir (H) is kept in an ice bath. The heterogeneous network on the surface of the gel (D) is constructed by the projection (B) of a disk pattern generated by a computer (A) via a mirror assembly (C). Images were captured with a digital camera fitted with a 455-nm narrow-bandpass interference filter (I).

Figure 4

Diode junction ($x\to y=0$, $y\to x=1$). Left column sequences show wave (signal) development compared alongside analogous numerical simulations on the right. (a) The signal propagates horizontally ($x\to y$). A narrow-band pore (4 SP) at the second (right-angle) junction prohibits propagation downward to $y$. (b) The signal propagates vertically ($y\to x$). A broad-band pore (6 SP) at the second (right-angle) junction connection permits the signal to expand horizontally toward $x$. The simulation-to-gel projection ratio is 56 SP to 10 mm diameter [42].

Figure 5

Two input nand gate ($z=\overline{x•y}$). Left column sequences show wave (signal) development compared alongside numerical simulations on the right. (a) $z=1,(x,y)=[0,0]$. (b) $z=0,(x,y)=[0,1]\left|\right[1,0]$. (c) $z=0,(x,y)=[1,1]$. Pores of 6 SP interconnect 56 SP diameter disks with simulation-to-gel projection ratio of 56 SP to 10 mm diameter [42].

Figure 6

Two input xor gate ($z=x\oplus y$). Left-column sequences show wave (signal) development compared alongside numerical simulations on the right for the input sets (a) $z=0,(x,y)=[0,0]$, (b) $z=1,(x,y)=[0,1]\left|\right[1,0]$, and (c) $z=0,(x,y)=[1,1]$. Pores of 6 SP interconnect 56 SP diameter disks with simulation-to-gel projection ratio of 56 SP to 10 mm diameter [42].

Figure 7

One-bit half-adder circuit ($S=x\oplus y,C=x•y$). The central reactor disk performs both xor and and logic functions to create the desired logic. (a) xor component, $S=1,C=0,(x,y)=[0,1]\left|\right[1,0]$. (b) and component, $S=0,C=1,(x,y)=[1,1]$. Pores of 4 SP interconnect disks of 56 SP diameter scaled by $1.2$ for $S$ and $0.8$ for $x$ and $y$, with simulation-to-gel projection ratio of 56 SP to 10 mm diameter. The circuit exploits three types of wave modulation: connection angle, disk size, and pore efficiency [42].