Logic gates in excitable media

Article abstract of DOI:10.1063/1.469732

The interaction of chemical waves propagating through capillary tubes is studied experimentally and numerically.Certain combinations of two or more tubes give rise to logic gates based on input and output signals in the form of chemical waves and wave ini

Full text of DOI:10.1063/1.469732

Logic gates in excitable media
Ágota Tóth and Kenneth Showalter
Citation: The Journal of Chemical Physics 103, 2058 (1995); doi: 10.1063/1.469732
View online: http://dx.doi.org/10.1063/1.469732
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/103/6?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Noise Enhanced Logic Gates
AIP Conf. Proc. 1339, 67 (2011); 10.1063/1.3574845
Logic gates with a single graphene transistor
Appl. Phys. Lett. 94, 073305 (2009); 10.1063/1.3079663
Spin-wave logical gates
Appl. Phys. Lett. 87, 153501 (2005); 10.1063/1.2089147
Fiberoptic logic gate
Appl. Phys. Lett. 46, 317 (1985); 10.1063/1.95615
Optical gating and logic with pyroelectric crystals
Appl. Phys. Lett. 24, 81 (1974); 10.1063/1.1655103
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59

Logic gates in excitable media
´
a)
Agota T o´ th and Kenneth Showalter
Department of Chemistry, West Virginia University, Morgantown, West Virginia 26506-6045
͑
Received 26 January 1995; accepted 4 May 1995͒
The interaction of chemical waves propagating through capillary tubes is studied experimentally and
numerically. Certain combinations of two or more tubes give rise to logic gates based on input and
output signals in the form of chemical waves and wave initiations. The geometrical configuration,
the temporal synchronization of the waves, and the ratio of the tube radius to the critical radius of
the excitable medium determine the features of the logic gates. © 1995 American Institute of
Physics.
I. INTRODUCTION
which yields an output of 1 ͑0͒ for an input of 0 ͑1͒, with the
OR, AND, and XNOR gates.
Nonlinear chemical reactions provide simple models for
excitable media in biological systems. Studies of spiral
waves in the Belousov–Zhabotinsky ͑BZ͒ reaction, for ex-
A number of studies have appeared on the use of non-
linear chemical systems for computational devices. Logic
operations in a bistable chemical system were described by
1
R o¨ ssler over 20 years ago.12 Recently, Ross and
ample, have contributed important insights into the nature of
tachycardia and fibrillation in the heart muscle.2 Chemical
model systems for signal transmission might similarly offer
insights into the excitable dynamics of neurons and neuronal
,3
13–15
co-workers
have proposed methods for constructing
chemical computers with reactor systems coupled by mass
flow. Various logic gates as well as simple computational
devices were demonstrated. Schneider and co-workers
4
16,17
networks. Narrow channels such as capillary tubes, for ex-
ample, can be considered as signal carriers in excitable
chemical systems.5 For a single capillary tube, an input
wave gives rise to a response of either 1 ͑an output wave͒ or
have reported numerical and experimental studies of logic
gates constructed from bistable reactions in coupled CSTRs,
with the coupling externally controlled to satisfy different
neural networks corresponding to particular gates. Okamoto
and co-workers18 have considered certain enzymatic reac-
tions as basic switching elements in neural networks.
We describe in this paper numerical and experimental
studies of logic gates based on chemical waves propagating
through capillary tubes in excitable BZ solutions. Traveling
waves of excitation are the basis of the binary coding: The
states 1 and 0 are defined by the presence or absence of a
wave at certain fixed locations. The features of various gates
are examined with a generic model for excitable media in
Sec. II, and experimental examples of logic gates in the ex-
citable BZ reaction are described in Sec. III. Combinations
of basic gates to yield higher order functions are described in
Sec. IV, and we conclude in Sec. V with a look at possible
connections to neural behavior.
–7
0
͑no output wave͒, as well as complex resonance patterns
for successive input waves. In the case of a neuron, however,
one input signal is usually not sufficient to produce an output
signal, i.e., multiple inputs are typically necessary for a neu-
ron to fire. In this paper, we study the output signals arising
from two or more input signals, in the form of chemical
waves propagating through narrow capillary tubes. The mul-
tiple inputs give rise to logic gates, where the particular gate
depends on the geometrical configuration and the properties
of the excitable medium.
Logic gates can be constructed by using physical quan-
tities involving binary states. In electronic devices these
states are generally represented by different voltages. In de-
vices based on fluid dynamics, the pressure of fluid flow
8
compared to atmospheric pressure determines the state. In
chemical systems, systematically designed single molecules
can serve as OR and AND gates, with the change in fluores-
cence of a receptor molecule on binding different ions pro-
viding the basis of the binary coding.9
II. NUMERICAL STUDY
,10
Logic gates are the realization of Boolean functions
A. Model
11
͑
ٙ,ٚ,Ј͒ operating on binary input sets. We will consider
binary coded elements with the values 0 ͑FALSE͒ and 1
TRUE͒ in two input channels. The response of different
We use the Barkley model,19 a computationally efficient
algorithm for describing excitable media, to examine various
multiple-channel arrays numerically. The spatiotemporal
evolution of the fast and slow variables u and v is governed
by the reaction-diffusion equations
͑
gates to all four combinations of the input sets are summa-
rized in Table I. An OR gate, for example, yields the value 1
as the output if the value of either or both input channels is 1.
The AND gate gives a response of 1 only when both inputs
are 1. The response of an XNOR gate is 1 if both inputs are
the same and 0 if they are different. The NOR, NAND, and
XOR gates can be realized by combining the NOT function,
ץu
ץt
ץv
ץt
2
2
ϭٌ uϩf͑u,v͒,
ϭٌ vϩg͑u,v͒,
͑1͒
where the functions f and g describe the local dynamics
1
f͑u,v͒ϭ u͑1Ϫu͓͒uϪu ͑v͔͒, g͑u,v͒ϭuϪv. ͑2͒
a͒To whom correspondence should be addressed.
th
⑀
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
058 J. Chem. Phys. 103 (6), 8 August 1995 0021-9606/95/103(6)/2058/9/$6.00 © 1995 American Institute of Physics
2
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59

A. Toth and K. Showalter: Logic gates in excitable media
2059
TABLE I. Output of logic gates for different input sets.
TABLE II. Critical radius as a function of parameter a.
I1
I2
OR
AND
XOR
NOR
NAND
XNOR
a
rc
1
0
1
0
0
1
1
0
1
1
1
0
0
0
1
0
1
1
0
0
0
0
0
1
1
1
0
1
0
0
1
1
0.8
0.7
0.6
0.55
0.0438Ϯ0.001
0.0683Ϯ0.0005
0.0938Ϯ0.0005
0.114Ϯ0.001
The threshold u (v) is defined as u (v) ϭ (v ϩ b)/a and
th
th
varied by varying the number of grid points in the gap ͑and
the overall system width͒. Planar waves were initiated peri-
odically by setting the value of u to 0.7 for one time step in
a strip three grid points wide at the left and right boundaries.
The waves were monitored close to each channel entrance
and far from each channel exit, as indicated in Fig. 1.
⑀
is a ‘‘small parameter’’ that determines the relative time
scales of u and v. The variables of this generic model are
analogous to the dimensionless concentrations of bromous
acid and ferriin in the Tyson–Fife model20 of the BZ reac-
tion. The specific behavior is determined by the parameters
a, b, and ⑀ ; to mimic an excitable BZ system we choose
bϭ0.01 and ⑀ϭ0.02 with 0.45ϽaϽ0.9.
The partial differential equations were integrated by an
explicit Euler method with the grid spacing hϭ0.065 and
the time step ⌬tϭ10 . The two-dimensional Laplacian was
B. Critical radius
Ϫ3
The logic gates described here rely on an essential char-
acteristic of excitable media: There exists a critical nucle-
ation size for the successful initiation of wave activity. This
is due to the dependence of wave velocity on front curvature,
which is described to a good approximation by an eikonal
approximated by a five-point formula, and no-flux boundary
conditions were applied at the walls and corners. To model
the experimental configuration for an AND or OR gate an
area of 51ϫ301 grid points was used, with channels of three
grid points in width ͑or ‘‘inner diameter’’͒ and 50 grid points
in length, as shown in Fig. 1. The channels were arranged in
the middle of the grid with a distance of five grid points
between the exits. In some calculations, this distance was
21,22
relation,
cϭc0ϩD␬,
͑3͒
where c is the normal wave velocity, c is the planar wave
0
velocity, D is the diffusion coefficient of the autocatalyst u,
and ␬ is the curvature of the front. In an expanding region of
excitation, the wave velocity is reduced due to the enhanced
diffusive dispersion of the autocatalyst in the convex wave
front ͑with ␬Ͻ0͒. The normal wave velocity approaches zero
as the radius of curvature (rϭ1/
value, which, from Eq. ͑3͒, is given by r ϭD/c . Regions of
͉␬͉͒ approaches a critical
c
0
excitation with rϾr expand to form outwardly propagating
c
waves, while regions with rϽr collapse. Although the ei-
c
konal equation is strictly applicable only for waves with
slight curvature, it provides a remarkably good estimation of
critical nucleation radius ͑where curvature is high͒. Experi-
mental measurements of the critical nucleation radius have
been made from BZ waves propagating through microcapil-
7
lary tubes, in which the hemisphere of excitation at the tube
exit was assumed to have the same radius as that of the tube.
The planar wave velocity in the Barkley model can be
varied by changing the parameter a. An increase in a results
in a higher wave velocity, and, based on the eikonal equa-
tion, a smaller critical radius. For a given grid spacing, the
critical radius can therefore be determined as a function of
the parameter a. Table II shows values of critical radius
determined for a single channel; we discuss below how the
critical radius also depends on the gap width in the AND/OR
configuration. The values show that the ͑fixed͒ channel ra-
dius of 0.065 is above the critical radius for aտ0.70. For
these values of a any wave propagating into the channel
gives rise to the initiation of a wave at the channel exit. ͑The
no-flux boundaries are defined by reflecting two rows of grid
points on either side of the boundary; the radius of a channel
three grid points in width is therefore ͑2ϫ0.065͒/2ϭ0.065.͒
FIG. 1. Assembly for OR gate ͑top panel͒. Dots show monitoring sites of
inputs I1 and I2 and output O. Time series show output of OR gate for
different input signals. Top and middle time series correspond to inputs I1
and I . Bottom time series shows output O for the corresponding inputs.
2
Wave input period ␶ϭ9, channel radius rϭ0.065, gap width wϭ0.195,
and aϭ0.6.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
J. Chem. Phys., Vol. 103, No. 6, 8 August 1995
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59

2060
A. Toth and K. Showalter: Logic gates in excitable media
We also note that the half-width of the gap between the chan-
nel exits must be above the critical radius for the wave to
expand into the middle chamber.
C. OR and AND gates
Figure 1 shows the configuration for the OR gate, where
the channel radius is greater than the critical radius for the
values of a and gap width. The top, middle, and bottom time
series correspond to the input channels on the left and right,
I and I , and the output, O. From the response at the output
1
2
to various input sets, we see that the system operates as an
OR gate. Hence, an input of 1 in either or both channels
results in an output of 1.
In the case of only one input, the wave initiated from the
channel can cross the output compartment and enter the sec-
ond channel, which gives rise to a spurious wave initiation in
the second compartment. To prevent this possibility, the half-
width of each channel at its entrance was reduced to a value
below the critical radius. This was accomplished by simply
decreasing the width at each channel entrance by one grid
point, as shown in Fig. 1. Thus when a wave is initiated in
the middle chamber and enters the other channel, it is unable
to propagate past the restricted channel entrance. The small
perturbations in the time series in Fig. 1 are due to such
waves collapsing at the restricted channel entrance.
FIG. 2. Assembly for AND gate ͑top panel͒. Time series show output of
On decreasing the value of a and thereby the wave ve-
locity such that the channel radii are just below the critical
radius, no output wave can be initiated from a single input
wave. Two simultaneous input waves, however, give rise to
the initiation of an output wave in the middle compartment.
Figure 2 shows the configuration for the AND gate, which is
the same as the OR gate except the value of a is slightly
smaller. The output O, shown in the bottom time series, is 1
AND gate for different input signals. The parameters are the same as in Fig.
1
except aϭ0.55.
termined by measuring the width of the u concentration pro-
file at half its maximum amplitude.͒ The maximum value of
this ratio for AND gate behavior is less than 0.5, indicating
that substantial overlap of the regions of excitation is neces-
sary for wave initiation.
only if both inputs, I1 and I , shown in the top and the
2
middle time series, are also 1. For a single input, the wave
collapses at the channel exit in the middle compartment,
which is seen in the other time series as a small perturbation.
At still lower values of the parameter a, wave initiation does
not occur at all in the central chamber, i.e., the output is 0 for
all inputs.
Previous studies of wave propagation through single
capillary tubes have shown that resonance patterns in the
input–output response occur on varying the period of suc-
7
cessive input waves. We therefore anticipate the appearance
Figure 3 shows the input-output response as a function
of parameter a and the gap width w. The boundary between
the AND and OR gate behavior represents values of a and w
for which the channel radii are equal to the critical radius.
For large gap widths, the value of a corresponding to the
critical radius is similar to that for a single channel. For
smaller gap widths, OR gate behavior is exhibited for values
of a below that corresponding to the single channel critical
radius. The restricted diffusion in the gap ͑compared to the
unrestricted diffusion at the exit of a single channel͒ gives
rise to smaller values of the effective critical radius. We also
see AND gate behavior at lower values of a for small gap
widths. At the opposite extreme of large gap widths, w must
be small enough for two subcritical regions of excitation to
combine to initiate a wave; no AND gate behavior is ob-
served for wϾ3. Another factor that influences the output is
the width of the wave, which increases with increasing a.
Figure 4 shows the input–output response as a function of
the ratio of gap width to wave width. ͑Wave width was de-
FIG. 3. Input–output response as a function of parameter a and gap width
w. OR gate behavior shown by ᮀ, AND gate behavior by ᭿, and no waves
initiated in output compartment by ϫ.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
J. Chem. Phys., Vol. 103, No. 6, 8 August 1995
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59

A. Toth and K. Showalter: Logic gates in excitable media
2061
FIG. 4. Input–output response as a function of parameter a and ratio of gap
width to wave width w/wwave . OR gate behavior shown by ᮀ, AND gate
behavior by ᭿, and no waves initiated in output compartment by ϫ.
FIG. 6. Phase diagram showing resonance patterns for AND gate as a func-
tion of wave input period ␶ and gap width w. Patterns indicated by 1:1 ͑᭹͒,
1:2 ͑᭺͒, 0:1 ͑ϫ͒. The parameters are the same as in Fig. 5.
Figure 6 shows a phase diagram of the behavior for dif-
ferent input periods and gap widths, where only the 0, 1/2,
and 1/1 resonance patterns are indicated. The bifurcation dia-
gram in Fig. 5 appears in this diagram as the column at
wϭ0.2. More complex patterns were also observed, where
of resonance patterns in the output response of the AND gate
for successive input waves. The patterns can be characterized
by the firing number fn,23 defined as the ratio of the number
of output waves to the number of input waves. On decreasing
the period of the input waves, the system undergoes the bi-
furcations shown in Fig. 5. Each resonance pattern is char-
acterized by fnϭ(nϪ1)/n, where n is the number of input
waves and (nϪ1) the number of output waves. As the input
period is increased, the firing number rapidly approaches
unity through a series of steps. This behavior is typical of
forced excitable systems and can be characterized in terms of
the firing numbers are characterized by Farey arithmetic
sums of neighboring patterns.2
9,30
For logic functions, we
have restricted our study to the regime of simple behavior
where fnϭ1. To ensure a firing number of unity, an input
period of ␶ϭ9 was used for each of the logic gates.
The timing of the input signals in the AND gate is of
critical importance: Waves must enter the output chamber
almost simultaneously for the successful initiation of a wave.
If the arrival time differs by more than about 1% of the
period, no output wave is observed. A limited study of the
input–output response as a function of the arrival time dif-
ference revealed complex behavior over a narrow range of
this parameter. Resonance patterns much like those in Fig. 5
are exhibited between the simple 1/1 response and the 0/1
response, where no output waves are observed.
an interrupted circle map.7,24 If the input period is decreased
below the 1/2 resonance, no output waves are observed, i.e.,
fnϭ0/1. Wave initiation is inhibited at the channel exits
because the system does not recover sufficiently to support
wave behavior. The restricted diffusion in the small gap and
small wave velocities near the critical radius cause the recov-
ery period to be longer than the wave input period. This
behavior is reminiscent of propagation failure, which has
been found in certain discrete25,26 and forced27,28 excitable
systems.
D. NOT function and XNOR gate
Other types of gates can be created from assemblies with
more than two channels. These are based on the observation
that a phase difference between the input signals of an AND
gate gives rise to an output signal of zero. We first consider
the construction of a NOT function, in which an input of 1
results in an output of 0 and vice versa. This useful function
can be connected in series with another gate to yield the
opposite output for every input. Thus an AND gate followed
by a NOT function generates a NAND gate, etc.
The configuration of a NOT function with three wave
channels is shown in Fig. 7. The NOT function requires an
internal periodic source to serve as a clock; therefore, two
synchronized pacemakers, P, are incorporated in the left and
right compartments. ͑A single periodic source would be suf-
ficient in a configuration in which these compartments were
connected.͒ The two pacemakers and coaxial channels con-
FIG. 5. Bifurcation diagram showing firing number fn as a function of wave
input period ␶ for AND gate. The parameters are the same as in Fig. 2.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
J. Chem. Phys., Vol. 103, No. 6, 8 August 1995
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59

2062
A. Toth and K. Showalter: Logic gates in excitable media
FIG. 7. Assembly for NOT function ͑top panel͒. Dots show monitoring sites
of input I, output O, and periodic waves P. Time series show output of NOT
function for different input signals. Top and middle time series correspond
to periodic waves P and input I. Bottom time series shows output O for the
corresponding inputs. The parameters are the same as in Fig. 2.
FIG. 8. Assembly for XNOR gate ͑top panel͒. Dots show monitoring sites
of inputs I1 and I , output O, and periodic waves P. Time series show
2
output of XNOR gate for different input signals. Top and middle time series
correspond to inputs I1 and I . Bottom time series shows output O for the
2
corresponding inputs. The parameters are the same as in Fig. 2.
stitute a periodic AND gate, with wave initiation occurring
on the simultaneous arrival of the waves in the middle com-
partment. The input of the assembly is via the side channel.
This is positioned so that the distance from the input to the
exit in the middle compartment is shorter than the distance
from each periodic source to the corresponding channel exit.
The input signal is synchronized with the periodic sources so
the signals respond in a 1:1 manner. The input wave there-
fore annihilates the signal from the left source and arrives at
the middle chamber earlier than the signal generated by the
right source. This temporal decoupling of the AND gate re-
sults in an output signal of 0 for an input signal of 1. With no
input signal, the periodic sources give rise to wave initiation;
hence, an input of 0 generates an output of 1.
The time series in Fig. 7 illustrates the action of the NOT
function, showing the synchronous periodic sources P ͑top͒,
the input signal I ͑middle͒, and the output signal O ͑bottom͒.
The configuration qualitatively reverses the input signal: The
result of an input wave is the absence of an output wave and
vice versa. The calculations were carried out as described
above, except the size of the domain was increased to 91
ϫ301 grid points.
produces an output wave if both of the inputs are the same.
Thus the output is 1 for inputs 1,1 or 0,0, but 0 for inputs 1,0
or 0,1.
III. EXPERIMENTAL STUDY
A. Procedures
The reaction mixtures were prepared as described
7
previously, with the exception of the metal catalyst. Instead
of the anionic bathophenanthrolate complex used in our pre-
vious study, we used ferroin, prepared by mixing iron II-
sulfate and 1,10-phenanthroline in a 1:3 mole ratio. The fer-
roin solution was standardized by measuring its optical
Ϫ1
Ϫ1 31
absorbance at 510 nm ͑⑀ϭ11 100 M cm ͒. The polyim-
ide coating of the capillary tubes ͑100 ␮m inner diameter,
Polymicro Technologies͒ was removed by gentle heating
over a Bunsen burner. The capillaries were then inserted
through holes, with inner diameters matching the outside di-
ameters of the tubes, drilled in Plexiglas blocks. Two
Plexiglas-block barriers, each with mounted capillary tubes,
were placed in a polycarbonate petri dish ͑10 cm diameter,
Nalgene͒ so the gap between the coaxial tubes was ϳ0.5 cm.
The Plexiglas barriers were sealed into place with acetone
solvent. The capillary tubes were then carefully positioned
such that the gap between them was ϳ200 ␮m.
An XNOR gate can be constructed by adding one more
channel, as shown in Fig. 8. Based on the construction of the
NOT function, two input channels are added to the AND gate
in a symmetrical fashion. The time series show the output O
for various combinations of the inputs, I and I , where the
A 12.5 ml solution was prepared with the concentrations
given in Table III and swirled until an oxidation cycle was
observed. The reaction mixture was then poured into the pe-
1
2
calculations were carried out as in Fig. 7. The XNOR gate
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
J. Chem. Phys., Vol. 103, No. 6, 8 August 1995
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59

A. Toth and K. Showalter: Logic gates in excitable media
2063
TABLE III. Reagent concentrations.
NaBrO3
H SO
Malonic acid
Ferroin
0.10–0.14 M
0.14 M
2
4
Ϫ2
2.0ϫ10
1.0ϫ10
M
M
Ϫ2
tri dish, which was thermostated at 25.0Ϯ0.1 °C. Gaps be-
tween the Plexiglas barriers and the petri dish wall allowed
the solution levels in the different compartments to quickly
equalize. Absorption spectra of ferroin and ferriin showed
that the largest difference in their molar absorptivities occurs
at 510 nm. We therefore used a broad band interference filter
with a 500 nm peak transmittance ͑Oriel͒ to monitor the
waves. Details of the wave initiation and image processing
techniques are described in Ref. 7.
FIG. 9. Subtraction images showing a chemical wave at 10 s intervals
traveling inside a capillary tube of 50 ␮m radius. Inputs of 0 ͑left tube͒ and
B. Results
1
͑right tube͒ with output of 1 demonstrate OR gate behavior. Concentra-
Ϫ
tions given in Table III with ͓BrO ͔ϭ0.14 M. Field of view showing
3
The critical radius rc was first determined for an
AND/OR gate assembly with a fixed gap width and capillary
tube size. Because the planar wave velocity depends on
middle compartment of AND/OR assembly: 0.38 mmϫ2.42 mm for top
three panels and 0.82 mmϫ2.42 mm for bottom panel. The gap between the
tube exits is 180 ␮m.
Ϫ
3
32
͓
BrO ͔, the critical radius can be varied continuously by
changing the bromate concentration in the reaction mixture.
Ϫ
3
Above a critical concentration, ͓BrO ͔ ,waves initiated in
c
The timing of the input waves and the gap width are of
critical importance in these experiments. Figure 12 shows the
AND gate where the time between the appearance of the
exiting waves is too large to support wave initiation. The first
excitation collapses before the second develops, and no out-
put wave is initiated. The outcome is similar if the tube exits
are too far apart: The two hemispheres of excitation collapse
in the gap even for simultaneous input waves. For successive
wave initiations, no output signal is observed with high fre-
quency input signals even when the timing and gap width are
appropriate, since the recovery period for the excitable me-
dium at each exit is longer than the wave initiation period.
the right or left compartment give rise to wave initiation in
the middle compartment at the tube exit. Below this concen-
tration, no wave initiation is observed in the middle compart-
ment for a single input wave. The critical bromate concen-
Ϫ
tration ͓BrO ͔ ϭ 0.11 M was determined for tube radii of
3
c
50 ␮m and a gap width of 180 ␮m.
The radius of the capillary tube is above the critical ra-
Ϫ
3
dius for solutions with ͓BrO ͔ Ͼ 0.11 M. If a wave enters
one of the tubes, it propagates to the tube exit and a wave is
initiated in the middle compartment, as shown in Fig. 9.
When initiating waves simultaneously in the left and right
compartments, we again observe an output in the form of
wave initiation in the central compartment. This tube con-
figuration therefore functions as an OR gate, provided the
bromate concentration is above the critical value. An output
signal of 1 is registered for input signals of ͑1,0͒, ͑0,1͒, or
͑1,1͒. The wave initiation in the central compartment is
monitored far from the tube exits to ensure that the output
signal represents a propagating wave.
The radius of the capillary is smaller than the critical
Ϫ
3
radius for ͓BrO ͔ Ͻ 0.11M. In solutions with bromate
concentrations just below the critical value, no wave is initi-
ated in the central compartment when a wave reaches the
tube exit. As shown in Fig. 10, the small hemisphere of ex-
citation collapses at the tube exit. Wave initiation in the cen-
tral compartment becomes possible, however, with two si-
multaneous input waves. As shown in Fig. 11, waves
initiated simultaneously in the left and right compartments
result in an output signal of 1. Thus the tube configuration
now serves as an AND gate, provided the bromate concen-
tration is only slightly below the critical value. An output
signal of 1 is registered only for input signals of 1,1; for
input signals of 1,0 or 0,1 an output signal of 0 is registered.
FIG. 10. Subtraction images showing a chemical wave at 10 s intervals
traveling inside a capillary tube. Inputs of 0 ͑left tube͒ and 1 ͑right tube͒
with output of 0 demonstrate AND gate behavior. Solution composition,
distance between the tube exits and the tube radii are the same as in Fig. 9,
Ϫ
3
except ͓BrO ͔ ϭ 0.1085M. Field of view showing middle compartment
of AND/OR assembly: 0.38 mmϫ2.58 mm in each panel.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
J. Chem. Phys., Vol. 103, No. 6, 8 August 1995
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59

2064
A. Toth and K. Showalter: Logic gates in excitable media
the NOT function and the XNOR gate can be constructed by
adding periodic wave sources and additional wave channels
to the AND/OR assembly. These gates are based on the input
signals interfering with the synchronized internal periodic
waves.
Other types of gates can be constructed by combining
the above gates in various configurations. For example, the
serial combination of an OR gate and a NOT function—
where the output of the OR is the input of the NOT—yields
a NOR gate. This composite gate gives an output of 0 for
inputs of ͑0,1͒, ͑1,0͒, or ͑1,1͒ and an output of 1 for an input
of 0,0. A similar serial combination of an AND gate and a
NOT function yields a NAND gate, with outputs opposite of
the AND gate. Combining the XNOR gate with the NOT
function yields the XOR gate to complete the basic gates
listed in Table I.
The key element in all the gates, with the exception of
the OR gate, is the union of two subcritical regions of exci-
tation to initiate an output wave—the basis of AND gate
behavior. The parameters affecting AND gate behavior are
gap width, wave width, and the period and timing of the
waves exiting into the output compartment. For a range of
solution compositions ͑or values of a in the Barkley model͒,
AND gate behavior is exhibited between a minimum and a
maximum gap width. The minimum gap width is defined by
twice the critical radius. Below this value, waves are unable
to exit from the gap. Examples of this behavior are shown by
FIG. 11. Subtraction images showing chemical waves at 10 s intervals trav-
eling inside capillary tubes. Inputs of 1 in both tubes with output of 1
demonstrate AND gate behavior. Conditions are the same as in Fig. 10. Field
of view showing middle compartment of AND/OR assembly: 0.38 mm
ϫ2.58 mm in top four panels and 0.64 mmϫ2.58 mm in bottom panel.
IV. DISCUSSION
The spatial and temporal coupling of excitable elements
can be used as the basis for logic gates. We have used chemi-
cal waves exiting from the ends of capillary tubes with radii
near the critical nucleation radius as excitable elements. Two
coaxial tubes through barriers separating a central output
chamber from two input chambers form the basis of AND
and OR gates. A slight adjustment in the reaction mixture
composition is sufficient to convert the assembly from an OR
gate to an AND gate and vice versa. Chemical analogues of
the first two columns of values in Fig. 6, where wϽ2r . The
c
maximum gap width is highly dependent on the wave width.
For wave widths that are small compared to the critical ra-
dius we would anticipate the maximum gap width to be ap-
proximately 2ϫr , based on wave initiation arising from the
c
combination of two subcritical hemispheres of excitation.
Examination of Fig. 3 shows, however, that the gap width
may be much greater than this for AND gate behavior, par-
ticularly as the channel ͑or tube͒ radius approaches the criti-
cal radius. This is because the wave width increases with
increasing velocity ͑or value of a), and for very wide waves
the region of excitation at the tube exit is substantially ex-
tended. Figure 4 shows that the ratio of gap width to wave
width is less than 0.5 for the maximum value of AND gate
behavior, indicating that significant overlap of the excitation
regions must occur for wave initiation.
The arrival of the input signals at the channel exits must
be nearly synchronous for AND gate behavior. Wave initia-
tion does not occur when the arrival time differs by more
than about 1%. The basis of the NOT and XNOR gates is a
deliberate dephasing of the excitations via interference by
the input signals. In the case of the NOT function, the input
signal arrives at the output compartment slightly before the
pacemaker signal, and no wave initiation can occur. This is
also the case for the XNOR gate with inputs of 0,1 or 1,0;
however, with an input of 0,0 or 1,1 the signals are synchro-
nized ͑even though from different sources͒ and the result is
wave initiation in the output compartment.
FIG. 12. Subtraction images showing chemical waves at 10 s intervals trav-
eling inside capillary tubes. Inputs are 1 in both left and right tubes but with
a delay of ϳ10 s, resulting in an output of 0. Conditions are the same as in
Fig. 10. Field of view showing middle compartment of AND/OR assembly:
The timing requirement of the AND gate can be used to
generate an additional function—namely a frequency selec-
tor. For two input signals with different periods, output sig-
nals are observed only for the least common multiples of the
0
.38 mmϫ2.58 mm in each panel.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
J. Chem. Phys., Vol. 103, No. 6, 8 August 1995
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59

A. Toth and K. Showalter: Logic gates in excitable media
2065
periods ͑provided the input signals are in phase͒. Thus the
unknown frequency of one input signal could be determined
by scanning through the frequency of the other input signal
and monitoring the output signal.
assemblies based on two channels; however, it is easy to
imagine AND type behavior arising in assemblies with many
channels. The gate fires when a threshold is reached, deter-
mined by a sufficient region of excitation for the initiation of
a wave. As a specific example of a multichannel gate, one
can imagine adding a third channel to the AND gates in Figs.
2 or 11 to form a T-shaped assembly. The simultaneous input
of three waves would transform parameter regions in Fig. 3
from no response to AND gate behavior ͑e.g., at aϭ0.65,
wϭ2.0).
Another example of multiple input signals for triggering
an output signal is seen in the complex (nϪ1)/n resonance
patterns with successive waves. The fnϭ1/2 pattern, for
example, requires two input waves for every output wave.
Thus n oscillations are necessary to trigger a burst of nϪ1
oscillations in an output wavetrain. This behavior is reminis-
cent of bursting behavior seen in certain single and coupled
nerve cell assemblies.33
Finally, we note that while the above analogies apply to
excitatory connections there is also behavior analogous to
inhibitory connections. The ‘‘dephasing’’ caused by an input
signal in the NOT and XNOR gates gives rise to an inhibi-
tion of wave initiation. The input channels can therefore be
considered as ‘‘inhibitory connections’’ in these gates.
Logic gate assemblies become increasingly difficult to
construct as multiple channels with special features are re-
quired. Other techniques, such as those developed by Nosz-
ticzius and co-workers,34 where the catalyst is bound to a
membrane in a desired pattern, might allow the construction
of more complex gates. We also note that miniaturized as-
semblies could provide logic gates that operate on much
faster time scales than the tube assemblies. Recent studies
have shown that narrow channels can be created by using
photolithography in the CO–platinum system.35 It should be
possible using similar methods to construct miniature logic
gates based on excitable media in this system, thus providing
faster chemical based computing devices.
The period of successive input waves determines
whether the input and output signals have a simple 1:1 rela-
tionship or whether complex resonance patterns occur, as
shown in Fig. 5. At high periods where fnϭ1, the system
has sufficient time to relax to the steady state to function
consecutively as an AND or OR gate. At lower periods, the
firing number varies through a sequence of (nϪ1)/n pat-
terns. The complex behavior may allow further interesting
input–output processing. The (nϪ1)/n patterns occur in a
stepwise fashion, each appearing over a range of input fre-
quency, as shown in Fig. 5. As described in Ref. 7, each is a
distinct pattern; for example, the 3/4 resonance is a repeating
train of three ‘‘quarter notes’’ followed by a ‘‘quarter rest.’’
Because ranges of the input frequency correspond to particu-
lar output patterns, the assembly could be used as a fre-
quency coder.
At high frequencies, no output waves are generated be-
yond a threshold frequency of input waves. Because the
wave velocity approaches zero as the channel radius ap-
proaches the critical radius, the relaxation to the steady state
is slower at the channel exit than in the case of an unre-
stricted wave. Above the frequency threshold, wave initiation
is inhibited because the system has no time to recover to its
excitable state. The gap width also plays an important role in
this behavior, since the diffusive dispersion of the autocata-
lyst is restricted as the gap width is decreased. This behavior
is similar to propagation failure in discrete and forced excit-
2
5–27
able systems.
Computational devices can be created by combining
logic gates in particular arrays. For example, a binary adding
machine can be made by operating an AND gate and an
XOR gate in parallel. Thus inputs of 1,1 yield an output of 1
for the AND gate and 0 for the XOR gate, which interpreted
as the binary output ͑10͒ yields the decimal sum of the inputs
͑
1ϩ1ϭ2͒. Similarly, inputs of ͑0,0͒, ͑0,1͒, and ͑1,0͒ yield
ACKNOWLEDGMENTS
binary outputs of ͑00͒, ͑01͒, and ͑01͒, or the decimal sums 0,
1
14
We thank Oliver Steinbock for helpful discussions on
logic gates in excitable media and for critically reading the
manuscript. We are grateful to the National Science Founda-
tion ͑CHE-9222616͒ and the Office of Naval Research
, and 1 of the inputs. Hjelmfelt, Weinberger, and Ross
18
have used the switching enzymatic model of Okamoto to
develop examples of a binary decoder, binary adder, and
stack memory. Logic gates based on excitable media can be
similarly combined to yield more complex computing de-
vices.
͑N00014-95-1-0247͒ for supporting this research. Acknowl-
edgment is made to the donors of The Petroleum Research
Fund, administered by the American Chemical Society, for
partial support of this research.
V. CONCLUSION
We have studied logic gates based on the properties of
excitable media, where the signals are coded by the presence
or absence of chemical waves. Intriguing analogies can be
drawn between the various gates we have considered and the
simplest descriptions of nerve cell firing. Signals arriving on
multiple axons are transferred via synaptic connections to the
dendrites, which carry the excitation to the nerve cell body.
The nerve fires, sending an action potential down its axon,
when a threshold is reached from the accumulated signal
input. Similarly, multiple signals are necessary for ‘‘firing’’
in an AND gate based on excitable media. We have studied
1
A. N. Zaikin and A. M. Zhabotinsky, Nature ͑London͒ 225, 535 ͑1970͒.
A. T. Winfree, The Geometry of Biological Time ͑Springer, Berlin, 1980͒.
J. M. Davidenko, A. M. Pertsov, R. Salomonsz, W. Baxter, and J. Jalife,
Nature ͑London͒ 355, 349 ͑1992͒.
E. R. Kandel and J. H. Schwartz, Principles of Neural Science ͑Elsevier,
New York, 1985͒.
A. Babloyantz and J. A. Sepulchre, Physica D 49, 52 ͑1991͒.
2
3
4
5
6
A. Babloyantz and J. A. Sepulchre, Phys. Rev. Lett. 66, 1314 ͑1991͒.
7
8
´
A. T o´ th, V. G a´ sp a´ r, and K. Showalter, J. Phys. Chem. 98, 522 ͑1994͒.
D. Bouteille and C. Guidot, Fluid Logic Controls and Industrial Automa-
tion ͑Wiley & Sons, New York, 1973͒.
A. P. de Silva, H. Q. N. Gunaratne, and K. R. A. S. Sandanayake, Tetra-
hedron Lett. 31, 5193 ͑1990͒.
9
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
J. Chem. Phys., Vol. 103, No. 6, 8 August 1995
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59

2066
A. Toth and K. Showalter: Logic gates in excitable media
1
0
23
A. P. de Silva, H. Q. N. Gunaratne, and C. P. McCoy, Nature ͑London͒
64, 42 ͑1993͒.
E. Mendelson, Theory and Problems of Boolean Algebra and Switching
Circuits ͑McGraw-Hill, New York, 1970͒.
O. E. R o¨ ssler, in Physics and Mathematics of the Nervous System, edited
by M. Conrad, W. G u¨ ttinger, and M. Dal Cin ͑Springer, Berlin, 1974͒, pp.
L. Glass and M. C. Mackey, From Clocks to Chaos ͑Princeton, New
Jersey, 1988͒.
J. Finkeov a´ , M. Dolnik, B. Hrudka, and M. Marek, J. Phys. Chem. 94,
4110 ͑1990͒.
J. P. Keener, J. Theor. Biol. 148, 49 ͑1991͒.
J.-P. Laplante and T. Erneaux, J. Phys. Chem. 96, 4931 ͑1992͒.
J. Kosek and M. Marek, J. Phys. Chem. 97, 120 ͑1993͒.
J. Kosek, I. Schreiber, and M. Marek, Philos. Trans. R. Soc. London A
347, 643 ͑1994͒.
J. Farey, Philos. Mag. J. ͑London͒ 47, 385 ͑1816͒.
J. Maselko and H. L. Swinney, J. Chem. Phys. 85, 6430 ͑1986͒.
A. A. Schilt, Analytical Applications of 1,10-phenanthroline and Related
Compounds ͑Pergamon, London, 1969͒.
R. J. Field and R. M. Noyes, J. Am. Chem. Soc. 96, 2001 ͑1974͒.
M. Colding-Jørgensen, in Complexity, Chaos, and Biological Evolution,
edited by E. Mosekilde and L. Mosekilde ͑Plenum, New York, 1991͒.
A. L a´ z a´ r, Z. Noszticzius, H.-D. F o¨ rsterling, and Zs. Nagy-Ungv a´ rai,
3
11
24
25
12
2
2
6
7
3
99–418.
1
1
3
4
28
A. Hjelmfelt, E. D. Weinberger, and J. Ross, Proc. Natl. Acad. Sci. USA
8, 10983 ͑1991͒.
A. Hjelmfelt, E. D. Weinberger, and J. Ross, Proc. Natl. Acad. Sci. USA
9, 383 ͑1992͒.
8
29
30
8
1
1
1
5
6
7
31
A. Hjelmfelt, F. W. Schneider, and J. Ross, Science 260, 335 ͑1993͒.
D. Lebender and F. W. Schneider, J. Phys. Chem. 98, 7533 ͑1994͒.
K.-P. Zeyer, G. Dechert, W. Hohmann, R. Blittersdorf, and F. W.
Schneider, Z. Naturforsch. Teil A 49, 953 ͑1994͒.
M. Okamoto, T. Sakai, and K. Hayashi, Biol. Cybern. 58, 295 ͑1988͒.
D. Barkley, Physica D 49, 61 ͑1991͒.
J. J. Tyson and P. C. Fife, J. Chem. Phys. 73, 2224 ͑1980͒.
J. J. Tyson and J. P. Keener, Physica D 32, 327 ͑1988͒.
V. S. Zykov, Biophysics 25, 906 ͑1980͒.
32
33
1
1
2
2
2
8
9
0
1
2
34
35
Chaos ͑to be published͒.
M. D. Graham, I. G. Kevrekidis, K. Asakura, J. Lauterbach, K. Krischer,
H.-H. Rotermund, and G. Ertl, Science 264, 80 ͑1994͒.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
J. Chem. Phys., Vol. 103, No. 6, 8 August 1995
138.251.14.35 On: Fri, 19 Dec 2014 17:37:59