CHIN. PHYS. LETT. Vol. 29, No. 5 (2012) 054203
Wave Optics in Discrete Excitable Media
*
GU Guo-Feng(顾国锋), WEI Hai-Ming(韦海明), TANG Guo-Ning(唐国宁)**
College of Physics Science and Technology, Guangxi Normal University, Guilin 541004
(Received 1 September 2011)
Refraction and reflection of planar waves in a discrete excitable medium is numerically investigated by using
the Greenberg–Hasting model. It is found that the medium is anisotropic because the speed of the planar wave
depends on the excitability of the medium and the direction of wave propagation. The reflection, diffraction,
refraction, double refraction and delayed refraction are observed by using the correct choice of model parameters.
When the incident angle is larger than the critical angle, the reflection, which is a back refraction, takes place.
The reflection angle changes with the incident angle. The refraction in certain situations obeys Snell’s law. Also,
our results demonstrate that the incident, refracted and reflected waves can have different periods. The reflected
and refracted waves can disappear.
PACS: 42.25.Gy, 47.54.−r
DOI: 10.1088/0256-307X/29/5/054203
Wave propagation and pattern formation in excitable media are widely studied in chemistry and biology. The most basic properties of excitation waves in
spatially uniform reaction-diffusion systems and some
insights into the spatiotemporal pattern formation
in biological excitable media have been established
in various experimental and theoretical studies.[1,2]
To obtain further insights into the important features of biological excitable media, investigations on
nonuniform reaction-diffusion systems are needed because biological media are strongly nonuniform. Wave
propagation in nonuniform media has attracted much
attention.[3−14] It is found that these waves can be
treated by following a geometrical approach since they
exhibit behavior similar to light waves in the same situations.
Refraction was first studied theoretically by
Mornev[15] in a system consisting of two regions with
different diffusion coefficients and identical local chemical kinetics. The refraction and reflection of chemical waves in a Belousov–Zhabotinsky (BZ) reactiondiffusion medium was experimentally investigated by
Zhabotinsky et al.[3] Accordingly, refraction takes
place if the wave goes from a medium that supports
a higher wave speed to a medium with a lower wave
speed. It also occurs when the wave crosses the interface from a lower speed medium to a higher speed
medium, provided that the incidence angle is smaller
than the critical angle. The refraction of chemical
waves obeys Snell’s law. Reflection takes place if the
incident wave leaves the low-velocity medium with an
angle larger than the critical value. The reflection angle is always equal to the critical angle, i.e., chemical
wave reflection does not obey the law of light reflection (i.e., the angle of reflection must equal the angle of incidence). Sainhas and Dilão[8] showed that
chemical wave reflection is a back refraction, which
can be simply explained by Snell’s law. It is also
shown that, when the period of incident chemical
waves from the high-velocity medium is shorter than
the refractory period in the low-velocity medium, wave
propagation across the boundary causes a change of
frequency.[16] The above geometrical wave theory enables us to represent the development of the wave
fronts easily. Based on the theory, a chemical lens was
created and studied experimentally.[17] However, wave
propagation in anisotropic reaction-diffusion systems
has not yet been studied. Whether double refraction
and diffraction can occur in excitable media remains
unknown.
The Greenberg–Hasting (GH) model[18] is a kind
of excitable cellular automata which can be used to
model an excitable medium. It can emulate the behavior of a wide range of complex, coherent, periodic
wave phenomena in space. Therefore, we use the twodimensional GH model to study the refraction and
reflection of excitation waves at the interface between
two media with different update rules. Cells are uniformly distributed on a regular square lattice, and we
take the nearest neighbor distance between cells to be
1. The size of the lattice is 600 × 400. We adopt the
extended Moor structure. 𝑅 denotes the neighbor radius. Thus there are 𝑁𝑅 = (2𝑅 + 1)2 − 1 cells around
the central cell. Here 𝑢(𝑡) is the state function of a cell
at time 𝑡, and 𝑢(𝑡) ∈ {0, 1, 2, · · · , 𝐾 − 1}. The value
of 𝐾 is the number of states of a cell; 𝑢(𝑡) = 0, 1 correspond to the rest and excited states, respectively;
𝑢(𝑡) = 2, 3, · · · , 𝐾 − 1 correspond to the refractory
states. The update rules for the state of the cell labeled by (𝑖, 𝑗) are as follows:
⎧
⎨(𝑢𝑖,𝑗 (𝑡)+1)mod𝐾, if 1 ≤ 𝑢𝑖,𝑗 (𝑡) ≤ 𝐾−1,
𝑢𝑖,𝑗 (𝑡+1)= 1,
if 𝑂(𝑢𝑖,𝑗 (𝑡) = 0) ≥ 𝑂th ,
⎩
0,
if 𝑂(𝑢𝑖,𝑗 (𝑡) = 0) < 𝑂th ,
(1)
where 𝑂th is the excitation threshold of cells.
𝑂(𝑢𝑖,𝑗 (𝑡) = 0) denotes the number of excited cells in
the neighborhood of the cell (𝑖, 𝑗) which is in a rest
* Supported
by the National Natural Science Foundation of China under Grant Nos 11165004 and 10765002.
author. Email: [email protected]
© 2012 Chinese Physical Society and IOP Publishing Ltd
** Corresponding
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CHIN. PHYS. LETT. Vol. 29, No. 5 (2012) 054203
state. 𝐾, 𝑅, 𝑄th and 𝑡 take integer values. Near the
medium boundary, the excitation threshold of a cell
takes the integer part of (𝑂th 𝑁𝑅′ /𝑁𝑅 ), where 𝑁𝑅′ is
the number of neighbor cells of the cell.
y
0
0
0
7
7
7
7
0
0
0
0
7
7
7
1
0
0
0
0
7
7
1
1
0
0
0
7
7
1
1
0
0
7
1
B 0
1
0
1
0
0
0
7
1β
1
1
0
0
0
0
1
1
1
1
0
0
0
2
1
1
1
0
0
0
2A
2
1 h
1
1
1
0
0
2
1
1
1
1
0
2
2
2
1
1
1
0
2
2 C
2
1
1
1
1
2
2
2 L1 2
1
1
1
3
3
ϕ
2
2
2
1
1
1
2
2
2
2
1
1
the positive 𝑥 axis. The media I and II are indexed by
(𝐾𝑖 , 𝑂th,𝑖 ) and (𝐾𝑟 , 𝑂th,𝑟 ), respectively. The incident
planar wave is generated by setting the status of the
rectangle area (𝑅 × 400) near the left boundary of the
system to be excited at each time step, and at the same
time the initial status of else area to be rest. Hence,
the period of the incident wave 𝑇𝑖 satisfies 𝑇𝑖 = 𝐾𝑖 .
The angle of incidence 𝜃𝑖 equals 90∘ − 𝜃 since 𝜃𝑖 is always defined as the angle between the direction of the
incident wave propagation and the interface normal.
3.0
th=4
th=5
th=6
2.7
th=7
th=8
2.4
th=9
th=10
th=11
2.1
1.8
x
0
Fig. 1. Propagation of a planar wave. The planar wave
moves from point A on wave front 𝐿1 to point B on wave
front 𝐿2 in a time step.
In this Letter, no-flux boundary conditions and
𝑅 = 3 are adopted. We show firstly that the GH
model exhibits anisotropic behavior. Let 𝜙 be the angle between the 𝑥 axis and the direction of planar wave
propagation, and ℎ the height difference between two
adjacent points on the wave front (see Fig. 1). From
Fig. 1 we obtain wave speed 𝑉 ≈ BC cos 𝛽, where BC
is the displacement in the 𝑦-direction of a point on
the wave front in a time step and 𝑡𝑔𝛽 = ℎ; 𝛽 does not
equal 90∘ − 𝜙 generally since ℎ is an integer. In Fig. 2
we show the speed of a planar wave, 𝑉 , versus 𝜙 for
𝐾 = 8 and different values of 𝑂th . It is observed that
𝑉 depends on the excitation threshold 𝑂th and the direction of planar wave propagation (i.e., 𝜙). Moreover,
such dependence is non-monotonic since the reduction
of 𝛽 is not proportional to the reduction of 90∘ − 𝜙.
Wave speed 𝑉 reduces monotonously as 𝑂th increases
for a given value of 𝜙. When 𝜙 > 20∘ and 𝑂th = 11,
the planar wave vanishes due to the low excitability
of the medium. The desired planar wave can not be
generated due to the boundary effect for 0 ≤ 𝜙 < 5∘
and 𝑂th = 8. If 𝜙 = 0, one can obtain[19]
[︁
𝑂th ]︁
𝑉 = int 𝑅 + 1 −
,
(2)
2𝑅 + 1
where int[·] denotes the integer part of [·]. The numerical results suggest that the speed of planar wave
is independent of 𝐾.
Now we investigate the refraction and reflection of
a planar wave. The medium is divided into two uniform regions. The corresponding media in these two
regions are called media I and II, respectively. The
borderline between media I and II is a straight line
which is inclined at angle 𝜃 to the 𝑥 axis. We suppose that the incident planar wave propagates along
10
20
30
40
50
(deg)
Fig. 2. Speed of the planar wave as functions of 𝜙 and
𝑂th at 𝐾 = 8.
Because the wave speed is independent of 𝐾, the
refraction has nothing to do with 𝐾. In order to confirm this prediction, 𝐾𝑖 = 8 and 𝐾𝑟 = 4 (or 𝐾𝑖 = 4
and 𝐾𝑟 = 8) are applied in our simulation. Let 𝑂th
vary in the interval [4,11]. The symbol 𝜃𝑟 (𝜃𝑠 ) denotes
the angle between the direction of the refracted (reflected) wave propagation and the interface normal.
We fix firstly 𝜃𝑖 = 30∘ and investigate the effect of 𝐾
and 𝑂th on the refraction and reflection of the planar wave. We find that the system exhibits different patterns when the related parameters are properly
chosen. Normal refraction, double refraction, delayed
refraction, diffraction and wave propagation without
refraction are observed. Wave propagation without
refraction occurs when the wave speeds in the media I
and II are the same, otherwise refraction takes place.
The period of the refracted wave 𝑇𝑟 depends on the
difference between 𝐾𝑟 and 𝐾𝑖 . 𝑇𝑟 is equal to 𝑇𝑖 when
𝐾𝑖 ≥ 𝐾𝑟 , otherwise 𝑇𝑟 = 2𝑇𝑖 . The results can be easily understood. When two incident waves successively
approach the interface, the first wave can initiate a
wave, but the second one fails to initiate a wave in
medium II because the medium is in the refractory
state when the second wave arrives at the interface.
The appearances of diffraction and double refraction depend on the wave speed in medium II. Let 𝑉𝑏 denote the speed of the intersection point of the refracted
wave front and the upper (or lower) boundary of
medium II. Simple analysis gives 𝑉𝑏 = 𝑉𝑟 / cos(𝜃𝑟 − 𝜃𝑖 ),
where 𝑉𝑟 denotes the speed of the refracted wave.
𝑂th,𝑟
When 𝑉𝑏 ̸= int[𝑅 + 1 − 2𝑅+1
], the incident wave that
just passes the upper edge of the interface between media I and II is slightly deflected. Waves propagating
almost along the direction of the incident wave prop-
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CHIN. PHYS. LETT. Vol. 29, No. 5 (2012) 054203
agation are generated at the upper edge of medium
II. The phenomenon closely resembles the bending of
light into a region of an obstacle’s geometrical shadow,
and it is known as diffraction. If the incident wave
that just passes the interface is deflected, an extraordinary wave is generated at the lower edge of medium
II, while an ordinary wave (i.e., the refracted wave)
is generated in medium II. At this time, the system
exhibits the double refraction effect. Refraction that
is contiguous to the interface between two media is
called ordinary refraction. The other refraction is
called extra-ordinary refraction. The appearance of
extra-ordinary refraction just makes the wave speed
at the lower boundary of medium II equal to 2.5.
This leads to the occurrence of delayed refraction [see
Fig. 4(c)]. In this case, the diffracted wave does not
propagate along the 𝑥 axis to make 𝑉𝑏′ equal to 2.5.
𝑂th,𝑟
If 𝑉𝑏′ = int[𝑅 + 1 − 2𝑅+1
], the diffracted wave will
propagate along the 𝑥 axis [see Fig. 4(b)].
33
(b)
(a)
θi
θr
3
(c)
1
11
1
2
1
5
(d)
4
10
6
th,
9
8
(e)
(f)
7
6
5
4
4
5
6
7
8
9 10 11
th,
Fig. 3. Phase diagram revealing the different behaviors of
wave propagation. The incident angle is 30∘ . Circle: the
wave propagation without refraction. Cross: the desired
planar wave is not generated. Down-triangle: normal refraction that obeys Snell’s law. Left-triangle: coexistence
of the diffraction and the normal refraction. Square: coexistence of the diffraction and the delayed refraction that
does not obey Snell’s law. Right-triangle: double refraction that does not obey Snell’s law.
In Fig. 3, we present a phase diagram revealing different wave propagation behaviors. The results shown
in Fig. 3 do not depend on 𝐾. Some behaviors of wave
propagation are shown in Fig. 4. Direct measurements
confirm that the normal refractions shown in Fig. 4
obey Snell’s law, whereas the double refraction and
delayed refraction do not obey Snell’s law. For instance, when 𝑂th,𝑖 = 10 and 𝑂th,𝑟 = 4, we obtain
𝑉𝑖 = 2 from Eq. (2) because the incident wave propagates along the positive 𝑥 axis. Directly measured values of the speed and angle are 𝑉𝑟 ≈ 2.88 and 𝜃𝑟 ≈ 46∘ .
According to Snell’s law (i.e., sin 𝜃𝑟 / sin 𝜃𝑖 = 𝑉𝑟 /𝑉𝑖 ),
one obtains 𝜃𝑟 = 46.05∘ . These results show that the
refraction shown in Fig. 4(a) obeys Snell’s law.
In order to understand the formation mechanism
of the delayed refraction, the wave speed 𝑉𝑏′ at the upper and lower boundaries of the system is measured.
𝑂th
]
It is found that 𝑉𝑏′ is 2.5 instead of int[𝑅 + 1 − 2𝑅+1
when 𝑂th = 8. On the other hand, the wave speed in
𝑂th
the medium is int[𝑅+1− 2𝑅+1
]. Thus the desired planar wave is not generated for 𝑂th,𝑖 = 8 [see Fig. 4(f)].
When 𝑂th,𝑖 = 10 and 𝑂th,𝑟 = 8, the incident wave
can cross the interface between media I and II with
its direction unchanged due to the same wave speed
𝑂th,𝑟
𝑂th,𝑖
(i.e., int[𝑅 + 1 − 2𝑅+1
] = int[𝑅 + 1 − 2𝑅+1
] = 2).
Fig. 4. Propagation of a planar wave corresponding to the
symbols shown in Fig. 3. We set 𝐾𝑖 = 8, 𝐾𝑟 = 4, 𝜃𝑖 = 30∘ .
(a) Closed down-triangle: 𝑂th,𝑖 = 10, 𝑂th,𝑟 = 4. (b)
Closed left-triangle: 𝑂th,𝑖 = 5, 𝑂th,𝑟 = 10. (c) Closed
square: 𝑂th,𝑖 = 10, 𝑂th,𝑟 = 8. (d) Open right-triangle:
𝑂th,𝑖 = 5, 𝑂th,𝑟 = 8. (e) Open circle: 𝑂th,𝑖 = 4,
𝑂th,𝑟 = 6. (f) Cross: 𝑂th,𝑖 = 8, 𝑂th,𝑟 = 4. Numbers
1 and 3 represent the incidence and diffraction regions,
numbers 2 and 4 represent the normal and delayed refraction regions, and numbers 5 and 6 indicate the regions of
the ordinary and extra-ordinary refraction. The directions
of the wave propagation are indicated by arrows.
We now study the effect of the incident angle on
the propagation of a planar wave. Without loss of
generality, we fix 𝐾𝑖 = 8, 𝐾𝑟 = 4, 𝑂th,𝑖 = 10 and
𝑂th,𝑟 = 4. When the incident angle is properly chosen, the system exhibits different types of refraction
and reflection (see Fig. 5). Normal refraction is observed when 𝜃𝑖 ≤ 33∘ [see Fig. 5(a)]. Double refraction takes place when 34∘ ≤ 𝜃𝑖 ≤ 45∘ [see Fig. 5(b)].
The ordinary refraction obeys Snell’s law, and the area
occupied by the ordinary refraction in medium II increases with the incident angle. The extra-ordinary
refraction does not obey Snell’s law. Let 𝑉er be the
wave speed of the extra-ordinary refraction and 𝛿 be
the angle between the direction of the extra-ordinary
refracted wave propagation and the 𝑥 axis. We find
that 𝑉er and 𝛿 do not depend on 𝜃𝑖 since the speed of
the intersection point of the extra-ordinary refracted
wave front and the lower boundary of medium II must
𝑂th,𝑟
] = 3. The measured values
be equal to int[𝑅+1− 2𝑅+1
of speed and angle are 𝑉er ≈ 2.83 and 𝛿 ≈ 19∘ , respectively. The predicted angle 𝛿 is cos−1 (𝑉er /3) = 19.3∘ .
Thus the angle of extraordinary refraction 𝜃er is equal
to 𝜃𝑖 + 𝛿.
For 𝜃𝑖 ≥ 46∘ , the coexistence of reflection and dou-
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CHIN. PHYS. LETT. Vol. 29, No. 5 (2012) 054203
ble refraction is observed [see Fig. 5(c)], and moreover,
the double refraction does not obey Snell’s law. Reflection is a back refraction. This indicates that reflection occurs just when the incidence angle is larger
than the critical angle. With the increase of 𝜃𝑖 , the
area of the reflective region gradually increases, and
the area of the ordinary refraction region gradually decreases. When 𝜃𝑖 exceeds 59∘ , the ordinary refraction
disappears. The coexistence of reflection and extraordinary refraction arises [see Fig. 5(d)]. The angle of
reflection 𝜃𝑠 increases with the incident angle 𝜃𝑖 when
46∘ ≤ 𝜃𝑖 ≤ 60∘ . For 𝜃𝑖 ≥ 61∘ , the angle of reflection
𝜃𝑠 begins to decrease. When 𝜃𝑖 goes beyond 74∘ , the
reflected wave disappears because it can not be generated by the refracted wave due to low excitability [see
Fig. 5(f)].
1
2
δ
(c)
3
(d)
θs
3
θi
1
2
2
(e)
Fig. 6. Refracted wave disappears at 𝐾𝑖 = 8, 𝐾𝑟 = 4,
𝑂th,𝑖 = 4, 𝑂th,𝑟 = 10 and 𝜃𝑖 = 60∘ .
In summary, we have investigated refraction and
reflection in anisotropic, discrete, excitable media, and
observe their diffraction, normal refraction and double
refraction. These behaviors are similar to those found
in light waves. In addition, we find that with properly chosen parameters the incident, refracted and reflected waves can have different periods, and delayed
refraction can occur. This investigation may broaden
our understanding of excitation waves and transmission in heterogeneous media, with potential implications for material and biological systems as well.
Hopefully, our results can be confirmed experimentally.
(b)
(a)
curs repeatedly, and the pattern shown in Fig. 5(e)
is generated. Refraction can disappear due to the
low excitability for the right choice of parameters (see
Fig. 6).
(f)
References
3
3
2
2
Fig. 5. Refraction and reflection phenomena for different
values of 𝜃𝑖 . The simulation parameters are the same as
Fig. 4(a), except for the angle of incidence: (a) 𝜃𝑖 = 33∘ ,
(b) 𝜃𝑖 = 45∘ , (c) 𝜃𝑖 = 50∘ , (d) 𝜃𝑖 = 60∘ , (e) 𝜃𝑖 = 68∘ , (f)
𝜃𝑖 = 75∘ . The numbers 1 and 2 represent the regions of
the ordinary and extra-ordinary refraction, and the number 3 indicates the reflection region. The directions of the
wave propagation are indicated by the arrows.
The refraction and reflection shown in Figs. 5(c)
and 5(d) have the same period. However, there is
other type of refraction and reflection. When 65∘ ≤
𝜃𝑖 ≤ 73∘ , the periods of incident, refracted and reflected waves exhibit different values [see Fig. 5(e)].
The period of the reflected wave 𝑇 𝑟 is 16 (i.e., 𝑇 𝑟 =
2𝐾𝑖 = 2𝑇𝑖 ), while the period of the refracted wave
changes alternatively between 4 and 12. The formation process of the pattern is as follows: when two
incident waves arrive successively at the interface, the
two refracted waves and two reflected waves are successively generated. The medium between the second
refracted and first reflected waves is excited after the
two refracted waves are generated. Thus the second
refracted wave is attracted to one side of the first refracted wave, leading to the breakup and disappearance of the second reflected wave. The process oc-
[1] Davidenko J M, Pertsov A M, Salomontsz R, Baxter W and
Jalife J 1992 Nature 355 349
[2] Lechleiter J, Girard S, Peralta E and Clapham D 1991 Science 252 123
[3] Zhabotinsky A M, Eager M D and Epstein I R 1993 Phys.
Rev. Lett. 71 1526
[4] Kosek J and Marek M 1995 Phys. Rev. Lett. 74 2134
[5] Hwang S C and Timothy H H 1996 Phys. Rev. E 54 3009
[6] Brazhnik P K and Tyson J J 1996 Phys. Rev. E 54 1958
[7] Pechenik L and Levine H 1998 Phys. Rev. E 58 2910
[8] Sainhas J and Dilão R 1998 Phys. Rev. Lett. 80 5216
[9] Skaar J 2006 Phys. Rev. E 73 026605
[10] Manz N, Ginn B T and Steinbock O 2006 Phys. Rev. E 73
066218
[11] Zhang R, Yang L, Zhabotinsky A M and Epstein I R 2007
Phys. Rev. E 76 016201
[12] Lü Y P, Gu G F, Lu H C, Dai Y and Tang G N 2009 Acta
Phys. Sin. 58 2996 (in Chinese)
[13] Lü Y P, Gu G F, Lu H C, Dai Y and Tang G N 2009 Acta
Phys. Sin. 58 7573 (in Chinese)
[14] Gu G F, Lü Y P and Tang G N 2010 Chin. Phys. B 19
050515
[15] Mornev O A 1984 Self-organization, Autowaves and Structures far from Equilibrium ed Krinsky V I (Berlin: Springer)
pp 111–118
[16] Oosawa C, Fukuta Y, Natsume K and Kometani K 1996 J.
Phys. Chem. 100 1043
[17] Kullai K K, Roszol L and Volford A 2005 Chem. Phys. Lett.
414 326
[18] Greenberg J M and Hastings S P 1978 SIAM J. Appl. Math.
34 515
[19] Zhang L S, Deng M Y, Kong L J, Liu M R and Tang G N
2009 Acta Phys. Sin. 58 4493 (in Chinese)
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