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Application of the activator inhibitor principle to physical systems
Volume 136. number 9
PHYSICS LETTERS A
24 April I 9119
APPLICATION OF THE ACTIVATOR INHIBITOR PRINCIPLE TO PHYSICAL SYSTEMS H.-G. PURWINS, C. RADEHAUS, T. DIRKSMEYER. R. DOHMEN, R. SCHMELING arid H. WILLEBRAND Institute for Applied Phj’sics, University of Munster, (‘orreussirasse 2/4, D-440() MOnster, ERG
Received 30 January 1989: accepted for publication 27 February 1989 Communicated by A.P. Fordv
The transition from a spatially homogeneous state into a spatially periodic state and the development of solitary filaments are observed experimentally in an electrical network and in a dc-gas discharge system, respectively. The explanation for these phenomena is done in terms ofthe biomathematical activator inhibitor principle with the help ofa recently proposed model.
The activator inhibitor principle has been used at vartous places in biology and biomathematics to describe pattern formation of living structures [1]. Recentlv this principle has been applied to pattern formation in spatially extended phystcal systems [2— 101. Later work on bifurcation theory and pattern formation in ferromagnetic resonance [11] and gas discharge systems [121 from another point of view are related to the present subject. The general activator inhibitor equation used so far to describe pattern formation in physical systems. and suchsemiconductor as electrical networks, systems devices, isgas an discharge extended version of a two-component reaction diffusion equation on a spatial region ~ which can be written in normalized components and variables as v =aAu+f(v, w)-4-,uVvVsv,
f(r’, u’)r=h(v)—w. =
g(t’. tt’=
a,
0. w) =0,
t (vu, sv0) >0. g
(Va. si’)>
g(v0,
=0
14~)
1, (v0, ~
<0
0, g, (s’~,si’0) <0
,
(2)
we may interpret v as an activator favouring the cxcitation of v and ii’, and si as inhibitor damping both components. Such activator inhibitor systems arc discussed physically in more detail in refs. [7] and [8]. In our case h(o) in eq. (1) is monotonically decreasing for large ci and increasing i’~. 3, A>0. for eq. small (1) is the For j~,K~=0 andoriginally h(v)=Ac—i’ system treated in biomathematics. The special case a, ó>> 1 is the Fitz-Hugh—Nagumo model for nerve axon pulse transmission. Eq. (1) may have stationary homogeneous, inhomogeneous, periodic and nonperiodic solutions. Also nonstationary homogeneous osctllations and waves are observed. Eq. (1) can be visualized in an electrical equivalent circuiL this circuit has actually been built. Experimental results will be presented in the following. Fig. Ia presents a single element of a one-dimen-
Asi’+g(v, sv) +K~W.
w)=c—.w—K
f(L
.
,
1 ~ —h-J w dQ,
I
sional chain [9]. The circuit contains as nonltnear element a resistance with S-shaped characteristic. This characteristic can be shifted by the current I~.
~
The circuit voltage U~is and driven internal by a voltage resistance source R(. with For no-load current
Considering a certain neighbourhood (c stationary solution of eq. (1) with requiring 480
ji,
0, 5t’~) of a 1(7=0 and
structures I, and voltage structures L~with characteristic wave lengths much larger than the distance of adjacent single elements the discrete network of
Volume 136. number 9
PHYSICS LETTERS A
°...°
~°
U~0~UR
L
p...
B
24 April 1989
~
~
R
0..
(a)
(b)
Fig. 1. A single element of the one-dimensional version of an electrical network obeying eq. (1) with 1~=0 (a) and an electrical Continuum device (b) being described by eq. (1) with p0 in general.
fig. I a is well described by the continuum equation (1) with fl=O, h(v)=2v—v3, the components v(x, 1) ~ w(x, t) —Ui and the parameters ,c~ U 5, R0. We mention that the current coupling via the resistance R1 gives rise to the term A v and the coupling via R5 to A w. For further details see refs. [2,6,9]. Interpreting fig. la in its two-dimensional version as an equivalent circuit for a continuum system we obtain fig. lb with a linear material L on top of a nonlinear material N with S-shaped electrical characteristic. L and N are connected via a metallic layer M to the voltage source. For the continuum model we obtain in eq. (1) v”-j(x, .v, t) and w—~ U(x, v, /) being the current density vertical to the interface and the potential in this interface, respectively. To take account of charge carrier drift we add a phenomenological term jiVvVw. L and N may be e.g. a semiconductor and a gas [4,6,7,10] or both semiconductors as is the case in pin diodes [3,5]. Typical experimental results of a one-dimensional electrical network are shown in fig. 2a. Decreasing the parameter a by decreasing R~we observe a bifurcation of the homogeneous state into a stationary inhomogeneous pattern. The agreement with numerical calculation on the basis of eq. (1) shown in fig. 2b is very good. From bifurcation analysis we cxpect a sinusoidal stationary spatial structure below the bifurcation value with amplitude I obeying 2~~-R I 1, From figs. 2 and 3 we see that experiment, theory and calculation are again in good agreement. In fig. 4 we report experimental results when in—
creasing the driving voltage U~.The interesting feature is the appearance of current filaments of almost identical size in the range where no filament interaction takes place. These structures are also described by eq. (1) with p~= 0. For details see ref. [9]. Changing parameters we observe all kinds of structures mentioned above [2,6—9]. Typical results for measurements in a gas discharge system described in ref. [4] are shown in fig. 5a for various values of U~.Again we observe current filaments of well defined size. They are in good qualitative agreement with calculations on the basis of eq. (1) when choosing ~ The latter is required to explain the splitting of filaments. Physically the term jNvVw is due to the possibility of charge carncr drift in the gas which cannot take place in the networks described above. We note that at present exact quantitative calculations cannot be done because of lack of accurate parameters. The gas discharge system also exhibits inhomogeneous oscillatiorts. All these structures can be dealt with by using eqs. (1) [4,6,7,10]. Also chaotic behaviour has been observed experimentally. Meanwhile filaments have been observed also in two-dimensional space [12]. We note that all calculations done by using eq. (1) account also for the stability of the experimentally observed structure. The investigations ofpattern formation in ferromagnetic resonance give results similar to those observed in gas discharge systems [11]. From the present work we see that a reaction diffusion activator inhibitor model originally developed in biology and biomathematics provides also a 481
:
Volume 136. number 9
PHYSICS LETTERS A
a
o
b
‘
0
20
,
40
60
80
100
020
20
40
60
0
~
0
20
40
60
50
100
iao
4~f~W~J
~~IVV~ 0” 0
24 April 1989
60
.~ 100
120
0
20
40
000
oao
0
20
40
60
60
100
020
100
020
~
0L. O
20
40
60 80 cell number
60 80 cell number
Fig. 2. Measured (a) and calculated (b) current I, as a function of cell number of the one-dimensional electrical network indicated in fig. I a for voltage coupling R1~decreasing from bottom to top. The normalized parameters used in thecalculation are: ~= 0.043 0.064. ó= 1.85. j,=0.423 and ic1 =K~=p=0. For details see ref. [91.
R1
=
4.02 k~)
20000
\~ \x
\x
OJ
10000
x
-
tOO
200
Fig. 3. Square of the amplitude I of thesinusoidal modulation of the current of fig. 2 of R1 — o. The parameters are as in fig. 2. For details see ref. (91.
482
300
(~experiment,
X calculation) plotted asa function
Volume 136. number 9
PHYSICS LETTERS A
_____ 0—
o
20
J1I\ 0
C
5
0
20
40
60
I 80
100
120
____________________________ 40
60
24 April 1989
80
100
(2) (1) 120
.J~\TL1\iLT\._J~\_.1\J~T’L~ft 0
20
40
60
80
100
120
a
~‘‘1j’1[1J1J1J1jT1f~\ 14)
C O
20
40
60
80
100
120
5 (5) C
0
40
60
80
100
120
~
5 C
20
0
16) 20
40
60
80
100
120
~~ C
0
20
40
60
60
cell number
100
120
Fig. 4. Current I, as a function of cell number for the one-dimensional electrical network for increasing (1—5) and decreasing (5 7) voltage U5. The normalized parameters of the network are in this case: a=0.64, ~= 1.59, K1 = —1.93 0.97 and K2_7X ~ For details see ref. (9].
good basis for a mathematical description of the evolution and stability of patterns in physical systems. The model has various general features allowing for wide spread application: In many circumstances physical systems can be reduced to two-component systems, in which one component can be interpreted as an activator and the other as an inhibitor, The spatial coupling is mainly due to diffusion. Besides convection diffusion is the most important transport process. Also many other types of spatial coupling can be described by diffusion terms. The coupling is linear and should be a good first —
—
b
(~)
Fig. 5. Measured (a) and calculated (b) light intensity as a function of lateral extension x of a quasi-one-dimensional gas discharge system for increasing voltage U5. The parameters used in the calculation are a=0. I, p= 1.7 x I 0~, 15=r 0.1, 0.68 5.76, see ref.ic2=0.92, (10]. ~=7.89 and a special form of 1(t). For details
approximation for the spatial coupling in a large number of cases. The ansatz for relaxation is the same as that used successfully to describe linear systems. We believe that it is a good approximation for a large class of nonlinear systems, too. We conclude that the activator inhibitor reaction diffusion model is of fundamental importance for pattern formation in a large class of spatially extended physical systems. —
The authors are grateful to A. Böckmann and G. Cornelsen for help in preparing the manuscript.
—
483
Volume 136. number 9
PHYSICS LETTERS A
References [I] P. Fife. Lecture notes in biomathematics, Vol. 28. Mathematical aspects of reaction and diffusion systems (Springer. Berlin. 1979); J. Murray. Lectures on nonlinear differential equation models in biology (Clarendon, Oxford. 1977): 1-1. Meinhardt, Models of biological pattern formation (Academic Press, London. 1982); I-I. Fujii. M. Mimura and Y. Nishiura, Physica D 5 (1982) 1. (2] J. Berkemeier, T. Dirksmeyer, G. Klempt and H.-G. Purwins. Z. Phys. B 65(1986) 255. [31 Ch. Radehaus. K. Kardell, H. Baumann, D. Jager and H.0. Purwins, Z. Phys. B 65 (1987) 515. [4] Ch. Radehaus, T. Dirksmeyer, H. Willebrand and 1-1-0. Purwins. Phys. Len. A 125 (1987) 92.
484
24 April 1989
[5] H. Baumann. R. Symanczyk, Ch. Radehaus, H.-G. Purwins and D.Jhger,Phys.Lett.A 123 (1987) 421. [6] H.-G. Purwins, G. Klempt and J. Berkemejer. Festkbrperprobleme 27 (1987) 27. (71 H.-G. Purwins, Ch. Radehaus and J. Berkemeier. Z. Naturforsch. 43a (1988) 17. 18] H.-G. Purwins and Ch. Radehaus, in: Springer series in synergetics, Vol. 42. Neural and synergetic computer. ed. H. Haken (Springer, Berlin). [91 R. Schmeling, Diplomarbeit Universität MOnster (1988): T. Dirksmeyer, R. Schmeling. J. Berkemeier and 1-1-0. Purwins. to be published. [101 Ch. Radehaus, R. Dohmen, H. Willebrand, J. Niedernostheide and H.-G. Purwins. to be published. [11] F.J. Elmer, Z. Phys. B 68 (1987)105: Physiea D 30 (1988) 321. 112] KG. Muller. Phys. Rev. A 37 (1988) 4836.
PHYSICS LETTERS A
24 April I 9119
APPLICATION OF THE ACTIVATOR INHIBITOR PRINCIPLE TO PHYSICAL SYSTEMS H.-G. PURWINS, C. RADEHAUS, T. DIRKSMEYER. R. DOHMEN, R. SCHMELING arid H. WILLEBRAND Institute for Applied Phj’sics, University of Munster, (‘orreussirasse 2/4, D-440() MOnster, ERG
Received 30 January 1989: accepted for publication 27 February 1989 Communicated by A.P. Fordv
The transition from a spatially homogeneous state into a spatially periodic state and the development of solitary filaments are observed experimentally in an electrical network and in a dc-gas discharge system, respectively. The explanation for these phenomena is done in terms ofthe biomathematical activator inhibitor principle with the help ofa recently proposed model.
The activator inhibitor principle has been used at vartous places in biology and biomathematics to describe pattern formation of living structures [1]. Recentlv this principle has been applied to pattern formation in spatially extended phystcal systems [2— 101. Later work on bifurcation theory and pattern formation in ferromagnetic resonance [11] and gas discharge systems [121 from another point of view are related to the present subject. The general activator inhibitor equation used so far to describe pattern formation in physical systems. and suchsemiconductor as electrical networks, systems devices, isgas an discharge extended version of a two-component reaction diffusion equation on a spatial region ~ which can be written in normalized components and variables as v =aAu+f(v, w)-4-,uVvVsv,
f(r’, u’)r=h(v)—w. =
g(t’. tt’=
a,
0. w) =0,
t (vu, sv0) >0. g
(Va. si’)>
g(v0,
=0
14~)
1, (v0, ~
<0
0, g, (s’~,si’0) <0
,
(2)
we may interpret v as an activator favouring the cxcitation of v and ii’, and si as inhibitor damping both components. Such activator inhibitor systems arc discussed physically in more detail in refs. [7] and [8]. In our case h(o) in eq. (1) is monotonically decreasing for large ci and increasing i’~. 3, A>0. for eq. small (1) is the For j~,K~=0 andoriginally h(v)=Ac—i’ system treated in biomathematics. The special case a, ó>> 1 is the Fitz-Hugh—Nagumo model for nerve axon pulse transmission. Eq. (1) may have stationary homogeneous, inhomogeneous, periodic and nonperiodic solutions. Also nonstationary homogeneous osctllations and waves are observed. Eq. (1) can be visualized in an electrical equivalent circuiL this circuit has actually been built. Experimental results will be presented in the following. Fig. Ia presents a single element of a one-dimen-
Asi’+g(v, sv) +K~W.
w)=c—.w—K
f(L
.
,
1 ~ —h-J w dQ,
I
sional chain [9]. The circuit contains as nonltnear element a resistance with S-shaped characteristic. This characteristic can be shifted by the current I~.
~
The circuit voltage U~is and driven internal by a voltage resistance source R(. with For no-load current
Considering a certain neighbourhood (c stationary solution of eq. (1) with requiring 480
ji,
0, 5t’~) of a 1(7=0 and
structures I, and voltage structures L~with characteristic wave lengths much larger than the distance of adjacent single elements the discrete network of
Volume 136. number 9
PHYSICS LETTERS A
°...°
~°
U~0~UR
L
p...
B
24 April 1989
~
~
R
0..
(a)
(b)
Fig. 1. A single element of the one-dimensional version of an electrical network obeying eq. (1) with 1~=0 (a) and an electrical Continuum device (b) being described by eq. (1) with p0 in general.
fig. I a is well described by the continuum equation (1) with fl=O, h(v)=2v—v3, the components v(x, 1) ~ w(x, t) —Ui and the parameters ,c~ U 5, R0. We mention that the current coupling via the resistance R1 gives rise to the term A v and the coupling via R5 to A w. For further details see refs. [2,6,9]. Interpreting fig. la in its two-dimensional version as an equivalent circuit for a continuum system we obtain fig. lb with a linear material L on top of a nonlinear material N with S-shaped electrical characteristic. L and N are connected via a metallic layer M to the voltage source. For the continuum model we obtain in eq. (1) v”-j(x, .v, t) and w—~ U(x, v, /) being the current density vertical to the interface and the potential in this interface, respectively. To take account of charge carrier drift we add a phenomenological term jiVvVw. L and N may be e.g. a semiconductor and a gas [4,6,7,10] or both semiconductors as is the case in pin diodes [3,5]. Typical experimental results of a one-dimensional electrical network are shown in fig. 2a. Decreasing the parameter a by decreasing R~we observe a bifurcation of the homogeneous state into a stationary inhomogeneous pattern. The agreement with numerical calculation on the basis of eq. (1) shown in fig. 2b is very good. From bifurcation analysis we cxpect a sinusoidal stationary spatial structure below the bifurcation value with amplitude I obeying 2~~-R I 1, From figs. 2 and 3 we see that experiment, theory and calculation are again in good agreement. In fig. 4 we report experimental results when in—
creasing the driving voltage U~.The interesting feature is the appearance of current filaments of almost identical size in the range where no filament interaction takes place. These structures are also described by eq. (1) with p~= 0. For details see ref. [9]. Changing parameters we observe all kinds of structures mentioned above [2,6—9]. Typical results for measurements in a gas discharge system described in ref. [4] are shown in fig. 5a for various values of U~.Again we observe current filaments of well defined size. They are in good qualitative agreement with calculations on the basis of eq. (1) when choosing ~ The latter is required to explain the splitting of filaments. Physically the term jNvVw is due to the possibility of charge carncr drift in the gas which cannot take place in the networks described above. We note that at present exact quantitative calculations cannot be done because of lack of accurate parameters. The gas discharge system also exhibits inhomogeneous oscillatiorts. All these structures can be dealt with by using eqs. (1) [4,6,7,10]. Also chaotic behaviour has been observed experimentally. Meanwhile filaments have been observed also in two-dimensional space [12]. We note that all calculations done by using eq. (1) account also for the stability of the experimentally observed structure. The investigations ofpattern formation in ferromagnetic resonance give results similar to those observed in gas discharge systems [11]. From the present work we see that a reaction diffusion activator inhibitor model originally developed in biology and biomathematics provides also a 481
:
Volume 136. number 9
PHYSICS LETTERS A
a
o
b
‘
0
20
,
40
60
80
100
020
20
40
60
0
~
0
20
40
60
50
100
iao
4~f~W~J
~~IVV~ 0” 0
24 April 1989
60
.~ 100
120
0
20
40
000
oao
0
20
40
60
60
100
020
100
020
~
0L. O
20
40
60 80 cell number
60 80 cell number
Fig. 2. Measured (a) and calculated (b) current I, as a function of cell number of the one-dimensional electrical network indicated in fig. I a for voltage coupling R1~decreasing from bottom to top. The normalized parameters used in thecalculation are: ~= 0.043 0.064. ó= 1.85. j,=0.423 and ic1 =K~=p=0. For details see ref. [91.
R1
=
4.02 k~)
20000
\~ \x
\x
OJ
10000
x
-
tOO
200
Fig. 3. Square of the amplitude I of thesinusoidal modulation of the current of fig. 2 of R1 — o. The parameters are as in fig. 2. For details see ref. (91.
482
300
(~experiment,
X calculation) plotted asa function
Volume 136. number 9
PHYSICS LETTERS A
_____ 0—
o
20
J1I\ 0
C
5
0
20
40
60
I 80
100
120
____________________________ 40
60
24 April 1989
80
100
(2) (1) 120
.J~\TL1\iLT\._J~\_.1\J~T’L~ft 0
20
40
60
80
100
120
a
~‘‘1j’1[1J1J1J1jT1f~\ 14)
C O
20
40
60
80
100
120
5 (5) C
0
40
60
80
100
120
~
5 C
20
0
16) 20
40
60
80
100
120
~~ C
0
20
40
60
60
cell number
100
120
Fig. 4. Current I, as a function of cell number for the one-dimensional electrical network for increasing (1—5) and decreasing (5 7) voltage U5. The normalized parameters of the network are in this case: a=0.64, ~= 1.59, K1 = —1.93 0.97 and K2_7X ~ For details see ref. (9].
good basis for a mathematical description of the evolution and stability of patterns in physical systems. The model has various general features allowing for wide spread application: In many circumstances physical systems can be reduced to two-component systems, in which one component can be interpreted as an activator and the other as an inhibitor, The spatial coupling is mainly due to diffusion. Besides convection diffusion is the most important transport process. Also many other types of spatial coupling can be described by diffusion terms. The coupling is linear and should be a good first —
—
b
(~)
Fig. 5. Measured (a) and calculated (b) light intensity as a function of lateral extension x of a quasi-one-dimensional gas discharge system for increasing voltage U5. The parameters used in the calculation are a=0. I, p= 1.7 x I 0~, 15=r 0.1, 0.68 5.76, see ref.ic2=0.92, (10]. ~=7.89 and a special form of 1(t). For details
approximation for the spatial coupling in a large number of cases. The ansatz for relaxation is the same as that used successfully to describe linear systems. We believe that it is a good approximation for a large class of nonlinear systems, too. We conclude that the activator inhibitor reaction diffusion model is of fundamental importance for pattern formation in a large class of spatially extended physical systems. —
The authors are grateful to A. Böckmann and G. Cornelsen for help in preparing the manuscript.
—
483
Volume 136. number 9
PHYSICS LETTERS A
References [I] P. Fife. Lecture notes in biomathematics, Vol. 28. Mathematical aspects of reaction and diffusion systems (Springer. Berlin. 1979); J. Murray. Lectures on nonlinear differential equation models in biology (Clarendon, Oxford. 1977): 1-1. Meinhardt, Models of biological pattern formation (Academic Press, London. 1982); I-I. Fujii. M. Mimura and Y. Nishiura, Physica D 5 (1982) 1. (2] J. Berkemeier, T. Dirksmeyer, G. Klempt and H.-G. Purwins. Z. Phys. B 65(1986) 255. [31 Ch. Radehaus. K. Kardell, H. Baumann, D. Jager and H.0. Purwins, Z. Phys. B 65 (1987) 515. [4] Ch. Radehaus, T. Dirksmeyer, H. Willebrand and 1-1-0. Purwins. Phys. Len. A 125 (1987) 92.
484
24 April 1989
[5] H. Baumann. R. Symanczyk, Ch. Radehaus, H.-G. Purwins and D.Jhger,Phys.Lett.A 123 (1987) 421. [6] H.-G. Purwins, G. Klempt and J. Berkemejer. Festkbrperprobleme 27 (1987) 27. (71 H.-G. Purwins, Ch. Radehaus and J. Berkemeier. Z. Naturforsch. 43a (1988) 17. 18] H.-G. Purwins and Ch. Radehaus, in: Springer series in synergetics, Vol. 42. Neural and synergetic computer. ed. H. Haken (Springer, Berlin). [91 R. Schmeling, Diplomarbeit Universität MOnster (1988): T. Dirksmeyer, R. Schmeling. J. Berkemeier and 1-1-0. Purwins. to be published. [101 Ch. Radehaus, R. Dohmen, H. Willebrand, J. Niedernostheide and H.-G. Purwins. to be published. [11] F.J. Elmer, Z. Phys. B 68 (1987)105: Physiea D 30 (1988) 321. 112] KG. Muller. Phys. Rev. A 37 (1988) 4836.