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INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
A CNN-BASED EXPERIMENTAL FRAME FOR PATTERNS
AND AUTOWAVES
PAOLO ARENA*, MARCO BRANCIFORTE AND LUIGI FORTUNA
Dipartimento Elettrico, Elettronico e Sistemistico, Universita+ di Catania, Viale A. Doria, 6, 95125 Catania, Italy
SUMMARY
An experimental board is discussed in this paper, in which pattern formation and active wave propagation phenomena
can be observed. In particular, the CNN paradigm has been considered for the rules which drive connections among
circuits in the whole frame. It is therefore organized as a CNN array of 25 cells arranged into a matrix of dimesion 5;5,
where each cell is a second-order non-linear system characterized by a piecewise linear output non-linearity. Calculations
and experiments are compared in order to see that such spatio-temporal phenomena can take place in arrays of
non-linear second-order circuits locally connected, strictly resembling what obtained in simulations, in spite of noise
arising from a real analog frame. 1998 John Wiley & Sons, Ltd.
KEY WORDS: cellular neural networks; patterns; autowaves; two layer CNN array
1. INTRODUCTION
Recently, the scientific community has been greatly involved in the study of complex phenomena, such as
travelling wave fronts and autowaves.\ Spatio-temporal pattern formation is of great interest in developmental biology, and in particular in Morphogenesys; the possibility to deeply study such dynamic in order
to draw its main rules, opens the way to develop geometrically perfect structures characterized by intrinsic
robustness against disturbances and noise, as it commonly happens in nature. Indeed a heavy mathematical
framework, as well as a large number of simulation results have been provided in the literature.\.
The possibility to qualitatively reproduce such phenomena by using arrays of non-linear circuits has also
allowed the simulation and artificial experimentation. Travelling wave trains have been observed in
reaction—diffusion systems with oscillatory kinetics, while, in other, completely diverse situations and under
suitable conditions, chemicals can react and diffuse in such a way so as to produce steady-state heterogeneous spatial patterns of chemical concentration. Therefore, both the phenomena can be conceptually seen as
particular reaction—diffusion mechanisms.
In this paper an experimental, circuit-based setup to physically observe and ‘measure’ such phenomena is
presented. Such a frame has been thought and built to work as a development system for the study of
complexity. In fact, as it will be seen in the following sections, such a system allows to realize a great number
of different connections in a matrix 5;5, in which the elements can be non-linear systems of different order,
each one showing a different dynamics and mutually and locally interacting by simply connecting any of their
variables to any other one. In this paper some original experiments are reported by using recent results
which state that the same second-order circuit can generate, if used as a cell in a CNN array, autowaves or
turing patterns only by a suitable modulation of its parameters.
All the phenomena considered are reproduced by employing a simple two-layer CNN array with constant
templates..
*Correspondence to: Paolo Arena, Dipartimento Elettrico, Elettronico e Sistemistico, Universita’ di Catania, Viale A. Doria, 6, 95125
Catania, Italy. Email: [email protected]
CCC 0098—9886/98/060635—16$17.50
1998 John Wiley & Sons, Ltd.
Received 15 May 1997
Revised 19 November 1997
636
P. ARENA, M. BRANCIFORTE AND L. FORTUNA
The experimental board presented can be seen as a practical device to study different examples of
reaction—diffusion spatio-temporal phenomena, that have been traditionally observed in structurally different circuit arrays. The paper is organized in two main parts: in the first part the model of the single cell, as
well as the behaviour of the whole CNN array are briefly discussed in relation to the complex phenomena
they are able to generate. In the second part the experimental board in which spatio-temporal dynamics can
be observed is presented through a set of experiments.
2. THE TWO-LAYER CNN MODEL FOR WAVE PROPAGATION AND PATTERN FORMATION
In this Section a two-layer reaction—diffusion CNN with constant templates is introduced.
Let us consider the system
xR
"!x
#(1#k#e) y
!s y
#i
GH
GH
GH
GH
xR
"!x
#s y
#(1#k!e)y
#i
GH
GH
GH
GH
(1)
y "0)5 (" x #1 "!" x !1 ")
G
G
G
(2)
with
and
i"0, 1,2, M!1, j"0, 1,2, N!1
Some propositions were proved in References 12, 15 to show that the above system, while used as a cell in
a M;N CNN array, for suitable choices of its parameters, is able to show pattern formation or autonomous
wave propagation.
In particular, for the parameter set: k"0)7, e"0, s "s "1, i "!0)3 and i "0)3, autowaves can be
generated, while for: k"!0)6, e"1)82, s "2, s "2)5, i "i "0, the cell satisfies all the conditions for
pattern formation. The parameters s and s can take on the same value, but the sufficient conditions to
obtain ¹uring patterns, applied to this cell allow the s to vary independently and in a quite broad set.
G
If a discrete Laplacian template is introduced to modulate local interactions among cells, system (1)
becomes
#y ;
#y ;
!4y ; )#i ;
xR ; "!x ; #(1#k)y ; !sy ; #D (y ; #y ;
GH
GH
GH
H
G\H
GH\
GH>
GH
GH
xR ; "!x ; #sy ; #(1#k)y ; #D (y ;
#y ;
#y ;
#y ;
GH
GH
GH
GH
G>H
G\H
GH\
GH>
!4y
) #i ;
; GH
(3)
1)i)M; 1)j)N
These represent the state equations of a two-layer autonomous reaction—diffusion standard CNN. In
particular, for autowave formation the cloning templates are:
A"
A
A
,
A
A
i
I" ,
i
where
D
0
0 0 0
G
A " D !4D #k#1 D , A "!A " 0 s 0
GG
G
G
G
0
D
0
0 0 0
G
1998 John Wiley & Sons, Ltd.
0
(4)
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
637
A CNN-BASED FRAME FOR PATTERNS AND AUTOWAVES
with D "D "0)1, while, for pattern formation, the following templates have been used:
0
D
0
0
D
0
A " D !4D #k!e#1 D ,
I"0, A " D !4D #k#e#1 D ,
0
D
0
0
D
0
0
0
0
A " 0 !s 0 ,
0
0 0
(5)
0
0
0
A " 0 s 0 ,
0 0 0
with D "1, D "0)01. The simulated CNN structure, with templates (4), and with the so-called Zero-Flux
(Neumann) boundary conditions gave rise to several complex phenomena of wave propagation, in
particular, auto-wave, spirals, multi-armed vortexes, self-organizing patterns arising from random initial
conditions. Such phenomena have been fully explained in References 12, 18. In particular, in Reference 12
simulation results showed the robustness of this CNN model to parametric uncertainties and noise. It was
shown through simulations that, in spite of significant amplitude of uncertainties in the capacitance values,
(20% of the nominal value), the wave fronts succeed in propagating even if slightly out of shape. Moreover,
uncertainties into the template values around 1% were able to generate spurious wave fronts, which however
disappeared when colliding with the actual wave front. Therefore, suitable component tolerances (10% for
capacitors and 1% for resistors) have been used for the circuit realization.
As regards pattern formation, the necessary conditions, which define the so-called ¹uring space of the
CNN parameters, are the following:
k(0; k#s s 'e
(6)
D #D
,
e'!k k(D !D )#e(D #D ) '4s s D D
(7)
D !D
Briefly, conditions (6) satisfy the stability constraints for each isolated cell, in the absence of diffusion, while
equation (7) guarantees the instability of the whole CNN once the diffusion process onsets. In fact, Turing
patterns can develop only if at least one temporal eigenvalue has positive real part. Temporal modes are
a function of some spatial eigenvalues k , corresponding to the eigenfunctions of the discrete Laplacian:
KL
' (m, n, i, j)"!k ' (m, n, i, j), where M, N are the CNN dimensions, m, n the summation indexes
+,
KL +,
and: i"0, 1,2, M!1; j"0, 1,2, N!1. The relation between temporal modes and spatial eigenvalues is
represented by the Dispersion Curve. Such a curve depends on the system parameters and not on the
boundary conditions. On the other hand, the spatial eigenvalues depend on the CNN dimensions and on the
boundary conditions. At this point, if, for the particular boundary conditions adopted, there exist spatial
eigenvalues K lying in the region of the dispersion curve corresponding to positive temporal modes, (the
KL
so-called Band of unstable modes B ), Turing instability onsets and patterns do arise. The number of different
S
geometries is determined by the number of K inside B . For the zero-flux boundary conditions, the spatial
KL
S
eigenfunctions and eigenvalues assume the following form:
' (m, n, i, j)"cos
+,
(2j#1) nn
mn
nn
(2i#1)mn
cos
and k "4 sin
#sin
KL
2M
2N
M
N
(8)
with m"0,2, M!1; n"0,2, N!1
Moreover, the limits of B are the values k , k :
S
1
k "
[(D (k#e)#D (k!e))$(D (k#e)!D (k!e))!4D D s s ]
2D D
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
638
P. ARENA, M. BRANCIFORTE AND L. FORTUNA
that, taking into consideration the chosen CNN cell parameters and the diffusion coefficients, become:
k"1)96; k"109. From equation (8), in the case M"N"3, the following unstable spatial eigenvalues
can be found:
!1)14 2)5063 2)5063
k "
KL
2)5063 5)6326 5)6326
(9)
5)6326 5)6326 5)6326
The temporal eigenvalues associated with the spatial modes already determined can be obtained using the
following relation:
D #D
#
Rj(k )"R k!k KL
KL
2
D !D !s s
!e#k KL
2
In our case they take on the following values:
0)0000 0)6463 0)6463
j " 0)6463 0)7726 0)7726
KL
0)6463 0)7726 0)7726
(10)
Therefore, for our parameter choice, eight spatial modes are contained into B , and thus it is possible to
S
obtain eight possible pattern configurations, as it appears from Figure 1, which depicts all patterns arising
from the spatial eigenfunctions (8) related to their eigenvalues k .
KL
Since each pattern presents two available polarities, the whole number of different pattern configurations
is sixteen. Of course, the configuration corresponding to ' arising from k cannot take place, since
k does not lie inside B .
S
When more than one are unstable, a kind of competition between modes onsets. It has to be outlined that
the conditions for the onset of Turing patterns are based on a linearization of the CNN state equations. In
particular, an analysis of the cells in the linear region around the origin has been performed. Therefore, once
Figure 1. Turing pattern configurations for M"N"3
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
A CNN-BASED FRAME FOR PATTERNS AND AUTOWAVES
639
the diffusion process onsets, the system, sooner or later, will become non-linear, and the final pattern
geometry will be influenced by the non-linear iteration among other temporal modes. Therefore, it can be
predicted only how patterns begin to evolve.
It is intuitive to think that the spatial mode, and therefore its associated eigenfunction, corresponding to
the temporal eigenvalue with the most positive real part, will most probably appear; but initial conditions, as
well as unavoidable unbalancies among the circuit parameters play a fundamental role in the pattern
formation phenomenon in the circuit realization, as it will be seen in the experiments reported in the
following. This part strictly reflects what commonly happens in living structures.
3. THE EXPERIMENTAL SETUP
In this second part of the paper the circuit implementation of the CNN cells, as well as of the main board
realizing their physical connections, are presented. In particular, in the following subsections the circuit
realization of the differential equations of the two cell prototypes for the generation of autowaves and
patterns are reported together with a description of the couplings among the cells and of the main board so as
to form a CNN 5;5. The realization of the Laplacian operator will be considered when dealing with the
couplings implementation.
The realization of each of equations (1), scaled with a coefficient k"10\, suitable for circuit implementation, can be easily performed by using an algebraic summing operational amplifier, whose general scheme is
reported in Figure 2(a).
The equation derived from such a scheme is the following:
R
R
1
(11)
C xR "! x# y ! y
G
R R
R R H
R
In Figure 2(b) the circuit scheme is shown for the implementation of the non-linear output function (2). An
operational amplifier has been used in such a way so as to exploit its natural output saturation. In particular,
R , R , R and R are chosen such that the amplifier output saturates when "x"'1. Moreover, the output
voltage divider scales the amplifier output in order to match the correct signal level. The following equations
therefore hold:
R
R
x for !1(x(1
y"
R #R R
y"1 for x'1
(12)
y"!1 for x(1
Figure 2. (a) realization of one of equations (1); (b) the circuit scheme is shown for the implementation of the non-linear output
function (2)
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
640
P. ARENA, M. BRANCIFORTE AND L. FORTUNA
Figure 3. Realization of the cell for autowave generation
Figure 4. Simulation and the circuit measures of the two state variables trajectories for the cell for autowave generation
3.1. Realization of the cell for autowave generation
In the present case the parameter values, already shown in Section 2, are: k"0)7, s"1, i "!0)3 and
i "0)3. The corresponding parameter values and the whole circuit scheme are reported in Figure 3.
The bias values i , i have been realized by means of a unique voltage divider on the power supply » ,
AA
since their absolute values are the same. The simulation of the cell dynamics as well as the visualization, via
the labview program, of the measures made on the realized circuit are shown in Figure 4 and the simulation
and the circuit measures of the two state variables trajectories of the cell are depicted. It can be drawn that the
state variables, during the time evolution of the circuit, undergo a stable limit cycle, proven in Reference 12
and confirmed by their simulations [Figure 4(a)].
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
A CNN-BASED FRAME FOR PATTERNS AND AUTOWAVES
641
Figure 5. Circuit realization of the cell for pattern formation
Figure 6. Simulation and measured state variables x , x for circuit of Figure 5
3.2. Realization of the cell for pattern formation
Following the considerations made in the previous sections, it is very easy to obtain a circuit realization for
the cell realising Turing patterns from that of one implementing the cell for autowave propagation. In fact,
from the circuit scheme reported in Figure 5 and by comparison with Figure 3, it can be seen that the two
structures match, except for the lack of the input voltage divider that realizes the bias currents, and for the
change of values of a few resistors. Therefore both the circuits can be realized on the same board and some
jumpers can swich from one-cell configuration to the other one. In Figure 6 the temporal trends of the state
variables x , x are reported from a Spice simulation of the circuit shown in Figure 5, together with the
measures of the same state variables. In both cases the initial conditions for the two state variables have been
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
642
P. ARENA, M. BRANCIFORTE AND L. FORTUNA
Figure 7. Circuit realization of the laplacian couplings
set to 1 V. In particular, the measurements has been performed after locking the cell state variables to their
initial values an then releasing them after 400 ms from the measure beginning. Notice that the time scales
difference is due to the presence in the circuit of a capacitance C"220 lF (see Figure 3), which modifies the
circuit time constant with respect to the simulation, in which a unit capacitance had been assumed.
The circuit realization of the devices for the initial condition locking-releasing has been built by means of
some analog mosfet low-output impedance switches digitally driven.
3.3. Realization of the ¸aplacian couplings and boundary conditions
The implementation of the Laplacian couplings has been performed simply using some operational
amplifiers in summer configuration. A schematic diagram is shown in Figure 7. The output of such a circuit
will be connected to the non-inverting input of the op. amp. in the cells previously discussed by means of
suitable resistors whose values will represent the diffisivity coefficients D , D and varying correspondently
the offset resistor R . Looking at equations (3) and at this circuit structure, it should be emphasized that while
L
the two layers interact within each single cell generating the oscillatory slow—fast behaviour (for autowave
generation) or the stable focus (for pattern formation), conversely, interaction with the neighbouring cells,
once the cells are connected, is obtained separately by means of the Laplacian templates with diffusion
coefficients D (for the first layer) and D (for the second one), respectively. Therefore, there is no direct
interaction between layer 1 of a cell and layer 2 of its neighbours and vice versa. The zero-flux boundary
conditions can be easily realized. In fact considering equations (3) the influence of the boundary cells is
realized by the discrete Laplacian. When a cell C belongs for example to the right boundary edge of the
GH
array, the dynamics of the missing cell, say C
is assumed to match the C dynamics. Therefore, in this
GH>
GH
case the discrete Laplacian simply modifies to: D ) (y ;
#y ;
#y ;
!3y ; ) , K"1, 2. Similar
I I G>H
I G\H
I GH\
I GH
arguments also hold for cells belonging to a corner position, for example to the upper right one for which it
results: D ) (y ;
#y ;
!2y ; ), K"1, 2. Such considerations circuitally translate into a suitable
I I G>H
I GH\
I GH
modulation of the ratio R/R (Figure 7) in the realization of the Laplacian couplings for the boundary cells.
3.4. Realization of the main board
All the circuits previously discussed have to be ‘put’ together in order to realize the CNN array and to
show self-organization phenomena. To this aim a main board has been designed and built, able to realize all
the necessary connections between the circuits, which can be easily located on to the board with suitable
slots. Moreover, in the whole frame it has to be possible to easily modify or change the resistors realizing the
couplings between circuits. A particular section in the board has been therefore implemented, in which all the
resistive couplings could take place. The board has also been provided with a section to perform all measures
on the state and output variables. In Figure 8 a schematic diagram showing the main board is reported.
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
A CNN-BASED FRAME FOR PATTERNS AND AUTOWAVES
643
Figure 8. Main board scheme
Moreover, for the direct visualization of the phenomena, a LED matrix of dimension 5;5 has been built. In
this way each led diode represents a cell in the CNN array. The LED used are able to change their emitted
light colour from green to red as the applied voltage changes from !1 V to #1 V. In such a way it is
possible to gain enough information about the CNN system state. In order to report the results of the
experiments in the clearest way, a data acquisition system, based on a Personal Computer in Windows O.S.
and a data acquisition board National AT-MIO-16E-10 has been used, together with the LabView program
for data managing and visualization. The experiments on auto waves will be presented by means of a virtual
Lab View intrument panel, showing the temporal trends of certain variables useful for the experiment
comprehension, together with a table, shown in the right top margin, in which the index of the monitorized
cells is reported. A matrix scheme of the cells, standing for the LED matrix, is also represented in the bottom
right margin. The virtual instrument panel for turing pattern experiments will present the temporal trends of
certain variables belonging to the CNN cells. Moreover patterns will be also presented as they appear in the
first layer output of the CNN matrix. In particular, the white colour represents #1 V, black !1 V, while
the brushed cells are set to 0 V. The cells will be identified as belonging to a matrix. Therefore, for example,
A will represent the cell positioned in the first row and second column in the CNN matrix.
3.5. Autowave experiments
The setup previously discussed has been adopted to perform experiments about autonomous phenomena
regarding propagation of waves in active media. Indeed, several papers appeared in the literature, showing
different models of arrays of non-linear circuits, identical to one another and discretely coupled by means of
diffusion rules, and able to show active wave propagation and other spatio-temporal phenomena. In
this paper the same second-order structure is shown to be able to generate, used a building block in a CNN
carray, propagating active waves or turing patterns by suitably modulating its parameters. Such a result is
presented with particular emphasis to the circuit implementation, in order to directly see if tolerances usually
met in commercially available discrete components as well as the presence of noise could heavily affect the
results of the experiments. As previously outlined, these phenomena revealed their robustness in a number of
simulations reported in Reference 12. The first experiment performed regards the propagation of an
autowave through the CNN matrix. Initially all cells were tuned in order to show the same dynamics, namely
a stable limit cycle with slow—fast regime, compatibly with the component tolerances (10% for capacitors and
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
644
P. ARENA, M. BRANCIFORTE AND L. FORTUNA
Figure 9. Autowave propagating through the CNN matrix rows
Figure 10. Time evolution of the first and fourth rows of the CNN matrix during the propagation of an autowave
1% for resistors). The initial conditions for this experiment consisted in setting each cell of the first row of the
matrix at the following voltages: x "#1 V, x "!1 V. If such initial conditions hold, all the other cells,
left to approximately zero initial conditions, will soon synchronize with the propagation of the wave
generated by the first row of the matrix. Of course, if no other initial conditions are imposed into the matrix,
at the end of the propagation through the fifth row, the autowave vanishes and a competition between waves
induced by each cell starts. Such competing wave fronts mutually annihilate, leading to the onset of an
autowave propagating in a particular direction, which depends on the positions of some ‘dominant’ circuits
which bias the whole phenomenon. In our case, the favoured steady-state direction is the one along the
matrix diagonal. In order to re-obtain the autowave propagation along the matrix rows (i.e. the desired
direction), the initial conditions have to be set to the cells of the first row at the end of each propagation
through the CNN matrix. Figure 9 shows the LabView acquired image of the time evolution of the first layer
output of the cells belonging to the first and fourth row of the CNN matrix, as it is also outlined by the
schematic picture representing the CNN matrix reported at the bottom right margin of Figure 9. In Figure 10
the time evolutions of the first and fourth row of the matrix are reported in a 3-D graphic plot, clearly
showing the time required for the auto wave to propagate through the CNN matrix, which corresponds to
about 0.66 circuits/s. Taking into consideration also the legend reported in the top right margin if Figure 9, it
can be concluded that the cells of the two rows depicted (and therefore all the cells belonging to each row)
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
A CNN-BASED FRAME FOR PATTERNS AND AUTOWAVES
645
Figure 11. Concentric wave from the CNN matrix central cell
proceed with unchanged amplitude and shape and well synchronized in time, as typical of the autowaves. The
slight imbalance of the rising and falling fronts is mainly due to the circuit component tolerances.
Figure 11 shows the propagation of an autowave from the central cell of the CNN matrix. In this
experiment the initial conditions x "#1 V and x "!1 V were locked for the cell A , which,
therefore, acts as a source of concentric waves. The wave front propagates well synchronizing the cells
positioned at the same distance from the pace maker cell A . Taking into consideration the simulation
results reported in Reference 12 as well as the experiments above explained, it results that, in spite of the
influences of the circuit component tolerances, the underlying reaction—diffusion phenomenon takes place
and the generated fronts show the typical behaviour of auto waves.
4. PATTERN FORMATION AND PROPAGATION
Reaction—diffusion dynamics has significant applications in the patterning process, especially in biology. In
Referenmce 1 it has been shown that patterns of animal coat markings, for example on the zebra, could be
generated using a reaction—diffusion mechanism. Several paper appeared in literature, in which the same
results have been obtained starting from different theoretical approaches. In this second part of the paper
such phenomena will be shown taking into consideration a real CNN based circuit which realizes reaction—diffusion rules.
At the beginning of the experiments, all the cells were tuned in such a way to realize circuits where the
origin should be a stable focus, according to the parameter set presented in Section 2. Initial conditions on all
cells are fundamental in this case in order to perform successfully the experiment. Actually pattern formation,
in the simulation results shown in the literature, is generally studied in large circuit arrays. When a circuit
realization is considered, the possibility to obtain patterns reflecting a certain geometry is greatly affected by
noise. Therefore, such task can be reached mainly in small circuit arrays. In the present case, it has been
observed that, given an initial condition in a certain set of cells, during the earliest part of the spatio-temporal
evolution, the cells lying at a certain distance from the set of cells chosen for the initial settings, are affected by
a spurious, noise-induced, diffusion-driven instability. Such stimuli preceed the actual diffusion induced by
the right signal propagation. Therefore, even if a steady-state configuration is found, the ‘geometry saving’
propagation, typical of Turing patterns can be prevented. This set of experiments was therefore carried out in
a 3;3 CNN, and a comparison between the calculations previously performed and the experimental results
will be reported.
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
646
P. ARENA, M. BRANCIFORTE AND L. FORTUNA
Figure 12. Turing pattern from ‘near-zero’ initial conditions
Larger CNN structures can be considered while studying the propagation phenomenon in growing
structures, simulating living cell reproduction. Such an example will be presented at the end of the paragraph.
Referring to the considerations and calculations in Section 2, in the case M"N"3 eight possible pattern
configurations are allowed, according to the extention of B . Among such patterns, the most allowed ones are
S
those possessing the largest positive temporal associated eigenvalues, (such as ' through ' in Figure 1).
Moreover, such conditions are strictly dependent on the set of parameters employed. In fact, all the unstable
temporal eigenvalues are of the same magnitude order, and therefore no clean supremacy takes place among
them. However, such considerations can be made only referring to simulations and calculations based on
a linearization of the CNN state equations. In the real case, each cell realization has its own intrinsic noise
that biases its behaviour and therefore some particular pattern configurations are most allowed. This can be
easily seen in the following experiment, shown in Figure 12, in which the CNN cells are allowed to evolve
from zero initial conditions. Indeed, as it can be observed from the plots in Figure 12, each circuit starts from
a condition near zero, say »M . The final pattern configuration therefore is biased by the disturbances inside
each cell and due to the circuits tolerances, and only incidentally matches one of the patterns with the largest
positive temporal mode (' in Figure 1). Of course, changing the cell position inside the main board, the
final pattern configuration will change accordingly, as it can be seen from Figure 13. Moreover, some pattern
configurations arise even if they are not contained in those ones predicted by linear theory, since non-linear
competition between unstable modes takes place when nonlinearities onset. These last patterns can be
considered as ‘spurious’, or as defects in the CNN structure. Therefore, the possibility to drive the structure
itself to some desired geometry avoiding the ‘defects’ can be implemented introducing suitable initial
conditions so as to excite the modes responsible for the desired final patter geometry. Initial conditions have
to assume values greater enoughly than voltages »M , whose greatest absolute value is around 0)01 V (see
Figure 12), otherwise other spurious configurations can arise. In Figure 14 the desired checkerboard pattern
is obtained by imposing an initial condition » "!0.15 V on the central cell A , in order to compete and
‘win’ the other ‘noise-biased’ initial conditions. Smaller initial conditions lead to the onset of spurious
patterns. The possibility to control the final pattern configuration is further emphasized in the following
experiments. It is to be outlined that the pattern control is realized only with suitable initial conditions, and
not with the addition of forcing actions. Therefore the structure of the CNN is not further complicated.
In Figure 14 the time evolution for each cell is also reported, which clearly shows the role of diffusion in
inducing an instability in the stable focus of each cell, leading to a new steady state condition which shows the
checkerboard pattern depicted in the right-hand side of the figure.
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
A CNN-BASED FRAME FOR PATTERNS AND AUTOWAVES
647
Figure 13. Turing pattern from ‘near-zero’ initial conditions
Figure 14.
Checkerboard turing pattern
Figure 15 depicts the initial conditions leading to the formation of a strip, the corresponding time
evolution of the first layer outputs, as well as the final pattern configuration. Such an experiment can also be
seen as the propagation of a strip from the first to the third row of the CNN matrix.
Figure 16 shows the effect of a perturbation on a reached steady-state condition. Given some initial
conditions which lead to the formation of a strip pattern, if this steady state condition is perturbed setting the
central cell x variable from !1 V to #1 V (see the time trend of A in Figure 16), another state transition
takes place, and a new pattern geometry, a checkerboard one, is reached.
The next experiment refers to a phenomenon which can easily be observed in the animal coat, looking at
the particular parts linking the body with legs or tail. In these zones the coat growth takes place in two
different directions that cannot be recast into a unique (radial or linear) direction, as it commonly happens in
the other parts of the animal body: there are two main directions of growth, mutually orthogonal. Therefore,
the propagation of a strip, which proceeds in one direction, near such scapular sites , has to ‘change’ direction.
Such a situation has been reported in the next experiment, in which a strip is forced to propagate through
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
648
P. ARENA, M. BRANCIFORTE AND L. FORTUNA
Figure 15. Formation of a strip pattern
Figure 16. Pattern formation and further evolution after perturbing the reached steady state conditions of the central cell
Figure 17. Propagation of a strip through a corner: matrix view
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
A CNN-BASED FRAME FOR PATTERNS AND AUTOWAVES
649
Figure 18. Propagation of a strip through a corner: time evolution
a corner region. Figure 17(a) depicts the initial conditions, which are zero for all the cells, except for the two in
the bottom [namely A and A in Figure 17(a)], set to 1 V. Moreover, the two cells A and A are
physically disconnected to the other ones: they will be connected subsequently. Figure 17(b) shows the
steady-state conditions: the strip propagates through the corner cells. In fact the cells now set to 1 V are
A , A and A , which represent the ‘boundary’ between horizontal and vertical strips. All the other cells
are set to !1 V. If now A and A are connected to the CNN array with zero initial conditions, simulating
the growing process, they both move to 1 V, thus confirming the propagation of the strip horizontally
[Figure 17(c)]. Figure 18 shows the time evolution of the first state variable of the cells, during this
experiment.
5.
CONCLUSIONS
In this paper a 5;5 CNN matrix for the experimental modelling of complex phenomena such as active wave
propagation and pattern formation has been proposed. The number and the type of experiments reported is
no doubt a very little part of the possibilities offered by this experimental frame, which represents a useful
development system for the study of complexity. The feasibility of the approach is emphasized considering
some results previously obtained which allow the same cell structure to be able to generate autowaves or
patterns only slightly modulating its parameters. Therefore, the whole frame can be considered as a device for
observing the most different examples of reaction—diffusion phenomena that, until now, have been simulated
or implemented in structurally different systems. Autowaves and sources of concentric waves have been
1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 635—650 (1998)
650
P. ARENA, M. BRANCIFORTE AND L. FORTUNA
observed if each second-order cell behaves as a slow—fast oscillator, while Turing patterns arise if the same
cell is ‘tuned’ so as to show a stable focus at the origin. The self-organization phenomenon is completed via
a suitable Laplacian coupling among neighbouring cells. These experiments, related to the corresponding
calculations and simulations, give an idea of how and to what extent the unavoidable uncertainties and
imbalances, typical of real systems, can affect the onset of such phenomena, which appear to be fairly robust.
It is further emphasized that the CNN paradigm can represent a unified approach for modelling and gaining
further insight in complex behaviours.
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