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Local entropy patterns in continuous cellular automaton models

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Pawel Zgrzebnicki
Pawel Zgrzebnicki
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The article presents the results of the simulation of a cellular automaton model whose cells assume continuous values. The described experiment examined the behaviour of local entropy and showed the occurrence of its three qualitatively different images with their character resembling Turing patterns. The hypothesis, according to which the behaviour of local entropy might be identified with an image created in a reaction-diffusion process, was presented.
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Local entropy patterns in continuous cellular automaton models
Paweł Zgrzebnicki1
1University of Social Sciences and Humanities, Warsaw, Poland
pzgrzebnicki@st.swps.edu.pl
Abstract
The article presents the results of the simulation of a cellular
automaton model whose cells assume continuous values. The
described experiment examined the behaviour of local entropy
and showed the occurrence of its three qualitatively different
images with their character resembling Turing patterns. The
hypothesis, according to which the behaviour of local entropy
might be identified with an image created in a reaction-diffusion
process, was presented.
Problem
The research study concerning the mathematical foundations of
creating cultures (Zgrzebnicki, in press-a) considers
a continuous cellular automaton model as a hypothetical
framework. Although there have been many studies on the
properties of discrete cellular automata published so far, the
models whose cells assume any values limited only by floating
point precision have been much less discussed in scientific
works. The research presented herein assumed that the
discussed automaton may change values of its cells according
to any function transforming the values of neighbours in the
neighbourhood of a certain radius, and that this radius and
weight of the neighbours' contribution to a cell's change may
also be arbitrarily defined for the purposes of a certain
experiment. In comparison with discrete automata (Wolfram,
1984), such assumptions imply a much broader class of
solutions. The conducted research is aimed at finding an answer
to the question whether there are any general regularities or
patterns describing the dynamics of the assumed model.
Method
Let there exist a matrix !, whose components are called cells.
Let there exist a function ", which transforms matrix ! into
matrix !# in a way that:
$%
&' "()%
*$*+ ,-
, where $ stands for the cell's value, while % and.* mean
coordinates, and r means a radius within which the
neighbourhood of a certain cell has an influence on its value's
transformation. Furthermore, to eliminate the problem of
boundary values, let us assume that the boundaries of the matrix
are glued together so that topology of the described space is
identical with a torus.
Figure 1: The value of cells established on the basis of the
values of their neighbours located within the radius , ' /.
Topology of glued boundaries guarantees continuity of
solutions on the edges of the matrix.
Finally, let us consider process 0 in which values of all the
cells are changed so that a matrix obtained after the change
becomes a source matrix for the next step:
01 ! 2 !32 !33 2 !333 2 4
Experiment
The following experiment was done. In matrix ! of a size of
512 x 512 components, a random value from the range of [0,1]
was attributed to each cell. In each subsequent time step,
arithmetical mean 5 of a value of neighbouring cells within
radius , was calculated for each cell. Subsequently, on the basis
of the following functions, the weight with which the
neighbourhood contributes to the later change was calculated:
671 8 ' 9:5;.< =
>
>?>,..6@1.8 ' A9:5; <=
>
>?>B C,..6D1.8 ' C
, where $E stands for the value of the cell whose neighbourhood
is analysed, while F constitutes a parameter of the experiment
defined a priori.
Late-Breaking Abstracts Booklet of the 14th European Conference on Artificial Life (ECAL 2017), Lyon, France, 4-8 September 2017.
17
Figure 2: Normalized local entropy of the matrix (top) and local entropy in a time slice across the whole matrix and along cell no.
128 (bottom) in the experiments of the following parameters: ["7, 6D, ,=4, step of the process: 150] (left),
["@, 6@, 5=0.5, ,=4, step of the process: 800) (middle), ["@, 6@, 5=0.1, ,=48, step of the process: 285) (right).
The value of each cell in a subsequent time step was calculated
with the use of the following functions:
"71 $E
3'G=H.IJ
7HI ,.."@1 $E
3'G=H.I(J:G=-
7HI
Results
By means of the aforementioned transformations, the value of
the cells of matrix ! in the subsequent steps of process 0 was
calculated. Subsequently, for each P process, proper K matrix
was obtained whose every cell assumes the value equal to local
entropy (MathWorks, 1994-2017) in the neighbourhood of an
analogous cell of matrix !. The process of qualitative
evaluation of the results distinguished three images of changes
occurring within the frameworks of local entropy. Each of the
obtained result could be attributed to one of these three groups.
The first image is a fine, tangled pattern, almost unchanged
over time; the second image is a pattern similar to broadly
scattered contour lines and it varies over time in a cyclical
fashion; while the third pattern is the two aforementioned
patterns — unchanged and cyclical — overlapping.
Discussion
The distinguished images resemble Turing patterns occurring,
among others, as the result of reaction-diffusion processes
(Turing, 1952). According to the proposed hypothesis, the
analysed systems might be treated as reaction-diffusion ones if
low local entropy is understood as a high concentration of one
component, while high local entropy as mapping of a near
homogeneous mixture. Observation of a time change of Turing
patterns in the dynamics of a continuous cellular automaton
may show that the mechanism of imitating the neighbourhood,
on which its operation relies, may imply waves of ordering
patterns recurrent over time.
References
MathWorks (1994-2017). Entropyfilt. Local entropy of grayscale image,
https://www.mathworks.com/help/images/ref/entropyfilt.html.
Turing, A. M. (1952). The Chemical Basis of Morphogenesis. In
Philosophical Transactions of the Royal Society of London B.
237(641):37–72.
Wolfram, S. (1984). Universality and complexity in cellular automata,
Physica D: Nonlinear Phenomena, 10(1-2):1-35.
Zgrzebnicki, P. (in press-a). Perspektywy modelowania numerycznego
w kulturoznawstwie [Perspectives on numerical modelling in cultural
studies]. In Stępkowska I. M., Stępkowska J. K.., editors, Innowacje
w nauce i społeczeństwie wczoraj i dziś – perspektywa
interdyscyplinarna [Innovations in Science and Society yesterday
and today – interdisciplinary prospect].
Zgrzebnicki, P. (in press-b). Wizualizacja w matematycznym modelowaniu
kultur [Visualisation in mathematical modelling of cultures]. In
Kowalska, K. and Osińska, V., editors, Wizualizacja informacji
w humanistyce [Information visualisation in humanities].
Late-Breaking Abstracts Booklet of the 14th European Conference on Artificial Life (ECAL 2017), Lyon, France, 4-8 September 2017.
18
ResearchGate has not been able to resolve any citations for this publication.
Perspektywy modelowania numerycznego w kulturoznawstwie
Chapter
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W ostatnich latach w wielu dziedzinach obserwujemy coraz szersze podej- ście interdyscyplinarne w badaniach naukowych – od studiów o profilu mieszanym do prób wydzielania nowych gałęzi wiedzy, takich jak neuronauka (neuroscience) czy nauka o złożoności (complexity science). Obecnie widać także inny wyraźny trend: do łączenia nauk humanistycznych z naukami ścisłymi. Tradycyjny podział na science and humanities ulega zatarciu zarówno w warstwie pojęciowej, jak i metodologicznej. To właśnie innowacyjność w podejściu metodologicznym, zawierającym nie tylko praktykę, lecz także zmienioną podstawę teoretyczną, daje nadzieję na nowe spojrzenie na podstawowe problemy kulturoznawstwa: czym jest kultura, jaka jest jej struktura, jakim prawom podlega jej rozwój? Czy w ogóle można mówić o rozwoju, czy może jedynie o zmianach? Wyniki tradycyjnego myślenia o tych problemach mogą w owym podejściu zostać zestawione z analizą matematyczną i modelowaniem komputerowym kultur oraz rewizją metafizycznego spojrzenia na byt i relację, takiego jak np. teoria aktora-sieci Bruno Latoura. Być może dopiero opracowanie modeli od ich ontologicznych podstaw pozwoli choć w części uniknąć błędów zarzucanych odhumanizowanej inżynierii.
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The Chemical Basis of Morphogenesis
Article
  • Jan 1952
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Article
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Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable.
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The Chemical Basis of Morphogenesis
Article
  • Feb 1990
It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns onHydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
View
  • Jan 1994
  • Mathworks
MathWorks (1994-2017). Entropyfilt. Local entropy of grayscale image, https://www.mathworks.com/help/images/ref/entropyfilt.html.