understanding organism growth and cellular diﬀerentiation ......cellular automata (see [44][17]...

Understanding Organism Growth and Cellular Differentiation Through

Evolutionary Selection of Dynamical Properties.

by

Yuri Cantor

Thesis Proposal for the degree of

Doctor of Philosophy

at the

Graduate Center of the City University of New York

Committee: Professor Bilal Khan (Advisor)Professor Nancy GriffethProfessor Matthew JohnsonProfessor Kirk Dombrowski

May, 2016

Contents

1 Abstract 1

2 Introduction 2

2.1 Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Area 1: Cellular Differentiation and Robustness 8

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3 Formal Hypothesis 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.5 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Area 2: Cellular Differentiation and Adaptivity 12

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13


4.4 Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14


5 Area 3: Symmetry and Growth 16

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18


5.4 Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

i


6 Obstacles 24

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.2 Candidate Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.2.1 Powers of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.2.2 Sampling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.2.3 Distributed processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 Summary 27

8 Timeline 28

ii

1 Abstract

Computer simulations of networks of cellular automata serve as an experimental approach to explor-

ing relationships between properties of a network and the dynamics of the network. Consequently,

these dynamical systems are commonly used to model biological organisms representing cells with

the abstraction of cellular automata.

Natural selection describes a process where traits or properties of the biological organism that are

better suited to the environment will be selected for. It follows that properties of the dynamics that

are better suited to the environment will also be selected for. This proposal studies the impact of

Darwinian selection on cellular differentiation and organism growth and focuses on the relationship

of organism properties of homogeneity and heterogeneity to the dynamical properties of robustness,

adaptivity, and chromatic symmetry.

I propose an experimental approach to show that cellular differentiation increases the expected

robustness in an organism’s dynamics, cellular differentiation leads to increasing adaptivity, and

organism growth by elongation preserves symmetry in the dynamics of an organism. Further, I will

also address practical concerns arising from computational limitations and big data in experimental

approaches to studying dynamical systems.

1

2 Introduction

Biological neural network behavior is difficult to model computationally. These networks are com-

posed of densely connected neurons, have a state of excitation that is continuous in degree, and

transmit state asynchronously [50]. There has been extensive research into biological networks and

cellular automata (see [44][17] for brief surveys). Cellular automata as described by Von Neumann

[40] can be connected to form a network which serves as an abstraction of a biological networks and

computational simulation of these simplified networks of cellular automata can lead to understand-

ing the behavior in correlated complex biological networks [28][33][46][19].

In an experimental approach abstractions are necessary to maximize the number of networks and

depth of simulation that can be modeled. Further, it is known that decomposing super networks into

sub networks can lead to understanding pathways and signaling in biological circuitry [53]. Random

Boolean Networks (RBNs) have been used to abstract from continuous state values to Boolean.

Kauffman’s NK network model abstracts from densely connected to K connections [20][21]. The

trivial case where K=1, restricts the nodes to connected along a line. While Conway’s "Game of

Life" is example of a Boolean NK network represented using a two dimensional grid where K=8

[18]. Asynchronous transmissions with small temporal tolerances can be modeled as synchronous

[39].

This proposal explores theoretical synthetic biology, using the more restricted class of synchronous

Boolean cyclic NK networks (where K=2) as a formal basis. This class of networks has received

considerable attention [47], and exhibits many of the phenomena seen in more general NK coun-

terparts [45]. Even with these reductions in complexity, fully simulating the dynamics remains

computationally intractable except for networks of relatively small size [51].

The relationship of network structural properties to dynamical properties is crucial to understand

2

how and why biological networks have evolved. Further, this relationship can be used to construct

networks with desirable properties [41]. Dynamical systems have been previously be evaluated for

their landscape ruggedness [35][34], redundancy [21], and reversibility and surjectivity (reachability

and Garden of Eden states [47]) [26][37]. I propose an experimental approach to evaluate the

relationship between network structural properties of heterogeneity, homogeneity, and growth to

dynamical properties of robustness [9], adaptivity [7], and symmetry [10].

This proposal is divided into four parts. The first three parts correspond to three hypotheses

intended to provide insight to the relationship between network structural properties and dynamical

properties: cellular differentiation increases the expected robustness in an organism’s dynamics,

cellular differentiation leads to increasing adaptivity, and organism growth by elongation preserves

symmetry in the dynamics of an organism. The fourth part will address practical concerns to

the experimental approach arising from computational limitations and big data in experimental

approaches to studying dynamical systems.

2.1 Preliminary Definitions

Throughout this proposal networks of cellular automata will be referred to as organisms where

each node in a network will be a cell of the organism. The application of biological terminology to

describe theoretical and synthetic networks is common practice as seen since Von Neumman’s seminal

work on cellular automata [40][30]. As noted in the introduction the structure I am focusing on is

selected to decrease computational complexity in order to facilitate deeper exploration of state space

and exploration of larger organisms.

Structure. Informally, linear cyclic organisms are composed of cells that are directly connected

to their two neighboring cells such that the cells form a ring where an initial cell is connected to the

3

Table 1: Truth table mapping with inputs at time t and resulting output at time t+ 1

s(vi�1, t) s(vi, t) s(vi+1, t) s(vi, t+ 1)

0 ⇤ 0 b00 ⇤ 1 b11 ⇤ 0 b21 ⇤ 1 b3

Table 2: XOR b0b1b2b3 = 0110 truth table mapping

s(vi�1, t) s(vi, t) s(vi+1, t) s(vi, t+ 1)

0 ⇤ 0 00 ⇤ 1 11 ⇤ 0 11 ⇤ 1 0

next cell and the last cell. Formally, the linear cyclic structure is modeled as an undirected cyclic

graph C = (V,E) of size n. Vertices are the cells, and are enumerated V = {v0, . . . , vn�1} where each

cell vi in V is connected in cyclic order to two neighbors such that edges E = {(vi, vi+1 (mod n))|i =

0, . . . , n � 1}. The set of possible linear cyclic organisms of size N are a subset of Kaufman’s NK-

networks [12] of size N where for each cell the number of inputs is K = 2.

Cell state is Boolean, either 0 or 1, and deterministic by fixing a function f : V �! F that

assigns to each cell v 2 V , a function f(v) from F = {g : {0, 1}{0, 1} �! {0, 1}}, the set of all

binary Boolean functions. The action of f at a vertex vi can be thought of as a truth table mapping

from the two inputs K, vi’s left and right neighbors’ current state, to vi’s state at the next time

step. This function differs from the traditional Wolfram model [51][55] in that Wolfram’s model

used K = 3 and considered the self-input or state of cell vi as well. The bits b0, b1, b2, b3, as in

Table 3, must be either 0 or 1 and the 4-bit binary string b0b1b2b3 is used to name the function f.

Note that because there are 2 choices for each of the 4 possible input a cell’s state must be 0 or 1

as a result of the possible the inputs from its neighbors, |F | = 222 = 16. For example, Table 2 is

the XOR function truth table. Note, that the space of possible functions for Wolfram’s model is the

more expansive set such that |F | = 223 = 256 [52]. In Table 3 these rules are defined using Boolean

4

Table 3: Table of update rules.Rule numbers expressed as Boolean logic with 2 inputs:

The value of the left neighbor vi�1, and the value of the right neighbor: vi+1Rule Boolean Logic

Rule 0= 0Rule 1= ¬vi�1 ^ ¬vi+1Rule 2= ¬vi�1 ^ vi+1Rule 3= ¬vi�1Rule 4= vi�1 ^ ¬vi+1Rule 5= ¬vi+1Rule 6= (vi�1 ^ ¬vi+1) _ (¬vi�1 ^ vi+1)Rule 7= ¬(vi�1 ^ vi+1)Rule 8= vi�1 ^ vi+1Rule 9= (vi�1 ^ vi+1) _ (¬vi�1 ^ ¬vi+1)Rule 10= vi+1Rule 11= vi+1 _ (¬vi�1 ^ ¬vi+1)Rule 12= vi�1Rule 13= vi�1 _ (¬vi�1 ^ ¬vi+1)Rule 14= vi�1 _ vi+1Rule 15= 1

logic and ordered in ascending order according to the Boolean input values. For reference, rule 6 in

Table 3 corresponds to the earlier example truth table shown in Table 2.

The organisms in this proposal are synchronous where every cellular state is instantaneously

determined at each time interval according to:

s(vi, t+ 1) = f(vi)�s(vi�1 (mod n), t), s(vi+1 (mod n), t)

�

for each i = 0, . . . n�1 and t > 0. Note that results from synchronous models can be transformed and

be realized in asynchronous models with synchronization stays within small tolerances locally [39].

This implies that research into synchronous organisms has implications for asynchronous organisms.

Homogeneity and heterogeneity have previously been studied with respect to connectivity [32] and

functions [23]; however, in this proposal they will refer only to functions. If every cell in an organism

has the same function assigned to it, then the organism is homogeneous. There are 16 unique

5

homogeneous organisms of any given size. However, if not every cell in an organism has the same

function assigned to it, then the organism is heterogeneous. There are 16n � 16 heterogeneous

organisms of size n. Formally, an organism is homogeneous if |Im(f)| = 1; otherwise it is hetero-

geneous. There has been research into the distance between functions using hamming distance as

a measure of proximity [27][55]. However, here I simplify and look only at the number of different

functions instead of the specificity of what those functions are and their distance from each other.

An organism state, as in Figure 3, is the ordered set of its cell states at time t. This set of

states for example in Figure 1, can be expressed as the sequence of states v0v1v2v3v4 = 00010.

Dynamics. The dynamics of the organism are represented as a directed graph S connecting

organism states, as in Figure 1, to their successor states, seen in Figure 2, representing the

organism’s phase space, in which (X, Y) is an edge if it can be said that whenever the organism

is in state X at time t, it is necessarily (absent noise) in state Y at time t + 1. Formally, the

dynamics is a directed graph S = (2V , D) whose vertex set consists of all possible states of the

organism (i.e. the power set of V), and whose edge set D includes every ordered pair (X, Y) for

which s+(t) = X =) s+(t + 1) = Y . If an organism state X is connected to organism state Y by

a directed edge from X to Y, then Y is the successor state of X. Not only have successor functions

have been studied for extensively, but their resulting dynamics and dynamical components have been

the focus of research problems [52].

0

1

00

0

Figure 1: Initial state.

1

0

10

0

Figure 2: Successor state.

v

v

vv

v

2

3

40

1

Figure 3: Size 5 network.

6

In the dynamics, a Garden of Eden state is a state that cannot be reached from any other state

by a directed edge in the dynamics space of an organism. In other words, no state in the dynamics

of the organism has as its successor a Garden of Eden state. This dynamical component has drawn

attention as part of the problem of reachability: to determine if a state is a Garden of Eden state

[47]. Research into Garden of Eden states involves developing algorithms for inverting the successor

function or reversibility [26][37], and Wuensche and Lesser introduced a reverse algorithm [59].

An attractor state is an organism state that occurs in a cycle in the dynamics space of an

organism. In other words, if X is an attractor state then there exists some discrete number of time

intervals j such that when starting at state X and time t and computing the successor states at

times t+1, . . . , t+ j then at the state at time t = t+ j will be X. An attractor is the set of all states

that are in the same cycle in the dynamics space of an organism and attractor length is the number

of states in an attractor. Here the term attractor encompasses the classes of attractors defined by

Wolfram including: fixed point, periodic, and strange [51]. In Hopfield networks attractors are the

content-addressable memory [25].

A tributary state is a state that is not in an attractor. When starting at a tributary state and

computing the successor states eventually an attractor state will be encountered because the state

space has a discrete number of possible states. A tributary is the set of states composing a path of

tributary states that lead to the first encountered attractor state when computing forward from a

Garden of Eden state. An attractor state can have many tributaries or no tributaries. Basins of

attraction [56] are formed by these tributaries.

An attractor with its tributaries composes a single connected component of the dynamics

graph of an organism. The graph of an organism’s dynamics space is made up of these components.

Figures 4, 5, 6 show an example decomposition of the size 12 homogeneous XOR organism. The

7

complete dynamics graph for the homogeneous XOR organism of size 12 has 6 attractors of length

2 (Figure 4), 60 attractors of length 4 (Figure 5), and 4 attractors of length 1 (Figure 6). In the

Figures, nodes are labeled with organism state which is composed of the Boolean state of the cells.

Directed edges lead from an organism state to its successor state. Black edges connect tributaries

to their successor state and blue edges connect attractor states to their successor state [10].

3 Area 1: Cellular Differentiation and Robustness

3.1 Introduction

The evolutionary trend in organisms has been away from homogeneity and towards heterogeneity

and complex structures. Since natural selection is a driving force for evolution, there must be

some properties of the underlying networks [49] and the dynamics of those networks that are being

selected for [13] which drive evolution towards heterogeneity. Consequently, I propose the following

hypothesis: Cellular differentiation increases the expected robustness in an organism’s

dynamics.

When exploring the dynamics of an organism over infinite time, most of the time will be spend

cycling in attractors. As such, in this proposal the focus is only on attractor state perturbations

and the effects of those perturbations. Note, this assumption decreases the number of states whose

perturbations are computed and explored thereby also minimizing the computational overhead of

000001010110

000010000111

100101001101

111000111101

001101100101

000011010011

100111001111

111101111000

000011010100

000111000010

000101000111

101000101101

000110010110

001111100111

000111000101

101101101000

001001110100

010111010010

001010100101

110000011000

001011110001

110010011010

001011110110

010010010111

001110110100

011010110010

001111100000

011000110000

010000010010

010010010000

010011000011

010100000011

010110000001

010110000110

011010110101

011011100001

011100100001

011110100011

011110100100

011111110000

100000001111

100001011011

100001011100

100011011110

100100011110

100101001010

101001111001

101001111110

101011111100

101100111100

101101101111

101111101101

110000011111

110001001011

110100001001

110100001110

110101011010

110110001011

111000111010

111001101001

111010111000

111100101011

111100101100

111110101001

Figure 4: Attractor length 4.

000001111110

000011000011

100111100111

111100111100

000101101111

101001101001

000111010010

001101001101

001001011100

010110010110

001011100001

110010110010

001111110000

011000011000

010000111010

010010000111

010100101011

011010100101

011100001001

011110110100

100001001011

100011110110

100101011010

101011010100

101101111000

101111000101

110000001111

110100011110

110110100011

111000101101

111010010000

111110000001

Figure 5: Attractor length 2.

000000000000

000100010001

101010101010

001000100010

010101010101

001100110011

111111111111

010001000100

011001100110

011101110111

100010001000

100110011001

101110111011

110011001100

110111011101

111011101110

Figure 6: Attractor length 1.8

exploring both the state space and the perturbations.

3.2 Definitions

A mutation is the perturbation of a random cell state in the attractor state. In other words, it is a

bit flip of a single bit in the organism state [27]. The organism state that results from a mutation of

a single random cell state is a mutation state. Wuensche describes graphs of these perturbations

as attractor jump-graphs [57]. Mutation states are connected to the originating attractor state

by a mutation edge. Each organism state of an organism of size n has n possible resulting

mutations states and outgoing mutation edges. This type of perturbation is one of many that have

been explored. Perturbations can also be made in the cell function [54] resulting in a change in the

wiring or circuitry of the organism or in the timing or synchronicity of the signal [8]. There has

been research into the distance between cell states and cell functions, using the hamming distance

between two binary values [27][55]. For example, the distance D between a state 00000 and another

state 00111 is D = 3 since there are three bits with differing values.

A mutation state is an organism state that is either in an attractor or connected to an attractor

through directed edges in the dynamics graph. If that attractor or connected attractor is the

originating attractor then the mutation edge is called robust. Otherwise the mutation edge is not

robust. Robustness is the ability to resist external influences [6][11], and in the case of the organism

dynamics, robustness is the ability to conserve the dynamical topology [14][22][43]. Research where

the perturbation is in the synchronicity of signals applies the definition of robustness to be qualitative

changes resulting from perturbations in synchronicity in the update scheme [8].

Attractor robustness is computed by dividing the number of robust mutation edges by the

total number of mutation edges of the attractor states. An attractor of length L in the dynamics

9

space of an organism of size n has n ⇤L mutation edges. Therefore, the robustness of that attractor

would be the number of robust mutation edges r/n ⇤ L.

Organism robustness ⇢ is the average attractor robustness of all the attractors in an organism’s

dynamics space. If the total number of attractors in the dynamics space of an organism is ↵, and

the robustness of attractor A is rA then

⇢ = (1/↵)

↵X

A=1

rA.

Note, that organism robustness could be computed as the average number of edges that are robust or

R/E where R is the number of robust mutation edges and E is the total number of mutation edges.

However, in this proposal, I apply the former definition such that organism robustness is the average

attractor robustness for all attractors in an organism’s dynamics space.

3.3 Formal Hypothesis 1

Applying the definitions, hypothesis 1 can be posed formally as follows. Let the average robustness

of the set of all heterogeneous organisms X at any fixed size N be Avg(⇢(X(N)))) and let the average

robustness of the set of all homogeneous organisms Y at the fixed size N be Avg(⇢(Y (N))) then for

N > 2

Avg(⇢(X(N))) > Avg(⇢(Y (N))).

3.4 Research Plan

The experimental methodology for testing this hypothesis will follow entail 3 components: program-

ming, simulation and data collection, and analysis.

The first stage is to develop a C program that sequentially computes the full organism state

10

space, the number of attractors, and organism robustness. The program will also generate data files

of dynamics graph with mutation edges that can be rendered using graphviz [15]. There is existing

software, in particular Discrete Dynamics lab [58] is a tool that has been extensively used to model,

simulate, and study dynamical systems. Since the focus in this proposal is to explore properties that

differ from the properties studied before, the benefit of building my own code base is that it will be

highly customize-able. Developing a program instead of using the existing software further enables

optimization for larger organisms (exhaustive search of state space is limited, n 6 12 [55]) and to

develop novel algorithms for simulation (e.g. the later discussed simulation by multiple successor

steps).

The second stage is to simulate and collect data for analysis. The program will be run on organism

sizes that can be fully explored. The program will explore all 16 unique homogeneous organisms

for sizes n = 2 . . . 16. For organism sizes n = 2 . . . 16 the space of possible heterogeneous organisms

(n

16 � 16) is too large to fully explore. Therefore the program will also be run on a randomly

sampled set of 1000 heterogeneous organisms for sizes n = 2 . . . 16.

Finally, in the third stage data collected from the simulations will be analyzed at each size n

for homogeneous organism and for heterogeneous organisms to compare average robustness ⇢ and

average number of attractors ↵. Finally, the data will be used to compare dynamical properties

between the sets of homogeneous and heterogeneous organisms

3.5 Preliminary results

The number of attractors found from fully exploring all 16 homogeneous organisms of sizes = 2 . . . 16

can be divided into two classes: Class 1 organisms where the number of attractors remains bound

by some integer b as organism size increases (see figure 1), and Class 2 organisms where the

11

Figure 7: Homogeneous organisms with bounded (left) and unbounded (right) numbers of attractors.

number of attractors grows unbounded (see figure 2). Given this preliminary result, the proposal

is to compare each of these classes of homogeneous organisms separately to the randomly sampled

heterogeneous organisms.

4 Area 2: Cellular Differentiation and Adaptivity

4.1 Introduction

Starting from a single cell, every organism is homogeneous. Therefore, this proposal seeks to identify

a property that when selected for would lead to the initial cellular differentiation from a homogeneous

organism to a "minimally" heterogeneous organism. In biological networks, specialization of cells

in heterogeneous organisms helps facilitate their adaptivity to environments. In an organism’s

dynamics the corollary to a group of specialized cells is an attractor. Consequently, the informally

proposed hypothesis is that cellular differentiation results in an increase in number of

attractors and in turn to an increase in adaptivity.

12

Note that robustness and adaptivity are related properties both in nature and as dynamical

properties. Biologically, an organism can either be robust to a stimulus or adapt and evolve; as

such, there has been research into how an organism can be robust, while still evolving and adapting

[3]. In the previous area I propose exploring the dynamical property of robustness by varying the

organism property of homogeneity and heterogeneity while organism size is static. In this area, I

propose to focus on the dynamical property of adaptivity with respect to the organism property of

homogeneity and heterogeneity during growth.

Related work has involved researching spread [4] and influence [24] in the dynamics of social

networks. In particular the standing ovation model [38] and similar consensus problems are of

interest because the correlation in the dynamics of the organism is a collapse of adaptivity. However,

it is know that searching the dynamics space for specific attractor structures is computationally

challenging. Specifically, finding singleton attractors which have attractor length 1 is known to be

NP-hard [42][60][1].

4.2 Definitions

As mentioned in the preliminary definitions and introduction, there has been research into the ham-

ming distance between functions in an organism [27][55]. However, here I use the simple count of

different functions as a measure of homogeneity and heterogeneity of an organism. Therefore, a

homogeneous organism would have 0 different functions or formally |Im(f)| = 1, while a heteroge-

neous organism would have at least 1 different function or |Im(f)| > 1. Given these two distinct

classes of organisms, the border case is of interest. Specifically, an organism with 1 different func-

tion, formally |Im(f)| = 2, where every cell determines its state using the same function except for

one cell. I refer to this border case class of organism as minimal heterogeneous.

13

Given the earlier definition of an attractor attractor, I can now define Adaptivity as the number

of attractors ↵(X) in the dynamics space of an organism. Attractor count is a measure of interest

that has been extensively studied [28][51][17]. I use the term adaptivity due to its relationship to

evolvability, defined by Aldana et al. to mean that an organism can acquire new funcitons and adapt

to new environments [3]. Further, if attractor count is considered memory [25], then organisms with

greater memory are more likely to be adaptive to new environments that require shifting memory.


Applying the definitions, hypothesis 2 can be posed formally as follows. Let the robustness of

a single homogeneous organism Y at size N be ⇢(YN ) and the robustness of a single minimally

heterogeneous organism X at size N be ⇢(XN ). Then as the organisms X and Y increase in size

from N = 2 . . .1,

9 i | i > 1 and ⇢(XN+i) > ⇢(YN+i).

4.4 Research Plan

The experimental methodology for testing this hypothesis will follow entail 3 components: program-

ming, simulation and data collection, and analysis.

The first stage will be to extend the C program from area 1 that sequentially explores the

organism state space to compute the number of attractors. The program will also generate data

files of dynamics graph with mutation edges that can be rendered using graphviz [15] and charted

using gnuplot [36]. In order to explore larger networks, I will expand the program to sampling the

state space and estimate attractor counts.

The second stage will simulate and collect data for analysis. The program will be run on organism

sizes that can be fully explored and sample the state space of for larger sizes. The program will

14

0

1

2

3

Figure 8: Homogeneous XOR size 2.

0

1

10

2

5

3

15

4

6

7

8

9

11

12

13

14



explore the homogeneous XOR organisms for sizes n > 2 . . . 16. The program will also explore every

possible minimally heterogeneous organisms of the same size as the homogeneous XOR organisms

simulated.

In the third stage, analysis of the resulting number of attractors discovered for the homogeneous

XOR organism will be compared to the number of attractors found for each individual minimally

heterogeneous organism at every size organism whose dynamics space is fully explored and every at

every size organism whose dynamics space is sampled.


The initial results of the dynamics space of homogeneous XOR organism exhibit a pattern of collapse

in adaptivity at sizes which are a power of 2 (see figures 8,9,10 and the chart in figure 11). Therefore,

for sizes n > 16 simulations will focus on organism sizes which are powers of 2.

15

Figure 11: ↵ for homogeneous XOR organisms size N = 2 . . . 20.

5 Area 3: Symmetry and Growth

5.1 Introduction

In areas 2, adaptivity is explored with the assumption has been that organisms will grow incre-

mentally by a single cell from size n to n+1. However, growth rate is a network property that will

have its own relationship to dynamical properties. If growth rate impacts the dynamical properties

then natural selection would prefer a rate that increased or at least retained the desirable dynam-

ical properties. In area 2, the preliminary results show that for the homogeneous XOR organism

adaptivity is adversely impacted by incremental growth towards sizes that are powers of 2.

To introduce the idea of how organism growth could preserve symmetry, let’s look at the example

of a starfish. The starfish has clear rotational or radial symmetry. How can a starfish grow while

preserving rotational symmetry? There is the possibility to remain the same size while growing

another arm. And, there is the possibility to simply grow larger, where each part of the organism is

elongated. Both of these preserve its rotational symmetry. This proposal will look at both possible

16

symmetry preserving patterns of growth.

In this area, I propose to explore how dynamical properties of symmetry can be preserved during

growth [5] in order to understand which patterns of growth might be selected for in organisms.

Previous work in symmetry of cellular automata has been explored by focusing on symmetry of

the functions [29][52], time symmetry referring to the successor function computing forwards and

backwards [16], and an organism’s dynamic state pattern symmetry resulting from an initial state

[48].

In area 1 and 2, I take a perspective of the organism as a whole and observe the global dynamics.

Since the focus in this area is on symmetry of the organism, in this area I will focus on the cell

dynamics as they relate to the global organism dynamics. As before in the assumption posited

in area 1, I will focus only on attractor states since from any intial state after an infinite number

of discrete time intervals, the organism will spend the majority of its time in attractor states.

Observing each individual cell of the organism as the organism orbits in an attractor, the individual

cells flip on and off in a period that divides the length of the attractor. Now, rather than the

instantaneous state of an organism in this area I consider the state of each cell over unbounded time

by examining the periodicity of its binary oscillations as the organism orbits in an attractor. The

periodic transition from 0 to 1 of the cell states may result in symmetric pattern of state changes

of cells in the organism cells during the orbit of an attractor. In this area, I propose to explore this

symmetry of cell periodicity in an attractor as a dynamical property in relation to organism growth

[10].

The focus in this area is symmetry. Here we will focus on rotational symmetry. Using the example

of a starfish that has clear rotational symmetry, I propose to look at two forms of growth which

could preserve that rotational symmetry. The first method of growth is cell by cell elongation of

17

each "arm" simultaneously. The second method of growth is the reduplication or growth of an entire

new "arm" all at once. Taking inspiration from the biological organism, a starfish grows through

elongation rather then adding another arm: informally the proposed hypothesis is that growth by

elongation is more likely to preserve symmetry in dynamical properties then growth

by reduplication.

5.2 Definitions

Color is determined by the periodicity of a cell’s state as the organism traverses an attractor. Color

is assigned to an individual cell. Note that periodicity of cell state is dependent upon the attractor

being traversed; as such, for every attractor, each cell can have a different periodicity and coloring.

Because the periodicity has an upper bound of the length l of the attractor and a lower bound of 1

the possible colors for a cell will vary dependent upon the length of an attractor. As defined earlier,

the organisms are cyclic. Therefore, for each attractor in the dynamics of an organism, as each cell

is colored with respect to its periodicity, the organism itself takes on a pattern of corresponding

colors. In Figure 12 on the left, the homogeneous XOR organism of size 12 orbits an attractor of

length 4 and the. In Figure 12 in the middle, each of the 12 cells’ states are expanded outwards

towards infinity as the organism orbits in that attractor of length 4. In Figure 12 on the right, the

periodicity of the cells of the homogeneous XOR organism of size 12 can be seen as the organism

orbits an attractor of length 4, and corresponding colors are assigned to the organism.

The pattern of colors arranged in a ring can result in symmetry. In particular, the coloring could

be rotationally symmetry. Rotational symmetry in the case of coloring means that after a certain

amount of rotation the coloring would be the same. Segment size l is the number of cells which

when rotated result in the same coloring of an organism. Since rotation by segment size l results in

18

the same coloring, segment size divides organism size. Foldedness k is the number of times segment

size divides into organism size k = N/l. Trivial rotational symmetry is defined as a segment size

l = 1 and foldedness k = n because it is the case where all cells have the same periodicity resulting

in an organism where all cells have the same color. This proposal is only interested in non-trivial

cases of symmetry.

In this area,Chromatic symmetry is defined to be non-trivial rotational symmetry in the

coloring of the organism cells as it traverses an attractor. A rotationally symmetric organism has

k segments of size l > 1 where each segment has the same pattern of coloring. In Figure 12 (on

the right), the coloring of the homogeneous XOR organism of size n = 12 as it orbits the attractor

of length 4 shown in Figure 12 (on the left) exhibits chromatic symmetry with segment size l = 6

and foldedness k = 2. This means that any the organism coloring can be rotated from any initial

position by a segment of 6 cells to the right or left and the resulting organism coloring will be the

same as the initial organism coloring [10].

4

1

244

42441

4

4

1

0

111

11100

1

0 1

0

001

00110

1

1

1

0

110

11110

0

10

0

011

10010

1

1

11101110...

10101010...

10111011...

11011101...

00000000...

11011101...

10111011...

10101010...

11101110...

01110111...

00000000...

01110111...

2381

3645869

3960

Figure 12: Attractor of size 4 in dynamics of homogeneous XOR organism size n=12.

Growth. Growth patterns have been extensively studied [2], for example for modeling morpho-

genesis in plants using L-systems [31]. Growth in areas 1 and 2 focused on incremental organism

19

growth by a single cell from size n to size n+1. Reduplication is growth by incrementally increas-

ing the number of segments from k to k + 1. This increase in segments means that the organism

growth is from size n to size n+ l. Elongation is growth by incrementally increasing the segment

length from l to l+1. This increase in segment length means that every segment increases in length

by 1 and in turn the organism growth is from size n to size n + k. In Figure 13 the example of a

homogeneous XOR organism of size n = 10 in the orbit of an attractor of length 6 in its dynamics

graph Elongation gives a starting point from which the two forms of growth can be explored.

Note, that in the example the chromatic symmetry has segment size l = 5 and foldedness k = 2.

Figure 14 represents growth that occurs by segment elongation from a size n = 10 organism to a size

n = 12 organism, where the growth is an increased by increasing the segment size l = 5 to l = 6.

Figure 15 represents growth that occurs by segment reduplication from a size n = 10 organism to a

size n = 15 organism, where the growth is an increased by increasing the foldedness k = 2 to k = 3.

Note that in Figure 15 the segment size remains l = 5 [10].


Applying the definitions, hypothesis 3 can be posed formally as follows. Let A be a homogeneous

XOR organism of size N which has nontrivial rotational chromatic symmetry of segment length

LAi as the organism A traverses attractor i. Let Probability(sym(R)) be the probability that

there exists a homogeneous organism R of size N + LAi which has nontrivial rotational chromatic

symmetry of segment length LAi and let Probability(sym(E)) be the probability that there exists a

homogeneous organism E of size N + (N/LAi) which has nontrivial rotational chromatic symmetry

20

0

0

000

00101

1

0

000

010001

1

010

01000

0

0

100

10000

0

1

010

00000

0

0

100

10101

6

6

661

66616

5

520

277

160

272

680

Figure 13: Coloring in orbit of attractor length 6 in dynamics of homogeneous XOR organism ofsize n=10

of segment length LAi + 1, then:

Probability(sym(E)) > Probability(sym(R))

5.4 Research Plan

The first stage will further extend the C program from area 2 to explore the dynamics of organisms

compute the chromatic symmetry of the organism across its attractors.

Because symmetry will be tested at larger organism sizes, the program will have to be able to

sample the dynamics space and handle larger numbers. Therefore, the program will be written

21

4

4

414

44144

4

4

1

1

000

00000

0

0 0

0

101

00000

1

1

0

0

000

10001

0

00

0

000

01010

1

1

1345

3485

2176


3

13

33

31

33

13

33

33

1

10

10

01

01

00

00

01

0

10

11

01

11

00

10

10

1

11

00

11

00

01

01

01

8081

14453

11472


using the GNU MP big number library.

The second stage will simulate the organism dynamics space and collect data. The program will

explore the dynamics space of all homogeneous organisms and identify their possible colorings. Next,

selecting from the homogeneous organisms a subset that exhibit non-trival chromatic symmetry, the

22

program will be run focusing on sizes which correspond to patters of growth that entail elongation

and reduplication.

In the third stage analyzing the data collected will require comparing the probability of preserving

symmetry by each growth pattern (elongation and re-duplication) in the sampled dynamics across

the simulated organism sizes.


Attractor length bounds periodicity; therefore, organisms with attractor lengths of 1 are restricted to

the trivial case of chromatic symmetry. By sampling the dynamics of all the homogeneous organism

from 1000 random start states for sizes n = 2 . . . 50, preliminary data collection of attractor lengths

can be used to identify organisms whose coloring will be restricted to the trivial case of chromatic

symmetry. Figure 16 shows that only a subset of homogeneous organisms have increasing attractor

lengths and are therefore more likely to exhibit non-trivial chromatic symmetry.

1

10

100

1000

10000

100000

0 5 10 15 20 25 30 35 40 45 50

Attra

ctor

Len

gth

Organism Size

Attractor Lengths at Organism Size

All FunctionsFunction 6

Figure 16: Attractor lengths of homogeneous organisms of sizes N = 2 . . . 50

23

The next step is to determine the relative frequency of the chromatic symmetry for these organ-

isms. The simulation and data collection will focus on only those homogeneous organisms that have

a non-zero relative frequency of chromatic symmetry.

6 Obstacles

6.1 Introduction

The experimental approach I propose taking faces obstacles that accompany big data. As the size

of a network grows linearly, the state space grows exponentially. Consequently, even at relatively

small network sizes, exhaustive search of the state space becomes computationally intractable. Be-

ing unable to exhaustively search the state space results in the inability to generate the complete

dynamics including identifying attractors, robustness, and chromatic symmetry. As the size of a

network grows linearly, the possible organisms grow exponentially. Further, the time and compu-

tational limitations of simulating each of these networks becomes computationally intractable even

at small network sizes.

In developing the programs, I propose the following three approaches to mitigating big data ob-

stacles: First, using algorithms which make use of patterns to increase efficiency in memory and

search by computing, storing, and comparing the successor state at multiplicative or exponential

time increments instead of single increments. Second, using sampling techniques to estimate prop-

erties of the dynamics and the organism. Third, using distributed processing to explore the state

space and sets of possible organisms in parallel

24

6.2 Candidate Approaches

6.2.1 Powers of 2

Successor states are computed sequentially. Starting from a state, every next state is compared

against every previously viewed successor state of the initial state. Attractors are determined once

a previously seen state is seen again.

I propose to implement a more efficient algorithm then the trivial sequential successor computa-

tion with every state computed being compared with every previously computed successor state. If

P is the number of states from an initial state to the first state in an attractor + the length of the

attractor L, then the memory required is P states and the number of comparisons to discover an

attractor is (P 2)/2.

It is possible to lower the memory required for detecting an attractor by only storing a state if

it is at a successor step that is a power of 2. A side effect of the lower memory fewer results in

fewer comparison operations (log2p where p is the path explored so to the current successor state)

to check if the state has been visited before for each successor computation since there are fewer

states being stored. This algorithm requires P (log2P ) comparisons and log2 P memory.

For XOR organisms, I propose to work on developing an algorithm for simulating more then one

successor state at a time with a goal of computing forward in powers of 2 successor states at once.

The basis for this algorithm is the pattern observed in the preliminary results for area 2. The core

component of this algorithm relies on a proof that organisms of size n is a power of 2 in any state

will collapse to a single attractor after a set number C of successor computations. With a proof

of what the state of the organism will be after a set number of successor computations, then the

algorithm can be modified such that the incremental successor computations to operate in C steps

25

instead of steps of size 1. Additional steps in the algorithm will be necessary to compute states that

need to be computed at intervals that differ from exact powers of C.

6.2.2 Sampling techniques

Random sampling is accomplished by picking a random state between 0 and (2N ) � 1, instead of

sequentially starting at each subsequent state in the state space and tracing it to an attractor.

Attractors are determined in the same way however, by computing the successor states from the

initial state until a previously seen state is seen again. For random sampling, I select a set D of

random states and trace them until I identify the attractor they fall into to.

For capture recapture, the program will randomly sample a number of states D, twice as 2

separate intervals. The set of attractors identified from the first set of D starting states are labeled

as Capture 1. The set of attractors identified from the second set of D starting states are labeled

as Capture 2. And, the set of attractors seen in both Capture 1 and Capture 2 are labeled as

Recapture.

Capture Recapture has limitations which occur as the size of the state space and the number of

attractors increase to such a degree that the sampled Capture 1 and Capture 2 have no overlap.

Further Capture Recapture assumes that the size and distribution of the basin of the attractors are

the same.

I propose developing a capture recapture technique which takes into account the varying attractor

sizes to estimate attractor counts of differing sized attractors and the total number of attractors in

an organisms dynamics space.

26

6.2.3 Distributed processing

I propose to use a message passing interface to distribute sets of initial states to trace into attractors

to machines in a cluster. I will implement the distributed processing in the forensics lab cluster at

John Jay College of The City University of New York. Data computed by the simulation will then

be returned to a centralized server where it can be compressed and bulk inserted to a centralized

database that keeps track of the states explored. The bottleneck occurs at the centralized database.

Future work would be to decentralize the database to improve access time to the database for

reading and writing.

7 Summary

This proposal studies the impact of Darwinian selection on cellular differentiation and organism

growth. The proposal seeks a greater understanding of natural selection through exploring the re-

lationship of cellular differentiation to robustness and adaptivity and the relationship of growth to

chromatic symmetry. This proposal will follow an experimental approach to testing three hypothe-

ses:

• Cellular differentiation increases the expected robustness in an organism’s dynamics.

• Cellular differentiation leads to increasing adaptivity

• Organism growth by elongation preserves symmetry in the dynamics of an organism.

The proposed experimental approach faces obstacles related to big data. I propose three approaches

to addressing these obstacles: algorithms, sampling and distributed processing.

27

8 Timeline

Date TaskOctober 2011 Research Plan 1November 2014 Research Plan 2November 2015 Research Plan 3May 2016 Dissertation ProposalJune-September 2016 Thesis writingSeptember 2016 Thesis defense

Table 4: Schedule of Tasks

Table 4 describes the time-line to complete the dissertation.

28

References[1] Tatsuya Akutsu, Satoru Kuhara, Osamu Maruyama, and Satoru Miyano. Identification of Gene

Regulatory Networks by Strategic Gene Disruptions and Gene Overexpressions. In Proceedingsof the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’98, pages 695–702, Philadelphia, PA, USA, 1998. Society for Industrial and Applied Mathematics.

[2] V. Z. Aladjev. Classical Cellular Automata. Homogeneous Structures. Fultus Corporation,September 2010.

[3] Maximino Aldana, Enrique Balleza, Stuart Kauffman, and Osbaldo Resendiz. Robustness andevolvability in genetic regulatory networks. Journal of Theoretical Biology, 245(3):433–448,April 2007.

[4] Elizabeth Ball. Dynamic Spread of Social Behavior in Boolean Networks. BiblioBazaar, May2012. 00000.

[5] Gregory Bateson. A re-examination of âĂĲBatesonâĂŹs ruleâĂİ. Journal of Genetics,60(3):230–240, September 1971.

[6] Hedi Ben Amor, Jacques Demongeot, and Sylvain Sené. MICAI 2008: Advances in Artifi-cial Intelligence: 7th Mexican International Conference on Artificial Intelligence, Atizapán deZaragoza, Mexico, October 27-31, 2008 Proceedings, chapter Structural Sensitivity of Neuraland Genetic Networks, pages 973–986. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008.

[7] Yuri Cantor Bilal Khan. Attractor-Based Obstructions to Growth in Homogeneous CyclicBoolean Automata. Journal of Computer Science & Systems Biology, 08(06), 2015.

[8] Olivier Bouré, Nazim Fatès, and Vincent Chevrier. Unconventional Computation: 10th In-ternational Conference, UC 2011, Turku, Finland, June 6-10, 2011. Proceedings, chapter Ro-bustness of Cellular Automata in the Light of Asynchronous Information Transmission, pages52–63. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011.

[9] Yuri Cantor, Bilal Khan, and Kirk Dombrowski. Heterogeneity and its impact on ThermalRobustness and Attractor Density. Procedia Computer Science, 6:15–21, 2011.

[10] Yuri Cantor, Bilal Khan, and Kirk Dombrowski. Towards a Formal Understanding of Bateson’sRule: Chromatic Symmetry in Cyclic Boolean Networks and its Relationship to OrganismGrowth and Cell Differentiation. Procedia Computer Science, 36:476–483, 2014.

[11] J. Demongeot, E. Goles, M. Morvan, M. Noual, and S. Sené. Attraction basins as gauges of therobustness against boundary conditions in biological complex systems. PLoS One, 5(8):e11793,2010.

[12] E. Dubrova, M. Teslenko, and A. Martinelli. Kauffman networks: Analysis and applications.In Proceedings of the 2005 IEEE/ACM International Conference on Computer-aided Design,ICCAD ’05, pages 479–484, Washington, DC, USA, 2005. IEEE Computer Society.

[13] Marc Ebner, Mark Shackleton, and Rob Shipman. How Neutral Networks Influence Evolvabil-ity. 2001. 00090.

29

[14] A. Elena and J. Demongeot. Interaction motifs in regulatory networks and structural robust-ness. In Complex, Intelligent and Software Intensive Systems, 2008. CISIS 2008. InternationalConference on, pages 682–686, March 2008.

[15] John Ellson, Emden Gansner, Eleftherios Koutsofios, Stephen North, and Gordon Woodhull.Graphviz - Open Source Graph Drawing Tools. Graph Drawing, pages 483–484, 2001. 00001.

[16] AnahÃŋ Gajardo, Jarkko Kari, and AndrÃľs Moreira. On time-symmetry in cellular automata.Journal of Computer and System Sciences, 78(4):1115 – 1126, 2012.

[17] Niloy Ganguly, Biplab K. Sikdar, Andreas Deutsch, Geoffrey Canright, and P. Pal Chaudhuri.A survey on cellular automata. 2003.

[18] M Gardner. The fantastic combinations of John Conway’s new solitaire game "life”. ScientificAmerican, 223:120–123, October 1970.

[19] Felix Gers, Hugo de Garis, and Michael Korkin. CoDi-1bit : A simplified cellular automatabased neuron model. In Jin-Kao Hao, Evelyne Lutton, Edmund Ronald, Marc Schoenauer, andDominique Snyers, editors, Artificial Evolution, number 1363 in Lecture Notes in ComputerScience, pages 315–333. Springer Berlin Heidelberg, October 1997. DOI: 10.1007/BFb0026610.

[20] Carlos Gershenson. Classification of Random Boolean Networks. arXiv:cs/0208001, August2002. arXiv: cs/0208001.

[21] Carlos Gershenson, Stuart A. Kauffman, and Ilya Shmulevich. The Role of Redundancy in theRobustness of Random Boolean Networks. arXiv:nlin/0511018, November 2005.

[22] E. Goles and L. Salinas. Comparison between parallel and serial dynamics of boolean networks.Theor. Comput. Sci., 396(1-3):247–253, May 2008.

[23] Alireza Goudarzi. On the Effect of Heterogeneity on the Dynamics and Performance of Dy-namical Networks. Dissertations and Theses, January 2012. 00000.

[24] Michel Grabisch and Agnieszka Rusinowska. A model of influence in a social network. Theoryand Decision, 69(1):69–96, July 2010.

[25] J J Hopfield. Neural networks and physical systems with emergent collective computationalabilities. Proceedings of the National Academy of Sciences, 79(8):2554–2558, 1982.

[26] Jarkko Kari. Reversibility and surjectivity problems of cellular automata. Journal of Computerand System Sciences, 48(1):149–182, February 1994.

[27] S. A. Kauffman. Metabolic stability and epigenesis in randomly constructed genetic nets.Journal of Theoretical Biology, 22(3):437–467, March 1969.

[28] S.A. Kauffman. The Origins of Order: Self Organization and Selection in Evolution. OxfordUniversity Press, 1993.

[29] MaÅĆgorzata J. Krawczyk. New aspects of symmetry of elementary cellular automata. Chaos,Solitons & Fractals, 78:86 – 94, 2015.

[30] R.A. Laing, University of Michigan. Dept. of Computer, and Communication Sciences. For-malisms for biology: technical report. University of Michigan Technical Reports. Computer andCommunication Sciences Department, University of Michigan, 1969.

30

[31] A. Lindenmayer. Developmental algorithms for multicellular organisms: a survey of L-systems.Journal of Theoretical Biology, 54(1):3–22, October 1975.

[32] Min Liu and Kevin E. Bassler. Emergent criticality from coevolution in random booleannetworks. Phys. Rev. E, 74:041910, Oct 2006.

[33] Torbjorn Lundh. Cellular Automaton Modeling of Biological Pattern Formation: Characteriza-tion, Applications, and Analysis Authors: Andreas Deutsch and Sabine Dormann, Birkhauser,2005, XXVI, 334 p., 131 illus., Hardcover. ISBN:0-8176-4281-1, List Price: $89.95. GeneticProgramming and Evolvable Machines, 8(1):105–106, February 2007.

[34] Thomas E Malloy and Gary C Jensen. Dynamic constancy as a basis for perceptual hierarchies.Nonlinear dynamics, psychology, and life sciences, 12(2):191–203, April 2008. 00005 PMID:18384716.

[35] Thomas E Malloy, Gary C Jensen, and Timothy Song. Mapping knowledge to boolean dynamicsystems in Bateson’s epistemology. Nonlinear dynamics, psychology, and life sciences, 9(1):37–60, January 2005. 00007 PMID: 15629067.

[36] Thomas Williams and Colin Kelley and {many others}. Gnuplot 4.4: an interactive plottingprogram, March 2010.

[37] Maurice Margenstern. About the Garden of Eden Theorems for Cellular Automata in theHyperbolic Plane. Electronic Notes in Theoretical Computer Science, 252:93–102, October2009.

[38] John H. Miller and Scott E. Page. The standing ovation problem. Complexity, 9(5):8–16, 2004.

[39] Chrystopher L. Nehaniv. Evolution in Asynchronous Cellular Automata. pages 201–209. MITPress, 2002. 00000.

[40] John Von Neumann. Theory of Self-Reproducing Automata. University of Illinois Press, firstedition edition edition, 1966.

[41] Ranadip Pal, Ivan Ivanov, Aniruddha Datta, Michael L. Bittner, and Edward R. Dougherty.Generating Boolean networks with a prescribed attractor structure. Bioinformatics,21(21):4021–4025, January 2005. 00114 PMID: 16150807.

[42] Yushan Qiu, Takeyuki Tamura, Wai-Ki Ching, and Tatsuya Akutsu. On control of singletonattractors in multiple Boolean networks: integer programming-based method. BMC systemsbiology, 8 Suppl 1:S7, 2014.

[43] Adrien Richard and Jean-Paul Comet. Necessary conditions for multistationarity in discretedynamical systems. Discrete Appl. Math., 155(18):2403–2413, November 2007.

[44] Palash Sarkar. A brief history of cellular automata. ACM Comput. Surv., 32:80–107, 2000.

[45] Cosma Rohilla Shalizi and Kristina Lisa Shalizi. Quantifying Self-Organization in Cyclic Cel-lular Automata. In in Noise in Complex Systems and Stochastic Dynamics, Lutz Schimansky-Geier and Derek Abbott and Alexander Neiman and Christian Van den Broeck, Proceedings ofSPIE, vol 5114, 2003. 00019.

[46] J. E. S. Socolar and S. A. Kauffman. Scaling in ordered and critical random boolean networks.Physical Review Letters, 90(6):068702, February 2003.

31

[47] Klaus Sutner. Linear cellular automata and the garden-of-eden. The Mathematical Intelligencer,11(2):49–53, January 2009.

[48] Ikuko Tanaka. Effects of initial symmetry on the global symmetry of one-dimensional legalcellular automata. Symmetry, 7(4):1768, 2015.

[49] D’Arcy Wentworth Thompson. On growth and form. Cambridge : University Press ; New York: Macmillan, 1945. 00009.

[50] Joaquin J. Torres and Pablo Varona. Modeling Biological Neural Networks. In GrzegorzRozenberg, Thomas Back, and Joost N. Kok, editors, Handbook of Natural Computing, pages533–564. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. 00005.

[51] Stephen Wolfram. Cellular automata as models of complexity. Nature, 311(5985):419–424,October 1984. 01308.

[52] Stephen Wolfram. A new kind of science, 2002.

[53] D. M. Wolpert, R. C. Miall, and M. Kawato. Internal models in the cerebellum. Trends inCognitive Sciences, 2(9):338–347, September 1998.

[54] A. Wuensche. Genomic regulation modeled as a network with basins of attraction. PacificSymposium on Biocomputing. Pacific Symposium on Biocomputing, pages 89–102, 1998.

[55] Andrew Wuensche. Discrete Dynamical Networks and their Attractor Basins, February 1998.00063.

[56] ANDREW WUENSCHE. Basins of attraction in network dynamics: A conceptual frameworkfor biomolecular networks. Modularity in development and evolution, page 288, 2004.

[57] Andrew Wuensche. Artificial Life Models in Software, chapter Discrete Dynamics Lab: Toolsfor Investigating Cellular Automata and Discrete Dynamical Networks, pages 263–297. SpringerLondon, London, 2005.

[58] Andrew Wuensche. Discrete dynamics lab, tools for investigating cellular automata and discretedynamical networks. In Artificial life models in software, pages 215–258. Springer London, 2009.

[59] Andrew Wuensche and Mike Lesser. The Global Dynamics of Cellular Automata: An Atlas ofBasin of Attraction Fields of One-dimensional Cellular Automata. Andrew Wuensche, January1992.

[60] Shu-Qin Zhang, Morihiro Hayashida, Tatsuya Akutsu, Wai-Ki Ching, and Michael K. Ng.Algorithms for Finding Small Attractors in Boolean Networks. EURASIP Journal on Bioin-formatics and Systems Biology, 2007:1–13, 2007.

32

understanding organism growth and cellular diﬀerentiation ......cellular automata (see [44][17]...

Documents