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Morphozoic, Cellular Automata with Nested Neighborhoods as a Metamorphic Representation of Morphogenesis

  • Gulf Specimen Marine Laboratory

Abstract and Figures

A cellular automaton model, Morphozoic, is presented. Morphozoic may be used to investigate the computational power of morphogenetic fields to foster the development of structures and cell differentiation. The term morphogenetic field is used here to describe a generalized abstraction: a cell signals information about its state to its environment and is able to sense and act on signals from nested neighborhood of cells that can represent local to global morphogenetic effects. Neighborhood signals are compacted into aggregated quantities, capping the amount of information exchanged: signals from smaller, more local neighborhoods are thus more finely discriminated, while those from larger, more global neighborhoods are less so. An assembly of cells can thus cooperate to generate spatial and temporal structure. Morphozoic was found to be robust and noise tolerant. Applications of Morphozoic presented here include: (1) Conway's Game of Life, (2) cell regeneration, (3) evolution of a gastrulation-like sequence, (4) neuron pathfinding, and (5) Turing's reaction-diffusion morphogenesis.
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Morphozoic, Cellular Automata with Nested
Neighborhoods as a Metamorphic
Representation of Morphogenesis
Thomas Portegys
Dialectek, USA
Gabriel Pascualy
University of Michigan, USA
Richard Gordon
Gulf Specimen Marine Laboratory & Aquarium, USA & Wayne State University, USA
Stephen P. McGrew
New Light Industries, USA
Bradly Alicea
Orthogonal Research, Champaign, IL, USA
A cellular automaton model, Morphozoic, is presented. Morphozoic may be used to investigate the
computational power of morphogenetic fields to foster the development of structures and cell
differentiation. The term morphogenetic field is used here to describe a generalized abstraction: a cell
signals information about its state to its environment and is able to sense and act on signals from nested
neighborhood of cells that can represent local to global morphogenetic effects. Neighborhood signals
are compacted into aggregated quantities, capping the amount of information exchanged: signals from
smaller, more local neighborhoods are thus more finely discriminated, while those from larger, more
global neighborhoods are less so. An assembly of cells can thus cooperate to generate spatial and
temporal structure. Morphozoic was found to be robust and noise tolerant. Applications of Morphozoic
presented here include: (1) Conway’s Game of Life, (2) cell regeneration, (3) evolution of a gastrulation-
like sequence, (4) neuron pathfinding, and (5) Turing’s reaction-diffusion morphogenesis.
Keywords: Morphogenesis, Cellular automata, Moore neighborhoods
Morphogenesis is a biological process by which cells move and differentiate into organs and tissues
through genetic expression and collaborative, often physical mechanisms. They become different kinds
of cells, perhaps as many as 7000 kinds in our bodies (Gordon, 1999). One of the most persistent
concepts of morphogenesis is the morphogenetic field (Beloussov, Opitz & Gilbert, 1997; Alberts et al.,
2002; Levin, 2011, 2012; Morozova & Shubin, 2012; Vecchi & Hernández, 2014; Beloussov, 2015) with
clinical significance for human birth defects (Opitz & Neri, 2013). A morphogenetic field is a region of an
embryo that has the potential to develop into a specific structure. How this happens has been subject to
much investigation and debate. Some mechanisms are better understood than others. As discussed
in (Tyler, 2014), which reviews the full panoply of models of morphogenetic fields, we have added the
idea that a morphogenetic field is the trajectory of a two dimensional differentiation wave that triggers
a step of differentiation in each cell it traverses (Gordon, 1999; Gordon & Gordon, 2016b, a).
While the mechanisms behind the transformation of an egg into an embryo (embryogenesis) and then
to an adult organism have traditionally been of great interest to biologists, as a pattern formation
problem it is equally intriguing to computer scientists. Part of the allure involves the spontaneous
attainment of order of great complexity from geometrically simple beginnings. Even though one of the
founding fathers of computer science (Leavitt, 2006; Hodges, 2014; Tyldum, 2014), Alan Turing
proposed a plausible model for understanding one level of the self-organizing aspect of
morphogenesis (Turing, 1952; Gordon, 2015), the process of phenotype-building has a “ghost in the
machine” (Koestler, 1967), a cybernetic aspect (Gordon & Gordon, 2016a; Gordon & Stone, 2016) that
has gone underappreciated.
Recent chemical experiments have revealed that while Turing’s original reaction-diffusion equations
portray certain aspects of morphogenesis, they do not account for heterogeneity (Tompkins et al., 2014)
or the multistep hierarchical differentiation of cells into different types (Gordon, 2015). In this study, we
propose that given the right representation, simulated morphogenesis can yield solutions that are
biologically plausible. Our approach, Morphozoic, models a hierarchical structure of cellular
communities. Computationally, these communities are nested versions of Moore-like neighborhoods. A
Moore neighborhood is the set of cells that are immediate neighbors to a cell, so in a two-dimensional
square array the Moore neighborhoods contain eight cells (Weisstein, 2016b). In Morphozoic, a single
higher-level cell houses an entire lower-level Moore neighborhood, down to single cells, and a set of
lower-level neighborhoods compose a higher-level neighborhood (Gordon & Rangayyan, 1984a, b).
While this serves as a constraint on cell-cell communication, it also serves as top-down information. This
top-down information, when coupled with local, bottom-up information at different spatial scales,
provides us with a mechanism for strongly emergent phenomena (Holland, 1992).
We originally called such nested neighborhoods “adaptive neighborhoods” in the context of image
processing, because around each pixel a Moore neighborhood size was chosen that best fit the local
features of the picture (Gordon & Rangayyan, 1984a) (Gordon & Rangayyan, 1984b). The field of
adaptive neighborhood image processing now has over 400 publications, and an independent discovery
of the idea has extended the idea in many additional directions (Katkovnik, Egiazarian & Astola, 2006). In
the field of cellular automata, adaptive neighborhoods have been used in the sense of changing
neighborhood type rather than size (Mofrad et al., 2015). Our approach of nested neighborhoods has
been combined with cellular automata for edge detection (Liu et al., 2012) and rule
identification (Adamatzky, 1997; Sun, Rosin & Martin, 2011; Zhao, Wei & Billings, 2012). Irregular (Batty,
2003) and “extended” (Guan & Clarke, 2010) Moore neighborhoods have also been used for geographic
cellular automata, though not with the nesting idea in mind. Irregular, grown adaptive neighborhoods
have been called coalitions in cellular automata (Burguillo, 2013). Morphozoic appears to be unique in
using nested neighborhoods, not to find an optimal neighborhood of a cell, but to provide information
at many scales to that cell. It thus permits the study of local/global interactions.
Computer modeling of biological systems is widespread (Wyczalkowski et al., 2012; Tanaka, 2015). The
Morphozoic approach is based on the Cellular Automaton (CA) architecture which exhibits
computational universality (Dobrescu & Purcarea, 2011) (Wolfram, 2001) that is not well understood in
the context of biological development. The aim of the Morphozoic project is to build an abstract model
of morphogenetic fields to explore its computational capabilities. While the model may lead to some
insights into biology, this is not a central goal of the project. Simulations presented here suggest that
the model can be used to produce general self-organizing structures. In particular, many aspects of
modern human life involve local/global interactions, so Morphozoic may contribute to the social
sciences (Batten, 2001). We also show how Morphozoic may be used to reverse engineer a sequence of
state changes of a system and derive an approximation to the rules governing that system. Morphozoic
may therefore be used for reverse engineering (Gordon & Melvin, 2003; Deutsch, Maini & Dormann,
2007; Elmenreich & Fehervari, 2011; Lobo & Levin, 2015).
Because of its local/global construction, Morphozoic may be a step towards meeting the challenge
posed by Russ Abbott:
...when a glider appears in the Game of Life, it has no effect on the how the system behaves. The agents don’t see a glider
coming and duck. More significantly we don’t know how to build systems so that agents will be able to notice gliders and duck.
It would be an extraordinary achievement in artificial intelligence to build a modeling system that could notice emergent
phenomena and see how they could be exploited. Yet we as human beings do this all the time” (Abbott, 2006).
The Morphozoic platform can model a wide range of natural and artificial phenomena, but the question
remains whether or not the software can exhibit biological realism. Certainly in terms of approximating
morphogenesis, it is not clear whether patterns formed and identified by Morphozoic are produced by
biologically-realistic mechanisms. However, in the realm of biological realism, Morphozoic is consistent
with similar approximations of biological complexity. Particularly at the level of interacting cells,
morphogenesis and Morphozoic alike exhibit what Abbott (Abbott, 2006) calls ‘epiphenomena’. These
epiphenomena, or emergent outcomes of collective interactions between agents, are as biologically
realistic at the macro-scale as gene action is at the micro-scale. However, as morphogenesis is non-
reductive (Abbott, 2006; Gordon & Gordon, 2016a), it becomes difficult to make exact predictions of
behavior in existing biological systems. What makes for an excellent naturalistic pattern replication
mechanism (Wolfram, 1984) might make for a poor descriptor of unfolding processes in the frog
Biological realism in modeling involves the degree to which selectively adding in features of the system
you are attempting to model produces a useful representation. At a macro-level of description,
Morphozoic exhibits a priori biological realism (Bourgine & Lesne, 2010) that captures the higher-order
dynamics of a biological system rather than the lower-level causal mechanisms of complexity (Ruxton &
Saravia, 1998). A priori realism incorporates known features in a minimal and abstract fashion (Bourgine
& Lesne, 2010). In cases where the underlying system has many moving parts and layers of complexity, a
high degree of abstraction is required to avoid transcomputational limits (Bremermann, 1967; Ashby,
1968; Gordon, 1970; Bower, 2005). Transcomputational limits are of particular concern in biological
systems, where the myriad sources of variation can produce a very high dimensional problem space.
While abstraction in the face of transcomputational limits is a supposed requirement for biologically-
inspired simulations, nevertheless many models of collective behavior (Resnick, 1994) and
evolution (Adami, 1998; Alicea & Gordon, 2014) also utilize highly abstract representation while
producing realistic biological system dynamics. This is one reason why Morphozoic can reproduce
patterns seen in morphogenesis without invoking mechanisms of gene expression.
Another reason why Morphozoic can exhibit “lifelike” patterns is due to the nature of Cellular Automata.
In many cases what drives changes in the dynamics of the model are not lower-level control
mechanisms, but the order in which key spatial and temporal events play out. While this is important for
real biological systems as well, changes to spatial and temporal order of how events are executed lead
to synergistic effects in Cellular Automata dynamics (Ruxton, 1996). Furthermore, the timing of events in
a Cellular Automata model impact predictive ability, as Huberman and Glance (Huberman & Glance,
1993) have shown by implementing asynchronous behaviors in the Prisoner's Dilemma game. In
Morphozoic, we keep the standard features of cellular automata: discrete time steps, each cell changes
state simultaneously with all the others, and no cell moves from its initial position. These three aspects
are clearly not biological. Lifting these constraints is a topic for future research. Morphozoic provides a
rich starting point.
There are also parallels between biological and sociotechnical systems that demonstrate how adaptive
behavior might be as much a consequence of temporal evolution than formal biological mechanisms.
Sometimes this temporal evolution is intertwined with structural features of the system, such as when
scientists can “unboil” an egg and return proteins to their original conformational state (Bijelic et al.,
2015). The origin of evolutionary novelties also relies upon how temporal sequences interact with
existing variation to produce innovations. This, evolutionary novelties are generated from what is
already available through recombination and repurposing (Jacob, 1977). Assembling what already exists
into new combinations is called combinatorial innovation, part of something Wagner and
Rosen (Wagner & Rosen, 2014) refer to as innovability. Acting as a mechanism for new phenotypic
possibilities, characterizing the innovability of emergent systems like Morphozoic might provide a very
broad window into the systems-level mechanisms of developmental processes.
Innovability is a key driver in the functional diversity seen in Morphozoic, and can be analogous to the
types of developmental plasticity observed in biological systems (West-Eberhard, 2002). In phenomena
ranging from axonogenesis to generalized stress response (Bateson et al., 2004; Gluckman, Hanson &
Low, 2011; Low, Gluckman & Hanson, 2012), organisms can exhibit a morphogenetic adaptive response
to both local and global signals. In this sense, Morphozoic provides a means to explore how the timing
and phenotypic consequences of these processes might unfold (Moczek et al., 2011). While Morphozoic
does not possess the same mechanisms that drive biological plasticity, phenotypic changes can be
approximated using discrete computation. While Morphozoic may be able to approximate the pattern-
formation aspects of developmental plasticity, a concept known as the adjacent possible may be used to
bridge the gap between biology and simulation. The adjacent possible was originally proposed by Stuart
Kauffman as a way to describe possible new states for a system given historical contingency (Johnson,
2010). The adjacent possible is ever-expanding, and this non-random expansion into possibility space is
what drives subsequent innovations (Tria et al., 2014).
Fortunately, the cellular automata representation is well-suited to producing novelties that result from
interactivity. Identifying the specific mechanisms for the generation of novelties is a relatively
unexplored question (Kier, Bonchev & Buck, 2005). How can we discover these pathways and make
parallels with developmental and other biological systems? On the one hand, the adjacent possible can
be made salient using a method called fitness optimization. This allows us to understand patterns of
innovation that conform to the adjacent possible idea as a series of moves towards a fitness peak (or
functional optimum). On the other hand, many biological and technological systems also include
significant constraints on their evolution such that the states making up the so-called adjacent possible
are limited to relatively small portions of the whole system (Solé et al., 2013). One way to investigate
these non-innovable portions of the system is to construct a neutral network (van Nimwegen,
Crutchfield & Huynen, 1999) that defines all possible configurations of the system. To approximate the
full set of possible states (which may or may not be transcomputational), we may rely upon
computational information such as the neutral network, cellular automata rules, and knowledge about
the constraints a given system might pose. Applying the concept of the adjacent possible then allows us
to limit the range of plausible emergent mechanisms in biological and non-biological settings.
Coupling Morphozoic with other computational and data-driven models of embryogenesis may lead us
to a middle-out approach (Noble, 2002). The middle-out approach can be defined as a combination of
top-down and bottom-up approaches where specification begins at the level where the data are
sufficient. Using inferential and other methods, we can then move towards other levels of analysis. This
is ideal for systems that are too interactive for systems or reductionist approaches to be effective on
their own (Noble, 2002). The middle-out approach is particularly useful to data-driven research in that it
strikes a balance between reductionism and integration. As the reductionists have mechanisms, the
systems people have overarching descriptions (Kohl & Noble, 2009). Morphozoic occupies a middle-
ground between reductionism and integration in the sense that the states of single cells are influenced
by local neighborhoods, but that each local neighborhood is interlinked to form a potentially large
(global) problem space. This interlinking occurs at many levels, due to the use of nested neighborhoods.
By occupying this middle ground, defined by neighborhoods, we can approach multiscalar biological
processes in novel ways (Walker & Southgate, 2009). Multiscalarity can reach 8 orders of magnitude, as
in diatoms, for instance (Ghobara et al., 2016).
One of the inspirations for Morphozoic is the Morphone model (Portegys, 2002), a programmable,
modular signaling system for morphing complex patterns. One of the ways Morphozoic differs from
Morphone is in its leveraging of local vs. global signaling fields as computing mechanisms. Another
feature of Morphozoic, typical of computing systems but distinct from biological systems, is the use of
cell states as signals to other cells instead of supporting separate signaling objects.
Cellular automaton
Morphozoic is built upon a two-dimensional (2D) cellular automaton (CA) architecture. CAs generally
have these properties:
A cell has a state value (e.g. on or off). However, the value can be a more complex entity, such as a
real value or vector.
A cell senses the states of adjacent cells that makes up its neighborhood. One example is a 3x3
Moore neighborhood.
Action rules determine how a cell state changes based on the states of the cells in its neighborhood.
CAs date back to the 1940s and the work of Stanislaw Ulam, who, requiring a model to study the growth
of crystals, developed a simple lattice network (Ulam, 1962). At the same time, John von Neumann,
Ulam's colleague, was working on the problem of self-replicating systems (von Neumann & Burks, 1966).
Von Neumann's initial design was to have robots build new robots out of a “sea of parts”. This proved
problematic, and a more abstract and discrete model was later developed that became a foundation for
the CA approach. It was Ulam who suggested using a discrete system for creating a reductionist model
of self-replication. Von Neumann’s work in self-replication system is similar to what is probably the most
famous cellular automaton: the “Game of Life,” (Gardner, 1970) which is presented as an application of
Morphozoic in a later section. We introduced long-range interactions in cellular automata in a simple
model of two dimensional “snail” morphogenesis, showing they led to more robust pattern
formation (Gordon, 1966). Long-range effects are also incorporated into human evacuation CA
models (Kaji & Inohara, 2014).
In 2001, Stephen Wolfram’s book A New Kind of Science was published (Wolfram, 2001). The book
discusses how CAs are relevant to the study of biology, chemistry, physics, and all branches of science. In
addition, CAs are relatively easy to create in software and are straightforward in performance
evaluation. For these reasons the CA architecture was chosen as a platform for Morphozoic. As
demonstrated in Wolfram’s work, CAs can serve as a model of parallel distributed processing that fit
many natural systems well. Some of these properties of the physical world:
Contain a large number of simple parts, called cells, each of which is an automaton. This means
it acts on its own (autonomously).
Parallel operation.
Global, emergent effects from local interactions.
Each cell (automaton) can respond to external signals in “deciding” what to do. Those decisions are
restricted to certain choices, either finite in number or represented by continuous values. For example,
CAs can simulate fluid or gas dynamics by storing individual molecules in the cells and implementing
particle interactions by the local rules.
One of the main variations of CAs are in the number of dimensions they contain. A one-dimensional (1D)
CA is simply a row of cells. A two-dimensional (2D) CA is a grid of cells. Three-dimensional (3D) CAs
operate within a volume of discrete cells. The concept of a neighborhood is central to the description of
a CA, as the states of a cell’s neighborhood define the input to the local state transition rules. In a one-
dimensional CA, a cell’s neighborhood is usually its adjacent cells. In a 2D CA, typical neighborhood
configurations are shown in Figure 1. It is also common, as is done for Morphozoic, to include the center
cell in the definition of its neighborhood.
Figure 1 Typical two-dimensional CA neighborhoods. Adapted from Figure 2, Espinola, 2010.
CA cells contain a state that can take on values defined by the automaton. For example, a binary state
can be considered to be in a 0 (off)/1 (on) state, as shown in Figure 2.
Figure 2 A 2-dimensional grid of cells, each in a discrete state of “0” or “1”. Moore neighborhood of
cells shown inside red bounding box. Automaton for Moore neighborhood is denoted with a red circle.
State values change based on transition rules. A good way to understand how state transition rules work
is to consider a one-dimensional CA with adjacent cell neighborhood, as shown in Figure 3. Using
Wolfram’s terminology, the eight possible neighborhood state configurations and center cell state
transitions can be enumerated. So this rule set is number 30.
Figure 3 State transition rule set 30, where 30 base 10 = 00011110 base 2. From (Weisstein, 2016a)
with permission per Eric Wolfram's Notice of Copyright
Figure 4 demonstrates how a CA develops over time from a single “on” cell using rule 30. Time advances
from top to bottom.
Figure 4 Application of rule 30. A single cell is turned on (top row) and in the next time step (second
row) all of the cells are subjected to rule 30. Two of them change state as a result, leaving three cells
turned on in the second row. This process is iterated to generate the third row, etc. Patterns like these
are actually generated by the pigment depositing line of cells in marine cone snails (Waddington &
Cowe, 1969). The discrete version of such one dimensionally generated patterns became known as
Lindenmeyer patterns (de Koster & Lindenmayer, 1987). From (Weisstein, 2016a) with permission per
Eric Wolfram's Notice of Copyright
There are many other possible rules to compute a cell’s state from a group of cells. Consider blurring an
image. A pixel’s new state (i.e., its color) is the average of all of its neighbors’ colors. Most image
processing algorithms can be formulated as CAs. The rules define the functionality of the CA.
Variations of CAs
The following are some variations of the CA model:
1. Non-rectangular grids. There is no essential reason why a CA must be confined to a rectangular
grid or space.
2. Probabilistic. The rules of a CA need not necessarily work in a deterministic fashion (Gordon,
1966, 1980). An example is the Stochastic Game of Life (Monetti & Albano, 1997).
3. Continuous. The state of a cell can be something other than a discrete value, such as 0 or 1. For
example, the values could range between 0 and 1. Of course the rules must then reflect how to
calculate these continuous state values. Examples in materials science, traffic flow and
earthquake analysis abound (Olami, Feder & Christensen, 1992; Bubak & Czerwiński, 1999;
Kitakawa, 2004, 2005; Gosálvez et al., 2009; Ferrando et al., 2011; Ferrando, Gosálvez & Colóm,
2012; Li et al., 2015).
4. Historical. In the Game of Life CA, the current state configuration determines the next
configuration. However, taking into account historical states is also possible. For example,
Portegys and Wiles (Portegys & Wiles, 2004) describe a CA that self-repairs in the presence of
noise by use of historical cell states.
5. Moving cells. In these examples, cells have a fixed position on a grid, but can move to other grid
points (or, equivalently, their states can be transferred to other cells or switched with
them) (Gordon et al., 1972, 1975; Chopard, 1990; Fukui & Ishibashi, 1996; Hochberger,
Hoffmann & Waldschmidt, 1999; Halbach & Hoffmann, 2005; Moussa, 2005).
6. Nesting. Another feature of complex systems is that they can be nested into hierarchies. For
example, a city is a complex system of people, a person is a complex system of organs, an organ
is a complex system of cells, etc. Cell values that reflect this hierarchical arrangement are
possible (Weimar, 2001; Dunn & Majer, 2007; Kiester & Sahr, 2008; Dunn, 2010).
From a more speculative viewpoint (Ilachinski, 2001), researchers have raised the question of whether
the universe is a cellular automaton. For example, mathematical models have shown the emergence of
"particles" within CAs, such as the gliders in the Game of Life. This leads to conjectures that the natural
world, which is well described by physics with particle-like objects, could actually be a CA. This
hypothesis has led scholars to a perspective of nature existing within a discrete framework. Edward
Fredkin (Fredkin, 1992), a strong proponent, has proposed the "finite nature hypothesis", i.e., the idea
that "ultimately every quantity of physics, including space and time, will turn out to be discrete and
A main objective is to devise an abstraction that models morphogenesis in a CA using a nested
neighborhood approach. As a type of multilayered scheme (Bandini & Mauri, 1999; Dascalu et al., 2011),
nested neighborhoods, defined more formally below, are simply neighborhoods contained within
neighborhoods, much like Russian matryoshka dolls. Nested neighborhoods provide a straightforward
representation of a morphogenetic field that contains a hierarchy of local vs. global information.
Information about a more local cell neighborhood having fewer cells is more precise and finer grained
than information about a larger, more global neighborhood.
The scheme must be computationally plausible. Neighborhoods of increasing size contain increasing
numbers of cells. In order to constrain the potential information explosion, a specific number of bits are
used to represent each neighborhood, regardless of its size. The smallest neighborhood is represented
precisely; larger neighborhoods are increasingly “fuzzy” because they cannot be completely represented
by the available bits. This forms a precision gradient that decreases as the neighborhood grows in size.
An intended result of this plan is that more distant cells are sensed in aggregation, as described in the
next section.
Additional objectives:
Compact state change rules.
Noise tolerant.
Generalizable from exemplars.
Morphogenetic field specification
The morphogenetic field is specified in a CA by equipping cells with these properties:
A cell state is its type.
A cell emits, senses, and reacts to signals.
Signals carry information about the types of neighborhood cells.
A field is the confluence of signals sensed by each cell.
Rules are embodied in metamorphs, which encapsulate pattern-matching morphogens and cell state
change actions.
A “morphogen” abstracts many types of morphogenesis mechanisms: chemical (the classical definition),
physical, energy, etc. It also summarizes a morphogenetic field as a set of a cell’s nested neighborhoods
and their contents. This is shown in Figure 5. A neighborhood consists of an NxN set of sectors
surrounding a lower level neighborhood:
neighborhoodi =NxN(neighborhoodi-1)
where N is a fixed odd positive number chosen by the user of Morphozoic, and neighborhood0 is
composed of NxN elementary cell sectors.
Hence the number of cells in neighborhoodi = NixNi= N2i.
A morphogen is composed of a set of nested neighborhoods:
morphogen(cell) = { neighborhood0(cell), neighborhood1(neighborhood0), ...
neighborhoodn(neighborhoodn-1) }
The value of a sector is a vector representing a histogram of the cell type densities contained within it:
value(sector) = [ density(cell-type0), density(cell-type1), … density(cell-typen) ]
The number of cells contributing to the density histogram of a sector of neighborhoodi = Ni-1x Ni-1
Figure 5 Morphogen nested neighborhoods.
A metamorph represents a cellular automaton morphogen→action agent, defined as a mapping from a
morphogen to a cell type.
A set of metamorphs describing a pattern of cell activity can be generated from manual input or a
programmed sequence of cellular automaton transitions. For example, the Game of Life application uses
the programmed Game of Life rules to process the cell states. As the CA changes, the neighborhoods for
each cell are used to construct morphogens, and the cell type transitions associated with the
morphogens are actions. The morphogens and actions are composed into metamorphs.
Once generated, the metamorphs can be independently used to “execute” the application. Metamorph
execution consists of creating a morphogen for each cell in the grid and comparing each of these
morphogens to the stored set of morphogens contained in the generated metamorphs, where the
distance between them is given by:
 
     
 
The metamorph having the least morphogen distance is chosen as the cell action.
A unique feature afforded by the use of a distance metric to match morphogens is a noise-tolerant, self-
healing capability. Metamorphs act on cells according to neighborhood similarity, which steers cell
states toward patterns stored in the metamorphs. This feature is a hallmark of biological systems, and is
quite distinct from typical rigid CA rule formulations, such as the Game of Life (see applications), where
a minor introduction of noise often results in global disruptions.
Artificial neural network implementation
A compact, fast, and noise tolerant representation of metamorphs can be implemented by an artificial
neural network (ANN), a biologically-based learning machine that is particularly adept at classifying
input patterns into output categories (Haykin, 2011).
ANN background
ANNs are loosely modeled after the neuronal structure of biological nervous systems but on smaller
scales. A large ANN might have thousands of processor units and interconnections, whereas a human
brain, for example, can have 86 billion neurons. An African elephant brain contains 267 billion
neurons (Wikipedia, 2016b), and human cerebral cortex contains 150 trillion synaptic
interconnections (Drachman, 2005). ANN architectures often functionally diverge from their biological
counterparts in important ways, including how learning is implemented.
An ANN is a subset of a general computing model known as connectionism. A connectionistic model
features an interconnected network of simple units that produce emergent properties that are beyond
the capabilities of the individual units (Medler, 1998). In the emergent respect, that the proverbial
whole is greater than the sum of its parts, an ANN is similar to a cellular automaton.
A prevalent type of ANN is the multilayer perceptron (MLP). MLPs are organized in layers, shown in
Figure 6. Layers are made up of a number of interconnected neurons. Patterns are presented to the
network via the input layer, which connects to one or more hidden layers where the actual processing is
done via a system of weights associated with the connections. The hidden layers then connect to an
output layer where the output is represented by the activation of one or more neurons.
Figure 6 Multilayer perceptron. From (Cazala, 2015) with permission of Juan Cazala.
A weight value is associated with each connection in the network. The weights are multiplied by the
outputs of the source neurons, the sum of which is input to an activation function, which computes a
neuron’s output.
Activation functions are typically sigmoid shaped, such as the logistic function shown in Figure 7. Here
the output switches smoothly from 0 (off) to 1 (on) over an interval controlled by the β parameter. An
important property of an activation function is that is it differentiable, which allows a network to be
trained through learning.
Figure 7 Logistic activation function. From (Sayed, 2016) with kind permission of Saed Sayed.
Learning is the process of modifying the connection weights to produce outputs that differ least from
“correct” outputs, i.e. minimized error. When the correct outputs are known, this type of learning is
called supervised learning. Learning typically entails backpropagating the output error from the output
to input neurons to modify the weights of the connections such that the error is reduced in subsequent
computations. The most common modification algorithm for this is the delta rule.
Training is a process that involves repetitive runs of input-output patterns through the network with an
application of the learning rule performed for each pattern. A run through an entire training set is called
an epoch. ANNs, like their biological counterparts, are known for their ability to produce correct outputs
given noisy or similar inputs. After training, a separate test set of patterns with input variations can be
evaluated to assess the effectiveness of training.
There are a number of ANN variations. For example, the activation function can be a Gaussian function
instead of a sigmoid one. ANNs with many layers are also possible. In general these are known as deep
learning networks. In a convolutional network, an input layer neuron feeds only a subset of neurons in
the next layer. This resembles the architecture of the human retina. An important ANN variation,
capable of learning input-output sequences such as those found in speech patterns, are called recurrent
By squashing the morphogen sector values into an input vector and considering the cell types as a set of
outputs, an ANN can be trained to learn a set of metamorphs such that a morphogen derived from a cell
in a CA will map to the metamorph closest matching it. This is shown in Figure 8.
The advantages of using an ANN are threefold:
1. Speed: Instead of searching a set of metamorphs during execution for the closest morphogen,
an ANN performs a cascading set of arithmetic calculations to arrive at an output.
2. Compact representation: An ANN is capable of retaining a large number of input-output
mappings in the form of interconnection weights.
3. Generalization capability: An ANN is capable of classifying inputs that are similar to training
inputs. This capability will be exploited in subsequent applications.
Figure 8 Neural network implementation.
Various features of Morphozoic are illustrated by the following applications.
Conway’s Game of Life
This well-known CA (Gardner, 1970) was chosen as a baseline capability test for Morphozoic. In the
Game of Life (GoL), cells are in a rectangular array on a “game board”. Each cell is either in an “alive” or
“dead” state. The state change rules are as follows:
1. Any live cell with fewer than two live neighbors dies, as if caused by under-population.
2. Any live cell with two or three live neighbors lives on to the next generation.
3. Any live cell with more than three live neighbors dies, as if by overcrowding.
4. Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction.
A GoL “game” starts with an initial configuration of alive and dead cells. The rules are then applied to
the cells at each time step. Usually, if the initial configuration is random, the pattern of live and dead
cells appears to change chaotically for a while, then resolves itself into isolated clusters of cells that
cycle through a repeating series of states.
Figure 9 A Game of Life configuration.
A sample GoL configuration is shown in Figure 9. Despite its simple rules, the GoL produces many
dynamic self-sustaining patterns (and is capable of producing an unlimited number of such patterns).
One class of pattern, called “gliders”, move across the game board as they go through their state
changes and interact in various ways with other clusters when collisions occur (Rendell, 2002). More
complex configurations called “guns” are a special type of self-sustaining pattern that produce gliders at
a specific rate. Using gadgets such as these it has been shown that a single-tape Turing machine can be
simulated by GoL (Rendell, 2002). In complexity theory, a set of rules that manipulates the state of a
system is described as Turing complete if it can simulate any single-tape Turing Machine. To
demonstrate the Turing completeness of another computational system all one needs to do is show that
it is capable of simulating an existing Turing complete system.
Examples of Turing complete systems include most commonly used programming languages, which span
paradigms such as procedural (C), functional (Haskell), and object-oriented (Java). A language’s Turing
completeness is determined by whether it has the ability to branch conditionally and load/store an
arbitrary number of variables. These two features, inherent to general purpose languages, can be
implemented in other paradigms using analogous structures such as using recursion in Haskell to
implement repetition. Turing completeness research has resulted in the discovery of more and more
simple systems that hold these properties. In fact, GoL is not the only cellular automata that has been
proven to be Turing complete nor is it the simplest.
The elementary cellular automaton Rule 110, whose state transition rules are stated in Table 1, has also
been shown to be Turing complete (Cook, 2004). Unlike GoL, Rule 110 functions in only one dimension,
each cell’s next state results only from the current states of its two neighbors.
B0 B1 B2
Table 1. State transition from B1 to B1 given all possible starting states.
While proving that simple cellular automatons such as GoL and Rule 110 are Turing complete does not
allow us to compute in novel or more efficient ways, it does allow us to demonstrate the Turing
completeness of other systems such as Morphozoic. Proving that Morphozoic is Turing complete is
significant because “Turing completeness” means that it is a universal computer. Given enough memory,
Morphozoic would be able to perform any computation that any other Turing complete system is
capable of. This says nothing about how efficiently the computation will be done; it only says that the
computation can be done.
For Morphozoic, we “reverse engineer” a complete set of GoL rules, consisting of 512 3x3 Moore
neighborhood configurations. Generating a metamorph for each rule, the configurations were correctly
processed, both in the lookup and ANN implementations. Therefore, it follows that Morphozoic is also
Turing complete. From the perspective of modeling complex systems like embryos, this means that
Morphozoic is capable of modeling anything that can be computationally modeled.
Cell regeneration
The Morphozoic algorithm is capable of modeling cell regeneration. In order to demonstrate this, an
apparatus was devised that also highlights the functionality of nested neighborhoods. Figures 10 and 11
show the apparatus. The automaton is trained to regenerate either the horizontal bar on the right side
of Figure 10 or the vertical bar on the right side of Figure 11 beginning with the central block
configurations on the left side of the figures, respectively. The shaded borders are the distinguishing
features that determine regeneration direction: vertically shaded borders train for a horizontal bar and
horizontally shaded borders train for a vertical bar.
Figure 10 - Horizontal bar. Left: begin. Right: goal.
Figure 11 - Vertical bar. Left: begin. Right: goal.
To spotlight the results, only cells in the central 9x9 area in are allowed to create metamorphs during
training, and thus cells only in this area are allowed to regenerate by modifying their type values. Two
settings of the neighborhood dimension and number of neighborhoods were defined. The unnested
setting was a single 9x9 neighborhood. The nested setting was three nested 3x3 neighborhoods. The
unnested and nested automata thus equally contained 81 variable values. However, the nested
automaton neighborhoods span an area of 27 cells while the unnested automaton spans only 9 cells.
The nested automaton affords the extended range by aggregating cell values.
Figure 12 nested morphogen fields: left=3x3, center=9x9, right=27x27
The automata were trained and tested for a variety of border lengths, which are manipulated by
changing the dimensions of the grid: the larger the grid, the longer its borders. And as the border
lengths increase, the distances from the central block of regenerating cells to the borders increase,
eventually passing out of range of the morphogenetic fields associated with the metamorphs. This can
be seen in Figure 12, which from left to right shows the morphogenetic field sizes for the 3x3, 9x9, and
27x27 nested neighborhoods. It can be seen that only the 27x27 neighborhood intersects with the left-
right border, allowing morphogens associated with the central cells to generate the correctly aligned
bar. Morphogens associated with the 3x3 and 9x9 neighborhoods do not intersect the border cells, and
thus cannot determine the correct orientation.
For each border length setting, the automata were sequentially exposed to the correctly oriented bars
for the vertical and horizontal borders, causing the creation of metamorphs that generate appropriate
cell types. A test is correspondingly in two parts: the central block with vertical borders followed by the
central block with horizontal borders. A successful test is defined as the regeneration of the correct bars
for both parts.
The results are shown in Table 2 for border length increments of four cells. Both the unnested and
nested automata correctly process the first two border lengths. However, at the third length, the
unnested automaton cannot detect the borders and thus fails to regenerate both bars correctly.
Eventually the border recedes beyond the range of the nested automaton as well, indicated in the last
row of the table.
Table 2 - Cell regeneration results.
The cell regeneration presented here can also be understood as a type of tropism, wherein a directional
movement or growth takes place in response to an environmental stimulus. Phototropism, for example,
is a response to light (Goyal, Szarzynska & Fankhauser, 2013). Another biological counterpart involving
cell (re)generation stimulated by fields of chemical signals is paracrine signaling for wound
healing (Hocking & Gibran, 2010; Dittmer & Leyh, 2014). Stem cells migrating to wound sites release
bioactive factors that orchestrate wound healing.
When tissues are wounded, they are healed via a regenerative process called epithelialization (Hocking
& Gibran, 2010). During this process, it has been found that stem cells, particularly mesenchymal stem
cells (MSCs) aid in the speed of wound healing by releasing growth factors and other extracellular
factors that facilitate regeneration (Hocking & Gibran, 2010). In general, stem cells are functionally
plastic, and can differentiate into a variety of functional cell lineages depending on where they are
recruited and the functional context. During organ regeneration, bone marrow stem cells migrate to
sites that are damaged to provide various roles in the regeneration process (Morigi et al., 2004). It is this
functional context where the parallels with Morphozoic's generative process can be found. For example,
MSCs often act through paracrine signaling rather than direct replacement of the cells making up
damaged tissue (Bruno et al., 2013) .
Stem cells also have the capacity to self-renew, which allow them to proliferate and thus persist for
many more divisions than differentiated cells. Perhaps more importantly, stem cells can manipulate
their local environments to favor regeneration (Dittmer & Leyh, 2014). This is done by acquiring the
gene expression patterns of their new defined fates, as well as producing a host of secretion factors. It is
this secretome (Salgado & Gimble, 2013) that most strongly influence the extracellular environment,
and can act to coordinate the behavior of cells and tissues at multiple scales. Influence of the local
environment is also accomplished via paracrine signaling (Baraniak & McDevitt, 2010), which involves
short-range chemical signaling of growth factors between cells. In a very simple manner, Morphozoic
can mimic these signaling mechanisms, and the architecture of Morphozoic could be used to implement
these mechanisms at multiple spatial scales.
Cell regeneration is concerned with the restoral of missing information using nearby available
information. A related process, digital image inpainting (DII) is a computer algorithm that restores
missing information of images such as those of old oil paintings. A biological counterpart occurs in
human visual systems as well in the form of blind spots (Satoh, 2012). As an illustration of the
Morphozoic cell regeneration capability applied to image restoral, consider the problem of restoring an
image from its edges, as shown in Figure 13.
Figure 13 Lenna image with edges (Wikipedia, 2016a).
This is done by casting the original image and edge-detected image into a CA grid. The edge cells, shown
in the upper left part of Figure 14 (in reverse color for contrast), form an outline of the source pattern.
The original image, shown in the bottom right of Figure 14, forms the target pattern. Only the dark edge
cells are allowed to morph into target cells. With this restriction, five morphing steps are necessary to
complete the transition.
Figure 14 Step-by-step image restoral from edges.
Gastrulation evolution
There have been a number of attempts to model gastrulation and related developmental processes.
Developmental phenomena such as cleavage, blastulation, and gastrulation have been modeled using a
physical cellular model that indirectly assumes how genetic mechanisms effect the timing of these
processes (Drasdo and Forgacs, 2000). A more direct technique would involve the use of Genetic
Algorithms (GAs). GAs are a type of evolutionary computation that focuses on the construction and
optimization of programs using natural selection. In this case, natural selection act to select programs
that meet or exceed the criteria to reproduce and/or maintained in a population of programs. GAs have
been used in conjunction with Young's cellular automata to approximate the reaction-diffusion driven
biological pattern formation and replicate morphogenetic fields (Gravan and Lohoz-Beltra, 2004).
Because they are an instance of adaptive computing, there is the potential for broader application of
GAs to problems of biological development As an optimization procedure, GAs provide an adaptive
method that can find solutions despite a rugged search landscape (Bornholdt, 1998). This is analogous to
how biological populations evolve solutions to adaptive problems in a de novo fashion. We can
successfully approximate complex, emergent phenomena in development specifically and biology more
generally by more closely examining the relationship between computational representation and
biological complexity.
A GA consists of three parts: a genetic representation, a set of operators for mutation and/or crossover
operator, and a selection criterion (Holland, 1992). The algorithmic representation (genome) often
consists of one or more chromosomes, while each chromosome consists of several genes. A genetic
representation is a computational abstraction of a genome, each gene representing a compact encoding
of some aspect or feature in a given system. Each gene consists of serial bits and acts as a bit register,
which can be used to encode a wide variety of problems. This allows us to capture the power of
population dynamics and heredity rather than expression of genes, although specialized genetic
algorithms (Ferreira, 2001) can incorporate gene expression. In general, using genetic algorithms as a
form of theoretical inquiry allows us to replicate the dynamics of a given biological system, which in turn
allows us to better understand the complexity of the underlying system (Steventon and Arias, 2016).
The essential component of a genetic algorithm is the mutation and/or crossover operations. While
mutation consists of bit flipping, recombination consists of exchanging entire blocks of bits either within
or between programs. In both cases, an operator introduces variation in the function of a program. This
variation is then selected upon based on fitness criterion, but is also subject to historical constrains of
the problem space. This allows for the self-organization of patterns to emerge (Nizam and
Shanmugham, 2013) without breaking the code or destroying the structure of the initial program. One
way in which GAs avoid code fragility is to start with a population of programs that exhibit differential
reproduction. In this way, a genetic algorithm selects from a variety of potential solutions. A related
issue is the maintenance of structure and complexity in a naturally-selected program space is the
existence of building blocks (Forrest and Mitchell, 2014). This is similar to modularity in biological
evolution, in which parts of the program are protected from future evolutionary change (Wagner and
Altenberg, 1996).
All of these issues figure prominently into the approximation of gastrulation as a developmental
process. GAs are particularly good at finding heuristic solutions to problems residing in a large possibility
space, so they are likewise suited to simulating instances of biological self-organization where an exact
solution is not required. This is particularly true of simulating large problem spaces for where biological
experimentation would yield no clear solution. The problem of emergence in biological self-organization
can be viewed in terms of computational complexity (Grover, 2011). In gastrulation, we can see the
usefulness of both properties, in addition to the usefulness of incorporating gene expression into a
multi-scale model (Kaandorp, 2012). Biological self-organization also relies on rules and constraints
rather than explicit instructions.
Gastrulation is a process by which the cells of an embryo form an invagination in an early spheroid
shape in the course of tissue differentiation. A simple Morphozoic model of this is shown in Figure 15.
This configuration was formed through a progression of cell “divisions” starting from a single central cell.
Morphozoic generated metamorphs from a programmatically produced sequence. Two nested 3x3
neighborhoods were required; one neighborhood was found to be insufficient to reproduce the desired
output. Once generated, the metamorphs executed the sequence correctly.
Figure 15 Gastrulation configuration.
A model that purports to explain the workings of morphogenesis at any level of abstraction should be
evolvable. In order to demonstrate the mutability and evolvability of Morphozoic, a genetic algorithm
(GA) was used to evolve a population of “organisms” to gastrulate successfully. The fitness function was
a count of the number of matching cell types summed over the sequence of cell type transitions.
Each population member contained metamorphs that were initially constructed by randomly pairing
morphogens generated from a programmed gastrulation sequence with actions from the sequence. The
number of metamorphs in each member was determined by the number required to execute the
After a subset of the population was selected as fit members, the population was replenished with
mutants and offspring of fit members. Offspring were created by mating randomly chosen parents
members and performing a crossover operation on their metamorph sets. Crossover consisted of
supplying the child with a set of metamorphs randomly selected from its parents. Mutants were created
from individual fit members by discarding all metamorphs that were not executed and replacing them
with random ones.
Additional GA parameters:
Population size = 50
Fit population size = 10
Number of generations = 50
Number of offspring from matings = 20
Figure 16 Gastrulation evolution results.
As Figure 16 indicates, the GA produced the required gastrulation morph sequence. The computation
was done in approximately 12 hours on standard desktop computer.
Neuron pathfinding
Also called axon guidance, neuron pathfinding is a process by which axons are guided by chemical
signals to target neurons, a process essential for the formation of organized neural networks (Tessier-
Lavigne & Goodman, 1996). While cellular automata have been used to simulate similar “branching”
phenomena (Markus, Böhm & Schmick, 1999), the generalization of reverse engineered rules is a novel
One of the benchmarks presented here is that of neuron pathfinding. Morphozoic is able to accomplish
this gradually by exploiting the organizational feedback of cellular growth and spatial gradients
embodied in the metamorphs. In biology, neuron pathfinding is accomplished a bit differently, instead
using chemotropic mechanisms to guide axons to their target. Tropic cues are directional cues initiated
by various stimuli. In the case of chemotropic cues, chemical signals and gradients (e.g. the axonal
growth cone) guide regenerating tissues in the direction of its target (Huber et al., 2003). The
chemotropic mode of action has not only been shown to exist in developmental
morphogenesis (Tessier-Lavigne et al., 1988), but during tissue regeneration as well (Alto et al., 2009).
To simulate this, three types of cells are used: source neuron, target neuron, and axon. These can be
seen in Figure 17. In this application, source neurons are allowed to have multiple axons. Metamorphs
were generated from programmatic sequences of axon growth. For execution, however, the
metamorphs were used to train an ANN (see Figure 8), which classified actions based on pattern-
matching morphogens.
Figure 17 Neuron pathfinding example.
The training set was created by growing axons from source to target neurons. The source and target
neurons were randomly placed along the left and right sides of the grid, respectively. Metamorphs were
then generated from multiple random configurations.
For testing, source and target neurons were randomly placed, morphogens generated from cell
neighborhoods, and axons grown by pattern-matching the learned patterns. The challenge here is that
axons must be grown from source to targets in different positions than those for which training was
done, so training must generalize to handle novel neuron positions. The employment of an ANN as a
classification mechanism was useful to accomplish this.
This procedure was done for a single source neuron, one and two target neurons, and with one, five,
and ten training set exemplars. A successful trial is defined as axons connecting to the target neurons.
Ten trials were run for each parameter configuration.
The results are shown in Figure 18. As might be expected, performance for the single target neuron
exceeded that for two targets, the latter having a greater number of possible configurations. The
number of exemplars also had a significant beneficial effect: one exemplar was insufficient to produce
any successes for the two target neuron case.
Figure 18 Neuron pathfinding results.
Turing’s reaction-diffusion morphogenesis
Alan Turing is credited with pioneering a mathematical formulation of morphogenesis generally known
as a reaction-diffusion system (Turing, 1952). This system consists of a set of dynamically coupled
substances that, depending on parameters, are capable of producing various complex patterns, such as
stripes, spots, and spirals. Figure 19 shows a “cheetah coat” pattern generated from a random initial
pattern of three cell types.
Figure 19 Reaction-diffusion morphogenesis of a “cheetah coat” pattern from random values.
This application is a comparison with this well-known specialized morphogenesis algorithm.
Metamorphs were generated from the Turing reaction-diffusion morphogenesis. These were used to
train an ANN to learn appropriate cell type changes. To test, a freshly randomized pattern was
generated and the metamorph pattern-matching mechanism executed. A typical result is shown in
Figure 20. While not as smooth appearing as the original, the spotted pattern is distinctly visible.
Figure 20 Simulation of reaction-diffusion morphogenesis.
Biology presents us with numerous cases of morphogenetic fields as a morphogenesis mechanism. The
particular mechanisms and the depth of knowledge of these cases varies widely. Morphozoic is an
attempt to implement an abstraction of the functional commonality of morphogenetic fields as a
mechanism for self-organizing computation.
Morphozoic is a novel embodiment of a number of capabilities that are useful for morphogenesis:
flexibility, compact and economical computability, evolvability, and generalizability, especially in
association with the artificial neural network implementation. The cited applications were chosen to
demonstrate these properties.
Future directions
Neighborhood sector value: The assignment of a value to a sector could be any function of the
types of its cellular components: average, mode, winner take all, etc. An alternative is to look at the
change of cell type as an image processing operation, such as taking a Laplacian, Sobel and other
edge enhancements, starbyte transformations (Sivaramakrishna & Gordon, 1997; Sivaramakrishna,
1998), contrast enhancement (Gordon & Rangayyan, 1984b, a), etc.
Field signal strength and cell type variability: If cell types were continuous quantities instead of
discrete, a straightforward proportional mapping of morphogen similarity to cell type value would
be possible. On a related, more subtle possibility, morphogenetic fields are signaling constructs.
Signals can vary in some ways, such as amplitude, while retaining invariant signatures, such as
frequency spectra for electromagnetic waves. This suggests that variable action potentials
proportional to field strength could be a fruitful topic for research. For example, might scaled,
fractal-like structures be morphed?
Dynamic/temporal fields: Limiting morphogenetic fields exclusively to spatial representations
restricts the model to a state machine. Dynamic fields that incorporate temporal information could
also be explored (Portegys & Wiles, 2004; Martínez, Adamatzky & Alonso-Sanz, 2013). This could
take the form of metamorphs that contain past neighborhood patterns or hierarchical streams of
neighborhoods that embody context.
Fluid three-dimensional fields: A major reason that the cellular automaton architecture was chosen
was that it provides a straightforward mapping of local and global morphogenetic fields. However, a
fixed two dimensional grid is not a crucial feature of the Morphozoic model. The fluid three
dimensional medium that biological systems operate in points to an opportunity for the model to
Relevance to artificial intelligence
Morphozoic is also relevant for artificial intelligence (AI). AI research to a great degree focuses on the
brain and behaviors that the brain generates. But the brain, an extremely complex structure resulting
from millions of years of evolution, can be viewed as a solution to problems posed by the environment.
There is a common and somewhat ironic tendency to describe AI inputs and outputs in human cognitive
terms, i.e. post-processed brain output, such as symbolic variables (Hoffman, 2009).
An alternative approach, suggested by morphogenesis, is to view the environment as set of local/global,
spatial/temporal signal fields, and that the processing of fields is what spurs brain development.
Organisms are capable of performing amazing feats, such as navigation and nest-building, by the sensing
of unique environmental signals, such as polarized light, magnetism, and chemical scent trails.
Morphogenesis makes plain that signal fields can have powerful computing capabilities. Perhaps it is
worth exploring artificial intelligence as a solution to environments composed of these fields. As
Edmund Sinnott, author of Plant Morphogenesis (Sinnott, 1960), suggested, morphogenesis may be the
key to understanding intelligence (Sinnott, 1961; Sinnott, 1962b; Sinnott, 1962a; Sinnott, 1966), starting
with plant intelligence (Mancuso & Viola 2015).
In keeping with nature’s penchant for extending rather than replacing, the sponge-like shape of the
mammalian neocortex can be seen as symbolically apropos. For its purpose might be to soak up signals
from far reaches of time and space and render them, as though yet near and present, to the old brain
whose instinctual role has little changed over eons. The environmental gradients that clearly drive the
behavior of simpler creatures then became internalized in the nervous systems of more neurologically
complex ones.
The Morphozoic Java code is available here:
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... In addition, barring few exceptions (e.g. Bandini and Mauri, 1999;Dascalu et al., 2011;Portegys et al., 2016) most rule-based system also account for local interactions only and are thus not suited to study medium and long range interactions. This issue has been addressed in nested rule-based models wherein "nested neighborhoods provide a straightforward representation of a morphogenetic field that contains a hierarchy of local vs. ...
... This issue has been addressed in nested rule-based models wherein "nested neighborhoods provide a straightforward representation of a morphogenetic field that contains a hierarchy of local vs. global information" (Portegys et al., 2016). To the best of our knowledge however, rule-based systems are mostly 2D systems that seem to lack the mechanistic finess to address the simultaneous and reciprocal top-down and bottom-up multi-modal interactions that hold between macroscopic systems (e.g. ...
In order to understand living organisms, considerable experimental efforts and resources have been devoted to correlate genes and their expressions with cell, tissue, organ and whole organisms' phenotypes. This data driven approach to knowledge discovery has led to many breakthrough in our understanding of healthy and diseased states, and is paving the way to improve the diagnosis and treatment of diseases. Complementary to this data-driven approach, computational models of biological systems based on first principles have been developed in order to deepen our understanding of the multi-scale dynamics that drives normal and pathological biological functions. In this paper we describe the biological, physical and mathematical concepts that led to the design of a Computational Morphogenesis (CM) platform baptized Generic Modeling and Simulating Platform (GMSP). Its role is to generate realistic 3D multi-scale biological tissues from virtual stem cells and the intended target applications include in virtuo studies of normal and abnormal tissue (re)generation as well as the development of complex diseases such as carcinogenesis. At all space-scales of interest, biological agents interact with each other via biochemical, bioelectrical, and mechanical fields that operate in concert during embryogenesis, growth and adult life. The spatio-temporal dependencies of these fields can be modeled by physics-based constitutive equations that we propose to examine in relation to the canonical biological events that occur during embryogenesis.
... Introduced with several prototype tasks [30], Morphognosis has also modeled the locomotion and foraging of the C. elegans nematode worm [31] and the nest-building behavior of a pufferfish [32]. Morphognosis is a temporal extension of a spatial model of morphogenesis [29]. ...
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A bstract Honey bees are social insects that forage for flower nectar cooperatively. When an individual forager discovers a flower patch rich in nectar, it returns to the hive and performs a “dance” in the vicinity of other bees that consists of movements communicating the direction and distance to the nectar source. The bees that receive this information then fly to the location of the nectar to retrieve it, thus cooperatively exploiting the environment. This project simulates this behavior in a cellular automaton using the Morphognosis model. The model features hierarchical spatial and temporal contexts that output motor responses from sensory inputs. Given a set of bee foraging and dancing exemplars, and exposing only the external input-output of these behaviors to the Morphognosis learning algorithm, a hive of artificial bees can be generated that forage as their biological counterparts do.
... Developmental Dynamics currently involves using secondary data collected from embryos [52,53] along with bioinformatic and data science techniques to answer questions regarding the process of early embryogenesis and the timing of later morphogenesis. Cybernetics and Digital Morphogenesis has involved using cellular automata [54] or finite-element approaches [55] to model physical interactions during embryogenesis and morphogenesis. DevoWorm has also explored the use of cybernetic models and concepts to better understand the general process of embryogenesis [56]. ...
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The adoption of powerful software tools and computational methods from the software industry by the scientific research community has resulted in a renewed interest in integrative, large-scale biological simulations. These typically involve the development of computational platforms to combine diverse, process-specific models into a coherent whole. The OpenWorm Foundation is an independent research organization working towards an integrative simulation of the nematode Caenorhabditis elegans, with the aim of providing a powerful new tool to understand how the organism's behaviour arises from its fundamental biology. In this perspective, we give an overview of the history and philosophy of OpenWorm, descriptions of the constituent sub-projects and corresponding open-science management practices, and discuss current achievements of the project and future directions. This article is part of a discussion meeting issue ‘Connectome to behaviour: modelling C. elegans at cellular resolution’.
... To abstract this within a cellular automaton, the Morphozoic model (Portegys et al., 2017) has shown that tractable rule sets composed of hierarchically nested cellular neighborhoods are capable of producing both local and global information processing effects that can simulate reaction-diffusion morphogenesis (Turing, 1952), gastrulation, axon pathfinding, and other phenomena. Stephen Wolfram (2002) has demonstrated that simple rule sets can produce dynamic patterns within them, and that this complexity can be abstracted into hierarchical graphs (Wolfram, 2015). ...
The construction of an embryo from a single cell precursor is a highly complex process. Evolutionary emergence of the first embryos is even more complex, and involves both a transition to multicellularity along with the establishment of developmental mechanisms. We propose that embryogenesis relies on a community of cells conforming to a regulatory model of emergent multicellularity. This model draws together multiple threads in the scientific literature, from complexity theory to cybernetics, and from thermodynamic entropy to artificial life. All of these strands come together to inform a model of goal-oriented regulation for emergent structures in early life. This is an important step in the evolution of early life, as well as the emergence of complex life in the earliest habitats. Our model, called the cybernetic embryo, allows for a systems-level view of the embryogenetic process.
... Focusing on the cellular level, we have been able to approximate the shape and formation of complex multicellular structures. For example, we have introduced a hybrid cellular automata-neural network model called Morphozoic [12]. Morphozoic allows for nested two-dimensional neighborhoods (Moore neighborhoods nested inside of Metamorphs) to discover patterns in input data. ...
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Biological development is often described as a dynamic, emergent process. Yet beyond the observation of gene expression in individual cells, it is hard to conceptualize large-scale patterns that confirm this description. We provide an example of combining theoretical insights with a data science approach. The availability of quantitative data allows us to examine aggregate trends across development, from the spatial organization of embryo cells to the temporal trends as they differentiate. The first half of this paper lays out alternatives to the gene-centric view of development: namely, the view that developmental genes and their expression determine the complexity of the developmental phenotype. Caenorhabditis elegans biology provides us with a highly-deterministic developmental cell lineage and clear linkage between zygote and cells of the adult phenotype. These properties allow us to examine time-dependent properties of the embryonic phenotype. We utilize the unique life-history properties of C. elegans to demonstrate how these emergent properties can be linked together by relational processes and data analysis. The second half of this paper focuses on the process of developmental cell terminal differentiation, and how terminally-differentiated cells contribute to structure and function of the adult phenotype. An analysis is conducted for cells that were present during discrete time intervals covering 200 to 400 minutes of embryogenesis, providing us with basic statistics on the tempo of the process in addition to the appearance of specific cell types and their order relative to developmental time. As with ideas presented in the first section, these data may also provide clues as to the timing for the initial onset of stereotyped and autonomic behaviors of the developing animal. Taken together, these overlapping approaches can provide critical links across life-history, anatomy and function.
... To abstract this within a cellular automaton, the Morphozoic model (Portegys et al., 2017) has shown that tractable rule sets composed of hierarchically nested cellular neighborhoods are capable of producing both local and global information processing effects that can simulate reaction-diffusion morphogenesis (Turing, 1952), gastrulation, axon pathfinding, and other phenomena. Stephen Wolfram (2002) has demonstrated that simple rule sets can produce dynamic patterns within them, and that this complexity can be abstracted into hierarchical graphs (Wolfram, 2015). ...
Full-text available
The construction of an embryo from a single cell precursor is a highly complex process. Evolutionary emergence of the first embryos is even more complex, and involves both a transition to multicellularity along with the establishment of developmental mechanisms. In this chapter, we propose that embryogenesis relies on a community of cells conforming to a regulatory model of emergent multicellularity. This model draws together multiple threads in the scientific literature, from complexity theory to cybernetics, and from thermodynamic entropy to artificial life. All of these strands come together to inform a model of goal-oriented regulation for emergent structures in early life. This is an important step in the evolution of early life, as well as the emergence of complex life in the earliest habitats. Our model, called the cybernetic embryo, allows for a systems-level view of the embryogenetic process.
... Morphognosis is partially inspired by an abstract morphogenesis model called morphozoic (Portegys et al., 2017). Morphogenesis is the process of generating complex structures from simpler ones within an environment. ...
Conference Paper
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Artificial intelligence research to a great degree focuses on the brain and behaviors that the brain generates. But the brain, an extremely complex structure resulting from millions of years of evolution, can be viewed as a solution to problems posed by an environment existing in space and time. The environment generates signals that produce sensory events within an organism. Building an internal spatial and temporal model of the environment allows an organism to navigate and manipulate the environment. Higher intelligence might be the ability to process information coming from a larger extent of space-time. In keeping with nature's penchant for extending rather than replacing, the purpose of the mammalian neocortex might then be to record events from distant reaches of space and time and render them, as though yet near and present, to the older, deeper brain whose instinctual roles have changed little over eons. Here this notion is embodied in a model called morphognosis (morpho = shape and gnosis = knowledge). Its basic structure is a pyramid of event recordings called a morphognostic. At the apex of the pyramid are the most recent and nearby events. Receding from the apex are less recent and possibly more distant events. A morphognostic can thus be viewed as a structure of progressively larger chunks of space-time knowledge. A set of morphognostics forms long-term memories that are learned by exposure to the environment. A cellular automaton is used as the platform to investigate the morphognosis model, using a simulated organism that learns to forage in its world for food, build a nest, and play the game of Pong.
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Differentiation waves offer a different perspective on causality in embryogenesis from that of molecular developmental biology. Janus-faced cybernetic logic, with global and local top down/bottom up dynamics, eschews reductionism, is distinct from emergence, and outlines the process theoretically. Most aspects of differentiation waves require further investigation.
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A species of pufferfish builds fascinating circular nests on the sea floor to attract mates. This project simulates the nest building behavior in a cellular automaton using the morphognosis model. The model features hierarchical spatial and temporal contexts that output motor responses from sensory inputs. By considering the biological neural network of the pufferfish as a black box, and decomposing only its external behavior, an artificial counterpart can be generated. In this way a complex biological system producing a behavior can be filtered into a system containing only functions that are essential to reproduce the behavior. The derived system not only has intrinsic value as an artificial entity but also might help to ascertain how the biological system produces the behavior.
A species of pufferfish builds fascinating circular nests on the sea floor to attract mates. This project simulates the nest building behavior in a cellular automaton using the Morphognosis model. The model features hierarchical spatial and temporal contexts that output motor responses from sensory inputs. By considering the biological neural network of the pufferfish as a black box, decomposing only its external behavior, an artificial counterpart can be generated. In this way a complex biological system producing a behavior can be filtered into a system containing only functions that are essential to reproduce the behavior. The derived system not only has intrinsic value as an artificial entity but also might help to ascertain how the biological system produces the behavior.
Conference Paper
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Biological life is characterized by a robustness to noise often lacking in digital models. Although noise in the form of mutations has an important evolutionary purpose, it is also a threat that living systems are particularly adept at defending against. Artificial Life systems are frequently state-machines of some sort, requiring robustness mechanisms aimed at preserving and correcting state information. We explore a class of mechanisms based on robustness in transition dynamics rather than robustness in the emergent patterns of complex systems. We propose a Game of Life enhancement in which disrupting noise in the form of random state changes is corrected using a "light cone" of spatial and temporal context information. This enhancement does not change the original rules: a noiseless enhanced run is identical to an unenhanced run. We believe this general scheme could be applicable to other parallel computing systems in which contextual information determines state transitions.
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Organisms actively compete for environmental resources. They assess their surroundings, estimate how much energy they need for particular goals, and then realize the optimum variant. They take measures to control certain environmental resources. They perceive themselves and can distinguish between " self " and " non-self. " Current empirical data on all domains of life indicate that unicel-lular organisms such as bacteria, archaea, giant viruses, and protozoa as well as multicellular organisms such as animals, fungi, and plants coordinate and organize their essential life functions through signaling processes. Signaling allows for real life coordination and organization and is a communicative action in which species-specific behavioral patterns and sign repertoires are used. Cells, tissues, organs, and organisms that communicate share several key levels that are essential to all life forms and which serve as a uniform tool for investigating biocommunication. Complementary to this, active biocommunication depends on the deoxyribonucleic acid (DNA) storage medium and the agents that generate coherent content of nucleic acid sequences.
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A Monte Carlo method is proposed and demonstrated for obtaining an approximate algebraic solution to linear equations with algebraic coefficients arising from first order master equations at steady state. Exact solutions are hypothetically obtainable from the spanning trees of an associated graph, each tree contributing an algebraic term. The number of trees can be enormous. However, because of a high degeneracy, many trees yield the same algebraic term. Thus an approximate algebraic solution may be obtained by taking a Monte Carlo sampling of the trees, which yields an estimate of the frequency of each algebraic term. The accuracy of such solutions is discussed and algorithms are given for picking spanning trees of a graph with uniform probability. The argument is developed in terms of a lattice model for membrane transport, but should be generally applicable to problems in unimolecular kinetics and network analysis. The solution of partition functions and multivariable problems by analogous methods is discussed.
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The cell state splitter is a membraneless organelle at the apical end of each epithelial cell in a developing embryo. It consists of a microfilament ring and an intermediate filament ring subtending a microtubule mat. The microtubules and microfilament ring are in mechanical opposition as in a tensegrity structure. The cell state splitter is bistable, perturbations causing it to contract or expand radially. The intermediate filament ring provides metastability against small perturbations. Once this snap-through organelle is triggered, it initiates signal transduction to the nucleus, which changes gene expression in one of two readied manners, causing its cell to undergo a step of determination and subsequent differentiation. The cell state splitter also triggers the cell state splitters of adjacent cells to respond, resulting in a differentiation wave. Embryogenesis may be represented then as a bifurcating differentiation tree, each edge representing one cell type. In combination with the differentiation waves they propagate, cell state splitters explain the spatiotemporal course of differentiation in the developing embryo. This review is excerpted from and elaborates on “Embryogenesis Explained” (World Scientific Publishing, Singapore, 2016).
The formation of the spinal cord during early embryonic development in vertebrate embryos is a continuous process that begins at gastrulation and continues through to the completion of somitogenesis. Despite the conserved usage of patterning mechanisms and gene regulatory networks that act to generate specify spinal cord progenitors, there now exists two seemingly disparate models to account for their action. In the first, a posterior localized signalling source transforms previously anterior-specified neural plate into the spinal cord. In the second, a population of bipotent stem cells undergo continuous self-renewal and differentiation to progressively lay down the spinal cord and axial mesoderm by posterior growth. Whether this represents fundamental differences between the experimental model organisms utilised in the generation of these models remains to be addressed. Here we review lineage studies across four key vertebrate models: mouse, chicken, Xenopus and zebrafish and relate this to the underlying gene regulatory networks that are known to be required for spinal cord formation. We propose that by applying a dynamical systems approach to understanding how distinct neural and mesodermal fates arise from a bipotent progenitor pool, it is possible to begin to understand how differences in the dynamical cell behaviours such as proliferation rates and cell movements can map onto conserved regulatory networks to generate diversity in the timing of tissue generation and patterning during development.
The greatest mystery of life is how a single fertilized egg develops into a fully functioning, sometimes conscious multicellular organism. Embryogenesis Explained offers a new theory of how embryos build themselves, and combines simple physics with the most recent biochemical and genetic breakthroughs, based on the authors' prediction and then discovery of differentiation waves. They explain their ideas in a form accessible to the lay person and a broad spectrum of scientists and engineers. The diverse subjects of development, genetics and evolution, and their physics, are brought together to explain this major, previously unanswered scientific question of our time.As a follow up on The Hierarchical Genome, this book is a shorter but conceptually expanded work for the reader who is interested in science. It is useful as a starting point for the curious layman or the scientist or professional encountering the problem of embryogenesis without the formal biology background. There is also material useful for the seasoned biologist caught up in the new rush of information about the role of mechanics in developmental biology and cellular level mechanics in medicine.
This book deals with a wide class of novel and efficient adaptive signal processing techniques developed to restore signals from noisy and degraded observations. These signals include those acquired from still or video cameras, electron microscopes, radar, x rays, or ultrasound devices, and are used for various purposes, including entertainment, medical, business, industrial, military, civil, security, and scientific applications. In many cases useful information and high quality must be extracted from the imaging. However, often raw signals are not directly suitable for this purpose and must be processed in some way. Such processing is called signal reconstruction. This book is devoted to a recent and original approach to signal reconstruction based on combining two independent ideas: local polynomial approximation and the intersection of confidence interval rule. © 2006 The Society of Photo-Optical Instrumentation Engineers. All rights reserved.