# Reexamining a Geometric Theory of Biological Growth

Dominic Gastaldo  David Silverstein

(August 28, 2024)

###### Abstract

A first principles approach to the theoretical description of the development
of biological forms, from a fertilized egg to a functioning embryo, remains a
central challenge to applied physics and theoretical biology. Rather than
refer to principles of self-organization and non-equilibrium statistical
mechanics to describe a developing embryo from its active cellular
constituents, a purely geometric theory is constructed that references the
properties of the ambient space that the embryo occupies. In 1975 the Fields
laureate René Thom developed a system of techniques and local dynamical models
that are capable of reconstructing the local dynamic of an embryo at each new
growth event of the system. Each new growth event (the development of a limb,
for example) is a topological change in the dynamic of the system that can be
classified only according to the properties of space. The local models can be
non-conservative flows with robust attractor behavior that serve as organizing
centers for systems development. Hamiltonian flows can also be considered with
novel, self-reproducing vague attractor behavior. The set of growth events
become an unfolding space related to the differentiable manifold of states.
The set of growth events, which Thom refers to as the catastrophe set, has
special algebraic properties which permit these models to be low
dimensional–the local model contains few parameters. We present Thom’s work as
a research program outlining a framework for the construction of these local
models, and, ultimately, the synthesis of these models into a full theoretical
description of a developing biological organism. We give examples of the
application of selected models to key growth events in the process of
gastrulation.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

In 1973, René Thom published the first French edition of Structural Stability
and Morphogenesis: An Outline of a General Theory of Models, a monograph
outlining a research program to reconstruct the dynamical equations that
govern the geometric change in form of a developing embryo. Thom presents a
deeply philosophical meditation on the geometric principles of stability in
our universe. Rather than investigate biological structure as a system
composed of parts that have function, Thom articulates the view of a geometer
as he constructs the framework for a topologically centric theory. This theory
formulates complex biological change as derivative properties of space.
Spacetime naturally hosts systems that themselves have a dynamic that
describes their evolution, and the topological changes in the dynamic
qualitatively change its behavior, therefore the behavior of the system. These
dynamics may be transient; changes in the environment, perturbations and
thermodynamic fluctuations may disrupt a system and cause it to change how it
behaves. The stable ways that a system can change its dynamic are deeply
related to the geometry of space. Thom studies local differential models that
depend on the most general, stable way a dynamic can topologically change its
behavior. These are the so-called universal unfoldings of the dynamic. Thom
develops a research plan around selected differential equations that depend on
the catastrophe bifurcations in spacetime. Each developmental change in form
is associated with a catastrophe bifurcation. Thom’s program then becomes the
study of structurally stable bifurcations that locally determine a system of
differential equations describing each morphogenic event.

##  II Theoretical formulation

Thom’s method can be summarized in a general mathematical framework. The
general mathematical structure will take the form of a three-tuple $(M,X,K)$,
where $M$ is the state space differentiable manifold, $X$ is the vector field
dynamic encoding the solution to a system of differential equations governing
the time evolution of the embryo, and $K$ is the algebraic set of catastrophe
bifurcations. The state space $M$ is a finite dimensional space containing
position and momenta of the cells that make up an embryo. Beyond the scope of
Thom’s framework, one can consider a sequence of manifolds
$(M^{2},M^{4},...,M^{2^{n}})$, for example, that each preserve the local
dynamic of a particular growth process. Each member of $K$ represents a
morphogenesis of the system. Locally, $K$ depends on the dynamic $X$. This
permits $X$ to be locally reconstructed by the elements of $K$.

The catastrophe set $K$ has special algebraic properties that produce models
with few parameters. The algebraic properties of $K$ also permit $X$ to be
locally reconstructed even when all the parameters of $X$ are not directly
known. The possible catastrophe bifurcations of a dynamical system can be
classified according to the dimension and topology of the space that the
embryo occupies. The elements of $K$ are universal unfoldings of the
singularities in $X$. The singularities in $X$ encode each morphogenetic
event. Thom explores the ultimate consequence of this by considering a global
model of an organism as the universal unfolding of its egg. We examine each
local model as determined by the universal unfoldings of $X$ that make up $K$.

Each local differential model is determined by the elements of $K$. The
analytic structure of $K$ determines the structural stability of the dynamic.
If $K$ is everywhere dense, then $X$ is structurally stable. It’s also clear
that if $K$ is non-empty, then the dynamic must be non-conservative. However,
we present how local Hamiltonian models can still be used. Thom experiments
with conservative, Hamiltonian systems with vague attractors. Some models have
very interesting self-replicating features. The gradient dynamic implies a
Riemannian metric $g_{\mu\nu}$ equipped on the manifold $M$. This leads to the
possibility of making quantitative an otherwise qualitative theory. Finally,
certain phenomenological principles need to be mentioned. A thermodynamical
coupling may need to be defined between dynamical systems. This will define a
measure-theoretic entropy and permit the two systems to come into thermal
equilibrium. Also, a convention inspired from thermodynamics needs to be
adopted. If two attractors are in competition, the attractor at the lowest
potential is the attractor that determines the local dynamic. The bifurcation
of a degenerate critical point can result in a shock wave as the critical
points change their relative positions. The shock waves resulting from the
emergence or annihilation of critical points becomes embryological growth
waves. These phenomenological principles can be used together with the applied
mathematical model to develop robust descriptions of local growth.

###  II.1 Topological Classifications

This section, topological classifications, was first published in Thom’s
original Structural Stability and Morphogenesis: An Outline of a General
Theory of Models (chapter 3). We have here simplified and reorganized Thom’s
original classifications to elucidate meaning to the reader. In this section,
we follow Thom’s outline for constructing a family of structurally stable
unfoldings of degenerate critical points. We will begin by looking at the
specific topological classifications that are needed to satisfy the conditions
for the universal unfolding. The universal unfolding will define the
morphogenic effects that will allow one to set up a general theory of models.
Therefore it is crucial for one to realize when structural stability enters
the theory. We will begin by examining a theory which is going to have the
following homeomorphisms. Let us take a continuous family of geometric objects
$E_{s}$ where $s$ is a point within a space of parameters $S$. Here $S$ can be
represented as a Euclidean space or a differential manifold that is either
finite or infinite-dimensional. We can then do the same with a family of
geometric objects $E_{t}$ where $t$ is a point in $T$, with $T$ once again
representing what can be a Euclidean space or a differential manifold that is
either finite or infinite-dimensional. If the points $s$ and $t$ are
sufficiently close to one another, then the corresponds points or objects will
have the same topological form and can be classified as structurally stable.
If such points are structurally stable, they form an open subset of
structurally stable points classified as generic points. The corresponding
open subset has a compliment $K$ which is defined as the set of bifurcation
points. From here we can take a system of polynomial equations

$P_{j}(X_{i},S_{k})=0$ (1)

Here $X_{i}$ and $S_{k}$ are coordinates in Euclidean space $\mathbb{R}^{n}$
and $S$. Taking $S=(S_{k})$ being fixed we get an algebraic set $E_{s}$ in
$\mathbb{R}^{n}$ which contains a set of solutions of the equation, where the
coefficients could be both real or imaginary. With the given points $s$ and
$t$ being in the same component of $S-K$, our sets $E_{s}$ and $E_{t}$ are
homeomorphic and can be invariant under continuous deformations. Such
homeomorphic deformations play a critical role in defining the properties of
the universal unfolding. Let us now take two differential manifolds labeled
$X$ and $Y$, with $X$ being compact. This allows us to write down the function
space $L(X,Y)$ of our differential maps from $X$ to $Y$. Such a differential
mapping is denoted as the $C^{m}$ topology where the differential maps are
defined on a chart by the following metric

$d(f,g)=\sum_{|k|=0}^{m}\bigg{|}\frac{\partial^{|k|}}{\partial
x^{k}}(f-g)\bigg{|}$ (2)

Here we see that the infinite dimensional manifold is a Banach manifold due to
the labeling being done on a Banach space. Yet we specifically see that such a
labeling on the Banach space is a local labeling which sets the stage to allow
us to have a geometric theory of local models. With that said, the function
space $L(X,Y$) containing our two manifolds plays the role of the parameter
space $S$. If we were to take each point $F$ in the parameter space
$S=L(X,Y)$, we would see that this is a differentiable map $X\rightarrow Y$.
We know if we were to take two maps $f$ and $g$ pertaining to $X\rightarrow Y$
that they have the same topology type under homeomorphism. With both of these
being homeomorphic we can write down the two homeomorphisms $h:X\rightarrow X$
and $k:Y\rightarrow Y$. Because of this setup we now have in $L(X,Y)$ a
subspace $K$ being our closed bifurcation set. We now can have any two maps
$f$ and $g$ of $L(X,Y)$ which in turns give us a connected component $L-K$. If
our maps $f$ and $g$ that belong to $L(X,Y)$ in turn belong to the same
connected component $L-K$, they will have the same topology type. It is
important to note that the bifurcation set $K$ has a stratified structure
containing regular points that form an everywhere-dense open set in $K$. When
locally studying the stratum of the bifurcation set of codimension $K$, we
will see that such an analysis is going to play a key role in defining the
universal unfolding. To give a good idea of a stratified map, let us take the
following canonical map

$F\colon X\times L(X,Y)\rightarrow Y\times L(X,Y)\rightarrow L(X,Y)$ (3)

which is defined by

$(x,f)\rightarrow(y=f(x),f)\rightarrow f$ (4)

This map gives the existence of a closed subspace $H$ of infinite codimension
in $L(X,Y)$. Now the function space $L(X,Y)$ can now be written as $L(X,Y)-H$,
which gives the existence of a stratified map. However, we can recall that our
$L(X,Y)$ has the subspace of the closed bifurcation set $K$. Therefore the
stratification $L(X,Y)-H$ is defined by the bifurcation subset $K$. Since the
stratification of $K$ contains $H$ pertaining to the function space $L(X,Y)$,
we can rewrite our stratification of $L(X,Y)-H$ as $L(X,Y)-K$. Such maps are
structurally stable maps and can therefore be labeled as generic maps. Due to
the maps in $L(X,Y)-K$ being generic, $L(X,Y)-H$ can then be embedded in what
Thom calls a $q-parameter$ family where the map
$X\times\mathbb{R}^{q}\rightarrow Y\times\mathbb{R}^{q}$ will be defined as
generic. Perturbations leave the parameter space $\mathbb{R}^{q}$ invariant.
To see the proof of this, we first need to study the problem locally. We can
begin by taking the following germ
$g\colon\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ of a differential map which
has the following origin $O$. Such a germ allows one to send the origin to $O$
in $\mathbb{R}^{m}$ to the origin $O_{1}$ in $\mathbb{R}^{n}$ which gives an
expansion of the map. With such an expansion it will be important to go over
the jet space that defines the expansion. We can take the following jet space
$z$ of order $r$ and an integer $r+p$ with $p$ depending on parameters $m$,
$n$ and $r$. We can then extend the jet to order $r+p$ which allows one to
determine the local topology of the germ. Such a property is contained within
the algebraic subset which is the bifurcation set $K$. By taking the expansion
of the origin for $O$ to $O_{1}$ pertaining to the germ with such proprieties
in the bifurcation set allows one to take the stratification of the jet space
$z$. Just like before, such a stratification will give rise to a structurally
stable jet which will once again be invariant under perturbations. While we
have been carefully examining the topological classifications, we should now
turn on our attention to what actually controls morphogenesis. With that said,
we should now focus on a more detailed example of a bifurcation that relates
to both a stratified jet space and the expansion of the origin. With the
expansion and stratification of a jet space, we can easily consider a
differentiable function that is zero at the origin, which will have the
following Taylor series expansion

First order:

$f_{1}=a_{1}x+b_{1}y$ (5)

Second order:

$f_{2}=a_{1}x+b_{1}y+a_{2}x^{2}+2b_{2}xy+c_{2}y^{2}$ (6)

Third Order:

$f=a_{1}x+b_{1}y+a_{2}x^{2}+2b_{2}xy+c_{2}y^{2}+a_{3}x^{3}+3b_{3}x^{2}y+3c_{3}xy^{2}+d_{3}y^{3}$
(7)

Now lets write down the successive space of coefficients
$J^{1}(a_{1},b_{1}),\>J^{2}(a_{1},b_{1},a_{2},b_{2},c_{2}),\>J^{3}(a_{1},...,c_{3},d_{3})$.
To first order in $J^{1}$ we have the following topological classification
where either both coefficients $a_{1}$, $b_{1}$ are not equal to zero or
$a_{1}=b_{1}=0$. By taking a careful look at the first case with $a_{1}$ and
$b_{1}$ being nonzero at the origin, and due to the equations being expressed
locally, we can easily use the implicit function theorem. By using the
implicit function theorem as noted by Thom, we see that we can locally change
to curvlinear coordinates which will give us a linear function, and will in
turn remain invariant regardless of higher coefficients. If we look at the
second case where $a_{1}=b_{1}=0$; very little can be said regarding the
topological nature of the function due to everything being centered at the
singularity. Yet we have indeed seen such topological properties go through
such processes regarding the expansions of origins due the expansion and
stratification of the jet spaces. We have also seen that the expansion of such
jet spaces satisfies the properties of a bifurcation set. Hence, we can now
restrict our attention to the expansion of a singularity of finite
codimension. Begin by taking the local maps $f$ and $g$ being two local maps
in Euclidean space $\mathbb{R}^{n}\rightarrow\mathbb{R}^{p}$, which map the
origin $O$ on $\mathbb{R}^{n}$ the origin $O_{1}$ in $\mathbb{R}^{p}$. These
maps can have the same local jet of order $r$ at $O$, and we can in turn have
the set of such jets $J^{r}(n,p)$ forming a vector space containing the
coordinates of the coefficients of the Taylor series expansion. Due to this
having a bifurcation subset, we can write $J^{r}(n,p)$ as $J^{r}(n,p)-K^{r}$,
and the maps $f$ and $g$ will locally have the same topology type. Therefore
there will be a homeomorphism $h$ and $h_{1}$. We can then write out the
singularities as $h(O)=O_{1},\>h_{1}(O_{1})=O_{1}$. As stated before, because
the bifurcation set $K$ is a subspace, there is going to a corresponding
finite codimension. Like before, we can write our codimension as $q$ due to it
being able to be embedded in a map
$F\colon\mathbb{R}^{n}\times\mathbb{R}^{q}\rightarrow\mathbb{R}^{p}\times\mathbb{R}^{q}$,
is again defined as structurally stable, being invariant under perturbations.
Due to these all being apart of the $C^{m}$ topology which extends to the
parameter q, we have the expansions of our singularities extending from $m$ to
$q$. All these parameters have been shown to invariant under perturbations,
hence, it seems natural that we can write out a universal family pertaining to
the expansion of the singularities in the form of
$\mathbb{R}^{m}\rightarrow\mathbb{R}^{q}$. This allows us to examine the
unfolding of the singularities that pertain to the universal map
$\mathbb{R}^{m}\rightarrow\mathbb{R}^{q}$, which satisfies the properties of
the bifurcation set $K$. Such an unfolding is defined as the universal
unfolding, and from here we can clearly illustrate an example of the universal
unfolding. From the Taylor series, we have shown that the perturbations occur
at third-order. Therefore, we will study the universal unfolding of $y=x^{3}$.
Every deformation of $y=x^{3}$ is a differentiable function $y=f(x,v)$
depending on $m$ parameters denoted globally by $v$, where $f(x,0)=x^{3}$. We
write

$F(x,y,v)=f(x,v)-y$ (8)

Then, at the origin $x=y=v=0$ of $(x,y,v)-space$, $F$ satisfies

$F(0,0,0)=\frac{\partial}{\partial x}F(0,0,0)=\frac{\partial^{2}}{\partial
x^{2}}F(0,0,0)=0,\frac{\partial^{3}}{\partial x^{3}}F(0,0,0)\neq 0$ (9)

Due to the analytic nature of this function, it was pointed out by Thom, that
we can easily apply Weierstrass preparation theorem and get the following

$F(x,y,v)=h(x,y,v)[x^{3}-3a(y,v)x^{2}+b(y,v)x+c(y,v)]$ (10)

Here the functions $h,a,b$ and $c$ are differentiable functions that satisfy
$h(0,0,0)\neq 0\>\>\>\>a(0,0)=b(0,0)=c(0,0)=0$ and

$\frac{\partial c}{\partial y}(0,0)=\frac{\partial}{\partial
y}\bigg{[}\frac{F(x,y,v)}{h(x,y,z)}\bigg{]}_{{x={y=z}}}=\frac{1}{h(0,0,0)}\neq
0$ (11)

By taking the Taylor expansion, this allowed us to show how the universal
unfolding occurs at third order, which helps further illuminate in addition to
show the conditions needed to satisfy the universal unfolding. Now that the
topological conditions of the universal unfolding have been satisfied, we are
now able to move onto the next step which will be defining and classifying the
catastrophes.

###  II.2  Local Dynamical Models of Morphogenesis

In this section, we will examine and build upon Thom’s reconstruction of a
local gradient dynamic. Thom classifies elementary catastrophes in
$\mathbb{R}^{4}$. Elementary catastrophes are bifurcations of a gradient
dynamic $\dot{x}=-\nabla V(x)$. To fully realize Thom’s program, catastrophes
of more complex flows need to be studied along with their associated
morphologies. This may be very challenging, and we study only the gradient
dynamic in detail. These elementary catastrophes produce a bifurcation that
results in a structurally stable dynamic associated with a morphology. Each
point where both the derivative and the second derivative vanish can be
universally unfolded into a catastrophe that represents a structurally stable
dynamic (i.e. stable to perturbations). Each elementary catastrophe has a
codimention, which is the number of additional parameters that are added on to
the original function. The bifurcation then exists in the space defined by the
additional parameters. Elementary catastrophes in $\mathbb{R}^{4}$ are
associated with a morphology. Elementary catastrophes map onto a local
gradient dynamic.

We wish to construct models that are locally equivalent to the global dynamic
$X$ within the neighborhood of each point $x\in$ K. These models will depend
on the algebraic properties of the catastrophe set, and both the spacial
parameters and the parameters in the universal unfolding will depend on
properties of the critical point and its universal unfolding. Though each
model can be defined on $\mathbb{R}^{n}$, the relevant models reduce down to
the corank of the critical point, either one or two dimensions. These are the
minimum dimensions necessary to describe the topology of the underlying
dynamic. There is nothing preventing the dynamic from being embedded in
$\mathbb{R}^{3}$, and indeed, this is appropriate to endow these models with
further quantitative predictive power.

####  II.2.1 The Gradient Model

We first consider the gradient model, and we examine the classifications of
elementary catastrophes associated with a local gradient dynamic.

####  II.2.2 Fold

The fold is the simplest non trivial catastrophe. It has codimension one, and
it is the universal unfolding of $V(x)=\frac{1}{3}x^{3}$.

$V(u,x)=\frac{1}{3}x^{3}+ux$ (12)

The fold has no morphogenetic effect, and it can’t generate shock waves.

####  II.2.3 Cusp

The cusp catastrophe is defined as

$V(u,v,x)=\frac{1}{4}x^{4}+\frac{1}{2}ux^{2}+vx$ (13)

The gradient model determines the time evolution equation

$\frac{dx}{dt}=-\frac{d}{dx}(\frac{1}{4}x^{4}+\frac{1}{2}ux^{2}+vx)$

$\frac{dx}{dt}=-x^{3}-ux-v$

This differential equation undergoes a cusp catastrophe for the variation of
the parameters $(u,v)$. It has a set of fixed points $-x^{3}-ux-v=0$.

This elementary catastrophe becomes the structurally stable cuspidal of
caustics.

####  II.2.4 Swallow tail

The swallow tail catastrophe can produce growth waves as the stable and
unstable regions of its bifurcation set coincide. In optics, it can be
interpreted as the formation of caustics. In embryological development it
becomes the blastopore fold. During gastrulation, the embryo folds in through
invagination, and the swallow tail is the catastrophe that captures the local
dynamic of the formation of a stable fold.

Figure 1: The invagination of Drosophila. This begins the gastrulation process
and the formation of the digestive system. 222[image credit: Conte V, Ulrich
F, Baum B, Muñoz J, Veldhuis J, Brodland W, Miodownik M]

The potential is the universal unfolding of $V(x)=\frac{1}{5}x^{5}$

$V(u,v,w,x)=\frac{1}{5}x^{5}+\frac{1}{3}ux^{3}+\frac{1}{2}vx^{2}+wx$ (14)

With local model

$\dot{x}=-x^{4}-ux^{2}-vx-w$

This implies, if indeed the swallow tail models invagination as Thom suggests,
that invagination is a one dimensional topological dynamical system.

####  II.2.5 Hyperbolic umbilic

The hyperbolic umbilic is given by the unfolding of $V(x,y)=x^{3}+y^{3}$.

$V(u,v,w,x,y)=x^{3}-y^{3}+wxy-ux-vy$ (15)

Thom doesn’t go into detail about the embryological effect of the Hyperbolic
umbilic. Rather, he relates it to the geometry of the breaking of waves.

####  II.2.6 Elliptic umbilic

The elliptic umbilic can be used to locally model hair, spikes or protrusions

$V(u,v,w,x,y)=x^{3}-3xy^{2}+w(x^{2}+y^{2})-ux-vy$ (16)

This model will produce the system of first order non-linear equations

$\dot{x}=-3x^{2}+3y^{2}-2wx+u$

$\dot{y}=6xy-2yw+v$

####  II.2.7 Butterfly

We can unfold a $V(x)=x^{6}$ potential to describe the elementary catastrophe
resulting in an exfoliating blister. This is known as the butterfly
catastrophe.

$V(u,v,w,s,x)=\frac{1}{6}x^{6}+\frac{1}{4}sx^{4}+\frac{1}{3}ux^{3}+\frac{1}{2}vx^{2}+wy$
(17)

The butterfly catastrophe can generate an exfoliating blister.

####  II.2.8 Parabolic umbilic

The parabolic umbilic describes a mushroom morphology. Its potential is
$V(x,y)=x^{2}y+\frac{y^{4}}{4}$, and its universal unfolding is

$V(u,v,w,s,x,y)=x^{2}y+\frac{y^{4}}{4}+sx^{2}+wy^{2}-ux-vy$ (18)

The equations of motion describing the local dynamic are then

$\dot{x}=2x(y+s)-u$

$\dot{y}=x^{2}+y^{3}+2wy-v$

##  III Applications

###  III.1 The Construction of the Catastrophe Set

For morphogenetic events that require more structure than elementary
catastrophes, the catastrophe set must be constructed. The catastrophe set is
composed of elementary (or general) catastrophes that cascade into one another
to give rise to complex growth events. Local differential models capture the
dynamic in the neighborhood of a catastrophe. The corank of a degenerate
critical point determines the minimum spacial parameter necessary to preserve
the relevant topology of the dynamic. The dynamic, however, can be embedded in
arbitrarily higher spacial dimensions for the purposes of a direct model of
embryogenesis. Thom remarks on the theoretical advantage of this property
saying,

The dynamical origin of a catastrophe can be described even when all the
internal parameters of the system are not explicitly known. Thom

We consider two examples. The first is a tree without leaves, where the
catastrophe set contains only elliptic umbilic catastrophes. The second is a
Hamiltonian model with self-reproducing vague attractors and an application of
Maxwell’s principle.

####  III.1.1 Tree without leaves

Perhaps one of the most elementary applications of this program is the body of
a bare tree. Each growth initiation of a branch is an element of the
catastrophe set. The structurally stable bifurcation of the dynamic that will
generate a branch is the elliptic umbilic
$V(u,v,w,x,y)=x^{3}-3xy^{2}+w(x^{2}+y^{2})-ux-vy$. The catastrophe set is then
composed of elliptic umbilics at locations in the manifold of states that
correspond to the topology of tree branches. Examining the gradient model
$\dot{x}=-\nabla V(u,x)$

$\dot{x}=-3x^{2}+3y^{2}-2wx+u$

$\dot{y}=6xy-2yw+v$

If indeed the branch corresponds to the structurally stable elliptic umbilic,
the underlying dynamical system of the growth of a tree branch is
2-dimensional, and we expect cell migration to follow a gradient flow in the
neighborhood of each element of $K$. At each new growth event, a
thermodynamical coupling can be defined measure theoretically.

The phase space is $(x,y,\dot{x},\dot{y})$, and the unfolding space (parameter
space) is $(u,v,w)$. A catastrophe occurs for certain values of $(u,v,w)$. The
equilibrium surface is given by

$-3x^{2}+3y^{2}-2wx+u=0$

$6xy-2yw+v=0$

####  III.1.2 Self-Reproducing Singularities

Thom considers a local Hamiltonian dynamic. Even though Hamiltonian dynamics
can’t have conventional attractors, they can still have what are known as
vague attractors, which we will discuss in more detail in the next section.
Therefore, we can still study a universal unfolding of a singularity as a
potential. We will define the Hamiltonian to be
$H(p,q)=\frac{p^{2}}{2m}+V(q).$

$\dot{q}=\frac{\partial H}{\partial p}$ (19)

$\dot{p}=-\frac{\partial H}{\partial q}$ (20)

We can examine a codimention three singularity unfolded at the origin.
According to Thom,

$V_{1}=V(q)+ug_{1}(q)+vg_{2}(q)+wg_{3}(q)$

A universal unfolding of this type is what Thom calls a self reproducing
singularity. Self reproducing in the sense that if the singularity $V(x)$ has
the potential $V$ within a space of variables $x$ has the same topology type
as $V_{2}$. Configurations with such self reproducing singularities occur in
what Thom describes as Maxwell’s convention. In Maxwell’s convention one would
have a bifurcation set $H$ of the function space $L(U,V)$, which has
attractors of a local dynamic corresponding to a minima of the map $g$. Like
before, this bifurcation set $H$ is a strata. Here the catastrophe points are
going to be counter images of the strata with $H^{{}^{\prime}}$ of $H$. Here
$H^{{}^{\prime}}$ is going to be the set of functions with two local minima.
These strata tend to have a minima that decompose during bifurcation into two
distinct minima, hence, self reproducing. The most straightforward example
would be the Riemann-Hugoniot catastrophe $V=x^{4}/4$, which has the universal
unfolding

$V_{1}=\frac{x^{4}}{4}-\frac{ux^{2}}{2}+vx+v^{2}$ (21)

When looking at the equation, there is a curve $x^{4}/4-x^{2}/2$, which gives
an even curve containing two symmetrical minima where a curve defined on
$V_{2}$ is going to have the same topology type after applying Maxwell’s
convention. Maxwell’s convention removes all of the critical points except
that which has the minimum potential. Multiple critical points may remain if
there are several equivalent local minima all lower than their neighbors in a
potential well.

##  IV Hamiltonian Dynamics

In this section we continue to carefully follow Thom’s original outline in
addition to discussing new original mathematical and physical results. On a
differential manifold $M$, the dual of the tangent bundle
($T^{*}M=\cup_{p}T^{*}_{p}M$) has canonical coordinates $(p_{i},q_{i})$ that
permits the definition of a symplectic form $\omega=dq^{i}\wedge dp_{i}$,
endowing an abstract Hamiltonian flow on $M$. Our field $X_{H}$ is given by
the equation $dH=\iota(X_{H})\cdot\omega$, where $\iota$ is the interior
product. In Hamiltonian dynamics there are no attractors specifically due to
the invariance of the Liouville measure under the Hamiltonian dynamic.
However, there are specific types of attractors known as vague attractors.
Unlike regular attractors, vague attractors are able to be defined within a
Hamiltonian system, and these vague attractors maintain stability under small
perturbations of the Hamiltonian. With ordinary attractors, one can have
several basins of attraction competing with one another, and while this is
topologically very complicated, one is still able to retain structural
stability. Yet when such a competition between two basins of attractors
occurs, the final conditions will be very sensitive to perturbations. In
Hamiltonian dynamics it could be very difficult to speak of basins and you can
never be certain regarding the dominance of an attractor. This is why in the
case of vague attractors, the situation is subject to being even more
indeterminate, and these vague attractors are going to have stable properties
specifically under very small perturbations of the Hamiltonian. Therefore,
such properties are what make the system so indeterminate when working with
Hamiltonian systems. Interestingly enough, these realizations were pointed out
by Thom, who stated how this is very reminiscent of the probabilistic nature
of quantum mechanics. In return, such properties of vague attractors in the
Hamiltonian aren’t going to lead to a topological stability, but a global
stability of the Hamiltonian. To begin examining these properties, we can take
a field containing the local Hamiltonian dynamics on the open set $W$, which
is going to admit the fibration $p:F\rightarrow P$. From here we can also have
the open subset $U$ of $W$ contained in the vague attractor $c(x)$. If this is
the case, we can say the attractor is contained in the fiber of fibration
$F\rightarrow P$. Generally there would be a continuum of attractors $c(x)$
within the fiber, where such a continuum would be a sympletic manifold where a
Liouville measure $A$ can be defined. With this we now have a Liouville
measure $A$ and an entropy $S$ pertaining to $c$, which allows us to write out
the formula $S=log\>A$. With the entropy we can now define the temperature $T$
by $T^{-1}=dS/dE$. Let us now take a group locally containing the same local
dynamical equivalence $G$, which is also associated with an attractor $c$.
Here $G$ operates specifically as a subgroup on $W$, and due to $G$ also
acting on the continuum of attracts $c(x)$ in the fiber $F$, we are going to
have an isotropy group of $G$. Here we have a transitive relation over the
open subset $U$, where in turn we get the dynamic being homogeneous and
isotropic in $U$. Due to these conditions, it would seem we would be able to
have a local equilibrium, and if we have an equilibrium, we know that it must
not exchange energy with the rest of the system. Naturally, based on the
conditions, we would suppose the system’s invariant, where we would have a
proportionality between the exchange in energy and the temperature difference
resulting in $dE=0$. Yet it was pointed out by Thom that such conditions are
satisfied if and only if $\int_{D}(T-T_{0})d^{2}x=0$ on a disc $D$, where
$T_{0}$ is the temperature at the center of the disc. Thom gives the condition
for equilibrium in terms of the microcanonical entropy. If we define $m(x)$ as
the Liouville volume and let $a(x)=\frac{dm}{dx}$, then for two thermally
coupled systems $S_{1}$ and $S_{2}$, thermal equilibrium is achieved if

$\frac{1}{a_{1}}\frac{da_{1}}{dt}=\frac{1}{a_{2}}\frac{da_{2}}{dt}$ (22)

where $S=k_{b}$ log$(a(x))$. At equilibrium we see that the temperature acts
as a harmonic function $\Delta T=0$, and when not in equilibrium the system
will evolve based on $dE/dt=k/\Delta T(E)$. When acting as a linear
oscillator, $T$ and $E$ act as a scalar, which allows us to recover the heat
equation. However, when dealing with quantum fields, $E$ can be complex, which
will allow one to recover the Schrodinger equation. The interesting thought of
there being a natural processes that approaches thermal equilibrium through a
superposition proposed by Thom, is indeed something well worth investigating
further in the future. Returning to the dynamics of the Hamiltonian systems,
we established the existence of a fiber $(M,X)$ over the open set $W$ where
there would be a hypersurface pertaining to $E$. This would allow us to have a
finite number of vague attractors $C_{1},C_{2},...,C_{k}$, where such vague
attractors can contain the corresponding Liouville measures
$M_{1},M_{2},...,M_{k}$ of the hypersurface $E$. With the Liouville measures
acting as functions on $E$ we can then write the entropy as $S_{j}=log(m_{j})$
and the corresponding temperature $T_{j}^{-1}=\partial S/\partial E$. When the
phases $X_{j}$ are of the same temperature, there could be an equilibrium, and
there could also be a thermodynamical coupling of the attractors $C_{j}$. With
these conditions met, this would give rise to a universal thermodynamical
coupling of the attractors $C_{j}$. It is interesting to note that if the
functions $T_{j}(E)$ are invertible, we would see that $E_{j}(T)$ would then
act as the corresponding energies of the attractors $C_{j}$. We can take the
volumes $V_{1}+V_{2}+,...,V_{K}$ act as the volumes of the corresponding
phases described above. We can then write the phases out as
$X_{1}+X_{2}+,...X_{k}$, and we then get the following $V_{1}+V_{2}+...V_{K}$
= total volume of $W$. In turn we then can write up
$E_{1}V_{1}+E_{2}V_{2}+...E_{K}V_{K}$ = total energy of the system. While this
is a very convenient way to write out and express the total energy of the
system, this tells us very little about the volume of the attractors $V_{j}$
and spatial division of the phase $X_{j}$. It would be suspected just like in
Maxwell’s convention, there is a minimality condition that plays a role. If
such a condition plays a role we can have $A_{ij}$ donate the area of a shock
wave or perturbation that separates $X_{1}$ and $X_{j}$ would have some
expression in the form of

$P=\sum K_{ij}A_{ij}(K_{ij}>0)$ (23)

Such an expression would be expected to be minimized and the topology of the
fiber dynamic would seem to have little effect leaving it invariant. It was
suspected by Thom, that many patterns and biological forms such as butterfly
wings, shell patterns and formations would be expected to depend on a very
similar type of mechanism due to the variety and complexity of the patterns in
such formations.

###  IV.1 Thermodynamic Unfoldings

While we have shown many different ways to express the Hamiltonian, there is
still an elephant in the room, and with the setup we have, that particular
elephant begs for attention. This happens to be the study and formation of a
partition function. We know that a partition function describes a
thermodynamical system in equilibrium. When working out the dynamics of the
Hamiltonian we saw that we were able to define a Liouville measure $A$ on a
symplectic manifold when examining the vague attractors $c(x)$. We also saw
that there is a continuum of vague attractors that are contained within a
fiber where the continuum of the fiber is the corresponding symplectic
manifold. This symplectic manifold was shown that the Liouville measures $A$
are defined. From the Liouville measure we saw that it was possible to write
out the entropy $S=log\>A$ and the temperature as $T_{j}^{-1}=\partial
S/\partial E$. There also happened to be a subgroup $G$ that had the
equivalent dynamics locally, which also operates on the continuum of
attractors in addition to the open set $U$ that acts as a subgroup on $W$.
There was an isotropy group of $G$ that was transitive on the open subset $U$
on $W$ where we saw that the corresponding dynamic was isotropic and
homogeneous on $U$. Due to such homogeneity it was easy to write out the
exchange of energy in equilibrium expressed as $dE=0$. As we know, the
partition function is used to express the statistical properties in
thermodynamical equilibrium. Taking a Hamiltonian, the partition function can
be expressed as

$Z(\beta)=\sum_{x_{i}}exp(-\beta H(x_{1},x_{2},...,x_{n}))$ (24)

where $H$ is the Hamiltonian, $X_{i}$ represents a set of random variables and
$\beta$ being the inverse temperature. We have it written as a sum over
$X_{i}$ to express it as the sum over all the possible values the variables
$X_{i}$ could take on. Yet when examining the local dynamics of the
Hamiltonian corresponding to the vague attractors, there is going to be a
continuum of attractors $c(x)$. So for our purposes we can rewrite the
partition function as

$Z(\beta)=\int exp(-\beta H(x_{1},x_{2},...,x_{n}))dx_{1}dx_{2}...dx_{n}$ (25)

where we simply replace the sum by an integral, making the partition
continuous rather than discrete. Due to Thom’s theory being a topologically
based theory, it would be most useful to express the partition function
canonically in the form

$Z=\frac{1}{h^{3}}\int e^{-\beta H(q,p)}d^{3}qd^{3}p$ (26)

Canonically, the energies are Boltzmannian distributed Landau1 . This is a
maximization of the Shannon entropy subject to a constant average energy. It
is a distribution that is as “disorganized” as it could be while keeping the
average energy constant. A further study of the energy distribution should be
done measure theoretically and keeping in mind a thermodynamical coupling
between dynamical systems. So we expect a new growth event to, at least
provisionally, be thermally coupled to the system that it grew from. These
distributions may need to be derived circumstantially. Proceeding with the
canonical partition function, we can stop for a moment and look back at the
canonical map which gave the existence of the stratified map

$F\colon X\times L(X,Y)\rightarrow Y\times L(X,Y)\rightarrow L(X,Y)$ (27)

which was defined as

$(x,f)\rightarrow(y=f(x),f)\rightarrow f$ (28)

which was one of the key requirements needed to satisfy the universal
unfolding. We are beginning to see very strong similarities between the
universal unfolding and the Hamiltonian dynamics. Yet this poses the question,
is there an analog of the universal unfolding with respect to thermodynamical
systems? The answer seems to be yes. When analyzing the Hamiltonian dynamics
of the fiber $(M,X)$ we saw that the Liouville measures $m$ formed the
relative entropy $S_{j}=log(m_{j})$ and the corresponding temperatures
$T_{j}^{-1}=\partial S/\partial E$. If the phases $X_{j}$ of the system are
the same temperature there will be thermodynamical equilibrium $dE=0$, and if
we have thermodynamical equilibrium we will in turn have a universal coupling
between the attractors $c_{j}$. Such attractors have the Liouville measure
that form the entropy $S_{j}=log(m_{j})$ and temperature $T_{j}^{-1}=\partial
S/\partial E$. It was shown that the phases $X_{j}$ controlled by the
attractors are required to be in thermal equilibrium. In return this shows the
manifestation of a canonical partition function from a purely geometric
theory. There is one other key component to which we should turn our
attention: the unfolding of the singularities of the Hamiltonian. Luckily we
already have a fibration $F\rightarrow P$. Having a fibration makes it
possible to study the mapping of the singular points of the symplectic
manifold that is formed by the continuum of attractors $c(x)$. Since we have a
continuum and have the existence of a fiber bundle, this tells us that the
mapping is going to be a surjective submersion. Having surjective properties
would tell us that we can have a function $f$ where we can take the domain $X$
and codomain $Y$, where $X$ would have elements $x$ and $Y$ containing
elements $y$. Also due to there being surjectivity, $Y$ would contain elements
$y$, and for every element of $y$ in $Y$, there would be at least one element
$x$ in $X$. So we would get the function $f(x)=y$. In turn, having a
submersion tells us that we would have a map between two differentiable
manifolds $N$ and $M$ where the differentiable is everywhere surjective.
Specifically, we can take the map $f:M\rightarrow N$ where the map $f$ would
have a submersion at a point $p$ which would be an element of $M$. Therefore
taking the differential of the function at the point would give

$Df_{p}:T_{p}M\rightarrow T_{f(p)}N$ (29)

For the canonical map

$F\colon X\times L(X,Y)\rightarrow Y\times L(X,Y)\rightarrow L(X,Y)$ (30)

we had a very similar setup with two differential manifolds $X$ and $Y$ which
were expressed as the function space $L(X,Y)$. Yet looking back at our
solution, we saw that the canonical map was defined as

$(x,f)\rightarrow(y=f(x),f)\rightarrow f$ (31)

If we carefully look at the above equation, we are able to see that there is a
surjective function $f(x)=y$. Yet we have also shown that the canonical map
gives the existence of a stratified map, which was needed to satisfy the
universal unfolding. When we turned our attention to the vague attractors of
the Hamiltonian dynamics there was the emergence of a fibration. The fibration
allowed us to examine the mapping of the points on the symplectic manifold.
However, what we would really be interested in are more specifically the
singularities. Therefore, we would be interested in a map that is a fibration
that would allow us to study said singularities. Such a map would be a Milnor
map, which is also known as a Milnor fibration. To define a Milnor map we can
take $f(Z_{0},...,Z_{n})$ representing a polynomial function of complex
variables, where the vanishing of a locus would be at the origin. One can also
have a Milnor fiber of an isolated hypersurface containing singularities. So
taking the singularity at the origin, we would have $f=0$. It is important to
state that the polynomial function $f(Z_{0},...,Z_{n})$ is going to contain
$n+1$ complex variables. So we are going to have a $\mathbb{C}^{n+1}$
topology. So we will have $f_{t}:\mathbb{C}^{n+1}\rightarrow\mathbb{C}$ where
the polynomial becomes $(Z_{0},...,Z_{n})\rightarrow f(Z_{0},...,Z_{n})-t$,
where $f=0$ is now $f_{t}\neq 0$. Here we have a polynomial that had a
singularity at the origin, which went under a type of unfolding and became a
polynomial. We saw extremely similar situations when analyzing the jet space.
Recall that we began studying the jet space $z$ by analyzing the problem
locally where we had the germ $g\colon\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
of a differential map, which had the corresponding origin $O$. The germ
allowed one to send the origin $O$ in $\mathbb{R}^{m}$ to the origin $O_{1}$
in $\mathbb{R}^{n}$, which gave the expansion of the map. From there we looked
at the jet space that defined the expansion. Let us recall that such an
expansion allowed us to define the algebraic subset, which was our bifurcation
set $K$. Seeing how the topological space containing the germ was invariant
under deformations, rather than just focus on the Milnor maps, we can now look
at Milnor numbers. The advantage of Milnor numbers is that in singularity
theory, they are the invariant of a germ $f$ and it is a complex-valued
holomorphic function that is non-negative or infinite. This can be defined as
either a geometric or algebraic invariant where there is once again a
hypersurface singularity. A germ of the kind can simply be defined as
$f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)$. When we looked at the field
of the Hamiltonian dynamics, we saw that there was a local field of the
Hamiltonian. In return, we also had a fiber that indeed had a hypersurface,
which had the finite number of vague attractors, that formed the relative
entropies and attractors. We then saw that if the phases were the same
temperature there would be equilibrium, which would give us a universal
thermodynamical coupling. The conditions for the universal thermodynamic
coupling satisfied the conditions for us to derive the partition function.
However, we now see that this allows us to derive much more. With the
universal unfolding, we had the singularities
$\mathbb{R}^{m}\rightarrow\mathbb{R}^{q}$ pertain to the universal map
$\mathbb{R}^{m}\rightarrow\mathbb{R}^{q}$. The formation of the universal map
began with the germ that gave the expansion of the singularities. This then
gave the formation of the jet space that defined the expansion. In the
Hamiltonian dynamics there exists the universal thermodynamical coupling with
the fibration that allowed us to study the mapping of singular points on the
symplectic manifold. On the sympletic manifold there existed the continuum of
vague attractors, which gave us our fiber bundle. From the fiber bundle we had
the surjective submersion where we took the mapping of the two differentiable
manifolds $M\rightarrow N$, which simply gave rise to the solution of the
canonical map that had the surjective function $f(x)=y$. This also satisfies
the conditions for a stratified map, which had the subspace of the closed
bifurcation set. The very proof of such a stratified space was the study of
the universal family, which arose from the germ of the differential map that
ending up giving topological invariance. Thanks to the existence of the
fibration of the Hamiltonian, we were able to replicate this by taking both
the Milnor map and Milnor numbers.

The Milnor map or the Milnor fibration is a specific type of fibration that is
used to study singularities. We saw that the mapping of such singularities of
a differential function that gave the emergence of a polynomial in the same
way when studying the properties of the universal unfolding. The Milnor map
that satisfies the properties of the universal unfolding also satisfied the
properties of the universal thermodynamic coupling due to being a fibration of
the hypersurface containing the vague attractors. From the analysis of the
Milnor maps we saw that the Milnor numbers; just like the formation of the
universal unfolding, they are the invariant of a germ which shows us that
there is a geometric invariance. Hence, we have shown that the universal
thermodynamical coupling satisfies the conditions of the universal unfolding.
Therefore, the universal thermodynamical coupling gives rise to the
Hamiltonian that is an analog of the universal unfolding.

##  V Conclusion

Thom’s morphogenetic models remain open to investigation in an epoch of
optical experimental techniques that make possible precise measurements of
local embryological development. A reexamination of these methods by the
scientific community may be timely.

While Thom’s approach doesn’t offer a direct quantitative model, it does
outline a unified approach to reconstructing the phase space profile. This
means that experimental verification must be done in tandem with modern
techniques in data science. Empirically, and for appropriately small distances
and times, the trajectory of cell migration must be tracked until a phase
space profile can be reconstructed. We can verify that the underlying dynamic
of morphogenesis is indeed determined by the structural stability of
topological dynamical systems permitted by the structure of our spacetime, if
the empirical phase space profile is diffeomorphic to Thom’s respective local
models. Thom’s local models, however, are phenomenological; that is, upon
observing the initiation of a new growth event, a catastrophe is selected
appropriate to the final form that the event will produce. In that sense, the
theory is finalistic (i.e. we must already know the final state to apply it).
However, the selection of a final state is not as free as Lagrangian
mechanics. The final states are already classified and determined by the
dimension and topology of space. With this framework, we can begin to address
larger concerns. This classification of morphologies within Thom’s framework
defines the question: what can be grown in our universe?

Furthermore, if structural stability is indeed the final principle in these
developmental events, the existence of low dimensional models in otherwise
high dimensional data is verified. This represents an encouraging step forward
in our search for simple models of biophysical processes. To what extend a
complete theory of morphogenesis can be deduced from more fundamental
principles remains on open question, and theoretical approaches to biology
remain almost as diverse as the phenomena they wish to describe Reed . Unlike
theoretical physics, where complexity, in a gas or a solid for example, is
treated statistically and therefore reductionist, the open door to biological
theory may be through finding simplicity in the final product of biological
interactions. We then use that framework to construct theories of constituent
complex behavior.

We have shown and reviewed the mathematical framework of Thom’s theory of
morphogenesis. We went over necessary requirements needed to be made for the
theory to emerge. From the mathematical framework we also went over the
emergence of the physical applications that manifest. Such applications appear
to give rise to new realizations and insights into areas such as Hamiltonian
mechanics and thermodynamics. These physical applications have also been shown
to be dual to a purely geometric and topological theory. Much investigation is
still required, and many new avenues have been opened for one to explore. We
expect these tools to be useful not only in the direct modeling of biological
development, but also in the motivation of new theoretical methods. At the
very least we can conclude that there are exciting new mathematical and
physical realizations to be discovered in the future.

## Acknowledgements

We thank Prof. Robert Alfano (Institute for Ultrafast Spectroscopy and Lasers
of the City University of New York) for discussing the content and the
presentation of this work. We thank Johannes Zwanikken (University of
Massachusetts Lowell), Rahul Kashyap (Penn State University) and Christopher
Bresten (Ajou University, Suwon, Korea) for helpful discussions. We thank
Chris Peter (University of Massachusetts Dartmouth) for help in the
professional editing of this material. We also thank Mailde S. Ozório
(University of São Paulo) for help in the final edit of our manuscript.

## Appendix

To further realize Thom’s original program, a deeper look at the partition
function is needed. A discussion of the search for lower dimensional behavior
within biological systems is done in Bialek , and the author contextualizes
the role of dimensionality in modeling biological systems. A probability
distribution over trajectories is given by

$P[x(t)]=\frac{1}{Z}e^{-S[x(t)]}$

for a dynamical system with a Gaussian noise term. Consider the gradient
dynamic for invagination

$\dot{x}=-x^{4}-ux^{2}-vx-w$

Since this dynamical system is by derivation structurally stable, it is
diffeomorphic to

$\dot{x}=-x^{4}-ux^{2}-vx-w+\eta(t)$

where $\eta(t)$ is a Gaussian noise term. $S[x(t)]$ is defined as

$S[x(t)]=\frac{1}{2}\int\int x(t)K(t-t^{\prime})x(t^{\prime})dtdt^{\prime}$

The Gaussian kernel is

$K(t+t^{\prime})=\sum^{D}_{n=1}a_{n}\phi_{n}(t)\phi_{n}(t^{\prime})$

Hidden dimensions are identified according to the structure of correlations in
the noise source. The dimension $D$ of the dynamical system specifies the
number of initial conditions needed to determine the system. While a partition
function of this kind may not have the topological structure necessary to
satisfy the universal unfolding, the use of a partition function of this kind
may be useful when examining the physical applications. These applications
would include the thermodynamic properties of the system, such as probability
distributions and the entropy of the system.

## References

  * (1) Thom, René. Structural Stability and Morphogenesis: A General Theory of Models. Thom, R. (1975). Structural stability and Morphogenesis: An outline of a general theory of models. Reading: Benjamin.
  * (2) W Bialek, What do we mean by the dimensionality of behavior? arXiv:2008.09574 [q–bio.NC] (2020).
  * (3) L. D. Landau, E. M. Lifshitz, Statistical Physics, Third Edition, Part 1, Pergamon Press, Oxford (1980).
  * (4) M. Reed, Why Is Mathematical Biology So Hard? Notices of the American Mathematical Society (2004).