Alessandro Russo

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number = {7},
pages = {195-210},
}

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\begin{frontespizio*}
\ateneo{Politecnico di Torino}
\corsodilaurea{i}
\CorsoDiLaureaIn {Modelli di sistemi fisiologic}
\titolo{\emph{Mathematical models of neural growth}}
\CandidateName{Alessandro Russo, Fabio Rossi}
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\sottotitolo{}
\sedutadilaurea{Dicembre 2114}
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\begingroup
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\tableofcontents
\endgroup
\chapter{Biology}
\section {Nervous system overview}
The nervous system, referring to vertebrate animals, is a complex structure that coordinates voluntary and involuntary actions and transmits signals between body’s different parts; it is divided in two main sections, the central nervous system (CNS) and the peripheral nervous system (PNS).

The central nervous system is primarily composed of glial cells and neurons. The glial cells are divided into oligodendrocytes, astrocytes and microglia. They have nutritional and support function for neurons ensuring the isolation of nerve tissues and protection from foreign particles in case of injury.
The peripheral nervous system is composed by nerves (bundles of axons), ganglia (nerve cells clusters) and the spinal cord (a structure that physically links the CNS to the rest of the body); it mainly connects the CNS to the limbs and organs, working as a communication relay.
Neurons are the cells that receive and transmit nerve impulses \cite{Art10}. They can be classified according to their morphology, to their function and the direction of propagation of the nerve impulse. Concerning the morphology they are classified in pyramidal cells, stellate cells and fusiform cells. About direction of propagation we distinguish afferent neurons (sensory neurons), efferent neurons (motor neurons) or interneurons. Neuron consists of the cell body, called soma, from which the neurites ( generical projections of neurons, which can become an axon or a dendrite after differentiation) originate. The axon transmits impulses to other neurons, while dendrites form trees and receive signals from neighboring cells \cite{art11},\cite{art34}.
\section {Neural growth}

Genesis and development of the nervous system starts from neurite outgrowth, a process where the growth cone, a specialized and transient structure that constitute the advance front of a growing neurite, guides the extention of the new axon or dendrite from the soma in a precise direction; the last stage of this process is the formation of a new synapsis \cite{art3}.

The dynamics of the intracellular cytoskeletons are fundamental to neurite initiation, elongation and branching \cite{art2},\cite{art1}. The direction of neurite outgrowth is also determined by extracellular signals to the growth cone \cite{art8}.
Microtubules provide structural support being part of the cytoskeleton \cite{art9}, \cite{art10}. Microtubules are dynamic polymers composed of alpha and beta tubulin subunits that are assembled into heterodimers with the help of cofactors \cite{art18}, \cite{art26}. The tubulin assembly and disassembly rate are influenced by the following factors: actin in the growth cone, by microtubule associated proteins (MAPs) and by changes in the calcium concentration \cite{art32}. MAPs have stabilizing ability and promote branching \cite{art4}, \cite{art5}. Tubulin monomers are produced in the cell body and are transported through the neurite to the growth cone. \cite{art35}, \cite{art36}, \cite{art33}. In the growth cone, assembling of tubulin monomers elongates the microtubules and therefore the neurite too \cite{art12}. The function of microtubules is crucial in the morphology of neurons \cite{art19}. The incorrect microtubules assembling and transport mechanisms have long been recognized as common denominators of neurodegenerative diseases \cite{art26}.

\chapter {Mathematical models review}
\section {Introduction}
In the following sections, a description of some mathematical models about neurite outgrowth is given; the first three models will be reported as a review of sector’s literature, the fourth one (next chapter) is also implemented in MATLAB, so the result of the simulation will be discussed.
\section {Compartimental model}
Aim of this model is to describe analytically the morphological development of growing neurons, axons or dendrites, and is based on compartments which are added, deleted and modified over time; the fundamental role of some kind of molecules produced in the soma and transported along the growing neurite(such as tubulin) is also modeled \cite{art6}.

Consider a single, unbranched neurite elongating at a rate proportional to the concentration of a substance at the growth cone; this particular molecule is produced in the cell body and diffuses along the growing neurite, and its concentration changes due to decay (in the soma and at the tip) and consumption in the developing process. The following equation represent concentration of the substance in different neurite’s compartments (soma, intermediate and terminal respectively), and the rate of change in length:

\[
PROTECTED3
\]
With $k$ distance between cell body and growth cone compartment when $L_i=0$.

The numerical simulation highlight the fact that, if one of the neurites has a higher assembly rate or a smaller disassembly rate, it can slow down or even stop for a considerable period of time, the growth of other neurites, by consuming all the tubulin produced by the soma; only when the dominant neurite has reached a certain length, the concentration of tubulin in the other growth cones (called “dormants”) can increase, causing them to elongate after a certain delay (this depend on the constant $g$, the amount of tubulin produced in the soma). Also, with the simulation, we understand that when there are more growing neurites (increasing the total number $N$) at the same time, the outgrowth rate is smaller than with only one neurite, and the competition effect is smaller the higher is the contribute $f$ of the active transport.

\section {Stochastic model}
Some models show virtual dendritic trees, to study the mechanisms of growth of neurons \cite{art16}, \cite{art17}, \cite{art18}. Morphological features of such dendritic structures are sampled from distributions derived from real neurons.
The empirical evidence is described by rules \cite{art20}, \cite{art21}, \cite{art22}. Elongation and branching are considered as joint processes. They showed how the increase of the dendrites length in time depends on the branching process. They indicated that the probability of branching may be modulated by the total number of terminals, however they do not identify the underlying biophisical mechanisms.\\
In \cite{art22} dendritic branching is considered as a stochastic process occuring randomly in time. The average branching probability of a terminal segment consists of a time-dependent baseline component, and a competition component depending on the momentary number of terminal segments. The dendritic branching proces is described in the following equation: \\
%_________EQUAZIONI_____________
\begin{equation}
\label{eq:1}
\bar{p}(t)~=~D(t)\bar{n}(t)^{-E}
\end{equation}\\
%————————————
where $\bar{n}(t)$ is the mean number of terminal segments in the dendritic tree at time $t$, $D(t)$ a baseline branching rate function and $E$ a competition parameter.
The branching rate $D(t)$ could be a constant or a decreasing exponential as in the following expressions
%————————-
\begin{equation}
\label{eq:2}
D(t)~=~D_{c}
\end{equation}\\
%————————————
\begin{equation}
\label{eq:3}
D(t)~=~ce^{-t/\tau}
\end{equation}\\
%———————————–
The dendritic elongation process is described assuming that the total length of the tree $L(t)$ depends on the average rate of elongation $v(t)$ and on $n(t)$
\begin{equation}
\label{eq:4}
\frac{\textrm{d}L(t)}{\textrm{d}t}~=~n(t)v(t)
\end{equation}
Stochastic models are capable of accurately producing the morphology of a wide range of neuron types. They have also predicted interesting growth differences between cell types, such as the degree of competition between growth cones \cite{art20}. These models gave basis to the recently developed simulation tool \textit{NETMORPH} \cite{art23} for the stochastic generation of large-scale neuronal networks with realistic neuron morphologies.
\chapter {Continuum model}
A more complex, PDE-based model is here presented\cite {art13}; neurite elongation is seen as a function of the production, transport and assembly/disassembly of tubulin into microtubules. The importan difference between this more complete model and the compartmental one is the presence of a term describing protein degradation and the gradient of tubulin along the growing neurite. The governing equatione is also coupled with an ODE, which describe the elongation ratio as time-dependant. Other terms, such as diffusion of tubulin and active transport, are the same as in the previous models. In the following sections, governing equation, initial condition and boundary conditions are presented, in their canonical form and nondimensional form, which is the one used for numerical solution in MATLAB.
\section {Governing equation}
Tubulin concentration is considered a one-dimensional parameter, varying along a single,growing neurite; it is carried alongside the neurite by diffusion (coefficient $D$), active transport (constant and positive, with coefficient $a$) and may also degrade at constant rate $g$. The governing equation is the following:

\begin{equation}
\frac{\partial c}{\partial t}+a\frac{\partial c}{\partial x}=D\frac{\partial^{2}c}{\partial x^{2}}-gc
\end{equation}
Where the considered domain is $\Omega_{xt}\equiv\{(x,t):x \in [0,l], t \geq0\}$, with soma-neurite interface ad $x=0$, and growth cone at $x=l$ where there is a flux of tubulin across the boundary, which is used to elongate the neurite. As said before, length $l$ is also a function of time described by an ordinary differential equation:
\begin{equation}
\frac{dl}{dt}=r_gc|_{x=l}-s_g, \qquad l(0)=l_0
\end{equation}
with $r_g$ elongation rate and $s_g$ is the constant retraction rate, caused by microtubules disassembly.
\section {Boundary conditions and initial conditions}
Steady flux of tubulin on the domain $\Omega_{xt}$ is assumed, for all instants $t \geq0$, and the flux of tubulin at $x=l$ is proportional to the quantity of substance there; the flux at $x=0$ from left to right is fixed.

\begin{equation}
\frac{\partial c}{\partial x}=-\varepsilon_0c_0\quad \mathrm{at}\quad x=0\qquad \mathrm{and} \qquad \frac{\partial c}{\partial x}=\zeta_l-\varepsilon_lc \quad \mathrm{at} \quad x=l
\end{equation}
where $c_0$ is the tubulin scale, $\zeta_l$ is microtubules disassembly rate, $\varepsilon_0$ flux source-rate (positive constant) and $\varepsilon_l$ microtubules assembly rate (positive constant).
A simple linear initial condition in $x$ is chosen, which satisfies the boundary conditions:
\begin{equation}
c(x,0)=c_0\varepsilon_0( l_0+\frac{1}{\varepsilon_l}-x)+\frac{\zeta_l}{\varepsilon_l}
\end{equation}
\section {Model implementation}
For numerical solution in MATLAB, which has been obtained with a finite difference approach, the previous equations were nondimensionalised, and a coordinate change was applied, in order to work on a new domain $\Omega_{yt}\equiv \{(y,t): y\in [0,1], t\geq0\}$. With this transformation, governing equation’s parameters are function of the length $l$ of the neurite. The previous equations in nondimensional form become:
\begin{equation}
\mathrm{Governing\quad equation:}\qquad \frac{\partial C}{\partial t}+ \frac{\alpha}{l}\frac{\partial C}{y}=\frac{1}{l}\frac{\partial^2C}{\partial y^2}-\beta C+\frac{y}{l}\frac{\partial C}{\partial y}([C]_{y=1}-\gamma)
\end{equation}
sda
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