Abstract
This paper will look at neurogenesis in mice and the mechanisms by which this
process can recover brain size and/or organization following neuronal death. Neurogenesis
creates neurons from a pool of progenitor cells via sequences of different types of cell divisions
in time. When neurons die, there is extra propagation of “intermediate progenitor” (IP) cells
which can compensate for this cell death. These IP cells can help maintain normal brain size
and/or normal brain organization (separation of cells into upper and deep layer neurons).
Two foundational experimental papers each claim one type of recovery but not the other.
Dr. Touboul has undertaken the development of a stochastic model, whose preliminary
analysis tends to show that both outcomes are possible depending on the specific pattern
of cell death that occurred or was induced with different intensities and durations during
their respective experiments. The model reconciles these two observations by employing two
types of biological compensation: “1) an increase in the probability and maximal number
of intermediate progenitor proliferative divisions; 2) a delay in the switching time between
upper- and deep-layer neurons generation by a maximum of 24h.”[1]
The goal of this thesis is threefold: first, develop extensive simulations of the stochastic
model. Second, develop a deterministic model for the same process, if possible by deriving it as the limiting ODE system of the stochastic model from when the number of cells diverges.
Third, delineate the boundaries of brain homeostasis as a function of the cell death
parameters.
Relative to the second objective, the stochastic model used in the literature has the only
free parameters being the probabilities of different types of cell divisions, the patterns of induced
cell death, and parameters specific to the given mutant (e.g., when neurogenesis begins
in wild-type mice, where gliogenesis begins, etc.).[2] We will work to simplify the model into
a dynamical system from a stochastic one, first using simulations and then (time-allowing)
with a limit theorem, using the classical theory of fluid limits for pure jump processes.[3] By
taking the infinite limit of the initial number of progenitor cells, the resulting rescaling in
space and time is expected to have the stochastic system approaching the dynamical system
at this limit.
[1] Cui, Y. (2018). Neurogenesis regulation and homeostasis: role of Pax6 signalling and mathematical
modelling. Doctoral Thesis for the University of Paris VI, Doctoral School of Brain-Cognition-Behavior, 3.
[2] Hsu, L. C., et al. (2015). Lhx2 regulates the timing of β-catenin-dependent cortical neurogenesis. PNAS,
112(39), 1219912204. www.pnas.org/cgi/doi/10.1073/pnas.1507145112
[3] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York. Van Kampen, N.
G. (1981). Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam. Darling, R.W.R.
(2002). Fluid Limits of Pure Jump Markov Processes: A Practical Guide. National Security Agency, MD,
USA. https://arxiv.org/pdf/math/0210109.pdf