Connecting Stochastic and Deterministic Models with Applications to the Homeostasis of Neurogenesis in Mice
Creators: name: Kristal, Rebekah Fernanda
Publisher: name: Brandeis University
Year: 2023
Subjects: Neurogenesis
Stochastic Models

description: This paper will look at neurogenesis in mice and the mechanisms by which thisprocess can recover brain size and/or organization following neuronal death. Neurogenesiscreates neurons from a pool of progenitor cells via sequences of different types of cell divisionsin time. When neurons die, there is extra propagation of “intermediate progenitor” (IP) cellswhich can compensate for this cell death. These IP cells can help maintain normal brain sizeand/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 preliminaryanalysis tends to show that both outcomes are possible depending on the specific patternof cell death that occurred or was induced with different intensities and durations duringtheir respective experiments. The model reconciles these two observations by employing twotypes of biological compensation: “1) an increase in the probability and maximal numberof intermediate progenitor proliferative divisions; 2) a delay in the switching time betweenupper- and deep-layer neurons generation by a maximum of 24h.”[1]The goal of this thesis is threefold: first, develop extensive simulations of the stochasticmodel. 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 deathparameters.Relative to the second objective, the stochastic model used in the literature has the onlyfree parameters being the probabilities of different types of cell divisions, the patterns of inducedcell death, and parameters specific to the given mutant (e.g., when neurogenesis beginsin wild-type mice, where gliogenesis begins, etc.).[2] We will work to simplify the model intoa 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] Bytaking the infinite limit of the initial number of progenitor cells, the resulting rescaling inspace and time is expected to have the stochastic system approaching the dynamical systemat this limit.&lt;br&gt;[1] Cui, Y. (2018). Neurogenesis regulation and homeostasis: role of Pax6 signalling and mathematicalmodelling. Doctoral Thesis for the University of Paris VI, Doctoral School of Brain-Cognition-Behavior, 3.&lt;br&gt;[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&lt;br&gt;[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