Large deviation theory in stochastic processes: applications to biological modeling

Author

Moshe C. Silverstein, New Jersey Institute of TechnologyFollow

Author ORCID Identifier

0000-0001-7050-6267

Document Type

Dissertation

Date of Award

8-31-2024

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

James MacLaurin

Second Advisor

Amitabha Koshal Bose

Third Advisor

Victor Victorovich Matveev

Fourth Advisor

David Shirokoff

Fifth Advisor

Etienne Tanre

Abstract

This dissertation delves into developing and applying stochastic models to analyze complex biological systems. It leverages Large Deviation Theory (LDT) to gain insights into these systems, focusing on two key examples: neural networks and calcium signaling dynamics. Traditional deterministic methods frequently fail to capture biological processes' randomness and inherent variability. Meanwhile, many stochastic approaches struggle to be mathematically tractable or provide accessible insights. The approach introduced in this study provides rigorous mathematical frameworks to enhance understanding of these stochastic behaviors while remaining tractable and insightful.

A stochastic model for a random biological neural network is constructed that addresses the dependencies and variabilities in neural connectivity. Applying LDT, significant theoretical results are derived from the large deviations in the system's dynamics, providing a deeper understanding of the probabilistic behaviors and events in neural activity.

The study next focuses on calcium signaling in biological cells, where a one-dimensional stochastic model is developed to simulate calcium dynamics. A Piecewise-deterministic Markov process (PDMP) model is implemented to capture the system's stochastic and deterministic nature. This model is validated by comparing experimental data from in vitro and in vivo studies via Maximum Likelihood Estimation and stochastic simulations. LDT is used to derive the Euler-Lagrange equations and identify optimal trajectories in calcium signaling, offering predictive insights into the system's behavior under stochastic mechanisms.

The findings of this study demonstrate the power of LDT in biological modeling, providing a robust framework for analyzing the probabilistic nature of complex biological systems. While the models incorporate several simplifications, such as one-dimensional assumptions in calcium signaling, they pave the way for more sophisticated and accurate representations of biological processes.

This work advances the application of stochastic processes and LDT in mathematical biology, offering enhancements to methodologies and insights that can be extended to other complex systems. The proposed approach opens up new avenues for understanding and predicting the behavior of stochastic biological systems, with potential applications in fields such as neuroscience, cell biology, and systems biology.

Recommended Citation

Silverstein, Moshe C., "Large deviation theory in stochastic processes: applications to biological modeling" (2024). Dissertations. 1779.
https://digitalcommons.njit.edu/dissertations/1779