Introduction to Stochastic Models in Biology

Abstract

Why use mathematical or stochastic models to study something as complicated and often poorly understood as dynamics in physiology? Hopefully, this book can provide some partial answers and point to the exciting problems that still remain to be solved, where mathematical and stochastic tools can be useful. In this book we treat basics of stochastic process theory represented by a stochastic differential equation directed towards biological modeling and review the field of neuronal models. Theoretical models must be relevant physiologically to be useful and interesting, and their analysis can provide biological insight and help summarize and interpret experimental data. Predictions can be extracted from the model, and experiments verifying or invalidating the model can be suggested, thereby enhancing physiological understanding. Even if the mechanisms are well understood, simulations from models can explore the consequences of extreme physiological conditions that might be unethical or impossible to reproduce experimentally. The process of building a theoretical model forces one to consider and decide on the essential characteristics of the physiological dynamics, as well as which variables and mechanisms to include. Analysis and numerical simulations of the model illustrate quantitatively and qualitatively the consequences of the assumptions implied in the model. The unifying aim of theoretical modeling and experimental investigation is the elucidation of the underlying biological processes that result in a particular observed phenomenon. Many biological systems are highly irregular, and experiments under controlled conditions show a large trial-to-trial variability, even when keeping the experimental setup fixed. This calls for a stochastic, as opposed to deterministic, modeling approach, especially because ignoring the stochastic phenomena in the modeling may hugely affect the conclusions of the studied biological system. In linear systems the noise might only blur the underlying dynamics without qualitatively affecting it, but in nonlinear dynamical systems corrupted by noise, the corresponding deterministic dynamics can be drastically changed. In general, stochastic effects influence the dynamics and may enhance, diminish, or even completely change the dynamic behavior of the system. In certain biological systems, e.g.

Estimulación de un modelo de efectos mixtos usando un proceso de difusión parcialmente observado

“…Como los modelos definidos en (1), y (5) son no lineales, las distribuciones a posteriori no son explícitas, entonces se requiere de un procedimiento de estimación recursivo, en este trabajo se propone usar algoritmo Monte Carlo por cadenas de Markov (muestreador de Gibbs).…”

“…Recientemente, en la literatura estadística se desarrollan metodologías basadas en los modelos de difusión de efectos mixtos donde el proceso de difusión satisface la ecuación diferencial estocástica tipo Itô; es decir, se tiene una clase de procesos estocásticos de tiempo continuo que modelan la solución de la SDE como un proceso de Markov de primer orden. Esta metodología ha sido aplicada en dinámica de sistemas térmicos [1], experimentos en la industria farmacéutica ( [4], [13], [9], entre otros), en neurociencia [8,6], en pronóstico de energía solar y eólica [12], en dinámica neuronal [5] y en crecimiento de grietas [11]. Otras aplicaciones incluyen análisis de enfermedades de transmisión, tales como epidemias, series financieras, dinámica de crecimiento de población, y procesos intracelulares [16].…”

Approximate maximum likelihood estimation using data-cloning ABC

“…Here we consider a state-space model with stochastic Gompertz dynamics. Donnet et al (2010) and Ditlevsen and Samson (2013) used a hierarchical (mixed-effects) version of this model to study chicken growth. We do not use a mixed-effects model so our results cannot be directly compared to the cited references.…”

Stochastic Models to Estimate Population Dynamics

“…Under these assumptions, it is assumed that the dynamic system of the process is partially driven by noise. The model includes a random noise in the measurement of dynamic system observations, and also includes factors that can not be controlled such as epidemiological diseases, variations in blood pressure, hormonal oscillations, respiration, neuronal control of variables that model activity muscle, enzymatic processes, cellular metabolism, nerve activity, molecule interactions, or individual characteristics such as body mass index, genes, smoking, stress impact ( [8]).…”