Fast approximate Bayesian computation for inference in non-linear differential equations

Abstract

Complex biological systems are often modelled using non-linear differential equations which provide a rich framework for describing the dynamic behaviour of many interacting physical variables representing quantities of biological importance. Approximate Bayesian computation (ABC) using a sequential Monte Carlo (SMC) algorithm is a Bayesian inference methodology that provides a comprehensive platform for parameter estimation, model selection and sensitivity analysis in such non-linear differential equations. However, this method incurs a significant computational cost as it requires explicit numerical integration of differential equations to carry out inference. In this thesis we propose a novel method for circumventing the requirement of explicit integration, within the ABC-SMC algorithm, by using derivatives of Gaussian processes to smooth the observations from which parameters are estimated. We evaluate our methods using synthetic data generated from model biological systems described by ordinary and delay differential equations. Upon comparing the performance of our method to existing ABC techniques, we demonstrate that it produces comparably reliable parameter estimates at a significantly reduced execution time. To put emphasis on the practical applicability of our fast ABC-SMC algorithm we have used it extensively in the task of inverse modelling of a phenomenon pertaining to plant electrophysiology. Particularly we model the electrical responses in higher plants subjected to periods of ozone exposure. We investigate the generation of calcium responses at local sites following a stimulation and model electrical signals as a plant-wide manifestation of such responses. We propose a novel mathematical model that describes the experimentally observed responses to ozone. Furthermore, we pose the modelling task as an inverse problem where much of our insight is gained from the data itself. We highlight throughout the inverse modelling process the usefulness of the proposed fast ABC-SMC method in fitting, discriminating and analysing models described as non-linear ordinary differential equations. We carry out all these tasks using noisy experimental datasets, that provide limited information, to derive novel insights about the underlying biological processes.

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