Machine Learning and Statistical Analysis of Complex Mathematical Models: An Application to Epilepsy
Date: 18 February 2019
University of Exeter
PhD in Mathematics
The electroencephalogram (EEG) is a commonly used tool for studying the emergent electrical rhythms of the brain. It has wide utility in psychology, as well as bringing a useful diagnostic aid for neurological conditions such as epilepsy. It is of growing importance to better understand the emergence of these electrical rhythms and, ...
The electroencephalogram (EEG) is a commonly used tool for studying the emergent electrical rhythms of the brain. It has wide utility in psychology, as well as bringing a useful diagnostic aid for neurological conditions such as epilepsy. It is of growing importance to better understand the emergence of these electrical rhythms and, in the case of diagnosis of neurological conditions, to find mechanistic differences between healthy individuals and those with a disease. Mathematical models are an important tool that offer the potential to reveal these otherwise hidden mechanisms. In particular Neural Mass Models (NMMs), which describe the macroscopic activity of large populations of neurons, are increasingly used to uncover large-scale mechanisms of brain rhythms in both health and disease. The dynamics of these models is dependent upon the choice of parameters, and therefore it is crucial to be able to understand how dynamics change when parameters are varied. Despite they are considered low-dimensional in comparison to micro-scale neural network models, with regards to understanding the relationship between parameters and dynamics NMMs are still prohibitively high dimensional for classical approaches such as numerical continuation. We need alternative methods to characterise the dynamics of NMMs in high dimensional parameter spaces. The primary aim of this thesis is to develop a method to explore and analyse the high dimensional parameter space of these mathematical models. We develop an approach based on statistics and machine learning methods called decision tree mapping (DTM). This method is used to analyse the parameter space of a mathematical model by studying all the parameters simultaneously. With this approach, the parameter space can efficiently be mapped in high dimension. We have used measures linked with this method to determine which parameters play a key role in the output of the model. This approach recursively splits the parameter space into smaller subspaces with an increasing homogeneity of dynamics. The concepts of decision tree learning, random forest, measures of importance, statistical tests and visual tools are introduced to explore and analyse the parameter space. We introduce formally the theoretical background and the methods with examples. The DTM approach is used in three distinct studies to: • Identify the role of parameters on the dynamic model. For example, which parameters have a role in the emergence of seizure dynamics? • Constrain the parameter space, such that regions of the parameter space which give implausible dynamic are removed. • Compare the parameter sets to fit different groups. How does the thalamocortical connectivity of people with and without epilepsy differ? We demonstrate that classical studies have not taken into account the complexity of the parameter space. DTM can easily be extended to other fields using mathematical models. We advocate the use of this method in the future to constrain high dimensional parameter spaces in order to enable more efficient, person-specific model calibration.
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