PhD Defense Adam Lund
Title: Spatio-temporal modeling of neuron fields
Abstract:
The starting point and focal point for this thesis was stochastic dynamical modelling of neuronal imaging data with the declared objective of drawing inference, within this model framework, in a large-scale (high-dimensional) data setting. Implicitly this objective entails carrying out three separate but closely connected tasks; i) probabilistic modelling, ii) statistical modeling and iii) implementation of an inferential procedure. While i) - iii) are distinct tasks that range over several quite different disciplines, they are joined by the premise that the initial objective can only be achieved if the scale of the data is taken into consideration throughout i) - iii).
The strategy in this project was, relying on a space and time continuous stochastic modelling approach, to obtain a stochastic functional differential equation on a Hilbert space. By decomposing the drift operator of this SFDE such that each component is essentially represented by a smooth function of time and space and expanding these component functions in a tensor product basis we implicitly reduce the number of model parameters. In addition, the component-wise tensor representation induce a corresponding component-wise tensor structure in the resulting statistical model. Especially, the statistical model is design matrix free and facilitates an efficient array arithmetic. Using proximal gradient based algorithms, we combine this computationally attractive statistical framework with non-differentiable regularization to form computationally efficient inferential procedure with minimal memory foot prints. As a result we are able to fit large scale image data in a mathematically sophisticated dynamical model using a relatively modest amount of computational resources in the process.
The contributions presented in this thesis are computational and methodological. The computational contribution takes the form of solution algorithms aimed at exploiting the array-tensor structure in various inferential settings. The methodological contribution takes the form of a dynamical modelling and inferential framework for spatio-temporal array data. This framework was developed with neuron field models in mind but may in turn be applied to other settings conforming to the spatio-temporal array data setup.
Supervisors:
Prof. Niels Richard Hansen, University of Copenhagen
Co-Supervisor: Per Roland, Department of Neuroscience and Pharmacology
Assessment committee:
Ass. Prof Anders Tolver (Chairman), University of Copenhagen
Senior Lecture Johan Lindström, Lund University
Ass. Prof. Bo Martin Bibby, Aarhus University