PhD Defense Adam Lund – University of Copenhagen

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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