Identification of Dynamical Models of Spatio-temporal Pattern Formation with Applications to Slime Mould

Spatio-temporal systems are non-linear dynamical systems which include both time and space information. Many natural phenomena and processes can be described by spatio-temporal models, such as pattern formation, chemical reactions and physical dynamics. The main focus of this thesis is on the investigation of analysis and identification methods for spatio-temporal systems and the application of these methods to the dynamics of slime mould.

This thesis starts with a review of recent developments of spatio-temporal systems. Three general classes of spatio-temporal systems, Cellular Automata (CA), Coupled Map Lattices (CML) and Partial Differential Equations (PDE), which can be applied to modelling the behaviours of slime mould are discussed. Some basic problems associated with the identification of these models are addressed from various viewpoints. The main objective of this thesis is to develop the previous work in this area and to derive effective models from observed spatio-temporal data.

The dynamics of slime mould at the aggregation stage can be viewed as a spatio-temporal system. Three models which represent different types of spatio-temporal models respectively are introduced to model pattern formation of slime mould. All models can produce similar spirals and circle patterns with the observed patterns in experiments.
For the identification problem of the above mentioned models, one commonly used method is the orthogonal least squares (OLS) algorithm or the orthogonal forward regression (OFR) algorithm. However, when this classical identification method is applied to spatio-temporal data it may select spurious model terms in some cases, so a new algorithm called Orthogonal Forward Regression using Mutual Information (OFR-MI) algorithm is proposed. A new criterion of selecting model terms based on mutual information (MI) is employed in the new identification method and this effectively avoids the problem in the classical OLS or OFR algorithm.

Inspired from a Reaction-Diffusion-Chemotaxis (RDC) model for the gathering problem of slime mould, a new CA model which is called probabilistic multi-rule CA model is proposed. Unlike general CA models, this new model has two or more transition rules with associated probabilities, so that it has the potential to be used to model random processes in some spatio-temporal systems.

The identification of the probabilistic multi-rule CA system is a challenging topic, because of the stochastic character of this model. Based on the OLS algorithm and statistical methods, a new identification algorithm for probabilistic multi-rule CA models is proposed. Simulation results show that this new algorithm can work well either on the noise-free patterns in one-dimensional and two-dimensional cases or on spatio-temporal patterns with static noise or dynamic noise.