David Broomhead: Iterated Function Systems and Randomly Forced PDEs
Abstract:
The cable equation is a well-known partial differential equation model of an imperfectly insulated uniform conductor which is coupled to its surroundings by capacitive effects. In this work we study the dynamics of this system when it is subjected to randomly selected discrete input pulses. Generally, these pulses can be seen as representing a spatio-temporal coding of a finite alphabet of possible input symbols. We develop the theory of iterated function systems for this model, proving that it has a unique finite-dimensional attractor. We prove that the Hausdorff dimension of this attractor can be found using results of Falconer and Solomyak and more recent developments of Jordan, Pollicott and Simon. This latter work enables consideration of an additional underlying noise process to be included.
The context of this work is our development of an approach to signal processing which is not based on linear systems analysis. We note that this work could be applied to a simple physical model of the transmission of digital signals along a cable. However, we note also that the cable equation is much used as a basic model in neurobiology. We suggest that our results could provide the foundation of a theory of the response of neurons to spatio-temporally coded input pulse sequences. In particular, the symbolic coding of the attractor that this work implies, suggests an approach to the analysis of the computational capabilities of single neurons.