Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models

The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.

'An Artificial Immune System as a Recommender System for Web Sites'

Artificial Immune Systems have been used successfully to build recommender systems for film databases. In this research, an attempt is made to extend this idea to web site recommendation. A collection of more than 1000 individuals' web profiles (alternatively called preferences / favourites / bookmarks file) will be used. URLs will be classified using the DMOZ (Directory Mozilla) database of the Open Directory Project as our ontology. This will then be used as the data for the Artificial Immune Systems rather than the actual addresses. The first attempt will involve using a simple classification code number coupled with the number of pages within that classification code. However, this implementation does not make use of the hierarchical tree-like structure of DMOZ. Consideration will then be given to the construction of a similarity measure for web profiles that makes use of this hierarchical information to build a better-informed Artificial Immune System.

'An Artificial Immune System as a Recommender System for Web Sites'

Artificial Immune Systems have been used successfully to build recommender systems for film databases. In this research, an attempt is made to extend this idea to web site recommendation. A collection of more than 1000 individuals' web profiles (alternatively called preferences / favourites / bookmarks file) will be used. URLs will be classified using the DMOZ (Directory Mozilla) database of the Open Directory Project as our ontology. This will then be used as the data for the Artificial Immune Systems rather than the actual addresses. The first attempt will involve using a simple classification code number coupled with the number of pages within that classification code. However, this implementation does not make use of the hierarchical tree-like structure of DMOZ. Consideration will then be given to the construction of a similarity measure for web profiles that makes use of this hierarchical information to build a better-informed Artificial Immune System.