We have mathematically shown that cortical maps in the primary sensory cortices can be reproduced by using three hypotheses which have physiological basis and meaning. Here, our main focus is on ocular.dominance Monte Carlo simulations on the segregation of ipsilateral and contralateral afferent terminals are carried out. Based on these, we show that almost all the physiological experimental results concerning the ocular dominance patterns of cats and monkeys reared under normal or various abnormal visual conditions can be explained from a viewpoint of the phase transition phenomena. In order to describe the use-dependent self-organization of neural connections {Singer,1987 and Frank,1987}, we have proposed a set of coupled equations involving the electrical activities and neural connection density {Tanaka, 1988}, by using the following physiologically based hypotheses: (1) Modifiable synapses grow or collapse due to the competition among themselves for some trophic factors, which are secreted retrogradely from the postsynaptic side to the presynaptic side.

The elastic net was introduced as a heuristic algorithm for combinatorial optimisation and has been applied, among other problems, to biological modelling. It has an energy function which trades off a fitness term against a tension term. In the original formulation of the algorithm the tension term was implicitly based on a first-order derivative. In this paper we generalise the elastic net model to an arbitrary quadratic tension term, e.g. derived from a discretised differential operator, and give an efficient learning algorithm. We refer to these as generalised elastic nets (GENs). We give a theoretical analysis of the tension term for 1D nets with periodic boundary conditions, and show that the model is sensitive to the choice of finite difference scheme that represents the discretised derivative. We illustrate some of these issues in the context of cortical map models, by relating the choice of tension term to a cortical interaction function. In particular, we prove that this interaction takes the form of a Mexican hat for the original elastic net, and of progressively more oscillatory Mexican hats for higher-order derivatives. The results apply not only to generalised elastic nets but also to other methods using discrete differential penalties, and are expected to be useful in other areas, such as data analysis, computer graphics and optimisation problems.