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.

"Topographic" mappings occur frequently in the brain. A popular approach to understanding the structure of such mappings is to map points representing input features in a space of a few dimensions to points in a 2 dimensional space using some selforganizing algorithm. We argue that a more general approach may be useful, where similarities between features are not constrained to be geometric distances, and the objective function for topographic matching is chosen explicitly rather than being specified implicitly by the self-organizing algorithm. We investigate analytically an example of this more general approach applied to the structure of interdigitated mappings, such as the pattern of ocular dominance columns in primary visual cortex. 1 INTRODUCTION A prevalent feature of mappings in the brain is that they are often "topographic". In the most straightforward case this simply means that neighbouring points on a two-dimensional sheet (e.g. the retina) are mapped to neighbouring points in a more central two-dimensional structure (e.g. the optic tectum). However a more complex case, still often referred to as topographic, is the mapping from an abstract space of features (e.g.

"Topographic" mappings occur frequently in the brain. A popular approachto understanding the structure of such mappings is to map points representing input features in a space of a few dimensions to points in a 2 dimensional space using some selforganizing algorithm.We argue that a more general approach may be useful, where similarities between features are not constrained tobe geometric distances, and the objective function for topographic matching is chosen explicitly rather than being specified implicitlyby the self-organizing algorithm. We investigate analytically an example of this more general approach applied to the structure of interdigitated mappings, such as the pattern of ocular dominance columns in primary visual cortex. 1 INTRODUCTION A prevalent feature of mappings in the brain is that they are often "topographic". In the most straightforward case this simply means that neighbouring points on a two-dimensional sheet (e.g. the retina) are mapped to neighbouring points in a more central two-dimensional structure (e.g. the optic tectum). However a more complex case, still often referred to as topographic, is the mapping from an abstract space of features (e.g.

This is motivated by experimental evidencethat these phenomena may be subserved by the same mechanisms. An important aspect of this model is that ocular dominance segregationcan occur when input activity is both distributed, and positively correlated between the eyes. This allows investigation of the dependence of the pattern of ocular dominance stripes on the degree of correlation between the eyes: it is found that increasing correlation leads to narrower stripes. Experiments are suggested to test whether such behaviour occursin the natural system.

This is motivated by experimental evidence that these phenomena may be subserved by the same mechanisms. An important aspect of this model is that ocular dominance segregation can occur when input activity is both distributed, and positively correlated between the eyes. This allows investigation of the dependence of the pattern of ocular dominance stripes on the degree of correlation between the eyes: it is found that increasing correlation leads to narrower stripes. Experiments are suggested to test whether such behaviour occurs in the natural system.