  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.
