Fields are required to be physically realizable, which places restrictions on the allowable functions. I have already mentioned that fields are continuous functions over a bounded domain that take their values in a bounded subset of a linear space. Furthermore, it is generally reasonable to assume that fields are uniformly continuous finite-energy (i.e. square integrable) functions. Among other things, these assumptions imply that fields belong to a Hilbert space of functions. (See Pribram 1991 and MacLennan 1990, 1993b for more on Hilbert spaces as models of continuous knowledge representation in the brain; see MacLennan 1990 for more on the physical realizability of fields.)
A field transformation is any continuous (linear or nonlinear) function that maps one or more input fields into one or more output fields. Since a field comprises an uncountable infinity of points, the elements of a field cannot be processed individually in a finite number of discrete steps, but a field can be processed sequentially by a continuous process, which sweeps over the input field and generates the corresponding output sequentially. Normally, however, a field transformation operates in parallel on the entire input field and generates all elements of the output at once. Many useful information processing tasks can be implemented by a composition of field transformations, which feeds the field(s) through a fixed series of processing stages. (One might expect sensory systems to be implemented by such feed-forward processes, but in fact we find feedback at almost every stage of sensory processing, so they are better treated as recurrent computations, discussed next.)
In many cases we are interested in the dynamical properties of fields: how they change in time. The changes are usually continuous, defined by differential equations, but may also proceed by discrete steps. As with the fields treated in physics, we are often most interested in dynamics defined by local interaction processes, although nonlocal interactions are also used in field computation (several examples are considered later). One reason for dynamic fields is that the field may be converging to some solution by a recurrent field computation; for example, the field might be relaxing into the most coherent interpretation of perceptual data, or into an optimal solution of some other problem. Alternately, the time-varying field may be used for some kind of real-time control, such as the motor control.
An interesting question is whether there can be a universal field computer, that is, a general purpose device (analogous to a universal Turing machine) that can be programmed to compute any field transformation (in a large, important class of transformations, analogous to the Turing-computable functions). In fact, we have shown (Wolpert & MacLennan submitted) that any Turing machine, including a universal Turing machine, can be emulated by a corresponding field computer, but this does not seem to be the concept of universality that is most relevant to field computation. Another notion of universality is provided by an analog of Taylor's theorem for Hilbert spaces. It shows how arbitrary field transformations can be approximated by a kind of ``field polynomial'' computed by a series of products between the input field and fixed ``coefficient'' fields (MacLennan 1987, 1990).
Adaptation and learning can be accomplished by field computation versions of many of the common neural network learning algorithms, although some are more appropriate to field computation than others. Learning typically operates by computing or modifying ``coefficient fields'' or connection fields in a computational structure of fixed architecture.