Motivation for Field Computation

In this paper we discuss the applications of field computation to natural and artificial intelligence. (More detailed discussions of field computation can be found in prior publications, e.g. MacLennan 1987, 1990, 1993b, 1997.) For this purpose, a field is defined to be a spatially continuous arrangement of continuous data. Examples of fields include two-dimensional visual images, one-dimensional continuous spectra, two- or three-dimensional spatial maps, as well as ordinary physical fields, both scalar and vector. A field transformation operates in parallel on one or more fields to yield an output field. Examples include summations (linear superpositions), convolutions, correlations, Laplacians, Fourier transforms and wavelet transforms. Field computation may be nonrecurrent (entirely feed-forward), in which a field passes through a fixed series of transformations, or it may be recurrent (including feedback), in which one or more fields are iteratively transformed, either continuously or in discrete steps. Finally, in field computation, the topology of the field (that is, of the space over which it is extended) is generally significant, either in terms of the information it represents (e.g. the dimensions of the field correspond to significant dimensions of the stimulus), or in terms of the permitted interactions (e.g. only local interactions).

Field computation is a theoretical model of certain information processing processes that take place in natural and artificial systems. As a model, it is useful for describing certain natural systems and for designing certain artificial systems. The theory may be applied regardless of whether the system is actually discrete or continuous in structure, so long as it is approximately continuous. We may make an analogy to hydrodynamics: although we know that a fluid is composed of discrete particles, it is nevertheless worthwhile to treat it as a continuum for most purposes. So also in field computation, an array of data may be treated as a field so long as the number of data elements is sufficiently large to be treated as a continuum, and the quanta by which an element varies are small enough so that it can be treated as a continuous variable.

Physicists sometimes distinguish between structural fields, which describe phenomena that are physically continuous (such as gravitational fields), and phenomenological fields, which are approximate descriptions of discontinuous phenomena (e.g. velocity fields of fluids). Field computation deals with phenomenological fields in the sense that it doesn't matter whether their realizations are spatially discrete or continuous, so long as the continuum limit is a good mathematical approximation to the computational process. Thus, we have a sort of ``Complementarity Principle,'' which permits the computation to be treated as discrete or continuous as convenient to the situation (MacLennan 1993a).

Neural computation follows different principles than conventional, digital computing. Digital computation functions by long series of high-speed, high-precision discrete operations. The degree of parallelism is quite modest, even in the latest ``massively parallel'' computers. We may say that conventional computation is deep but narrow. Neural computation, in contrast, functions by the massively parallel application of low-speed, low-precision continuous (analog) operations. The sequential length of computations is typically short (the ``100 Step Rule''), as dictated by the real-time response requirements of animals. Thus, neural computation is broad but shallow. As a consequence of these differences we find that neural computation typically requires very large numbers of neurons to fulfill its purpose. In most of these cases the neural mass is sufficiently large (15 million neurons/${\rm cm}^2$) that it is useful to treat it as a continuum.

To achieve by artificial intelligence the levels of skillful behavior that we observe in animals, it is not unreasonable to suppose that we will need a similar computational architecture, comprising very large numbers of comparatively slow, low precision analog devices. Our current VLSI technology, which is oriented toward the fabrication of only moderately large numbers of precisely-wired, fast, high-precision digital devices, makes the wrong tradeoffs for efficient, economical neurocomputers; it is unlikely to lead to neurocomputers approximating the 15 million neurons/${\rm cm}^2$ density of mammalian cortex. Fortunately, the brain shows what can be achieved with large numbers of slow, low-precision analog devices, which are (initially) imprecisely connected. This style of computation opens up new computing technologies, which make different tradeoffs from conventional VLSI. The theory of field computation shows us how to exploit relatively homogeneous masses of computational materials (e.g. thin films), such as may be generated by chemical manufacturing processes. We need such a theory to guide our design and use of such radically different computers.