Not all the fields of interest are in the brain. When investigating motor activity we also have to consider the musculo-skeletal system as well as fields external to the animal. Further, for sensory-motor coordination we have to include various sensory fields (e.g., visual, proprioceptive, auditory, vestibular). Here I'll look briefly at three examples (discussed in more detail in section 6).
First, premotor circuits in the frog spinal column have associated convergent force fields in the vicinity of the frog's leg; the activation of multiple circuits creates a linear superposition (sum) of these fields, and the resulting convergent force field guides the leg to a fixed destination independently of its current position (Bizzi & Mussa-Ivaldi 1995). This is a kind of field computation, except that the purpose is not the computation of abstract quantities, but the generation of concrete physical forces. Nevertheless, the mathematics of field computation can be used to describe and analyze the motor system.
One way to understand (nondiscursive) action planning is in terms of environmental potential fields, an approach which has been useful in both robotics (e.g., Khatib 1986, Rimon & Koditschek 1989) and neuroscience (e.g., Hogan 1984). In moving from one place to another we naturally select a path that minimizes some notion of work. We avoid obstacles, of course, and generally try to have a minimum path length, but this strategy may be modified by judgments of the ease of passage, etc. For example, we may go around a hedge even though the shortest path is through it; the path around minimizes work (broadly defined). Our knowledge of a region of space can be represented by a potential field in which the height of the potential at a location reflects the difficulty in going through that location. As will be described later, field operations can be used to find (in parallel) an inexpensive path through the potential field, and to revise the path dynamically if the potential field is discovered to be inaccurate (e.g. we find a large mud puddle in our path).
The potential field is not limited to encoding environmental difficulty; it can also represent internal constraints, such as the range or facility of motion of joints and limbs. Further, the potential field can be defined over nonspatial continua, to allow planning paths through more abstract ``spaces.''
Finally, Sanger (submitted) has explained how neural population codes can be interpreted in terms of conditional probability density fields (CPDFs) defined over possible stimuli. Each neuron has a CPDF that corresponds to its receptive field; the CPDF of a population over s short time interval is given by the product of the CPDFs of the neurons firing in that interval.