Realization in the Brain

Computational maps are ubiquitous in the brain. For example, there are the well-known maps in somatosensory and motor cortex, in which the neurons form a topological image of the body. There are also the retinotopic maps in the vision areas, in which locations in the map mirror locations on the retina, as well as other properties, such as the orientation of edges. Auditory cortex contains tonotopic maps, with locations in the map systematically representing frequencies in the manner of a spectrum. Auditory areas in the bat's brain provide further examples, with systematic representations of Doppler shift and time delay, among other significant quantities.

In the presence of multiple stimuli, such maps typically represent the presence of all the stimuli. For example, if several tones are present in a sound, then a tonotopic map will show corresponding peaks of activity. Similarly, if there are patches of light (or other visual microfeatures, such as oriented grating patches) at many locations in the visual field, then a retinotopic map will have peaks of activity corresponding to all of these microfeatures. In this way the form of the stimulus may be represented as a superposition of microfeatures.

Computational maps such as these are reasonably treated as fields, and it is useful to treat the information processing in them as field computation. Indeed, since the cortex is estimated to contain at least 146,000 neurons per square millimeter (Changeux 1985, p. 51), even a square millimeter has enough neurons to be treated as a continuum, and in fact there are computational maps in the brain of this size and smaller (Knudsen et al. 1987). Even one tenth of a square millimeter contains sufficient neurons to be treated as a field for many purposes. The larger maps are directly observable by noninvasive imaging technique, such as NMR.

We refer to these fields as axonal fields, because the field's value at each location corresponds to the axonal spiking (e.g. rate and/or phase) of the neuron at that location. If only the rate is significant, then it is appropriate to treat the field as real-valued. If both rate and phase are significant (Hopfield 1995), then it is more appropriate to treat it as complex-valued.

Another place where field computation occurs in the brain is in the dendritic trees of neurons (MacLennan 1993a). The tree of a single pyramidal cell may have several hundred thousand inputs, and signals propagate down the tree by passive electrical processes (resistive and capacitive). Therefore, the dendritic tree acts as a large analog filter operating on the neuron's input field, which may be significant in dendritic information processing. In this case, the field values are represented by neurotransmitter concentrations, electrical charges and currents in the dendritic tree; such fields are called dendritic fields. They may have a complicated topology, since it is determined by the morphology of the dendritic tree over which it's spread.

Axonal and dendritic fields are comparatively dynamic, since their patterns of activity change on millisecond or faster time scales. There are also more static fields in the brain, which change on slower time scale or not at all. Examples include connection fields that describe patterns of connection between brain regions and synaptic fields that describe the transmission efficacies of masses of synapses. In the former case, we often find that the pattern of connections computes a convolution $\rho \otimes \phi$ with the input field $\phi$, where the field $\rho$ describes the common receptive field profile of all the output neurons. More generally, the connections may compute a linear transformation $L\phi = \int L(u,v) \phi(v) \d v$, where the kernel L of the operation is a connection field. In the case of synaptic fields, the transmitted signal is given by a pointwise product $\sigma(u)\phi(u)$ between the synaptic field $\sigma$ and the input field $\phi$. Connection fields and synaptic fields change comparatively slowly under the control of neurological development and learning.