There are several levels of neural activity that can be viewed as field computation.
The most obvious fields, which are measured by multiple electrode recording or by noninvasive imaging, such as NMR, are those comprising the spiking activity of neurons. Since, as we have seen, there are 146 thousand neurons per square millimeter of cortex, regions of cortex of this size are more than big enough to be treated as continua (reasonably, a tenth of a square millimeter is more than large enough). Indeed, Knudsen et al. (1987) observe that computational maps in the brain may be as small as a square millimeter, and perhaps smaller.
In cortical regions where the information is represented by
impulse rate, the field is real-valued; thus 
or 
represents the instantaneous impulse rate at
location p and time t.
Recently Hopfield (1995) has argued that
information may be represented by a combination of impulse
frequency and phase (relative to a global ``clock'' field or
to other neurons); in some cases at least, the phase
represents an analog value and the amplitude represents its
importance.
In such cases it's natural to treat the field as
complex-valued, with the complex number's
phase angle representing the impulse phase
and its magnitude representing the impulse amplitude.
Thus we write
,
where 
is the time-varying amplitude and
the time-varying phase.
Synapto-dendritic transmission of such a field, which affects
both its amplitude and phase, can be represented as
multiplication by a constant complex number.
For example, suppose a field 
results from
transmitting field 
through synapses
that introduce amplitude change 
and phase shift 
.
Then,
[ _p(t)
= [w_p e^i _p] ;a_p(t) e^i 
_p(t)
= [w_p a_p(t)] e^i[ 
_p(t) + _p] .
]
More compactly,
.
This encoding allows the soma potential to combine both the
analog values and the importance of signals arriving at the
synapses.
At the next level down we can consider the synaptic fields
associated with one neuron or a group of neurons.
For example, 
represents the time-varying activity
(measured, for example, by presynaptic potential or by
neurotransmitter flux across the synapse) of synapse p.
Certainly a pyramidal cell with 200 thousand synapses on its
dendritic tree can be said to have a synaptic field, and even
neurons with smaller numbers of inputs can treated as
processing fields.
The topology underlying the field is determined by the
dendritic tree, so in many cases the synaptic field cannot be
treated separately from the dendritic field (discussed next).
When we view the neuron at the level of the dendritic fields, we are concerned with the time-varying electrical potential field over the dendritic membrane. This varies continuously from point to point on the membrane and is determined by the detailed morphology of the dendritic tree. To a first approximation, field computation in the dendritic tree can be treated as a linear system (MacLennan 1993).
Finally, there are fields at larger scales.
For example, the phase delays discussed by
Hopfield (1995) may be relative to ``the phase of
an oscillating field potential'' in an area
(Ferster & Spruston 1995).
Further, there are global brain rhythms ( 
, 
etc.).
All the preceding fields are dynamic, changing on times
scales of milliseconds or faster.
It is often worthwhile to consider fields that are static or
that change on slower time scales (for example, through
learning or adaptation).
Such fields are represented in the connectivity patterns
between neurons and in patterns of synaptic efficacy.
For example, suppose that a topographic map A projects to a
topographic map B in such a way that the activity
of a neuron at location u in B depend on the
activities 
of neurons at locations v in A,
and that the strength of the dependence is given by 
.
In the simplest case we have a linear dependence,
[ _u(t) = _K_uv _v(t) v, ]
which we may write as a field equation,
.
The ``kernel'' K of this operator defines a connectivity
field between A and B.