There are many physical ``least action principles,'' in which
local behavior (of a particle in a field, for example) causes
the minimization of some global measure of ``action'' (e.g.,
time, distance, energy dissipation, entropy generation).
These processes are often governed by fields, and therefore
some optimization and constraint-satisfaction processes in
the brain may be implemented through corresponding field
computations.
One example will be discussed briefly.
In the same way that electromagnetic radiation (such as light) ``sniffs out'' in parallel a minimum-time path through space (Fermat's Principle), so also neural impulse trains can find a minimum-time path through a neural network. If transmission delays encode the difficulty of passage through a region of some (concrete or abstract) space, then the pulse train will follow the path of least difficulty, and it will automatically shift in parallel to a new optimum if regions change in difficulty; it is not necessary to reinitiate the path planning process from the beginning.
This works because, near an optimum path, the cost does not
vary, to a first approximation, with small perturbations of
the path, thus the impulses passing near to the optimal path
tend to stay in phase.
On the other hand, farther away from the optimum the cost
does vary, to a first approximation, with small perturbations,
so impulses on nearby paths tend to differ in phase.
As a result the signals along nonoptimal paths tend to cancel
each other out, so only the signals along near-optimal paths
have significant amplitude.
When difficulties change, the signals near the new optimum
tend to reinforce each other, while those that are no longer
near an optimum begin to cancel each other out.
Suppose the constant c represents the encoding of
difficulty in terms of time delay (in units of difficulty per
millisecond, for example), so a time difference of 
represents a difficulty difference of 
.
If the impulses have period T, then we can see that for
, signals will tend to cancel, whereas for
they will tend to reinforce.
Thus, impulses of period T will be sensitive to differences
in difficulty much greater than cT and insensitive to those
much less than cT; they will find paths within cT of the
optimum.
The sensitivity of search process can be adjusted by varying
the impulse frequency (higher frequency for a tighter
optimum).
Specifically, if the paths converging on a neuron represent a range of
difficulties of at least cT, then the neuron will be inactive, showing
that it's not near the optimal path. The neuron becomes more active,
reflecting its nearness to the optimum, as the range of input
difficulties decreases below cT.
Further, the amplitude of the impulses can be used to encode
the confidence in the difficulty estimate: regions of the
space for which this confidence is low will transmit signals
more weakly than high-confidence regions. In this way,
difficulty estimates are weighted by their confidence.
Specifically, the effect on the signal of passing through a
region of space is represented by multiplying by a complex
number 
, where d is the difficulty estimate
and k is the confidence of that estimate.
Such a complex multiplication could be accomplished by
synaptodendritic transmission, which introduces both an
amplitude shift k (reflecting confidence) and a time delay
d/c (representing difficulty).
Such amplitude/phase modulations would be relatively fixed,
subject to slow adaptive mechanisms.
However, the same can be accomplished more dynamically
(allowing, for instance, an environmental potential field to
be loaded into a brain region) by using an external bias to
control the phase shift dynamically (Hopfield 1995) and
a signal to a conjunctive synapse to control the amplitude
dynamically.