In the (approximately) linear case F can be separated into two linear operators L and M operating on the state and input, respectively; the time derivative of the state is a superposition of the results of these operations: [ . = L + M . ] Next we'll consider several important examples of linear field processes.
A diffusion process is defined by a linear differential
field equation:
[ . = k^2 ^2 , ]
where the Laplacian is defined:
[
^2 =
_k ^2 x_k^2 ,
]
and the summation is over all the dimensions 
of the
extent of 
.
Many useful computations can be performed by diffusion processes; for example chemical diffusion processes have been used for finding minimum-length paths through a maze (Steinbeck et al. 1995). Also, diffusion equations have been used to implement Boltzmann machines and simulated annealing algorithms, which have been used to model optimization and constraint-satisfaction problems, such as segmentation and smoothing in early vision, and correspondence problems in stereo vision and motion estimation (Miller et al. 1991, Ting & Iltis 1994).
In the brain, diffusion processes, implemented by the
spreading activation of neurons, could be used for planning
paths through the environment.
For example, a diffusion process is approximated by a network
in which each neuron receives activation from its neighbors,
without which its activity decays. Thus the change in
activity of neuron x is given by
[
._x =
k^2 (-_x
+ 1n _i _x_i ) ,
]
where 
are the activities of its n neighbors
.
More clearly, writing 
for the average
activity of its neighbors,
[ ._x = k^2 (_x_i - _x ) . ]
The averaging process can be accomplished by convolution with
a radial function, such as a Gaussian:
[ . = k^2 (- ) . ]
Constraints on the path (impassable regions) are represented
by neurons whose activity is inhibited; relatively impassable
regions can be represented by neurons that are only partly
inhibited.