  Finding optimal paths in connected graphs requires determining the smallest
total cost for traveling along the graph's edges. This problem can be solved by
several classical algorithms where, usually, costs are predefined for all
edges. Conventional planning methods can, thus, normally not be used when
wanting to change costs in an adaptive way following the requirements of some
task. Here we show that one can define a neural network representation of path
finding problems by transforming cost values into synaptic weights, which
allows for online weight adaptation using network learning mechanisms. When
starting with an initial activity value of one, activity propagation in this
network will lead to solutions, which are identical to those found by the
Bellman-Ford algorithm. The neural network has the same algorithmic complexity
as Bellman-Ford and, in addition, we can show that network learning mechanisms
(such as Hebbian learning) can adapt the weights in the network augmenting the
resulting paths according to some task at hand. We demonstrate this by learning
to navigate in an environment with obstacles as well as by learning to follow
certain sequences of path nodes. Hence, the here-presented novel algorithm may
open up a different regime of applications where path-augmentation (by
learning) is directly coupled with path finding in a natural way.
