  We associate learning and adaptation in living systems with the shaping of
the velocity vector field in the respective dynamical systems in response to
external, generally random, stimuli. With this, a mathematical concept of
self-shaping dynamical systems is proposed. Initially there is a zero vector
field and an "empty" phase space with no attractors or other non-trivial
objects. As the random stimulus begins, the vector field deforms and eventually
becomes smooth and deterministic, despite the random nature of the applied
force, while the phase space develops various geometrical objects. We consider
gradient self-shaping systems, whose vector field is the gradient of some
energy function, which under certain conditions develops into the
multi-dimensional probability density distribution (PDD) of the input.
Self-shaping systems are relevant to neural networks (NNs) of two types:
Hopfield, and probabilistic. Firstly, we show that they can potentially perform
pattern recognition tasks traditionally delegated to Hopfield NNs, but without
supervision and on-line, and without developing spurious minima of the energy.
Secondly, like probabilistic NNs, they can reconstruct the PDD of input
signals, without the limitation that new training patterns have to enter as new
hardware units. Thus, self-shaping systems can be regarded as a generalization
of the NN concept, achieved by abandoning the "rigid units" - "flexible
couplings" paradigm and making the vector field fully flexible and amenable to
external force. The new concept presents an engineering challenge requiring new
principles of hardware design. It might also become an alternative paradigm for
modeling of living and learning systems.
