  Several authors have recently highlighted the need for a new dynamical
paradigm in the modelling of brain working and evolution. In particular, the
models should include the possibility of non constant and non symmetric
synaptic weights $T_{ij}$ in the neuron-neuron interaction matrix, radically
overcoming the classical Hopfield setting. Krotov and Hopfield have proposed a
non constant, still symmetric model, leading to a vector field describing a
gradient type dynamics then including a Lyapunov-like energy function. In this
note, we first will detail the general condition to produce a Hopfield like
vector field of gradient type obtaining, as a particular case, the
Krotov-Hopfield condition. Secondly, we abandon the symmetry because of two
relevant physiological facts: (1) the actual neural connections have a marked
directional character and (2) the gradient structure deriving from the symmetry
forces the dynamics always towards stationary points, prescribing every pattern
to be recognized. We propose a novel model including a set of limited but
varying controls $|\xi_{ij}|\leq K$ used for correcting a starting constant
interaction matrix, $T_{ij}=A_{ij}+\xi_{ij}$. Besides, we introduce a
reasonable controlled variational functional to be optimized. This allows us to
reproduce the following three possible outcomes when submitting a pattern to
the learning system. If (1) the dynamics leads to an already existing
stationary point without activating the controls, the system has
\emph{recognized} an existing pattern. If (2) a new stationary point is reached
by the activation of controls, then the system has \emph{learned} a new
pattern. If (3) the dynamics is \emph{wandering} without reaching neither
existing or new stationary points, then the system is unable to recognize or
learn the pattern submitted. A further feature (4), appears to model
\emph{forgetting and restoring} memory.
