  We present a mathematical analysis of the effects of Hebbian learning in
random recurrent neural networks, with a generic Hebbian learning rule
including passive forgetting and different time scales for neuronal activity
and learning dynamics. Previous numerical works have reported that Hebbian
learning drives the system from chaos to a steady state through a sequence of
bifurcations. Here, we interpret these results mathematically and show that
these effects, involving a complex coupling between neuronal dynamics and
synaptic graph structure, can be analyzed using Jacobian matrices, which
introduce both a structural and a dynamical point of view on the neural network
evolution. Furthermore, we show that the sensitivity to a learned pattern is
maximal when the largest Lyapunov exponent is close to 0. We discuss how neural
networks may take advantage of this regime of high functional interest.
