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We present a mathematical analysis of the effects of Hebbian learning inrandom recurrent neural networks, with a generic Hebbian learning ruleincluding passive forgetting and different time scales for neuronal activityand learning dynamics. Previous numerical works have reported that Hebbianlearning drives the system from chaos to a steady state through a sequence ofbifurcations. Here, we interpret these results mathematically and show thatthese effects, involving a complex coupling between neuronal dynamics andsynaptic graph structure, can be analyzed using Jacobian matrices, whichintroduce both a structural and a dynamical point of view on the neural networkevolution. Furthermore, we show that the sensitivity to a learned pattern ismaximal when the largest Lyapunov exponent is close to 0. We discuss how neuralnetworks may take advantage of this regime of high functional interest.

A mathematical analysis of the effects of Hebbian learning rules on the dynamics and structure of discrete-time random recurrent neural networks