Hebbian学习规则对离散随机循环神经网络动力学和结构影响的数学分析
A mathematical analysis of the effects of Hebbian learning rules on the dynamics and structure of discrete-time random recurrent neural networks
摘要：
我们提出了一个数学分析的随机循环神经网络的赫布学习的影响，与一个通用的赫布学习规则，包括被动遗忘和不同的时间尺度的神经元活动和学习动力学。以前的数值研究表明，赫布学习驱动系统从混沌到一个稳定的状态，通过一系列的分叉。在这里，我们从数学上解释这些结果，并表明这些影响，涉及神经元动力学和突触图结构之间的复杂耦合，可以使用雅可比矩阵进行分析，它引入了结构和动力学的角度来看神经网络的进化。此外，我们发现，当最大的李雅普诺夫指数接近0时，对学习模式的灵敏度是最大的。我们将讨论神经网络如何利用这种高功能感兴趣的制度。
英文摘要：
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.
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