用数学方法揭示神经网络如何记与忘，统一解释记忆形成与灾难性遗忘。

Despite explosive expansion of artificial intelligence based on artificial neural networks (ANNs), these are employed as "black boxes'', as it is unclear how, during learning, they form memories or develop unwanted features, including spurious memories and catastrophic forgetting. Much research is available on isolated aspects of learning ANNs, but due to their high dimensionality and non-linearity, their comprehensive analysis remains a challenge. In ANNs, knowledge is thought to reside in connection weights or in attractor basins, but these two paradigms are not linked explicitly. Here we comprehensively analyse mechanisms of memory formation in an 81-neuron Hopfield network undergoing Hebbian learning by revealing bifurcations leading to formation and destruction of attractors and their basin boundaries. We show that, by affecting evolution of connection weights, the applied stimuli induce a pitchfork and then a cascade of saddle-node bifurcations creating new attractors with their basins that can code true or spurious memories, and an abrupt disappearance of old memories (catastrophic forgetting). With successful learning, new categories are represented by the basins of newly born point attractors, and their boundaries by the stable manifolds of new saddles. With this, memorisation and forgetting represent two manifestations of the same mechanism. Our strategy to analyse high-dimensional learning ANNs is universal and applicable to recurrent ANNs of any form. The demonstrated mechanisms of memory formation and of catastrophic forgetting shed light on the operation of a wider class of recurrent ANNs and could aid the development of approaches to mitigate their flaws.