Title: The Quenching-Activation Behavior of the Gradient Descent Dynamics for Two-layer Neural Network Models. Abstract: A  numerical and phenomenological study of the gradient descent (GD) algorithm for training two-layer neural network models is carried out for different parameter regimes. It is found that there are two distinctive phases in the GD dynamics in the under-parameterized regime: An early phase in which the GD dynamics follows closely that of the corresponding random feature model, followed by a late phase in which the neurons are divided into two groups: a group of a few (maybe none) “activated” neurons that dominate the dynamics and a group of ``quenched” neurons that support the continued activation and deactivation process.  In particular, when the target function can be accurately approximated by a relatively small number of neurons, this quenching-activation process biases GD to picking sparse solutions.  This neural network-like behavior is continued into the mildly over-parameterized regime, in which it undergoes a transition to a random feature-like behavior where the inner-layer parameters are effectively frozen during the training process.  The quenching process seems to provide a clear mechanism for  ``implicit regularization''. This is qualitatively different from the GD dynamics associated with the ``mean-field'' scaling where all neurons participate equally.