  In this work, we are concerned with a Fokker-Planck equation related to the
nonlinear noisy leaky integrate-and-fire model for biological neural networks
which are structured by the synaptic weights and equipped with the Hebbian
learning rule. The equation contains a small parameter $\varepsilon$ separating
the time scales of learning and reacting behavior of the neural system, and an
asymptotic limit model can be derived by letting $\varepsilon\to 0$, where the
microscopic quasi-static states and the macroscopic evolution equation are
coupled through the total firing rate. To handle the endowed flux-shift
structure and the multi-scale dynamics in a unified framework, we propose a
numerical scheme for this equation that is mass conservative, unconditionally
positivity preserving, and asymptotic preserving. We provide extensive
numerical tests to verify the schemes' properties and carry out a set of
numerical experiments to investigate the model's learning ability, and explore
the solution's behavior when the neural network is excitatory.
