  The noisy leaky integrate-and-fire (NLIF) model describes the voltage
configurations of neuron networks with an interacting many-particles system at
a microscopic level. When simulating neuron networks of large sizes, computing
a coarse-grained mean-field Fokker-Planck equation solving the voltage
densities of the networks at a macroscopic level practically serves as a
feasible alternative in its high efficiency and credible accuracy. However, the
macroscopic model fails to yield valid results of the networks when simulating
considerably synchronous networks with active firing events. In this paper, we
propose a multi-scale solver for the NLIF networks, which inherits the low cost
of the macroscopic solver and the high reliability of the microscopic solver.
For each temporal step, the multi-scale solver uses the macroscopic solver when
the firing rate of the simulated network is low, while it switches to the
microscopic solver when the firing rate tends to blow up. Moreover, the
macroscopic and microscopic solvers are integrated with a high-precision
switching algorithm to ensure the accuracy of the multi-scale solver. The
validity of the multi-scale solver is analyzed from two perspectives: firstly,
we provide practically sufficient conditions that guarantee the mean-field
approximation of the macroscopic model and present rigorous numerical analysis
on simulation errors when coupling the two solvers; secondly, the numerical
performance of the multi-scale solver is validated through simulating several
large neuron networks, including networks with either instantaneous or periodic
input currents which prompt active firing events over a period of time.
