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.

翻译：噪声泄漏积分放电（NLIF）模型在微观层面通过相互作用的多粒子系统描述神经元网络的电压构型。在模拟大规模神经元网络时，利用粗粒化的平均场福克-普朗克方程在宏观层面求解网络电压密度，凭借其高效性和可信精度，实际上已成为一种可行的替代方案。然而，当模拟具有活跃放电事件的显著同步性网络时，该宏观模型无法输出有效结果。本文针对NLIF网络提出一种多尺度求解器，该求解器兼具宏观求解器的低计算成本与微观求解器的高可靠性。在每个时间步中，当模拟网络的放电频率较低时，多尺度求解器采用宏观求解器；而当放电频率趋于激增时，则切换至微观求解器。此外，宏观与微观求解器通过高精度切换算法实现整合，以确保多尺度求解器的精度。我们从两个角度验证该多尺度求解器的有效性：首先，给出保证宏观模型平均场近似在实际中成立的充分条件，并对耦合两种求解器时的模拟误差进行严格数值分析；其次，通过模拟多个大规模神经元网络（包括存在瞬时或周期性输入电流从而引发一段时间内活跃放电事件的网络），验证了多尺度求解器的数值性能。