While backpropagation (BP) has been applied to spiking neural networks (SNNs) achieving encouraging results, a key challenge involved is to backpropagate a continuous-valued loss over layers of spiking neurons exhibiting discontinuous all-or-none firing activities. Existing methods deal with this difficulty by introducing compromises that come with their own limitations, leading to potential performance degradation. We propose a novel BP-like method, called neighborhood aggregation (NA), which computes accurate error gradients guiding weight updates that may lead to discontinuous modifications of firing activities. NA achieves this goal by aggregating finite differences of the loss over multiple perturbed membrane potential waveforms in the neighborhood of the present membrane potential of each neuron while utilizing a new membrane potential distance function. Our experiments show that the proposed NA algorithm delivers the state-of-the-art performance for SNN training on several datasets.