The information communicating among neurons in Spiking Neural Networks (SNNs) is represented as spiking signals. The outstanding energy efficiency of SNNs stems from the minimal computational cost on both the nonlinear calculations of the neurons and the communicating power between them. In this paper, we present a three-dimensional (3D) Memristive Spiking Neural Network (M-SNN) system which employs memristors not only as of the electronic synapse but also as the threshold function. Our memristors are fabricated three-dimensionally with two layers. The simulation results demonstrate our fabricated two-layer memristors outperform the one-layer configuration on design area, power consumption, and latency with the factors of 2, 1.48, and 2.58, respectively. Moreover, to alleviate the switching variation, the heat dissipation layers are added to our memristor resulting in a ~30% reduction in cycle-to-cycle variation. The performance of the 3D M-SNN system is evaluated through the benchmark dataset (CIFAR-10). Our memristive threshold function improves the power consumption by 36%, compared with other state-of-the-art memristor-based threshold functions. Our 3D low variation memristor-based synapse shows significant improvement (10% to 66%) on design area, power consumption, and latency, compared with the SRAM and other state-of-the-art memristive synapses.