Title: SPIDE: A Purely Spike-based Method for Training Feedback Spiking Neural Networks. Abstract: Spiking neural networks (SNNs) with event-based computation are promising brain-inspired models for energy-efficient applications on neuromorphic hardware. However, most supervised SNN training methods require complex computation or impractical neuron models, which hinders them from spike-based energy-efficient training. Among them, the recently proposed method, implicit differentiation on the equilibrium state (IDE), for training feedback SNNs is a promising way that is possible for generalization to locally spike-based learning with flexible network structures. In this paper, we study spike-based implicit differentiation on the equilibrium state (SPIDE) that extends the IDE method for supervised local learning with spikes, which could be possible for energy-efficient training on neuromorphic hardware. Specifically, we first introduce ternary spiking neuron couples that can realize ternary outputs with the common neuron model, and we prove that implicit differentiation can be solved by spikes based on this design. With this approach, the whole training procedure can be made as event-driven spike computation and weights are updated locally with two-stage average firing rates. Then to reduce the approximation error of spikes due to the finite simulation time steps, we propose to modify the resting membrane potential. Based on it, the average firing rate, when viewed as a stochastic estimator, achieves an unbiased estimation of iterative solution for implicit differentiation and the variance of this estimator is reduced. With these key components, we can train SNNs with either feedback or feedforward structures in a small number of time steps. Further, the firing sparsity during training demonstrates the great potential for energy efficiency. Meanwhile, even with these constraints, our trained models could still achieve competitive results on MNIST, CIFAR-10 and CIFAR-100. Our proposed method demonstrates the great potential for energy-efficient training of SNNs on neuromorphic hardware.