# Training Feedback Spiking Neural Networks by Implicit Differentiation on the Equilibrium State

arXiv: https://arxiv.org/abs/2109.14247 · HTML: https://mlsanity.com/paper/2109.14247/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 60 references that are in the map.
- First posted on arXiv 29 Sep 2021.

## Builds on (most cited first, 26 of 26)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Surrogate Gradient Learning in Spiking Neural Networks](https://mlsanity.com/paper/1901.09948/) (2019)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Spatio-Temporal Backpropagation for Training High-performance Spiking Neural Networks](https://mlsanity.com/paper/1706.02609/) (2017)
- [Converting Static Image Datasets to Spiking Neuromorphic Datasets Using Saccades](https://mlsanity.com/paper/1507.07629/) (2015)
- [Going Deeper With Directly-Trained Larger Spiking Neural Networks](https://mlsanity.com/paper/2011.05280/) (2020)
- [SLAYER: Spike Layer Error Reassignment in Time](https://mlsanity.com/paper/1810.08646/) (2018)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Going Deeper in Spiking Neural Networks: VGG and Residual Architectures](https://mlsanity.com/paper/1802.02627/) (2018)
- [Direct Training for Spiking Neural Networks: Faster, Larger, Better](https://mlsanity.com/paper/1809.05793/) (2018)
- [Training Deep Spiking Neural Networks using Backpropagation](https://mlsanity.com/paper/1608.08782/) (2016)
- [Long short-term memory and learning-to-learn in networks of spiking neurons](https://mlsanity.com/paper/1803.09574/) (2018)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Enabling Deep Spiking Neural Networks with Hybrid Conversion and Spike Timing Dependent Backpropagation](https://mlsanity.com/paper/2005.01807/) (2020)
- [Optimal Conversion of Conventional Artificial Neural Networks to Spiking Neural Networks](https://mlsanity.com/paper/2103.00476/) (2021)
- [Enabling Spike-based Backpropagation for Training Deep Neural Network Architectures](https://mlsanity.com/paper/1903.06379/) (2019)
- [Temporal Spike Sequence Learning via Backpropagation for Deep Spiking Neural Networks](https://mlsanity.com/paper/2002.10085/) (2020)
- [Brain-Like Object Recognition with High-Performing Shallow Recurrent ANNs](https://mlsanity.com/paper/1909.06161/) (2019)
- [Spiking Deep Networks with LIF Neurons](https://mlsanity.com/paper/1510.08829/) (2015)
- [Spike-Train Level Backpropagation for Training Deep Recurrent Spiking Neural Networks](https://mlsanity.com/paper/1908.06378/) (2019)
- [Hybrid Macro/Micro Level Backpropagation for Training Deep Spiking Neural Networks](https://mlsanity.com/paper/1805.07866/) (2018)
- [Unifying Activation- and Timing-based Learning Rules for Spiking Neural Networks](https://mlsanity.com/paper/2006.02642/) (2020)
- [Minimax Dynamics of Optimally Balanced Spiking Networks of Excitatory and Inhibitory Neurons](https://mlsanity.com/paper/2006.08115/) (2020)

## Built on by (most cited first, 30 of 40)

- [Spikformer: When Spiking Neural Network Meets Transformer](https://mlsanity.com/paper/2209.15425/) (2022)
- [Training High-Performance Low-Latency Spiking Neural Networks by Differentiation on Spike Representation](https://mlsanity.com/paper/2205.00459/) (2022)
- [Online Training Through Time for Spiking Neural Networks](https://mlsanity.com/paper/2210.04195/) (2022)
- [QKFormer: Hierarchical Spiking Transformer using Q-K Attention](https://mlsanity.com/paper/2403.16552/) (2024)
- [Ternary Spike: Learning Ternary Spikes for Spiking Neural Networks](https://mlsanity.com/paper/2312.06372/) (2023)
- [Towards Memory- and Time-Efficient Backpropagation for Training Spiking Neural Networks](https://mlsanity.com/paper/2302.14311/) (2023)
- [Spikformer V2: Join the High Accuracy Club on ImageNet with an SNN Ticket](https://mlsanity.com/paper/2401.02020/) (2024)
- [TorchDEQ: A Library for Deep Equilibrium Models](https://mlsanity.com/paper/2310.18605/) (2023)
- [Spiking Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.17046/) (2023)
- [Take A Shortcut Back: Mitigating the Gradient Vanishing for Training Spiking Neural Networks](https://mlsanity.com/paper/2401.04486/) (2024)
- [Spiking Transformer:Introducing Accurate Addition-Only Spiking Self-Attention for Transformer](https://mlsanity.com/paper/2503.00226/) (2025)
- [Efficient Federated Learning with Spike Neural Networks for Traffic Sign Recognition](https://mlsanity.com/paper/2205.14315/) (2022)
- [Enhancing the Performance of Transformer-based Spiking Neural Networks by SNN-optimized Downsampling with Precise Gradient Backpropagation](https://mlsanity.com/paper/2305.05954/) (2023)
- [SPIDE: A Purely Spike-based Method for Training Feedback Spiking Neural Networks](https://mlsanity.com/paper/2302.00232/) (2023)
- [A Comprehensive Review of Spiking Neural Networks: Interpretation, Optimization, Efficiency, and Best Practices](https://mlsanity.com/paper/2303.10780/) (2023)
- [Hebbian Learning based Orthogonal Projection for Continual Learning of Spiking Neural Networks](https://mlsanity.com/paper/2402.11984/) (2024)
- [Advancing Training Efficiency of Deep Spiking Neural Networks through Rate-based Backpropagation](https://mlsanity.com/paper/2410.11488/) (2024)
- [SNN2ANN: A Fast and Memory-Efficient Training Framework for Spiking Neural Networks](https://mlsanity.com/paper/2206.09449/) (2022)
- [Efficient ANN-Guided Distillation: Aligning Rate-based Features of Spiking Neural Networks through Hybrid Block-wise Replacement](https://mlsanity.com/paper/2503.16572/) (2025)
- [MK-SGN: A Spiking Graph Convolutional Network with Multimodal Fusion and Knowledge Distillation for Skeleton-based Action Recognition](https://mlsanity.com/paper/2404.10210/) (2024)
- [Temporal Spiking Neural Networks with Synaptic Delay for Graph Reasoning](https://mlsanity.com/paper/2405.16851/) (2024)
- [Efficient Parallel Training Methods for Spiking Neural Networks with Constant Time Complexity](https://mlsanity.com/paper/2506.12087/) (2025)
- [Should attention be all we need? The epistemic and ethical implications of unification in machine learning](https://mlsanity.com/paper/2205.08377/) (2022)
- [Exploring Extreme Quantization in Spiking Language Models](https://mlsanity.com/paper/2405.02543/) (2024)
- [Beyond accuracy: generalization properties of bio-plausible temporal credit assignment rules](https://mlsanity.com/paper/2206.00823/) (2022)
- [Online Pseudo-Zeroth-Order Training of Neuromorphic Spiking Neural Networks](https://mlsanity.com/paper/2407.12516/) (2024)
- [Spiking Diffusion Models](https://mlsanity.com/paper/2408.16467/) (2024)
- [ReverB-SNN: Reversing Bit of the Weight and Activation for Spiking Neural Networks](https://mlsanity.com/paper/2506.07720/) (2025)
- [Deep Unsupervised Learning Using Spike-Timing-Dependent Plasticity](https://mlsanity.com/paper/2307.04054/) (2023)
- [Efficient Logit-based Knowledge Distillation of Deep Spiking Neural Networks for Full-Range Timestep Deployment](https://mlsanity.com/paper/2501.15925/) (2025)

## Abstract (arXiv)

Spiking neural networks (SNNs) are brain-inspired models that enable energy-efficient implementation on neuromorphic hardware. However, the supervised training of SNNs remains a hard problem due to the discontinuity of the spiking neuron model. Most existing methods imitate the backpropagation framework and feedforward architectures for artificial neural networks, and use surrogate derivatives or compute gradients with respect to the spiking time to deal with the problem. These approaches either accumulate approximation errors or only propagate information limitedly through existing spikes, and usually require information propagation along time steps with large memory costs and biological implausibility. In this work, we consider feedback spiking neural networks, which are more brain-like, and propose a novel training method that does not rely on the exact reverse of the forward computation. First, we show that the average firing rates of SNNs with feedback connections would gradually evolve to an equilibrium state along time, which follows a fixed-point equation. Then by viewing the forward computation of feedback SNNs as a black-box solver for this equation, and leveraging the implicit differentiation on the equation, we can compute the gradient for parameters without considering the exact forward procedure. In this way, the forward and backward procedures are decoupled and therefore the problem of non-differentiable spiking functions is avoided. We also briefly discuss the biological plausibility of implicit differentiation, which only requires computing another equilibrium. Extensive experiments on MNIST, Fashion-MNIST, N-MNIST, CIFAR-10, and CIFAR-100 demonstrate the superior performance of our method for feedback models with fewer neurons and parameters in a small number of time steps. Our code is avaiable at https://github.com/pkuxmq/IDE-FSNN.