# Equivalence of Equilibrium Propagation and Recurrent Backpropagation

arXiv: https://arxiv.org/abs/1711.08416 · HTML: https://mlsanity.com/paper/1711.08416/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Nov 2017.

## Built on by (most cited first, 30 of 32)

- [Brain-inspired Computational Intelligence via Predictive Coding](https://mlsanity.com/paper/2308.07870/) (2023)
- [Beyond backpropagation: bilevel optimization through implicit differentiation and equilibrium propagation](https://mlsanity.com/paper/2205.03076/) (2022)
- [Backpropagation at the Infinitesimal Inference Limit of Energy-Based Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive Hebbian Learning](https://mlsanity.com/paper/2206.02629/) (2022)
- [Signal Propagation: A Framework for Learning and Inference In a Forward Pass](https://mlsanity.com/paper/2204.01723/) (2022)
- [Predictive Coding beyond Gaussian Distributions](https://mlsanity.com/paper/2211.03481/) (2022)
- [Brain-Inspired Machine Intelligence: A Survey of Neurobiologically-Plausible Credit Assignment](https://mlsanity.com/paper/2312.09257/) (2023)
- [Training Dynamical Binary Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2103.08953/) (2021)
- [Improving equilibrium propagation without weight symmetry through Jacobian homeostasis](https://mlsanity.com/paper/2309.02214/) (2023)
- [Short-Term Plasticity Neurons Learning to Learn and Forget](https://mlsanity.com/paper/2206.14048/) (2022)
- [A Review of Neuroscience-Inspired Machine Learning](https://mlsanity.com/paper/2403.18929/) (2024)
- [Sequence Learning Using Equilibrium Propagation](https://mlsanity.com/paper/2209.09626/) (2022)
- [Toward Lifelong Learning in Equilibrium Propagation: Sleep-like and Awake Rehearsal for Enhanced Stability](https://mlsanity.com/paper/2508.14081/) (2025)
- [Towards training digitally-tied analog blocks via hybrid gradient computation](https://mlsanity.com/paper/2409.03306/) (2024)
- [Biologically-inspired neuronal adaptation improves learning in neural networks](https://mlsanity.com/paper/2204.14008/) (2022)
- [Near-Equilibrium Propagation training in nonlinear wave systems](https://mlsanity.com/paper/2510.16084/) (2025)
- [A brain basis of dynamical intelligence for AI and computational neuroscience](https://mlsanity.com/paper/2105.07284/) (2021)
- [Training a Hopfield Variational Autoencoder with Equilibrium Propagation](https://mlsanity.com/paper/2311.15047/) (2023)
- [Learning by the F-adjoint](https://mlsanity.com/paper/2407.11049/) (2024)
- [Textual Equilibrium Propagation for Deep Compound AI Systems](https://mlsanity.com/paper/2601.21064/) (2026)
- [Heterogeneous Time Constants Improve Stability in Equilibrium Propagation](https://mlsanity.com/paper/2603.03402/) (2026)
- [Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization](https://mlsanity.com/paper/2604.05042/) (2026)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2101.05536/) (2021)
- [Contribute to balance, wire in accordance: Emergence of backpropagation from a simple, bio-plausible neuroplasticity rule](https://mlsanity.com/paper/2405.14139/) (2024)
- [Harnessing uncertainty when learning through Equilibrium Propagation in neural networks](https://mlsanity.com/paper/2503.22810/) (2025)
- [Ubiquity of Emergent Hebbian Dynamics in Regularized Learning](https://mlsanity.com/paper/2505.18069/) (2025)
- [Scalable Equilibrium Propagation via Intermediate Error Signals for Deep Convolutional CRNNs](https://mlsanity.com/paper/2508.15989/) (2025)
- [Deep Feedback Models](https://mlsanity.com/paper/2509.15905/) (2025)
- [VertAX: a differentiable vertex model for learning epithelial tissue mechanics](https://mlsanity.com/paper/2604.06896/) (2026)
- [Dendritic Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2605.08135/) (2026)
- [Score Broadcast and Decorrelation: A General Framework for Broadcast-Based Credit Assignment](https://mlsanity.com/paper/2605.30638/) (2026)

## Abstract (arXiv)

Recurrent Backpropagation and Equilibrium Propagation are supervised learning algorithms for fixed point recurrent neural networks which differ in their second phase. In the first phase, both algorithms converge to a fixed point which corresponds to the configuration where the prediction is made. In the second phase, Equilibrium Propagation relaxes to another nearby fixed point corresponding to smaller prediction error, whereas Recurrent Backpropagation uses a side network to compute error derivatives iteratively. In this work we establish a close connection between these two algorithms. We show that, at every moment in the second phase, the temporal derivatives of the neural activities in Equilibrium Propagation are equal to the error derivatives computed iteratively by Recurrent Backpropagation in the side network. This work shows that it is not required to have a side network for the computation of error derivatives, and supports the hypothesis that, in biological neural networks, temporal derivatives of neural activities may code for error signals.