# Generalization of Equilibrium Propagation to Vector Field Dynamics

arXiv: https://arxiv.org/abs/1808.04873 · HTML: https://mlsanity.com/paper/1808.04873/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Aug 2018.

## Built on by (most cited first, 23 of 23)

- [Predictive Coding: a Theoretical and Experimental Review](https://mlsanity.com/paper/2107.12979/) (2021)
- [Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations](https://mlsanity.com/paper/2209.00530/) (2022)
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- [A deep learning theory for neural networks grounded in physics](https://mlsanity.com/paper/2103.09985/) (2021)
- [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)
- [Dual Propagation: Accelerating Contrastive Hebbian Learning with Dyadic Neurons](https://mlsanity.com/paper/2302.01228/) (2023)
- [Improving equilibrium propagation without weight symmetry through Jacobian homeostasis](https://mlsanity.com/paper/2309.02214/) (2023)
- [A Review of Neuroscience-Inspired Machine Learning](https://mlsanity.com/paper/2403.18929/) (2024)
- [Equilibrium Propagation for Learning in Lagrangian Dynamical Systems](https://mlsanity.com/paper/2505.07363/) (2025)
- [Sequence Learning Using Equilibrium Propagation](https://mlsanity.com/paper/2209.09626/) (2022)
- [Correlative Information Maximization: A Biologically Plausible Approach to Supervised Deep Neural Networks without Weight Symmetry](https://mlsanity.com/paper/2306.04810/) (2023)
- [Equilibrium Propagation for Non-Conservative Systems](https://mlsanity.com/paper/2602.03670/) (2026)
- [Toward Practical Equilibrium Propagation: Brain-inspired Recurrent Neural Network with Feedback Regulation and Residual Connections](https://mlsanity.com/paper/2508.11659/) (2025)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2101.05536/) (2021)
- [Ubiquity of Emergent Hebbian Dynamics in Regularized Learning](https://mlsanity.com/paper/2505.18069/) (2025)
- [Finite-Nudge Equilibrium Propagation in Thermal Ensembles](https://mlsanity.com/paper/2511.22024/) (2025)
- [The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks](https://mlsanity.com/paper/2604.10272/) (2026)
- [Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machines](https://mlsanity.com/paper/2606.13454/) (2026)
- [Unifying Physical Backpropagation](https://mlsanity.com/paper/2608.11585/) (2026)
- [An all-transistor physical-learning transformer can be trained to perform in-context learning](https://mlsanity.com/paper/2610.12571/) (2026)

## Abstract (arXiv)

The biological plausibility of the backpropagation algorithm has long been doubted by neuroscientists. Two major reasons are that neurons would need to send two different types of signal in the forward and backward phases, and that pairs of neurons would need to communicate through symmetric bidirectional connections. We present a simple two-phase learning procedure for fixed point recurrent networks that addresses both these issues. In our model, neurons perform leaky integration and synaptic weights are updated through a local mechanism. Our learning method generalizes Equilibrium Propagation to vector field dynamics, relaxing the requirement of an energy function. As a consequence of this generalization, the algorithm does not compute the true gradient of the objective function, but rather approximates it at a precision which is proven to be directly related to the degree of symmetry of the feedforward and feedback weights. We show experimentally that our algorithm optimizes the objective function.