# Latent Equilibrium: A unified learning theory for arbitrarily fast computation with arbitrarily slow neurons

arXiv: https://arxiv.org/abs/2110.14549 · HTML: https://mlsanity.com/paper/2110.14549/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 55 references that are in the map.
- First posted on arXiv 27 Oct 2021.

## Builds on (most cited first, 17 of 17)

- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Assessing the Scalability of Biologically-Motivated Deep Learning Algorithms and Architectures](https://mlsanity.com/paper/1807.04587/) (2018)
- [Dendritic cortical microcircuits approximate the backpropagation algorithm](https://mlsanity.com/paper/1810.11393/) (2018)
- [Predictive Coding Approximates Backprop along Arbitrary Computation Graphs](https://mlsanity.com/paper/2006.04182/) (2020)
- [Deep Learning without Weight Transport](https://mlsanity.com/paper/1904.05391/) (2019)
- [Towards deep learning with segregated dendrites](https://mlsanity.com/paper/1610.00161/) (2016)
- [Fast and energy-efficient neuromorphic deep learning with first-spike times](https://mlsanity.com/paper/1912.11443/) (2019)
- [Beyond Backprop: Online Alternating Minimization with Auxiliary Variables](https://mlsanity.com/paper/1806.09077/) (2018)
- [Large-Scale Neuromorphic Spiking Array Processors: A quest to mimic the brain](https://mlsanity.com/paper/1805.08932/) (2018)
- [Two Routes to Scalable Credit Assignment without Weight Symmetry](https://mlsanity.com/paper/2003.01513/) (2020)
- [Deep learning with asymmetric connections and Hebbian updates](https://mlsanity.com/paper/1812.07965/) (2018)
- [Versatile emulation of spiking neural networks on an accelerated neuromorphic substrate](https://mlsanity.com/paper/1912.12980/) (2019)
- [Demonstrating Advantages of Neuromorphic Computation: A Pilot Study](https://mlsanity.com/paper/1811.03618/) (2018)
- [Activation Relaxation: A Local Dynamical Approximation to Backpropagation in the Brain](https://mlsanity.com/paper/2009.05359/) (2020)
- [Spike-based causal inference for weight alignment](https://mlsanity.com/paper/1910.01689/) (2019)
- [Characterization and Compensation of Network-Level Anomalies in Mixed-Signal Neuromorphic Modeling Platforms](https://mlsanity.com/paper/1404.7514/) (2014)
- [Dendritic error backpropagation in deep cortical microcircuits](https://mlsanity.com/paper/1801.00062/) (2017)

## Built on by (most cited first, 15 of 15)

- [Single-phase deep learning in cortico-cortical networks](https://mlsanity.com/paper/2206.11769/) (2022)
- [Self-Contrastive Forward-Forward Algorithm](https://mlsanity.com/paper/2409.11593/) (2024)
- [Is artificial consciousness achievable? Lessons from the human brain](https://mlsanity.com/paper/2405.04540/) (2024)
- [Backpropagation through space, time, and the brain](https://mlsanity.com/paper/2403.16933/) (2024)
- [Learning efficient backprojections across cortical hierarchies in real time](https://mlsanity.com/paper/2212.10249/) (2022)
- [Ornstein-Uhlenbeck Adaptation as a Mechanism for Learning in Brains and Machines](https://mlsanity.com/paper/2410.13563/) (2024)
- [Beyond accuracy: generalization properties of bio-plausible temporal credit assignment rules](https://mlsanity.com/paper/2206.00823/) (2022)
- [Neural Langevin Machine: a local asymmetric learning rule can be creative](https://mlsanity.com/paper/2506.23546/) (2025)
- [Toward Practical Equilibrium Propagation: Brain-inspired Recurrent Neural Network with Feedback Regulation and Residual Connections](https://mlsanity.com/paper/2508.11659/) (2025)
- [Spike-based alignment learning solves the weight transport problem](https://mlsanity.com/paper/2503.02642/) (2025)
- [A method for the ethical analysis of brain-inspired AI](https://mlsanity.com/paper/2305.10938/) (2023)
- [Prospective Messaging: Learning in Networks with Communication Delays](https://mlsanity.com/paper/2407.05494/) (2024)
- [Does Feedback Alignment Work at Biological Timescales?](https://mlsanity.com/paper/2510.18808/) (2025)
- [A Variational Latent Equilibrium for Learning in Neuronal Circuits](https://mlsanity.com/paper/2603.09600/) (2026)
- [Prospective Coding Improves Learning in Deep Continuous-Time Recurrent Networks](https://mlsanity.com/paper/2609.04134/) (2026)

## Abstract (arXiv)

The response time of physical computational elements is finite, and neurons are no exception. In hierarchical models of cortical networks each layer thus introduces a response lag. This inherent property of physical dynamical systems results in delayed processing of stimuli and causes a timing mismatch between network output and instructive signals, thus afflicting not only inference, but also learning. We introduce Latent Equilibrium, a new framework for inference and learning in networks of slow components which avoids these issues by harnessing the ability of biological neurons to phase-advance their output with respect to their membrane potential. This principle enables quasi-instantaneous inference independent of network depth and avoids the need for phased plasticity or computationally expensive network relaxation phases. We jointly derive disentangled neuron and synapse dynamics from a prospective energy function that depends on a network's generalized position and momentum. The resulting model can be interpreted as a biologically plausible approximation of error backpropagation in deep cortical networks with continuous-time, leaky neuronal dynamics and continuously active, local plasticity. We demonstrate successful learning of standard benchmark datasets, achieving competitive performance using both fully-connected and convolutional architectures, and show how our principle can be applied to detailed models of cortical microcircuitry. Furthermore, we study the robustness of our model to spatio-temporal substrate imperfections to demonstrate its feasibility for physical realization, be it in vivo or in silico.