# Backpropagation through space, time, and the brain

arXiv: https://arxiv.org/abs/2403.16933 · HTML: https://mlsanity.com/paper/2403.16933/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 111 references that are in the map.
- First posted on arXiv 25 Mar 2024.

## Builds on (most cited first, 30 of 32)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers](https://mlsanity.com/paper/2110.13985/) (2021)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Diagonal State Spaces are as Effective as Structured State Spaces](https://mlsanity.com/paper/2203.14343/) (2022)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Griffin: Mixing Gated Linear Recurrences with Local Attention for Efficient Language Models](https://mlsanity.com/paper/2402.19427/) (2024)
- [Hello Edge: Keyword Spotting on Microcontrollers](https://mlsanity.com/paper/1711.07128/) (2017)
- [Dendritic cortical microcircuits approximate the backpropagation algorithm](https://mlsanity.com/paper/1810.11393/) (2018)
- [Deep Learning without Weight Transport](https://mlsanity.com/paper/1904.05391/) (2019)
- [A Survey of Neuromorphic Computing and Neural Networks in Hardware](https://mlsanity.com/paper/1705.06963/) (2017)
- [Event-Based Backpropagation can compute Exact Gradients for Spiking Neural Networks](https://mlsanity.com/paper/2009.08378/) (2020)
- [Fast and energy-efficient neuromorphic deep learning with first-spike times](https://mlsanity.com/paper/1912.11443/) (2019)
- [Random feedback weights support learning in deep neural networks](https://mlsanity.com/paper/1411.0247/) (2014)
- [A Unified Framework of Online Learning Algorithms for Training Recurrent Neural Networks](https://mlsanity.com/paper/1907.02649/) (2019)
- [Surrogate gradients for analog neuromorphic computing](https://mlsanity.com/paper/2006.07239/) (2020)
- [Universality of Linear Recurrences Followed by Non-linear Projections: Finite-Width Guarantees and Benefits of Complex Eigenvalues](https://mlsanity.com/paper/2307.11888/) (2023)
- [NeuroBench: A Framework for Benchmarking Neuromorphic Computing Algorithms and Systems](https://mlsanity.com/paper/2304.04640/) (2023)
- [Online learning of long-range dependencies](https://mlsanity.com/paper/2305.15947/) (2023)
- [full-FORCE: A Target-Based Method for Training Recurrent Networks](https://mlsanity.com/paper/1710.03070/) (2017)
- [Latent Equilibrium: A unified learning theory for arbitrarily fast computation with arbitrarily slow neurons](https://mlsanity.com/paper/2110.14549/) (2021)
- [Demonstrating Advantages of Neuromorphic Computation: A Pilot Study](https://mlsanity.com/paper/1811.03618/) (2018)
- [Predicting non-linear dynamics by stable local learning in a recurrent spiking neural network](https://mlsanity.com/paper/1702.06463/) (2017)
- [Minimizing Control for Credit Assignment with Strong Feedback](https://mlsanity.com/paper/2204.07249/) (2022)
- [WaveSense: Efficient Temporal Convolutions with Spiking Neural Networks for Keyword Spotting](https://mlsanity.com/paper/2111.01456/) (2021)
- [Learning efficient backprojections across cortical hierarchies in real time](https://mlsanity.com/paper/2212.10249/) (2022)

## Built on by (most cited first, 10 of 10)

- [Self-Contrastive Forward-Forward Algorithm](https://mlsanity.com/paper/2409.11593/) (2024)
- [Accelerated Predictive Coding Networks via Direct Kolen-Pollack Feedback Alignment](https://mlsanity.com/paper/2602.15571/) (2026)
- [Spike-based alignment learning solves the weight transport problem](https://mlsanity.com/paper/2503.02642/) (2025)
- [Recurrent Joint Embedding Predictive Architecture with Recurrent Forward Propagation Learning](https://mlsanity.com/paper/2411.16695/) (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)
- [Dynamics and Representation Structure of Local Approximations to Gradient-Based Learning in Linear Recurrent Neural Networks](https://mlsanity.com/paper/2606.00243/) (2026)
- [Wave Function Backpropagation with Explicit Temporal-Interval Dynamics](https://mlsanity.com/paper/2609.00503/) (2026)
- [Prospective Coding Improves Learning in Deep Continuous-Time Recurrent Networks](https://mlsanity.com/paper/2609.04134/) (2026)
- [Increasing Width Allows Greedy Layer-wise Training to Rival End-to-End Backpropagation in Self-Supervised Learning](https://mlsanity.com/paper/2610.00753/) (2026)

## Abstract (arXiv)

How physical networks of neurons, bound by spatio-temporal locality constraints, can perform efficient credit assignment, remains, to a large extent, an open question. In machine learning, the answer is almost universally given by the error backpropagation algorithm, through both space and time. However, this algorithm is well-known to rely on biologically implausible assumptions, in particular with respect to spatio-temporal (non-)locality. Alternative forward-propagation models such as real-time recurrent learning only partially solve the locality problem, but only at the cost of scaling, due to prohibitive storage requirements. We introduce Generalized Latent Equilibrium (GLE), a computational framework for fully local spatio-temporal credit assignment in physical, dynamical networks of neurons. We start by defining an energy based on neuron-local mismatches, from which we derive both neuronal dynamics via stationarity and parameter dynamics via gradient descent. The resulting dynamics can be interpreted as a real-time, biologically plausible approximation of backpropagation through space and time in deep cortical networks with continuous-time neuronal dynamics and continuously active, local synaptic plasticity. In particular, GLE exploits the morphology of dendritic trees to enable more complex information storage and processing in single neurons, as well as the ability of biological neurons to phase-shift their output rate with respect to their membrane potential, which is essential in both directions of information propagation. For the forward computation, it enables the mapping of time-continuous inputs to neuronal space, effectively performing a spatio-temporal convolution. For the backward computation, it permits the temporal inversion of feedback signals, which consequently approximate the adjoint variables necessary for useful parameter updates.