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Learning efficient backprojections across cortical hierarchies in real time

A preprint version of the article is available at arXiv.


Models of sensory processing and learning in the cortex need to efficiently assign credit to synapses in all areas. In deep learning, a known solution is error backpropagation, which requires biologically implausible weight transport from feed-forwards to feedback paths. We introduce phaseless alignment learning, a bio-plausible method to learn efficient feedback weights in layered cortical hierarchies. This is achieved by exploiting the noise naturally found in biophysical systems as an additional carrier of information. In our dynamical system, all weights are learned simultaneously with always-on plasticity and using only information locally available to the synapses. Our method is completely phase-free (no forwards and backwards passes or phased learning) and allows for efficient error propagation across multi-layer cortical hierarchies, while maintaining biologically plausible signal transport and learning. Our method is applicable to a wide class of models and improves on previously known biologically plausible ways of credit assignment: compared to random synaptic feedback, it can solve complex tasks with fewer neurons and learn more useful latent representations. We demonstrate this on various classification tasks using a cortical microcircuit model with prospective coding.

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Fig. 1: Sensory processing over cortical hierarchies.
Fig. 2: Cortical microcircuit setup with one hidden layer.
Fig. 3: PAL aligns weight updates with BP in deep networks.
Fig. 4: PAL improves learning on teacher–student and classification tasks compared to fixed random synaptic feedback.
Fig. 5: PAL learns useful latent representations, where FA fails to do so.
Fig. 6: PAL outperforms (D)FA on CIFAR-10 using a continuous-time convolutional neural network.

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Data availability

The datasets analysed during the current study are publicly available in the following repositories: (Yin-Yang dataset40), (MNIST dataset71) and (CIFAR-10, ref. 72).

Code availability

The simulations used custom code written in Python using numpy (v.1.26.2) and PyTorch (v.2.0.1+cu117). Some simulations were made using a custom module for the GeNN simulation suite. All code is made available under (ref. 73).


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We thank J. Jordan, A. Meulemans and J. Sacramento for valuable discussions. We gratefully acknowledge funding from the European Union under grant agreement nos. 604102, 720270, 785907 and 945539 (Human Brain Project) and the Manfred Stärk Foundation. Additionally, our work has greatly benefited from access to the Fenix Infrastructure resources, which are partially funded from the European Union’s Horizon 2020 Research and Innovation programme through the ICEI project under the grant agreement no. 800858. This includes access to Piz Daint at the Swiss National Supercomputing Centre, Switzerland. Further calculations were performed on UBELIX, the High Performance Computing cluster at the University of Bern.

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K.M. derived, with contributions by L.K. and M.A.P., the PAL algorithm. K.M. and L.K. adapted the dendritic microcircuit model to include PAL for learning the feedback weights. G.P.G. and T.N. developed a dendritic microcircuit module for the GeNN simulator. L.K. added the latent equilibrium and PAL mechanisms to the module. K.M. and L.K. performed the simulation experiments. I.J. and K.M. worked on scaling the algorithm to a larger benchmark during the revision process. The paper was mainly written by K.M., aided by L.K. and M.A.P. M.A.P. and W.S. provided supervision and funding to this project.

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Correspondence to Kevin Max.

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Nature Machine Intelligence thanks Ben Lansdell and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Derivations, Extensions of theory, Supplementary Figs. 1–3, Algorithm 1 and Tables 1 and 2.

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Max, K., Kriener, L., Pineda García, G. et al. Learning efficient backprojections across cortical hierarchies in real time. Nat Mach Intell 6, 619–630 (2024).

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