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. 2020 Jul 17;11:3625. doi: 10.1038/s41467-020-17236-y

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© The Author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 1 — a RSNN with network inputs x, neuron spikes z, hidden neuron states h, and output targets y^*, for each time step t of the RSNN computation. Output neurons y provide a low-pass filter of a weighted sum of network spikes z. b BPTT computes gradients in the unrolled version of the network. It has a new copy of the neurons of the RSNN for each time step t. A synaptic connection from neuron i to neuron j of the RSNN is replaced by an array of feedforward connections, one for each time step t, which goes from the copy of neuron i in the layer for time step t to a copy of neuron j in the layer for time step t + 1. All synapses in this array have the same weight: the weight of this synaptic connection in the RSNN. c Loss gradients of BPTT are propagated backwards in time and retrograde across synapses in an offline manner, long after the forward computation has passed a layer. d Online learning dynamics of e-prop. Feedforward computation of eligibility traces is indicated in blue. These are combined with online learning signals according to Eq. (1).