{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T07:37:52Z","timestamp":1777621072315,"version":"3.51.4"},"update-to":[{"DOI":"10.1371\/journal.pcbi.1010221","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2022,7,1]],"date-time":"2022-07-01T00:00:00Z","timestamp":1656633600000}}],"reference-count":42,"publisher":"Public Library of Science (PLoS)","issue":"6","license":[{"start":{"date-parts":[[2022,6,21]],"date-time":"2022-06-21T00:00:00Z","timestamp":1655769600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union Horizon 2020 Research and Innovation","award":["SGA3 n. 945539"],"award-info":[{"award-number":["SGA3 n. 945539"]}]},{"name":"European Union Horizon 2020 Research and Innovation","award":["SGA3 n. 945539"],"award-info":[{"award-number":["SGA3 n. 945539"]}]},{"name":"European Union Horizon 2020 Research and Innovation","award":["SGA2 n. 785907"],"award-info":[{"award-number":["SGA2 n. 785907"]}]},{"name":"European Union Horizon 2020 Research and Innovation","award":["SGA2 n. 785907"],"award-info":[{"award-number":["SGA2 n. 785907"]}]},{"DOI":"10.13039\/501100013168","name":"Istituto Nazionale di Fisica Nucleare","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100013168","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"The field of recurrent neural networks is over-populated by a variety of proposed learning rules and protocols. The scope of this work is to define a generalized framework, to move a step forward towards the unification of this fragmented scenario. In the field of supervised learning, two opposite approaches stand out, error-based and target-based. This duality gave rise to a scientific debate on which learning framework is the most likely to be implemented in biological networks of neurons. Moreover, the existence of spikes raises the question of whether the coding of information is rate-based or spike-based. To face these questions, we proposed a learning model with two main parameters, the rank of the feedback learning matrix R<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula> and the tolerance to spike timing \u03c4<\/jats:italic>\u22c6<\/jats:sub>. We demonstrate that a low (high) rank R<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula> accounts for an error-based (target-based) learning rule, while high (low) tolerance to spike timing promotes rate-based (spike-based) coding. We show that in a store and recall task, high-ranks allow for lower MSE values, while low-ranks enable a faster convergence. Our framework naturally lends itself to Behavioral Cloning and allows for efficiently solving relevant closed-loop tasks, investigating what parameters (<\/mml:mo> R<\/mml:mi> ,<\/mml:mo> \u03c4<\/mml:mi> \u22c6<\/mml:mo><\/mml:msub> )<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula> are optimal to solve a specific task. We found that a high R<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula> is essential for tasks that require retaining memory for a long time (Button and Food). On the other hand, this is not relevant for a motor task (the 2D Bipedal Walker). In this case, we find that precise spike-based coding enables optimal performances. Finally, we show that our theoretical formulation allows for defining protocols to estimate the rank of the feedback error in biological networks. We release a PyTorch implementation of our model supporting GPU parallelization.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1010221","type":"journal-article","created":{"date-parts":[[2022,6,21]],"date-time":"2022-06-21T17:40:52Z","timestamp":1655833252000},"page":"e1010221","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":6,"title":["Error-based or target-based? 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