{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:21:18Z","timestamp":1760059278502,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T00:00:00Z","timestamp":1748908800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)","doi-asserted-by":"publisher","award":["UIDB\/50021\/2020","Novos Talentos em Matem\u00e1tica programme"],"award-info":[{"award-number":["UIDB\/50021\/2020","Novos Talentos em Matem\u00e1tica programme"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000348","name":"Calouste Gulbenkian Foundation","doi-asserted-by":"publisher","award":["UIDB\/50021\/2020","Novos Talentos em Matem\u00e1tica programme"],"award-info":[{"award-number":["UIDB\/50021\/2020","Novos Talentos em Matem\u00e1tica programme"]}],"id":[{"id":"10.13039\/501100000348","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>Equilibrium Propagation (EP) offers a biologically inspired alternative to backpropagation for training recurrent neural networks, but its reliance on symmetric feedback connections and stability limitations hinders practical adoption. The DirEcted EP (DEEP) model relaxes the symmetry constraint, yet suffers from convergence issues and lacks a principled learning guarantee. In this work, we generalize DEEP by incorporating neuronal leakage, providing new convergence criteria for the network\u2019s dynamics. We additionally propose a novel local learning rule closely linked to the objective function\u2019s gradient and establish sufficient conditions for reliable learning in small networks. Our results resolve longstanding stability challenges and bring energy-based learning models closer to biologically plausible and provably effective neural computation.<\/jats:p>","DOI":"10.3390\/math13111866","type":"journal-article","created":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T06:21:51Z","timestamp":1748931711000},"page":"1866","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Directed Equilibrium Propagation Revisited"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-7116-6293","authenticated-orcid":false,"given":"Pedro","family":"Costa","sequence":"first","affiliation":[{"name":"Instituto Superior T\u00e9cnico, University of Lisbon, 1049-001 Lisbon, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1369-0085","authenticated-orcid":false,"given":"Pedro A.","family":"Santos","sequence":"additional","affiliation":[{"name":"Instituto Superior T\u00e9cnico, University of Lisbon, 1049-001 Lisbon, Portugal"},{"name":"INESC-ID, 1000-029 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Shalev-Shwartz, S., and Ben-David, S. (2014). Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press.","DOI":"10.1017\/CBO9781107298019"},{"key":"ref_2","unstructured":"Silver, D., Hubert, T., Schrittwieser, J., Antonoglou, I., Lai, M., Guez, A., Lanctot, M., Sifre, L., Kumaran, D., and Graepel, T. (2017). Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm. arXiv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1038\/nature24270","article-title":"Mastering the game of Go without human knowledge","volume":"550","author":"Silver","year":"2017","journal-title":"Nature"},{"key":"ref_4","unstructured":"Bishop, C.M. (2006). Pattern Recognition and Machine Learning, Springer."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Scellier, B., and Bengio, Y. (2017). Equilibrium Propagation: Bridging the Gap between Energy-Based Models and Backpropagation. Front. Comput. Neurosci., 11.","DOI":"10.3389\/fncom.2017.00024"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Laborieux, A., Ernoult, M., Scellier, B., Bengio, Y., Grollier, J., and Querlioz, D. (2021). Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing Its Gradient Estimator Bias. Front. Neurosci., 15.","DOI":"10.3389\/fnins.2021.633674"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"102222","DOI":"10.1016\/j.isci.2021.102222","article-title":"EqSpike: Spike-driven equilibrium propagation for neuromorphic implementations","volume":"24","author":"Martin","year":"2021","journal-title":"iScience"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Foroushani, A.N., Assaf, H., Noshahr, F.H., Savaria, Y., and Sawan, M. (2020, January 12\u201314). Analog Circuits to Accelerate the Relaxation Process in the Equilibrium Propagation Algorithm. 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