{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T16:00:58Z","timestamp":1791475258680,"version":"4.3.4"},"reference-count":46,"publisher":"National Academy of Sciences","issue":"41","license":[{"start":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T00:00:00Z","timestamp":1791417600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,10,13]]},"abstract":"<jats:p>Animal brains exhibit remarkable efficiency in perception and action, while being robust to both external and internal perturbations. The means by which brains accomplish this remains, for now, poorly understood, hindering our understanding of animal and human cognition, as well as our own implementation of efficient algorithms for control of dynamical systems. A potential candidate for a robust mechanism of state estimation and action computation is the free energy principle, but existing implementations of this principle have largely relied on conventional, biologically implausible approaches without spikes. We propose an efficient and robust spiking control framework with realistic biological characteristics. The resulting networks function as free energy constrainers, in which neurons only fire if they reduce the free energy of their internal representation. The networks offer efficient operation through highly sparse activity while matching performance with other similar spiking frameworks, and have high resilience against both external (e.g., sensory noise or collisions) and internal perturbations (e.g., synaptic noise and delays or neuron silencing) that such a network would be faced with when deployed by either an organism or an engineer. Overall, our work provides a mathematical account for spiking control through constraining free energy, providing both better insight into how brain networks might leverage their spiking substrate and a route for implementing efficient control algorithms in neuromorphic hardware.<\/jats:p>","DOI":"10.1073\/pnas.2602161123","type":"journal-article","created":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T15:42:01Z","timestamp":1791474121000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["Efficient and robust control with spikes that constrain free energy"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-9299-4497","authenticated-orcid":false,"given":"Andr\u00e9","family":"Urbano","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02gfc7t72","id-type":"ROR","asserted-by":"publisher"}],"name":"Neuro AI and Robotics, Cajal Neuroscience Center, Spanish National Research Council","place":["Madrid, Spain"]},{"id":[{"id":"https:\/\/ror.org\/04n0g0b29","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Brain and Cognition, Pompeu Fabra University","place":["Barcelona, Spain"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9154-0798","authenticated-orcid":false,"given":"Pablo","family":"Lanillos","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02gfc7t72","id-type":"ROR","asserted-by":"publisher"}],"name":"Neuro AI and Robotics, Cajal Neuroscience Center, Spanish National Research Council","place":["Madrid, Spain"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5043-6724","authenticated-orcid":false,"given":"Sander","family":"Keemink","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/016xsfp80","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Machine Learning and Neural Computing, Don-ders Institute for Brain Cognition and Behaviour, Radboud University","place":["Nijmegen, The Netherlands"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,10,8]]},"reference":[{"key":"e_1_3_4_1_2","unstructured":"D. Purves Ed. Neuroscience (Sinauer Associates Publishers Sunderland MA ed. 3 2004)."},{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1109\/10.486289"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1113\/jphysiol.2004.080903"},{"key":"e_1_3_4_4_2","first-page":"70","article-title":"A free energy principle for the brain","volume":"100","author":"Friston K.","year":"2006","unstructured":"K. Friston, J. Kilner, L. Harrison, A free energy principle for the brain. J. Physiol. 100, 70\u201387 (2006).","journal-title":"J. Physiol."},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.7551\/mitpress\/12441.001.0001"},{"key":"e_1_3_4_6_2","unstructured":"P. Lanillos Active inference in robotics and artificial agents: Survey and challenges. arXiv [Preprint] (2021). https:\/\/doi.org\/10.48550\/arXiv.2112.01871 (Accessed 9 April 2025)."},{"key":"e_1_3_4_7_2","doi-asserted-by":"publisher","DOI":"10.3389\/fnbot.2018.00035"},{"key":"e_1_3_4_8_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1003258"},{"key":"e_1_3_4_9_2","doi-asserted-by":"publisher","DOI":"10.24072\/pcjournal.69"},{"key":"e_1_3_4_10_2","unstructured":"C. Eliasmith Neural Engineering (Bradford Books The MIT Press; Cambridge Massachusetts London England 2004)."},{"key":"e_1_3_4_11_2","doi-asserted-by":"crossref","unstructured":"F. S. Slijkhuis S. W. Keemink P. Lanillos \u201cClosed-Form Control With Spike Coding Networks \u201d in IEEE Transactions on Cognitive and Developmental Systems vol. 16 pp. 1677\u20131687.","DOI":"10.1109\/TCDS.2023.3320251"},{"key":"e_1_3_4_12_2","doi-asserted-by":"crossref","unstructured":"F. Huang J. Riehl S. Ching \u201cOptimizing the dynamics of spiking networks for decoding and control\u201d in 2017 American Control Conference (ACC) (IEEE Seattle WA 2017) pp. 2792\u20132798.","DOI":"10.23919\/ACC.2017.7963374"},{"key":"e_1_3_4_13_2","doi-asserted-by":"crossref","unstructured":"F. Huang S. Ching \u201cDynamical spiking networks for distributed control of nonlinear systems\u201d in 2018 Annual American Control Conference (ACC) (IEEE Milwaukee WI 2018) pp. 1190\u20131195.","DOI":"10.23919\/ACC.2018.8430996"},{"key":"e_1_3_4_14_2","doi-asserted-by":"publisher","DOI":"10.7554\/eLife.73276"},{"key":"e_1_3_4_15_2","unstructured":"A. W. N\u2019dri Predictive coding with spiking neural networks: A survey. arXiv [Preprint] (2024). https:\/\/doi.org\/10.48550\/arXiv.2409.05386."},{"key":"e_1_3_4_16_2","doi-asserted-by":"publisher","DOI":"10.1038\/s42256-020-00257-z"},{"key":"e_1_3_4_17_2","unstructured":"A. Mancoo S. Keemink C. K. Machens \u201cUnderstanding spiking networks through convex optimization\u201d in Advances in Neural Information Processing Systems H. Larochelle M. Ranzato R. Hadsell M. F. Balcan H. Lin Eds. (Curran Associates Inc. 2020) vol. 33 pp. 8824\u20138835."},{"key":"e_1_3_4_18_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.tics.2021.05.007"},{"key":"e_1_3_4_19_2","doi-asserted-by":"publisher","DOI":"10.1162\/neco_a_01627"},{"key":"e_1_3_4_20_2","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2020.2974451"},{"key":"e_1_3_4_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2023.120310"},{"key":"e_1_3_4_22_2","doi-asserted-by":"publisher","DOI":"10.1098\/rsif.2016.0616"},{"key":"e_1_3_4_23_2","doi-asserted-by":"crossref","unstructured":"P. Lanillos J. Pages G. Cheng Robot self\/other distinction: Active inference meets neural networks learning in a mirror. Version Number 1. arXiv [Preprint] (2020). https:\/\/doi.org\/10.48550\/arXiv.2004.05473 (Accessed 9 April 2025).","DOI":"10.3233\/FAIA200372"},{"key":"e_1_3_4_24_2","doi-asserted-by":"publisher","DOI":"10.1038\/s42003-021-02994-2"},{"key":"e_1_3_4_25_2","doi-asserted-by":"publisher","DOI":"10.7554\/eLife.12454"},{"key":"e_1_3_4_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuron.2017.05.016"},{"key":"e_1_3_4_27_2","doi-asserted-by":"publisher","DOI":"10.1109\/JPROC.2021.3067593"},{"key":"e_1_3_4_28_2","doi-asserted-by":"publisher","DOI":"10.1038\/s44182-023-00001-w"},{"key":"e_1_3_4_29_2","doi-asserted-by":"publisher","DOI":"10.7551\/mitpress\/9780262514620.001.0001"},{"key":"e_1_3_4_30_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-024-08253-8"},{"key":"e_1_3_4_31_2","doi-asserted-by":"publisher","DOI":"10.3390\/s120403831"},{"key":"e_1_3_4_32_2","unstructured":"P. Agliati Spiking neurons as predictive controllers of linear systems. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2507.16495 (Accessed 15 September 2025)."},{"key":"e_1_3_4_33_2","doi-asserted-by":"crossref","unstructured":"D. Casanueva-Morato C. Wu G. Indiveri J. P. Dominguez-Morales A. Linares-Barranco \u201cSpace-time smooth control of closed-loop neuromorphic robotic arms using spiking neural networks\u201d in 2025 International Joint Conference on Neural Networks (IJCNN) (IEEE Rome Italy 2025) pp. 1\u20139.","DOI":"10.1109\/IJCNN64981.2025.11228735"},{"key":"e_1_3_4_34_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0893-6080(01)00078-8"},{"key":"e_1_3_4_35_2","doi-asserted-by":"publisher","DOI":"10.1162\/neco_a_01658"},{"key":"e_1_3_4_36_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1004895"},{"key":"e_1_3_4_37_2","unstructured":"A. Alemi C. Machens S. Den\u00e8ve J. J. Slotine Learning arbitrary dynamics in efficient balanced spiking networks using local plasticity rules. Version number 2. arXiv [Preprint] (2017). https:\/\/doi.org\/10.48550\/arXiv.1705.08026 (Accessed 11 March 2025)."},{"key":"e_1_3_4_38_2","doi-asserted-by":"crossref","unstructured":"B. Scellier Y. Bengio Equilibrium propagation: Bridging the gap between energy-based models and backpropagation. arXiv [Preprint] (2017). https:\/\/arxiv.org\/abs\/1602.05179 (Accessed 11 March 2025).","DOI":"10.3389\/fncom.2017.00024"},{"key":"e_1_3_4_39_2","doi-asserted-by":"crossref","unstructured":"G. Pourcel D. Basu M. Ernoult A. Gilra Lagrangian-based equilibrium propagation: Generalisation to arbitrary boundary conditions & equivalence with Hamiltonian echo learning. arXiv [Preprint] (2025). https:\/\/arxiv.org\/abs\/2506.06248 (Accessed 11 March 2025).","DOI":"10.1088\/2634-4386\/aeac80"},{"key":"e_1_3_4_40_2","unstructured":"P. O\u2019Connor E. Gavves M. Welling \u201cTraining a spiking neural network with equilibrium propagation\u201d in Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research K. Chaudhuri M. Sugiyama Eds. (PMLR 2019) vol. 89 pp. 1516\u20131523."},{"key":"e_1_3_4_41_2","first-page":"780","article-title":"World models and predictive coding for cognitive and developmental robotics","volume":"37","author":"Taniguchi T.","year":"2023","unstructured":"T. Taniguchi , World models and predictive coding for cognitive and developmental robotics. Front. Challenges 37, 780\u2013806 (2023).","journal-title":"Front. Challenges"},{"key":"e_1_3_4_42_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00422-010-0364-z"},{"key":"e_1_3_4_43_2","unstructured":"C. L. Buckley C. S. Kim S. McGregor A. K. Seth The free energy principle for action and perception: A mathematical review. arXiv [Preprint] (2017). https:\/\/arxiv.org\/abs\/1705.09156 (Accessed 9 April 2025)."},{"key":"e_1_3_4_44_2","doi-asserted-by":"publisher","DOI":"10.1038\/nn.4243"},{"key":"e_1_3_4_45_2","doi-asserted-by":"publisher","DOI":"10.3389\/fninf.2013.00048"},{"key":"e_1_3_4_46_2","unstructured":"A. Urbano P. Lanillos S. Keemink Spiking-Free-Energy-Constrainer. GitHub. https:\/\/github.com\/neuro-ai-robotics\/Spiking-Free-Energy-Constrainer. Deposited 2 March 2026."}],"container-title":["Proceedings of the National Academy of Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pnas.org\/doi\/pdf\/10.1073\/pnas.2602161123","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T15:42:34Z","timestamp":1791474154000},"score":1,"resource":{"primary":{"URL":"https:\/\/pnas.org\/doi\/10.1073\/pnas.2602161123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,8]]},"references-count":46,"journal-issue":{"issue":"41","published-print":{"date-parts":[[2026,10,13]]}},"alternative-id":["10.1073\/pnas.2602161123"],"URL":"https:\/\/doi.org\/10.1073\/pnas.2602161123","relation":{},"ISSN":["0027-8424","1091-6490"],"issn-type":[{"value":"0027-8424","type":"print"},{"value":"1091-6490","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,8]]},"assertion":[{"value":"2026-03-20","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-07-30","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-10-08","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e2602161123"}}