{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:44:48Z","timestamp":1760060688699,"version":"build-2065373602"},"reference-count":71,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T00:00:00Z","timestamp":1757548800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MICIU\/AEI\/10.13039\/501100011033","award":["PID2023-148793NB-I00","PID2023-149249NB-I00","SGR2021-00453"],"award-info":[{"award-number":["PID2023-148793NB-I00","PID2023-149249NB-I00","SGR2021-00453"]}]},{"name":"Commission for Universities and Research of the Department of Innovation, Universities, and Enterprise of the Generalitat de Catalunya","award":["PID2023-148793NB-I00","PID2023-149249NB-I00","SGR2021-00453"],"award-info":[{"award-number":["PID2023-148793NB-I00","PID2023-149249NB-I00","SGR2021-00453"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The population activity of grid cells from a single module is topologically constrained to a toroidal manifold. Our work proposes an improved version of Gardner\u2019s earlier model, which can account for both geometric properties and force field dynamics. Employing methods from Differential Geometry, we have derived Lagrangian densities that\u2014under very general assumptions and avoiding dimensionful constants\u2014provide a rationale for the trajectories associated with the synaptic spacetime as a global solution to the Einstein\u2013Maxwell field equations. Then, we investigate the helical solutions to show that the synaptic toroidal topological space, as a locally flat Minkowski spacetime, with a Lorentzian metric is geodesically complete and, therefore, exhibits maximal stability. Finally, we consider a Lorentzian metric with curved spacetimes that give rise to Lorentzian tori admitting curvature spacetime singularities.<\/jats:p>","DOI":"10.3390\/sym17091511","type":"journal-article","created":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T07:51:46Z","timestamp":1757577106000},"page":"1511","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lagrangian Field Approach to Einstein\u2013Maxwell Equation for Brain Toroidal Topology"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1261-4408","authenticated-orcid":false,"given":"Manuel","family":"Rivas","sequence":"first","affiliation":[{"name":"Departament d\u2019Enginyeria Qu\u00edmica, EEBE, Universitat Polit\u00e8cnica de Catalunya, Carrer d\u2019Eduard Maristany 10-14, 08930 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0701-200X","authenticated-orcid":false,"given":"Manuel","family":"Reina","sequence":"additional","affiliation":[{"name":"Celltec-UB, Department of Cell Biology, Physiology and Immunology, University of Barcelona, Avinguda Diagonal, 643, 08028 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1038\/s41586-021-04268-7","article-title":"Toroidal topology of the population activity in grid cells","volume":"602","author":"Gardner","year":"2022","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"27249","DOI":"10.1038\/srep27249","article-title":"The topology of large Open Connectome networks for the human brain","volume":"6","author":"Gastner","year":"2016","journal-title":"Sci. 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