{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T14:01:29Z","timestamp":1765288889797,"version":"3.46.0"},"reference-count":23,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T00:00:00Z","timestamp":1765238400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Brain Sciences"],"abstract":"<jats:p>Background\/Objectives: This study presents a time-dependent mathematical model that describes how progressive amyloid-\u03b2 (A\u03b2) accumulation drives the gradual decline of cerebral electrical conductivity during Alzheimer\u2019s disease (AD). Methods: The formulation captures the coupled evolution of molecular burden and electrophysiological function through a pair of interconnected dynamical processes, enabling a mechanistic link between early biochemical alterations and large-scale neural degradation. Results: Simulations reveal a characteristic pattern in which A\u03b2 levels rise steadily toward a pathological plateau, while conductivity follows a delayed but persistent downward trajectory that stabilizes at an impaired state consistent with advanced neurodegeneration. The model reproduces key phenomena reported in experimental and clinical studies, including the slow, irreversible reduction in cortical conductivity and the strong inverse relationship between amyloid burden and electrophysiological integrity. Conclusions: Although intentionally minimal, the framework offers a tractable basis for interpreting disease progression and can be extended to incorporate additional pathological pathways such as tau aggregation, inflammatory responses, or spatial heterogeneity. By providing a compact yet biologically meaningful representation of the interplay between molecular pathology and electrical dysfunction, the model supports the development of computational biomarkers and contributes to a more integrated understanding of AD progression.<\/jats:p>","DOI":"10.3390\/brainsci15121313","type":"journal-article","created":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T13:51:46Z","timestamp":1765288306000},"page":"1313","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Mathematical Model for the Variation of Cerebral Electrical Conductivity and the Amount of \u03b2-Amyloid Protein Values Due to Alzheimer\u2019s Disease"],"prefix":"10.3390","volume":"15","author":[{"given":"Emmanouil","family":"Perakis","sequence":"first","affiliation":[{"name":"Bioinformatics and Human Electrophysiology Laboratory, Department of Informatics, Ionian University, 49100 Corfu, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0053-7847","authenticated-orcid":false,"given":"Panagiotis","family":"Vlamos","sequence":"additional","affiliation":[{"name":"Bioinformatics and Human Electrophysiology Laboratory, Department of Informatics, Ionian University, 49100 Corfu, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hao, W., and Friedman, A. (2016). Mathematical model on Alzheimer\u2019s disease. BMC Syst. Biol., 10.","DOI":"10.1186\/s12918-016-0348-2"},{"key":"ref_2","first-page":"534","article-title":"Evidence of oxidative stress in Alzheimer\u2019s disease brain: Central role for amyloid beta-peptide","volume":"8","author":"Butterfield","year":"2002","journal-title":"Trends Mol. Med."},{"key":"ref_3","first-page":"3609","article-title":"Proteolytic processing and cell biological functions of the amyloid precursor protein","volume":"120","year":"2007","journal-title":"J. Cell Sci."},{"key":"ref_4","first-page":"484","article-title":"NIA-AA Research Framework: Toward a biological definition of Alzheimer\u2019s disease","volume":"15","author":"Jack","year":"2019","journal-title":"Nat. Rev. Neurol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"595","DOI":"10.15252\/emmm.201606210","article-title":"The amyloid hypothesis of Alzheimer\u2019s disease at 25 years","volume":"8","author":"Selkoe","year":"2016","journal-title":"EMBO Mol. Med."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1038\/nm1782","article-title":"Amyloid-beta protein dimers isolated directly from Alzheimer\u2019s brains impair synaptic plasticity and memory","volume":"14","author":"Shankar","year":"2008","journal-title":"Nat. Med."},{"key":"ref_7","first-page":"101","article-title":"Amyloid-beta oligomers: Their production, toxicity and therapeutic intervention","volume":"8","author":"Walsh","year":"2002","journal-title":"Trends Mol. Med."},{"key":"ref_8","unstructured":"Fr\u00f6hlich, F. (2016). Network models of brain disorders: Linking structure and function. Front. Comput. Neurosci., 10."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"a006338","DOI":"10.1101\/cshperspect.a006338","article-title":"Neurotoxicity of amyloid \u03b2-protein: Synaptic and network dysfunction","volume":"2","author":"Mucke","year":"2012","journal-title":"Cold Spring Harb. Perspect. Med."},{"key":"ref_10","first-page":"2639","article-title":"Activity-dependent degeneration explains hub vulnerability in Alzheimer\u2019s disease","volume":"140","author":"Mott","year":"2017","journal-title":"Brain"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1126\/science.1072994","article-title":"The amyloid hypothesis of Alzheimer\u2019s disease: Progress and problems on the road to therapeutics","volume":"297","author":"Hardy","year":"2002","journal-title":"Science"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1006\/bulm.2002.0304","article-title":"A Mathematical Model of the Impact of Novel Treatments on the A\u03b2Burden in the Alzheimer\u2019s Brain, CSF and Plasma","volume":"64","author":"Craft","year":"2002","journal-title":"Bull. Math. Biol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"111194","DOI":"10.1016\/j.mehy.2023.111194","article-title":"A mathematical model on A\u03b2 blood\u2013brain transport: Simulations of plaques\u2019 formation in Alzheimer\u2019s disease","volume":"181","author":"Stura","year":"2023","journal-title":"Med. Hypotheses"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Jafari, N., Sarbaz, Y., Ebrahimi-kalan, A., and Golabi, F. (2022). Novel mathematical model based on cellular automata for study of Alzheimer\u2019s disease progress. Netw. Model. Anal. Health Inform. Bioinform., 11.","DOI":"10.1007\/s13721-022-00366-2"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2777","DOI":"10.62347\/UJQF5204","article-title":"Mathematical modelling of Alzheimer\u2019s disease biomarkers: Targeting Amyloid beta, Tau protein, Apolipoprotein E and Apoptotic pathways","volume":"16","author":"Patel","year":"2024","journal-title":"Am. J. Transl. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"89","DOI":"10.3233\/JAD-230938","article-title":"How can we use mathematical modeling of amyloid-\u03b2 in Alzheimer\u2019s disease research and clinical practices?","volume":"97","author":"Chu","year":"2024","journal-title":"J. Alzheimer\u2019s Dis."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/j.plrev.2025.01.004","article-title":"Mathematical models on Alzheimer\u2019s disease and its treatment: A review","volume":"52","author":"Maji","year":"2025","journal-title":"Phys. Life Rev."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bertsch, M., Franchi, B., Tesi, M.C., and Tora, V. (2023). The role of A\u03b2 and Tau proteins in Alzheimer\u2019s disease: A mathematical model on graphs. J. Math. Biol., 87.","DOI":"10.1007\/s00285-023-01985-7"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.brainresbull.2016.08.018","article-title":"Tau and tauopathies","volume":"126","author":"Arendt","year":"2016","journal-title":"Brain Res. Bull."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1038\/nrd3505","article-title":"The amyloid cascade hypothesis for Alzheimer\u2019s disease: An appraisal for the development of therapeutics","volume":"10","author":"Karran","year":"2011","journal-title":"Nat. Rev. Drug Discov."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Park, S., Jung, S.M., Lee, M.B., Rhee, H.Y., Ryu, C.W., Cho, A.R., Kwon, O.I., and Jahng, G.H. (2022). Application of high-frequency conductivity map using MRI to evaluate it in the brain of Alzheimer\u2019s disease patients. Front. Neurol., 13.","DOI":"10.3389\/fneur.2022.872878"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1016\/j.cell.2015.12.056","article-title":"The cellular phase of Alzheimer\u2019s disease","volume":"164","author":"Karran","year":"2016","journal-title":"Cell"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1038\/nrn3801","article-title":"Modern network science of neurological disorders","volume":"15","author":"Stam","year":"2014","journal-title":"Nat. Rev. Neurosci."}],"container-title":["Brain Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3425\/15\/12\/1313\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T13:56:55Z","timestamp":1765288615000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3425\/15\/12\/1313"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,9]]},"references-count":23,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["brainsci15121313"],"URL":"https:\/\/doi.org\/10.3390\/brainsci15121313","relation":{},"ISSN":["2076-3425"],"issn-type":[{"value":"2076-3425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,9]]}}}