{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:50227"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:50227","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Homöostase in Neuronen","abstract":"Neurons maintain their electrical activity patterns over long periods of time despite of ongoing channel turnover, cell growth and varying extracellular conditions. In order to maintain these fixed patterns Abbott et al. suggested in 1993 that neurons may regulate the maximal conductances for various currents, which leads to a variant of the classical Hodgkin-Huxley model (1952), where the maximal conductance of each ionic current was assumed to be a fixed parameter rather than a dynamical. The regulation process requires feedback systems capable of reacting to changes of electrical activity on different time scales. In the model under consideration, the intracellular calcium concentration serves as such a regulatory feedback element for the regulation of maximal conductances, because this concentration links neuronal conductances to electrical activity. If the activity pattern leaves the equilibrium state the calcium sensors ensure that the values of the conductances are adapted to the modified activity level. Abbott et al. investigated this model and several variants, mostly with the help of numerical simulations. The purpose of this work is a mathematical analysis of these types of models. Based on the models of Abbott et al. a five-dimensional system of differential equations is developed. Intermediate steps of independent interest are(1) Reduction of dimension, that is common approaches to reduce the four-dimensional system of Hodgkin and Huxley to a system of dimension two, as already done by FitzHugh and Nagumo, are described and confronted by a new approach based on the criterion of nearly-invariance. A related approach from singular perturbation theory is also discussed.(2) Qualitative analysis of the reduced system, that is the functions in the model equations are generally characterized by properties like positivity or monotonicity, rather than by concrete functional expressions and the analysis proceeds accordingly. Results as well for the existence and the number of stationary points and their stability as global properties can be received. (3) Maximal conductances are treated as parameters with regard to a possible Hopf bifurcation.(4) Critical analysis of the Abbott model and the alternative model for calcium target concentrations, that is the Abbott model is analysed and due to some inconsistencies an alternative model for the control of homeostasis in neurons is developed.","abstract_html":"Neurons maintain their electrical activity patterns over long periods of time despite of ongoing channel turnover, cell growth and varying extracellular conditions. In order to maintain these fixed patterns Abbott et al. suggested in 1993 that neurons may regulate the maximal conductances for various currents, which leads to a variant of the classical Hodgkin-Huxley model (1952), where the maximal conductance of each ionic current was assumed to be a fixed parameter rather than a dynamical. The regulation process requires feedback systems capable of reacting to changes of electrical activity on different time scales. In the model under consideration, the intracellular calcium concentration serves as such a regulatory feedback element for the regulation of maximal conductances, because this concentration links neuronal conductances to electrical activity. If the activity pattern leaves the equilibrium state the calcium sensors ensure that the values of the conductances are adapted to the modified activity level. Abbott et al. investigated this model and several variants, mostly with the help of numerical simulations. The purpose of this work is a mathematical analysis of these types of models. Based on the models of Abbott et al. a five-dimensional system of differential equations is developed. Intermediate steps of independent interest are(1) Reduction of dimension, that is common approaches to reduce the four-dimensional system of Hodgkin and Huxley to a system of dimension two, as already done by FitzHugh and Nagumo, are described and confronted by a new approach based on the criterion of nearly-invariance. A related approach from singular perturbation theory is also discussed.(2) Qualitative analysis of the reduced system, that is the functions in the model equations are generally characterized by properties like positivity or monotonicity, rather than by concrete functional expressions and the analysis proceeds accordingly. Results as well for the existence and the number of stationary points and their stability as global properties can be received. (3) Maximal conductances are treated as parameters with regard to a possible Hopf bifurcation.(4) Critical analysis of the Abbott model and the alternative model for calcium target concentrations, that is the Abbott model is analysed and due to some inconsistencies an alternative model for the control of homeostasis in neurons is developed.","abstract_has_math":false,"creators":["Dossing, Daniela"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Walcher, Sebastian"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2008,"date_issued":"2008","date_published":"2008","updated_at":"2026-07-30T19:40:16Z","subjects":["info:eu-repo/classification/ddc/510","Hodgkin, Alan L.","Differentialgleichungssystem","Homöostase","Ionentheorie der Erregung","Mathematik","Reduktion","Fast-Invarianz","Poincaré-Bendixson","Tikhonov","Steuerung","Abbott","Reduction","Nearly-Invariance"],"languages":["ger"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112781%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112781%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112781%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/50227","outbound_label":"Repository record","outbound_source":"dc:identifier"},"source_record":{"url":"https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A50227","prefix":"oai_dc"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Walcher, Sebastian"]},{"key":"dc:creator","label":"Author","values":["Dossing, Daniela"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2008"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-24858"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/510","Hodgkin, Alan L.","Differentialgleichungssystem","Homöostase","Ionentheorie der Erregung","Mathematik","Reduktion","Fast-Invarianz","Poincaré-Bendixson","Tikhonov","Steuerung","Abbott","Reduction","Nearly-Invariance"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["ger"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/50227","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112781%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Neurons maintain their electrical activity patterns over long periods of time despite of ongoing channel turnover, cell growth and varying extracellular conditions. In order to maintain these fixed patterns Abbott et al. suggested in 1993 that neurons may regulate the maximal conductances for various currents, which leads to a variant of the classical Hodgkin-Huxley model (1952), where the maximal conductance of each ionic current was assumed to be a fixed parameter rather than a dynamical. The regulation process requires feedback systems capable of reacting to changes of electrical activity on different time scales. In the model under consideration, the intracellular calcium concentration serves as such a regulatory feedback element for the regulation of maximal conductances, because this concentration links neuronal conductances to electrical activity. If the activity pattern leaves the equilibrium state the calcium sensors ensure that the values of the conductances are adapted to the modified activity level. Abbott et al. investigated this model and several variants, mostly with the help of numerical simulations. The purpose of this work is a mathematical analysis of these types of models. Based on the models of Abbott et al. a five-dimensional system of differential equations is developed. Intermediate steps of independent interest are(1) Reduction of dimension, that is common approaches to reduce the four-dimensional system of Hodgkin and Huxley to a system of dimension two, as already done by FitzHugh and Nagumo, are described and confronted by a new approach based on the criterion of nearly-invariance. A related approach from singular perturbation theory is also discussed.(2) Qualitative analysis of the reduced system, that is the functions in the model equations are generally characterized by properties like positivity or monotonicity, rather than by concrete functional expressions and the analysis proceeds accordingly. Results as well for the existence and the number of stationary points and their stability as global properties can be received. (3) Maximal conductances are treated as parameters with regard to a possible Hopf bifurcation.(4) Critical analysis of the Abbott model and the alternative model for calcium target concentrations, that is the Abbott model is analysed and due to some inconsistencies an alternative model for the control of homeostasis in neurons is developed."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University 200 S. : Ill., graph. Darst. (2008). = Aachen, Techn. Hochsch., Diss., 2008"]},{"key":"dc:title","label":"Title","values":["Homöostase in Neuronen"]}]}],"canonical_facts":{"dc:contributor":["Walcher, Sebastian"],"dc:coverage":["DE"],"dc:creator":["Dossing, Daniela"],"dc:date":["2008"],"dc:description":["Neurons maintain their electrical activity patterns over long periods of time despite of ongoing channel turnover, cell growth and varying extracellular conditions. 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The purpose of this work is a mathematical analysis of these types of models. Based on the models of Abbott et al. a five-dimensional system of differential equations is developed. Intermediate steps of independent interest are(1) Reduction of dimension, that is common approaches to reduce the four-dimensional system of Hodgkin and Huxley to a system of dimension two, as already done by FitzHugh and Nagumo, are described and confronted by a new approach based on the criterion of nearly-invariance. A related approach from singular perturbation theory is also discussed.(2) Qualitative analysis of the reduced system, that is the functions in the model equations are generally characterized by properties like positivity or monotonicity, rather than by concrete functional expressions and the analysis proceeds accordingly. Results as well for the existence and the number of stationary points and their stability as global properties can be received. (3) Maximal conductances are treated as parameters with regard to a possible Hopf bifurcation.(4) Critical analysis of the Abbott model and the alternative model for calcium target concentrations, that is the Abbott model is analysed and due to some inconsistencies an alternative model for the control of homeostasis in neurons is developed."],"dc:identifier":["https://publications.rwth-aachen.de/record/50227","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112781%22"],"dc:language":["ger"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-24858"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University 200 S. : Ill., graph. Darst. (2008). = Aachen, Techn. 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