{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T18:48:37Z","timestamp":1778525317260,"version":"3.51.4"},"reference-count":29,"publisher":"American Mathematical Society (AMS)","issue":"301","license":[{"start":{"date-parts":[[2017,1,8]],"date-time":"2017-01-08T00:00:00Z","timestamp":1483833600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00c3\u00bcr Bildung und Forschung","doi-asserted-by":"publisher","award":["FKZ 01GQ1001B"],"award-info":[{"award-number":["FKZ 01GQ1001B"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We consider spatially extended conductance based neuronal models with noise described by a stochastic reaction diffusion equation with additive noise coupled to a control variable with multiplicative noise but no diffusion. We only assume a local Lipschitz condition on the non-linearities together with a certain physiologically reasonable monotonicity to derive crucial\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript normal infinity\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^\\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -bounds for the solution. These play an essential role in both the proof of existence and uniqueness of solutions as well as the error analysis of the finite difference approximation in space. We derive explicit error estimates, in particular, a pathwise convergence rate of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartRoot 1 slash n EndRoot minus\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msqrt>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\/<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msqrt>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sqrt {1\/n}-<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and a strong convergence rate of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1 slash n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">1\/n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in special cases. As applications, the Hodgkin-Huxley and FitzHugh-Nagumo systems with noise are considered.\n                  <\/p>","DOI":"10.1090\/mcom\/3068","type":"journal-article","created":{"date-parts":[[2015,10,30]],"date-time":"2015-10-30T11:11:10Z","timestamp":1446203470000},"page":"2457-2481","source":"Crossref","is-referenced-by-count":10,"title":["Analysis and approximation of stochastic nerve axon equations"],"prefix":"10.1090","volume":"85","author":[{"given":"Martin","family":"Sauer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wilhelm","family":"Stannat","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,1,8]]},"reference":[{"key":"1","unstructured":"A. Andersson and S. Larsson, Weak Convergence for a Spatial Approximation of the Nonlinear Stochastic Heat Equation, arXiv preprint arXiv:1212.5564, 2012."},{"key":"2","doi-asserted-by":"publisher","first-page":"Art. 10, 50","DOI":"10.1186\/2190-8567-2-10","article-title":"Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons","volume":"2","author":"Baladron, Javier","year":"2012","journal-title":"J. Math. Neurosci."},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1093\/imanum\/drs035","article-title":"Full discretization of the stochastic Burgers equation with correlated noise","volume":"33","author":"Bl\u00f6mker, Dirk","year":"2013","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"4","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511666223","volume-title":"Stochastic equations in infinite dimensions","volume":"44","author":"Da Prato, Giuseppe","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0521385296"},{"key":"5","series-title":"Interdisciplinary Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-87708-2","volume-title":"Mathematical foundations of neuroscience","volume":"35","author":"Ermentrout, G. Bard","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387877075"},{"key":"6","doi-asserted-by":"crossref","unstructured":"R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J., 1 (1961), 445\u2013466.","DOI":"10.1016\/S0006-3495(61)86902-6"},{"key":"7","unstructured":"R. FitzHugh, Mathematical Models of Excitation and Propagation in Nerve, In Biological Engineering, McGraw-Hill, New York, 1969."},{"issue":"8","key":"8","doi-asserted-by":"publisher","first-page":"2355","DOI":"10.1016\/j.jfa.2012.07.001","article-title":"Strong solutions for stochastic partial differential equations of gradient type","volume":"263","author":"Gess, Benjamin","year":"2012","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"issue":"11","key":"9","doi-asserted-by":"publisher","first-page":"e1002247, 9","DOI":"10.1371\/journal.pcbi.1002247","article-title":"The what and where of adding channel noise to the Hodgkin-Huxley equations","volume":"7","author":"Goldwyn, Joshua H.","year":"2011","journal-title":"PLoS Comput. 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