Stability and Hopf Bifurcation Analysis of Hopfield Neural Networks with a General Distribution of Delays

dc.comment.hiddenThe results of Chapter 3 are published in the Neural Networks Journal, whose publisher is Elsevier. Elsevier does not require permission to include in a thesis. On their website, one of the author's rights includes: "the right to include the journal article, in full or in part, in a thesis or dissertation;" http://www.elsevier.com/wps/find/authorsview.authors/rights The results of Chapter 2 are published in the Math Mod of Nat Phenom Journal whose publisher is EDP Sciences. I received a permission from them through email to publish the results in my thesis: Dear Raluca Jessop, thank you for your message. We hereby grant permission for the requested use expected that due credit is given to the original source. If material appears within our work with credit to another source, authorisation from that source must be obtained. Credit must include the following components: Author(s) Name(s): Title of the Article. Name of the Journal. Publication year. Volume. Page(s). Copyright EDP Sciences, reproduced with permission. Best regards, Sophie Hosotte" I complied with their request and cited the article as they required. Thank you very much Ralucaen
dc.contributor.authorJessop, Raluca
dc.date.accessioned2011-12-16T21:38:48Z
dc.date.available2011-12-16T21:38:48Z
dc.date.issued2011-12-16T21:38:48Z
dc.date.submitted2011
dc.description.abstractWe investigate the linear stability and perform the bifurcation analysis for Hopfield neural networks with a general distribution of delays, where the neurons are identical. We start by analyzing the scalar model and show what kind of information can be gained with only minimal information about the exact distribution of delays. We determine a mean delay and distribution independent stability region. We then illustrate a way of improving on this conservative result by approximating the true region of stability when the actual distribution is not known, but some moments or cumulants of the distribution are. We compare the approximate stability regions with the stability regions in the case of the uniform and gamma distributions. We show that, in general, the approximations improve as more moments or cumulants are used, and that the approximations using cumulants give better results than the ones using moments. Further, we extend these results to a network of n identical neurons, where we examine the stability of a symmetrical equilibrium point via the analysis of the characteristic equation both when the connection matrix is symmetric and when it is not. Finally, for the scalar model, we show under what conditions a Hopf bifurcation occurs and we use the centre manifold technique to determine the criticality of the bifurcation. When the kernel represents the gamma distribution with p=1 and p=2, we transform the delay differential equation into a system of ordinary differential equations and we compare the centre manifold computation to the one we obtain in the ordinary differential case.en
dc.identifier.urihttp://hdl.handle.net/10012/6403
dc.language.isoenen
dc.pendingfalseen
dc.publisherUniversity of Waterlooen
dc.subjectdelay differential equationsen
dc.subjectdistributed delayen
dc.subjectdelay independent stabilityen
dc.subjectlinear stabilityen
dc.subjectcentre manifold computationen
dc.subjectHopf bifurcationen
dc.subject.programApplied Mathematicsen
dc.titleStability and Hopf Bifurcation Analysis of Hopfield Neural Networks with a General Distribution of Delaysen
dc.typeDoctoral Thesisen
uws-etd.degreeDoctor of Philosophyen
uws-etd.degree.departmentApplied Mathematicsen
uws.peerReviewStatusUnrevieweden
uws.scholarLevelGraduateen
uws.typeOfResourceTexten

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