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• W4254218374 abstract "Journal of Integrative NeuroscienceVol. 09, No. 01, pp. 11-30 (2010) Research ReportsNo AccessNEURONAL MODELS IN INFINITE-DIMENSIONAL SPACES AND THEIR FINITE-DIMENSIONAL PROJECTIONS: PART ISTANISłAW BRZYCHCZY and ROMAN R. POZNANSKISTANISłAW BRZYCHCZYDepartment of Differential Equations, Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Cracow, PolandCorresponding author. Search for more papers by this author and ROMAN R. POZNANSKIDepartment of Artificial Intelligence, Faculty of Computer Science and Information Technology, University of Malaya, Kuala Lumpur 50603, Malaysia Search for more papers by this author https://doi.org/10.1142/S0219635210002391Cited by:4 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractMethods of comparing discrete and continuous cable models of single neurons and dynamical phenomena observed in neurobiology can be described with infinite-coupled systems of semilinear parabolic differential-functional equations of the reaction-diffusion-convection type or infinite systems of ordinary integro-differential equations. It is known that numerous problems in computational neuroscience use finite systems of equations based on the so-called compartmental model. It seems a natural idea to extend the results obtained in the theory of finite systems onto infinite systems. However, this requires stringent assumptions to be adopted to achieve compatibility. In most instances the dynamics of infinite systems behave differently to their finite-dimensional projections. The truncation method applied to infinite systems of equations and presented herein yields a truncated system consisting of the first N equations of the infinite system in N unknown functions. A solution of infinite system is defined as the limit when N → ∞ of the sequence of approximations {zN}N=1,2,…, where are defined as solutions of suitable finite truncated systems with corresponding initial-boundary conditions. Geometrically, it may be described as the projection of an infinite system of differential equations considered in a function abstract space of infinite dimension (such as Banach or Hilbert space) onto its finite-dimensional subspaces.Keywords:Banach spaceinfinite-countable systeminfinite-uncountable systemtruncation methodtruncated systemprojection operatordiscrete modelcontinuous modelcable modelcompartmental modelneurons References H. Amann, Math. Nachr. 186, 5 (1997). Crossref, ISI, Google ScholarH. Amann, Arch. Ration. Mech. 151, 339 (2000), DOI: 10.1007/s002050050200. Crossref, ISI, Google Scholar D. G. Aronson and H. F. Weinberger , Lecture Notes in Mathematics 446 , ed. J. A. Goldstein ( Springer-Verlag , New York , 1975 ) . Google ScholarJ. M. Ball and J. Carr, J. Stat. Phys. 61, 203 (1990), DOI: 10.1007/BF01013961. Crossref, ISI, Google Scholar S. 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Poznanski30 September 2012 | Journal of Integrative Neuroscience, Vol. 11, No. 03Continuous and discrete models of neural systems in infinite-dimensional abstract spacesStanisŁaw Brzychczy and Lech Górniewicz1 Oct 2011 | Neurocomputing, Vol. 74, No. 17 Recommended Vol. 09, No. 01 Metrics History Received 23 November 2009 Accepted 5 March 2010 KeywordsBanach spaceinfinite-countable systeminfinite-uncountable systemtruncation methodtruncated systemprojection operatordiscrete modelcontinuous modelcable modelcompartmental modelneuronsPDF download" @default.
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