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Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i

Methods 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 ordin...

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Main Authors: Brzychczy, S.L., Poznanski, R.R.
Format: Article
Published: Imperial College Press 2010
Subjects:
Q Science (General)
Online Access:http://eprints.um.edu.my/15153/
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spelling my.um.eprints.151532015-12-22T01:39:46Z http://eprints.um.edu.my/15153/ Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i Brzychczy, S.L. Poznanski, R.R. Q Science (General) Methods 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 -> infinity of the sequence of approximations {z(N)} (N=1,2,...,) where z(N) = (z(N)(1), z(N)(2),..., z(N)(N)) 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. Imperial College Press 2010 Article PeerReviewed Brzychczy, S.L. and Poznanski, R.R. (2010) Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i. Journal of Integrative Neuroscience, 9 (1). pp. 11-30.
institution Universiti Malaya
building UM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Malaya
content_source UM Research Repository
url_provider http://eprints.um.edu.my/
topic Q Science (General)
spellingShingle Q Science (General)
Brzychczy, S.L.
Poznanski, R.R.
Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i
description Methods 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 -> infinity of the sequence of approximations {z(N)} (N=1,2,...,) where z(N) = (z(N)(1), z(N)(2),..., z(N)(N)) 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.
format Article
author Brzychczy, S.L.
Poznanski, R.R.
author_facet Brzychczy, S.L.
Poznanski, R.R.
author_sort Brzychczy, S.L.
title Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i
title_short Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i
title_full Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i
title_fullStr Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i
title_full_unstemmed Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i
title_sort neuronal models in infinite-dimensional spaces and their finite-dimensional projections: part i
publisher Imperial College Press
publishDate 2010
url http://eprints.um.edu.my/15153/
_version_ 1643689994041163776
score 13.211869

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