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BIOMATH108_3-8_lecture - 319 trl'mmothematins “$1206...

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Unformatted text preview: 319 trl'mmothematins “$1206 Spring. 2007 CMN Modellng Oendrttlc Trees WWEWG i.“- li'lh . '. - ' __..‘.-i_i .a‘ile slit. The mathematicel modetlng ol voltage as a function at time and mom on __ mnmhrene-enctosed cyllnders With constant diameters uniform manor-one prom. and uniform 'lnternel conductences origineted with Wilfrid Rel end omen early in the evolution of neurons! modeling, Unlike the memoranes on the wrfaoe of an axon. these membrenrte were assumed to have no voltage-dependent charm crossing them They are called "pesslve membranes". The models were developed to portray components of dendritlc trees. which then were assumed to have no such channels. For meny dendrites this Is a good essumption. But recent research has revealed loceltone on certain dendrlttc trees that cen have voltage-dependent channels - a subject tor some lnteresting model-hosed research today. We she! discuss this later. Until recently. we heve included a review of this classical W «modeling in Biomethemetlcs 208. and sometimes 108. over. today's research on realism complex dendntlc trees requires the computatione modeling approaches that we she! discuss now. For those who are interested. there is a very substantial coverage oi the cluelcel methemeticel approaches in Chepter 2. 'Cehle Theory for We Neurone“. euthoredm Wiltrld Roll end Hegel Agmon-Snir. pp. 27 - 92. in the second edttlon of Koch end Segev’s “Methods of Neuronal Modding“. to which you have access in the Computer Leo. The history at this is interesting. Why “cable theory"? These early modelers edepted the equettons used to model current flow end loss in cyllndricel components at the Irene-Atlantic cable. The basic equations ere the sense as those you see in the Hodgkin Huxley model all he axon. rt you omit the terms there thet ere releted to the voltage-dependent channels. sonthenext pagesuggestemetwhilesome olemnts in dendrtttc trees might be spproximeted reasonably tool by owners of undorm dierneter. others would not. For instance. tooketthe‘motornwrontromepnel cord'. at the lowerlelt in Flgure 1. There are substantial decree.» lnthetlrldlh of We silo need to deal with boundery conditions at the ls the end at the Drench "seeted‘. so that no current flows W it? Or. for a gmen‘ exfienmt. hes ttbeericut. sothetttoontoctsthefltudou (it ere. ...
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