Model neurons!!the membrane equation!

Transcription

1 Modl nurons!!th bran quation! Suggstd rading:! Chaptr in Dayan, P. & Abbott, L., Thortical Nuroscinc, MIT Prss, 2001.! Modl nurons: Th bran quation! Contnts:!!!!!! Ion channls Nnst quation Goldan-Hodgkin-Katz quation Ion concntrations Mbran capacity RC-circuit

2 Th nurons in th brain! Modl nurons: Th bran quation! 3! Exapls of nurons in th brain.! Nurons ar alrady xtrly coplicatd dvics.! Thy transfr signals by ans of ion xchangs through channls ad of protins.! Modl nurons: Th bran quation! 4! Trs and dfinitions!

3 Cll bran with ion channls! Modl nurons: Th bran quation! 5! Th cll bran kps a! voltag diffrnc btwn! th cll and th surrounding by slctivly allowing diffrnt ions in and out of th cll.! Mbran Potntial:! i t)! ( Typical ions:! Calciu [Ca 2+ ]! Potassiu [K + ]! Sodiu [Na + ]! Clorid [Cl - ]! Magnsiu [Mg 2+ ]! Modl nurons: Th bran quation! 6! Ion transportrs and ion channls! Ion transportrs and ion channls ar rsponsibl for ionic ovnts across brans. Transportrs crat ion concntration diffrncs by activly transporting ions against thir chical gradints.! Channls tak advantag of ths concntration gradints, allowing slctd ions to ov, via diffusion, down thir chical gradints.!

4 Modl nurons: Th bran quation! 7! Elctrochical quilibriu! (A) A bran prabl only to K+ sparats copartnts 1 and 2, which contain th indicatd concntrations of KCl.! Potassiu [K + ]! Clorid [Cl - ]! (B) Incrasing th KCl concntration in copartnt 1 to 10 M initially causs a sall ovnt of K+ into copartnt 2 until th lctrootiv forc acting on K+ balancs th concntration gradint, and th nt ovnt of K+ bcos zro.! Nrnst quation! Each ion has an quilibriu potntial associatd with whrby th diffusiv forcs and th lctrical forcs balanc. This can b! xprssd by qual probabilitis of an ion to cross th bran:! P concntration gradint P thral nrgy Modl nurons: Th bran quation! 8! q i! RT zf out ln in kbt qz ln out in T: th absolut tpratur ( C)! R: th univrsal gas constant ( jouls/ol K )! F: Faraday constant ( C/ol)! z: valnc of th ion! k B : Boltzan constant!

5 Nrnst quation! Each ion has an quilibriu potntial associatd with whrby th diffusiv forcs and th lctrical forcs balanc. This can b! xprssd by qual probabilitis of an ion to cross th bran:! P concntration gradint P thral nrgy Modl nurons: Th bran quation! 9! q i! RT zf out ln in kbt qz ln out in Th Nrnst quation only applis whn th channls that gnrat a particular conductanc allow only on typ of ion pass through th.! Explanation of th Nrnst quation! Fro th thory of throdynaics, it is known that th probability that a olcul taks a stat of nrgy E is proportional to th Boltzann factor.! Elctrical nrgy:! E ( x) zqu( x) Modl nurons: Th bran quation! 10! p( E) "! E k T Intrprt probability as! ion dnsity:! B u(x) +! zqu( x1 )! zqu( x2 )! k T B in out E " u u( x1 )! u( x2)

6 Modl nurons: Th bran quation! 11! Goldan-Hodgkin-Katz quation! Th Nrnst quation only applis whn th channls that gnrat! a particular conductanc allow only on typ of ion pass through! th. In th prsnc of svral diffrnt ions, th quilibriu! of th cll dpnds on th rlativ prability P of th ions.! Potassiu! Sodiu! Chlorid! q RT F ln P K [K + ] out + P Na [Na + ] out + P Cl [Cl " ] in P K [K + ] in + P Na [Na + ] in + P Cl [Cl " ] out Th prability of an ion dpnds on a nubr of factors such as! th siz of th ion, its obility, tc.! E.g. Squid giant axon: P k : P Na : P Cl 1 : 0.03 : 0.1! 1(10) (460) + 0.1(40) q 58 log! 70 1(400) (50) + 0.1(540) Modl nurons: Th bran quation! 12! Th Goldan-Hodgkin-Katz quation can b linarizd using! conductancs and individual ion potntials.! q g K EK + g K g NaE + g Na + g + g Na Cl Cl E Cl Oftn this quilibriu potntial is not coputd xplicitly but dfind as an indpndnt lakag potntial E L and dtrind by th ing potntial givn th xprintal data (fr paratr).! I ( ) L g L! EL

7 Modl nurons: Th bran quation! 13! Extracllular and intracllular ion concntrations! E Na pos.! Modl nurons: Th bran quation! 14! Siplifid bran capacity! Q C I C C d dt

8 Modl nurons: Th bran quation! 15! Th capacitanc and bran rsistanc of a nuron considrd as a singl copartnt with ara A! Th insulation of th bran is odld as a capacitanc and th pors ar dscribd by a conductanc.! Modl nurons: Th bran quation! 16! RC-Circuit if w injct a currnt I!

9 RC-Circuit! Modl nurons: Th bran quation! 17! I R! R Kirchhoffs law:! C d dt +! R I Mbran Equation:! d " dt! + + RI! RC with units!!f sc Modl nurons: Th bran quation! 18! Exapl: Injct currnt I 0 at t0:!! t " t) v0 + ( v v + 1 RI 0 ( t 0) v0 + v1 1 d " dt! Solution of th ODE! Equilibriu!(t " #) 0 " + + RI 0 (t) ( t 0) + + RI Df.:!! RI 0! t # " (1! ) +

10 Modl nurons: Th bran quation! 19! Exapl: Injct currnt I 0 :! Synaptic input into RC-Circuit:!! t # " (1! ) + "!( t! t # off ) + Appndix: Modl nurons!!!!!nurical intgration!

11 Appndix: Modl Nurons, Nurical intgration! Analytic solution of th ODE! Solution of an ordinary diffrntial quation! Sparat th variabls! dr i dt I " r i # dr # i I " r i dt # ("1) # ln I " r i t $ $ " ln k I " r i k " t # r i I " k " t # Th constant k is dtrind fro th boundary conditions! Nuric solution of th ODE! r(t 1 ) (t 1 " t 0 ) + r(t 0 ) Or or gnral using th stpsiz h:! r(t + h) h + r(t) Th Eulr-intgration thod approxiats th unknown function r(t) picwis by lins, by coputing th slop of th function in ach intrval. It thn stiats th nxt valu fro th prvious and th slop.! Th Eulr thod is sipl, but allows only for sall stpsizs to b sufficintly xact.! dr dt Rul of th thup for h:! Appndix: Modl Nurons, Nurical intgration! r(t + h) " r(t) h h! in 10

12 Exapl! " d dt r i #r i + I Appndix: Modl Nurons, Nurical intgration! " r i (t + h) # r i (t) h #r i + I r i (t + h) h ( " #r (t) + I i ) + r i (t)