Quantitative Physiology: Cells and Tissues Lecture 26: November 8, 2006(Volume 2: 3.1-3.2.1 3.3-3.4.2.1)MyelinatedaxonDendritictreeCell bodyInput cellOutputcellAxonal treeSynapseSynapseNormaldirectionof signalflowNerveterminalFigure 1.22− 5051015− 5 0 5 10 15 20 25 30Time (ms)1.01.000.890.440.220.11− 0.11− 0.22− 0.44− 0.89− 1.000I+−SeawaterSeawaterAxonie(t)ie(t)vo(t)vo(t)(mV)Figure 3.1+−+−+−+−MembraneIntracellularExtracellularJCJKJN aJLGK(Vm, t) GN a(Vm, t)GLVKVN aVLVmJmCmFigure 4.6If ∆Vmsmall → ∆m, ∆n, and ∆h small ... ignore!Jm= CmdVmdt+ GN a(Vm− VN a) + GK(Vm− VK) + GL(Vm− VL)= CmdVmdt+ XnGn!Vm− Gm(XnGnGmVn)= CmdVmdt+ GmVm− GmVomJm= CmdVmdt+ Gm(Vm− Vom)+−VmoVmAGmie(t)ACmAJmsmall, uniform,area Aie(t) = AJm= ACmdVmdt+ AGm(Vm− Vom)ACmAGmdVmdt+ Vm= Vom+ie(t)AGmVmAGmIetVmoGmCmie(t)IetτM = membrane time constantindependent of cell sizeCore Conductor ModelInner conductorOuter conductorMembrane+−z+−(a) (b)(d) (c)Ii(z, t)Io(z, t)Vi(z, t)Vo(z, t)Vm(z, t)Ii(z+∆z, t)Io(z+∆z, t)Vi(z+∆z, t)Vo(z+∆z, t)Vm(z+∆z, t)ri∆z ri∆zri∆zro∆z ro∆zro∆zKm(z+∆z, t)∆zKe(z+∆z, t)∆zKm(z, t)∆zKe(z, t)∆zz + ∆zFigure 2.7For ∆Vmsmall:Km= 2πaJm= 2πaCmdVmdt+ 2πaGm(Vm− Vom) = cmdVmdt+ gm(Vm− Vom)Combine with core-conductor model:∂2Vm∂z2= (ro+ ri)Km− roKe= (ro+ ri)"cm∂Vm∂t+ gm(Vm− Vom)#− roKeVm+cmgm∂Vm∂t−1gm(ro+ ri)∂2Vm∂z2= Vom+rogm(ro+ ri)KeVm+ τM∂Vm∂t− λ2C∂2Vm∂z2= Vom+ roλ2CKeLet Vm= vm+ Vom:vm+ τM∂vm∂t− λ2C∂2vm∂z2= roλ2CKe(Cable Equation)Km(z,t)∆zVm(z,t)gm∆zcm∆zVmo++−− V+−InsulationConductorTransmitter ReceiverOceanFigure 3.8SeawaterSeawaterAxon0 zIeFigure 3.9zAzzvm(z)dvm(z)dzd2vm(z)dz2AλC−AλCAλ2Cµ−2AλC¶Figure 3.10zSpace constantvm(z)λCroλC2IeFigure 3.11−1.5−1−0.500 1 2 3 4z (mm)− log10elog10(vo(z)/vo(0))Figure 3.20 00.20.40.60.810 1 2 3e−z/λCvo(z)/vo(0)z/λCFigure
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