Cable Theory, Cable Equation, passive conduction, AP propagation The following assumes passive conduction of voltage changes down an axon or dendrite. Consider a cylindrical tube as a model for a dendrite or axon process. The wall of the tube will be a high resistance membrane and the inside of the tube will be low resistance axoplasm. Say the inside of the tube has resistance/length = r-in, How will g-in depend on cable diameter? Conductance will
increases as the square of the diameter. Consider the material property we called rho = in strain gauge development: Here it appears again, as specific axial resistance of "axoplasm," and its value is about 100-cm; result: r-in is (100-cm)/Area with units of /cm imagining the end of the cable is grounded. Now consider the leakage current i-m going out of
the membrane. Per "compartment" of length we have a picture like Differentiate again and substitute in previous equation: where the minus signs cancel out. Consider the capacitance of the membrane:
the membrane current is now to be expressed as: (thats r-in connecting the nodes on the top rail Solving PDEs with Matlab http://www.mathworks.com/help/matlab/math/partial-differential-equations.html
pdeval, pdepe sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan) solves initial-boundary value problems for systems of parabolic and elliptic PDEs in one space variable and time. pdefun, icfun, and bcfun are function handles. The ordinary differential equations (ODEs) resulting from
discretization in space are integrated to obtain approximate solutions at times specified in tspan. The pdepe function returns values of the solution on a mesh provided in xmesh. >> help pdepe Help from Node Admittance Matrix Repeating component of lumped cable finite element model
Finding derivative of node voltage (state) N in terms of states Matlab function, script
function [sdot] = CableCktNAM(tt, ss, G1, G2, CC, ndes ) exer_CableCkt with key line: [tt, ss] = ode23t(@ (tt, ss) CableCktNAM(tt, ss, G1, G2, CC, ndes ), tspan, s0, options); How are r-in, r-m and c-m calculated from physical properties of the cell and its membrane? Recall from the strain gauge lecture that resistance R, in ohms, for a rod of length L and cross-section A is
Where is the resistivity of the material of the rod. The units of resistivity are -cm. Resistance per unit length, by a "dimensional analysis," is therefore It is known that resistivity of axoplasm is 100 -meter (From Neuron to Brain , p. 141) Compare to the resistivity of metal, like copper: about 10-8 -m!
What's the resistance of a 1 cm long axon, diameter 10 microns? about 108 Ohms! No label neuron wikidoc.org/index.php/Multiple_sclerosis mwsu-bio101.ning.com/forum/topics/distinct-human-celltypes membrane capacitance, conductance Next, consider membrane as a sheet of material specified by
Cm = capacitance/cm^2, and by mho/cm^2 = Gm. Smaller membrane capacitance, faster propagation time Conductance has units of mhos. Why feature conductance? It's proportional to the area of the membrane under consideration. For a cable of diameter d, the circumference is d. So d. So d. So d*1cm is the area of a unit length (cm) of membrane. Capacitance per unit length c-m = Cmd and conductance per unit length is Gmd. So d. So d. therefore
Capacitance / unit length and Resistance / unit length can be calculated from material properties of membrane Lipid membrane has 1 F /cm2 and RESISTANCE of about 2000 /cm2 These factors allow calculation of time constant and length constant of membrane A passive voltage change will decay toward zero with length constant .
Solutions to the cable equation: See D. J. Aidley, The Physiology of Excitable Cells, page 50 ff and B. Katz, Nerve, Muscle and Synapse, Oxford Univ Press (1970) Myelination significantly increases Rm and therefore increases the length constant of a axon, typically from 10 to 2000 microns! The nodes of Ranvier are closer than one length constant. A myelin wrap can also reduce membrane capacitance (remember capacitors in series?) so the effective time constant of the membrane is
about the same; the result is faster propagation of an action potential in a myelinated axon. Myelination and length constant
Myelination is a wrap of insulating sheath around an axon It significantly increases Rm and therefore increases the length constant of a axon, typically from 10 to 2000 microns! The nodes of Ranvier (breaks in myelin) are closer than one length constant
At the nodes, the action potential regenerates to full size, due to Na+ channels opening A myelin wrap can also reduce membrane apparent capacitance (remember capacitors in series?) Result: faster propagation of an action potential in a myelinated axon
http://www.bio.miami.edu/~cmallery/150/neuro/myelinated.axon.jpg Range of axon speeds: -type motoneuron axons: 100m/sec 1 meter from spine to foot 20 msec RT type-IV warmth receptor axons: 1m/sec
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