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 = ρ in strain gauge development: Here it appears again, as specific axial resistance of "axoplasm," and its value is about 100Ω-cm;result: r-in is (100Ω-cm)/Area with units of Ω/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:(that’s 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) Nin 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 ρ is the resistivity of the material of the rod. The units of resistivity are Ω-cm. Resistance per unit length, by a "dimensional analysis," is therefore

It is known that resistivity of axoplasm is ≈ 100 Ω-meter (From Neuron to Brain , p. 141)Compare to the resistivity of metal, like copper: about 10-8 Ω-m!

What's the resistance of a 1 cm long axon, diameter 10 microns? about 108 Ohms!

mwsu-bio101.ning.com/forum/topics/distinct-human-celltypes

No label neuron wikidoc.org/index.php/Multiple_sclerosis

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 π·d. So π·d*1cm is the area of a unit length (cm) of membrane. • Capacitance per unit length c-m = Cm·π·d and • conductance per unit length is Gm·π·d. therefore

• Capacitance / unit length and Resistance / unit length can be calculated from material properties of membrane

• Lipid membrane has 1 μF /cm2 and RESISTANCE of about 2000 Ω/cm2

• These factors allow calculation of time constant and length constant of membrane

• A passive voltage change will decay toward zero with length constant λ.

• 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:

• α-type motoneuron axons: 100m/sec

• 1 meter from spine to foot » 20 msec RT

• type-IV “warmth” receptor axons: 1m/sec