Axon Propagation

1. The influx of sodium ions into the axon at a Node of Ranvier increases the potential there, as well as at the nodes on either side, and to a negligible degree, at nodes further away.  When the potential at the downstream neighbor node exceeds the threshold value, the process begins there again, and so on down the axon, propagating the nerve impulse.  The value of the potential at the next node depends on how the charge inside the axon distributes itself as the sodium ions rush in.

We will model two charge distributions.  One treats the sodium influx as if it forms a line down the center of the axon.  The other treats it as if it forms a disk of negligible thickness across the cross section of the axon at the node.  The former model is parameterized by the ion drift velocity (vel, below) and the latter is parameterized by the radius of the axon.

The propagation speed depends on the radius of the axon, as does the internodal distance.  Research the drift velocity and using your value, determine the dependence of speed on axon radius for each model.  Vary r from 1 to 10 microns.  Compare the models in your report.

Reevaluate the entire notebook each time you change r.

r=5*(10^-6);

2. Do not change anything in the next two cells!

The ion channel density at each node is assumed to be constant, so that the influx is scaled with the radius.  The internodal distance is assumed to depend linearly on the radius, varying between 0.4 mm at r = 1 micron, to 2 mm at r = 10 microns.  The sodium influx is based on a node capacitance of 1440 pF, and represents the total number of sodium ions which have entered the axon as a function of time.  It is modelled as a Gaussian function (bell curve) with a peak at 1 ms and a width calibrated to experimental data.

We adjust the animation start times, stop times and increments (between frames) to gain accuracy for the faster axons.

Influx=6.3*(10^8);

NaPeak=.001;

NaWidth=.000763;

Q=N[Influx*(r/(5*(10^-6)))*(E^(-4((t(10^-6)-NaPeak)/NaWidth)^2))];

Print["Internodal distance is ",nextnode=0.0004+((r/(10^-6))-1).0016/9," meters."];

start=If[r<=4.(10^-6),25,15];

stop=If[r<=4.(10^-6),70,25];

inc=If[r<=4.(10^-6),2.5,0.5];

Plot[Q,{t,0,3000},PlotRange->All];

Plot[Q,{t,start,stop},PlotRange->All];

3. This cell computes the thin line charge distribution model.  The default ion drift velocity is 1 cm / s (the 10^-6 factor is a time saving factor for the calculation of V, and is there because all times are in microseconds).

The blue line represents the electrical potential between the currently firing node and the next node (far right of the graph).  The red line represents the threshold; it is currently set to - 55 mV.  When the blue line crosses the red line at the right side of the graph, the next node will automatically fire.

Consider the next node to have fired when there is no more blue below the red line.

vel=N[.01(10^-6)];

V:=N[1.6(10^-19)Q*Log[(x+vel*t)/Abs[x-vel*t]]/
		(8*Pi*78.54*8.85*(10^-12)vel*t)-.07];

Do[Plot[{V,-0.055},{x,0.0001,nextnode},PlotRange->{{0,nextnode},{-.071,.05}},
	Prolog->{Text[FontForm[StringForm["t = `` microseconds",t],
			{"Times",12}],Scaled[{0.5,0.9}],{1,0}]},
	PlotStyle->{RGBColor[0,0,1],RGBColor[1,0,0]}],{t,start,stop,inc}];

4. This cell computes the disk charge distribution model.  See (3) above.

V:=N[1.6(10^-19)Q*(Sqrt[x^2+r^2]-x)/
		(2*Pi*78.54*8.85*(10^-12)r^2)-.07];

Do[Plot[{V,-0.055},{x,0.0001,nextnode},PlotRange->{{0,nextnode},{-.071,.05}},
	Prolog->{Text[FontForm[StringForm["t = `` microseconds",t],
			{"Times",12}],Scaled[{0.5,0.9}],{1,0}]},
	PlotStyle->{RGBColor[0,0,1],RGBColor[1,0,0]}],{t,start,stop,inc}];