// based on calcrxc.hoc,v 1.3 2008/12/02 22:03:59
/*
This code does two things:
1. Calculates the transfer resistances between a monopolar
extracellular recording electrode and a model neuron.
Assumes that the intervening bath and tissue can be treated
as linear.
Suppose a patch of membrane centered at (x,y,z) generates
transmembrane current Im(x,y,z). The membrane will produce
a potential Vel that can be recorded at the extracellular
electrode and is given by
Vel = rx(x,y,z) Im(x,y,z)
where rx(x,y,z) is the transfer resistance between the
recording electrode and (x,y,z).
A conductive sphere of radius r0 is suspended in an infinite
volume of solution that has resistivity rho [ohm cm]. Ignoring
electrochemical effects at the electrode|solution interface,
what is the resistance between the surface of the sphere and
an infinitely distant ground electrode?
The surface area of a sphere with radius r is 4 PI r^2.
The resistance of a shell with thickness dr is
rho dr / 4 PI r^2
and the resistance is therefore
inf
INTEGRAL rho dr / 4 PI r^2
r0
inf
= - rho / 4 PI r | = rho / 4 PI r0
r0
So to a first approximation, a source of current I produces a
field in which potential V is given by
V = I rho / 4 PI r
where r is the distance from the center of the current source.
The approximation is best when r is large compared to the
physical size of the current source.
2. Calculates the effect of a uniform extracellular field on
the extracellular potential adjacent to the membrane of a model
neuron.
Assume a uniform field with orientation specified by polar coordinates:
phi is the angle between the field vector and the z axis.
theta is the angle between the x axis
and the projection of the field vector onto the xy plane
The isopotential surfaces are perpendicular to the orientation
of the field, and uniformly spaced.
Without loss of generality, we may assume that the potential is 0
for all points on the isopotential surface that passes through (0,0,0).
Let this be called the zero potential plane.
The potential at any point xyz is proportional to the distance of
that point from the zero potential plane. The distance is given by
the dot product of the vector from (0,0,0) to (x,y,z)
p = x*ux + y*uy + z*uz
with the field's unit vector (i.e. the unit vector that is orthogonal
to the zero potential plane):
ufield = sin(phi)*cos(theta)*ux + sin(phi)*sin(theta)*uy + cos(phi)*uz
where ux, uy, and uz are the unit vectors along the x, y, and z axes.
Thus the extracellular potential at (x,y,z) is
V(x,y,z) = -E * p dot ufield
= -E * (x*sin(phi)*cos(theta) + y*sin(phi)*sin(theta) + z*cos(phi))
= -E * d
where
d is the distance of the point from the zero potential plane
E is the field intensity in units of potential/distance, and is >= 0.
Note: why the minus sign in front of E?
Because a positive charge _loses_ potential energy as it moves in the
direction of the field ("electrical field lines move away from the
positive electrode and toward the negative electrode").
So the parameters are
Orientation
phi angle from the z axis
theta angle of projection onto the xy plane,
measured counterclockwise from the x axis
Strength
E field intensity in (volt/m)
How this information is used to simulate extracellular stimulation:
Insert the extracellular and xtrau mechanisms in all sections that are
subject to the extracellular field.
Compute the transfer resistance rx for every internal node of each
section that contains xtrau, as illustrated below.
Also compute the distance d of each of these nodes from the zero
potential plane.
Construct a Vector that specifies the time course of E.
For each internal node, use this Vector to drive E_xtrau(x).
The xtrau mechanism uses the d values to convert the stimulus field strength
waveform into the proper amplitude and sign of the local extracellular field,
and it uses the rx values to calculate the contribution of the local membrane
current to the potential that would be detected by the extracelluar recording
electrode.
If rho is changed, setrx() must be invoked to make the change take effect.
*/
// set up the transfer resistances
// what is the approximate resistivity of tissue anyway?
rho = 35.4 // ohm cm, squid axon cytoplasm
// for squid axon, change this to seawater's value
// for mammalian cells, change to brain tissue or Ringer's value
// assume monopolar recording
// electrode coordinates:
// for this test, default location is 50 microns horizontal from the cell's 0,0,0
XE = 50 // um
YE = 0
ZE = 0
proc setrx() {
forall {
// if (ismembrane("xtra")) {
if (ismembrane("xtrau")) {
// avoid nodes at 0 and 1 ends, so as not to override values at internal nodes
for (x,0) {
// r = sqrt((x_xtra(x) - xe)^2 + (y_xtra(x) - ye)^2 + (z_xtra(x) - ze)^2)
r = sqrt((x_xtrau(x) - xe)^2 + (y_xtrau(x) - ye)^2 + (z_xtrau(x) - ze)^2)
// 0.01 converts rho's cm to um and ohm to megohm
// rx_xtra(x) = (rho / 4 / PI)*(1/r)*0.01
rx_xtrau(x) = (rho / 4 / PI)*(1/r)*0.01
}
}
}
}
create sElec // bogus section to show extracell stim/rec electrode location
objref pElec // bogus PointProcess just to show stim location
objref gElec // will be a Shape that shows extracellular electrode location
gElec = new Shape(0) // create it but don't map it to the screen yet
// gElec.size(-245.413,275.413,-250,270)
//gElec.view(-245.413, -250, 520.827, 520, 629, 104, 201.6, 201.28)
proc drawelec() {
sElec {
// make it 1 um long
pt3dclear()
pt3dadd($1-0.5, $2, $3, 1)
pt3dadd($1+0.5, $2, $3, 1)
pElec = new IClamp(0.5)
}
gElec.point_mark(pElec, 2) // make it red
}
proc setelec() {
xe = $1
ye = $2
ze = $3
// setrx(xe, ye, ze)
setrx()
drawelec(xe, ye, ze)
}
setelec(XE, YE, ZE) // put stim electrode at (x, y, z)
// print "Use setelec(x, y, z) to change location of extracellular recording electrode"
/*
xpanel("Extracellular Electrode Location", 0)
xlabel("xyz coords in um")
xvalue("x", "XE", 1, "setelec(XE,YE,ZE)", 0, 1)
xvalue("y", "YE", 1, "setelec(XE,YE,ZE)", 0, 1)
xvalue("z", "ZE", 1, "setelec(XE,YE,ZE)", 0, 1)
xpanel(855,204)
*/