// $Id: calcrxc.hoc,v 1.7 2014/08/17 16:27:20 ted Exp $
/* Calculates the transfer resistances between
extracellular stimulating|recording electrode(s) and a
model neuron. Relies on the principle of reciprocity,
which assumes that the intervening bath and tissue can
be treated as linear. Suppose a stimulus current of
amplitude Is, applied to a particular configuration of
extracellular electrode(s), produces a potential Vext(x,y,z)
at location (x,y,z). Then the transfer resistance between
the electrode(s) and (x,y,z) is
rx(x,y,z) = Vext(x,y,z)/Is
According to the principle of reciprocity, a transmembrane
current Im(x,y,z) generated by membrane at (x,y,z) will
produce a potential Vel that can be recorded at the
extracellular electrode(s) and is given by
Vel = rx(x,y,z) Im(x,y,z)
-----------------------------------------
How to simulate extracellular stimulation
-----------------------------------------
Insert the extracellular and xtra mechanisms in all sections that are
subject to the extracellular field.
Compute the transfer resistance rx for every section that contains
xtra, as illustrated below.
Construct a stimulus waveform template and copy it to a Vector.
For each internal node along the axon, use this Vector to drive
is_xtra(x). The xtra mechanism uses the rx values to convert the
stimulus current waveform into the proper amplitude and sign of the
local extracellular field.
If rho, b, or c is changed, new_elec() must be invoked
to make the changes take effect.
-----------------------------------------
Monopolar electrode in an infinite medium
-----------------------------------------
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 monopolar stimulating electrode
that delivers 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 electrode.
The principle of superposition may be applied to deal with an
arbitrary number of monopolar electrodes, or even surface
electrodes with different shapes and areas, which are located
at arbitrary positions, and deliver arbitrary stimulus currents.
However, there are some noteworthy special cases.
---------------------------------------------
Special case: bipolar stimulation of an axon
---------------------------------------------
Imagine a pair of stimulating electrodes that lie along
a line parallel to an axon, like so:
===================== ---
c
o o ---
| b |
1 2
where b is the separation between the electrodes
and c is the perpendicular distance from them to the axon.
For the sake of this example, assume that the electrodes
straddle the midpoint of the axon.
The extracellular potential at location x produced by electrode
1 is
V1 = I rho / 4 PI r1
where r1 is the distance from electrode 1 to x. This distance is
r1 = sqrt( ((x-0.5)*L) + 0.5*b)^2 + c^2 )
Likewise the potential at x produced by electrode 2 is
V2 = -I rho / 4 PI r2
where r2 is the distance from electrode 2 to x, i.e.
r2 = sqrt( ((x-0.5)*L) - 0.5*b)^2 + c^2 )
The net extracellular potential at x is V1 + V2, i.e.
Vnet = (I rho / 4 PI)*((1/r1) - (1/r2))
so the transfer resistance that converts the stimulus current I to
an extracellular potential Vnet is simply
rx = (rho / 4 PI)*((1/r1) - (1/r2))
--------------------------------------------------------
Special case: uniform field between two parallel plates
--------------------------------------------------------
A uniform field has the same intensity and orientation at all
points in space, and the extracellular potential at any point
is a linear function of displacement in the direction of the
orientation of the field.
If an entire neuron lies in such a field, then without loss of
generality we may assert that the extracellular potential is 0
for all points that lie on some plane that is perpendicular to
the field. For this "zero potential surface" it is convenient
to choose the plane that passes through a particular location
in the cell, such as the 0 end of the soma.
*/
// 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
/*
b = 400 // um between electrodes
c = 100 // um between electrodes and axon
proc setrx() {
forall {
if (ismembrane("xtra")) {
// avoid nodes at 0 and 1 ends, so as not to override values at internal nodes
for (x,0) {
r1 = sqrt( ((x-0.5)*L + 0.5*b)^2 + c^2 )
r2 = sqrt( ((x-0.5)*L - 0.5*b)^2 + c^2 )
// 0.01 converts rho's cm to um and ohm to megohm
axon.rx_xtra(x) = (rho / 4 / PI)*((1/r1) - (1/r2))*0.01
// print r1, r2
}
}
}
}
*/
// assume monopolar stimulation and recording
// electrode coordinates:
// for this test, default location is 50 microns horizontal from the cell's 0,0,0
XE = soma.x_xtra // um
YE = soma.y_xtra
ZE = soma.z_xtra
proc setrx() { // now expects xyc coords as arguments
forall {
if (ismembrane("xtra")) {
// 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_xtra(x) - $1)^2 + (y_xtra(x) - $2)^2 + (z_xtra(x) - $3)^2)
// 0.01 converts rho's cm to um and ohm to megohm
// if electrode is exactly at a node, r will be 0
// this would be meaningless since the location would be inside the cell
// so force r to be at least as big as local radius
if (r==0) r = diam(x)/2
rx_xtra(x) = (rho / 4 / PI)*(1/r)*0.01
}
}
}
}
create sElec // bogus section to show extracell stim/rec electrode location
proc put_sElec() {
sElec {
// make it 1 um long
pt3dclear()
pt3dadd($1-0.5, $2, $3, 1)
pt3dadd($1+0.5, $2, $3, 1)
}
}
put_sElec(XE, YE, ZE)
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.view(-245.413, -250, 520.827, 520, 629, 104, 201.6, 201.28)
objref pElec // bogus PointProcess just to show stim location
sElec pElec = new PointProcessMark(0.5) // middle of sElec
gElec.point_mark(pElec, 2) // make it red
proc setelec() {
xe = $1
ye = $2
ze = $3
setrx(xe, ye, ze)
put_sElec(xe, ye, ze)
}
// setrx(50, 0, 0) // put stim electrode at (x, y, z)
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)