CA1 pyramidal neurons: effect of external electric field from power lines (Cavarretta et al. 2014)

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The paper discusses the effects induced by an electric field at power lines frequency.
1 . Cavarretta F, Carnevale NT, Tegolo D, Migliore M (2014) Effects of low frequency electric fields on synaptic integration in hippocampal CA1 pyramidal neurons: implications for power line emissions. Front Cell Neurosci 8:310 [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type: Dendrite; Extracellular;
Brain Region(s)/Organism: Hippocampus;
Cell Type(s): Hippocampus CA1 pyramidal cell;
Channel(s): I Na,t; I A; I K; I h;
Gap Junctions:
Receptor(s): AMPA;
Transmitter(s): Glutamate;
Simulation Environment: NEURON;
Model Concept(s): Activity Patterns; Dendritic Action Potentials; Active Dendrites; Detailed Neuronal Models; Action Potentials; Synaptic Integration; Extracellular Fields;
Implementer(s): Carnevale, Ted [Ted.Carnevale at]; Migliore, Michele [Michele.Migliore at]; Cavarretta, Francesco [francescocavarretta at];
Search NeuronDB for information about:  Hippocampus CA1 pyramidal cell; AMPA; I Na,t; I A; I K; I h; Glutamate;
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// 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
  INTEGRAL rho dr / 4 PI r^2
   = - rho / 4 PI r |     = rho / 4 PI 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 

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
  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
    phi    angle from the z axis
    theta  angle of projection onto the xy plane,
             measured counterclockwise from the x axis
    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

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
		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)
	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)

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