// based partly on calcrxc.hoc,v 1.3 2008/12/02 22:03:59 /* 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 x axis. 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, phi, or theta is changed, new_elec() must be invoked to make the changes take effect. */ // field orientation // default is parallel to y axis, positive direction upward PHIDEG = 90 // degrees away from the z axis THETADEG = 90 // degrees counterclockwise from the x axis // note that PHIDEG and THETADEG are in degrees, but phi and theta are in radians proc setd() { local ex, ey, ez // calculate the x, y, and z components of the field's unit vector ex = sin(phi)*cos(theta) ey = sin(phi)*sin(theta) ez = cos(phi) forall { if (ismembrane("xtrau")) { // avoid nodes at 0 and 1 ends, so as not to override values at internal nodes for (x,0) d_xtrau(x) = (x_xtrau(x)*ex + y_xtrau(x)*ey + z_xtrau(x)*ez) } } } create sField // bogus section to show orientation of extracell field objref pField // bogus PointProcess to show "positive" orientation of field RED = 2 BLUE = 3 // reuse gElec for this // objref gElec // a Shape that shows extracellular electrode location // argument is length of the section used to indicate the field orientation proc drawfield() { sField { // draw it from origin to a user-assigned distance from the origin pt3dclear() pt3dadd(0, 0, 0, 1) pt3dadd(\$1*sin(phi)*cos(theta), \$1*sin(phi)*sin(theta), \$1*cos(phi), 1) doNotify() gElec.color(RED) // make the field indicator section red doNotify() pField = new IClamp(1) // so its 1 end can be marked by a dot gElec.point_mark(pField, BLUE) // a blue dot } } proc setfield() { theta = \$2*PI/180 phi = \$3*PI/180 setd() drawfield(\$1) } FIELDLENGTH = 100 // for starters setfield(FIELDLENGTH, THETADEG, PHIDEG) // set orientation of field proc redrawfield() { sField { // draw it from origin to a user-assigned distance from the origin pt3dchange(1, \$1*sin(phi)*cos(theta), \$1*sin(phi)*sin(theta), \$1*cos(phi), 1) doNotify() gElec.color(RED) // make the field indicator section red } } proc changefield() { theta = \$2*PI/180 phi = \$3*PI/180 setd() redrawfield(\$1) } /* xpanel("Extracellular Field Orientation", 0) xlabel("polar coords in um and degrees") xvalue("length of field indicator", "FIELDLENGTH", 1, "changefield(FIELDLENGTH,THETADEG,PHIDEG)", 0, 1) xvalue("theta", "THETADEG", 1, "changefield(FIELDLENGTH,THETADEG,PHIDEG)", 0, 1) xvalue("phi", "PHIDEG", 1, "changefield(FIELDLENGTH,THETADEG,PHIDEG)", 0, 1) xpanel(629,371)*/