Distributed longitudinal voltage

[michiro]

Hi Neuron guru's,

I'm trying to simulate Eddy currents induced in an axon by a changing electromagnetic field. Setting aside the validity of such a condition - which I know is questionable, I can't figure out a way to introduce distributed, longitudinal voltages induced by the Faraday's law. All I need is a distributed, axon-internal voltage (in addition to passive currents, which I think comes free with a standard axon definition).

Could someone help please?

[ted]

I'm trying to simulate Eddy currents induced in an axon by a changing electromagnetic field.

Malmivuo and Plonsey's book <b>Bioelectromagnetism</b><br>
http://butler.cc.tut.fi/~malmivuo/bem/bembook/
has an interesting chapter--Magnetic Stimulation of Neural Tissue<br>
http://butler.cc.tut.fi/~malmivuo/bem/bembook/22/22.htm

I can't figure out a way to introduce distributed, longitudinal voltages induced by the Faraday's law. All I need is a distributed, axon-internal voltage

Start by writing the cable equation in a way that takes intra- and extracellular voltage
gradients into account. Then rearrange the terms so that the effect of magnetic stimulation
is factored into a term that can be represented by an extracellular potential. Then use
NEURON's extracellular mechanism--which has an extracellular voltage source--to
represent the stimulation term.

[michiro]

Start by writing the cable equation in a way that takes intra- and extracellular voltage
gradients into account. Then rearrange the terms so that the effect of magnetic stimulation
is factored into a term that can be represented by an extracellular potential. Then use
NEURON's extracellular mechanism--which has an extracellular voltage source--to
represent the stimulation term.

Thank you. Let V (which, in the most general case V_k,j(t)) be a longitudinal voltage induced by the EMF, assumed to be the same in both intra- and extra-cellular space, and let superscripts ^I and ^E denote intra- and extra-cellular.

(1) c_j(d/dt)(v_j^I-v_j^E)+ion_j(v_j^I-v_j^E) = g_j-1,j^I(v_j-1^I+V-v_j^I) + g_j+1,j^I(v_j+1^I-V-v_j^I)
(2) -c_j(d/dt)(v_j^I-v_j^E) = g_j-1,j^E(v_j-1^E+V-v_j^E) + g_j+1,j^E(v_j+1^E-V-v_j^E)

First thing to notice is that V terms vanish if V is homogeneous and conductances don't change along the axon.

Now, when they don't vanish, (2) can be easily be expressed in terms of extracellular voltage and its associated conductance. I could eliminate V from (1), but that would mess up other variables. Wouldn't it be easier if I use soft voltage clamps for each, er, "segment"?

[ted]

The Forum is not configured to allow effective markup of equations that
demand complex formatting. Perhaps you should just email me a PDF of
the equations
ted dot carnevale at yale dot edu