Extracellular mech and theory vs gradient of potential

[gabrielggn]

Dear collegues,

I recently took a look at the forum, particularly on these topics: viewtopic.php?p=18961&hilit=extracellular+math#p18961 and viewtopic.php?p=19941&hilit=extracellul ... ion#p19941, since I am interested in the maths/equations behind extracellular mechanism when it is used for extracellular electrical stimulation.

Indeed, the maths and circuitry are similar to the cable equation with extrernal electric field described in Rattay's papers on the topic. But then, why a linear gradient of extracellular potential (then assuming uniform electric field in the range of the cell) can have an effect on the cell since the second derivative of the extracellular potential (first of the electric field) is zero ? Did I miss something in the way it is included within NEURON ?

Best regards

[ted]

Why? Because you don't understand the limitations of the "activating function" approach. Frank Rattay was quite explicit about those limitations in at least one of his many articles, but he didn't reiterate them in all. Your task, if you are willing to accept it, is to track down those limitations and report them in this thread.

Hint: the limitations arise from the simplifications that the activating function approach assumes.

[gabrielggn]

Dear Ted,
Thanks for the reply. Indeed I understand the limitations of the AF. Btw if anybody reads this post, I suggest the following for discussion about this topic and other interesting ones: 10.1016/j.jphysparis.2011.10.003

The limitations are such as only the polarization due to the onset of the pulse is properly represented by the AF estimate, and only if the electrotonic length is small enough to neglect longitudinal currents.
The thing that I do not understand is more about the mathematical translation of the HH equations with extracellular term within NEURON. Indeed, the differences that is represented by the second partial derivative of the extracellular potential is still zero even without taking the limit dx->0. Therefore I don't see how a linear gradient can affect the membrane potential.

best regards

[gabrielggn]

If we take back the circuit associated with the extracellular mechanism (here taken from the previously mentionned post), then we have:

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          Ra		
o/-- o --\/\/-- o --\/\/-- o --\/\/-- o--\o vext[0] + v     intracellular layer   
     |          |          |          |     
    ---        ---        ---        ---
   |   |      |   |      |   |  c_m =  | | g_m 
    ---        ---        ---        ---
     |          |          |          |     
     |          |          |          |    
     |   R_0    |          |          |
 /-- o --\/\/-- o --\/\/-- o --\/\/-- o --\ vext[0]             layer 0
     |          |          |          |     
    ---        ---        ---        ---  
   |   |      |   |      |   |  c_0 =  | | g_0 
    ---        ---        ---        ---
     |          |          |          |     
     |          |          |          |     
     |   R_1    |          |          |     
 /-- o --\/\/-- o --\/\/-- o --\/\/-- o --\ vext[1]             layer 1
     |          |          |          |     
    ---        ---        ---        ---
   |   |      |   |      |   |      |   | 
   |   |      |   |      |   |  c_1 =  | | g_1
   |   |      |   |      |   |      |   | 
   |  ---     |  ---     |  ---     |  ---
   |   -      |   -      |   -      |   -   e_extracellular
    ---        ---        ---        ---
     |          |          |          |     
-------------------------------------------- ground

Then, using Kirchhoff's law, the equation at the ith node for the intracellular is:

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R_a c_m dv(i)/dt + R_a g_m v(i)  - ( v(i+1) - 2v(i) +v(i-1) ) = 
- ( v_ext[0](i+1) -2 v_ext[0](i) +  v_ext[0](i-1) )    			(1)

for the extracellular layer 0:

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c_0 dv_ext[0](i)/dt + g_0 v_ext[0](i)  - 1/R_0 (v_ext[0](i+1) - 2v_ext[0](i) +v_ext[0](i-1)) =
= c_0 dv_ext[1](i)/dt + g_0 v_ext[1](i) + c_m dv(i)/dt + g_m v(i) 			(2)

for the extracellular layer 1:

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(c_0 + c_1) dv_ext[1](i)/dt + (g_0 + g_1) v_ext[1](i)  - 1/R_1 (v_ext[1](i+1) - 2v_ext[1](i) +v_ext[1](i-1)) =
= c_0 dv_ext[0](i)/dt + g_0 v_ext[0](i) + g_1 e_extracellular(i) 			(3)

Did I make any mistake ?

If I'm right, then taking all xc =0; xg =1e9; xraxial=1e9 is equivallent to the usual formulation, and the right hand side should be zero (or really close to zero) in the case of a gradient (first order) of extracellular potential.

Best regards


[Edit]

I replaced the values into the equations. Lets note g = g_0 = g_1 = 1e9, and taking c_0 = c_1 = 0 & R = R_0 = R_1 = 1e9, considering 1/R ~0, the equations become:

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R_a c_m dv(i)/dt + R_a g_m v(i)  - (v(i+1) - 2v(i) +v(i-1)) = - ( v_ext[0](i+1) -2v_ext[0](i) + v_ext[0](i-1) )     		(1bis)

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 c_m dv(i)/dt + g_m v(i)   = g (v_ext[0](i) - v_ext[1](i))		(2bis)

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2 v_ext[1](i)  =  v_ext[0](i) + e_extracellular(i)     	(3bis)

Inserting (3bis) in (2bis) gives:

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 c_m dv(i)/dt + g_m v(i)   =  g/2 (v_ext[0](i) - e_extracellular(i) )  

Combinning (4) with (1bis) results in:

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(v(i+1) - 2v(i) +v(i-1)) = ( v_ext[0](i+1) -2v_ext[0](i) + v_ext[0](i-1) )  + R_a g/2 (v_ext[0](i) - e_extracellular(i) )

Here I can see that even if the "differential" term is zero, we still have a term (v_ext[0](i) - e_extracellular(i) ). The question is now how big this difference is. In the simulations I previoulsy conducted, v_ext and e_extracellular were close.

[ted]

if anybody reads this post, I suggest the following for discussion about this topic and other interesting ones: 10.1016/j.jphysparis.2011.10.003

An excellent suggestion. For those who don't immediately recognize the DOI, the citation is

Joucla S, Yvert B. Modeling extracellular electrical neural stimulation: from basic understanding to MEA-based applications. Journal of Physiology, Paris 106: 146–158, 2012.

In another paper, these authors proposed a somewhat different approach for predicting and understanding the effects of extracellular fields on cells, which has some advantages over the activating function approach. It should be kept in mind that all methods for predicting excitation of cells by extracellular fields, including multicompartmental computational modeling, are based on assumptions and approximations, and therefore have their own strengths, weaknesses, and domains in which they are "good enough for the purpose at hand."

The limitations are such as only the polarization due to the onset of the pulse is properly represented by the AF estimate

and as Joucla and Yvert (and others) note, "onset of the pulse" means "within a very short interval" (on the scale of microseconds) after the start of the extracellular stimulus.

and only if the electrotonic length is small enough to neglect longitudinal currents.

which raises the question: how applicable is the AF approximation to axons with diameters on the order of microns, and lengths on the order of centimeters? It's OK as long as the extracellular field is "spatially compact" e.g. what you'd get in the near neighborhood of a small monopolar or bipolar stimulating electrode, but it won't work for a field that produces an extensive region (on the scale of hundreds of microns or more) in which extracelluar potential has a significant gradient, as might be produced between a parallel pair of plates driven by different voltages, or between a high voltage power line and ground.

The thing that I do not understand is more about the mathematical translation of the HH equations with extracellular term within NEURON. Indeed, the differences that is represented by the second partial derivative of the extracellular potential is still zero even without taking the limit dx->0. Therefore I don't see how a linear gradient can affect the membrane potential.

Theory is powerful, theory is impressive, but sometimes it is useful to consider a simplified practical example.

Suppose you have a neurite with spatially uniform anatomical and biophysical properties. Let it be so short that it can be represented by three compartments of identical size, as in this crude diagram

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Compartment
A      B      C
+--ra--+--ra--+
|      |      |
RC     RC     RC
|      |      |
G      G      G

where + is the center of a compartment, -- and | are merely "wires", G means ground, RC is a resistor in parallel with a capacitor, and ra is the longitudinal cytoplasmic resistance between the centers of adjacent compartments.

The initial conditions are:
At t==0 the extracellular potential is 0 everywhere, and the charge on all capacitors is 0 so the potential difference between each + and G is 0.
At t==1, an extracellular stimulus is applied that leaves compartment B's extracelluar potential == 0, but makes the extracellular potentials for compartments A and C jump to -1 and +1, respectively. (a nice linear gradient)

What happens to the membrane potentials of these three compartments?
And what happens after the stimulus ends, so that the extracellular potentials fall back to 0?

For convenience, assume that all R == 1, C == 1, and ra == 1.
For compatibility with NEURON, you may assume the following units

time ms
potential mV
resistance megohm
capacitance nanofarad

but you're free to use whatever consistent set of units that you like.

Hint: you could build the circuit and run the simulation with NEURON's Linear Circuit Builder

[ted]

The limitations are such as . . . only if the electrotonic length is small enough to neglect longitudinal currents

Not so. The mathematical derivation and discussion that are presented in justification of the activating function approach make the tacit assumption that the axon / nerve fiber / neurite / cable is infinitely long. An axon that is infinitely long has infinite electrotonic length.

"What??! By saying this is a tacit assumption, you admit that you can't point to anything in the mathematical derivation or accompanying discussion that says the axon is infinitely long, yet you claim that the derivation and discussion assume that it IS infinitely long! Why is this not a violation of Aristotelian logic??"

If the derivation or discussion assumed that the length of the axon is finite, there would have to be a statement about the boundary condition(s) at the axon's termination(s). Since termination boundary conditions are not considered, the derivation and discussion pertain to an axon that has no termination. It must be infinitely long, or at least so long that the termination can be ignored (which is another way of saying that the shortest distance between the stimulating electrode and the axon must be small compared to the distance between the electrode and the axon's termination).

But more important than the axon's electrotonic length is this: ignoring the finite length of neurites is why the activating function approach fails to account for the fact that a linear gradient of extracellular potential can perturb membrane potential of an excitable axon sufficiently to trigger a spike. For any real neuron, terminations can't be ignored, because terminations are where the local perturbation of membrane potential tends to be largest when the first spatial derivative of extracellular potential is large compared to its second spatial derivative.

[gabrielggn]

Thank you very much for the enlightenment on the concept of the AF.

Considering the previous example you mentionned, indeed the resolution of the EDP gives (assuming all values to be 1) :

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Va = 0.5 * exp(-2t) -0.5 -> - 0.5 mV
Vb = 0 mV
Vc = 0.5 - 0.5exp(-2t) ->  0.5 mV

So indeed we can find this linear interaction term (half the applied voltage here. But here we are in the case of boundary conditions as Va and Vc are terminals. In these terminals, there is not second partial derivative anymore but a first order one that is non zero. Do you suggest that it is the BC that causes the effects ?

[ted]

Wrapping up the example from my post of Thu Aug 22, 2024--during the stimulus, the membrane potentials for compartments A and C approach 0.5 mV and -0.5 mV, respectively. Immediately after the end of the stimulus, the membrane potentials reverse polarity and then decay toward 0 as time advances.

The model is an example of the use of the method of lines, in which spatial discretization is used to create a family of ODEs whose solution approximates the solution of a PDE that describes a physical system that is continuous in space. Spatial discretization eliminates the spatial second derivative term. The sealed end boundary condition at A and C is represented by the fact that these compartments, unlike B, have a neighbor on only one side.

If you implemented a multicompartmental model of an unbranched neurite with "real" anatomical and biophysical properties (think "cylindrical cable with nothing attached to either end") and sealed end boundary conditions (internal longitudinal current is 0 at the ends) exposed to an extracellular field with a linear potential gradient, you would find that the perturbation of membrane potential is largest near the ends, and falls off rapidly with distance from the ends. Clinical examples include stimulation of cortex by cortical surface electrodes or transcranial electrodes. Regions that are electrically remote from the ends respond to extracellular stimulation in much the same way that an infinitely long neurite would. In other words, the activating function approximation is best for long fibers with uniform properties in regions that are far from branch points or sealed end terminations. A clinical example would be transcutaneous stimulation of the brachial nerve near the elbow, or epidural stimulation of cervical or lumbosacral nerve roots.