Triangles and Conductances

Just a little post on something that’s really simple, not new, and might seem obvious to some people – but might help when choosing how to explore the effect of altering parameters on action potential model behaviour.

The reason I thought I’d post this is that people have expressed surprise that they don’t see symmetric sized effects on action potentials when they make a maximal conductance bigger or smaller. By which, I mean that adding/subtracting a constant percentage (e.g.) +/- 50% to a conductance doesn’t lead to the same ‘balanced’ changes in action potential properties. But I don’t think you should expect that, and why not is all to do with triangles!

Remembering our GCSE maths, the simple equation for a straight line is:

y(x) = mx + c. ...(1)

The equation of a straight line

Figure 1: The equation of a straight line

Where m is the gradient of the line, or dy/dx. If you want to consider where the triangle in Figure 1 hits the x-axis (y = 0), just re-arrange:

x = -c/m ...(2)

which is positive in Figure 1 since m is a negative gradient.

If you think of a cardiac action potential model, we usually write something that looks like

dV/dt = -(g*open_probability*driving_voltage + same for each other current)

where the ‘g’ is the maximum conductance of a current, proportional to the number of available ion channels in the cell membrane.

So thinking of the triangle as a simplified action potential*, the gradient is given by m = dV/dt and looking at the intercept with the x-axis as the action potential duration (APD), we would write (2) as

APD = -c/(dV/dt) = c/(g*stuff) ...(3)

(I will drop the ‘stuff’ for simplicity). From this simple equation, you can see that to double APD you would have to halve ‘g’, and to halve APD you would have to double ‘g’. This has some consequences for exploring model behaviour.

Two graphs of 1/x

Figure 2. Left: the effect of altering the gradient by up to + 80% (APD can be reduced by up to 45%), Right: the effect of altering the gradient by up to -80% (APD can increase by up to 500%). Red line is 1/x, black dashed line is the identity line y=x. You wouldn’t believe how long it took me to make matlab do that overlapping shading thing.

In Figure 2 I’ve highlighted a couple of things to note from equation (3). I’ve plotted ‘APD’ as a function of g (a plot of 1/x). If we increase g by 80%, by changing its value from 1 to 1.8, we can see the APD is reduced from 1 to 1/1.8, by about -45%. Decreasing ‘g‘ by 80%, from 1 to 0.2, has a much larger effect, with increase in APD from 1 to 1/0.2, a change of +500%.

A graph of 1/x with x=0.5 to 2 highlighted.

Figure 3: scaling a conductance by multiplying/dividing by the same factor allows you to explore the same changes to behaviour.

In Figure 3 I’ve showed an alternative way of exploring the space. By multiplying/dividing by a factor (of 2 in what I’ve shown) we get a symmetric plot, so we can see changes of 2x or 1/2 in APD by exploring the same range in conductance ‘g’.

By way of a more useful example I’ve varied the calcium current conductance g_CaL in the ten Tusscher 2006 (epi) model, and show the results in Figure 4.

ten Tusscher 2006 action potentials

Figure 4: An example with ten Tusscher 2006 (epi), in both plots black is the control AP, red is with decreased g_CaL and blue is with increased g_CaL. Left: g_CaL scaled by +/-50%. Right: g_CaL scaled by *2 or /2. Here, in contrast to my advice in this post, I’ve just looked at one pace of simulation to avoid too many other effects coming in to confound this!

Just as we expected from the triangle analogy, we get far more ‘symmetry’ in the response when scaling g_CaL rather than changing it by +/- a fixed percentage (not exactly symmetric because the nonlinear parts of the system come into play too, as different currents get affected by the alteration to voltage).

You might decide that biology has set up its machinery in such a way as to make changes in conductances of +/-50% more likely than +100%/-50%, so you are happy seeing the consequences of varying parameters like that, which is fine. But if you want to do an inverse problem, and see which parameters could describe a wide range of APDs, then I’d suggest it makes sense to explore the parameters by scaling instead of percentage changes.

*of course, this is pretending there is only one (constant) ion current. But as we can see in the example in Figure 4, the principle on the possible effects of conductance changes still applies with more currents. Incidentally, none of the above argument needs the triangle to be the whole action potential, if you zoom in on any little bit of action potential it will look like a triangle, and the same argument therefore applies to any action potential property you care to look at…

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1 Response to Triangles and Conductances

  1. Pingback: Numerical errors from ODE solvers can mess up optimisation and inference very easily | Mathematical Matters of the Heart

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