$latex Vout = \\frac{ Vcmd Rm + Vm Rs}{ Rs + Rm } &s=1$<\/div>\n
Where Vm is the resting membrane potential. As you can see, if Rs is much smaller than Rm, then Vout approaches Vcmd, likewise, if Rs is massive, then Vout approaches Vm. This is where we can see the how the Rs < Rm\/10 rule of thumb comes about. If we say Rs = Rm\/10, then the above equation becomes:<\/p>\n
$latex Vout = \\frac{ Vcmd Rm + Vm \\frac{Rm}{10}}{ \\frac{Rm}{10} + Rm } &s=1$<\/div>\n
Which simplifies to:<\/p>\n
$latex Vout = \\frac{ Vcmd + \\frac{Vm}{10}}{ 1+ \\frac{1}{10} } &s=1$<\/div>\n
Which basically says Vout = Vcmd plus a tiny fraction of Vm. As Rs becomes a smaller fraction of Rm, the less Vm contributes to the final membrane potential.<\/p>\n
But this voltage divider is a frequency dependent voltage divider, because capacitors let more current through when the voltage across them changes rapidly. And do you know the other word for a frequency dependent voltage divider? A filter, in this case a low-pass filter. The corner frequency of this filter is given by a pretty awful looking equation:<\/p>\n
$latex f_{c} = \\frac{ \\sqrt{ Rm^{2}-2 Rm Rs-Rs^{2} } }{2 \\pi Cm Rm Rs} &s=1$<\/div>\n
But if Rs is much smaller (about 10x less) than Rm, you can approximate the cut-off frequency of the filter with the much simpler<\/p>\n
$latex f_{c} = \\frac{ 1 }{2 \\pi Rs Cm} &s=1$<\/div>\n
(A little helpful note, if Rs is in M\u03a9 and Cm is in pF then the answer is in MHz).<\/p>\n
$\"How$ <\/a>
How series resistance creates a low-pass filter. Behaviour of voltage clamp on a cell with Rm=100M\u03a9, Cm=100pF<\/p><\/div>\n
To the right you see what this means. If we compare the ratio of the voltage across the cell membrane to the command voltage (Vout\/Vin), we can see the effect of Rs on how good out voltage clamp is. Perfect voltage clamp would produce a horizontal line at Vout\/Vin = 1, meaning that no matter how fast you tried to change the command voltage, the membrane potential would follow it perfectly. But instead what you see is what I described above: even at very low frequencies (e.g. constant voltages) when the series resistance is high the membrane potential does not even come close to the command voltage. You also see that as Rs increases, the filter becomes more aggressive, meaning that even relative slow voltage changes cannot be achieved. In the case illustrated, of an 100M\u03a9\/100pF cell, when Rs is 40M\u03a9, you are beginning to fail to voltage clamp at the relatively glacial frequency of 10 Hz.<\/p>\n
$\"Trying$ <\/a>
Trying to voltage clamp a synaptic current in a 100M\u03a9\/100pF cell<\/p><\/div>\n
This may still seem relatively academic at this point, so let me really try to show you what this means. To the right you can see what happens when you try to voltage clamp a modeled synaptic current as Rs increases (the synaptic event is current based, rather than conductance, so it is the same no matter how bad the voltage clamp is). I have flipped the synaptic current, so they’re easy to compare. You see that even in the relatively ideal case of Rs = 10M\u03a9 the measured current is a lot smaller than the real current. In part this is unfair because this current is instantaneously rising (and as the previous figure showed, fast changes are the enemy of any voltage clamp), so note how the decay of the current is relatively well captured in all cases. However, the point is, if you had been performing voltage clamp, and your Rs had changed from 10M\u03a9 to 20M\u03a9, the current you would have measured would have dropped by about 15%<\/em>. If you had been washing on a drug at the time, you might think that the drug caused that reduction (which is exactly what happened to me). Hence why I am always a lot more convinced by any long term voltage clamp experiment that shows Rs as a function of time, along with any other measurement (e.g. Fig 1 here<\/a>).<\/p>\n
$\"The$ <\/a>
The EPSP produced by our synaptic current (100pA peak) with different values of Rs<\/p><\/div>\n
Of course, not only does Rs effect the current you end up recording, but it has big effects on the amount of unclamped voltage generated by any transmembrane current. To the right you see the size of the EPSP our synaptic current would produce in the absence of voltage clamp. In a perfect world this would be a flat line, i.e. the voltage would be clamped. But when Rs = 40M\u03a9, you can see that the EPSP produced is nearly half the size of the unclamped potential, i.e. this is not voltage clamp at all, but really some weird thing half way between current clamp and voltage clamp.<\/p>\n
Finally, it’s worth mentioning that series resistance compensation allows us to minimize all of these effects (but not remove them). Specifically, if you have a real Rs of 20 M\u03a9, and then you perform series resistance compensation to 75%, then your series resistance will appear to be 5 M\u03a9. It’s also important to remember that in large cells, you run into space clamp problems, where you are unable to voltage clamp distal parts of the cell (you can think about the distal sites being behind another series resistor, this time formed by the resistance down the cytoplasm of the dendrite), but that is a whole other story.<\/p>\n
For me, voltage clamp is no different that most science: you are not measuring perfectly and your measurements are just an approximation of the real thing. The important thing is to know when the approximation is good, and when it is bad, e.g. Voltage clamp recordings from cereballar granule cell (Cm = 5 pF, Rm = 1000M\u03a9) give you a good approximation of transmembrane current. But even if those good cases, when Rs changes, even a constant transmembrane current will appear to change, and that is really bad.<\/p>\n
Below you can adjust the properties of your voltage clamp system, and see how good you voltage clamp should be.
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