# Thread: Neurons as RC circuits

1. I just read about an experiment that led scientists to conclude that neurons behave like RC circuits. In the experiment they used an electrode to inject current into a neuron while measuring the voltage change. From start to finish, they injected a fixed amount of current per second into the neuron. They found that the voltage changes gradually and ultimately reaches a maximum. Once they stopped the current, the voltage again attenuanted gradually. From this they concluded a neuron doesn't behave like a simple resistor, but is more like a resistor combined with a capacitor.

The way they reasoned was as follows. According to Ohm's law, V remains the same if I and R remain the same. The current changed when they started the injection but after that maintained a fixed value. Assuming that a neuron's resistance doesn't change, you would expect, they argued, the voltage to jump up the moment you start injecting the current and then, like the current, to maintain a fixed value. Instead, it increased gradually.

Now, here's what I don't understand. I don't know much about electricity so I may be overlooking something basic. As I understand voltage, it measures, in this case, the difference in total charge between the inside of the neuron and the outside. When you inject the current, you inject a certain amount of charged particles into the neuron, and consequently the total charge, the voltage, changes. But even if you keep pumping in the same amount of particles per second, the total amount of particles inside (and thus the total charge inside) keeps increasing. In other words, even if I remains the same, and R remains the same, I don't see why V wouldn't continue increasing. Does anyone understand why V would remain fixed?  2.

3. Originally Posted by DCC I just read about an experiment that led scientists to conclude that neurons behave like RC circuits. In the experiment they used an electrode to inject current into a neuron while measuring the voltage change. From start to finish, they injected a fixed amount of current per second into the neuron.
"Current per second" makes no sense. I assume you mean "a fixed amount of current into the neuron."

They found that the voltage changes gradually and ultimately reaches a maximum. Once they stopped the current, the voltage again attenuanted gradually. From this they concluded a neuron doesn't behave like a simple resistor, but is more like a resistor combined with a capacitor.
I'm not sure how old the reference you're citing is. Hodgkin and Huxley published their famous model many decades ago. They got a Nobel in the early 60's for their research. Their RC model of a neuron is a very old one.

That said, if you have a current source driving a parallel RC, the voltage across the RC of course rises (the functional form is [1-exp{-t/RC}], asymptotically approaching a constant value equal to I*R (that's just Ohm's law, of course). In words, if you wait long enough (meaning many RC time constants), the dynamics become negligible, and the circuit's "final" value can be deduced by ignoring the capacitance (no d/dt --> no capacitor current, so all of the current flows into the resistor alone), leaving you with a trivial Ohm's law problem to solve.

The way they reasoned was as follows. According to Ohm's law, V remains the same if I and R remain the same. The current changed when they started the injection but after that maintained a fixed value. Assuming that a neuron's resistance doesn't change, you would expect, they argued, the voltage to jump up the moment you start injecting the current and then, like the current, to maintain a fixed value. Instead, it increased gradually.
If that's what they reasoned, they are either assuming a different topology than Hodgkin and Huxley, or they're simply lousy scientists. A current feeding a parallel RC doesn't induce a sudden jump in voltage. The voltage starts from zero and rises as in the equation I gave above.

Now, here's what I don't understand. I don't know much about electricity so I may be overlooking something basic. As I understand voltage, it measures, in this case, the difference in total charge between the inside of the neuron and the outside. When you inject the current, you inject a certain amount of charged particles into the neuron, and consequently the total charge, the voltage, changes. But even if you keep pumping in the same amount of particles per second, the total amount of particles inside (and thus the total charge inside) keeps increasing. In other words, even if I remains the same, and R remains the same, I don't see why V wouldn't continue increasing. Does anyone understand why V would remain fixed?
I hope that the explanation I gave provides the answer. The current splits between the capacitor and resistor. Eventually, the resistor takes all the current. You can come to that conclusion by a simple reductio ad absurdum argument. If you were to assume that current keeps flowing into the capacitor, its voltage would rise indefinitely. Since the R and C are in parallel, the voltage across the capacitor and resistor are the same. An infinitely rising resistor voltage implies an infinitely rising resistor current, which eventually must exceed the input current! This absurd result tells you that that current cannot keep flowing into the capacitor, as initially assumed.

Or you could just solve the first-order differential equation (or simply derive the equation and verify that the solution I gave you is correct).  4. Sometimes a mechanical analogy can help people who are not used to working with electrical.
If you had a positive displacement pump pumping water through a hose, that's like a current source and a resistor. The flow (current) is constant and the pressure (voltage) is proportional to the resistance. Now add a reservoir (analogous to the capacitor) at the outlet of the positive displacement pump. When you start the pump, the flow mostly goes easily into the reservoir, and there isn't much back pressure. As the reservoir fills up to a certain level, it develops some back pressure and more flow goes into the hose. Eventually an equilibrium level is reached, the reservoir level and back pressure remain constant, and all the flow is going to the hose.  5. Originally Posted by tk421 "Current per second" makes no sense. I assume you mean "a fixed amount of current into the neuron."

Okey, maybe I misunderstand the term. The amount of ampere remained the same throughout the injection of the current, which I took to mean that same amount of charged particles flows into the neuron per time unit.

I'm not sure how old the reference you're citing is. Hodgkin and Huxley published their famous model many decades ago.

It's about the Hodgkin and Huxley model and he wants to show why they concluded a neuron can be compared to an RC circuit.

The current splits between the capacitor and resistor. Eventually, the resistor takes all the current. You can come to that conclusion by a simple reductio ad absurdum argument. If you were to assume that current keeps flowing into the capacitor, its voltage would rise indefinitely. Since the R and C are in parallel, the voltage across the capacitor and resistor are the same. An infinitely rising resistor voltage implies an infinitely rising resistor current, which eventually must exceed the input current! This absurd result tells you that that current cannot keep flowing into the capacitor, as initially assumed.

Thanks, that explains it.  6. Originally Posted by Harold14370 Now add a reservoir (analogous to the capacitor) at the outlet of the positive displacement pump. When you start the pump, the flow mostly goes easily into the reservoir, and there isn't much back pressure. As the reservoir fills up to a certain level, it develops some back pressure and more flow goes into the hose. Eventually an equilibrium level is reached, the reservoir level and back pressure remain constant, and all the flow is going to the hose.
Nice analogy.   Bookmarks
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