The Membrane Equation
Professor David Heeger
September 5, 2000
RC Circuits
Figure 1A shows an RC (resistor, capacitor) equivalent circuit model for a patch of passive neural
membrane. The capacitor represents the fact that cellular membranes are good electrical insulators.
The battery represents the sodium-potassium pump that acts to hold the electrical potential of the
inside of the cell below that of the outside. This voltage difference is called the
resting potential
of
the neuron. The resistor represents the leakage of current through the membrane.
To understand the behavior of this circuit, we first need to review the behavior of the indi-
vidual electrical elements. We’ll use a water balloon as an analogy to help develop our intuition.
Figure 1B shows a leaky water balloon connected to a water pump. When the pump is off, the
balloon is empty. When the pump is first turned on, the balloon starts to fill with water. The bal-
loon expands rapidly at first because the force from the incoming water far overpowers the elastic
force of the unexpanded balloon. As the balloon is stretched more and more, the elastic force gets
greater and greater, so the balloon’s size increses ever more slowly, until it reaches equilibrium.
At this point, the current flowing into the balloon from the pump is equal to the current spurting
out through the holes, and the water pressure inside the balloon is equal to the pressure exerted by
the pump. Increasing the number (or size) of the holes would cause more water current to spurt
out. Also, if there were more (or bigger) holes, then the balloon would not inflate quite as much.
The behavior of the water balloon is expressed in terms of: (1) water volume, (2) water current, (3)
water pressure inside the balloon, (4) the size/number of holes, and (5) the elasticity of the balloon.
In the electrical RC circuit, the analogous quantities are: (1) electrical charge, (2) electrical
current, (3) electrical potential, (4) electrical conductance, and (5) capacitance. Electrical charge
(analogous to water volume), is measured in coulombs. Electrical current is the rate of flow of
charge (analogous to the rate of flow of the water in and out of the balloon), and is measured in
amperes or amps (1 amp = 1 coul/sec). Electrical resistance (analogous to the size/number of holes
in the balloon) is measured in ohms. Electrical conductance is the reciprocal of resistance, and is
measured in siemens (siemens = 1/ohms). Electrical potential (analogous to the water pressure) is
measured in volts. One volt will move 1 amp of current through a 1 siemen conductor.
When we say that a neuron is “at rest”, it is actually in a state of dynamic equilibrium. There is
always some current leaking out of the cell. But when at rest, that leak current is exactly balanced
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