Electrical Response ofthe Cell Membrane to Injected Current

Many electrical signals in nerve cells arise when ion channels open in the plasma membrane, allowing a flow of electrical current, carried by ions, to move across the membrane and alter the membrane potential of the cell. This situation can be mimicked experimentally by placing a microelectrode inside a cell and injecting charge into the cell through the microelectrode. Figure C-2 shows the response of a cell to injected current, considering only the capacitance of the cell membrane. If a constant current, I, is injected into the cell, then charge, q, is added to the membrane capacitor at a constant rate (I = dq/dt). Because q = CV for a capacitor, we obtain the result:

In other words, dV/dt is a constant, and voltage changes linearly (that is, at a constant rate) during injection of constant current.

The response of the cell to injected current is different, however, if we take into account the presence of ion channels in the cell membrane. Ion channels

Figure C-2 The rise of voltage during injection of constant current in a cell. Only the contribution of the membrane capacitance is considered, and the effect of membrane resistance is neglected. During injection of charge at a constant rate, the resulting voltage on the membrane capacitor rises linearly.

Figure C-2 The rise of voltage during injection of constant current in a cell. Only the contribution of the membrane capacitance is considered, and the effect of membrane resistance is neglected. During injection of charge at a constant rate, the resulting voltage on the membrane capacitor rises linearly.

provide a path for injected charge to move across the membrane, instead of being added to the charge on the membrane capacitor. The electrical analog of the current path provided by the ion channels is an electrical resistor. Figure C-3 illustrates the effect of adding a resistive path for current flow in the cell membrane, in parallel with the capacitance of the cell membrane. In a spherical cell, the injected current has equal access to all parts of the cell membrane at the same time. Therefore, we can combine all of the resistors and all of the capacitors for each patch of cell membrane, resulting in the analogous electrical circuit shown in Figure C-3, consisting of the combined, parallel resistance R

Voltmeter

Voltmeter

Membrane capacitor

Membrane resistor

Membrane capacitor

Membrane resistor

Exponential rise

Exponential decay

Constant current (/)

Figure C-3 The rise of voltage during injection of constant current into a spherical cell, considering both the capacitance and the resistance of the cell membrane. The membrane capacitors represent the insulating portion of the cell membrane, and the membrane resistors represent open ion channels that allow charge to move across the membrane. At the onset of the injected current, all of the injected charge initially flows onto the membrane capacitance. As the voltage on the capacitor builds up, progressively more of the current flows through the resistance. Finally, all of the current flows through the membrane resistance, and the asymptotic voltage is governed by Ohm's law ( V = IR ).

and the combined parallel capacitance C. The injected current now consists of two components: iC, the component that flows onto the capacitor, and iR, the component that flows through the membrane resistor, R. The capacitative current is given by Equation C-2, and the resistive current is given by Ohm's law: iR = V/R. Hence, the total current is

R dt

Solving Equation (C-3) for V yields:

Thus, voltage rises exponentially during injection of a constant current, I. The product, RC, is the exponential time constant of the voltage rise, which is abbreviated t. The asymptotic value of the voltage is IR, which is the voltage expected when all of the current is flowing through the membrane resistance. Initially, all of the injected charge flows onto the membrane capacitor, but as charge accumulates, more and more charge flows instead through the resistor, until finally all of the current flows through the resistive path. When the current injection terminates, the accumulated charge on the capacitor discharges through the parallel resistance, R. This decay of voltage is also exponential, with the same time constant, t, given by RC.

Figure C-4 The equivalent electrical circuit for a long cylindrical cell. A constant current is injected at one end of the cell. At each position along the cell, current divides into a membrane component, im, flowing onto the membrane resistance and capacitance at that point, and a longitudinal component, i, that flows through the resistance of the cell interior to more distant portions of the membrane. The amount of current remaining at each position along the cell is indicated by the thickness of the arrows.

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