T

20 30

Time (msec)

Stimulating _|-

Current

FIGURE 2 Measurement of the membrane time constant. (A) Experimental setup; (B) change in membrane potential as a function of time after the delivery of a constant step of depolarizing current. (Modified from Kandel ER. The cellular basis of behavior. San Francisco: Freeman, 1976.)

function of time. Just as the temperature in the block changed exponentially as a function of time, the change in potential in the neuron produced by an applied depolarization (or hyperpolarization) also follows an exponential time course. For such exponential functions of time it is possible to define what is known as a time constant (r). The time constant refers to the time it takes for the potential change to reach 63% of its final value.

The general equation for a response that changes as an exponential function of time is:

where t is time and r is the time constant. At time 0, e—0 is equal to 1, and the % response is zero. At infinite time, e—1 is equal to zero, so the % response is 100. A special case occurs when t = r. Then, e—1 = 0.37, and the % response = 63%. A detailed understanding of the mathematics is not important. What is important is that the time constant is a simple index of how rapidly a membrane will respond to a step stimulus.

The time constant for the cell illustrated in Fig. 2B is 10 msec. Thus, in 10 msec the potential has changed from —60 mV to —53.7 mV (63% of its final displacement). The smaller the time constant of the cell, the more rapidly the cell will respond to an applied stimulus. If the time constant is 1 msec, the potential change would occur very rapidly, and the potential would reach —53.7 mV in just 1 msec. If the time constant is 40 msec, the potential change would occur very slowly, and the potential would reach —53.7 mV in 40 msec.

Here, Rm simply reflects the resistive properties of the membrane and is equivalent to the inverse of the permeability, because the less permeable the membrane the higher the resistance. Cm represents the membrane capacitance. This is a physical parameter that describes the ability of a membrane to store charge. It is equivalent to the ability of the metal block to store heat. The larger the size of a block, the better able it is to store heat. Similarly, the larger the membrane capacitance, the better able it is to store charge.

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