The Response to Current Injection in a Cylindrical Cell

In a spherical cell, as in Figure C-3, the injected current flows equally to the resistors and capacitors in all parts of the membrane at the same time. However, neurons typically give rise to many long, thin neurites that extend long distances to make contact with other cells. Current injected in the cell body of the neuron, for example, must flow along the interior of a neurite to reach the portion of the cell membrane located in the neurite at a distance from the cell body. In this situation, then, current does not have equal access to all parts of the membrane.

Figure C-4 illustrates the analogous electrical circuit for a long cylindrical cell. To reach the parallel resistor and capacitor at progressively more distant portions of the cell membrane, current injected at one end of the cell must flow through the resistance provided by the interior of the cell. This resistance can be quite large for cylindrical neurites of neurons. The resistance of a cylindrical conductor is given by

r4l n d2

Outside

Outside

and so on

Inside and so on

Inside where r is the specific resistance of the conducting material, I is the length of the cylinder, and d is the diameter of the cylinder. For the cytoplasm of a neurite, r is approximately 100 Q cm, which is about 107 times worse than copper wire. Thus, a neurite 1 |m in diameter would have an internal resistance of approximately 1.3 x 106 Q per |m of length.

The current at the site of injection divides into two components. Some of the current (designated im, for membrane current) flows onto the parallel membrane resistance and capacitance at the injection site. The remainder of the current (indicated by il, for longitudinal current) flows through the internal resistance of the neurite. At the next portion of the neurite, the current again divides into membrane and longitudinal components. Thus, the amount of current declines with distance along the neurite. In addition, current entering the parallel RC circuit at each position changes with time, because the voltage on the capacitance at each local position builds up as described previously for the spherical cell. As a result, the change in membrane voltage produced by current injection in the cylindrical cell varies as a function of both time and distance from the injection site.

Analysis of the electrical circuit shown in Figure C-4 leads to the following equation for membrane voltage:

V + TdV/dt = X2d2V/dx2, where T = rmcm and X = ^/imA (C-6)

In this second-order, partial differential equation, rm and cm are the resistance and the capacitance of the amount of membrane in a 1 cm length of the cylindrical cell, and ri is the internal resistance of a 1 cm length of the cylindrical cell. For an infinitely long cylindrical cell, the solution of Equation (C-6) is the cable equation:

In this equation, X = x/X and T = t/T. That is, both distance and time are normalized with respect to X and T, which are defined in Equation (C-6). As in the exponential equation governing rise of voltage during current injection in a spherical cell, T is the time constant of the cylindrical cell. The constant factor, X, is called the length constant of the cylindrical cell.

The function erf in Equation (C-7) is the error function, which is defined as erf (z) = 1/Vrcj—/ - ^dy (C-8)

The error function, erf(z), is the integral under a Gaussian probability distribution from —z to +z, as illustrated graphically in Figure C-5. Note that as z increases from 0, the integral of the Gaussian function first increases rapidly,

Figure C-5 The error function represents the area under a Gaussian curve. (a) The bell-shaped curve represents a Gaussian function. The error function (erf) of a variable, z, is the integral of the Gaussian function from -z to +z. (b) The time-course of rise of voltage with time after onset of a constant current. The error function rises more steeply than an exponential function. Time is normalized with respect to the time constant, t, in both cases. When t = t (that is, T = 1), the error function has reached 0.84 of its final value, V0, but the exponential function has reached 0.63 of its final value.

then progressively more slowly. The rise of erf(T) with increasing T is shown in Figure C-5b, compared on the same time scale with an exponential rise. When t = t (that is, when T = 1), the exponential function rises to 0.63 (that is, 1 - 1/e) of its final, asymptotic value, whereas the error function rises to 0.84 of its asymptotic value.

Although Equation (C-7) may seem daunting, it reduces to simpler relations under certain circumstances. For example, the steady-state decay of voltage with distance from the injection site (that is, V(x) at t = <*>) can be obtained by recognizing that dV/dt eventually becomes zero a long time after the onset of current injection. Thus, when dV/dt = 0, Equation (C-6) becomes

which has an exponential solution:

Figure C-6 The steady-state decay of voltage with distance when a constant current is injected at X = 0 in an infinitely long cylindrical cell. Distance is normalized with respect to the length constant, X. At x = X (that is, X = 1), steady-state voltage is 37% of the steady-state voltage at the site of current injection, V0.

In this equation, V0 is the steady-state voltage at the injection site at t = <™. Thus, in the steady state, voltage declines exponentially with distance from the injection site, and the spatial decay is governed by the length constant, X. Figure C-6 summarizes the decline of voltage along a cylindrical neurite. At a distance X (that is, one length constant) from the injection site, the steady-state voltage declines to 1/e (that is 0.37) of the voltage at the injection site.

Another special case is the rise of voltage with time at the site of current injection (that is, V(t) at x = 0). With x = 0, Equation (C-7) reduces to

In other words, voltage at the injection site rises with a time-course given by the error function, as shown in Figure C-7. At a distance x = X from the injection site, the asymptotic voltage at t = <™ is 0.37V0, as described above, and the time-course of the rise is given by the cable equation (Equation (C-7)) with X = 1. This time-course is also shown in Figure C-7. Note that unlike the rapid rise at x = 0, the voltage at x = X rises with a pronounced delay, which represents the time for the injected current to begin to reach the membrane distant from the injection site. Because of the appreciable internal resistance to current flow, injected charge will flow first onto the membrane capacitance at the injection site and then in the intervening portions of membrane, before reaching more distant parts of the membrane. Thus, the rise of voltage is not only smaller but also slower at progressively greater distance from the point where current is injected into a cylindrical cell.

Figure C-7 The rise of voltage with time after the onset of current injection at two locations along an infinitely long cylindrical cell. Time is normalized with respect to the time constant, t. At the site of injection (x = 0), the voltage rises according to the error function. At a distance of X from the injection site (x = X), the voltage rises with an S-shaped delay to its final value, which is 37% of the steady-state voltage at the site of current injection, V0.

In the nervous system, the passive cable properties of neurites have functional significance for the influence exerted by a particular synaptic input on action potential firing in a postsynaptic neuron. A synaptic input located on a dendrite at a distance from the cell body of the neuron would produce a smaller, slower change in membrane potential in the cell body than a synaptic input located near the cell body. Thus, nearby synaptic inputs have greater influence on the activity of postsynaptic cells.

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