Once an electrical signal is generated in part of a neuron, for example in response to a synaptic input on a branch of a dendrite, it is integrated with the other inputs to the neuron and then propagated to the axon initial segment, the site of action potential generation. During signaling, when a stimulus generates action potentials or local (synaptic or sensory generator) potentials in a neuron, the membrane potential changes frequently.
What determines the rate of change in potential with time or distance? What determines whether a stimulus will or will not produce an action potential? Here we consider the neuron's passive electrical properties and geometry and how these relatively constant properties affect the cell's electrical signaling. In the next five chapters we shall consider the properties of the gated channels and how the active ionic currents they control change the membrane potential.
Neurons have three passive electrical properties that are important for electrical signaling. We have already considered the resting membrane conductance or resistance (gr = 1/Rr) and the membrane capacitance, Cm. A third important property that determines signal propagation along dendrites or axons is their intra cellular axial resistance (ra). Although, as mentioned above, the resistivity of cytoplasm is much lower than that of the membrane, the axial resistance along the entire length of an extended thin neuronal process can be considerable. Because these three elements provide the return pathway to complete the electrical circuit when active ionic currents flow into or out of the cell, they determine the time course of the change in synaptic potential generated by the synaptic current. They also determine whether a synaptic potential generated in a dendrite will depolarize the trigger zone on the axon initial segment enough to fire an action potential. Finally, the passive properties influence the speed at which an action potential is conducted.
Membrane Capacitance Slows the Time Course of Electrical Signals
The steady-state change in a neuron's voltage in response to subthreshold current resembles the behavior of a simple resistor, but the initial time course of the change does not. A true resistor responds to a step change in current with a similar step change in voltage, but the neuron's membrane potential rises and decays more slowly than the step change in current (Figure 6–15). This property of the membrane is caused by its capacitance.
The rate of change in the membrane potential is slowed by the membrane capacitance.
The upper plot shows the response of the membrane potential (ΔVm) to a step current pulse (Im). The shape of the actual voltage response (red line) combines the properties of a purely resistive element (dashed line a) and a purely capacitive element (dashed line b). The time taken to reach 63% of the final voltage defines the membrane time constant, τ. The lower plot shows the two elements of the total membrane current (Im) during the current pulse: the ionic current (Ii) across the resistive elements of the membrane (ion channels) and the capacitive current (Ic).
To understand how the capacitance slows down the voltage response, recall that the voltage across a capacitor is proportional to the charge stored on the capacitor. To alter the voltage, charge (Q) must be added to or removed from the capacitor (C):
To change the charge across the capacitor (the neuron's lipid bilayer), current must flow across the capacitor (Ic). Since current is the flow of charge per unit time (Ic = ΔQ/Δt), the change in voltage across a capacitor is a function of the magnitude of the current and the time that the current flows:
Thus the magnitude of the change in voltage across a capacitor in response to a current pulse depends on the duration of the current, because time is required to deposit and remove charge across the capacitor.
If the membrane had only resistive properties, a step pulse of outward current would change the membrane potential instantaneously. Conversely, if the membrane had only capacitive properties, the membrane potential would change linearly with time in response to the same step of current. Because the membrane has both capacitive and resistive properties in parallel, the actual change in membrane potential combines features of the two pure responses. The initial slope of the relation between Vm and time reflects a purely capacitive element, whereas the final slope and amplitude reflect a purely resistive element (Figure 6–15, upper plot).
In the simple case of the spherical cell body of a neuron, the time course of the potential change is described by the following equation:
ΔVm (t) = Im Rm (1 − e−t/τ),
where e is the base of the system of natural logarithms and has a value of approximately 2.72, and τ is the membrane time constant, given by the product of the membrane resistance and capacitance (Rm Cm). The time constant can be measured experimentally as the time it takes the membrane potential to rise to 1 – 1/e, or approximately 63% of its steady-state value (Figure 6–15, upper plot). Typical values of for neurons range from 20 to 50 ms. We shall return to the time constant in Chapter 10 where we consider the temporal summation of synaptic inputs in a cell.
Membrane and Axoplasmic Resistance Affect the Efficiency of Signal Conduction
So far we have considered the effects of the passive properties of neurons on signaling only within the cell body. Distance is not a factor in the propagation of a signal in the neuron's soma because the cell body can be approximated as a tiny sphere whose membrane voltage is uniform. However, a subthreshold voltage signal traveling along extended structures such as dendrites, axons, and muscle fibers decreases in amplitude with distance from the site of initiation. To understand how this attenuation occurs, we will consider how the geometry of a neuron influences the distribution of current.
If current is injected into a dendrite at one point, how will the membrane potential change along the length of the dendrite? For simplicity, consider how membrane potential varies with distance after a constant-amplitude current pulse has been on for some time (t ≫ τ). Under these conditions the membrane capacitance is fully charged, so membrane potential reaches a steady value. The variation of the potential with distance thus depends solely on the relative values of the membrane resistance in a unit length of dendrite, rm (units of Ω · cm), and the axial resistance per unit length of dendrite, ra (units of Ω/cm). The change in membrane potential becomes smaller with distance along the dendrite away from the current electrode (Figure 6–16A). This decay with distance is exponential and expressed by
where λ is the membrane length constant, x is the distance from the site of current injection, and ΔV0 is the change in membrane potential produced by the current at the site of injection (x = 0). The length constant is the distance along the dendrite to the site where ΔVm has decayed to 1/e, or 37% of its initial value (Figure 6–16B). It is a measure of the efficiency of electrotonic conduction, the passive spread of voltage changes along the neuron, and is determined by the values of membrane and axial resistance as follows:
The change in membrane potential along a neuronal process during electrotonic conduction decreases with distance.
A. Current injected into a neuronal process by a microelectrode follows the path of least resistance to the return electrode in the extracellular fluid. (The thickness of the arrows represents the magnitude of membrane current.)
B. The change in Vm decays exponentially with distance from the site of current injection. The distance at which ΔVm has decayed to 37% of its value at the point of current injection defines the length constant, λ.
The better the insulation of the membrane (that is, the greater rm) and the better the conducting properties of the inner core (the lower ra), the greater the length constant of the dendrite. That is because current is able to spread farther along the inner conductive core of the dendrite before leaking across the membrane at some point x to alter the local membrane potential:
The length constant is also a function of the diameter of the neuronal process. Neuronal processes vary greatly in diameter, from as much as 1 mm for the squid giant axon to 1 μm for fine dendritic branches in the mammalian brain. For neuronal processes with similar ion channel densities (number of channels per unit membrane area) and cytoplasmic composition, the larger the diameter, the longer the length constant. Thus thicker axons and dendrites have longer length constants than do narrower processes and hence can transmit passive electrical signals for greater distances. Typical values for neuronal length constants range from 0.1 to 1.0 mm.
To understand how the diameter affects the length constant, we must consider how the diameter (or radius) of a process affects rm and ra. Both rm and ra are measures of resistance for a unit length of a neuronal process of a given radius. The axial resistance ra of the process depends inversely on the number of charge carriers (ions) in a cross section of the process. Therefore, given a fixed cytoplasmic ion concentration, ra depends inversely on the cross sectional area of the process, 1/(π · radius2). The resistance of a unit length of membrane rm depends inversely on the total number of channels in a unit length of the neuronal process. Channel density, the number of channels per μm2 of membrane, is often similar among different-sized processes. As a result, the number of channels per unit length of a neuronal process increases in direct proportion to increases in membrane area, which depends on the circumference of the process times its length; therefore, rm varies as 1/(2 π radius). Because rm/ra varies in direct proportion to the radius of the process, the length constant is proportional to the square root of the radius.
The efficiency of electrotonic conduction has two important effects on neuronal function. First, it influences spatial summation, the process by which synaptic potentials generated in different regions of the neuron are added together at the trigger zone at the axon hillock (see Chapter 10). Second, electrotonic conduction is a factor in the propagation of the action potential. Once the membrane at any point along an axon has been depolarized beyond threshold, an action potential is generated in that region. This local depolarization spreads passively down the axon, causing successive adjacent regions of the membrane to reach the threshold for generating an action potential (Figure 6–17). Thus the depolarization spreads along the length of the axon by local current driven by the difference in potential between the active and resting regions of the axon membrane. In axons with longer length constants local current spreads a greater distance down the axon, and therefore the action potential propagates more rapidly.
Electrotonic conduction contributes to propagation of the action potential.
A. An action potential propagating from right to left causes a difference in membrane potential in two adjacent regions of the axon. The difference creates a local-circuit current that causes the depolarization to spread passively. Current spreads from the more positive active region (2) to the less positive resting region ahead of the action potential (1), as well as to the less positive area behind the action potential (3). However, because there is also an increase in membrane K+ conductance in the wake of the action potential (see Chapter 7), the buildup of positive charge along the inner side of the membrane in area 3 is more than balanced by the local efflux of K+, allowing this region of membrane to repolarize.
B. A short time later, the action potential has traveled down the axon and the process can be repeated.
Large Axons Are More Easily Excited Than Small Axons
The influence of axonal geometry on action potential conduction plays an important role in a common neurological exam. In the examination of a patient for diseases of peripheral nerves, the nerve often is stimulated by passing current between a pair of external cutaneous electrodes placed over the nerve, and the population of resulting action potentials (the compound action potential) is recorded farther along the nerve by a second pair of cutaneous voltage-recording electrodes. In this situation the total number of axons that generate action potentials varies with the amplitude of the current pulse.
To drive a cell to threshold, a stimulating current from the positive electrode must pass through the cell membrane into the axon. There it travels along the axoplasmic core, eventually exiting the axon into the extracellular fluid through the membrane to reach the second (negative) electrode. However, most of the stimulating current does not even enter the axon, moving instead through neighboring axons or through the low-resistance pathway of the extracellular fluid. Thus, the axons into which current enters most easily are the most excitable.
In general, axons with the largest diameter have the lowest threshold for excitation. The greater the diameter of the axon, the lower the axial resistance to the flow of current down the axon because the number of charge carriers (ions) per unit length of the axon is greater. Because more current enters the larger axon, the axon is depolarized more efficiently than a smaller axon. For these reasons, larger axons are recruited at low values of current; axons with smaller diameter are recruited only at relatively greater current strengths.
The fact that larger axons conduct more rapidly and have a lower current threshold for excitation aids in the interpretation of clinical nerve-stimulation tests. Neurons that convey different types of information (eg, motor versus sensory) often differ in axon diameter and thus conduction velocity (see Table 22–1). In addition, a specific disease process may preferentially affect a subset of the functional classes of axons. Thus, using conduction velocity as a criterion to determine which classes of axons have defective conduction properties can help one infer the neuronal basis for the neurological deficit.
Passive Membrane Properties and Axon Diameter Affect the Velocity of Action Potential Propagation
The passive spread of depolarization during conduction of the action potential is not instantaneous. In fact, electrotonic conduction is a rate-limiting factor in the propagation of the action potential. We can understand this limitation by considering a simplified equivalent circuit of two adjacent segments of axon membrane connected by a segment of axoplasm.
An action potential generated in one segment of membrane supplies depolarizing current to the adjacent membrane, causing it to depolarize gradually toward threshold (see Figure 6–17). According to Ohm's law, the larger the axoplasmic resistance, the smaller the current between adjacent membrane segments (I = V /R) and thus the longer it takes to change the charge on the membrane of the adjacent segment.
Recall that, since ΔV = ΔQ/C, the membrane potential changes slowly if the current is small because ΔQ, equal to current multiplied by time, changes slowly. Similarly, the larger the membrane capacitance, the more charge must be deposited on the membrane to change the potential across the membrane, so the current must flow for a longer time to produce a given depolarization. Therefore, the time it takes for depolarization to spread along the axon is determined by both the axial resistance ra and the capacitance per unit length of the axon cm (units F/cm). The rate of passive spread varies inversely with the product ra cm. If this product is reduced, the rate of passive spread increases and the action potential propagates faster.
Rapid propagation of the action potential is functionally important, and two adaptive strategies have evolved to increase it. One is an increase in the diameter of the axon core. Because ra decreases in proportion to the square of axon diameter, whereas cm increases in direct proportion to diameter, the net effect of an increase in diameter is a decrease in ra cm. This adaptation has been carried to an extreme in the giant axon of the squid, which can reach a diameter of 1 mm. No larger axons have evolved, presumably because of the competing need to keep neuronal size small so that many cells can be packed into a limited space.
The second strategy to increase conduction velocity is the wrapping of a myelin sheath around an axon (see Chapter 4). This process is functionally equivalent to increasing the thickness of the axonal membrane by as much as 100 times. Because the capacitance of a parallel-plate capacitor such as the membrane is inversely proportional to the thickness of the insulation material (see Appendix A), myelination decreases cm and thus ra cm. Myelination results in a proportionately much greater decrease in ra cm than does the same increase in the diameter of the axon core because the many layers of membrane wrapped in the myelin sheath produce a very large decrease in cm. For this reason, conduction in myelinated axons is faster than in nonmyelinated axons of the same diameter.
In a neuron with a myelinated axon the action potential is triggered at the nonmyelinated initial segment of membrane just distal to the axon hillock. The inward current that flows through this region of membrane is available to discharge the capacitance of the myelinated axon ahead of it. Even though the capacitance of the axon is quite small (because of the myelin insulation), the amount of current down the core of the axon from the trigger zone is not enough to discharge the capacitance along the entire length of the myelinated axon.
To prevent the action potential from dying out, the myelin sheath is interrupted every 1 to 2 mm by bare patches of axon membrane approximately 1 μm in length, the nodes of Ranvier (see Chapter 4). Although the area of membrane at each node is quite small, the nodal membrane is rich in voltage-gated Na+ channels and thus can generate an intense depolarizing inward Na+ current in response to the passive spread of depolarization down the axon. These regularly distributed nodes thus boost the amplitude of the action potential periodically, preventing it from decaying with distance.
The action potential, which spreads quite rapidly along the internode because of the low capacitance of the myelin sheath, slows down as it crosses the high-capacitance region of each bare node. Consequently, as the action potential moves down the axon it jumps quickly from node to node (Figure 6–18A). For this reason, the action potential in a myelinated axon is said to move by saltatory conduction (from the Latin saltare, to jump). Because ionic membrane current flows only at the nodes in myelinated fibers, saltatory conduction is also favorable from a metabolic standpoint. Less energy must be expended by the Na+-K+ pump in restoring the Na+ and K+ concentration gradients, which tend to run down as the action potential is propagated.
Action potentials in myelinated nerves are regenerated at the nodes of Ranvier.
A. The densities of capacitive and ionic membrane currents (membrane current per unit area of membrane) are much higher at the nodes of Ranvier than in the internodal regions. (In the figure the density of membrane current at any point along the axon is represented by the thickness of the arrows.) Because of the higher capacitance of the axon membrane at the nodes, the action potential slows down as it approaches each node and thus appears to skip rapidly from node to node as it propagates from left to right.
B. In regions of the axon that have lost their myelin, the spread of the action potential is slowed down or blocked. The local-circuit currents must discharge a larger membrane capacitance and, because of the shorter length constant (caused by the low membrane resistance in demyelinated stretches of axon), they do not spread well down the axon.
Various diseases of the nervous system are caused by demyelination, such as multiple sclerosis and Guillain-Barré syndrome. As an action potential goes from a myelinated region to a bare stretch of demyelinated axon, it encounters a region of relatively high cm and low rm. The inward current generated at the node just before the demyelinated segment may be too small to provide the capacitive current required to depolarize the segment of demyelinated membrane to threshold. In addition, this local-circuit current does not spread as far as it normally would because it is flowing into a segment of axon that has a short length constant because of its low rm (Figure 6–18B). These two factors can combine to slow, and in some cases actually block, the conduction of action potentials, causing devastating effects on behavior (see Chapter 7).