Medical Physiology, 3rd Edition

Signal Conduction in Dendrites

The word dendrite is derived from the Greek word dendron [tree], and indeed some dendrites resemble tree branches or roots. Inspired by trees, no doubt, the anatomist Camillo Golgi suggested in 1886 that the function of dendrites is to collect nutrients for the neuron. The truth is analogous but more interesting: dendrites arborize through a volume of brain tissue so that they can collect information in the form of synaptic input. The dendrites of different types of neurons exhibit a great diversity of shapes. Dendrites are often extensive, accounting for up to 99% of a neuron's membrane. The dendrites of a single neuron may receive as many as 200,000 synaptic inputs. The electrical and biochemical properties of dendrites are quite variable from cell to cell, and they have a profound influence on the transfer of information from synapse to soma.

Dendrites attenuate synaptic potentials

Dendrites tend to be long and thin. Their cytoplasm has relatively low electrical resistivity, and their membrane has relatively high resistivity. These are the properties of a leaky electrical cable, which is the premise for cable theory (see p. 201). Leaky cables are like leaky garden hoses; if ionic current (or water) enters at one end, the fraction of it that exits at the other end depends on the number of channels (or holes) in the cable (hose). A good hose has no holes and all the water makes it through, but most dendrites have a considerable number of channels that serve as leaks for ionic current (see Fig. 7-22).

Cable theory predicts how much current flows down the length of the dendrite through the cytoplasm and how much of it leaks out of the dendrite across the membrane. As summarized in Table 7-3, we can express the leakiness of the membrane by the resistance per unit area of dendritic membrane (specific membrane resistance, Rm), which can vary widely among neurons. The intracellular resistance per cross-sectional area of dendrite (specific resistivity of the cytoplasm, Ri) is also important in determining current flow inasmuch as a very resistive cytoplasm forces more current to flow out across the membrane rather than down the axis of the dendrite. Another important factor is cable diameter; thick dendrites let more current flow toward the soma than thin dendrites do. Figure 7-22C illustrates the consequences of a point source of steady current flowing into a leaky, uniform, infinitely long cable made of purely passive membrane. The transmembrane voltage generated by the current falls off exponentially with distance from the site of current injection. The steepness with which the voltage falls off is defined by the length constant (λ; see p. 201), which is the distance over which a steady voltage decays by a factor of 1/e (~37%). Estimates of the parameter values vary widely, but for brain neurons at rest, reasonable numbers are ~50,000 Ω ⋅ cm2 for Rm and 200 Ω ⋅ cm for Ri. If the radius of the dendrite (a) is 1 µm (10−4 cm), we can estimate the length constant of a dendrite by applying Equation 7-8.

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(12-1)

Because dendrite diameters vary greatly, λ should also vary greatly. For example, assuming the same cellular properties, a thin dendrite with a radius of 0.1 µm would have a λ of only 354 µm, whereas a thick one with a radius of 5 µm would have a λ of 2500 µm. Thus, the graded signal voltage spreads farther in a thick dendrite.

Real dendrites are certainly not infinitely long, uniform, and unbranched, nor do they have purely passive membranes. Thus, quantitative analysis of realistic dendrites is complex. The termination of a dendrite decreases attenuation because current cannot escape farther down the cable. Branching increases attenuation because current has more paths to follow. Most dendrites are tapered. Gradually expanding to an increased diameter progressively increases λ and thus progressively decreases attenuation. Real membranes are never completely passive because all have voltage-gated channels, and therefore their Rm values can change as a function of voltage and time. Finally, in the working brain, cable properties are not constant but may vary dynamically with ongoing brain activity. For example, as the general level of synaptic input to a neuron rises (which might happen when a brain region is actively engaged in a task), more membrane channels will open and thus Rm will drop as a function of time, with consequent shortening of dendritic length constants. However, all these caveats do not alter the fundamental qualitative conclusion: voltage signals are attenuated as they travel down a dendrite.

So far, we have described only how a dendrite might attenuate a sustained voltage change. Indeed, the usual definition of length constant applies only to a steady-state voltage shift. An important complication is that the signal attenuation along a cable depends on the frequency components of that signal—how rapidly voltage changes over time. When Vm varies over time, some current is lost to membrane capacitance (see p. 158), and less current is carried along the dendrite downstream from the source of the current. Because action potentials and EPSPs entail rapid changes in Vm, with the fastest of them rising and falling within a few milliseconds, they are attenuated much more strongly than the steady-state λ implies. If Vm varies in time, we can define a λ that depends on signal frequency (λAC, where AC stands for “alternating current”). When signal frequency is zero (i.e., Vm is steady), λ = λAC. However, as frequency increases, λAC may fall sharply. Thus, dendrites attenuate high-frequency (i.e., rapidly changing) signals more than low-frequency or steady signals. Another way to express this concept is that most dendrites tend to be low-pass filters in that they let slowly changing signals pass more easily than rapidly changing ones.

Figure 12-2A shows how an EPSP propagates along two different dendrites with very different length constants. If we assume the synapses trigger EPSPs of similar size in the end of each dendrite, then the dendrites with the longer λ deliver a larger signal to the axon hillock. How do leaky dendrites manage to communicate a useful synaptic signal to the soma? The problem is solved in two ways. The first solution deals with the passive properties of the dendrite membrane. The length (l) of dendrites tends to be relatively small in comparison to their λ; thus, none extends more than one or two steady-state length constants (i.e., the l/λ ratio is <1). One way that dendrites achieve a small l/λ ratio is to have a combination of diameter and Rm that gives them a large λ. Another way is that dendrites are not infinitely long cables but “terminated” cables. Figure 12-2B shows that a signal is attenuated more in an infinitely long cable (curve a) than in a terminated cable whose length (l) is equal to λ (curve b). The attenuation of a purely passive cable would be even less if the terminated cable had a λ 10-fold greater than l (curve c). Recall that in our example in Figure 12-2A, such a 10-fold difference in λ underlies the difference in the amplitudes of the EPSPs arriving at the axon hillock.

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FIGURE 12-2 Effect of λ on propagation of an EPSP to two different axons. A, The neuron at the top fires an action potential that reaches the left and right neurons below, each at a single synapse. The EPSPs are identical. However, the left neuron has a thin dendrite and therefore a small length constant (λ = 0.1 mm). As a result, the signal is almost completely attenuated by the time it reaches the axon hillock, and there is no action potential. In the right neuron, the dendrite is thicker and therefore has a larger length constant (λ = 1 mm). As a result, the signal that reaches the axon hillock is large enough to trigger an action potential. B, The graph shows four theoretical plots of the decay of voltage (logarithmic plot) along a dendritic cable. The voltage is expressed as a fraction of maximal voltage. The length along the cable is normalized for the length constant (λ). Thus, an l/λ of 1.0 corresponds to one length constant along the dendrite. Curve a: If the cable is infinitely long and passive, the voltage decays exponentially with increasing length, so that the semilog plot is linear. Curve b: If the cable is terminated at a length that is equal to one length constant, then voltage decays less steeply. Curve c: If the cable is terminated at a length that is equal to 10% of the length constant, the voltage decays even less steeply. Curve d: If the membrane is not passive but has a slow voltage-gated conductance, the dendritic attenuation will be much smaller. (Data from Jack JJB, Noble D, Tsien RW: Electrical Current Flow in Excitable Cells. Oxford, UK, Oxford University Press, 1975.)

The second solution to the attenuation problem is to endow dendrites with voltage-gated ion channels (see pp. 182–199) that enhance the signal more than would be expected in a purely “passive” system (curve d). We discuss the properties of such “active” cables in the next section.

Dendritic membranes have voltage-gated ion channels

All mammalian dendrites have voltage-gated ion channels that influence their signaling properties. Dendritic characteristics vary from cell to cell, and the principles of dendritic signaling are being studied intensively. Most dendrites have a relatively low density of voltage-gated channels (see pp. 182–199) that may amplify, or boost, synaptic signals by adding additional inward current as the signals propagate from distal dendrites toward the soma. We have already introduced the principle of an active cable in curve d of Figure 12-2B. If the membrane has voltage-gated channels that are able to carry more inward current (often Na+ or Ca2+) under depolarized conditions, a sufficiently strong EPSP would drive Vm into the activation range of the voltage-gated channels. These voltage-gated channels would open, and their additional inward current would add to that generated initially by the synaptic channels. Thus, the synaptic signal would fall off much less steeply with distance than in a passive dendrite. Voltage-gated channels can be distributed all along the dendrite and thus amplify the signal along the entire dendritic length, or they can be clustered at particular sites. In either case, voltage-gated channels can boost the synaptic signal considerably, even if the densities of channels are far too low for the generation of action potentials.

An even more dramatic solution, used by a few types of dendrites, is to have such a high density of voltage-gated ion channels that they can produce action potentials, just as axons can. One of the best-documented examples is the Purkinje cell, which is the large output neuron of the cerebellum. As Rodolfo Llinás and colleagues have shown, when the dendrites of Purkinje cells are stimulated strongly, they can generate large, relatively broad action potentials that are mediated by voltage-gated Ca2+ channels (Fig. 12-3). Such Ca2+ spikes can sometimes propagate toward—or even into—the soma, but these Ca2+ action potentials do not continue down the axon. Instead, they may trigger fast Na+-dependent action potentials that are generated by voltage-gated Na+ channels in the initial segment. The Na+ spikes carry the signal along the axon in the conventional way. Both types of spikes occur in the soma, where the Na+ spikes are considerably quicker and larger than the dendritic Ca2+ spikes. The faster Na+ spikes propagate only a short distance backward into the dendritic tree because the rapid time course of the Na+ spike is strongly attenuated by the inherent filtering properties of the dendrites (i.e., the λAC is smaller for the rapid frequencies of the Na+ action potentials than for the slower Ca2+ action potentials). The dendrites of certain other neurons of the CNS, including some pyramidal cells of the cerebral cortex, can also generate spikes that are dependent on Ca2+, Na+, or both.

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FIGURE 12-3 Ca2+ action potentials in dendrites of Purkinje cells. Usually, dendrites do not fire action potentials; however, in these Purkinje cells of the cerebellum (left), the high density of voltage-gated Ca2+ channels in the dendrites allows the generation of slow dendritic Ca2+ spikes (records a, b, and c on the right), which propagate all the way to the axon soma. In the axon soma, these Ca2+ action potentials trigger fast Na+ action potentials (record d on the right). Moreover, the fast Na+ spikes back-propagate into the dendritic tree but are attenuated. Thus, these fast Na+ spikes appear as small spikes in the proximal dendrites (record c) and even smaller blips in the midlevel dendrites (record b). (Data from Llinás R, Sugimori M: Electrophysiological properties of in vitro Purkinje cell dendrites in mammalian cerebellar slices. J Physiol 305:197–213, 1980.)

Dendritic action potentials, when they exist at all, tend to be slower and weaker, and with higher thresholds, than those in axons. The reason is probably that one of the functions of dendrites is to collect and to integrate information from a large number of synapses (often thousands). If each synapse were capable of triggering an action potential, there would be little opportunity for most of the synaptic inputs to have a meaningful influence on a neuron's output. The cell's dynamic range would be truncated; that is, a very small number of active synapses would bring the neuron to its maximum firing rate. However, if dendrites are only weakly excitable, the problem of signal attenuation along dendritic cables can be solved while the cell still can generate an output (i.e., the axonal firing rate) that is indicative of the proportion of its synapses that are active.

Another advantage of voltage-gated channels in dendrites may be the selective boosting of high-frequency synaptic input. Recall that passive dendrites attenuate signals of high frequency more than those of low frequency. However, if dendrites possess the appropriate voltage-gated channels, with fast gating kinetics, they will be better able to communicate high-frequency synaptic input.