We have seen that analytical solutions can be given for the voltage along a passive cable with uniform geometrical and electrical properties. If we want to apply the above results in order to describe the membrane potential along the dendritic tree of a neuron we face several problems. Even if we neglect `active' conductances formed by non-linear ion channels a dendritic tree is at most locally equivalent to an uniform cable. Numerous bifurcations and variations in diameter and electrical properties along the dendrite render it difficult to find a solution for the membrane potential analytically (Abbott et al., 1991).
Numerical treatment of partial differential equations such as the cable equation requires a discretization of the spatial variable. Hence, all derivatives with respect to spatial variables are approximated by the corresponding quotient of differences. Essentially we are led back to the discretized model of Fig. 2.16, that has been used as the starting point for the derivation of the cable equation. After the discretization we have a large system of ordinary differential equations for the membrane potential at the chosen discretization points as a function of time. This system of ordinary differential equations can be treated by standard numerical methods.
In order to solve for the membrane potential of a complex dendritic tree numerically, compartmental models are used that are the result of the above mentioned discretization (Bower and Beeman, 1995; Yamada et al., 1989; Ekeberg et al., 1991). The dendritic tree is divided into small cylindric compartments with an approximatively uniform membrane potential. Each compartment is characterized by its capacity and transversal conductivity. Adjacent compartments are coupled by the longitudinal resistance that are determined by their geometrical properties (cf. Fig. 2.19).
Once numerical methods are used to solve for the membrane potential along the dendritic tree, some or all compartments can be equipped with nonlinear ion channels as well. In this way, effects of nonlinear integration of synaptic input can be studied (Mel, 1994). Apart from practical problems that arise from a growing complexity of the underlying differential equations, conceptual problems are related to a drastically increasing number of free parameters. The more so, since almost no experimental data regarding the distribution of any specific type of ion channel along the dendritic tree is available. To avoid these problems, all nonlinear ion channels responsible for generating spikes are usually lumped together at the soma and the dendritic tree is treated as a passive cable. For a review of the compartmental approach we refer the reader to the book of Bower and Beeman (Bower and Beeman, 1995). In the following we illustrate the compartmental approach by a model of a cerebellar granule cell.
As an example for a realistic neuron model we discuss a model for cerebellar granule cells in turtle developed by Gabbiani and coworkers (Gabbiani et al., 1994). Granule cells are extremely numerous tiny neurons located in the lowest layer of the cerebellar cortex. These neurons are particularly interesting because they form the sole type of excitatory neuron of the whole cerebellar cortex (Ito, 1984).
Figure 2.20 shows a schematic representation of the granule cell model. It consists of a spherical soma and four cylindrical dendrites that are made up of two compartments each. There is a third compartment at the end of each dendrite, the dendritic bulb, that contains synapses with mossy fibers and Golgi cells.
One of the major problems with multi-compartment models is the fact that the spatial distribution of ion channels along the surface of the neuron is almost completely unknown. In the present model it is therefore assumed for the sake of simplicity that all active ion channels are concentrated at the soma. The dendrites, on the other hand, are described as a passive cable.
The granule cell model contains a fast sodium current INa and a calcium-activated potassium current IK(Ca) that provide a major contribution for generating action potentials. There is also a high-voltage activated calcium current ICa(HVA) similar to the IL-current discussed in Section 2.3.4. Finally, there is a so-called delayed rectifying potassium current IKDR that also contributes to the rapid repolarization of the membrane after an action potential (Hille, 1992).
Cerebellar granule cells receive excitatory input from mossy fibers and inhibitory input from Golgi cells. Inhibitory input is conveyed by fast GABA-controlled ion channels with a conductance that is characterized by a bi-exponential decay; cf. Section . Excitatory synapses contain both fast AMPA and voltage-dependent NMDA-receptors. How these different types of synapse can be handled in the context of conductance-based neuron models has been explained in Section 2.4.
Figure 2.21 shows a simulation of the response of a granule cell to a series of excitatory and inhibitory spikes. The plots show the membrane potential measured at the soma as a function of time. The arrows indicate the arrival time of excitatory and inhibitory spikes, respectively. Figure 2.21A shows nicely how subsequent EPSPs add up almost linearly until the firing threshold is finally reached and an action potential is triggered. The response of the granule cell to inhibitory spikes is somewhat different. In Fig. 2.21B a similar scenario as in subfigure A is shown, but the excitatory input has been replaced by inhibitory spikes. It can be seen that the activation of inhibitory synapses does not have a huge impact on the membrane potential. The reason is that the reversal potential of the inhibitory postsynaptic current of about -75 mV is close to the resting potential of -68 mV. The major effect of inhibitory input therefore is a modification of the membrane conductivity and not so much of the membrane potential. This form of inhibition is also called `silent inhibition'.
A final example shows explicitly how the spatial structure of the neuron can influence the integration of synaptic input. Figure 2.22 shows the simulated response of the granule cell to an inhibitory action potential that is followed by a short burst of excitatory spikes. In Fig. 2.22A both excitation and inhibition arrive on the same dendrite. The delay between the arrival time of inhibitory and excitatory input is chosen so that inhibition is just strong enough to prevent the firing of an action potential. If, on the other hand, excitation and inhibition arrive on two different dendrites, then there will be an action potential although the timing of the input is precisely the same; cf. Fig. 2.22B. Hence, excitatory input can be suppressed more efficiently by inhibitory input if excitatory and inhibitory synapses are closely packed together.
This effect can be easily understood if we recall that the major effect of inhibitory input is an increase in the conductivity of the postsynaptic membrane. If the activated excitatory and inhibitory synapses are located close to each other on the same dendrite (cf. Fig. 2.22A), then the excitatory postsynaptic current is `shunted' by nearby ion channels that have been opened by the inhibitory input. If excitatory and inhibitory synapses, however, are located on opposite dendrites (cf. Fig. 2.22B), then the whole neuron acts as a `voltage divider'. The activation of an inhibitory synapse `clamps' the corresponding dendrite to the potassium reversal potential which is approximately equal to the resting potential. The excitatory input to the other dendrite results in a local depolarization of the membrane. The soma is located at the center of this voltage divider and its membrane potential is accordingly increased through the excitatory input.
The difference in the somatic membrane potential between the activation of excitatory and inhibitory synapses located on the same or on two different dendrites may decide whether a spike is triggered or not. In cerebellar granule cells this effect is not very prominent because these cells are small and electrotonically compact. Nevertheless, the influence of geometry on synaptic integration can be quite substantial in neurons with a large dendritic tree. Effects based on the geometry of the dendritic tree may even have important implications for the computational ``power'' of a single neuron (Koch and Segev, 2000).
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