Neurons in the cortex and other areas of the brain often exhibit highly developed dendritic trees that may extend over several hundred microns (Fig. 3.4). Synaptic input to a neuron is mostly located on its dendritic tree. Disregarding NMDA- or Calcium-based electrogenic ‘spikes’, action potentials are generated at the soma near the axon hillock. Up to now we have discussed point neurons only, i.e., neurons without any spatial structure. What are the consequences of the spatial separation of input and output?
The electrical properties of point neurons have been described as a capacitor that is charged by synaptic currents and other transversal ion currents across the membrane. A non-uniform distribution of the membrane potential on the dendritic tree and the soma induces additional longitudinal current along the dendrite. We are now going to derive the cable equation that describes the membrane potential along a dendrite as a function of time and space. In Section 3.4 we will see how geometric and electrophysiological properties can be integrated in a comprehensive biophysical model.
Consider a piece of dendrite decomposed into short cylindric segments of length each. The schematic drawing in Fig. 3.5 shows the corresponding circuit diagram. Using Kirchhoff’s laws we find equations that relate the voltage across the membrane at location with longitudinal and transversal currents. First, a longitudinal current passing through the dendrite causes a voltage drop across the longitudinal resistor according to Ohm’s law,
where is the membrane potential at the neighbouring point . Second, the transversal current that passes through the RC-circuit is given by where the sum runs over all ion channel types present in the dendrite. Kirchhoff’s law regarding the conservation of current at each node leads to
The values of the longitudinal resistance , the capacity , the ionic currents as well as the externally applied current can be expressed in terms of specific quantities per unit length , , and , respectively, viz.
These scaling relations express the fact that the longitudinal resistance and the capacity increases with the length of the cylinder. Similarly, the total amount of transversal current increases with the length simply because the surface through which the current can pass is increasing. Substituting these expressions in Eqs. (3.9) and (3.10), dividing by , and taking the limit leads to
Eq. (3.13) is called the general cable equation.
In the following we will concentrate on the equation for the voltage and start our analysis by a discussion of the Green’s function for a cable extending to infinity in both directions. The Green’s function is defined as the solution of a linear equation such as Eq. (3.17) with a Dirac -pulse as its input. It can be seen as an elementary solution of the differential equation because – due to linearity – the solution for any given input can be constructed as a superposition of these Green’s functions.
Suppose a short current pulse is injected at time at location . As we will show below, the time course of the voltage at an arbitrary position is given by
where is the Green’s function. Knowing the Green’s function, the general solution for an infinitely long cable is given by
The Green’s function is therefore a particularly elegant and useful mathematical tool: once you have solved the linear cable equation for a single short current pulse, you can write down the full solution to arbitrary input as an integral over (hypothetical) pulse-inputs at all places and all times.
Previously, we have just ‘guessed’ the Green’s function and then shown that it is indeed a solution of the cable equation. However, it is also possible to derive the Green’s function step by step. In order to find the Green’s function for the cable equation we thus have to solve Eq. (3.17) with replaced by a impulse at and .
Fourier transformation with respect to the spatial variable yields
This is an ordinary differential equation in and has a solution of the form
with denoting the Heaviside function. After an inverse Fourier transform we obtain the desired Green’s function ,
In the context of a realistic modeling of ‘biological’ neurons, two non-linear extensions of the cable equation have to be discussed. The obvious one is the inclusion of non-linear elements in the circuit diagram of Fig. 3.5 that account for specialized ion channels. As we have seen in the Hodgkin-Huxley model, ion channels can exhibit a complex dynamics that is in itself governed by a system of (ordinary) differential equations. The current through one of these channels is thus not simply a (non-linear) function of the actual value of the membrane potential but may also depend on the time course of the membrane potential in the past. Using the symbolic notation for this functional dependence, the extended cable equation takes the form
A more subtle complication arises from the fact that a synapse can not be treated as an ideal current source. The effect of an incoming action potential is the opening of ion channels. The resulting current is proportional to the difference of the membrane potential and the corresponding ionic reversal potential. Hence, a time-dependent conductivity as in Eq. (3.1) provides a more realistic description of synaptic input than an ideal current source with a fixed time course.
If we replace in Eq. (3.17) the external input current by an appropriate synaptic input current with being the synaptic conductivity and the corresponding reversal potential, we obtain44We want outward currents to be positive, hence the change in the sign of and .
This is still a linear differential equation but its coefficients are now time-dependent. If the time course of the synaptic conductivity can be written as a solution of a differential equation, then the cable equation can be reformulated so that synaptic input reappears as an inhomogeneity to an autonomous equation. For example, if the synaptic conductivity is simply given by an exponential decay with time constant we have
Here, is a sum of Dirac functions which describe the presynaptic spike train that arrives at a synapse located at position . Note that this equation is non-linear because it contains a product of and which are both unknown functions of the differential equation. Consequently, the formalism based on Green’s functions can not be applied. We have reached the limit of what we can do with analytical analysis alone. To study the effect of ion channels distributed on the dendrites nummerical approaches in compartmental models become invaluable (Section 3.4).
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