- 2.5.1 Derivation of the Cable Equation

- 2.5.2 Green's Function (*)

- 2.5.3 Non-linear Extensions to the Cable Equation

2.5 Spatial Structure: The Dendritic Tree

Neurons in the cortex and other areas of the brain often exhibit
highly developed dendritic trees that may extend over several hundreds
of m. Synaptic input to a neuron is mostly located on its
dendritic tree, spikes, however, are generated at the soma near the
axon hillock. What are the consequences of the spatial separation of
input and output? Up to now we have discussed point neurons only,
i.e., neurons without any spatial structure. 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
passive dendrite as a function of time *and* space. In
Section 2.6 we will see how geometric and
electrophysiological properties of a certain type of neuron can be
integrated in a comprehensive biophysical model.

2.5.1 Derivation of the Cable Equation

Consider a piece of a dendrite decomposed in short cylindric segments
of length
d*x* each. The schematic drawing in Fig. 2.16 shows the corresponding circuit diagram. Using Kirchhoff's
laws we find equations that relate the voltage *u*(*x*) across the
membrane at location *x* with longitudinal and transversal
currents. First, a longitudinal current *i*(*x*) passing through the
dendrite causes a voltage drop across the longitudinal resistor
*R*_{L} according to Ohm's law,

where

The values of the longitudinal resistance

R_{L} = r_{L} dx , R^{-1}_{T} = r^{-1}_{T} dx , C = c dx , I_{ext}(t, x) = i_{ext}(t, x) dx . |
(2.25) |

These scaling relations express the fact that the longitudinal resistance and the capacity increase with the length of the cylinder, whereas the transversal resistance is decreasing, simply because the surface the current can pass through is increasing. Substituting these expressions in Eqs. (2.24) and (2.25), dividing by d

Taking the derivative of these equations with respect to

We introduce the characteristic length scale =

After a transformation to unit-free coordinates,

x = x/ , t = t/ , |
(2.29) |

and rescaling the current variables,

i = i , i_{ext} = r_{T} i_{ext} , |
(2.30) |

we obtain the cable equations (where we have dropped the hats)

in a symmetric, unit-free form. Note that it suffices to solve one of these equations due to the simple relation between

The cable equations can be easily interpreted. These equations
describe the change in time of voltage and longitudinal current. Both
equations contain three different contributions. The first term on the
right-hand side of Eq. (2.32) is a diffusion term that is
positive if the voltage (or current) is a convex function of *x*. The
voltage at *x* thus tends to decrease, if the values of *u* are lower
in a neighborhood of *x* than at *x* itself. The second term on the
right-hand side of Eq. (2.32) is a simple decay term that
causes the voltage to decay exponentially towards zero. The third
term, finally, is a source term that acts as an inhomogeneity in the
otherwise autonomous differential equation. This source can be due to
an externally applied current, to synaptic input, or to other
(non-linear) ion channels; cf.Section 2.5.3.

In order to get an intuitive understanding of the behavior of the
cable equation we look for stationary solutions of Eq. (2.32a), i.e., for solutions with
*u*(*t*, *x*)/*t* = 0. In that
case, the partial differential equation reduces to an ordinary
differential equation in *x*, viz.

u(t, x) - u(t, x) = - i_{ext}(t, x) . |
(2.32) |

The general solution to the homogenous equation with

u(t, x) = c_{1} sinh(x) + c_{2} cosh(x) , |
(2.33) |

as can easily be checked by taking the second derivative with respect to

Solutions for non-vanishing input current can be found by standard
techniques. For a stationary input current
*i*_{ext}(*t*, *x*) = (*x*)
localized at *x* = 0 and boundary conditions
*u*(±) = 0 we find

cf. Fig. 2.17. This solution is given in units of the intrinsic length scale = (

For arbitrary stationary input current
*i*_{ext}(*x*) the
solution of Eq. (2.32a) can be found by a superposition of
translated fundamental solutions (2.35), viz.,

u(t, x) = dx' e^{-x - x'} i_{ext}(x') . |
(2.35) |

This is an example of the Green's function approach applied here to the stationary case. The general time-dependent case will be treated in the next section.

2.5.2 Green's Function (*)

In the following we will concentrate on the equation for the voltage and start our analysis by deriving 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. (2.32) 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.

In order to find the Green's function for the cable equation we thus
have to solve Eq. (2.32a) with
*i*_{ext}(*t*, *x*)
replaced by a impulse at *x* = 0 and *t* = 0,

Fourier transformation with respect to the spatial variable yields

u(t, k) + k^{2} u(t, k) + u(t, k) = (t)/ . |
(2.37) |

This is an ordinary differential equation in

u(t, k) = exp - 1 + k^{2} t/ (t) |
(2.38) |

with (

The general solution for an infinitely long cable is therewith given through

We can check the validity of Eq. (2.40) by substituting
*G*_{}(*t*, *x*) into the left-hand side of Eq. (2.37). After a
short calculation we find

where we have used (

exp - t - = (x) , |
(2.42) |

which proves that the right-hand side of Eq. (2.42) is indeed equivalent to the right-hand side of Eq. (2.37).

Having established that

- + 1 G_{}(t, x) = (x) (t) , |
(2.43) |

we can readily show that Eq. (2.41) is the general solution of the cable equation for arbitrary input currents

Real cables do not extend from - to + and we have to
take extra care to correctly include boundary conditions at the ends.
We consider a finite cable extending from *x* = 0 to *x* = *L* with sealed
ends, i.e.,
*i*(*t*, *x* = 0) = *i*(*t*, *x* = *L*) = 0 or, equivalently,
*u*(*t*, *x* = 0) = *u*(*t*, *x* = *L*) = 0.

The Green's function *G*_{0, L} for a cable with sealed ends can be
constructed from *G*_{} by applying a trick from electro-statics
called ``mirror charges'' (Jackson, 1962). Similar techniques can also
be applied to treat branching points in a dendritic tree
(Abbott, 1991). The cable equation is linear and, therefore, a
superposition of two solutions is also a solution. Consider a
current pulse at time *t*_{0} and position *x*_{0} somewhere along the
cable. The boundary condition
*u*(*t*, *x* = 0) = 0 can be satisfied if we add a second, virtual
current pulse at a position *x* = - *x*_{0} *outside* the interval
[0, *L*]. Adding a current pulse outside the interval [0, *L*] comes for
free since the result is still a solution of the cable equation on
that interval. Similarly, we can fulfill the boundary condition at
*x* = *L* by adding a mirror pulse at
*x* = 2 *L* - *x*_{0}. In order to account
for both boundary conditions simultaneously, we have to compensate for
the mirror pulse at - *x*_{0} by adding another mirror pulse at
2 *L* + *x*_{0} and for the mirror pulse at
*x* = 2 *L* - *x*_{0} by adding a fourth
pulse at -2 *L* + *x*_{0} and so forth. Altogether we have

We emphasize that in the above Green's function we have to specify both (

u(t, x) = dt_{0} dx_{0} G_{0, L}(t_{0}, x_{0};t, x) i_{ext}(t_{0}, x_{0}) . |
(2.44) |

An example for the spatial distribution of the membrane potential along the cable is shown in Fig. 2.18A, where a current pulse has been injected at location

2.5.3 Non-linear Extensions to the Cable Equation

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. 2.16 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
*i*_{ion}[*u*](*t*, *x*) for this functional dependence the
extended cable equation takes the form

u(t, x) = u(t, x) - u(t, x) - i_{ion}[u](t, x) + i_{ext}(t, x) . |
(2.45) |

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. (2.19) provides a more realistic description of synaptic input than an ideal current source with a fixed time course.

If we replace in Eq. (2.32a) the external input current
*i*_{ext}(*t*, *x*) by an appropriate synaptic input current
- *i*_{syn}(*t*, *x*) = - *g*_{syn}(*t*, *x*)[*u*(*t*, *x*) - *E*_{syn}]
with
*g*_{syn} being the synaptic conductivity and
*E*_{syn} the corresponding reversal potential, we
obtain^{2.2}

u(t, x) = u(t, x) - u(t, x) - g_{syn}(t, x)[u(t, x) - E_{syn}] . |
(2.46) |

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,

Cambridge University Press, 2002

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