Next examined are the dynamic properties of a uniform diameter, core-conductor (dendrite), using the notation of Lathi (1965). The passive, cylindrical dendrite is characterized by the functions:

Characteristic impedance : Zo (s) = ^(ro + ri))(gm + scm) 4.2-15

Propagation function : y(s) = .y(ro + ri)(gm + scm) 4.2-16

Lathi (1965) derives voltage and current transfer functions for a uniform, finite transmission line:

The characteristic impedance of the line, Zo, can be written:

Zo (s) = V(o + ri )/( + scm ) = ( + ri V (1 + st) 4.2-20

where X is the space constant derived above, and the time constant of Zo is t = cm/gm s. If an ideal voltage source, vin, sets vm at x = 0, then in general:

vm(x, t) = L-1(Vin(s) Hv(x, s)], 0 ô x ô L 4.2-21

Now let vfa(t) = S(t) at the left end (x = 0) of the line, terminate the line in its characteristic impedance, Zo(s), so p = 0, and examine vm(t) at x = L/2. Needed is vm(L/2, t) = L- } = L 1 {e-YW2} = L"1 {exp[-aV(7^} 4.2-23

(note that the same result will be obtained if the line is half-infinite, i.e., it ends at

L ^ x) where a = (L/2) (ri + ro)cm , and c = gm/cm r/s. The inverse Laplace transform for Equation 4.3-23 is not known. One means of finding vm(L/2, t) is to let s = jra in Equation 4.2-23, and compute the inverse discrete Fourier transform of {Vi (jra)exp[-a (jo) + c) }. (Note that in polar form, the vector, v = a ^(jo + c) , can be written v = a ^(o2 + c2) Z {[tan-1(ra/c)]/2}.)

Many interesting, complex dendritic architectures have been analyzed by Rall (in Koch and Segev, 1989). Several principles of dendritic behavior emerge from Rall's calculations. (1) The farther a psp site is out along the dendritic "tree," the smaller and more delayed the corresponding psp will be at the SGL. (2) There is also a loss of high frequencies from the psp as it propagates toward the soma and SGL. Thus, the peak of the psp at the soma is lower and delayed with respect to the psp at the synapse; it is also more rounded due to the high frequency attenuation (see Figure 4.2-3).

Impulse input

Impulse input

FIGURE 4.2-3 Figure illustrating how an impulse input at the tip of a dendrite propagates toward the soma and thence out a neurite (an axon is not shown). Arrows mark the peaks of the wave; note the progressive delay and attenuation with distance from the source.

FIGURE 4.2-3 Figure illustrating how an impulse input at the tip of a dendrite propagates toward the soma and thence out a neurite (an axon is not shown). Arrows mark the peaks of the wave; note the progressive delay and attenuation with distance from the source.

The analysis above has considered the psp stimulus to be a voltage source applied at some point on the x axis of a dendrite model. More realistically, the input to the dendrite can be considered to be a local conductance increase at the SSM for a specific ion (Na+ for epsps) caused by the arrival of neurotransmitter molecules at receptor sites in the SSM. The inrush of Na+ locally depolarizes the SSM, generating the epsp considered to be a voltage source. In a more realistic simulation of dendrite function, one might apply a transient increase of gm at a point. For this parametric input to be effective, the core-conductor dendrite model must include a dc bias voltage to represent the -65 mV resting potential. Still another approach is to inject some charge (a current pulse) at the SSM site to represent the transient Na+ inrush.

The bottom line about dendrites is that their analysis in terms of neuron function is exceedingly complex, and any meaningful detailed description of the function of dendrites with complex geometry necessarily must rely on tedious computational means. Medium-scale neural modeling can avoid the detailed simulation of dendrite effects by passing a unit impulse representing the action potential of a presynaptic neuron's through a two-time-constant, low-pass ("ballistic") filter to generate the psp, then pass the psp through an attenuating low-pass filter followed by a pure delay operation to emulate the psp popagation along the dendrite to the SGL. The resulting processed epsps (or ipsps) are literally summed at the SGL to determine whether the neural model will generate a spike.

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