The Neurobiology Laboratory, Institute for Biomedical Research, The
Department of Physiology and The School of Mathematics and Statistics,
University of Sydney, New South Wales 2006, Australia
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INTRODUCTION |
The motor-nerve terminal in amphibia consists of a
number of terminal branches ranging in length from ~20 µm to >100
µm, generally arranged in parallel on the surface of the muscle, and
containing tens to hundreds of active zones (Bennett et al.
1986
; Katz 1969
). The release of transmitter
from different sites on these terminal branches by depolarizing pulses
applied with an external electrode in the absence of impulses has been
used to determine the relationship between the size of the passive
membrane depolarization (the electrotonus) (Hodgkin and Rushton
1946
) and the resulting number of quanta released (Dudel
1984
; Katz and Miledi 1967
). In addition, the extent to which transmitter release at a test electrode is modified by
prior stimulation at a conditioning electrode placed elsewhere on a
terminal branch has been used to analyze possible interactions between
the release process at different active zones (Dudel et al.
1993
). However, to this time there has been no cable analysis of how an electrotonus is likely to be generated and to propagate in
the branches of the terminal, so there is no analytic formulation to
guide the interpretation of experimental results. The problem concerns
how the electrotonus propagates in a branch that is sealed at both ends
and wrapped tightly in a Schwann cell sheath (Heuser and Reese
1977
), following the application of current at a chosen site by
an external electrode placed on the sheath. Although there has been
extensive analysis of the spread of current in dendritic trees
following the initiation of a synaptic potential (for a review see
Rall et al. 1992
), as well as of the propagation of action potentials in branching nerve terminals (Lindgren and
Moore 1989
; Lüscher and Shiner 1990
;
Manor et al. 1991
), there has been little attention paid
to the spread of electrotonus in nerve terminals (Jackson
1993
). In the present work, an analytic description is given of
the membrane potential changes in a motor-nerve terminal branch,
wrapped in Schwann on the surface of a muscle fiber, in response to
current injection from one or more external electrodes placed on the
branch. This theory is used to interpret the results of how quantal
release by an electrotonus at one electrode on a branch can be modified
by a preceding electrotonus at another site on the same or a contiguous
branch of the motor-nerve terminal.
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METHODS |
Experimental
PREPARATION AND SOLUTIONS.
The iliofibularis muscle of the toad Bufo marinus was used
in all experiments. Animals were collected between the months of April
and February, measuring between 47 and 73 mm in length, and were killed
by pithing. Muscles were pinned on a silicone elastomer (Sylgard) bed
in a perspex organ bath of 3 ml capacity. Ringer solution of the
following composition was used to perfuse the preparation (in mM):
111.2 NaCl, 2.5 KCl, 1.5 NaH2PO4, 16.3 NaHCO3, 7.8 glucose, and 1.2 MgCl2.
CaCl2 was present at 1.8 mM, unless otherwise stated, in
both the bath and the electrodes. The Ringer solution was continually
bubbled with carbogen gas (95% O2-5% CO2) and
maintained at a pH of 7.2-7.4. The temperature was kept between 18 and
20°C. Tetrodotoxin in sodium citrate buffer and cadmium were
dissolved directly in water and used at 0.2 and 50 µM, respectively.
Nifedipine was used at 10 µM in the presence of 0.02% ethanol.
DODC IODIDE AND FM1-43 FLUORESCENCE.
The motor nerve terminal was visualized using epifluorescence following
treatment with either the styryl dye N-(3-(triethyl ammonium) propyl)-4-(4-dibutylaminostyryl pyridium), dibromide (FM1-43) (Betz and Bewick 1990
; Betz et al.
1992
) or 3,3'-diethyloxadicarbocyanine iodide (DODC iodide)
(Bennett et al. 1986
; Yoshikami and Okun 1984
). DODC iodide revealed the position of the axon and
Schwann cell nucleus, whereas FM1-43 most accurately defined the
extent of the endplate by showing the location of presynaptic vesicle clusters. In some instances the often complex branching of the nerve
terminal could only be clarified using FM1-43.
The iliofibularis was exposed to 0.1 µM DODC iodide in 0.001% DMSO
in Ringer solution for 40 s and then washed in Ringer solution for
5 min. DODC iodide fluorescence was observed during excitation at 540 nm using an Olympus microscope (BH-2) with fluorescence attachment and
rhodamine filter set. An Olympus WPlanFL40XUV water immersion objective
(0.7 NA) was used to view the terminals, and the image was displayed on
a video monitor (National WV-5470) using a low light TV camera
(National WV-1900/B). Images were captured and saved using a Scion
Corporation LG3 framegrabber with a 7200/90 Power Macintosh. Terminals
were stained with FM1-43 by exposure to 2 µM FM1-43 in a modified
Ringer solution (53.7 mM NaCl, 60 mM KCl) for 5 min followed by a
minimum of 30 min washing with Ringer solution. FM1-43 fluorescence
was observed during excitation using an Olympus fluorescein filter set.
ELECTROPHYSIOLOGICAL RECORDINGS.
In each experiment a pair of electrodes was placed on the nerve
terminal under direct visual control. Each electrode was capable of
passing a stimulating current of up to 2 µA as well as recording evoked release in response to the stimulus. Axon Instruments HS-2A headstages (X10MG) were used with two Axoclamp-2A amplifiers. Recording
electrodes were manufactured in the following way. Glass micropipettes
were pulled to form tips of 2 µm diam and were then chipped to form a
tip with an internal diameter of 15 µm, angled at ~50-60° to the
long axis of the pipette. Each pipette was heat polished in a
microforge to yield a final tip diameter of ~12 µm. Electrodes were
filled with the tissue bathing solution. The position of the electrode
tips relative to each other, to the muscle surface, and to the nerve
terminal as revealed by staining and epifluorescence was determined by
viewing the images on the video monitor. Electrodes were placed
directly on selected regions of nerve terminals. Light downward
pressure was applied at each electrode site to form a slight seal.
Negative pressure was often applied to the solution within the
electrodes, equivalent to 100 mm of H2O, to improve the
seal. In experiments where the bathing solution contained no added
calcium, positive pressure (equivalent to 100 mm of H2O)
was applied to the solution within the electrodes that contained
calcium (1.8 mM). Particular attention was paid to the level of
spontaneous activity that may indicate mechanical irritation. At any
pair of sites, the duration of stimulating pulses was the same while
the current amplitude was adjusted at each electrode to yield a quantal
content close to 0.2. Quantal content was calculated using the method
of failures [m = ln(N/N0), where N is
number of impulses and N0 is the number of
failures] (del Castillo and Katz 1954
). If quantal
content rose or fell by >25% over the course of a recording trial,
the observations were not included in the results. If more than two
spontaneous events were observed over the 4.8 s (384 × 12.5 ms)
of recorded activity at each site during a trial, the results were
discarded. Tetrodotoxin (0.2 µM) was always present in both the bath
and the electrodes.
Each trial consisted of 384 stimulation pulses delivered at the rate of
1 Hz to each electrode. The order and delay of current pulses delivered
to different sites on the terminal by the two electrodes was varied
using a three-phase cycle, and this cycle was repeated 128 times.
Figure 1 shows a cycle in a typical trial: 1) a pair of pulses delivered simultaneously at each site
(zero delay); 2) a pulse delivered at electrode 1 ~4 ms before a pulse at electrode 2 (4-ms delay);
3) a pulse delivered at electrode 2 ~4 ms
before a pulse at electrode 1. This alternating system of
stimuli was used to avoid any problems relating to nonstationarity as
the conditioning pulses in the second and third pairs could be used as
the unconditioned references for the test pulses. In the case of the
first pair of pulses in Fig. 1, where pulses were delivered
simultaneously to each site, referring to one electrode as the test
electrode and the other as the conditioning electrode is done for ease
of explanation only. Where there is zero delay between pulses, either
electrode can be regarded as the test electrode or the conditioning
electrode and mT/m at zero delay can
be calculated for both sites, where
mT/m refers to the measure of spatial
facilitation or depression (Dudel et al. 1993
), as the
average quantal content measured at the test site when a pulse at the
test site is preceded by a pulse at the conditioning site
(mT), divided by the average quantal content
produced by a pulse at the test site in the absence of any stimulus at
the conditioning site (m).

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Fig. 1.
Stimulation of the nerve terminal and evoked responses. Three
consecutive pairs of traces from the start of a train of 384 pairs of
pulses delivered at 1 Hz. Pulses of 80 µs duration were used in this
example, and the delay between the start of the conditioning pulse and
the start of the test pulse was set at 4 ms in pairs number 2 and
number 3.
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Data were collected using a MacLab/4s data acquisition system.
Theoretical
This section contains only a summary of the theoretical model; a
detailed account can be found in Bennett et al. (1999)
.
A motor-nerve terminal branch is modeled as a one-dimensional cable
with external resistance and leakage to earth (see Fig. 1C
in Bennett et al. 1999
). The extracellular longitudinal
resistance is primarily the resistance to longitudinal current flow
between the nerve-terminal and its associated Schwann-cell sheath,
whereas the resistance to earth is primarily that provided by the
tortuosities in the Schwann-cell sheath. Let Vi
be the intracellular potential, Vo the
extracellular potential, and Vm = Vi
Vo the membrane potential. In the continuum limit these satisfy the differential equations
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(1)
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(2)
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where
= ro/ri, µ = rm/re,
=
,
X =
, T
= t/
m,
m = rmcm. The quantities ro, ri,
rm, re,
cm, and
m are defined in Table
1; x (µm) is distance along
the cable, and t (ms) is time. The injected current is
Iin(T) and the delta-function
(X) indicates that injection takes place at the point
X = 0.
In the case of no earth leakage (µ = 0), Eqs. 1 and 2 can be combined to give a single equation for the membrane
potential
|
(3)
|
which is just the standard cable equation and is readily soluble
for a range of inputs and boundary conditions. However, for µ
0, Eqs. 1 and 2 do not uncouple, and their solution
is more difficult. For the infinite cable, some analytic solutions are
possible (Bennett et al. 1999
), but for the finite
cable, which is the case of interest here, it is best to proceed
straight to a numerical solution. A numerical method previously
employed to solve similar equations describing a bidomain model of
smooth muscle tissue (Henery et al. 1997
) was adapted to
the present case (again, details are in Bennett et al.
1999
). This allows the solution of the present cable model with
sealed ends and current injection at arbitrary positions at arbitrary times.
 |
RESULTS |
Experimental
The evoked quantal release at a test stimulating electrode on a
terminal branch as a result of injecting current with a conditioning electrode on the same or a contiguous terminal branch, at different test-conditioning intervals, was determined. Tetrodotoxin was always
present to block voltage-dependent sodium conductance changes and
cadmium sometimes present to block voltage-dependent calcium conductance changes.
Two stimulating/recording electrodes were placed in the loose-patch
mode at different positions along the length of a single terminal
branch that had previously been visualized with either DODC iodide or
FM1-43. The electrode proximal to the site of nerve entry was used to
deliver a conditioning impulse at an interval before determination of
the quantal release produced by a test impulse at the distal site
(mtd). This was compared with the quantal release evoked at the distal site in the absence of a conditioning stimulus at the proximal site (md), so giving
mtd/md. Data from the
reverse stimulating protocol (conditioning stimulus at a distal site
and the test stimulus at a proximal site) was also available from the
same recording trial and allowed calculation of
mtp/mp. The three phase
alternating cycle of stimuli illustrated in Fig. 1 allows calculation
of mtd/md and
mtp/mp for delays of
between 0 and 4 ms. Pulses were either 80 or 200 µs in duration. The
extent to which release was evoked from sites beyond the rim of the
stimulating electrode was checked by stimulating with the loose-patch
electrode and simultaneously recording all the quantal release from the nerve terminal using an intracellular electrode. It was found that all
releases recorded with the intracellular electrode were accompanied by
negative-going signs of quantal release at the loose-patch electrode
(data not shown).
The values of mtd/md and
mtp/mp observed for
different conditioning-test delays when both electrodes were placed on
the same motor-nerve terminal branch are shown in Fig.
2. Results have been pooled for 11 different
terminal branches in which the distance between the 2 stimulating/recording electrodes varied between 28 and 85 µm and the
branch length from 57 to 154 µm. The average value of
mtp/mp at zero delay was
0.56 ± 0.06 (mean ± SE, n = 9);
and mtd/md was 0.38 ± 0.08 (n = 11; Fig. 2D); in both cases no
depression in release was observed when the delay between stimuli was 2 or 4 ms. The average quantal content at the proximal electrode was
0.33 ± 0.06 (n = 9) and at the distal electrode
0.21 ± 0.03 (n = 11). Experiments were also
carried out when each of the two stimulating/recording electrodes were
placed on different terminal branches. Figure
3A shows the results when each of
these branches belong to the same motor-nerve terminal. Depression in
the average release of quanta by a test impulse at one electrode
following a conditioning impulse at the other electrode when the delay
was zero occurred in much the same way as that observed when both electrodes were on the same terminal branch (compare Fig. 3A
with Fig. 2D); this occurred independently of which
electrode produced the conditioning stimulus. These results indicate
that the electrotonus generated in one terminal branch can propagate to
a contiguous terminal branch.

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Fig. 2.
Degree of depression of quantal release at a test site following
stimulation at a conditioning site. A: schematic diagram
showing placement of 2 electrodes ( and ) on a
single branch of a nerve terminal for which data are shown in
B-D. B: abscissa shows the time delay between the start
of the conditioning pulse and the start of the test pulse. Ordinate
gives the ratio of the average quantal content at the test-site
electrode following a conditioning pulse
(mtp or mtd) to
the average quantal content at the test-site electrode without a
conditioning pulse (mp or
md); because the results were pooled, the
ordinate is labeled mT/m.
Individual estimates of mT/m
from every recording trial at each of 11 different motor-nerve
terminals are included. Each point represents an estimate of
mT/m at either a proximal or
distal site for a single set of sites. Two estimates of
mT/m are gained from each
pair of sites because of the alternating but symmetrical stimulus
paradigm (Fig. 1). , test electrode positioned proximally to
the site of axon entry relative to the conditioning electrode;
, test electrode more distal. Recording trials were not
included where m was <0.05. C: average
results for each of the 11 different motor-nerve terminals. Each
average estimate was derived from between 1 and 4 recording trials at a
set of sites. D: average estimates of
mT/m for all sets of sites on
the same terminal branch. Estimates of
mT/m gained from up to 4 recording trials at a particular motor-nerve terminal were given equal
weighting as those gained from a single trial (a train of 384 paired
pulses) at another motor-nerve terminal. Vertical bars, means ± SE. All pulses were of 80 or 200 µs duration and delivered at 1 Hz.
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Fig. 3.
Degree of depression with different electrode configurations.
A: average value of
mT/m when the 2 electrodes
are on different branches of the same motor-nerve terminal. Estimates
of mT/m from sites on both
branches were combined when calculating the average at each delay
(n = 25 when delay was 0). B:
average value of mT/m when
the 2 electrodes are on different motor-nerve terminals on the same
muscle fiber (n = 14 when delay was 0).
C: estimates of
mT/m when the test electrode
is placed on a motor-nerve terminal branch and the conditioning
electrode is placed either directly on the muscle surface
( ), or the conditioning electrode is placed above the
same motor-nerve terminal branch out of direct contact
( ) as shown in the insets. Where the
conditioning electrode was placed directly on the muscle surface, the
distance between the electrodes was 65 µm with the electrode on the
muscle displaced from the end of the terminal branch by ~40 µm.
D: averages of data shown in C.
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Although the loose-patch electrodes were placed visually on the same or
contiguous terminal branches, further checks were made to ensure that
current injection from the electrodes only gave rise to an electrotonus
in the injected branch(s). When muscle fibers that received a
polyneuronal innervation were observed, the stimulating/recording
electrodes were placed on different branches, one belonging to one
motor-nerve terminal and the other to the second terminal (Fig.
3B). In this case no depression was observed in the average
release of quanta evoked by a test impulse at one electrode when
delivered at zero delay following a conditioning impulse at the other
electrode. The average distance between the two electrodes (Fig.
3B) was 81 µm, whereas electrodes on the same branch (Fig.
3) were an average of 52 µm apart, and electrodes on separate
branches of the same terminal were on average 70 µm apart (Fig. 3,
A and B). In another series of experiments, one electrode was placed in the loose-patch mode on a terminal branch and
the other electrode on the underlying muscle fiber, ~65 µm away, as
shown schematically in Fig. 3C (top panel). A
conditioning impulse on the muscle fiber had no affect on quantal
release at the test electrode on the nerve terminal (Fig. 3,
C and D). Alternatively, one electrode was placed
on a terminal branch in the loose-patch mode, and the other was
displaced 70 mm vertically above the same terminal branch so that it
was no longer in direct contact with the branch. Again there was no
effect of a conditioning pulse on quantal release by a test pulse at
the electrode in loose-patch mode on the motor nerve terminal branch
(Fig. 3, C and D). The current injection is then
restricted to the terminal branch(s) on which the electrodes are placed.
To check that the modifications in quantal release at the test
electrode due to current injection at the conditioning electrode were
due to passive propagation of an electrotonus, checks were made to
ensure that there were no voltage-dependent calcium conductances involved, the sodium ones having been eliminated with tetrodotoxin. The
effect of decreasing [Ca2+] in the bathing solution on
test quantal secretion was therefore examined. Pulses of 500 µs
duration were used when calcium was not added to the bath, because
under these circumstances it is difficult to evoke quanta with current
pulses of 80 or 200 µs durations. In experiments where electrodes
were either on the same branch or on different branches of the same
terminal, the average value of mT/m
at zero delay when 500-µs pulses were used with 1.8 mM
[Ca2+] in both the bath and electrodes was 0.62 ± 0.18 (n = 8; Fig. 4A). If calcium was not added to
the bathing Ringer solution but calcium was maintained at 1.8 mM in
both electrodes, then mT/m was
0.76 ± 0.10 (n = 10; Fig. 4B), so that
the depression in release was not contingent on a propagating potential
dependent on voltage-dependent conductance changes. A further check on
this was made by introducing 50 µM cadmium into the normal bath
solution. The depression in mT/m when
a zero delay was used was not affected by cadmium (0.49 ± 0.08, n = 4) compared with 0.56 ± 0.05 (n = 35; Fig. 4C). Nifedipine (10 µM) also
had little effect (Fig. 4D).

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Fig. 4.
Effect of pulse duration and of Ca2+, cadmium, and
nifedipine on depression. A: average values of
mT/m for different values of
pulse duration. Abscissa gives the pulse duration used for both
conditioning and test pulses with 1.8 mM [Ca2+] in the
bath and electrodes (n = 35, 14, 8, and 2, respectively). B: average values of
mT/m using 500 µs duration
pulses in the presence of Ca2+ (1.8 mM) ( )
and in its absence ( , n = 10, 10, and 4) while maintaining 1.8 mM [Ca2+] in the electrodes.
C: average values of
mT/m when 50 µM cadmium is
present in both the bath and electrodes ( ,
n = 4) and when it is not ( ,
n = 4). D: average values of
mT/m when 10 µM nifedipine
is present in both the bath and electrodes ( ,
n = 8, 7, and 8) and when it is not
( , n = 8). Diagrams show the
arrangement of the electrodes in each case.
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Theoretical
The cable model was used to determine the effects of injecting a
depolarizing current pulse of 200 µs duration with an external electrode at the middle of terminal branches of different length. The
resulting membrane potential changes were then determined for branches
of length 2,000, 400, 300, and 200 µm. Figure
5A shows that the spatial
distribution of the depolarization just before the end of the current
pulse extends, in the case of a cable 200 µm long, for a distance of
~30 µm on either side of the electrode, before passing into a
hyperpolarization that extends out much greater distances to reach the
end of the branches for all but the longest branch considered. The time
courses of the potential changes at different distances along the
length of branches of different lengths, shown in Fig. 5,
B-D, indicate that the membrane potential change close to
the site of current injection (within ~30 µm for the 200-µm-long
cable) is always in the depolarizing direction, but that at further
distances out the initial depolarization changes to a hyperpolarization
that outlasts the time course of the current pulse of 200 µs duration
by periods that are longer than the pulse duration.

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Fig. 5.
Electrotonic potentials in the cable model of a terminal branch
following injection of a current pulse of duration 200 µs and
magnitude 15 nA at the midpoint. Aa-Ad:
spatial distribution of the membrane potential displacement at the end
of the current pulse for branches of different total lengths, as
indicated; the horizontal broken line is zero potential.
B-D: temporal changes in the membrane potential at the
3 positions indicated by the vertical dotted lines in the corresponding
graph in A.
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The cable model was next used to determine the effects of injecting
depolarizing currents pulses of 200 µs duration simultaneously with
two external electrodes placed at different distances apart but
symmetrically in the middle of terminal branches of different length.
The spatial distributions of the membrane potential change just before
the end of the current pulse when the electrodes are 60 µm apart are
shown in Fig. 6A for terminal
branches of different length. The membrane is depolarized between the
electrodes and for some distance beyond on either side of them at all
branch lengths, but then passes into a hyperpolarization that reaches to the end of the branches. Determination of the time course of membrane potential change at one electrode due to current injection alone at the other shows how the potential changes from being predominantly a depolarization to a hyperpolarization as the length of
the branch is decreased (Fig. 6B). On the other hand,
determination of the time course of the membrane potential change at
one electrode due to current injection alone at that electrode shows
that this is of course always a depolarization, but of smaller
amplitude as the length of the branch decreases (Fig. 6C),
due to the end effects caused by the termination of the branch. The
result is that the time course and amplitude of the membrane potential
change at an electrode when both electrodes inject current
simultaneously is due to the interaction of the propagating
electrotonus from the other electrode as well as the current flow from
the electrode in question in relation to the end of the branch (Fig.
6D).

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Fig. 6.
Electrotonic potentials in the cable model of a terminal branch
following synchronous current injection by 2 electrodes
(E1 and E2) at 2 sites symmetrically
placed on the branch. Each injection is a pulse of duration 200 µs
and magnitude 15 nA. Aa-Ad: spatial distribution of the
membrane potential displacement at the end of the current pulses, for
branches of the total lengths indicated, when there is simultaneous
injection from E1 and E2 at the positions
indicated by the vertical dotted lines; the horizontal broken line is
zero potential. B: time course of the membrane potential
at E2 when there is current injection at
E1 only. C: time course of the membrane
potential at E2 when there is current injection at
E2 only. D: time course of the membrane
potential at E2 when there is current injection at both
electrodes. In each case, results are given for the 4 cable lengths in
A.
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The changes in amplitude of the membrane potential at one of
these two electrodes near the end of the 200-µs pulse for
simultaneous current injections from the electrodes at different
distances apart on branches of different length are shown in Fig.
7A. As the separation between the
electrodes increases, there is at first a substantial decrease in the
amplitude of the membrane potential change beneath an electrode due to
the increasing hyperpolarizing effects produced by current flow from
the other electrode. With further separation, the membrane potential
change increases as the effects of current flow from the electrode on
the ends of the branch become important. The changes in amplitude of
the membrane potential at one of the two electrodes near the end of the
200-µs pulse when current is injected from the electrode alone at
different positions on branches of different length is given in Fig.
7B. This shows clearly the effects of current flowing into
the ends of the branch, gradually increasing the size of the potential at the electrode as it approaches the end of the branch.

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Fig. 7.
Spatial distribution of the peak amplitude of the electrotonic
potentials in the cable model of a terminal branch following
synchronous current injection by 2 electrodes (E1 and
E2) at sites symmetrically placed on the branch. Each
injection is a pulse of duration 200 µs and magnitude 15 nA.
A: amplitude of the membrane potential at
E2 as a function of the electrode separation when there
is simultaneous injection of current at E1 and
E2 for the 3 terminal lengths indicated.
B: amplitude of the membrane potential at
E2 as a function of the electrode separation when
current is injected only at E2. The 4 curves are for
terminal branches of lengths 2,000, 200, 160, and 120 µm, as
indicated.
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The effects of asynchronous injection of current pulses from two
electrodes on the cable model of a terminal branch were also investigated. In this case, current injection with a pulse of 200 µs
duration at one electrode (the conditioning electrode) was followed by
a 200-µs pulse at the other electrode (the test electrode) at
different intervals. The spatial distribution of the membrane potential
at different positions along the length of branches of different length
at the end of the second pulse when the interval between the beginning
of one pulse and that of the other is 200 µs is shown in Fig.
8A. In this case the
depolarization at the test electrode is still affected by the
propagating hyperpolarization from the conditioning electrode 60 µm
away for all terminal lengths. This can be seen to be the case by
reference to Fig. 8, B-D, which shows the time course of
the propagating electrotonus from the conditioning electrode to the
test electrode, the size of the depolarization at the test electrode
when it alone injects current, and the cumulative effects of both when
current is injected asynchronously from both electrodes, respectively.
Clearly, the time course of the propagating electrotonus from the
conditioning electrode is such that it can still effect the
depolarization at the test electrode when this occurs at the end of the
conditioning depolarization.

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Fig. 8.
Electrotonic potentials in the cable model of a terminal branch
following asynchronous current injection by 2 electrodes
(E1 and E2) at 2 sites 60 µm apart
symmetrically placed on the branch. Each injection is a pulse of
duration 200 µs and magnitude 15 nA. Aa-Ad: spatial
distribution of the membrane potential displacement at the end of 400 µs, for branches of the different total lengths indicated, when there
is a current pulse of 200 µs at E1 followed
immediately by a 20-µs pulse at E2; the positions of
E1 and E2 are indicated by the vertical
dotted lines, and the horizontal broken line is zero potential.
B: time course of the membrane potential at
E2 when there is current injection at E1
only. C: time course of the membrane potential at
E2 when there is current injection at E2
only. D: time course of the membrane potential at
E2 when there is asynchronous current injection at both
electrodes. In each case, results are given for the 4 cable lengths in
A.
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Qualitative comparisons between the cable model of propagating
electrotonus in terminal branches and experiments
A comparison was made between the extent to which quantal
release at a test electrode is depressed by simultaneous injection of
current at a conditioning electrode at different distances away on
different sized terminal branches and the results of the cable
analysis. Figure 9 shows the results for 24 different experiments on terminal branches ranging in length from 120 to 200 µm. Shown is the extent of the depression produced at one
electrode in response to stimulation at the other for distances between
the electrodes of between 40 and 130 µm for these branches and the
theoretical predictions of the extent to which the depolarization at
the test electrode is decreased by the injection of current at the
conditioning electrode.

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Fig. 9.
Comparison between experimental results and theoretical predictions for
the effects of a conditioning impulse at one site (electrode
E1) on quantal release by a test impulse at another site
(electrode E2) for different distances between the
electrodes. The electrodes were placed symmetrically on the terminal
branches and synchronous pulses of 200 µs duration were applied at
each electrode. The continuous lines are solutions of the cable model
for the peak amplitude of the membrane potential at E2,
normalized by dividing by the amplitude of the membrane potential at
the same electrode position when only E2 injects
current. Results are given for 3 branch lengths, as indicated. The
points give the experimental results for the ratio of quantal release
at E2 when there is simultaneous injection of current at
E1 and the quantal release due to E2
current injection alone.
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The predictions of the cable model were also compared with the
experimental observations on the different extents to which quantal
secretion due to current injection at the test electrode was depressed
by current injection at the conditioning electrode at different
preceding intervals. Figure 10 shows such a
comparison for an electrode separation of 70 µm. There is good
agreement between the theoretical predictions and the observations,
with very little depression in the quantal release at the test
electrode occurring for intervals greater than ~0.25 ms.

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Fig. 10.
Comparison between experimental results and theoretical predictions for
the effects of a conditioning impulse at one site (electrode
E1) on quantal release by a test impulse at another site
(electrode E2) for different delays between the pulses.
Asynchronous pulses of 200 µs duration and magnitude 15 nA were
applied at E1 and E2. The continuous line
is the solution of the cable model for the peak amplitude of the
membrane potential at E2, normalized by dividing by the
amplitude of the membrane potential when only E2 injects
current. The branch length was 160 µm, and the electrodes were
symmetrically placed with a separation of 72 µm. The points give the
experimental results for the quantal release at E2 when
there is injection of current at E1 after the time
interval indicated on the abscissa; these results are normalized to the
quantal release when E2 alone injects current.
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DISCUSSION |
The use of quantal release at a site on a branch as a measure of
the size of the electrotonus at that site under different conditions
that lead to changes in the electrotonus is at most qualitative. There
is a nonlinear relationship between the depolarization due to external
current injection at a site and the resultant quantal release, at least
in the range of quantal releases most commonly observed in this work
(namely, 0.1-0.5) (see Katz and Miledi 1967
). Another
problem involves the extent to which the varying resistance between the
tip of the external electrode and the underlying Schwann cell sheath
will alter the amount of current reaching the terminal branch. The
consequence is that there is poor control over the size of the currents
injected by two electrodes on a branch. This is supported by the
observation that reversing the test/conditioning sequence between the
electrodes gave rise, in general, to different extents of depression of
quantal release at the test electrode. Some comment should also be made
about the size of the microelectrodes used to inject current and record the quantal release. These were typically of ~12 µm diam and so cannot be treated as a point source as has been done in the theory. [However, an investigation of the effect of a distributed step current
input into an infinite cable indicates that the hyperpolarization effect resulting from an extended electrode is considerably enhanced over that found for a point source (see Bennett et al.
1999
, Fig. 7).] The use of such electrodes is akin to those
used by Katz and Miledi (1967)
(~4 µm) and by
Dudel et al. (1993)
(~10 µm). These electrodes are
generally used under a slight negative pressure and so tend to form a
loose-patch seal. The effect of this is to increase the resistance
between the inside of the electrode and the extracellular solution,
therefore increasing the current that crosses the membrane (Katz
and Miledi 1967
). This has not been allowed for in the cable analysis.
The effect of the Schwann cell sheath is to greatly increase the
resistance to both longitudinal and transverse current flow around the
nerve terminal branch in comparison to that that would be the case if
the terminal was considered to be simply in a volume conductor. Thus
the ratio of the external to internal longitudinal resistance of the
terminal branch is ~1 as is the ratio of the membrane resistance to
external transverse resistance to earth. This would arise because of
the very close proximity of the Schwann cell to the terminal branch as
well as to the tortuous path the current must take through the
interstices of the Schwann cell sheath to reach the volume conductor
provided by the bathing solution. The problem of calculating the
voltage drop across the sheath or epineurium around nerves and nerve
trunks when determining the results of injection of current from
external electrodes is an old one, and early calculations show that the
drop is substantial (Rashbass and Rushton 1949
), as the
present investigation also suggests.
It should be noted that the simulation results presented here are not
highly sensitive to the specific choice of parameter values. Because
Eqs. 1 and 2 are linear in the potentials, the effect of varying
Iin, for a constant input
current Iin, is simply to rescale these
potentials so the choice of value for
=
is not
critical for the results presented here. On the other hand, the
parameters
= r0/ri
and µ = rm/re are
involved in a nontrivial way. In the absence of accurate experimental
measurements, both these parameters have been taken to be equal to 1. The effect of other choices has been investigated in Bennett et
al. (1999)
. There it was shown that varying
over two orders
of magnitude, from
= 0.1 to
= 10, causes quite a large change
in the magnitude of the membrane potential but a much smaller change in
the ratio of the maximal hyperpolarization to the maximum
depolarization, which decreases from 0.47 to 0.22 (see Bennett
et al. 1999
, Fig. 6A). Similarly, varying µ over
the range µ = 0.1 to µ = 10 changes the membrane potential
considerably, but the polarization ratio only changes from 0.45 to 0.30 (Bennett et al. 1999
, Fig. 6B). These facts,
plus the enhancement of the hyperpolarization when a more realistic
distributed input is used (Bennett et al. 1998
, Fig. 7),
suggest that the results from the model are robust under reasonable
parameter variation.
Katz and Miledi (1965)
first estimated the DC length
constant of the motor-nerve terminal at ~250 µm, based on a
diameter of 1.5 µm, Rm of 3,000
cm2 and Ri of 200
cm; this DC
length constant is similar to that in the present work. They also
calculated an AC length constant of ~60 µm for the case of an
action potential with a characteristic frequency of 1 kHz,
Rm of 5,000
cm2 and a membrane
capacitance of 1 µF cm
2 (Katz and Miledi
1968
). This then lead to the argument that if the action
potential failed to propagate through the last node of Ranvier, then
most terminal branches that are substantially longer than 60 µm would
fail to be depolarized sufficiently to release transmitter. Using
pulses of 200 µs duration gives a characteristic frequency of 5 kHz,
which according to the calculation of Katz and Miledi
(1968)
would give an AC length constant of ~30 µm, not much
different to that obtained in the present work.
The question arises as to whether the propagation of the electrotonus
along terminal branches is modified by accompanying changes in
voltage-dependent channels, as might occur between different regions of
a branching and excitable terminal system (Moore et al.
1988
). Motor-nerve terminals possess voltage-dependent sodium,
calcium, and potassium channels as well as calcium-dependent potassium
channels (Angaut-Petit et al. 1989
). The spatial
distribution of these along single terminal branches at the amphibian
endplate is heterogeneous, with sodium channels decreasing in number in the proximo-distal direction along the terminal branches, whereas potassium channels reach a peak density about the center of the branches (Mallart 1984
). The sodium channels were always
blocked in the present work with tetrodotoxin, and blocking the calcium channels with cadmium did not change the characteristics of the effects
of conditioning electrotonus on the quantal release by the test
electrotonus, when the calcium block was excluded from the vicinity of
the electrodes. It seems then that neither sodium, calcium, nor
calcium-activated potassium channels are involved in the changes in
quantal secretion observed with these stimulus paradigms.
The question arises as to whether activation of KA channels
at the test electrode contributes substantially to the inhibition observed there on a conditioning pulse. Release at the test electrode, due to current injection there, is maximally depressed when there is
simultaneous current injection at the conditioning electrode; this
depression is much less (or nonexistent) when injection at the two
electrodes is separated in time. But any contribution from
KA channels would be expected to be greatest when the
conditioning current injection precedes the test current injection. It
thus seems unlikely that an IKA contributes
substantially to the inhibition at the test electrode due to current
injection at the conditioning electrode.
The experimental results presented here confirm many of the
observations made with a similar technique on nonvisualized terminal branches by Dudel et al. (1993)
. At short intervals of
<1 ms between the pulses given through the two electrodes on a branch,
depression of quantal release was always observed at the test electrode
as in the present work. The cable analysis shows that this can be explained by the hyperpolarizing effects that the conditioning electrode has on the electrotonus generated at the test electrode, given that the extent of quantal release is dependent on the absolute level of depolarization at the test electrode (Katz and Miledi 1967
). These explain both the changes in the extent of
depression in quantal release at the test electrode for different
distances between this electrode and the conditioning electrode as well as the extent of depression at the test electrode for different test-conditioning intervals at a fixed separation between the two
electrodes. Furthermore, as in Dudel et al. (1993)
, at
no time during the thousands of measurements of evoked and spontaneous release made at the test electrode was this release detected at the
conditioning electrode, when the latter was not used to inject current.
However, one discrepancy between the present observations and those of
Dudel et al. (1993)
is that the latter observed a facilitation of the test quantal release following 2 ms after the
conditioning stimulus. We have not observed any effects on quantal
release for intervals between the conditioning/test stimuli greater
than ~0.25 ms. The present cable theory does not provide an
explanation for such an enhanced release either.
As the conditioning/test electrodes were placed on visualized terminal
branches, it was possible to show that even when these were on
different branches there was a depression in quantal release at the
test electrode. This indicates that the entire terminal branching
system at the endplate should be viewed as a branching system in
electrotonic continuity. In this case the extensive theory of cable
analysis of branching dendritic trees is applicable (see, for example,
Rall et al. 1992
). However, there does not seem to be
any treatment of the problem considered here, namely that of the
injection of current from extracellular electrodes at two sites on a
short cable in a volume conductor. This analysis is a simplification in
that it deals with the case of two electrodes on the same or contiguous
branches as if no other branches existed, including the parent axon. At
its present stage of development, the theory can at best be used to
provide a qualitative explanation of how a conditioning depolarizing
current on a motor-nerve terminal gives rise to a depression of quantal
release at a test electrode on the terminal.
Address for reprint requests: M. R. Bennett, Neurobiology
Laboratory, Dept. of Physiology, University of Sydney, N.S.W. 2006, Australia.
The costs of publication of this article were defrayed in part by the
payment of page charges. The article must therefore be hereby marked
"advertisement" in accordance with 18 U.S.C. Section
1734 solely to indicate this fact.