Department of Physiology, University of Wisconsin School of
Medicine, Madison, Wisconsin 53706
 |
INTRODUCTION |
The extensive neuronal
connections through axons or dendrites provide the structural basis for
signaling and information processing in the nervous system. Branching
in the axonal tree greatly enriches spatial information processing.
Both nonmyelinated and myelinated nerve fibers in the CNS and
peripheral nervous system (PNS) exhibit extensive branching.
For example, before the sensory fiber reaches the dorsal column, there
is a single branch that leads to the dorsal root ganglion neuron.
Branching also occurs as the terminal is reached, as in the case of the
nerve terminals of motor and sensory fibers, and the terminal axonal
arbors in thalamocortical and pyramidal fiber tracts (Deschenes
and Landry 1980
; Quick et al. 1979
). Do all
impulses faithfully invade branch points or nerve terminals?
Physiological experiments have shown that the safety factor for action
potential propagation is usually lowered at branch points
(Stockbridge 1988
; Stockbridge and Stockbridge 1988
). This could be manifested by either branch point failures or differential conduction into one of several daughter branches. These
mechanisms allow a branch point to be actively involved in action
potential routing and temporal editing of neuronal information, which
could provide a structural basis for some forms of plasticity. A
well-characterized example is the T-junction in the dorsal root ganglion, where a myelinated sensory fiber branches and bifurcates to
the spinal cord and the dorsal root ganglion (DRG) neuron. Experimental
results have shown that not all afferent action potentials could reach
the DRG cell body (Luscher et al. 1994
; Stoney
1990
), with the branch point at the T-junction of the DRG
functioning like a switch or a low-pass filter, protecting the soma
from excessive input.
Branch points in myelinated and nonmyelinated nerve fibers are very
different. In the nonmyelinated nerve fibers, branches could bifurcate
from any point along the axon. For myelinated nerve fibers, however,
branches only emerge from the nodes of Ranvier (Quick et al.
1979
). The myelin sheath enwrapping the axon not only forms an
electrical insulation for the axon, but also alters the axonal membrane
properties including ion channel distribution. In mammalian myelinated
nerve fibers, the specific conductance of the nodal membrane is much
higher than that of the internodal axonal membrane. In the
nonmyelinated nerve fibers, ion channels appear to be evenly
distributed on the axonal membrane. In contrast, in the mammalian
myelinated nerve fibers, most sodium channels, and slow K channels are
concentrated at the nodes of Ranvier, whereas fast K channels are
concealed underneath the myelin in the juxtaparanodal region. Based on
these morphological and physiological differences, the propagation of
action potentials on myelinated nerve fibers is fast and saltatory and
incurs low metabolic costs, which is in contrast to the slower and
continuous nature of propagation in the nonmyelinated fibers. In a
recent computer simulation of the myelinated fiber (Halter and
Clark 1991
), during saltatory conduction, most of the voltage
drop in the internode occurs across the myelin sheath with little or no voltage drop across the internodal axolemma.
There have been many empirical and theoretical studies on action
potential propagation through axonal branch points, but all of them
have focused on nonmyelinated nerve fibers (Hines 1984
; Mascagni and Sherman 1998
; Parnas and Segev
1976
). Even though there exist many excellent modeling programs
(NEURON, SPICE, etc.) available for simulation of very complex neuronal
structures such as dendritic tress, all of these are single
compartmental models, with each compartment (segment) having only one
layer that represents the axonal membrane (Segev et al.
1998
). For realistic simulation of the myelinated nerve, we
need a model with at least two layers for each computer segment, one
for the axonal membrane and the other for the myelin sheath.
No computer model is yet available for simulating branch points in a
myelinated fiber. Considering the basic differences between the
myelinated and nonmyelinated nerve fibers, it is important to provide a
mathematical model for bifurcations in myelinated nerve that allows a
study to be made of the role played by the branch point in action
potential propagation. To provide a realistic simulation of branch
points in myelinated fibers, it is essential to uncouple the
calculation of intra-axonal voltage signal from peri-axonal voltage in
a two-layer model. Such a distributed-parameter compartment
mathematical model was published by Halter and Clark (1991)
for a single unbranched myelinated nerve fiber. In that model, each segment has multiple layers, and the electrical activity in
the periaxonal space was calculated separately from that of the axonal
signal. In this paper, we developed a computer model for a single
branch point in a myelinated fiber with one parent branch and two
daughter branches, based on Halter and Clark's unbranched model
(Halter and Clark 1991
). We then used our model to
examine various factors that might affect local excitability at the
branch point, including geometrical parameters such as internodal
length, myelin thickness and periaxonal space thickness, as well as K
ion accumulation at the paranodal regions.
 |
THEORY |
The branch point is reported by three segments: N0, N1, and
N2 (Fig. 1). Segment N0 belongs to the
parent branch and is shared by two identical daughter branches. With
the application of Kirchoff's law to segment N0 (Fig.
1C), we obtained Eq. 1, which
describes the currents in N0 with second-order temporal accuracy
(see Halter and Clark 1991
)
|
(1)
|
Notation
Note that Iam has two components: a
capacitative component and an ionic component (see the right hand side
of Eq. 2). The ionic component, designated
iionic, has three components (Na current,
K current, and leak current), which has been explicitly explained in
Halter and Clark (1991)
as well as in our previous work
(Zhou et al. 1999
).

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Fig. 1.
Schematic drawing of the simulation model for branch point in
myelinated nerve. A: the model describes a single
myelinated fiber (parent branch) bifurcating into 2 smaller daughter
branches (1 and 2). The arrows showed the direction of action potential
propagation. The daughter branch is half the size of the parent branch.
All geometrical parameters are in µm. B: the Ranvier
node is represented by one segment. The internodal region can be
sepatared into MYSA, FLUT, STIN, FLUT and MYSA. They are represented
with 4, 4, 5, 4, and 4 segments, respectively (Zhou et al.
1999 ). C: the branch point is represented by 3 segments and named as N0, N1, and N2. The intra-axonal voltages of
these segments are V0,
V1, and V2. With
application of Kirchoff's law to segment N0, the transmembrane
current Iam is equal to the difference
between I0 and the sum of
I1 and I2. MYSA,
myelin sheath attachment region; FLUT, fluted internodal region; STIN,
stereotype internodal region; I0,
inyra-axonal current into segment 0;
I1/I2,
intra-axonal currents flowing from segment N0 to segment N1 or N2;
p, periaxonal; i, intra-axonal.
|
|
By substituting the currents with voltage-resistance expression, we
obtained Eq. 2 for N0 with second-order spatial and temporal accuracies
|
(2)
|
Notation
The equation above can be re-organized into
|
(3)
|
where
When all the equations for each segment are combined, we obtained
the following expression
where Vp and
Vi are the two vectors representing the
periaxonal and intra-axonal voltages of all segments
The above equations are novel to this paper, as they are developed
for a single branch point of a myelinated nerve fiber. For equations
and matrix for other segments along the parent and daughter branches
away from the branch point, we use those already developed for
unbranched myelinated fibers by Halter and Clark (1991)
.
To couple both daughter branches with segment N0 of the parent branch,
certain modifications have to be made to the matrix according to
Eqs. 1-3.
2N0l and
N2i were moved to specific
position to couple N0 with N2, and the original positions were replaced
with 0 (see the following matrix). The following diagram shows the
intra-axonal part of the matrix after modification; similar
modifications were made for the periaxonal part. In our model, there is
only one branch point that connects the parent branch with two
daughter branches. This model could be modified in future studies
to simulate multiple branches by adding more couplings to the matrix,
an understanding that is similar to the treatment for multiple branches
in nonmyelinated nerve fibers (Hines 1984
)
The settings for the morphological and electrophysiological
parameters for the parent branch were based on two published papers
(Halter and Clark 1991
; Zhou et al.
1999
). The node of Ranvier was represented by 1 segment, and
the internode was responded by 21 segments using the same terminology
as in Halter [4 for each myelin sheath attachment region (MYSA) on
either end of an internode; 4 for each fluted internodal region (FLUT)
on either end of an internode; 5 for stereotype internodal region
(STIN), the region in between]. Figure 1B shows the
structure of each internode, showing the MYSA, the FLUT, and the STIN.
Each daughter branch is half the size of the parent branch (axon
diameter, segment length, myelin thickness, etc.). The same
physiological parameters were used for the parent branch and the
daughter branches (specific axon/myelin membrane
conductance/capacitance, Na/Fast K/Slow K channel distributions).
In our model, we assumed that K ion concentration in the periaxonal
space is dynamically linked only to K ion efflux through fast K
channels on the axonal membrane. The K concentration is calculated
according to the volume of periaxonal space and an assumed time
constant for potassium clearance (Eq. 4). Based on previous
studies, fast K conductance in the mammalian myelinated nerve is
assumed to be localized underneath the myelin in the paranodal or
juxtaparanodal region in our model (Chiu and Ritchie 1980
; Mi et al. 1995
). In the present study,
these fast K channels provide the major source for the extracellular K
ion accumulation. The average thickness of the periaxonal space at the
paranodal region is 5 nm in our standard model. Previous theoretical
and experimental results provided different values for K clearance in
the squid giant axon, ranging from 10 ms (Astion et al.
1988
) to 25 ms (Inoue et al. 1997
). In our
standard model, the time constant for K clearance was set at 10 ms at
37°C or 20 ms at 20°C
|
(4)
|
where iKf is the K current through
fast K channels, Vol is the volume of the periaxonal space
at each segment, F is the Faraday constant, 5.9 is the
resting extracellular K concrentration (5.9) mM,
Kout is the extracellular K concentration,
and
is the K ion clearance time constant in the periaxonal space.
 |
RESULTS |
Transmission of action potentials from parent branch to daughter
branches
In our model, the branch point is represented by three segments,
each having the same specific membrane properties and channel distributions as the nodes of Ranvier along the rest of the fiber. N0
is the last segment of the parent branch as it approaches the branch
point. N1 and N2 are the first segment of each daughter branch (Fig.
1C). The diameter of N1 or N2 is equal to that of the node
of Ranvier along the rest of the daughter branch. Since the size of the
branch point determines the impedance matching between the daughter
branches and the parent branch, we adjusted the lengths of the three
segments (N0, N1, N2) until a single action potential could
successfully invade both daughter branches. At this setting, the total
axonal membrane area at the branch point is 27.5 µm2. For comparison, the nodal membrane in the
rest of the parent and daughter branches is 23.5 and 5.9 µm2, respectively.
The model was stimulated by injecting a brief current (0.1 A, 25/50
µs at 37/20°C) into the left-most node in the parent branch with
eight internodes. This stimulation is large enough to generate an
action potential that propagates toward the branch point. In most of
the following figures, the voltage signal across the nodal membrane was
used to show the activity in the nerve fiber. Temporal responses from
several nodes in the parent and daughter branches near the branch point
were illustrated in Fig. 2. The shifts
between curves reflect the delayed activation of nodes along the nerve fiber in the direction of impulse propagation. The dashed line in each
figure showed responses of the three segments at the branch point (N0,
N1, or N2; from top to bottom). The action
potentials of N1 and N2 were identical, suggesting that N1 and N2 were
activated simultaneously. As mentioned in THEORY,
expressions for N1 and N2 were physically separated in the matrix, and
modifications were made to couple both segments with N0 in the parent
branch. The simultaneous activation of N1 and N2 suggested that
modifications to the matrix successfully transferred activity from the
parent branch to both daughter branches.

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Fig. 2.
Action potential propagates across branch point from parent to daughter
branches. An action potential was evoked by injecting current into the
leftmost node of the parent branch, and propagated from the parent
branch to both daughter branches (the arrow shows the direction).
Trans-axon membrane voltages at Ranvier nodes near the
branch point were shown verse time. Dotted curves showed the responses
of 3 segments representing the branch point, N0, N1, and N2 (from
top to bottom). Note the time scale is
different between left (37°C) and right
(20°C).
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One major feature of this model is that the myelin sheath is
incorporated into the simulation. In this model of the myelinated nerve
fiber, action potential actually travels on the myelin sheath with
little or no depolarization occurring on the main part of the
internodal axon membrane (Halter and Clark 1991
). Using
the nerve fiber length as the x-axis, Fig.
3 illustrates a spatial view of action
potential distribution on the myelinated nerve fiber at a given time.
Trans-axon and trans-myelin-sheath voltages were
shown along the fiber axis in the figure. Two time points were
selected. The first one is chosen just prior to the arrival of the
impulse at the branch point, and the second one 1 ms later when the
impulse has crossed the branch point. The sharp upward peaks in the
trans-axon voltage curve and downward peaks in the trans-myelin voltage curve correspond to the positions of
the nodes of Ranvier. Comparing the trans-axon voltage with
trans-myelin sheath voltage, it is clear that the impulses
are actually jumping from node to node, using the myelin sheath as a
bridge (Halter and Clark 1991
). Two points need to be
stressed. First, our simulation result can only be achieved by
completely separating the myelin sheath from the axon membrane in our
two-layer model for each segment, a major advantage of our model.
Second, the sharp voltage peaks at the nodes of Ranvier have a certain
spatial spread to them, suggesting that the depolarization at the node
passively spreads into the paranodal region. This depolarization of the paranodal axon membrane will lead to activation of fast K channels at
the paranodal region, a conclusion corroborated by intracellular recordings in rat myelinated axons (Barrett and Barrett
1982
; David et al. 1995
). The electrotonic
coupling between the nodal and the paranodal membranes becomes
important as factors affecting branch point excitability are explored,
as will be evident below.

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Fig. 3.
Spatial distribution of propagating impulse on parent and daughter
branches. Trans-axon membrane voltage
(top) or trans-myelin sheath voltage
(bottom) were shown with nerve length as
x-axis. Since responses of the 2 daughter branches are
identical in the model, only 1 is shown in the figure. The time points
selected corresponded to the time just before the action potential
reached the branch point and 1 ms later when the impulse passed the
branch point. The open arrow shows the position of branch point.
T = 20°C.
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Branch point failures at high frequency
The active role of branch points in filtering neural information
was tested with a train of high-frequency impulses (Fig. 4). Comparing spikes on the parent branch
upstream from the branch point with spikes on both daughter branches
(*) downstream from the branch point, it can be seen that there is a
certain failure in the parent-to-daughter branch when the frequency of
stimulation was increased. In other words, intermittent failures were
found on the daughter branches. At 37°C (left), 400-Hz
action potentials could propagate on the parent branch without
failures. After the branch point, about 60% of the impulses are seen
on the daughter branches (calculated with a time window of 50 ms as
shown in Fig. 4). At 20°C, the daughter branch can only follow
~50% of the 250-Hz stimulation delivered to the parent branch. There
are two possible origins of this intermittent failure in the daughter
branches. One possibility is that the daughter branch, being of a
smaller diameter than the parent branch, has a correspondingly longer refractory period. Thus, whereas the parent branch can follow high-frequency stimulation, the daughter branch cannot, even though there is 100% transmission at the branch point. To examine this possibility, we directly stimulated the daughter branch at the same
frequency as applied to the parent branch (400 Hz at 37°C, 250 Hz at
20°C). The results showed that the daughter branch could follow this
high rate of stimulation with no conduction failures (data not shown).
This suggests that the intermittent failures in the daughter branches
are due to failures of transmission at the branch point. We now used
our model to examine several factors that might affect the transmission
efficacy at the branch point.

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Fig. 4.
Intermittent branch point failures occurred at high frequency. The
simulation model was tested with a train of stimulus at different
frequencies (400 Hz at 37°C, 250 Hz at 20°C) to the parent branch.
Responses of the last node in parent branch and the 3rd node after the
branch point in one of the daughter branches are shown. Intermittent
conduction failures could be detected by comparing the responses of
parent branch to those of daughter branches.
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K accumulation in the periaxonal space
In mammalian myelinated axons, virtually all of the kinetically
fast K channels are localized underneath the myelin in the paranodal or
juxtaparanodal region. Considering the very restricted space between
the myelin sheath and the axonal membrane, a small amount of K ion
efflux through these channels might dramatically increase local K ion
concentration (Chiu 1991
). Is there any functional significance for this local K accumulation? We used our model to
examine the effect of K accumulation on branch point excitability. We
treated the intra-axonal [K] as constant (150 mM). In our standard model, we assumed that the K clearance time constant in the periaxonal space is 10 ms at 37°C or 20 ms at 20°C. We first examined the effect of K accumulation in electrogenesis on the unbranched portion of
the nerve. Figure 5A shows
that for the standard model subjected to a single stimulation, the
periaxonal K concentration rises to a peak of 15 mM at 37°C and to 38 mM at 20°C. Comparing the nodal action potentials with (
) and
without (· · ·) K accumulation, it can be seen that K
accumulation leads to a small afterdepolarization that is more
pronounced at 20°C (Fig. 5B). One mechanism by which local
K accumulation affects excitability is by altering the Nernstian K
equilibrium potential for the fast K channels, which is
82.3 mV at
20°C and
86.6 mV at 37°C. While the potassium current
typically flows outward during an action potential because the K
equilibrium potential is always more negative than the membrane
potential, this situation becomes different when there is K
accumulation, which makes the K equilibrium potential more depolarized.
In our simulations, we found that during an action potential, there is actually a period during which the K equilibrium potential (solid line)
is higher (i.e., more depolarized) than the local axon membrane potential (dashed line; Fig. 5C). During this period, K
current becomes inward at 20°C (Fig. 5). Around 5 ms after the peak
of action potential, fast K channels quickly deactivated, leading to an
abolishment of this inward current even though the driving force for
the fast K current still remains in the inward direction. At 37°C,
this inward driving force does not produce an inward K current, since
the fast K channels are rapidly deactivated (Fig. 5D).

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Fig. 5.
Activity-dependent K ion accumulation in the periaxonal space between
axon membrane and myelin sheath. K ion clearance time constant is set
as 10 ms at 37°C (left) or 20 ms at 20°C
(right) in the standard model. All simulations were
carried out in the parent branch. A: periaxonal K
concentration in the 1st segment of FLUT region. Dotted line shows the
resting K ion level (5.9 mM). B:
trans-axon membrane potential at the Ranvier node with
( ) or without (· · ·) K accumulation in the simulation.
C: K equilibrium potential was calculated according to
the Nernst equation based on the periaxonal [K] curve in
A, as shown by solid line. Dashed line shows the resting
K equilibrium potential ( 82.3 mV at 20°C, 86.6 mV at 37°C) and
the local trans-axon membrane potential (pointed by
arrow). D: potassium current density in the 1st segment
of FLUT region with or without K ion accumulation in the simulation
(37°C, left; 20°C, right).
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We next examined the effects of K accumulation on branch point
transmission. In our standard model with a K clearance time constant of
10/20 ms at 37/20°C, we have shown that intermittent failures at the
branch point occur at high frequency (400 Hz at 37°C or 250 Hz at
20°C). How does hastening or slowing K clearance affect the branch
point transmission? To address this issue, we apply the same fixed high
frequency of stimulation, but vary the K clearance time constant to 1 ms, 10 ms (standard model), and 100 ms at 37°C. The respective values
at 20°C are 1 ms, 20 ms (standard model), and 100 ms. The simulations
showed that varying the K clearance time constant produced complex
patterns of excitability changes at the branch point (Fig.
6A). At 37°C, progressively slowing K clearance (from 1 to 10 to 100 ms) progressively reduces the
transmission rate through the branch point (transmission is 100% pass
at 1 ms, 60% pass at 10 ms, and 58% pass at 100 ms). Thus at 37°C,
more K accumulation leads to more branch point failures. The results at
20°C were similar. As the K clearance time constant is progressively
increased from 1 to 20 ms, the branch point transmission was depressed
(from 100% transmission at 1 ms to 50% transmission at 20 ms).
Further retarding K clearance time to 100 ms leads to no further branch
point failures. Figure 6B shows the paranodal K accumulation
corresponding to these simulations. With fast K clearance rate (1 ms),
there was no sustained K ion buildup at both temperatures. With very
slow K clearance rate (
100 ms), local K concentration increases
quickly and reaches a ceiling (~35 mM). The existence of a
ceiling level of [K] accumulation is consistent with experimental
results in the cerebral cortex (Heinemann and Lux 1977
)
and the optic nerves (Connors et al. 1982
).

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Fig. 6.
Effects of local K ion accumulation in branch point excitability. The
effects of K ion accumulation were tested with impulse train at 400 Hz
(37°C, left) or 250 Hz (20°C, right).
Three different time constants for K ion clearance were tested [1 ms,
10 ms at 37°C or 20 ms at 20°C (standard model), 100 ms].
A: responses of the 3rd node after branch point in a
daughter branch at different K ion clearance time constants. From
top to bottom, = 1 ms, standard
model (10 ms at 37°C, 20 ms at 20°C), and 100 ms. B:
periaxonal K ion concentrations in the paranodal region near branch
point in the parent branch. The number at the right of
each curve shows the corresponding K ion clearance time constant.
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Impedance mismatch
As suggested by previous studies (Swadlow et al.
1980
; Waxman 1975
), a low safety factor for
action potential propagation at branch point might be due to impedance
mismatch. In other words, if the combined electrotonic loading of the
two daughter branches is bigger/smaller than that of parent branch, the
threshold for action potential to propagate forward is higher/lower at
the site of nonuniform geometry. We therefore examined the effects of
several geometrical adjustments that might affect the impedance
matching between the parent branch and the daughter branches. All
manipulations were tested in the standard model with an impulse train
(400 Hz at 37°C, or 250 Hz at 20°C).
INTERNODAL LENGTH.
Shortening the internodal length is one common solution to overcome
impedance mismatch in sites of abrupt morphological changes, as in the
case of the transition zone from myelinated to demyelinated axons
(Waxman and Brill 1978
) and the motor axon terminal
region (Zhou et al. 1999
). It would be interesting to
examine whether a foreshortening of the internodal length on the parent
branch prior to the branch point facilitates impulse propagation
through the branch point. We therefore shortened the length of the last internode in the parent branch just before the branch point by half.
This internodal shortening dramatically eliminates branch point
failures, allowing 100% transmission from the parent branch to the two
daughter branches at both 37°C and 20°C (Fig.
7, middle). Interestingly,
this reduction of internodal length proximal to the branch point
improves transmission through the branch point without penalizing the
conduction latency for the arrival time of the first action potential
at the daughter branch. Thus when the simulation traces for the
preinternodal shortening in Fig. 7 are examined more closely at
expanded time scale, the first action potential was found to arrive at
the daughter branch slightly sooner than in the case of the standard
model, at both 37°C and 20°C. Shortening the internodes in the
daughter branches, just after the branch point, has the opposite
effects of reducing the transmission rate through the branch point
(Fig. 7, bottom).

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Fig. 7.
Effects of internodal length irregularities on branch point
transmission. Top: standard model with regular
internodal lengths in both parent and daughter branches.
Middle: the parent branch internodal just before the
branch point was shortened by half. The shortened internode was marked
by the open arrow. Bottom: the daughter branch
internodal length just after the branch point was shortened by half.
The shortened internode was marked by the open arrow. In all cases, the
system was tested with impulse trans of 400 Hz (37°C,
left) or 250 Hz (20°C, right). The
response of the 3rd node after the branch point in the daughter branch
is shown with an asterisk in the figure. The adjustment was done to the
length of stereotype internodal region (STIN) without any other
parameter changes.
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PERIAXONAL SPACE.
Besides internodal length, another factor that may affect impedance
matching between the parent branch and the daughter branches is how
tightly the myelin is wrapped around the axons, i.e., the thickness of
the periaxonal space. During development and even in adulthood,
myelinated nerve structure undergoes continuous modification apparently
to adapt to different functional requirements (Kleitman and
Bunge 1995
; Schroder 1986
). It has been reported that the distance between myelin sheath and axon is smaller in adult
mice than in younger ones. Intuitively, a tight wrapping (reduced
periaxonal space) will reduce the excitation membrane area, which may
affect impedance matching between parent and daughter branches.
Further, a consequence of the tiny volume of the paranodal periaxonal
space is that any variation in the thickness of the periaxonal space
could have significant impact on the periaxonal K accumulation. We
therefore examined the effects on branch point transmission when the
periaxonal space thickness was selectively varied on either the parent
branch (Fig. 8) or the daughter branches (Fig. 9).

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Fig. 8.
Effects of varying periaxonal space width in the parent branch. The
distances between the axon membrane and the myelin sheath at the
paranodal region (MYSA and FLUT) were either decreased
(left) or increased (right) by a factor
of 2 from the standard model (5 nm). The system was tested with an
impulse train of 400 Hz at 37°C. The response of the 3rd node after
the branch point in a daughter branch was shown in the
top. The corresponding paranodal [K] in the periaxonal
space near the 4th node of Ranvier in parent branch was shown in the
bottom.
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Fig. 9.
Effects of varying the periaxonal space width of the daughter branches.
The distances between the axon membrane and the myelin sheath at the
paranodal region (MYSA and FLUT) were either decreased
(left) or increased (right) by a factor
of 2 from the standard model (5 nm). The system was tested with the
same impulse train as in Fig. 8. The response of the 3rd node after the
branch point in a daughter branch was shown in the top.
The corresponding paranodal [K] in the periaxonal space near the 4th
node of Ranvier in the daughter branch was shown in the
bottom.
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When the periaxonal space in the parent branch was made narrower by a
factor of 2 from the standard model (from 5 to 2.5 nm), there is an
increase in branch point transmission. There is a slight increase in
the ceiling for the periaxonal K accumulation from ~27 mM to ~31
mM. When the periaxonal space was widened from the standard model by a
factor of 2 (to 10 nm), more failures at the branch point occurred. A
more dramatic effect on the branch point transmission results when the
periaxonal space of the daughter branches is altered. On narrowing the
periaxonal space by a factor of 2, transmission failure through the
branch point is completely eliminated. When the daughter branch
periaxonal space was widened by a factor of 2, dramatic enhancement of
transmission failure results. There is a correspondingly larger change
in the periaxonal K accumulation, probably due to the smaller sizes of
the daughter branches compared with the parent branches.
Temperature
Empirical studies have shown that in both nonmyelinated
axons (Westerfield et al. 1977
) and myelinated axons
(Stoney 1990
), conduction failures at branch points are
very sensitive to small changes in temperatures. The general consensus
is that warming increases branch point conduction failures, and cooling
improves conduction through branch points (Luscher et al.
1983
; Swadlow et al. 1980
). We therefore
examined the effects of changing the temperature on branch point
failures, with particular attention paid to small temperature changes
around the physiological operating point of 37°C. Figure
10 shows the relationship between
transmission through the branch point versus temperature when the
parent branch was driven at 400 Hz. For the standard model (
),
~60% of the action potentials are transmitted through the branch
point into the daughter branches at 37°C. As the temperature is
increased from 37 to 40°C, the transmission is sharply reduced to
zero. On the other hand, simply lowering the temperature by 2°C, from 37 to 35°C, greatly improved transmission through the branch point from ~60% to near 100%. This improvement subsides as the
temperature is further reduced, so that by 27°C, the transmission
falls back to ~60%. In control simulations (data not shown), we
found that over the temperature range from 27 to 40°C, both the
parent and daughter branches can follow 400-Hz action potentials when
they are directly stimulated, thus validating the conclusion that
transmission in Fig. 10 reflects transmission through the branch point.
Interestingly, when the branch point was preceded by a shortened
internode, transmission through the branch point becomes totally
temperature insensitive, remaining at 100% over the entire temperature
range (27-40°C).

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Fig. 10.
Effects of temperature on branch point transmission. Transmission
through the branch point was measured by driving the parent branch at
400 Hz and measuring the % of action potentials that make it into the
daughter branches over a fixed time window (35 ms after initial
stimulation) over the temperature range 27-40°C. Control simulations
(data not shown) showed that all branches could follow 400-Hz
stimulations with no failures when stimulated directly. ,
results from the standard model with uniform internodal lengths for all
branches (see Fig. 7); , results when the parent branch
has a preshortened internode just preceding the branch point (see Fig.
7).
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DISCUSSION |
In this paper we provided the first computer simulation of action
potential propagation through a single branch point in myelinated nerve
fibers. The model is based on a previously published multiple-layer compartmental model for an unbranched myelinated nerve fiber
(Halter and Clark 1991
). Our model successfully
simulates action potential propagation from a parent myelinated branch
through a single branch point to two identical, but smaller, myelinated
daughter branches. Since in our model each segment has two layers
representing the axonal membrane and the myelin sheath, a very
realistic description to the action potential propagation across branch
point in a myelinated nerve fiber is achieved. Through independent and
separate calculation of the intra-axonal and periaxonal signals, it is
possible to theoretically examine various subtle changes in geometrical
and physiological parameters important for modulating branch point excitability in a myelinated fiber, such as longitudinal currents in
the periaxonal space thickness, internodal length, and K accumulation in the periaxonal space.
Most neuronal information conducted through an axon is encoded in the
pattern of impulses. In addition to passive transfer of information
from the parent branch to the daughter branches, a branch point might
also actively process the information by differential routing,
frequency filtering, and pattern editing. One major focus of this paper
is the morphological basis by which branch point excitability and
frequency filtering is affected. A future application of our model is
to study branch points in sensory fiber terminals, in which afferent
impulses propagate retrogradely from daughter to parent branches. The
results obtained in this paper on branch point regulation may shed
light on how afferent signal is processed as it originates from the
sensory endings. One major result of our theoretical analysis is that the branch point in a myelinated fiber acts like a low-pass filter, with the cutoff frequency for high-frequency transmission critically determined by various factors related to the local geometry in the
vicinity of the branch point.
K accumulation in the periaxonal space affects frequency filtering
of branch point
Electrical activity in nerve fibers could have short-term or
long-term effects on nerve fibers, allowing nerve fibers to adapt to
certain functional requirements. K accumulation in the space between
the myelin sheath and axolemma has been suggested to mediate signaling
between axons and glial cells, and this could be an important factor
affecting local excitability near branch points (Nicholson
1995
). Activation of fast K channels on the paranodal axon
leads to K accumulation in the periaxonal space (Chiu
1991
). In our standard model (Fig. 6A), when the
parent branch is driven at 400 Hz at 37°C, only 60% of the impulses
are transmitted to the daughter branches. What is the basis of these
transmission failures? This failure cannot be explained by the
inability of the daughter branches (which are smaller) to follow 400-Hz
stimulation, since direct stimulation of the daughter branches with 400 Hz revealed no failures. Hence, failures occur at the branch point. What is the cause of the branch point failure? Our computer simulations suggest that K accumulation in the periaxonal space is one main reason
for this failure. Hence, when the K clearance in the standards model is
hastened to
= 1 ms, branch point failures disappear (Fig.
6A). When the K clearance is retarded to
= 100 ms,
branch point failures increase (Fig. 6A). Notice that
varying K clearance time constant affects only the branch point
transmission; the conduction on either the parent or daughter branches
is unaffected. The reason for the branch point failures may be that the
paranodal depolarization induced by K ion accumulation passively
spreads to the nodal region and inactivates nodal Na channels. This
causes a conduction block at the branch point, a site of low safety
factor. Figure 6A shows that a branch point not only can act
like a low-pass filter, but that the branch point may edit the
information by altering the impulse pattern, as shown in Fig. 6
(37°C). For example, the responses of the daughter branches at
= 10 ms and
= 100 ms have similar transmission rate
(~60%) but have different patterns.
Our finding that branch point transmission is highly sensitive to K
accumulation in the periaxonal space suggests that information flow in
axon arbors of myelinated fibers with multiple branching is very
vulnerable to disregulation of K homeostasis under the myelin sheath.
It is noteworthy that the internodal axon is heavily stained for Na/K
ATPase (Mata et al. 1991
), suggesting that K homeostasis
under the myelin sheath is critical for normal signaling in myelinated
axons, particularly at branch points.
As shown by simulations at 20°C, K accumulation causes the K reversal
potential to be higher than the membrane potential during part of the
repolarizing phase. This, coupled with a slowing down of fast K channel
deactivation by cooling, allows a time window during the action
potential when inward K currents are generated (Fig. 5). This inward K
current may even enhance the excitability of the system, allowing K
accumulation in this special instance to actually facilitate
transmission through the branch point. Experimental results have shown
that under certain conditions, such as postischemic or posttetanic
firing, K accumulation in the periaxonal space can generate inward,
regenerative potassium currents (Bostock et al. 1991
;
Kiernan et al. 1997
). Collectively, these experimental
results and our theoretical results suggest that it is quite possible
that K accumulation at the periaxonal space will show
significant and unexpected physiological impact on branch point excitability.
Since the fast K channels concealed underneath the myelin are the major
source of K accumulation, our model suggests that these
myelin-concealed K channels may exert profound control on branch point
transmission via modulating periaxonal K accumulation in the branch
point vicinity. Our previous analysis of Kv1.1-null mice, where one K
channel subtype under the myelin was deleted, suggested that
myelin-concealed K channels are important modulators of transition zone
excitability at the motor nerve terminal (Zhou et al.
1998
, 1999
). Whether Kv1.1 plays an important
role in branch point excitability is an open question. Preliminary
experimental results on the sciatic nerve-DRG preparation showed that
in the Kv1.1-null mice, the percentage of high-frequency spikes
reaching the DRG neuron soma is higher than the wild type (unpublished observations). Our theoretical study here suggests that Kv1.1 may
modulate branch point failures via periaxonal K accumulation. In the
CNS, A-type fast K channels have been shown to play a critical role in
gating action potential propagation in CA3 pyramidal axons (Debanne et al. 1997
; Kopysova and Debanne
1998
).
Role of impedance mismatch in branch point failure
It has been shown that the safety factor for action potential
propagation is usually lower at regions of geometrical heterogeneity, such as sites of abrupt change in axonal diameter and axonal
bifurcation (Parnas et al. 1976
). How does geometrical
heterogeneity affect action potential propagation? A key factor is
impedance mismatch (Rall 1959
; Swadlow et al.
1980
). For the case of nonmyelinated axons, the GR ratio
of the two daughter branches (d1,
d2) to the parent branch
(d) is (d13/2 + d23/2)/d3/2. If this
ratio is larger than 1, invasion of the daughter branches is slowed
down or may even fail. If this ratio is smaller than 1, invasion of the
daughter branches is secure and without failures. In our model, the two
daughter branches each have half the diameter of the parent branch,
which yields a GR ratio of 0.71 if our axons were nonmyelinated.
However, other factors come into play in branch point of myelinated
fibers. For example, we found that the nodal area (N0 + N1 + N2) at the
branch point is critical. If it is too small, action potential cannot
invade the daughter branches. In our standard model, the area was
adjusted to give successful invasion of the daughter branches.
Our simulations also identified other important determinants of branch
point transmission. One of them is the internodal length of the parent
branch just prior to the branch point. By replacing the single
internode prior to the branch point with two half-size short
internodes, transmission failure is completely eliminated, allowing the
branch point to dramatically increase its cutoff frequency for
high-frequency signals. Reduction of the internodal length just
proximal to the branch point also dramatically abolished the
temperature sensitivity of conduction failures at the branch point.
With uniform internodal lengths, warming abruptly block transmission
through the branch point, and cooling first improves transmission then
reduces it (Fig. 10). With a single prebranch point internodal
shortening, fidelity of branch point transmission stays constant at
100% over a broad temperature range (27-40°C). In unbranched
myelinated fibers, previous theoretical and experimental studies have
confirmed that internodal shortening can dramatically increase the
safety factor at the transition zone near demyelinated regions
(Waxman 1972
). Our present study extends these results to branch points of myelinated fibers and shows that a shortened prebranching internodal length could improve the safety factor at the
branch point and increase the cutoff frequency for high-frequency signal transmission. Interestingly, shortening the internodes in the
daughter branches immediately after the branch point has the opposite
effect of shifting the cutoff frequency to a lower value, screening the
daughter branches from receiving high-frequency signals from the parent
branch (Fig. 7).
Our finding that internodal length irregularity around a branch point
can shift the cutoff frequency for signal transmission in opposite
direction, depending on the pre- or postbranching location of the
irregularity, clearly has significance in terms of signal integration
in an axonal tree. In a detailed analysis of the morphology of the
terminal arborization of thalamic and cortical neurons,
Deschenes and Landry (1980)
found that irregular spacing
is the rule, rather than the exception, for nodes of Ranvier near
branch points of these CNS myelinated fibers. They noted examples where
the parent branch undergoes internodal shortening just prior to the
branch point, which according to our simulations would shift the cutoff
frequency for parent-to-daughter transmission to the high-frequency
spectrum. Interestingly, internodal lengthening in the parent branch
prior to the branch point can also occur (Deschenes and Landry
1980
), which might suggest a functional requirement to reduce
high-frequency, parent-to-daughter, transmission in certain areas of
the axon arbors. Likewise, shortening of postbranch internodes has also
been observed in daughter branches, which according to our analysis
should be another mechanism for limiting high-frequency,
parent-to-daughter transmissions. As Deschenes and Landry
(1980)
have remarked, the regular and predictable spacing of
nodes of Ranvier on the basis of axonal diameter according to the
Rushton's Law (Rushton 1951
) for optimal conduction
appears to undergo a breakdown on arborization. Our theoretical
analysis suggests that internodal length irregularity and its deviation from Rushton's Law at branch point vicinity may serve an important functional role in filtering neural information in the spatial and
temporal domain in an axonal tree (Deschenes and Landry
1980
).
Besides internodal length irregularities, we also found that the width
of the periaxonal space is critical. We found that selectively altering
the periaxonal space in the daughter branches dramatically alters the
efficacy of high-frequency invasion of the daughter branches.
Tightening the myelin wrappings around the daughter branches
(decreasing the width of the periaxonal space) greatly enhances the
passage of high-frequency signal from the parent to the daughter
branches. Loosening the myelin wrapping (widening the periaxonal space)
screens the daughter branches from receiving high-frequency signals.
One reason for this effect of periaxonal space is that the thickness of
this space determines the electronic coupling between the nodal and the
internodal axon and hence the capacitative load of the daughter
branches as seen by an invading parent branch. For example, widening
the periaxonal space in the daughter branches would allow increased
electrotonic coupling between the nodal and the internodal axolemma.
Since the internodal axolemma has few sodium channels to support
conduction, this would have the effect of increasing the capacitative
load in the daughter branches, thereby reducing the chances of
successful parent-to-daughter branch invasion. Using periaxonal space
geometry to control information flow into daughter branches could have functional significance, particularly during development where immature
myelinated axons may have a slightly wider periaxonal space then adult
nerves. In a developmental study by Yamamoto et al.
(1996)
, the axoglial junctions where the terminal myelin loops
attach to the paranodal axolemma was examined in 10- and 31-day-old
rats. When the terminal loops attach to the axolemma, the extracellular
distance between the loop and the axolemma (i.e., the periaxonal space)
is 4.0 nm. This distance widens to 5.5 nm when the terminal loops do
not attach to the axolemma. Yamamoto et al. (1996)
further found that even though the paranodal length remains unchanged
during development, the frequency of terminal toops with attachment
increased with fiber grow. Hence, there is a progressive reduction in
the overall width of the periaxonal space as myelination proceeds.
Other examples of periaxonal space alteration can be found in various
demyelinating diseases. For example, a widening of the periaxonal space
was detected in myelinated fibers in the peripheral demyelinating
neuropathy associated with the toxins of the buckthorn shrub
(Heath et al. 1982
). One could imagine that the looser
periaxonal space in new branches formed during either development or
regeneration may screen them from potential excitotoxicity due to
high-frequency signaling in the parent branches.
Another possible function of a branch point is differential routing of
impulses into one and not the other daughter branches. In nonmyelinated
axons, this differential routing occurs in daughter branches of
different sizes, suggesting that the input impedance of individual
daughter branches is an important determinant of differential routing.
In our model, we used two identical daughter branches and therefore
have not systematically investigated the conditions for differential
routing at branch points of myelinated fibers. Nevertheless, we have
performed preliminary studies on differential routing with our model
and found that differential routing only occurs when the two daughter
branches are of very different morphological parameters. Another
essential requirement for differential routing is that the initial node
of two daughter branches at the branch point (N1 and N2) should have
different sizes (data not shown), otherwise the activities of the two
daughter branches are always coupled together, activating or failing at the same time. This scenario is quite different from branch points in
nonmyelinated nerve fibers, in which differential routing could be
achieved easily without dramatic geometrical difference between the two
daughter branches. Indeed, even random channel noise could affect
differential routing in branch points of nonmyelinated axons
(Horikawa 1993
). Our computer simulations highlight the importance of the membrane properties of the nodal membrane at the
branch point (N0, N1, N2) in determining branch point transmission. There is little or no direct measurement of the membrane properties of
the special node at the branch point. It is possible that the node of
Ranvier at the branch point may have specialized properties adapted for
branch point transmission.
Comparison with empirical studies on branch points of myelinated
axons
As discussed in the INTRODUCTION, a large body of work
already exists on empirical and theoretical analysis of conduction at
branch points in invertebrate nonmyelinated axons, including crayfish
and the squid giant axons (Grossman et al. 1973
,
1979
; Parnas and Segev 1979
; Smith
1980
, 1983
). These studies of nonmyelinated branch points all show that K accumulation, temperature, frequency of
stimulation, and local geometry affect conduction failures at the
branch point. Our study shows that branch point of myelinated axons are
similarly affected by these factors, but with new features unique to
regulating branch point excitability being the internodal length in the
vicinity of the branch point and the differentiation of the paranodal
structure. How does our paper compare with previous work? There is no
theoretical work on branch point excitability in myelinated axons prior
to our work. Empirical analysis of branch point transmission in
myelinated axons started with the work of Krnjevic and Miledi
(1958
, 1959
), who inferred, rather than directly proved, that failure of neuromuscular transmission in the phrenic nerve-diaphram preparation during high-frequency stimulation is due to
branch point failure. Subsequent empirical work on branch point
excitability of myelinated axons has relied on both using the dorsal
root ganglion preparation where three myelinated axons form the
T-junction, and the neuromuscular junction preparation (Sieck
and Prakash 1995
). The empirical study of Stoney
(1990)
on adult frog DRG is the closet to which our theoretical
work can be compared. Here, Stoney (1990)
observed that
the T-junction starts to block transmission into the dorsal root at
frequency above 363 Hz (21-23°C). Interestingly, Stoney
(1990)
demonstrated that in the frog, moderate warming above
22°C improves the ability of branch points of myelinated axons to
follow brief, high-frequency action potentials. Further warming to
above 37°C blocks branch point transmission (Stoney
1990
). Thus both empirical analysis (Stoney
1990
) and our theoretical analysis are in qualitative agreement
in demonstrating that branch point transmission has a bell-shaped
dependence on temperature in myelinated axons. The quantitative
difference between the empirical data on frog and our case may lie in
the physiological operating temperature, which is room temperature in
frog and 37°C in this study. In another empirical study on the
T-junction in dorsal root ganglion, Stoney (1985)
has
noticed the frequency filtering action of the branch point and has
suggested that internodal shortening in the immediate vicinity of the
branch point may be a mechanism for extending the useful frequency
range of the T-junction in information flow. Indeed, as Stoney
(1985)
has pointed out, some dorsal root ganglion in both
amphibians (Ito and Takahashi 1960
) and mammals
(Lieberman 1976
) have unusually short peribifurcation
internodes that may represent a functional adaptation in extending the
useful frequency range of the branch point. This is dramatically
confirmed in our computer simulations: a single short internode
proximal to the branch point confers not only high-frequency following
ability to the branch point, but allows this frequency following
ability to be maintained over a broad temperature range. Finally, in
the soleous nerve muscle preparation, Schiller and Rahamimoff
(1989)
observed neuromuscular transmission failure during
high-frequency stimulation that they attributed to axonal conduction
failures, presumably at branch points of these myelinated axons.
Interestingly, they found that diabetes rats have less failure than
normal ones. Further, elevating extracellular potassium increases
tetanic failures. They suggested that the reduced failure in diabetes
rat resulted from less activity-dependence potassium accumulation. Our
simulations clearly show that a reduction in potassium accumulation in
the periaxonal space (Fig. 6) can dramatically facilitate action
potential conduction through branch point of a myelinated axon.
Address for reprint requests: S. Y. Chiu, Dept. of Physiology,
University of Wisconsin School of Medicine, 277 Medical Sciences Bldg.,
1300 University Ave., Madison, WI 53706 (E-mail:
chiu{at}physiology.wisc.edu).