The Journal of Neuroscience, March 23, 2005, 25(12):3046-3058; doi:10.1523/JNEUROSCI.3064-04.2005
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Behavioral/Systems/Cognitive
A Model for Interaural Time Difference Sensitivity in the Medial Superior Olive: Interaction of Excitatory and Inhibitory Synaptic Inputs, Channel Dynamics, and Cellular Morphology
Yi Zhou,1
Laurel H. Carney,2 and
H. Steven Colburn1
1Department of Biomedical Engineering, Center for
Hearing Research, Boston University, Boston, Massachusetts 02215, and2
Department of Bioengineering and Neuroscience,
Institute for Sensory Research, Syracuse University, Syracuse, New York
13244-5290
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Abstract
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This study reports simulations of recent physiological results from the
gerbil medial superior olive (MSO) that reveal that blocking glycinergic
inhibition can shift the tuning for the interaural time difference (ITD) of
the cell (Brand et al., 2002
).
Our simulations indicate that the model proposed in the study by Brand et al.
(2002) requires precisely
timed, short-duration inhibition with temporal accuracy exceeding that
described in the auditory system. An alternative model is proposed that
incorporates two anatomic observations in the MSO: (1) the axon arises from
the dendrite that receives ipsilateral inputs; and (2) inhibitory synapses are
located primarily on the soma in adult animals. When the inhibitory currents
are activated or blocked, the model cell successfully simulates experimentally
observed shifts in the best ITD. The asymmetrical cell structure allows an
imbalance between the ipsilateral and contralateral excitatory inputs and
shifts the ITD curve such that the best ITD is not at zero. Fine adjustment of
the best ITD is achieved by the interplay of somatic sodium currents and
synaptic inhibitory currents. The shift of the best ITD in the model is
limited to 
0.2 ms, which is behaviorally significant with respect to ITDs
encountered in perceptual tasks. The model suggests a mechanism for
dynamically "fine-tuning" the ITD sensitivity of MSO cells by the
opponency between depolarizing sodium currents and hyperpolarizing inhibitory
currents.
Key words: interaural time differences; medial superior olive; inhibition; asymmetrical cell morphology; binaural hearing; neural modeling
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Introduction
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Interaural time difference (ITD) is the primary cue for localizing
low-frequency sounds in the horizontal plane
(Wightman and Kistler, 1992).
Jeffress (1948) hypothesized
that ITD information is carried by an array of coincidence detectors, each of
which discharges maximally when the external ITD is equal to the internal
delay difference of the cell.
Neural mechanisms of coincidence detection have been studied and confirmed
in avian nucleus laminaris (NL) neurons
(Joseph and Hyson, 1993;
Reyes et al., 1996) and
mammalian medial superior olive (MSO) neurons
(Goldberg and Brown, 1969;
Yin and Chan, 1990;
Spitzer and Semple, 1995;
Brand et al., 2002). However,
anatomical evidence of the delay-line structure is less clearly demonstrated
in the mammalian system (Smith et al.,
1993; Beckius et al.,
1999) than in the avian system
(Young and Rubel, 1983;
Carr and Konishi, 1990;
Overholt et al., 1992).
McAlpine and Grothe (2003)
hypothesized that inhibitory synaptic inputs can systematically shift the best
ITD (i.e., the ITD at which an MSO cell discharges maximally) and thus that
the best ITD can be manipulated independently of the physical delayline
structure. Their hypothesis is based on the experimental observation in
vivo that blocking glycinergic inhibition shifts the rate-ITD curve in
MSO cells of gerbils (Brand et al.,
2002). To interpret the experimental results, Brand et al.
(2002) proposed a model that
includes a precisely timed inhibition with a short duration (leading
excitation by 0.2 ms; 
= 0.1 ms for the synaptic conductance). However,
the observed time constant of the glycinergic inhibitory synapses in the MSO
is much longer than 0.1 ms [e.g., 2 ms
(Smith et al., 2000)].
Furthermore, no experimental studies have investigated the precision of the
relative timing of excitatory and inhibitory inputs to the MSO in response to
acoustic stimuli.
Here, we present an alternative explanation for the data of Brand et al.
(2002). This model incorporates
a morphological feature of MSO: the axons of some principal MSO neurons arise
from the dendrites instead of the soma [cat
(LaVilla, 1898;
Kiss and Majorossy, 1983),
gerbil (N. Golding, personal communication), guinea pig
(Smith, 1995), mouse
(Ramón y Cajal, 1909)]
(see Fig. 1).
Figure 1 illustrates this
asymmetry in sample MSO cells from guinea pig and gerbil. Brew
(1998) first suggested that the
asymmetry may cause different delays between two interdendritic excitatory
inputs en route to the axon; however, no studies have explored whether the
asymmetry could affect the interaction between excitation and inhibition. We
postulated that the observed shift of the ITD curves attributable to blocking
inhibition might be associated with the effective asymmetry of excitatory and
inhibitory inputs because of the relative locations of the axon and of the
inhibitory synapses. To examine our hypothesis, we used a multicompartment
model to study the interaction of excitation, inhibition, and ionic currents
on this asymmetrical structure. This model simulated the shift of the ITD
function by inhibition with a time constant comparable to available data. The
model results suggest a potential mechanism for dynamic
"finetuning" of ITD sensitivity at the single neuron level, which
has implications for our understanding of auditory spatial attention.
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Figure 1. The asymmetrical cell structure with an axon (a) emerging from the
ipsilaterally innervated dendrite. A, A cell from a guinea pig
MSO [reproduced from Smith
(1995) with permission].
B, A cell from a gerbil MSO (Golding, unpublished observation)
(scale bar not shown). Both cells have an axon emerging from the ipsilaterally
innervated dendrite.
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Materials and Methods
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Simulations presented here were generated by two types of neural models: a
point-cell model that was insensitive to spatial distribution of synapses and
a bipolar-cell model that incorporated asymmetric morphology
(Fig. 1), dendritic
glutaminergic excitation, somatic glycinergic inhibition, and somatic sodium
channels. We simulated both models in the NEURON environment (Hines and
Carnevale, 1997,
2000). The bipolar model is
the focus of this study and is presented in detail in the following section.
Both the point- and bipolar-cell models received input discharges from model
afferent fibers. A simplified input model was used to generate these
phase-locked discharge events according to the rate function of a
nonstationary Poisson process that mathematically described the time-varying
discharge properties of bushy cells in the anteroventral cochlear nucleus
(AVCN) and of cells in the medial nucleus of the trapezoid body (MNTB).
Input model. The input model consisted of two periodic rate
functions of Poisson-like processes: one described the characteristics of
bilateral excitatory inputs from the AVCN to the MSO, and the other described
the characteristics of contralateral inhibitory inputs from the MNTB to the
MSO. Each run of the program involved multiple independent instances for one
rate function. Each instance represented a discharge train for one excitatory
or inhibitory input fiber, which innervated one excitatory or inhibitory
synapse. The same input model was used for both cell models explored here.
A rate function had three main parameters: the period T of the
input stimulus, the average input rate Rave, and the input
vector strength rin. The parameters
Rave and rin independently controlled
the rate and the temporal aspects of input discharge trains. For tonal
stimuli, the period T in milliseconds is the inverse of the tone
frequency fin in Hertz (T =
1000/fin). Input discharges were phase-locked to the input
frequency fin, and the tightness of the phase-locking was
controlled by rin. The average input rate
Rave was the average number of input discharges per
second, which was less than or equal to fin.
Because we postulated that each input fiber had no more than one discharge
per input period, it was more efficient to specify the location of a discharge
in the period than to specify the occurrence of a discharge at each simulation
time step. Generating periodic discharge trains involved two steps. (1) For
each input period, an action potential was generated for each fiber with
probability p, where p =
min(Rave/fin,1). Usually,
Rave was less than fin so that no
discharge occurred in some cycles (p < 1). Thus,
Rave controlled the probability p and the average
input rate. (2) For any given input frequency fin, the
next discharge time Ti was determined relative to the
start time of the current period ti, where
ti = i x T. The discharge time
Ti was drawn from a Gaussian distribution,
Ti N(T/2, T/(2 x
F)), where T was the period of the input stimulus in
milliseconds, and F was a constant that determined the strength of
phase-locking. The associated vector strength r was given by
r = exp(-
2/(2F2)),
independent of input frequency fin. Additionally, if
Ti > T or Ti < 0, the
input model constrained Ti to be within the current period
by replacing Ti with its circular residue
T'i (i.e.,
T'i = TimodT
if Ti > T, or
T'i = Timod
T + T if Ti < 0). This method prevented
the occurrence of multiple discharges in a single period while maintaining the
shape and area of the rate density function.
In both excitatory and inhibitory input trains, a 0.5 ms absolute
refractory period was imposed after each input discharge and eliminated
subsequent discharge times within the refractory period. As the input
frequency increased, or the discharge distribution for periodic inputs
approached a uniform function, the number of input discharges eliminated by
refractory periods became significant; thus, Rave was the
upper bound of the actual average input rate.
Finally, the input model assumed that excitatory and inhibitory input
discharge trains had the same frequency fin under the
assumption that the model cell was innervated by AVCN bushy cells and cells
from MNTB, which have discharges that are phase-locked to the tone frequency
fin.
Simulations of rate-ITD responses. The rate-ITD responses for the
first part of the study were generated using the point-MSO model described by
Brand et al. (2002) with
fin = 500 Hz; simulations for the second part of the study
were generated using the bipolar MSO model with fin
ranging from 250 to 1000 Hz. The ITDs used to test these models were actually
interstimulus time differences, defined as the time difference between the
arrivals at the cell of two periodic excitatory input trains, one to each
dendrite of the bipolar cell. The contralateral inhibition and contralateral
excitation arrived at the same time statistically in the default setup. In
some tests, the spike times of contralateral inhibitory inputs were
artificially advanced or delayed to simulate the leading or lagging inhibition
relative to the contralateral excitation. The discharge count in response to
each ITD was measured at the center of the soma for the point model and at the
center of the axon for the bipolar model, collected within the simulation time
window of 100 ms for the point model and of 1000 ms for the bipolar model, and
averaged over 10 trials. Only the average rates are presented in the results
below.
All simulations in this study used the backward Euler integration method in
NEURON with a fixed numerical time step of 25 µs. The accuracy of the
integration was estimated with smaller time steps in some simulations, and the
shapes of rate-ITD curves were not affected.
Point-MSO model. The point-MSO model was identical to the MSO
model of Brand et al. (2002);
it had a single-compartment structure with ionic channels based on those
described for AVCN cells (Rothman et al.,
1993; Brughera et al.,
1996). In the model of Brand et al.
(2002), bilateral excitatory
inputs (2 x 24 fibers) and contralateral inhibitory inputs (24 fibers)
are all derived from tonal responses of an auditory nerve model
(Carney, 1993). For
simplicity, we used an input rate function with a limited number of parameters
to generate input discharge trains as described in the input model. The
synaptic inputs to the point model were implemented exactly as for the bipolar
model, as described below, except that they were not distributed spatially
along a model structure.
Bipolar MSO model. The bipolar MSO model cell had a
multicompartment structure that comprised one active soma, one active axon,
and two passive dendrites, one of which was innervated ipsilaterally and the
other contralaterally. The axon was connected with the dendrite ipsilaterally
innervated (45 µm from the soma), based on the drawings of MSO cells and
their axons in the study by Smith
(1995). The geometrical
specifications of the model are provided in
Table 1.
Synapses and ionic channels. Ten excitatory synapses were
distributed evenly along the proximal half-section of each dendrite. Ten
inhibitory synapses that received contralateral inputs were located on the
soma (Clark, 1969;
Kuwabara and Zook, 1992;
Grothe and Sanes, 1993;
Kapfer et al., 2002). Each
input discharge triggered an excitatory or inhibitory synaptic conductance
change. The synaptic conductance g{e,i}(t) was
simulated as a linear combination of two exponentials with the peak
conductance Ge or Gi at time
tp such that
where the normalization factor was g(tp) =
exp(-tp/{e,i}) -
exp(-tp/rise), and the time of
the peak was
with {e,i} > rise. In both the point and
bipolar models, e = 0.1 ms for all excitatory synapses. In the
point model, i varied from 0.1 to 1 ms; in the bipolar model,
i = 2 ms. When {e,i} = 0.1 ms, to avoid a zero
denominator in tp, rise = 0.0999 ms for
excitatory or inhibitory synapses; otherwise rise = 0.1 ms.
The reversal potentials for excitatory and inhibitory synapses were
Ee = 0 mV and Ei = -70 mV,
respectively.
Specified ionic channels were inserted into the axon to generate action
potentials: fast sodium INa, low-threshold potassium
ILTK, highthreshold potassium IHTK,
hyperpolarization-activated inward current Ih, and leakage
channel IL. The kinetics of all channels in the bipolar
model were based on the recent models of type II cells in the AVCN
(Rothman and Manis, 2003). The
soma had only sodium channels, which were turned on or off to study their
influence on the ITD sensitivity during the simulation.
The peak conductances Ge and Gi of
synapses, the average input rate Rave, and the input
vector strength rin were the main parameters used to
adjust the performance of the model cell, and their values are provided in the
corresponding results below. Parameters that specify the peak conductance of
ionic channels used for the bipolar model are listed in
Table 1.
Linear cable properties. We measured cable properties of the model
cell to ensure the numerical accuracy of the simulations and used them to
estimate the amount of delay and attenuation of synaptic potentials traveling
along a cable-like structure. We applied a linear cable theory
(Rall and Rinzel, 1973) to
define the following terms using model parameters in
Table 1: (1) steadystate space
constant
(µm); (2) membrane time constant m =
Cm/(1000 x GL) (ms); (3)
electrotonic length of a cylinder
(4) input resistance at the origin of a semi-infinite cylinder
(M
) and (5) input resistance at the origin of a finite cylinder with
a sealed end Rin = R
coth(L) (M). The calculations of
R and Rin were only applied to
the dendrites and the axon. The time constant m, the space
constant 
DC, and the input resistance
Rin for each section of the model are provided in
Table 1.
Additional simulations. We also used the point model to study the
time scale of the effective inhibition observed in the lateral superior olive
(LSO), which receives ipsilateral excitation from the AVCN and contralateral
inhibition from the MNTB. Contralateral excitation is less frequently observed
in the LSO (Cant and Casseday,
1986; Kuwabara and Zook,
1992; Wu and Kelly,
1994). We simulated responses in the experiment of Joris and Yin
(1995), in which pulse trains
with an interpulse interval of 10 ms were delivered to two ears, and the onset
time of one stimulus was advanced or delayed to change the timing between
excitation and inhibition in the LSO. In the simulation, we assumed that one
acoustic pulse triggered one ipsilateral EPSP and one contralateral IPSP in
the LSO. Because of the lack of knowledge of membrane properties of the MSO
and the LSO, we used the same point-cell model but excluded contralateral
excitatory inputs in this simulation. Each input fiber generated 100 input
pulses with interpulse intervals of 10 ms, and the number of model cell
discharges was measured.
For the bipolar model, the asymmetric cell structure imposed a traveltime
difference between contralateral and ipsilateral inputs to the axon. We
characterized this time difference in the model using two different
approaches. First, we measured computationally the compound EPSP in response
to unilateral synaptic inputs (fin = 500 Hz) at the origin
of the axon. The purpose of this calculation was to compare the net delay and
attenuation of the ipsilateral and contralateral synaptic potentials arriving
at two dendrites. Because the synaptic potential could be suprathreshold for
this test, action potentials were generated occasionally, which distorted the
shape of the underlying synaptic potential. To compare the relative shape of
synaptic potentials, we blocked the sodium, potassium, and inward current
channels on the axon by setting their conductance to zero for this test. The
compound EPSP was recorded over a 200 ms time window and folded into one input
period (T = 2 ms) to estimate the average EPSP.
Second, we explored the delay difference between two dendritic inputs based
on the theory of signal delay in a passive cable
(Agmon-Snir and Segev, 1993;
Zador et al., 1995;
Rall and Agmon-Snir, 1998;
Koch, 1999). We chose this
theory because it provides a closed-form solution to the transfer delay of a
signal (Agmon-Snir and Segev,
1993), which can be directly applied to the bipolar model.
The definitions and properties applied in our analysis are summarized
briefly here. The reference time 
of
a signal is defined as the centroid of the signal (in either current or
voltage). The transfer delay is the difference between the reference time of a
current signal injected at location y and the reference time of a
voltage signal measured at location x:
The propagation delay is the difference between the reference times of two
voltage signals at locations y and x:
Note that Py
x is dependent on
Dyx as well as the local delay at location
yDyy. Furthermore, it has been shown
(Agmon-Snir and Segev, 1993;
Zador et al., 1995) that the
transfer delay is symmetric between opposite directions (i.e.,
Dyx = Dxy) and that the
propagation delay is additive in the direction a signal travels (i.e.,
Pykx = Pyk +
Pkx).
In addition, Agmon-Snir and Segev
(1993, their Eq. 15) showed
that the transfer delay on a dendrite can be expressed analytically with the
boundary condition of an equivalent "soma" at the origin, which
lumps the rest of the structure together:
 | (1) |
In Equation 1, Dyx is in units of
d, L is the electrotonic length of the dendrite,
X and Y are the electrotonic distances of x and
y (i.e., x and y normalized by
DC of the dendrite) for 0 
X Y
L;
where 
and 
are the two factors that describe
the relationship in membrane properties between the "equivalent"
soma and the dendrite, specifically
and R's is the resistance of the
equivalent soma at the origin.
Using Equation 1, we estimated the transfer delay difference
(
TD) between two dendritic inputs under different membrane
conditions, as characterized by and . The axon
was placed on the ipsilaterally innervated dendrite at a distance x
from the soma. Because there was only one input on each dendrite,
TD could be calculated straightforwardly with the aid of some
assumptions. For simplicity, we assumed that the injection sites of synaptic
currents to the two dendrites were at equal distances from the soma
(y1 = y2 = y), that the two dendrites were
identical electrotonically, that the soma at the origin was isopotential with
zero propagation delay (P00 = 0), and that the axon
with its high input resistance had negligible influence on signal propagation
along the dendrites. Based on the definitions and properties of signal delays,
we calculated the TD between two excitatory inputs to the axon
as follows.
If the distance of dendritic inputs to the soma was longer than that of the
axon to the soma (y 
x):
 | (2) |
If the distance of dendritic inputs to the soma was shorter than that of
the axon to the soma (y x):
 | (3) |
TD was a function of the distance of the axon to the soma
(x) in Equation 2 and a function of the distance of the injection
site to the soma (y) in Equation 3. Transfer delays and local delays
in Equations 2 and 3 were then calculated using Equation 1, with the boundary
resistance of the equivalent soma described as
Because Equations 2 and 3 are equivalent if x and y are
interchanged, only the result of TD as a function of the
distance of the axon from the soma (Eq. 2) is shown in Results.
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Results
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Point-MSO model: temporal precision of inhibition is required
In this section, we extend the analysis of the model of Brand et al.
(2002) using the same cell
model and similar inputs. The results in the model of Brand et al.
(2002) suggest that a precisely
timed synaptic inhibition shifts the peak of the ITD function, a mechanism
that demands a short-duration inhibition.
Figure 2 shows the extent of
the shift of the ITD function with inhibition relative to that without
inhibition for different i values for the inhibitory synapses
(Fig. 2A) and
different lead times of inhibition with respect to excitation on the
contralateral side (Fig.
2B). In this test, fin = 500 Hz,
Rave = 150 Hz for bilateral excitation and contralateral
inhibition, rin = 0.926 for excitation, and
rin = 1.0 for inhibition. In
Figure 2A, the peak of
the ITD function indicates that inhibition with i = 0.1 ms was
most effective in moving the peak activity to ipsilateral delays. Inhibition
of relatively long duration, such as that of i = 1 ms, only
suppressed the rate-ITD response, whereas the peak of the ITD curve did not
shift. For the simulations, the value of Gi was adjusted
for longer i to preserve discharge rates. In addition,
shortduration inhibition (i = 0.1 ms) worked only if it
arrived at times closely adjacent to and ahead of excitation.
Figure 2B shows that
the inhibition with lead times 0.2 ms most effectively shifted the best
ITD to the contralateral-leading side. When inhibition lagged contralateral
excitation, the best ITD shifted to the ipsilateral-leading side. As the
timing between excitation and inhibition increased, the ITD sensitivity around
zero ITD was less affected by inhibition, such that the peak ITD shifted back
to the center. The maximum shift of the ITD function by the point model was
0.2 ms, which was constrained by the time scale of the excitation and
inhibition (e = i = 0.1 ms) and by the timing
of the inhibition.
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Figure 2. Effects of temporal precision of inhibition in the point model.
A, Effects of the time constant i of
inhibition. The peak position of the ITD function is greatly influenced by the
brief (i = 0.1 ms), but not by the relatively long,
inhibition. Parameter values are Ge = 2 nS for excitation
and Gi = 10, 2, and 2 nS for inhibition with
i = 0.1, 0.5, and 1 ms, respectively. Inhibition leads
excitation by 0.2 ms. B, Effects of relative timing of
inhibition. The shift of the ITD function is more prominent when inhibition
leads excitation by 0.1 ms. Parameter values are Ge =
2 nS, Gi = 10 nS, and i = 0.1 ms.
Parameters marked with asterisks are those used in the model of Brand et al.
(2002). ipsi, Ipsilateral;
contra, contralateral.
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The best ITD (best) shifted from the time of coincident
excitations (ITD = 0) to a different ITD (ITD =best),
because the increase in the summation of excitation at an ITD of 0 over that
at an ITD of best was reduced by the suppressive effect
of inhibition. Inhibition shortly preceding contralateral excitation coincided
with both excitations more at ITD = 0 than at ITD
=best. To suppress the summation of excitation
differentially between ITD = 0 and ITD = best,
inhibition also had to be transient with respect to
best. If inhibition was constant between ITD = 0 and
ITD = best, the maximum response would remain at ITD =
0. Thus, inhibitions of longer duration were not able to shift
best.
The model of Brand et al.
(2002) uses fast inhibitory
effects, i = 0.1 ms, which is much shorter than that of IPSCs
observed in MSO neurons of mammals (Smith
et al., 2000). Inhibition with a time constant in the observed
range in our test could not achieve the same result as their model.
It is possible that the IPSPs in the slice preparation have longer
durations than in vivo. Indeed, indirect evidence of shortduration
inhibitory effects lasting 1 ms in response to transient pulse stimuli in
the LSO have been reported by Joris and Yin
(1995), shorter than in in
vitro experiments (Wu and Kelly,
1992). To examine the effective time scale of inhibition, we
measured the discharge probability of the point model as a function of the
arrival time difference between an EPSP and an IPSP. The strengths of
excitatory and inhibitory synapses were adjusted such that excitatory inputs
could fully evoke an action potential and the discharge was suppressed when
inhibition coincided with excitation.
The LSO simulation showed that the time window of effective inhibition
could be shorter than the duration of the IPSP.
Figure 3 shows that the
suppression of the response by inhibition lasted 1 ms, although it was
generated by an inhibition with i = 2 ms. In
Figure 3 (right), shapes of the
unitary IPSP and EPSP are plotted. It is clear that the model neuron was
sensitive to a transient temporal onset disparity between excitation and
inhibition that was shorter than the duration of the inhibition. The disparity
in duration and the temporal waveform between EPSP and IPSP also resulted in
different slopes of release from inhibition, similar to that in experimental
observations (Wu and Kelly,
1992; Joris and Yin,
1995). Thus, LSO responses to pulse stimuli do not require the
existence of inhibition with a duration as short as i = 0.1
ms.
The point model results (Fig.
2B) also indicate that a shortduration inhibition
(i = 0.1 ms) shifted best only when it led
excitation by up to 0.2 ms. This result implies that temporal jitter in the
arrival time of inhibition in this model would weaken its role in shifting the
ITD function to an exact location. Increasing i in the
synaptic conductance function (Gi) delays the time of the
peak and can increase the limit on the lead time of inhibition. However,
inhibition with longer i inevitably lost the power to shift
the ITD function (Fig.
2A), especially at high frequencies. Because
physiological experiments have not reported synaptic inhibition with temporal
precision such as that required by the model of Brand et al.
(2002), alternative
explanations of the data are desirable.
Bipolar MSO model: interaction of inhibition and sodium channels
The conceptual scheme of the bipolar model is shown in
Figure 4, which illustrates the
asymmetrical placement of the axon and the hypothetical mechanism for the
shift in the ITD function attributable to inhibition. Assuming that the action
potential is initiated at the axon hillock, we made three hypotheses. First,
if the soma were purely passive and received no inhibition, the contralateral
excitation would be more attenuated than the ipsilateral excitation, because
the contralateral input has a longer path to the axon hillock. This difference
between the passive attenuations would cause the peak of the ITD curve to
shift toward the contralateral-leading side, even if the excitatory inputs
arrived with zero ITD at the dendrites of the MSO cell
(Fig. 4A, EE). Our
second hypothesis was based on the fact that blocking inhibition centers ITD
tuning in the in vivo data (Brand
et al., 2002). We hypothesized that a depolarizing mechanism on
the soma that overcomes the shunting effect of the passive membrane may
underlie the symmetric ITD curve that is observed after blocking inhibition.
We proposed that active sodium currents provide this depolarizing mechanism
(Fig. 4B, EE+Na).
Third, we hypothesized that inhibitory inputs to the soma counteract the
depolarizing force and move the peak of the ITD curve toward the
contralateral-leading side (Fig.
4C, EE+Na+I). We chose the sodium (Na+)
current as a candidate for the depolarizing force because its temporal profile
is similar to and its polarity is opposite to that of the glycinergic
inhibitory currents at the soma.
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Figure 4. Illustration of the mechanisms involved in the bipolar model. Three soma
membrane conditions that were studied were a simple RC circuit (EE)
(A), the RC circuit with active sodium channels (EE+Na)
(B), and the RC circuit with active sodium channels and with
inhibitory synapses (EE+Na+I) (C). Zero ITD corresponds to
zero arrival delay between the two excitatory inputs. ipsi, Ipsilateral;
contra, contralateral.
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Membrane potentials
In this model, the electrical properties of the soma differed for the three
conditions in which the model operated.
Figure 5 shows sample traces of
model responses to the same input patterns for different model configurations.
The membrane potentials were measured at the origin of the axon in
Figure 5, A and
B. In Figure
5A, EPSPs of contralateral inputs have smaller amplitudes
and delayed peak times than EPSPs of ipsilateral inputs, because of the
asymmetric structure. The membrane of the soma was passive in this test. In
Figure 5B, membrane
potentials of identical contralateral inputs are compared for the three
conditions to demonstrate the opponency between sodium currents and inhibitory
currents (i = 2 ms). The activation of sodium channels
increased the depolarization of the membrane (EE+Na), whereas the inhibition
hyperpolarized the membrane (EE+Na+I). Note that the amplification of EPSPs by
sodium channels is voltage dependent; in contrast, the suppression of EPSPs by
synaptic inhibition is less dependent on the membrane potential. These
differences were less noticeable in simulations with bilateral inputs, which
often resulted in suprathreshold EPSPs.
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Figure 5. Membrane potentials in response to 500 Hz inputs (A,
B) and current injection (C).
A, Contralateral (contra) EPSPs have smaller amplitudes and
delayed peak times with respect to ipsilateral (ipsi) EPSPs in the passive
membrane condition. B, The membrane is depolarized by active
sodium currents and hyperpolarized by inhibitory currents at the soma in
response to contralateral inputs at three membrane conditions. Parameter
values for inputs are in Table
2. C, Somatic and axonal action potentials have
different shapes and thresholds to current injection. A sustained current with
an amplitude of 1.0 nA and a duration of 20 ms was injected into the soma. The
peak conductance of sodium channels GNa were 0.1
S/cm2 on the soma and 0.3 S/cm2 on the axon.
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The response of the model to a somatic current injection is shown in
Figure 5C. The
potentials were measured at the center of the soma and of the axon. Only one
onset discharge was elicited by a sustained current attributable to
subthreshold activity of the low-threshold potassium channels ILTK
(Smith, 1995;
Rothman and Manis, 2003).
Blocking ILTK yielded multiple discharges (data not shown). On the
soma, the deactivation of sodium channels prevented repetitive discharges to
sustained currents, and accumulating charges on the membrane raised the
potential. Because the axon had a higher density of sodium channels, the
amplitude of an action potential was higher and the threshold of an action
potential was lower at the axon than at the soma. Because the soma membrane
did not contain any outward potassium currents, the width of an action
potential was broader at the soma.
Rate-ITD functions
In Figure 6, the rate-ITD
responses of the bipolar model in three conditions are shown for four
frequencies. The peak conductance of each synapse Ge and
Gi, the average input rate Rave, and
the input vector strength rin for each frequency are
listed in Table 2. These
results demonstrate that the mechanism for shifting the ITD function with
inhibition (EE+Na+I) relative to that without inhibition (EE+Na) is effective
over a range of input frequencies.
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Figure 6. Rate-ITD responses at four frequencies. The three rate-ITD functions in
each panel are results of the model with three configurations of the soma:
passive (EE), with active sodium channels (EE+Na), and with both active sodium
channels and inhibition (EE+Na+I). Parameters for excitation and inhibition at
each frequency are listed in Table
2. ipsi, Ipsilateral; contra, contralateral.
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The inhibitory synaptic conductance in these simulations had a time
constant i = 2 ms, which accumulated over cycles, and
contained limited timing information for fin 500 Hz
despite the highly phase-locked afferent inputs. In
Figure 7A, the shift
of the ITD function for 500 Hz is compared for inhibition with highly
synchronized inputs and with randomly timed inputs. The lack of effect of
synchronization shows that the mechanism used in the bipolar model does not
require accurate arrival times for inhibition. In contrast, for the point
model, inhibition with the same time constant (i = 2 ms) did
not shift the ITD function, nor did the shorter-duration inhibition
(i = 0.1 ms) when it did not arrive just before contralateral
excitation. MSO cells receive ipsilateral inhibitory inputs from the lateral
nucleus of the trapezoid body (LNTB) (Cant
and Hyson, 1992; Kuwabara and
Zook, 1992) as well as from the MNTB. It is not clear at present
whether inputs from the LNTB exhibit temporal accuracy similar to those from
the MNTB. Nevertheless, because the mechanism in the bipolar model for
shifting the ITD curve does not depend on the detailed timing or duration of
the inhibition, the effects of inhibitory inputs from the LNTB or from any
other source could resemble those of inputs from the MNTB. In the model of
Brand et al. (2002), LNTB
inputs are omitted.
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Figure 7. Temporal and spatial effects of inhibition in the bipolar model.
A, Both synchronized and random inhibitory inputs shifted the
ITD function. The asterisks mark parameter values used for 500 Hz inputs in
Figure 6. The average input
rate Rave was 240 Hz for inhibition in all three
conditions. B, Somatic inhibition was more effective than
dendritic inhibition in shifting the peak of the ITD function. Dendritic
inhibition was distributed evenly along the proximal section of the
contralaterally innervated dendrite. Input parameters are listed in
Table 2 (for 500 Hz
inputs).
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We also tested the influence of the distribution of inhibitory synapses on
the shift of the ITD function. Figure
7B compares the ITD functions of two model cells, one
with somatic inhibition and the other with dendritic inhibition collocated
with contralateral excitation. Results show that the magnitude of the shift
was slightly larger for somatic inhibition than for dendritic inhibition of
the same strength. Somatic inhibition "on the path" to
contralateral excitation can interact with sodium currents more directly and
reduce currents flowing from the contralateral dendrite to the axon more
effectively (Koch, 1999). This
result presents a possible functional explanation for the observation of
Kapfer et al. (2002) that the
locations of inhibitory synapses on MSO cells shift toward the soma during
development.
Opponency between sodium channels and inhibition
In comparison to the point-neuron model with short-duration inhibition
(Brand et al., 2002), the
active components that shift the ITD function in the bipolar model include
inhibition and ionic currents on the soma. The point model with short-duration
inhibition controlled the lateral position of the best ITD, positive or
negative, by leading or lagging inhibition on one side relative to excitation,
and the magnitude of the shift was determined by the strength of inhibition.
For the bipolar model, the direction and size of the shift of the best ITD was
produced by completely different mechanisms from those in the point-neuron
model. Figure 8 illustrates the
"push-pull" influence on the ITD function of the sodium current
and inhibition. Increasing the density of sodium channels gradually pushed ITD
tuning toward zero ITD, as shown in Fig.
8A. In contrast, increased inhibition in the presence of
sodium channels pulled the ITD tuning back to contralateral leads, as shown in
Fig. 8B. Because the
sodium current and inhibitory current were opponents, they did not modulate
the ITD tuning independently. The activation of sodium channels alone moved
the ITD function toward ipsilateral leads, but longduration inhibition
balanced the sodium channel activity to shift the best ITD back toward
contralateral leads. Moreover, the shift of the ITD function by inhibition was
bounded by the offset of the best ITD in the EE condition, and increasing the
sodium density did not push the best ITD to the ipsilateral-leading side. The
shift of the ITD function in the bipolar model relies on the asymmetry of the
cell structure, which creates the disparity in the temporal shapes of EPSPs by
the passive membrane (Fig.
5A, EE). To displace the best ITD to the
ipsilateral-leading side by the same mechanism would have required an axon
located on the dendrite that receives contralateral inputs. Best ITDs favoring
the ipsilateral-leading side are not frequently observed in the MSO
(Yin and Chan, 1990;
Spitzer and Semple, 1995), and
asymmetrical axons are typically observed on the dendrite that receives
ipsilateral inputs (Smith,
1995) (Golding, personal communication).
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Figure 8. Opponent effects of sodium channels and inhibition on rate-ITD tuning.
A, The increase of Na+ channel density
(GNa) moved the ITD function toward ipsilateral leading in
the EE+Na condition. B, The increase of the strength of
inhibition moved the ITD function toward contralateral leading in the EE+Na+I
condition; Gi on the soma was 0.1 S/cm2 for
these curves. Other input parameter values are listed in
Table 2 (for 500 Hz). The
asterisks mark values of GNa and Gi
used for 500 Hz inputs in Figure
6.
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Bounds of the shifts
Because the best ITD in the EE condition [i.e., the model with only passive
attenuation (Fig. 5A)]
bounded the maximal shift of the best ITD by the inhibition in the bipolar
model, the limitation in size of the ITD shift is a prominent question. The
position of the axon was varied from the soma to the midpoint of the
ipsilaterally innervated dendrite to answer this question, because,
intuitively, a more distal axon might be expected to enlarge the ITD
shift.
First, we measured computationally at the origin of the axon the average
compound EPSPs generated by either contralateral or ipsilateral inputs, while
the active ion channels were turned off on the axon. Results are shown in
Figure 9. Moving the axon
toward the distal end of the ipsilaterally innervated dendrite attenuated the
contralateral EPSP and delayed its peak
(Fig. 9A). On the
other hand, the strength of the ipsilateral EPSP was nonmonotonically related
to the location of the axon (Fig.
9B); the maximum EPSP occurred when the axon was placed
within the distribution of synapses. (Excitatory synapses covered the proximal
half-section of each dendrite.) Moreover, the peak time of the ipsilateral
EPSP came earlier as its amplitude increased. The dependence of the
ipsilateral EPSP size on the location of the axon and the distribution of the
synapses indicated that moving the axon distally did not result in a larger
shift of the peak ITD in the EE condition for the current model. As described
by Smith (1995), the location
of the axon in 14 of 15 principal MSO cells is reported to be within 45 µm
of the cell body. We chose a value of 45 µm for the model simulations shown
in Figures 5,
6,
7,
8 and
10,
11,
12. These results also show
that the maximum difference in peak times between the two EPSPs was 0.2
ms, similar to the shift of the peak ITD in the passive membrane condition
(EE), which resulted from the delay of the peak of the contralateral EPSP and
the advance of the peak of the ipsilateral EPSP in
Figure 9.
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Figure 9. Effects of the location of the axon on the shape of the EPSP. Increasing
the distance of the axon from the soma causes attenuation of the contralateral
(Contra) EPSP (A) and amplification of the ipsilateral (Ipsi)
EPSP (B), which is maximal for the 75 µm location. The
vertical dashed lines indicate the peak position of the EPSP when the axon was
at the center of the soma. Input parameter values are listed in
Table 2 (for 500-Hz). The
asterisks mark the location of the axon used in the simulations shown in
Figures 5,
6,
7,
8 and
10,
11,
12.
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Figure 10. The analysis of the transfer delay difference TD (Eq. 2)
between contralateral (contra) and ipsilateral (ipsi) excitations in the
passive condition. The two dendritic inputs were equidistant to the soma
(y), and the axon was at x of the ipsilaterally innervated
dendrite (y x). The transfer delay difference between
inputs to the axon was affected by electrotonic properties of the passive
membrane, particularly
as shown. The asterisk marks parameters used by the bipolar model with 500
Hz inputs.
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Figure 11. The effect of excitation level on the shift of the peak ITD.
A, Overall response rates were reduced when the strength of
excitation was decreased (Ge = 9 nS) compared with that in
Figure 6 for 500 Hz inputs, but
activations of sodium currents (GNa = 0.1
S/cm2) and inhibition (Gi = 6 nS) still shifted
the peak ITD. B, The disparity in bilateral levels of
excitation (Ge = 30 nS for the contralateral,
Ge = 6 nS for the ipsilateral) did not affect the peak ITD
in the EE condition. Here, Gi = 15 nS and
GNa on the soma was 0.07 S/cm2. Other input
parameter values are listed in Table
2 (for 500 Hz).
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Figure 12. The effect of somatic potassium channels on the shift of the peak ITD.
Activation of potassium currents reduced responses but did not affect the
position of the peak ITD (EE+K). Increasing the strength of sodium currents
moved the peak ITD to the ipsilateral-leading side (EE+Na+K); increasing
inhibition (Gi = 11 nS) moved the peak ITD to the
contralateral-leading side (EE+Na+K+I). Identical ionic channels were used for
the soma and the axon membrane, with the exception of the density of sodium
channels; GNa was 0.2 S/cm2 on the soma and 0.3
S/cm2 on the axon.
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Second, we explored the effect of electrical properties of the membrane on
the delay difference between contralateral and ipsilateral inputs. Only one
excitatory input was placed on each dendrite to simplify the analysis. The
transfer delay difference (TD) analyzed at the location of the
axon is described in Equations 2 and 3 (see Materials and Methods).
Figure 10 shows the result
of TD as a function of the distance from the axon to the soma
(Eq. 2) under different membrane conditions, as characterized by
The result shows that TD increased as the axon was moved
distally from the soma. Furthermore, the decrease of
or the increase of for a given axon location generated a larger
TD.It is clear that TD, directly related to
the peak ITD, was affected jointly by the properties of the soma and
dendrites. This analysis suggested means to increase the peak ITD shift in the
passive membrane condition (EE) (e.g., reducing the soma size to decrease
Rs or decreasing the dendrite diameter to increase
R produced a larger TD). Varying
the value of implied different cytoplasmic properties between the soma
and dendrites. These parameter effects were studied in modeling exercises.
The increase of TD with the distance of the axon to the
soma in Figure 10 conflicted
with the simulation results in Figure
9. The discrepancy resulted from the simplification in the delay
analysis we applied: each dendrite had one distal input in the analysis (Eq.
2) instead of distributed ones as in the computational model. For inputs
closer to the soma, TD was actually determined by the location
of the input (Eq. 3). For distributed inputs, the study of TD
between two compound potentials arriving at the axon would have to take into
account not only transfer delays but also transfer impedances of individual
inputs (Zador et al., 1995;
Koch, 1999). The latter
determines the contribution of a single input to the received signal.
Therefore, the result in Figure
10 merely indicates the effect of membrane electrical properties
on the relative size of TD.
It is possible that the asymmetrical cell structure we explored was only
one of several factors that could introduce a delay difference between
ipsilateral and contralateral inputs. In the bipolar model, the two dendrites
had identical electrical properties and spatial segregations of excitatory
synapses. However, the length and branching patterns of dendrites vary
considerably among cells in the MSO and NL
(Smith and Rubel, 1979;
Henkel and Brunso-Bechtold,
1990; Smith,
1995). A transfer delay difference could be generated simply by a
disparity in the location of excitation on the two dendrites relative to the
axon. It can be speculated that the bilateral difference in dendrite cytology,
the spatial locations of excitatory synapses, and the ionic channel
distributions might enable a virtual asymmetric system, regardless of the
actual cell shape. The more complicated ITD computations that would result
from such a system shall be explored as more detailed physiological
measurements become available.
Effects of excitation level
In general, discharge rates of cells in the auditory nerve and in the AVCN
increase monotonically with sound level
(Kiang, 1965;
Joris et al., 1994). It is
reasonable to speculate that in the MSO the strength of synaptic excitation
coming directly from the AVCN increases with level as well. Theoretically, the
average strength of EPSPs grows proportionally to the product of the peak
conductance of a synapse (Ge), the number of input fibers,
and the average input rate of each fiber (Rave). For
simplicity, we used Ge to explore changes in the
excitatory drive as the sound level varied at each ear. We tested the
robustness of the three-step mechanism proposed in this study to variations in
excitation level (Fig.
11).
In Figure 11A, the
strength of excitation (Ge = 9 nS) was 20% smaller than
that in Figure 6 for 500 Hz
inputs (Ge = 11 nS). The results show that the ITD
functions in this case exhibited similar patterns of shifts by activations of
sodium currents and inhibition. Compared with
Figure 6 for 500 Hz inputs, the
decrease of Ge reduced the overall discharge rates in
Figure 11A,
especially the trough rates, resulting in narrower response curves to ITDs.
The values of Gi and Ge were the same
as those in Figure 6 for 500 Hz
inputs. When the strength of excitation was increased above
Ge = 11 nS (data not shown), the overall rates increased,
and the patterns of shifts in the ITD functions still remained.
Although the model had identical parameters for excitation on the two
dendrites, the ipsilateral EPSPs were much larger than the contralateral ones
because of the difference in their path lengths
(Fig. 9). The strength of
ipsilateral EPSPs as a result determined the discharge rate at each ITD. This
observation raised the question of whether stronger contralateral EPSPs, which
could result from interaural level differences, would diminish the shift of
the peak ITD in the passive membrane condition and thereby nullify the role of
inhibition and sodium currents. In Figure
11B, the strength of contralateral excitation was
increased and the strength of ipsilateral excitation was decreased
(Ge = 30 nS for contralateral; Ge = 6
nS for ipsilateral) such that the contralateral and ipsilateral compound EPSP
measured at the axon had the same amplitude (data not shown). The shifts in
the ITD curves of Figure
11B indicate that the mechanism proposed here is not
affected by amplitude differences between bilateral excitations.
Moreover, the amount of the shift in
Figure 11 (EE) was insensitive
to the level of excitation. The analysis of the transfer delay difference in
Figure 10 provided a possible
explanation; the delay difference only depended on the structure and not the
shape of the input signal (Agmon-Snir and
Segev, 1993). In a passive membrane with closely distributed
inputs, the peak time of a compound EPSP, which is a sublinear
(Agmon-Snir et al., 1998)
summation of EPSPs with various shapes, can potentially vary with sound level,
although this was not observed in our simulations.
Because the position of the peak ITD under the influence of inhibition and
sodium currents depended on their strengths
(Fig. 8), the model could have
level-dependent ITD curves in Figure
11 if arbitrary level-dependent inhibition and sodium current
activities had been used. Hence, simulations here without the knowledge of
these dependencies cannot exclude the possibility that changes in sound level
will influence the ITD sensitivity of this model.
Somatic potassium channels
The somatic membrane of the bipolar model did not include outward potassium
channels. Both low-threshold (ILTK) and high-threshold
(IHTK) potassium channels, however, are present in the MSO
and the NL (Smith, 1995;
Grigg et al., 2000). To test
the robustness of the model, we added potassium channels and a
hyperpolarization-activated inward current (Ih) to the
soma with identical parameters to those on the axon. The density of sodium
channels on the soma was increased to offset these potassium currents.
Figure 12 shows rate-ITD
functions with potassium channels on the soma and with the three other
membrane conditions presented above. The activation of potassium channels
(EE+K) decreased the overall discharge rates, but the peak ITD remained the
same as in the passive condition (EE). Furthermore, sodium currents still
centered the peak ITD (EE+Na+K), and inhibition moved the peak back toward the
contralateral-leading side (EE+Na+K+I). Potassium currents and synaptic
inhibition, both of which hyperpolarized the membrane, exhibited similar
functional roles in balancing sodium currents on the soma. We concluded that
the direction of the peak ITD variation by active currents (EE+Na and EE+Na+I)
depends mainly on the total volume and the dynamics of anionic and cationic
flows across the somatic membrane. The exact complement of such currents
requires additional theoretical and experimental exploration and
confirmation.
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Discussion
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The present study focuses on whether coincidence detection in an MSO cell
can be influenced by its asymmetric morphology. The role of cell morphology on
ITD sensitivity was first computationally explored in the NL by Agmon-Snir et
al. (1998). Most modeling
studies have simplified MSO cells to point-neuron models, which ignore cell
morphology. Point models necessitate strong and potentially unrealistic
requirements on the timing and duration of inhibition to replicate
(Fig. 2) the in vivo
data of Brand et al. (2002).
Our alternative model (Fig. 4)
included the asymmetrical structure of the cell and the interactions of
excitation and inhibition within this structure and simulated the in
vivo data using realistic model parameters
(Fig. 6). In this model, active
sodium currents and synaptic inhibitory currents were opponent forces that
shifted the ITD function in opposite directions
(Fig. 8). These mechanisms
would also allow dynamic descending and/or ascending control over
ITD-sensitive cells in the MSO.
Time constant of synaptic inhibition
The point model relies on a short-duration inhibition (i =
0.1 ms, similar to excitation) to produce the observed shift in the ITD curve
(Brand et al., 2002). However,
for postsynaptic currents, the observed value of i is 2
ms and e is 1 ms, as estimated from voltage-clamp
measurements in the MSO (Smith et al.,
2000) (adjusted to 37°C; Q10 = 2.1). For
postsynaptic potentials, the observed duration of the IPSP is also on the
order of milliseconds and is longer than that of the EPSP
(Sanes, 1990;
Smith, 1995).
Of course, in vivo neurons may have more precise synaptic
responses through presynaptic and/or postsynaptic mechanisms than the slice
preparation. However, the time window suggested for the effective inhibition
(1 ms) in in vivo experiments
(Joris and Yin, 1995) is not
equal to the duration of the IPSP (Fig.
3). More direct evidence concerning the time course of the
inhibitory currents from in vivo experiments in the adult MSO is
needed.
Finally, even if short-duration inhibition in the MSO exists, the model of
Brand et al. (2002) also
requires precise timing between inhibitory projections from the MNTB and
excitatory projections from AVCN bushy cells contralaterally onto each MSO
cell (Fig. 2B).
Experiments are required to determine this timing relationship between
contralateral excitation and inhibition.
Asymmetrical cell structure
The potential for the asymmetrical axon of MSO cells to contribute
passively to ITD tuning was first raised by Brew
(1998). Our studies on the
propagation of EPSPs indicate that the passive properties of the cell body and
the location of the axon indeed affected delays and attenuations of the two
excitatory inputs (Fig. 9). As
a result, the best ITD attributable to the passive system was not zero ITD for
the asymmetric cell (Fig. 6,
EE). However, the passive model of the asymmetric cell can only explain the
peak-ITD shift by the transfer delay difference between excitations and is
thus not adequate to explain the observed responses of MSO cells after
blocking inhibition. The opponency between sodium currents and inhibitory
currents ad
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Abstract
Introduction
