1Howard Hughes Medical Institute, Computational Neurobiology Laboratory, The Salk Institute, La Jolla 92037; and 2Department of Physics, 3Neurobiology Unit, Scripps Institution of Oceanography, 4Department of Neuroscience, and 5Department of Biology, University of California, San Diego, La Jolla, California 92093
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ABSTRACT |
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Moortgat, Katherine T.,
Theodore H. Bullock, and
Terrence J. Sejnowski.
Gap Junction Effects on Precision and Frequency of a Model
Pacemaker Network.
J. Neurophysiol. 83: 984-997, 2000.
We investigated the precision of spike timing in a
model of gap junction-coupled oscillatory neurons. The model
incorporated the known physiology, morphology, and connectivity of the
weakly electric fish's high-frequency and extremely precise pacemaker nucleus (Pn). Two neuron classes, pacemaker and relay cells, were each
modeled with two compartments containing Hodgkin-Huxley sodium and
potassium currents. Isolated pacemaker cells fired periodically, due to
a constant current injection; relay cells were silent but slightly
depolarized at rest. When coupled by gap junctions to other neurons, a
model neuron, like its biological correlate, spiked at frequencies and
amplitudes that were largely independent of current injections. The
phase distribution in the network was labile to intracellular current
injections and to gap junction conductance changes. The model predicts
a biologically plausible gap junction conductance of 4-5 nS (200-250
M). This results in a coupling coefficient of ~0.02, as observed
in vitro. Network parameters were varied to test which could improve
the temporal precision of oscillations. Increased gap junction
conductances and larger numbers of cells (holding total junctional
conductance per cell constant) both substantially reduced the
coefficient of variation (CV = standard deviation/mean) of relay
cell spike times by 74-85% and more, and did so with lower gap
junction conductance when cells were contacted axonically compared with
somatically. Pacemaker cell CV was only reduced when the probability of
contact was increased, and then only moderately: a fivefold increase in the probability of contact reduced CV by 35%. We conclude that gap
junctions facilitate synchronization, can reduce CV, are most effective
between axons, and that pacemaker cells must have low intrinsic CV to
account for the low CV of cells in the biological network.
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INTRODUCTION |
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Individual neurons from the pacemaker nucleus (Pn)
of certain weakly electric fish generate spikes with far greater
temporal precision than any other neurons known (Moortgat et al.
1998, 2000
). In addition, the precision in the weakly electric
fish can change spontaneously and can be modulated by behavioral
stimuli (Moortgat et al. 1998
). It has been proposed
that the high precision results from gap junction coupling among
neurons in the Pn. In this study, we model the Pn neurons to test how
their precision and its modulation are affected by network coupling,
and what intrinsic cellular precision would be required. We constrain
our compartmental model with experimental data on Pn connectivity and
neuron morphology (Dye and Heiligenberg 1987
;
Elekes and Szabo 1985
) and responses to intracellular
current injections (Dye 1991
; Juranek and Metzner
1998
; Moortgat et al. 2000
).
Our study differs from previous research that has largely emphasized
mechanisms for frequency locking and synchronization between cell
units, often with the simplifying assumption of all-to-all coupling
(Chow and Kopell 1999; Ernst et al. 1995
;
Hansel et al. 1995
; Matthews and Strogatz
1990
; Sherman and Rinzel 1991
). Some modeling
studies have focused explicitly on the temporal precision of
oscillations and neural spiking (Enright 1980a
,b
;
Shinbrot and Scarbrough 1999
) but have concluded that
thousands of cells are required for high precision. The Pn model
presented here is based directly on a biological pacemaker network made
up of only 150 rhythmically firing, sparsely coupled neurons whose
behavioral importance is well understood.
The medullary Pn commands the timing of the weakly electric fish's
electric organ discharge (EOD), which is part of the fish's active
electrosensory system. The EOD is an electric dipole field which, in
"wave-type" fish, oscillates at a frequency of 60-2,000 Hz with a
coefficient of variation (CV = standard deviation/mean period) of
2 × 104 corresponding to a standard
deviation of the period (SD) as low as 0.14 µs (Bullock
1970
; Bullock et al. 1972
; Moortgat et
al. 1998
). Both the high frequency and the low CV are
maintained throughout the lifetime of a fish, but each can be modulated
in behavioral contexts as well as pharmacologically (frequency:
Dye 1987
; Hagedorn and Heiligenberg 1985
;
Heiligenberg et al. 1981
; Keller et al. 1991
; CV: Moortgat et al. 1998
). Weakly electric
fish determine the locations of objects by evaluating relative phase
and amplitude shifts at the electroreceptors that cover its body
(Heiligenberg 1991
; von der Emde et al.
1998
). The low SD of the emitted electric field oscillations
may be crucial to the fish's ability to make phase discriminations as
small as 0.40 µs (Carr et al. 1986
).
When the Pn is cut away from the brain stem, its neurons continue to
fire at the same frequency (Meyer 1984) and with the same precision (Moortgat et al. 2000
) as in vivo. The
adult Pn is comprised of 100-160 neurons, coupled solely via
axosomatic, axoaxonic, and axodendritic gap junctions (Dye and
Heiligenberg 1987
; Elekes and Szabo 1985
;
Moortgat et al. 2000
). The importance of different gap
junction locations is not known and cannot readily be tested physiologically.
A Pn network model can thus be particularly tractable, with just 150 resistively coupled neurons. We show that while relay cell CV is substantially reduced by network coupling, the same does not hold for pacemaker cells. Taking this result in combination with the Pn physiology, we conclude that the high precision of pacemaker cells must largely result from single-cell rather than network properties. We also compare axosomatic to axoaxonic gap junction coupling and find the latter to have enhanced effects on precision and frequency.
Earlier versions of this work were included in a PhD thesis
(Moortgat 1999).
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METHODS |
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Model neurons
Because Pn neurons showed important voltage activity both above
and below spike threshold (Moortgat et al. 2000), we
used model neurons described by Hodgkin-Huxley equations
(Hodgkin and Huxley 1952
). The neuronal simulation
package, NEURON (Hines 1993
), was used to model two Pn
neuron types: the pacemaker and relay cells, and their electrotonic
interconnections as seen in the Pn of the weakly electric fish
Apteronotus leptorhynchus. Each model cell had two
compartments: one somatic and one axonic. Dendritic compartments were
not included despite anatomic evidence for extensive relay cell
dendrites (Dye 1991
; Elekes and Szabo
1985
) with gap junctions (Moortgat et al. 2000
),
because the dendrites are thought to be involved primarily in synaptic
integration, being covered with chemical synaptic boutons from higher
brain centers (Elekes and Szabo 1985
). These higher
centers modulate spike frequency but are not required for the
continuous, phase-locked oscillations in the Pn.
The two model cell types are distinguished in part by their
morphology, as seen in fixed and live Pn tissue (Dye
1991; Elekes and Szabo 1985
; Moortgat et
al. 2000
). Model pacemaker cells have somata of 30 µm diam,
and cylindrical axons of 8 µm diam and 45 µm length, whereas the
larger model relay cells have somata of 65 µm diam and cylindrical
axons of 7 µm diam and 40 µm length. The length of the model
cell's axon was chosen to roughly match the length of the biological
cell's axon initial segment. We did not aim to model the full,
branched axon or action potential propagation through it.
The model parameters used in each compartment for each cell type are
listed in Table 1. Somatic leak currents
were set to give input resistances of 20 and 5 M, respectively, for
isolated pacemaker and relay cells, within the experimentally measured range for neurons in the intact Pn (Dye 1991
;
Juranek and Metzner 1998
). The current and voltage
dynamics of the two active conductances included in the model, sodium
and potassium, were calculated using the Hodgkin-Huxley equations
(Hodgkin and Huxley 1952
)
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(1) |
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To give each pacemaker cell an "intrinsic" spiking frequency, at which it fired without being coupled to other cells, a constant current of 1 nA was injected into each soma. The injected current took the place of a pacemaker current that would produce a pacemaker potential, as observed in the Pn pacemaker cells. Larger or smaller injected currents led to higher or lower spiking frequencies for these isolated cells. The 1-nA value was chosen so the spike frequency (~612 Hz) would fall in the biological range for the species studied (500-900 Hz). The relay cells also received a constant current injection of 0.5 nA, which depolarized them but did not bring them to spike threshold. Only with inputs from pacemaker cells did the relay cells spike at the pacemaker frequency. Larger current injections into relay cells further depolarized the membrane potential but did not cause repeated spiking; instead, the membrane potential resonated but did not repolarize fully, presumably because of a low K+ channel density. In some simulations, the current injected was randomized between cells (giving each cell a different but fixed intrinsic frequency or interspike interval) and/or randomized over time (making the interspike interval slightly different for each interval).
Model network
Model networks contained a 4:1 ratio of pacemaker and relay
cells. The total number of neurons in the model was varied from 50 to
200. Unless otherwise specified, the model contained 150 neurons, with
120 pacemaker and 30 relay cells. Model neurons were coupled by
resistive (gap junction-like) connections, such that the current
Igap between connected cells was
proportional to the difference in the cells' membrane potentials, and
pre: Igap = ggap × (
pre), where
ggap is the gap junction conductance. Each pacemaker axon contacted 35% of relay cells and 7% of other pacemaker cells, chosen randomly (Fig.
2A). For a network of 150 neurons, each pacemaker cell contacted an average of 18.5 cells total.
This resulted in a mean of 42 and 8 contacts, respectively, received by each relay and pacemaker cell (Fig.
2B).
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The gap junction contacts were rectified, with current flowing from a
pacemaker cell axon into a contacted cell's soma or axon. That is, gap
junction connections were axosomatic or axoaxonic, depending on the
simulation, and only passed current when the contacted cell's voltage
was lower than that of the cell contacting it. It is not known whether
gap junctions in the electric fish's Pn are rectified. The biological
gap junctions, however, occur at long distances along multiple axon
branches from the axon initial segment, and the rectified gap junction
would seem to best model this. An unrectified antidromic subthreshold
signal would decrement to insignificance over the long axonal distance.
Gap junction conductance varied between simulations from 0.5 to 10 nanosiemens (nS) per contact, equivalent to a resistance of 100-2000
M.
Simulations of the full network, using Euler integration with a time step of 1 µs, took ~100 s real time per ms of simulated time (Dec Alpha server 2100/300). The first 10 ms of each simulation was considered to be settling time and was discarded from our analysis. Voltage measurements were made at the soma, unless otherwise indicated.
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RESULTS |
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We tested how well the network model yielded the
electrophysiological results (Moortgat et al. 2000) and
used the model to make predictions for future physiological studies.
The general strategy was to explore the network properties with the
given parameter values, not to search for the best parameter values for
each simulation.
Spike shape and amplitude depend on cell type
Isolated model pacemaker cells fired periodically with a mean intrinsic frequency of 612.5 Hz. Relay cells had no spontaneous rhythm, but had a resting membrane potential close to spike threshold (Fig. 1A). When the pacemaker and relay cells were coupled together with axosomatic gap junctions of 5 nS conductance, they caused a voltage deflection in the passive relay cell soma that reached the axon with sufficient amplitude to cause the axon to spike (Fig. 1B). The relay cell action potential rose abruptly from the minimum membrane potential, whereas the pacemaker cell showed a prolonged "pacemaker potential" before the spike onset. Another difference between the cell types was the brief "shoulder" in the relay somatic waveform (Fig. 1C), which occurred at about one-half the oscillation amplitude in the form of a decrease in the voltage slope. Adding small somatic conductances, at least up to 10% of axonic conductances, did not remove the shoulder, but did increase the amplitude of the somatic voltage oscillations from 50 to 85 mV. However, model relay cells that had a more depolarized intrinsic membrane potential, did not show the shoulder. Also, when axosomatic gap junctions were replaced with axoaxonic ones, the shoulder disappeared. Thus the shoulder in the passive somatic waveform of these cells appeared to result from the delay between somatic gap junction input and the antidromic axonal spike arriving at the soma.
Nonzero phase lag between cells in the model network
The model relay cell in Fig. 1, B and C, fires with a phase delay after the pacemaker cell. We calculated the phase lag, relative to the average spike peak time of the pacemaker cells, for each neuron in the model network, and report it as a percent of the interspike period. Simulations began with all neurons at zero phase lag, but the phases changed to new and stable values within a few interspike periods for moderate values of the gap junction conductance (3-6 nS). With conductances <2 nS, the phases distributed only after tens of spike periods. As the phase distribution widened to its steady state, the amplitude of passive voltage oscillations in a relay cell increased (one example relay cell is shown in Fig. 3). Thus the widened phase distribution appeared to increase the gap junction current that the relay cell received. Gap junctions pass no current when the voltage across them is zero, as can occur when the phase lag is zero, and pass increasingly large currents as the voltage difference (and phase lag) increases. See DISCUSSION for one interpretation.
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Bimodal phase distribution narrows with increased gap junction conductance
All model relay cells fired with a delay relative to the pacemaker cells. The phase delay was cell specific, with a value that depended on the gap junction conductance (ggap; Fig. 4A). At low conductance values, spiking relay cells had larger phase delays, but not all relay cells fired action potentials. For example, for ggap = 2.5 nS, spiking relay cells lagged the pacemaker cell by an average 20% (range 18-21%), but only 9 of 30 relay cells were spiking. Doubling the gap junction conductance (ggap = 5.0 nS) reduced the phase lag between relay and pacemaker cells to an average 12.1% (range 9.5-15.7%) and recruited all relay cells into the network spiking rhythm (Fig. 4B). If the relay cells received an "intrinsic" depolarization that was larger than specified in METHODS, then a smaller ggap was required to recruit all relay cells with a 10-15% phase lag.
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Phase distribution: axoaxonic versus axosomatic coupling
The phase lag between model pacemaker and relay cells best matched
the biologically observed distribution when axosomatic gap junctions
had 5-nS conductance (equivalent to a resistance of 200 M). However,
networks coupled with axoaxonic coupling required only 4 nS (250 M
resistance) gap junction conductance to attain a similar phase lag
(mean of 11.4%). Axoaxonic coupling also recruited all relay cells
into the network oscillations at a lower conductance (Fig.
4C) than did axosomatic coupling (Fig. 4A). In
addition, axoaxonic coupling of 4-nS conductance narrowed the phase
distribution within the relay cells (9.7-12.8%, Fig. 4D)
relative to axosomatic coupling of 5 nS (9.5-15.7%, Fig.
4B). Further network simulations use the value of
ggap that gave the best match with
biological phase distributions; namely, 5 nS for axosomatic or 4 nS for
axoaxonic network coupling. Thus an average relay cell in a network
with axosomatic coupling will have a total gap junction conductance of
210 nS (mean of 42 contacts: see METHODS), whereas a
pacemaker cell will have on average 40-nS total gap junction
conductance. These values are relatively small compared with the
conductance between a relay cell's axon and soma (750 nS) or a
pacemaker cell's axon and soma (4,500 nS). The increased membrane
conductance through gap junctions leads to input resistances of 13 and
2.5 M
, respectively, for pacemaker and relay cells in a network with
axosomatic gap junctions of 5 nS.
Coupling coefficients between coupled cells are small
We measured the coupling coefficients between axosomatically
coupled pacemaker and relay cells for the gap junction conductance determined above. The coupling coefficient is the ratio of the passive
voltage deflection in one cell (V2)
in response to a voltage change
(
V1) in the coupled (presynaptic)
cell
(
V2/
V1). This measure could not be directly applied to the model cells because
they spiked rhythmically with short interspike intervals such that
their voltages continuously oscillated. We measured the coupling
coefficient as the ratio of the minimum somatic voltage during the
spike's repolarization phase (trough) before and during a constant
current injection in the presynaptic cell's soma. The minimum voltage
in one model pacemaker cell shifted with injected current. We expected
a substantial shift in a directly coupled cell's minimum voltage, but
found only a small one. The coupling coefficients were ~0.02 and
0.025 for two different pacemaker cells contacting a single relay cell.
Even simultaneously injecting current into two pacemaker cells that
were coupled to a relay cell caused the latter's minimum voltage to
shift by only 1.5 mV, whereas that of the directly injected pacemaker
cells changed by 14 and 42 mV, respectively. Similarly small coupling
coefficients are consistent with in vitro results (Moortgat et
al. 2000
) and were observed even when the model relay cell's
active conductances were removed, making the cell entirely passive.
Thus the currents from each individual gap junction make only small
voltage deflections in the postsynaptic cell. However, the sum of all
gap junction inputs to a cell can be substantial, as shown below.
Axosomatically coupled cells: responses to intracellular current injection
Pacemaker and relay cells, coupled by axosomatic gap junctions of
5-nS conductance, were injected somatically with step currents of
various amplitudes. Moderate somatic current injection (4 to +2 nA)
into one model relay cell linearly altered the phase lag of the
injected cell's spikes relative to those of a model pacemaker cell
(Fig. 5A) and changed the
spike amplitude from 69 to 50 mV (Fig. 5B). Currents of up
to ±2 nA varied the peak voltage of the spike by only 2.5 mV (Fig.
5C). Large depolarizing currents (>4 nA) reduced the slope
with which the phase lag changed as current increased, whereas large
hyperpolarizing currents (
4 to
10 nA) increased the slope. For
these large currents, the amplitude of relay cell oscillations
continued to decrease, though only slightly, with more positive
current. Hyperpolarizing currents larger than
13 nA caused
the relay cell's oscillation amplitude to drop to 48 mV, where it
remained, even with further increased hyperpolarizing current. These
massive hyperpolarizing currents also reduced the peak spike voltage
and caused it to decrease more rapidly with more negative current. The
current amplitude required for each of these effects varied from cell
to cell, with a relay cell reaching a constant amplitude oscillation
with as little as
7 nA or as much as
14 nA. The larger currents
were required for relay cells that received more gap junction contacts. Compared with relay cells, pacemaker cells had qualitatively similar but more sensitive responses to current injections. For example, a
pacemaker cell's oscillation amplitude reached a fixed value with only
2.5 nA. None of the intracellular current injections perturbed the
frequency of the injected cell's voltage oscillations away from the
model network frequency, just as observed in vitro (Moortgat et
al. 2000
).
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Axoaxonically coupled cells: responses to current injection
In the same network, now coupled with axoaxonic gap junctions of 4 ns conductance, the phase lag of the same relay cell was more resistant
to the somatic step current injections (Fig. 5D). That is,
the phase lag during injection of 10 nA was only 17.8%, compared
with 20.3% in the axosomatically coupled network. Also, currents of
4-10 nA did not alter the phase lag from 8% in the axoaxonically
coupled network. The relay cell in this network more than halved its
oscillation amplitude (from 68 to 31 mV; Fig. 5E), and
dropped its peak voltage from
39 to
96 mV (Fig. 5F) with
a shift in injected current from
10 to
14 nA. Despite these
differences between the relay cell in the two network configurations, the qualitative responses were similar.
Passive and active membrane voltage oscillations
The reason for the abrupt changes in the relay cell's current
response became clear by comparing the cell's normal (Fig. 5, E and F, ) to its passive response (Fig. 5,
E and F,
). The relay cell's response to
currents of
14 nA or more negative, and of 4 nA or more positive, was
predominantly passive. That is, the Na+ and
K+ conductances were not activated. During these
current injections, the somatic membrane voltage continued to
oscillate; for
14 nA, oscillation amplitudes were 48 mV for
axosomatic and 31 mV for axoaxonic coupling, roughly the same
amplitudes seen in the passive cell at rest
(Iinj = 0). Thus large membrane
oscillations in the highly hyperpolarized or depolarized cell were due
entirely to gap junction currents. The active membrane processes of
this relay cell shut down between
10 and
14 nA, causing the
transition from larger amplitude action potentials to smaller amplitude
(but still sizeable) voltage oscillations caused by gap junction
inputs. Apparently the axoaxonic gap junction inputs failed to bring
the cell to spike threshold when the cell received massive
hyperpolarizing current, and only brought about passive subthreshold
membrane oscillations. The amplitudes of the passive oscillations were significantly less voltage dependent than the active processes.
To understand the difference in the relay cell response between the two types of network coupling, consider the relative location of the recorded voltage signal and the gap junctions. The voltage was recorded in the soma. Gap junction inputs at the relay soma caused larger amplitude somatic membrane oscillations than inputs at the axon because the signal was not dampened by the axon's high axial resistance. Hence, the passive somatic oscillation amplitude was larger for axosomatic than axoaxonic coupling. Similarly, the peak somatic voltage of the passive oscillation was greatest when the currents driving the oscillation arrived directly at the soma, rather than resistively through the axon (compare Fig. 5, C and F).
Passive oscillations can alternate with spikes within a narrow range of hyperpolarizing current
In the narrow transition between the full amplitude spikes and passive oscillations (Fig. 5), model pacemaker and relay cells could spike in integral ratios of the network frequency. The most common and stable pattern was one spike for every two cycles of the model Pn network, in a 1:2 ratio (as shown in Fig. 6, A and B). Lower ratios, including 1:3, 1:4, and 1:5, were also observed in both pacemaker and relay cells. In some cases, lower ratios settled into a 1:2 ratio with continued (beyond ~20 ms) current injection. Other cases of low spiking ratios were sustained indefinitely, with each oscillation within the pattern having a lower amplitude until a minimum was reached. After the minimum amplitude oscillation, the next oscillation had the full spike amplitude. Only a narrow range of currents, typically within ±0.5 nA for a relay cell and ±0.1 nA for a pacemaker cell, caused the cells to alternate between passive oscillations and spikes in any ratio.
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To confirm that the lowest amplitude oscillations were indeed the cell's passive response to gap junction input, we removed the cell's active conductances. The membrane potential during the low-amplitude oscillation had the same amplitude and peak voltage when the cell's active properties were removed. Thus the low-amplitude membrane oscillations were truly "skipped spikes": the cell's passive response to gap junction input, and had no active membrane component.
Differences in peak voltage alternations between relay and pacemaker cells
Pacemaker and relay model cell waveforms differed somewhat during
these hyperpolarizing current injections. These differences mirrored in
vitro observations (see Moortgat et al. 2000, Figs. 5 and 6 and section
entitled Skipped spikes during massive current injection).
First, the pacemaker cell's passive membrane oscillation had a
significantly smaller amplitude than the relay cell's. This difference
reflected the larger number of gap junction contacts onto the relay
cell than onto the pacemaker cell, and hence the smaller gap junction
currents. Second, the model pacemaker and relay cell waveforms differed
in the minimum membrane potential reached during the repolarization
phase after active spikes and passive oscillations. Specifically, the
relay cell's minimum membrane potential was more hyperpolarized after
a passive membrane oscillation than after an action potential. In
contrast, pacemaker cells usually had the same minimum voltage after a
spike as after a passive oscillation. In a few model pacemaker cells,
the minimum membrane potential was more hyperpolarized after a full
spike than after a passive oscillation, the opposite of relay cells.
The cause for the differences between pacemaker and relay cell minimum
voltages is not known, but is consistent with biological observations.
Similar alternations between passive oscillations and active
action potentials were seen in a network with axoaxonic coupling. In
this case, the relay cells required more hyperpolarizing current (16
nA for a cell that required
12 nA with axosomatic coupling) to spike
in a 1:2 locking with the network oscillations. The range of
hyperpolarizing currents that caused the alternations was even narrower
in networks with axoaxonic coupling than in those with axosomatic coupling.
How should noise be added to the deterministic model?
We further tested which network parameters had the strongest
effect on the spike timing precision, as measured by the coefficient of
variation (CV = standard deviation/mean) of each cell's
interspike intervals. First, the time step of the simulation was set to
t = 1µs to allow detection of CVs >3 × 10
4, within the range observed physiologically
(Moortgat et al. 2000
). Then, a time-varying Gaussian
noise (updated every time step for each cell) was added to the
deterministic model. Two sites for the noise were considered: the
conductance of the leak current and the constant current that sets each
cell's intrinsic frequency. The former proved inappropriate because of
its limited dynamic range. That is, the standard deviation had to be
approximately equal to the mean leak conductance to produce a
sufficiently high CV. In addition, adding noise to the leak conductance
implied a voltage-dependent noise, an unnecessary complication.
Time-varying current noise increases cell CV
Time-varying Gaussian noise was added to the current that sets
each cell's intrinsic frequency. An isolated pacemaker cell, i.e., a
single cell outside the network, was injected with this noise current.
A range of standard deviations of the noise current was explored, and
the resulting CV of interspike intervals was measured (Fig.
7A). The CV was limited by
t = 1µs for noise currents with SD less than
~0.02 nA, but increased with larger SDs of the injected current. For
an isolated pacemaker cell to spike with CV = 10
3 or 3 × 10
3,
the injected noise had, respectively, SD = 0.1 and 0.2 nA.
|
Specification of the noise used in subsequent simulations
In the following simulations, pacemaker cells that had CVs of
~3 × 103, due to cell-dependent,
time-varying injected current noise of SD = 0.2 nA, were coupled
to relay cells injected with current of the same SD. These neurons had
independent noise that was updated every time step, and were coupled
with axosomatic or axoaxonic gap junctions.
Increased gap junction conductance: effects on CV and frequency
We investigated the effects of gap junction conductance on the
spike timing precision of relay and pacemaker cells. A network of 150 neurons (120 pacemaker and 30 relay cells, each with independent noise
of the same SD = 0.2 nA) was coupled with axosomatic gap junctions. Gap junction conductances
(ggap) of 1 nS failed to drive relay
cells to the firing frequency of the pacemaker cells (612.5 Hz). Some
relay cells fired every other pacemaker cycle in the same pattern
described above for the hyperpolarized cell in a network of normal
ggap (5 nS). For
ggap = 1.5 nS, many (20/30) relay
cells fired at the pacemaker frequency with CVs ranging 16-70 × 10
4 and a mean CV of 33 × 10
4 (Fig.
8A). As the gap junction
conductance increased, the minimum CV of all relay cells decreased, and
the mean and range of the CVs also shifted to lower values (mean of
4.9 × 10
4, range 3.6-6.9 × 10
4 at ggap = 5 nS). The relay cell CVs were not further reduced with
ggap values above 5 nS due to the
simulation time step, and therefore are not shown in the figure. The
pacemaker cells in the same network with axosomatic coupling (not
shown) had a minimum CV of 18 × 10
4 (mean
of 26 × 10
4) at 1-nS conductance. This
minimum reduced only to 16 × 10
4 (range
16-35 × 10
4, mean of 26 × 10
4) with a fivefold increase in conductance.
There was no concurrent change in the firing frequency of the pacemaker
or relay (Fig. 8B) cells. Giving pacemaker cells a
distribution of intrinsic frequencies (with at least 10% variance in
the current injected to set the cell frequency) did not change either
cell type's response to increased gap junction conductance.
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In a network with axoaxonic coupling and independent noise currents in
each model cell as described above, all relay cells fired at the
pacemaker frequency at a gap junction conductance as small as 1 nS.
Again, increased ggap reduced the
relay cell CVs (Fig. 8C) in their minimum (from 16 × 104 to 4 × 10
4)
and mean from 28 × 10
4 to 5 × 10
4). Beyond ~3 nS, the CV of relay cells no
longer decreased, having reached the simulation's limit of resolution.
The axoaxonic coupling among pacemaker cells only somewhat reduced
their CV from a mean of 26 × 10
4 to
22 × 10
4 (Fig. 8E), such that
the slope of CV against conductance was much lower than among the relay
cells in this network. Increased ggap
did increase the frequency of the relay cells (Fig. 8D), because of the pacemaker cells' increased frequency (Fig.
8F).
Other effects of gap junction conductance
The increased gap junction conductance, in both network
configurations, had other effects as well. As mentioned above, the phase lag between model pacemaker and relay cells decreased with increasing gap junction conductance. In addition, the relative phase
could shift between pairs of cells of the same type, even changing in
sign. That is, one pacemaker cell could phase lag another at one value
of ggap, but phase lead it at another
value. This type of shift in relative phase, with sign changes, was
observed in the biological network (Moortgat et al.
2000). Gap junction conductance also modulated the apparent
somatic spike amplitude. This effect was most pronounced in an
electrically passive model relay soma.
Network size: effects on CV
To determine whether the network size affects the firing precision or frequency of noisy cells (noise as described above), we studied networks with different numbers of cells, always in a 4:1 ratio of pacemaker to relay cells that were coupled with the usual probabilities. Increasing the numbers of neurons while maintaining constant coupling probabilities results in each cell receiving a larger number of contacts on average, causing a greater gap junction drive. We normalized the gap junction conductance for the average number of contacts received by relay cells to maintain the same conductance and thus distinguish the effects of increased numbers of inputs from the effects of increased gap junction current drive (previous section).
The total number of cells in a network was varied from 50 to 200, the
approximate range observed in the biological pacemaker nucleus. Each
cell in the model network was injected with an independent noisy
current of SD = 0.2 nA, and coupled with axoaxonic gap junctions of normalized conductance. The relay cells became more precise, with
the average CV decreasing from 19 × 104
to 5 × 10
4 as the network grew from 50 to
200 cells (Fig. 9A). This CV
decrease among relay cells was consistent with the law of large numbers (Helstrom 1991
) for the number of contacts received.
Namely, the CV decreases by
1/
, where
Nr is the average number of contacts
received by a relay cell. On the other hand, the pacemaker cells did
not become more precise with increased numbers of cells in the network
(Fig. 9C). Even taking into account the low probability of
pacemaker-to-pacemaker cell contacts (0.07) compared with the
probability of pacemaker-to-relay cell contact (0.35), and that the
pacemaker cells therefore receive fewer contacts, the law of large
numbers still predicted a larger CV decrease for pacemaker cells than
the model demonstrated (Fig. 9C). Varying the network size
from 50 to 200 cells did not impact the spiking frequency of either
relay (Fig. 9B) or pacemaker cells (Fig. 9D), which remained fixed between 612.5 and 612.9 Hz. This result reflects the successful normalization of ggap
by the number of cells in the network. Networks made up of <50 neurons
were not included in the figures because some cells received no
contacts and did not spike.
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The network size also failed to reduce pacemaker cell CV when cells were coupled with axosomatic gap junctions and when the SD of the injected noise was arbitrarily halved (SD = 0.1 nA). With axosomatic coupling, networks of all sizes had spiking frequencies of ~612.7 Hz, slightly lower than the 613 Hz seen with axoaxonic coupling. The relay cell CV decreased with the network size, but less so when cells were coupled axosomatically than when coupled axoaxonically.
Contact probability: effects on CV and frequency
Although increasing the two global network parameters, the gap junction conductance ggap, and the number of cells substantially reduced relay cell CV, they failed to reduce the CV of pacemaker cells. To determine why the pacemaker cells were not changing CV while the relay cells did, we considered the differences between the two cell types. One difference is the number of contacts that cells of each type receive. To examine the possibility that low numbers of contacts would account for the limited decrease in pacemaker cell CV, the probability of pacemaker cells contacting each other was increased by a factor of three and five times, such that each pacemaker cell received an average of 24 and 40 contacts, respectively. The CV was observed during increases in the gap junction conductance ggap in a network containing 150 neurons (120 pacemaker and 30 relay cells, each with the same mean current but independent noise of the same SD = 0.2 nA) that were coupled with axoaxonic gap junctions.
The relay cells' CVs reached the minimum resolvable value at
ggap as low as 2 nS in the network
with five times the contact probability between pacemaker cells (Fig.
10A), a somewhat lower ggap than required in the network with
normal contact probabilities (Fig. 8B). Also, the pacemaker
cells' CVs decreased more with the higher contact probabilities (Fig.
10C) than with the normal probability of axoaxonic contacts
(Fig. 8E). That is, with five times the normal probability
of axoaxonic contacts, the CVs of pacemaker cells decreased from a mean
of 23 × 104 to 15 × 10
4 with a gap junction conductance increase
from 1 to 10 nS, whereas with normal contact probability (see above)
the mean pacemaker cell CV decreased from 26 × 10
4 to 22 × 10
4
over the same ggap change. For every
conductance value, the CV was lower for higher contact probabilities,
and the CV reduced more rapidly (higher slope with conductance).
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Increasing contact probabilities affected network frequency more than did other parameters tested, raising it to 614.3 Hz, just above the pacemaker cells' intrinsic frequency (612.5 Hz). Thus multiplying by five the probability of contacts between pacemaker cells reduced the CV of pacemaker cells by <35% and allowed a 2-Hz increase of the network frequency above the intrinsic pacemaker frequency.
A network with high probabilities of axosomatic (rather than axoaxonic) contacts was also studied. The results were similar in both networks, although the effects on CV and frequency were less pronounced in the axosomatic network than even the moderate effects in the network with axoaxonic gap junctions. This was part of the general trend that networks with axoaxonic gap junctions have larger changes in CV and even frequency than seen in networks with axosomatic gap junctions.
Other network parameters have only limited effects on pacemaker cell CV and frequency
We sought other network parameters that would modulate the CV of pacemaker cells when they were coupled in a network. The primary remaining parameters that differ between pacemaker and relay cells are the high axial resistance Ra of the relay cell axon, and the passive membrane in the relay cell soma. Raising the Ra of the pacemaker cell axon did not help reduce the pacemaker cell CVs; and adding active conductances to the relay cells did not remove their ability to reduce CV with increasing gap junction conductance.
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DISCUSSION |
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Until now, no proposed mechanism has adequately explained the phenomenal precision of the weakly electric fish's Pn neurons, although the question and its significance have been recognized for three decades. We propose an explanation, based primarily on the above network model and in vitro physiology, in two steps. First, the pacemaker cells are intrinsically more precise (lower CV) than any previously known cellular biological clock. Second, these cells converge on relay cells with a ratio of ~40:1 and a conductance of 4 nS, which together reduce the relay cell CV by an average of ~10 times. Several possible mechanisms are suggested that could be under central control and explain the biologically observed modulations of regularity.
With a simple network model, we explored the genesis of the high
precision of the weakly electric fish's pacemaker nucleus and the
biologically observed responses of neurons in this nucleus to current
injection. The model was composed of 150 2-compartment neurons of 2 types, containing sodium and potassium Hodgkin-Huxley-type currents,
and coupled with gap junctions. The model reproduced many detailed
physiological results and made new predictions for the electric fish's
pacemaker nucleus, in particular, and for gap junction-coupled
oscillators in general. Also, the model predicted that not only
networks of neurons, but single neurons themselves are capable of the
extreme precision (CV = 6-25 × 104)
observed in the biological Pn.
Biologically reasonable values of cellular parameters gave the observed
low input resistance for both cell types, and, for pacemaker cells, a
high intrinsic spiking frequency and the ability to drive other cells
to fire. We included only sodium and potassium currents, although we
know that calcium also plays an important role in setting the firing
frequency (Dye 1991), possibly producing a pacemaker
potential in a role similar to that in many other pacemaker cells
(Hille 1992
). Here, the role of the pacemaker potential
was played by a continuous step current. The injected currents and
their distribution on the cell membrane produced spikes that had
similar waveform and relative amplitude to those seen in the biological
homologue. The absolute amplitudes, however, are significantly larger
in the model than in the biological neurons, probably because of
differences in the recording location. The relative amplitude decay
between the somatic and axonic compartments of a relay cell was largely
determined by the passive relay soma, and the high axial resistance of
the relay axon, as suggested for dendrites of neocortical pyramidal
neurons (Mainen and Sejnowski 1996
).
Model pacemaker cells were coupled to each other, and model pacemaker
cells were coupled to model relay cells roughly according to the
statistics of anatomic data (Dye and Heiligenberg 1987; Moortgat et al. 2000
), with a maximum of one gap
junction per cell pair. Interrelay cell coupling was not incorporated
in the model despite anatomic evidence of dendrosomatic gap junctions among relay cells (Moortgat et al. 2000
) for two
reasons. First, the model did not include dendritic compartments.
Second, the effect on the network frequency and precision due to
distant gap junctions on fine processes would be minimal. The dendritic
gap junctions may be more involved during modulations of network
frequency that are driven by higher brain centers by way of the
relay cell dendrites (Heiligenberg et al. 1996
;
Spiro 1997
).
Simulations began with all cells arbitrarily at the same phase (zero phase lag), but the phase rapidly changed to new stable values, and many cells simultaneously received increased gap junctional currents. The increased input arose from increased phase lags among cells. That is, a gap junction only passes current when the voltage across it is nonzero; in the case of coupled oscillators, the phase between cells must be nonzero. At the same time, an overly broad phase distribution may not drive the contacted relay cells above spike threshold. The network might effectively be maximizing, over the first oscillation cycles, the total current passed through all the gap junctions or some other function of phase lag.
The final distribution of phases depended on the gap junction
conductance, which was chosen to be 5 and 4 nS, respectively, in
networks with axosomatic and axoaxonic coupling, to fit the biologically observed 10-15% phase lags between pacemaker and relay
cells (Dye 1988; Moortgat et al. 2000
).
The factor of 4 or 5 higher conductance than, for example, the 1 nS
directly measured between rat cardiac cells (Spray et al.
1985
), could result from larger numbers of gap junction
channels, even at a single club ending, as well as higher unitary
conductance (conductance per gap junction channel). Unitary
conductances are connexin-specific and range as widely as 30-300 pS in
mammals (Spray 1997
). Thus the model predicts a
biologically plausible gap junction conductance.
Phase lags not only between but also within cell types shifted with gap
junction conductance. Also, the phase lag between any cell pair varied
with gap junction conductance: a cell that spiked after its neighbor at
one conductance could spike ahead of it at another conductance. Such
phase shifts were observed in vitro when gap junction blockers (aimed
at reducing the gap junction conductance) were applied (see Fig.
7B1 of Moortgat et al. 2000). Thus the in
vitro result is consistent with blockage of gap junctions. A cell's
phase also depended on injected current, both in this model and in the
biological preparation. Because the model used injected current to set
a cell's intrinsic firing frequency, shifting the injected current
effectively changed its intrinsic firing frequency. Thus another
explanation for the relative phase shifts seen when gap junction
blockers were applied to biological cells is that cells changed
relative to one another in intrinsic frequency. This could occur if the
gap junction blocker altered the intrinsic frequencies of cells in some
spatially dependent way: for example, different network locations could
have received different blocker concentrations. However, other lines of
evidence (frequency change, below) support the conclusion that gap
junction conductances were indeed decreasing with drug application.
Current injections to model neurons qualitatively replicated the
sometimes perplexing in vitro results. Namely, the currents never
altered the frequency of membrane voltage oscillations. Also, the
responses to moderate and large hyperpolarizing currents (0 to 10 nA)
had only limited effects on the spike amplitude while substantially
driving the peak spike voltage. A narrow range of hyperpolarizing
currents led a biological or model neuron in a network to alternate
between high and low oscillation amplitudes. The range of currents
required for this alternation pattern were quantitatively similar in
the biological and model cells. In the model cells, we could attribute
variation in the required current to the number and location of gap
junction contacts that the cell received, with larger currents required
for high conductance gap junction inputs located at axons (axoaxonic
coupling). Even the details of the minimum voltage achieved after high-
and low-amplitude oscillations were reproduced in the model (Fig. 6,
A and B). That these experimental results
were reproduced, without additional model parameter tuning, supports
the validity of our model.
Simulations revealed that the high-amplitude oscillations reflected
full spikes, whereas the low-amplitude oscillations are the passive
response to gap junction input. We could not test this possibility in
the biological cells, but we confirmed it in selected model pacemaker
and relay cells by removing all active conductances in those cells and
observing the same low-amplitude oscillations. The model showed that a
narrow range of somatic currents produced the biologically observed
spike skipping in a 1:2 ratio with cycles of the network, and predicted
that further hyperpolarizing a Pn neuron should stop its active spiking
altogether. The model also predicted that other ratios of skipped
spikes to network oscillations, including 1:3, 1:4, and 1:5, could
occur in relay cells and that spikes would have graded amplitude.
Neither the graded amplitudes nor the low ratios were observed in the biological relay cells (Moortgat et al. 2000), perhaps
because of the narrow range of currents that produce them. The
biological pacemaker cells did show the low ratios, without graded
spike amplitudes.
To study the effects of gap junctions on spiking precision, we included
a stochastic process in our deterministic model. Biological sources of
noise are typically differentiated between external synaptic noise, and
intrinsic noise of the spike initiation mechanism that includes
cellular morphology as well as ion channel fluctuations. The intrinsic
noise is thought to be dominated by the latter. Models have
incorporated noise in many ways: from a fluctuating spike threshold in
an integrate and fire model (Reich et al. 1997) to ion
channel-specific fluctuations (Wilders and Jongsma 1993
) in a modified Hodgkin-Huxley model.
We added intrinsic noise in another way, through a time-varying current
injection into each cell's soma. In networks of cells that each
received independent noise current, cells had intrinsic variability in
their spikes times of CV = 30 × 104. We tested
whether gap junctions among pacemaker cells and between pacemaker and
relay cells could reduce the CV of spiking, as has been reported for
coupled photoreceptors, for example (Lamb and Simon
1976
). Coupling the noisy Pn neurons with gap junctions, and
increasing the gap junction conductance from 0 to 4-5 nS caused rapid
and dramatic reductions of the CV of relay cell spike times, without
significantly altering the pacemaker cell CVs. Thus the model suggests
that coupling Pn cells with gap junctions of biologically plausible
conductance can reduce relay cell CV from ~30 × 10
4 by a factor of ~10. This may be part of the
explanation for the low CV observed in relay cells. Also, behavioral
modulations in the CV (Moortgat et al. 1998
) could be
achieved by changing gap junction conductance. Relevant sensitivities
of gap junction conductance include pH, voltage, and calcium
concentrations (Spray and Bennett 1985
). Calcium
concentrations in particular could be adjusted in the vicinity of gap
junctions by co-localized glutamate receptors, known to be involved in
the modulation of Pn frequency (Heiligenberg et al.
1996
; Kawasaki and Heiligenberg 1989
;
Keller et al. 1991
) and possibly involved in CV
modulation (Moortgat et al. 1998
). Alternatively, active
glutamate receptors may directly increase the CV by opening synaptic
current channels that would normally be closed. As mentioned above,
synaptic currents have long been thought to cause high CVs in some neurons.
We observed a frequency increase with increasing gap junction
conductance in both the model, and, we believe, in vitro. These results
are qualitatively consistent with other models of gap junction-coupled
oscillators that show that increased gap junction conductance can
change pacemaking frequency in a direction that depends on spike shape
(Chow and Kopell 1999; Kepler et al.
1990
). However, our results are quantitatively most plausible
for the Pn. Specifically, Kepler et al.'s model used gap junction
conductances that were a factor of 1,000 higher than the range we
considered for Pn coupling. Within the range of conductances we tested,
the frequency of model cells changed by <1%, well below the
experimentally observed 30-50% decrease when pharmacological gap
junction blockers were applied (Moortgat et al. 2000
).
On the other hand, other model neurons (Chow and Kopell
1999
) showed a fivefold decrease in frequency with a fivefold
reduction in gap junction conductance. We conclude that the effect on
frequency by the gap junction blockers was partially due to blocking
gap junctions, but was also influenced by drug side effects. For
example, the drugs may have reduced calcium and sodium conductances and
thereby substantially reduced the pacemaking frequency.
Distributing pacemaker cell intrinsic frequencies
(Iinj: mean = 1.0 nA, SD = 0.2),
did not obviously alter the frequency locking or CV of coupled model
neurons within the tested range of gap junction conductances. However,
substantially different choices of frequency distribution and coupling
strength (possibly outside the biological range) might lead to
significantly different spiking patterns, as described in oscillators
with mean-field coupling (Matthews and Strogatz 1990).
Adding cells to a network while normalizing the gap junction
conductance to the average number of contacts received by relay cells
also reduced only the relay cell CV, but only moderately (by
~1/, where
Nr is the average number of contacts
received by each relay cell). The CVs of pacemaker cells were not
changed between the same 50 to 200-cell networks. This result is
consistent with the biological observation that increasing the numbers
of neurons beyond ~50 in the Pn of one species of weakly electric
fish does not significantly reduce the CV of pacemaking
(Hagedorn et al. 1992
).
For the Pn to send a precisely timed drive to the fish's output tail
organ (the electric organ) only the Pn's relay cells need to have
particularly low CV. However, we only observe CVs in the range of 6 to
30 × 104 in the biological Pn, with most cells
within 10 to 20 × 10
4. There was no evidence for
the bimodal distribution suggested by the model. The only network
parameter that decreased pacemaker cell CV below its intrinsic values
(set by the noise current to 30 × 10
4) was an
increased probability of contact between pacemaker cells. Increasing
the probability to three to five times the numbers in the anatomic data
slightly reduced the CV of pacemaker cells.
What is the intrinsic CV of single, isolated pacemaker and relay cells?
This has not yet been directly measured in vitro. All simulations
assumed that the CV of model pacemaker cells is 30 × 104 (Fig. 8-10) or lower (not shown). However, most
reports of cellular precision (weakly electric fish's pacemakers and
circadian rhythms are notable exceptions) describe CVs of 0.01-0.1 at
a minimum. If model pacemaker cells had intrinsic CVs of this value,
then our model predicts we would have measured biological CVs of this same order. However, CVs in isolated or in vivo nuclei were below ~25 × 10
4, with a few minor exceptions that could
reflect poor intracellular recordings. Our model predicts that, unless
the reported anatomy of pacemaker to pacemaker contacts is wrong by a
factor of five or more, the observed biological CVs will only occur
when individual pacemaker cells have low intrinsic CVs.
We therefore conclude that biological pacemaker cells have an
intrinsically low CV; at least as low as 20 to 25 × 104, which is at the high end of the CVs observed in the
biological Pn, but well below the CV of any other known biological
system. The biological relay cells may have slightly lower CVs than
pacemaker cells, with the lowest CVs primarily in cells with more
independent gap junction contacts, and secondarily in those with more
axoaxonic compared with axosomatic contacts.
Further research will be required to determine whether these neurons
are "normal," with high precision that is in fact possible in many
neural systems but that has not yet been seen because of more
complicated synaptic circuitry. Alternatively, the individual Pn
neurons may be specialized for high precision firing. One
specialization that appears to improve fidelity in photoreceptors is
having a high density of ion channels whose 25 pA conductances are
blocked to 4 fA (for review see Yau and Baylor 1989).
The partial block reduces the noise from each ion channel, whereas the
high channel density allows enough current to enter the cells. Similar
investigation of the Pn cells and their ion channels will require new
experiments and techniques.
The correspondence between the experimental results for current injection and gap junction blockade and the model reported here is surprisingly good, given the simplicity of the model. Only two currents were included in the model and their kinetics were not constrained by detailed biophysical channel measurements, which have not been performed. This suggests that the qualitative and some quantitative (e.g., the range of hyperpolarizing current that is required for a neuron to skip spikes) properties of the model are dependent primarily on the connectivity and passive properties of the neurons and do not depend on the details of the channel kinetics.
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ACKNOWLEDGMENTS |
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We thank J. Enright, J. M. Fellous, and R. Traub for helpful discussions; J. M. Fellous, Z. Mainen, W. Wheeler, and M. Hines for assistance with NEURON; and W. Kristan and two anonymous reviewers for insightful critique of the manuscript.
K. T. Moortgat was supported by National Institute of Mental Health Predoctoral Fellowship MH-10864-03 and by the Sloan Foundation; T. H. Bullock by the National Institute of Neurological Disorders and Stroke; and T. J. Sejnowski by the Howard Hughes Medical Institute.
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FOOTNOTES |
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Present address and address for reprint requests: K. T. Moortgat, Sloan Center, Dept. of Physiology, University of California, San Francisco, Box 0444, 513 Parnassus Ave., San Francisco, CA 94143-0444.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 19 March 1999; accepted in final form 18 October 1999.
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REFERENCES |
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