1Cellular and Systems Neurobiology Section,
Butera, Robert J., Jr.,
John Rinzel, and
Jeffrey C. Smith.
Models of Respiratory Rhythm Generation in the
Pre-Bötzinger Complex. II. Populations of Coupled Pacemaker
Neurons.
J. Neurophysiol. 82: 398-415, 1999.
We have
proposed models for the ionic basis of oscillatory bursting of
respiratory pacemaker neurons in the pre-Bötzinger complex. In
this paper, we investigate the frequency control and synchronization of
these model neurons when coupled by excitatory amino-acid-mediated
synapses and controlled by convergent synaptic inputs modeled as tonic
excitation. Simulations of pairs of identical cells reveal that
increasing tonic excitation increases the frequency of synchronous
bursting, while increasing the strength of excitatory coupling between
the neurons decreases the frequency of synchronous bursting. Low levels
of coupling extend the range of values of tonic excitation where
synchronous bursting is found. Simulations of a heterogeneous
population of 50-500 bursting neurons reveal coupling effects similar
to those found experimentally in vitro: coupling increases the mean
burst duration and decreases the mean burst frequency. Burst
synchronization occurred over a wide range of intrinsic frequencies
(0.1-1 Hz) and even in populations where as few as 10% of the cells
were intrinsically bursting. Weak coupling, extreme parameter
heterogeneity, and low levels of depolarizing input could contribute to
the desynchronization of the population and give rise to quasiperiodic
states. The introduction of sparse coupling did not affect the burst
synchrony, although it did make the interburst intervals more irregular
from cycle to cycle. At a population level, both parameter
heterogeneity and excitatory coupling synergistically combine to
increase the dynamic input range: robust synchronous bursting persisted
across a much greater range of parameter space (in terms of mean
depolarizing input) than that of a single model cell. This extended
dynamic range for the bursting cell population indicates that cellular
heterogeneity is functionally advantageous. Our modeled system accounts
for the range of intrinsic frequencies and spiking patterns of
inspiratory (I) bursting cells found in the pre-Bötzinger complex
in neonatal rat brain stem slices in vitro. There is a temporal
dispersion in the spiking onset times of neurons in the population,
predicted to be due to heterogeneity in intrinsic neuronal properties,
with neurons starting to spike before (pre-I), with (I), or after
(late-I) the onset of the population burst. Experimental tests for a
number of the model's predictions are proposed.
In the preceding paper (Butera et al.
1999 All simulations were performed on Pentium-based UNIX/LINUX or
SGI IRIX workstations. Simulations were coded in the C programming language. Numerical integration of simulations of pairs of cells were
performed using the numerical integration package CVODE (Cohen and Hindmarsh 1996 Automated determination of modes of activity (silence, bursting or
beating) and a quantification of these dynamics (burst duration, burst
frequency, etc.) for simulations of pairs of cells required a high
degree of accuracy (see DISCUSSION), thus we adopted the
following procedure:
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
), we presented minimal biophysical models of oscillatory
bursting of respiratory neurons in the mammalian pre-Bötzinger
complex (pre-BötC); current evidence suggests that these neurons,
coupled via excitatory amino-acid (EAA)-mediated synaptic connections,
form a bilaterally distributed, hybrid pacemaker network that is the
origin of rhythm generation in vitro. The bursting frequency of these
neurons is voltage dependent, potentially regulated by a variety of
intrinsic and synaptic mechanisms that control membrane potential,
including synaptic input from a tonically firing population of beating
neurons (see Smith 1997
; Smith et al. 1995
for review). The dynamic
interactions of both the intrinsic membrane and synaptic mechanisms
underlie rhythm generation. In the present paper, we investigate these
dynamics in a model pacemaker network. We utilize one of the cellular
models presented in the companion paper (Butera et al.
1999
) to investigate how factors such as synaptic coupling
strength and heterogeneity of cellular properties affect the
oscillation frequency and synchrony of the population of coupled
bursting pacemaker neurons. We begin by studying a pair of model
neurons and analyze how excitatory synaptic coupling and excitatory
tonic drive affect the mode of activity (silence, bursting, beating)
and the frequency of bursting. We then consider a larger network
consisting of a population of burst-capable neurons (neurons that
exhibit intrinsic bursting at some level of depolarizing input) with
parameter heterogeneity and study the synchronization and frequency
control of this population with comparisons with experimental data. We
also study emergent network rhythms that arise from populations of
nonburst-capable neurons. Finally we consider an even larger case of a
population of tonically firing cells providing input to a large network
of coupled bursting neurons that provides synaptic drive to a
population of follower cells. The bursting neurons and follower cells
are heterogeneous, and the resulting spike-frequency histograms and voltage-clamp synaptic current measurements are related to experimental data. Preliminary reports of these modeling results have been presented
in condensed form (Butera et al. 1997
, 1998a
,b
).
METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
), available at
http://netlib.cs.utk.edu/ode/cvode.tar.Z. Numerical simulations of
large networks of cells were integrated using a fifth order
Runge-Kutta-Fehlberg method with Cash-Karp parameters and an adaptive
step size (Press et al. 1992
). For final simulations,
relative and absolute error tolerances were 10
6 or
smaller for all state variables.
1
Run the simulation for 60 s (of simulation time) to allow
transients to decay.
2
Collect interspike intervals (ISIs) for 60 s. Let p
equal the maximum of these ISIs. Let w define a window size
equal to 0.9p.
3
If there were no spikes, the cell is classified as silent.
4
Collect ISIs for 120 s. Collect the following statistics: ISI,
burst duration (BD or active phase duration), burst interval (BI or
silent phase duration), and burst period (BP). A sliding window of size
w is used to determine the transition between active and
silent phases of a burst cycle. The beginning of a burst is defined at
any spike following an interval of duration w or longer. The
end of a burst is defined at the last spike preceding an interval of
duration w or longer.
5
If over the collection period there was never a transition to a silent
phase, the cell is classified as beating and the mean, standard deviation, minimum, and maximum of the ISIs is computed.
6
Otherwise, record the mean, standard deviation, minimum, and maximum of
the ISIs, BDs, BIs, and BPs. Let c equal the coefficient of
variation (standard deviation divided by the mean) for both the ISIs
and BPs. If cISI < s × cBP, the cell is classified as beating. Otherwise, the cell is classified as
bursting. The value of s was determined
empirically, and for the present results, we used s = 0.2. We chose a value of s < 1 because in the case of
coupled populations beating solutions tend to show little variability, while solutions that we would visually classify as bursting still may
have an inherent degree of variability in burst period from cycle to
cycle. Qualitatively similar results were obtained for values of
s from 0.1 to 0.95.
The concept behind the above algorithm is that bursts
may be identified from spike trains by looking for particularly large ISIs. Thus ISIs that are "similarly large" (the window size
0.9w) are used to identify silent phases between bursts. For
simulations of large populations (50-500) of bursting neurons, we
adopted a simpler definition of the transition between the active and silent phase of the burst. A burst begins at the first spike following a window of 500 ms. From these spike times, the BD, BI, and BP are
calculated. For long time series of spike times, these values were
calculated for each cycle and averaged.
Model of synaptic dynamics
Throughout this paper we will use model 1, as
presented in the preceding paper (Butera et al. 1999).
During oscillatory bursting activity, the burst is initiated by a
persistent Na+ current (INaP-h) and
terminated by slow voltage-dependent inactivation of that current. We
have focused on this model because its dynamics are more consistent
with our experimental recordings; however, some of the general results
presented in this paper (such as those presented in Figs.
1 and 6) have been reproduced
qualitatively using model 2 as well.
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The equations and parameter sets are those presented in the companion
paper (Butera et al. 1999) unless otherwise noted. For simulations of large populations, some parameters were picked randomly
from a normal distribution with a given mean and standard deviation
(see Table 1). To model intercellular
coupling, we have added an additional current,
Isyn-e, to the model. Thus the equation for the
rate of change of the membrane potential is modified to be
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(1) |
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Isyn-e models the EAA-mediated coupling between
individual bursting neurons in all simulations. It also is used in
those simulations that explicitly consider a modeled population of
beating neurons, in lieu of Itonic-e. The
synaptic input to neuron j from the population of N neurons is
described as
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(2) |
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(3) |
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RESULTS |
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Pairs of cells
To study the effects of excitatory coupling on the dynamics of
bursting, we initially consider the simple case of a pair of neurons.
Each neuron of the pair uses the ionic current equations and parameters
of model 1 (Butera et al. 1999). These
neurons are coupled reciprocally by an excitatory synaptic current
(Isyn-e). Each neuron in the pair also receives
a mean level of tonic excitatory synaptic drive
(Itonic-e). As described in the companion paper, this current has a similar depolarizing effect on the burst dynamics of
single neurons as Iapp. We studied the dynamics
of this pair of bursters as
syn-e and
gtonic-e were varied to assess the control of
bursting frequency by excitatory tonic drive
(gtonic-e) and excitatory synaptic coupling
(
syn-e). In all cases,
gtonic-e and
syn-e
were identical for each cell. The time course of the membrane potential
oscillations for various values of gtonic-e and
syn-e are illustrated in Fig. 1. The
frequencies of bursting and beating, as well as burst duration and
burst period, at various values of
syn-e
as gtonic-e is varied are plotted in Fig.
2.
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Both gtonic-e and
syn-e activate excitatory currents,
although gtonic-e may be described as constant
and
syn-e as phasic. The cases of weak
coupling (
syn-e < 1 nS) and high tonic
drive (gtonic-e > 0.6 nS) will be considered
shortly. For the rest of the parameter space illustrated in Fig. 2
where bursting activity occurs, several trends are evident.
At a given level of gtonic-e,
syn-e increases both the burst duration
(Fig. 2A) and the burst interval (Fig. 2B), with
a net effect of increasing the burst period. These effects are not independent: increasing excitatory coupling, which is triggered by the
firing of action potentials, causes action potentials in each cell to
further excite the other cell, with a net increase in intensity of
firing in both cells. The higher spike frequency increases excitatory
synaptic input during the active phase of a burst. For the burst to
terminate, INaP-h must be inactivated further
than it would in the absence of synaptic input. Thus the burst duration
is extended. The additional inactivation of
INaP-h prolongs recovery from inactivation as
well, increasing the duration of the silent phase of the burst cycle.
Hence we obtain the net, counterintuitive, effect of reduced burst
frequency with stronger excitatory coupling (also noted by
Somers and Kopell 1993
for idealized slow-wave neural oscillators).
At a given level of syn-e, burst duration
increases with gtonic-e. Through most of this
range, the burst interval decreases with
gtonic-e. This decrease in burst interval is the
dominating effect and the overall burst period decreases. However, at
higher values of gtonic-e in this parameter
range, the burst interval changes minimally or slightly increases. In
this region, which is just before the region of complex dynamics
described later in this paper, the increase in burst duration is
dominant and the burst period increases.
An important effect of synaptic coupling (except possibly for very
strong syn-e) is that it extends the
range of values of gtonic-e over which bursting
is supported. This effect will be investigated further in the following
two sections, which consider the dynamics of larger populations of
bursting neurons. This increased dynamic range is maximal at
approximately
syn-e = 2 nS (Fig. 2). The
half-pear shaped regions extend rightward beyond their intersection
with the abscissa near gtonic-e = 0.4 nS, which
denotes the bursting-to-beating transition point for an isolated cell. The transition from silence to bursting is the vertical boundary (on
left) because there is a critical value of
gtonic-e necessary to depolarize the cell into a
bursting mode, and
syn-e is inactive when
both cells are silent. In contrast, the transition between bursting and
beating occurs at a voltage range where the coupling synapses are fully
turned on, and there is an approximate trade-off between
gtonic-e and
syn-e
that determines the transition, i.e., a critical amount of synaptic
input (both tonic or excitatory coupling) is necessary to maintain the
cells in a beating mode. Less gtonic-e requires
more
syn-e to maintain beating and vice versa. This explains the nearly linear slope of the bursting regime's right-side boundary, above the regime of weak coupling. This trend is
only applicable at high coupling strengths (>1 nS).
At low coupling strengths (syn-e < 1 nS), synaptic input during the firing of action potentials synchronizes
the two cells but does not significantly alter the dynamics of each
cell (compared with the uncoupled state). In this case, the bursts of
the two cells synchronize but the burst duration hardly changes with
gtonic-e, while the burst interval decreases, as
in the case of the uncoupled cell (
syn-e = 0); see Fig. 2, A-C. The net effect on burst period is
similar to that in the case of stronger coupling: an overall decrease
in the burst period. Also, unlike the case of strong coupling strengths
(
syn-e > 1), increasing
syn-e increases the value of
gtonic-e where the transition from bursting and
beating takes place.
At high values of gtonic-e (>0.6), regardless
of the value of syn-e, the model neurons
approach the parameter range at the interface between bursting and
beating and their dynamics become quite complicated and sometimes
irregular. This is most evident in the burst duration (Fig.
2A), and those quantities that depend on this measure (Fig.
2, C and F).
Effects of coupling on population burst dynamics
Real neurons (as opposed to deterministic model neurons) possess
considerable variability in their intrinsic membrane parameters, such
as maximal conductances. We investigated the role of coupling in
synchronizing a population of bursting neurons with heterogeneous properties. We chose as our heterogeneity parameters
EL, NaP-h, and
syn-e. EL
determines the intrinsic baseline level of depolarization of the
bursting cells, an important parameter for voltage-dependent frequency
control as described in Butera et al. (1999)
;
NaP-h determines the ability of a neuron
to intrinsically burst (see Figs. 6 and 7, Butera et al.
1999
); and
syn-e is the coupling strength between individual neurons. Tables 1 and 2 specify the
intrinsic and synaptic parameters that were allowed to vary and how
they were distributed for all of the population simulation results
presented. Typically parameters were chosen randomly for each cell from
a normal distribution with a specified mean and standard deviation. The
standard deviations of EL and
NaP-h were chosen so that the distributed
parameters fell within the range of dynamic behaviors studied in
Butera et al. (1999)
. For the random distribution of
synaptic conductances, we specified a standard deviation of 25% (or
more) of the mean value. A condition was enforced that all conductances
be greater than zero. For all simulations, initial conditions were
chosen randomly from a uniform distribution across the physiological
dynamic range of individual state variables. After any change in
parameters or initial values, a settling period of
60 s of simulation
time was allowed before data were collected.
Some of the simulations presented in this section were repeated where
all the intrinsic conductances of the model were selected randomly from
a normal distribution, using the nominal conductances for model
1 in Butera et al. (1999) as the mean and a CV of
20%. Results were similar to those presented in this section, where the intrinsic heterogeneity parameters are solely
EL and
NaP-h.
Figure 3 illustrates the dynamics of
individual neurons and overall network activity before and after
synaptic coupling for a typical simulation. The population consisted of
50 heterogeneous bursting neurons with all-to-all coupling. The
left column illustrates the dynamics of the
uncoupled network (syn-e = 0), and the
right column illustrates the dynamics of the coupled
network. When uncoupled, neurons in the heterogeneous population
exhibit a variety of activity modes, with some spiking tonically, some
bursting, and some remaining silent (raster plot, Fig. 3A1).
There is little correlated network activity (population spike
histogram, Fig. 3A2). This dispersion in intrinsic
activities is consistent across 50 simulations with different randomly
generated parameter sets and initial conditions (bar plots, Fig.
3A3). After coupling, most of the cells burst in relative
synchrony with a coordinated periodic burst of action potentials (Fig.
3B, 1 and 2). This finding is also consistent across a large number of simulations (B3).
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Among the population of burst-capable neurons, the excitatory coupling allows the bursting cells to recruit silent neurons by providing sufficient additional depolarization to trigger some of the silent cells to burst. Tonically spiking cells also are recruited into bursting via additional excitatory input from bursting cells, which transiently increases spike frequency. This increase in frequency leads to additional inactivation of INaP-h, which temporarily may terminate spiking, resulting in bursting behavior. The recruitment of both intrinsically silent and spiking cells to a bursting mode increases the net excitatory synaptic activity in the population, making bursting at a network level more robust (see following text).
The relatively synchronous burst dynamics were found to be quite
tolerant of parameter heterogeneity. We performed numerous additional
simulations and report several initial observations regarding the
effects of coupling and parameter heterogeneity on the dynamics of the
network. First, general synchrony of bursts is obtained if the coupling
is sufficiently strong. Greater parameter heterogeneity requires
stronger coupling to synchronize bursts (Pinsky 1994;
Pinsky and Rinzel 1994
). Second, we investigated the
effects of sparse coupling, where each cell-to-cell connection has a
probability of existing. We found that connectivity patterns did not
greatly affect the ability of the network to synchronize bursts (even
with connection probabilities as low as 3%), given a similar level of
mean synaptic input to each cell. However, although sparsity did not
greatly affect synchronization of the population, the interval between
burst episodes of the population became more irregular as the
connectivity was made sparser. Third, when synchrony starts to break
down, due to increased parameter heterogeneity, weaker coupling, or by
decreasing the mean level of EL such that less
cells are intrinsically bursting or spiking, the network usually makes
a transition to a mode where most cells burst every N
cycles, with some bursting at higher multiple frequencies (Fig.
4, A and B). In
this case, the N:1 phase-locking of the cell to the network
output is still quite regular. In some cases, hyperpolarizing the mean
level of EL resulted in irregular network-wide synchronized bursts (Fig. 4C).
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Johnson et al. (1994) performed a systematic study of
the effects of synaptic coupling on the dynamics of oscillatory
bursting neurons in the pre-BötC. Inspiratory neurons in the
pre-BötC were identified via extracellular recording, and those
that continued to burst rhythmically after switching to a
low-Ca2+ solution to block synaptic transmission were
classified as possessing intrinsic bursting properties. The burst
duration and burst frequency were quantified for each bursting cell in
the control solution (coupled network) and low-Ca2+
solution (uncoupled), allowing an assessment of the role of coupling on
the dynamics of bursting at a cellular level. Figure
5 is a replotted version of the data
presented in Fig. 8, A and B, from Johnson
et al. (1994)
. The general effects of excitatory coupling on
burst frequency are illustrated in the histogram: the cell-to-cell variance in burst frequency is reduced, and the distribution of burst
frequencies is compressed toward lower (slower) values. The mean burst
duration increases with coupling. These effects are similar to those
illustrated in the simulations with only two cells (Fig. 1).
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We performed numerous independent simulations of our network of 50 bursting cells using the heterogeneity parameters specified in Table
3. These simulations were run without and
with EAA coupling as previously illustrated in Fig. 3. For each cell in
each simulation that burst when uncoupled, we quantified the burst
frequency and burst duration before and after EAA coupling. This
yielded a large number of cells on which to quantify the effects of
coupling in a heterogeneous network. The pooled results (Tables
4 and 5), when quantified, show similar effects as those in Fig. 5. However, Johnson et al. (1994) typically only recorded from one
cell per slice preparation, thus having quantifiable data from <50
bursting neurons. To assess the role of sampling in potentially biasing the results, we randomly selected only one endogenously bursting neuron
from each of 50 simulations and pooled the results, which are
illustrated in Fig. 6. Similar trends are
evident even in the much reduced dataset: coupling decreases the mean
and standard deviation of the burst frequency, while coupling increases
the mean burst duration (see Table 5).
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We used various other parameter distributions (Table 3) for additional simulations to assess the robustness of these coupling effects. We chose different parameter distributions to vary the number of intrinsically silent, bursting, and beating neurons (Table 4). These distributions ranged from a nearly even distribution of silent, bursting, and beating neurons (set 1) to cases with a majority of silent neurons and <20% intrinsically bursting (sets 6 and 7). In general, we found that the coupling effects described earlier persisted across all the parameter distributions where synchronous bursting activity occurred (Table 5). In Table 5, set 1 corresponds to the results presented in Figs. 3, and set 1r corresponds to the results presented in Fig. 6.
The results obtained in this section were verified further by running
simulations of a population of 500 cells using a similar parameter
distribution as in parameter set 1 with 1/10 of the cell-to-cell
coupling strength (syn-e) because there
are 10 times as many cells. The parameters for this simulation are
referred to as parameter set 8 (Table 3), and the effects of coupling on activity mode and burst dynamics are specified in Tables 4 and 5,
respectively. Results are similar to those obtained for parameter set
1, thus the results reported in this section using a population of 50 cells do not appear to be altered by small population effects.
Frequency control of network activity
As analyzed in the companion paper (Butera et al.
1999), the oscillatory bursting neurons in the pre-BötC
exhibit voltage-dependent frequency control, where the burst frequency
of individual neurons is regulated by depolarizing input, whether
intrinsic (EL) or synaptic
(gtonic-e). We investigated how well this
mechanism of frequency control persisted in a large population of
neurons with the same types of parameter heterogeneity described in the
previous section. For each simulation, the mean level of
EL was set to
65 mV. The mean depolarizing
input to the population, gtonic-e was initially
zero and increased every 120 s of simulation time in 0.05-nS
increments. Figure 7A,
1-7, illustrates the aggregate population spike activity of
one simulation as a function of gtonic-e. Parameters were distributed as indicated in Tables 1 and 2.
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As gtonic-e is increased (Fig. 7A,
1-6), the following effects on population bursting are
exhibited: an increase in the population burst frequency; a decrease in
the amplitude (spike frequency) of the burst of network activity; an
increase in the spread of the onset of burst firing (the bursts of
network activity at low gtonic-e have sharp
rises/falls and large peak amplitudes, while the bursts of network
activity at higher levels of gtonic-e have more
gradual rises/falls and smaller peak amplitudes); and a decrease in the
signal to noise ratio, as evidenced by an increase in the mean and
variance of spike activity and a decrease in the amplitude of the
network burst. At a sufficient level of depolarization (Fig.
7A7), all of the neurons in the network are spiking
continuously with no coordinated bursting evident. A comparison of the
burst frequency of the network simulation and the range of burst
frequencies of a single cell using the mean values of
EL and NaP-h and
identical gtonic-e values is illustrated in Fig.
7C. This reveals that the dynamic range of
gtonic-e where bursting is supported is much larger for the population than for a single cell. Similar effects are
obtained in the two cell case (Fig. 2).
To assess which factors contributed to the
gtonic-e versus burst frequency relationship of
the heterogeneous network, we ran additional network simulations with
no parameter heterogeneity (all cells have identical parameters),
intrinsic heterogeneity only (EL and
NaP-h), and synaptic heterogeneity only.
The effects of increasing coupling strength also were investigated. The
results are illustrated in Fig. 8. The
homogeneous network had a dynamic range of
gtonic-e larger than the single cell but less
than that for the heterogeneous network (Fig. 8A). However,
the homogeneous network displayed a smaller range of burst
frequencies than both the single cell and the heterogeneous population.
We also found that synaptic parameter heterogeneity yielded a frequency
versus gtonic-e relationship similar to the
homogeneous network, whereas intrinsic parameter heterogeneity yielded
a frequency versus gtonic-e relationship similar
to the fully heterogeneous network (Fig. 8B). From these
results, we conclude that both intrinsic heterogeneity and excitatory
synaptic connectivity contribute toward the increased dynamic range of
gtonic-e where bursting occurs (as opposed to a
single mean-value cell); intrinsic heterogeneity is necessary for the
population to have a range of burst frequencies similar to that of the
average single cell response; and synaptic heterogeneity made little
difference as long as the coupling did not become effectively too
sparse.
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We also investigated the role of coupling strength on the burst
dynamics of the network (Fig. 8C). The primary effect is
similar to that observed in the two-cell simulation (Fig. 1):
increasing the coupling strength increases the mean burst duration and
decreases the mean burst frequency of the population. As the coupling
strength was increased, the ranges of both
gtonic-e and burst frequencies where synchronous
bursting occurred were decreased. We also found that higher coupling
strengths also decrease the temporal distribution of burst onset times
in heterogeneous populations (not shown). At very high coupling
strengths, the network showed only beating activity with no
subthreshold oscillations at the cellular level, analogous to the
phenomena known as "oscillator death" (Ermentrout and
Kopell 1990). This differs from the case of modeled bursting neurons with gap-junctional coupling (Sherman and Rinzel
1992
), where high coupling strengths lead to the dynamics of
the coupled cells approaching that of a "mean" cell.
We repeated the simulations of this section using the mean value of EL, instead of gtonic-e, as a frequency-control parameter. Similar results to those shown in Figs. 7 and 8 were obtained.
As gtonic-e is varied through the range where the network bursts rhythmically, the distribution of intrinsic cell firing properties changes (Fig. 9). At the low end, coupling is adequate to induce bursting even though nearly 90% of the cells are silent. At the upper end, bursting is maintained with a population that is >90% beaters. This graphic illustration emphasizes that this population is made up of voltage-dependent (conditional) bursters: cells with INaP-h that can be recruited readily to burst by collective depolarizing inputs even though few of the cells are spontaneous bursters when decoupled from each other. For each gtonic-e within the network's operational range, the synchronized population burst frequency is below the mean frequency of the individuals. Again, this behavior is the large-population analogue (including heterogeneity) for the counterintuitive effect of phasic excitatory synaptic input on burst frequency noted for a pair of identical cells in Fig. 2. Figure 9 also can be used to predict the results of a synaptic blocking experiment in which both interneuronal coupling among the bursting population as well as excitatory tonic drive are removed (see DISCUSSION).
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Emergent network oscillations from coupled cells with INaP-h
The examples above have used populations of burst-capable neurons
where at any given parameter set, some fraction of the neurons were
intrinsically bursting. Simulations presented in this section examined
network behavior with a population of 50 neurons with a low mean level
of NaP-h such that no neuron in the
population exhibited intrinsic oscillatory bursting properties at any
level of depolarization (EL or
Iapp), i.e., they were only capable of spiking
or silence. In the companion paper (see Fig. 7 in Butera et al.
1999
) we define the parameter space in
NaP-h and EL
determining the activity modes of the pacemaker cells, which indicates
regions of
NaP-h with only quiescent or
beating modes. We therefore wanted to determine if there were low
NaP-h parameter sets where synchronous
oscillation could emerge as a network property. In the simulation shown
in Fig. 10A, the uncoupled
population shows only five neurons exhibiting spiking activity, the
rest are silent. Coupling the population (Fig. 10B)
increases the level of spiking activity and recruited two more neurons
to spiking, but the rest of the population remains silent. An
additional 25% increase in
syn-e gave
rise to synchronous bursting (Fig. 10C) across the entire
population. Alternatively, an increase in the mean value of
NaP-h (Fig. 10E) also gave
rise to synchronous population bursting. Even at this higher mean value
of
NaP-h, none of the neurons in the
population exhibit intrinsic bursting activity (Fig. 10D).
In these cases (Fig. 10, B and E), a sufficient
amount of depolarizing subthreshold inward current from both intrinsic (
NaP-h) and synaptic
(
syn-e) sources will initiate a burst, and a decrease in one current can be offset by an increase in the
other. We identified two other necessary criteria for synchronous bursting activity to occur when
NaP-h is
low:
10% or so of the population must possess intrinsic spiking
activity and sufficient
NaP-h must be
present to inactivate and terminate the active phase of the burst. In
effect, the combined actions of
syn-e (synaptic) and
NaP-h (intrinsic)
depolarize the cells to a spiking state, and the spiking recruits
inactivation of INaP-h resulting in a transient
pause in the spiking. The role of slow processes (e.g., adaptation) in
generating network bursting behavior has been investigated in the case
of networks of spiking neurons with adaptation currents and excitatory
coupling (van Vreeswijk and Hansel 1998
).
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The oscillations that occur in these cases represent an emergent
property of the network of excitatory coupled neurons with low
NaP-h. The oscillatory frequencies
generated under these conditions tend to be slower and at the lower end
of those achieved where
NaP-h is high
enough for intrinsic burst generation. Preliminary simulations of
frequency control of the network under low
NaP-h via gtonic-e
(not shown) reveal a restricted range of frequencies where bursting
occurs, similar to the case of strong coupling in Fig. 8C
(×3).
Components of network activity
In this section, we investigate the dynamics of a more complete
model network that incorporates the pacemaker population kernel, a
population of beating neurons that regulate the excitability of the
kernel, and a follower cell population that is synaptically driven by
the pacemaker cells. These populations are hypothesized to be the
rudimentary elements generating, controlling, and transmitting the
rhythm in the pre-BötC in the in vitro slice preparation (Smith et al. 1991, 1995
). The model network consists of
the following components:
1 | |
50 bursting neurons with all-to-all EAA-mediated coupling. The
parameters ![]() ![]() |
|
2 | |
200 beating neurons, which represent the population of beating cells
providing tonic input to the bursting neurons (in lieu of the mean
tonic input parameter gtonic-e used in earlier
simulations). For simplicity, we use our bursting cell model with
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|
3 | |
50 follower neurons that receive non-NMDA EAA excitatory input from the
population of bursting neurons. The population activity of these
follower neurons transmit the rhythm to (pre)motoneurons and may be
considered to be indicative of integrated recordings of motor output
from ventral roots of the hypoglossal nerve in the in vitro slice
preparation (see Funk et al. 1993![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Figure 11 illustrates how these populations combine to produce rhythmic network activity. In Fig. 11A, the network is disconnected. The beating neurons are not providing input to the bursting neurons and the bursting neurons are uncoupled, some of which are firing bursts independently in an unsynchronized fashion. The raster plot shows that with the particular randomized parameter distribution used, 15 of the cells are bursting, 11 of the cells are beating, and 24 of the cells remain silent. In Fig. 11B, the bursting neurons still are uncoupled but receive depolarizing synaptic input from the population of beating cells. The bursting neurons are more excitable than in Fig. 11A and burst with less regularity due to the summated asynchronous synaptic input from the beating cell population. In this simulation, 30 neurons are bursting, 18 are beating, and 2 neurons remain silent. To assess how the effects of a "noisy" synaptic input contributed to the recruitment of bursting neurons, we reran the simulation using an identical assignment of intrinsic and synaptic parameters, only using a mean synaptic conductance (0.19 nS) via gtonic-e, in lieu of a population of beating neurons. In this case, only 18 of the neurons were classified as bursting, with 23 beating and 9 silent. We speculate that asynchronous "noise-like" ongoing input to the population of burst-capable neurons is an additional factor that promotes bursting behavior in individual neurons. The noise occasionally will kick cells out of either the silent or beating mode to execute transient bursts. However, such a role for asynchronous synaptic input is predicated on the assumption that the number of synaptic inputs to each cell is "finite" (i.e., not a very large number).
|
In Fig. 11C, the bursting neurons are coupled to each other
and the entire population bursts in a coordinated fashion (see raster
plot). The summated effect of this coordination is represented in the
regular burst firing of the follower neuron population as well. The
more depolarized bursting neurons (high EL)
spike earlier in the cycle, whereas the more hyperpolarized neurons burst later in the cycle. Raster plots of the spike activity of the
population of follower neurons and the integrated population activity
are illustrated in Fig. 11C. The main bursts of the coupled pacemaker and follower cell population activity exhibit a rapidly peaking and slowly decrementing time course characteristic of respiratory network activity in in vitro preparations (e.g.,
Funk et al. 1993; Smith et al. 1990
).
Heterogeneity and spike-frequency histograms
Respiratory neurons conventionally have been described by their
firing patterns, which are referenced to the epoch of the respiratory
cycle (inspiratory, expiratory) during which the neuron fires.
Johnson et al. (1994) classified the firing patterns of respiratory neurons with intrinsic bursting properties in
the pre-BötC with respect to the onset of XII motor output, which represents inspiratory phase activity (Smith et al. 1990
,
1991
). Spike-frequency histograms were computed for the neurons
using the onset of hypoglossal discharge as a reference when synaptic transmission was intact; intrinsic bursting properties of these cells
were identified after blocking synaptic transmission with low-Ca2+ medium. A majority of the neurons (34/67) were I
cells, i.e., they burst with an onset of spiking in synchrony with the
XII output. A minority of cells (11/67) were classified as pre-I cells. These pre-I cells fire action potentials at a low frequency before the
motor output and often increase in firing frequency immediately before
the motor output event. One cell was a late-I cell, where the onset and
peak firing was after the initiation XII motor output. Representative
histograms of these different three types are illustrated in Fig.
12 This temporal dispersion of spiking
onset has been hypothesized to result from heterogeneity of pacemaker
cell intrinsic and synaptic properties (Smith et al.
1995
). We analyzed spiking patterns within our heterogeneous
network model.
|
Figure 12 illustrates the spike-frequency histograms of the three
labeled bursting neurons (1-3) and the integrated output of the
follower population (F) in Fig. 11. These histograms were calculated as
described in METHODS, using the activity of the follower
population to generate the reference times for each burst. The shape
and timing of the spike-frequency histograms are qualitatively similar
to those of the data. The more depolarized cells fire before the onset
of the motor output (pre-I) and increase in firing frequency
immediately preceding the motor event. The raster plot in Fig.
11C indicates a substantial dispersion in the onset of low-frequency spiking of different pre-I cells with a few cells spiking
throughout the interburst interval, analogous to the phase-spanning I
cells (not shown in Figs. 12 and 13)
described by Johnson et al. (1994). The least
depolarized cells within the bursting neuron population fire in
synchrony with the motor output, whereas the most hyperpolarized cells
fire after a delay.
|
Figure 13, A-C, illustrates typical membrane potential
trajectories (B, 1-3) and the total synaptic current
(Isyn-e; C, 1-3) recorded from each
of the three cells in the simulation of Fig. 12 under a simulated
voltage clamp at a holding potential of 60 mV. Because the bursting
cells are coupled globally, the time courses of synaptic currents in
B, 1-3, are similar. A comparison of the membrane potential
trajectory with
syn-e for cell
1 reveals that Isyn-e has shown no
appreciable activation when the cell has started firing. Thus in the
case of the pre-I cells, intrinsic properties appear to determine the
onset of spiking, and the subsequent increase in spike frequency when a
majority of the population fires is due to a combination of both
synaptic and intrinsic (voltage-dependent activation of
INaP-h) mechanisms. In contrast in cells
2 and 3, the onset of the burst occurs after a
noticeable increase in Isyn-e, suggesting a
greater role for synaptic mechanisms in initiating the burst,
especially in cell 3.
The follower cell populations were heterogeneous in
EL, and each cell also received input from
different randomly selected neurons in the bursting population. Figure
13, D-F, illustrates the spike-frequency histograms
(D, 1-3), typical membrane potential trajectory (E,
1-3), and a typical (gray) and average (black) Isyn-e elicited by a simulated voltage clamp at
60 mV for each cell (F, 1-3).
These cells have no intrinsic bursting properties and fire action
potentials when synaptic input exceeds a given amount. The follower
cells simply "read-out" the dynamics of the population activity
of bursting cells providing their synaptic input, as reflected in the
time course of the synaptic drive current envelope (Fig.
13F). The spike-frequency histograms of these follower cells have a similar spectrum of shapes, ranging from pre-I
(D1-F1) and I (D2-F2) to late-I
(D3-F3), as those of the bursting cells in Fig. 12. These
spiking patterns have been found (Johnson et al. 1994)
for neurons without intrinsic bursting properties in the pre-BötC
of neonatal rat slice preparations. The rapidly peaking, slowly
decaying time course of the synaptic current envelope (Fig.
13F) is characteristic of follower-type neurons in the
slices (Funk et al. 1993
; Smith et al.
1991
) and the more en bloc in vitro preparations (Smith
et al. 1990
, 1992
). Our results suggest that a range of both
intrinsic (for bursters) and synaptic (for bursters and followers)
heterogeneous properties give rise to the spectrum of spike-frequency
histograms recorded from pre-BötC inspiratory neurons.
The amplitude of Isyn-e (Fig. 13C) in
bursting cells is smaller than that required to generate subthreshold
depolarization and the firing of action potentials in the follower
cells (Fig. 13F). The only difference between the follower
cells and the bursting cells is the absence of
INaP-h. The presence of
INaP-h not only allows intrinsic bursting to
occur but means that less additional depolarizing input (such as
Isyn-e) is required to elicit a burst. This
postulated role of INaP-h has been referred to
as synaptic amplification (e.g., Ramirez and Richter
1996), which has been hypothesized (Ramirez and Richter
1996
; Rekling and Feldman 1998
; Smith et
al. 1995
) to promote synchronized bursting activity in the
pacemaker population. Our simulations demonstrate this function of
INaP-h, which is particularly apparent in cases
where most of the burst-capable cells in the population are silent but
fully synchronous bursting occurs readily with synaptic input (e.g., Figs. 9 and 10).
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DISCUSSION |
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In this study, we have investigated the dynamics of populations of
model pacemaker cells incorporating postulated subthreshold mechanisms
for intrinsic burst generation and excitatory synaptic mechanisms for
the synchronization and control of respiratory rhythm. We have
hypothesized (Smith 1997; Smith et al.
1995
) that this hybrid of pacemaker and network mechanisms is
the basis of rhythm generation in the pre-BötC. It has not been
proven that a population of the type of cells we have modeled is the
actual rhythm-generating kernel in the pre-BötC although the
mechanisms we have incorporated in the model are consistent with in
vitro experimental data (reviewed in Rekling and Feldman
1998
; Smith et al. 1995
). Experimental data on
intrinsic currents and network architecture is lacking. Our approach
therefore has been to explore the dynamics of populations of minimal
models of cells with plausible burst-generating mechanism(s) and to
investigate the consequences of heterogeneous cellular properties and
connectivity schemes incorporating heterogeneous distributions of
connection strengths as well as varying degrees of sparsity of
connections for small and large populations of cells. Our objective was
to determine general principles of network operation that may account
for available experimental observations on patterns of bursting
pacemaker and follower cell activity in the network. These general
principles include:
1 | |
excitatory synaptic tonic drive increases the synchronous bursting frequency of the population, while phasic excitatory synaptic coupling decreases the frequency. | |
2 | |
in a population of burst-capable cells (some degree of
![]() ![]() |
|
3 | |
sparsity of excitatory coupling (as low as 3%) has little effect on the synchronization of individual network-wide bursting, however, sparse networks exhibit greater variability in burst-to-burst cycle length. | |
4 | |
except at very large values of ![]() |
|
5 | |
increasing the strength of excitatory coupling (for values of
![]() |
|
6 | |
asynchronous tonic input from a finite population of beating cells, as opposed to a mean level of tonic input, increases the number of cells that burst when uncoupled. | |
7 | |
both burster and follower cells may exhibit the range of spike-frequency histograms (pre-I, I, late-I) recorded from inspiratory neurons in pre-BötC slice preparations. | |
8 | |
bursting neurons with higher intrinsic burst frequencies tend to be more depolarized and fire earlier in the cycle than bursting neurons with lower intrinsic frequencies. |
Comparison with experimental data and model limitations
Johnson et al. (1994) studied the effect of
synaptic coupling on pre-BötC bursting neuron activity by
examining burster cell dynamics with synaptic transmission intact and
blocked by low-Ca2+ solution. We modeled these experiments
by analyzing the dynamics of a heterogeneous population of model
neurons before and after elimination of EAA-mediated synaptic coupling.
Our model predictions are consistent with experimental observations.
However, there are several possible flaws with the technique applied in
Johnson et al. (1994)
and our approach in modeling these
experiments. First, the use of a low-Ca2+ solution to
eliminate synaptic connectivity makes it possible that Johnson et al.
also observed the effect of altering the dynamics of Ca2+
currents in the bursting neurons they were measuring. Similar experiments have been performed using
6-cyano-7-nitroquinoxaline-2,3-dione (CNQX) (Smith et al.
1997
) to block EAA-mediated transmission, and similar
quantitative effects (Figs. 5 and 6) on burst dynamics have been
observed. Second, blocking synaptic transmission eliminates not only
synaptic coupling between bursting neurons but also synaptic input from
tonic cells onto the population of bursters. We have not modeled this
effect because the amount of excitatory tonic input received by a
bursting pre-BötC cell has not been quantified. The effect of
excitatory tonic input on single model cells is to depolarize them and
increase their frequency (see Fig. 1) (see also Fig. 6 of Butera
et al. 1999
), whereas the effects of EAA coupling between
bursting neurons is to decrease their frequency. Figure 9 illustrates
that simultaneous removal of synaptic coupling within the population as
well as a modest reduction in depolarizing input (modeled by
gtonic-e) still will result in an increase in the mean bursting frequency of the uncoupled population. Third, we have
not explicitly modeled inhibitory synaptic connections from a
population of tonic inhibitory cells that also may provide modulatory
input in vitro (see discussion in Smith et al. 1995
). The experimental data and our model predictions show that synaptic connectivity leads to a reduction in mean bursting frequency and increase in burst duration; thus we conclude that the effect of EAA-mediated coupling among bursting neurons is more significant (under
the experimental conditions of Johnson et al. 1994
) than the effect of EAA-mediated or inhibitory tonic drive to the bursting neurons. Fourth, bidirectional electronic coupling has been reported in
respiratory motoneurons (Rekling and Feldman 1997
).
Although it has not been identified in pre-BötC interneurons, it
remains a potential coupling mechanism. However, Koshiya and
Smith (1998
, 1999
) have shown using Ca2+imaging
that pre-BötC neurons with pacemaker properties continue to
burst, but desynchronize, in the presence of CNQX. Thus even if
electrotonic coupling is present, it may not be a major mechanism for
synchronizing the activity within the slice. Electrotonic coupling is
also unlikely to account for bilateral synchronization of the rhythm.
Our model neuron populations have temporal patterns of neuronal spiking
as shown by spike-frequency histograms, as well as patterns of synaptic
drive currents in individual cells, similar to those found
experimentally for burster and nonburster follower neurons in vitro.
However, our neurons are simplified one-compartment models designed to
examine how postulated subthreshold mechanisms for intrinsic burst
generation can account for the dynamics of both individual neurons and
populations of bursting neurons in the pre-BötC. We have
deliberately not attempted to model the role of other specific ionic
currents (such as IA, IT,
or IK,Ca) in shaping the firing patterns of
action potentials in pre-BötC neurons, although they certainly
can modulate the firing patterns of isolated respiratory neurons as
well as those embedded in a network (e.g., Rybak et al.
1997a,b
). Interestingly, in the case of follower-type cells in
vitro, we previously have shown that the shape of spike frequency
histograms and time course of synaptic drive potentials largely reflect
the time course of the excitatory synaptic drive current envelope
(e.g., Funk et al. 1993
; Liu et al.
1990
). The time course of excitatory drive current (rapidly peaking, slowly decaying) exhibited by follower cells in our model is
similar to that seen experimentally and is reflective of the overall
time course of the presynaptic cell population activity which is read
out by the follower cells. We have also not included in our model
dendritic compartments even though respiratory neurons may have a
significant number of proximal dendrites (e.g., Funk et al.
1993
), which could affect the ionic current distribution and
firing patterns in response to synaptic drive. More complex models that
include the effects of ionic currents that modulate firing frequency
and dendritic compartments will provide addition insight into the shape
and amplitude of the spike-frequency histograms. More quantitative
information is required from experimental measurements on the types and
kinetics of ionic currents as well as morphology intrinsic to
respiratory pre-BötC neurons.
Although our single cell models possess membrane capacitance values and
resting membrane impedances consistent with whole cell measurements
from respiratory neurons in vitro (Smith et al. 1991,
1992
), the magnitude of the synaptic or applied currents required to change membrane potential of the bursting cells may be
somewhat smaller in the model than typically observed experimentally (N. Koshiya and J. Smith, unpublished observations). We attribute this
mainly to INaP-h in the model, whose
voltage-dependencies and maximal conductances have not been determined,
and these factors greatly affect the amount of current input required
to elicit a burst.
Model predictions and interpretations
HETEROGENEITY AND DYNAMIC RANGE.
This study has illustrated that both excitatory synaptic coupling and
intrinsic parameter heterogeneity make rhythmic bursting more robust.
Figure 2 illustrates that with identical cells, EAA-mediated coupling
increases the range of gtonic-e where
synchronous bursting is supported as opposed to the case of a single
cell (syn-e = 0). Figures 7 and 8 show
similar effects when comparing the range of
gtonic-e where bursting is supported in
populations as compared with single cells. The functional consequences
of heterogeneity in bursting neuron networks have not been analyzed
previously. Smolen et al. (1993)
studied models of
clusters of heterogeneous populations of pancreatic
-cells. While
they considered a different form of connectivity (3-dimensional, local,
gap-junctional coupling), a similar finding was made: the range of a
key control parameter (a glucose sensing conductance) where bursting
occurred was larger for the coupled population than for individual
cells. Thus it is plausible that the inherent cell-to-cell variability
in intrinsic properties is a functionally advantageous factor in
maintaining robust rhythmic output from a network of cells with
intrinsic bursting properties. Information on the degree of
heterogeneity in the pre-BötC pacemaker cell network remains to
be obtained. We predict that in vitro, the range of values of a
depolarization parameter (e.g., [K+]o) over
which bursting occurs is broader for the respiratory network in slices
containing the pre-BötC than for individual isolated inspiratory
bursting neurons within the pre-BötC.
NETWORK FREQUENCY CONTROL.
Our model simulations have illustrated the principle of
voltage-dependent frequency control of both isolated inspiratory
pre-BötC pacemaker neurons (Butera et al. 1999)
and an excitatory-coupled network of modeled pre-BötC neurons
(Fig. 7). In both cases, the frequency of the bursting oscillation is
regulated by the amount of depolarizing input, such as
gtonic-e or EL. Similar effects have been shown at the level of a single cell via
Iapp (Smith et al. 1991
) or
perturbations in [K+]o (R. Butera, C. Del
Negro, and J. Smith, unpublished data). We predict that titrating the
level of excitation and frequency of bursting in the pre-BötC
slice (e.g., by controlling [K+]o) will show
the following effects: an increase in frequency with elevated
[K+]o, a decrease in overall amplitude of
burster cell and thus follower cell population activity, and a change
in the shape of the integrated population activity from a rapidly
peaking population burst at hyperpolarized potentials to a more gradual
incrementing-decrementing profile. Assuming that the integrated XII
motor output is an accurate measure of overall network spiking activity
in the slice, changes in XII motor output similar to the spike-activity
plots of Fig. 7A, 1-7, should be obtained as
pre-BötC neuron tonic excitation is increased.
BURST-CAPABLE NEURONS, SYNAPTIC AMPLIFICATION, AND EMERGENT NETWORK OSCILLATIONS. The simulations show that with a heterogeneous pacemaker cell population, many of the neurons may not be spontaneously bursting when isolated but are burst-capable, and these burst-capable cells may be transformed into phasically bursting neurons in the presence of phasic synaptic input. The firing of these cells further contributes to the robustness of population bursting by generating additional excitatory synaptic input to the population.
We predict that, in the extreme case, it is not necessary for any neurons to exhibit intrinsic oscillatory bursting for synchronous bursting to emerge in the network (Fig. 10). This possibility also has been suggested by Rekling and Feldman (1998)SYNCHRONIZATION OF BURSTING.
Synchronized bursting is predicted to occur with substantial
heterogeneity, but desynchronization can occur with very weak coupling,
extreme parameter heterogeneity, and/or low levels of depolarizing
input. Recent experiments on pre-BötC slice preparations in mice
(Ramirez et al. 1996, 1998
) have reported N:1
(such as 3:1) coupling patterns between pre-BötC neurons and
hypoglossal motor output in slices from juvenile mice. We predict that
a breakdown of 1:1 coupling would occur in the pre-BötC slice
preparations when synaptic coupling strength is progressively reduced
(e.g., by CNQX). Such a breakdown in synchronized bursting may be
tested for by using activity-dependent imaging techniques to visualize bursting activity of multiple pacemaker cells simultaneously
(Koshiya and Smith 1998
, 1999
).
PACEMAKER NEURON SPIKING PATTERNS.
We put forward the notion that the different spike-frequency histograms
recorded from oscillatory bursting neurons in pre-BötC slice
preparations may not be due to differences in the existence of specific
types of ion channels or synaptic connectivity but rather may be
accounted for by a single model with variable levels of expression of
ionic currents responsible for generating bursting dynamics. Pre-I, I,
and late-I cells simply may possess different levels of expression of
ion channels and/or synaptic input regulating intrinsic levels of
depolarization and the ability for burst initiation. The possible role
of pre-BötC neurons with pre-I spiking patterns in rhythm
generation has been discussed (e.g., Schwarzacher et al.
1995; Smith et al. 1997
). The early low
frequency spiking of the pre-I cells provides excitation and
contributes toward recruiting the bursts of the remaining cells in the
network; they do not dictate the overall frequency of the network,
however, because many other cells in the network are also intrinsic
pacemakers or burst-capable and collectively set the oscillation
frequency. Thus the pre-I cells cannot be considered as simple triggers
of the I-phase. The frequency of the coupled network is (in all the simulations we have studied) lower than the frequency of the pre-I cells when uncoupled. We predict that pre-I neurons should correspond to neurons with higher intrinsic bursting frequency or baseline levels
of depolarization (or both), while neurons which fire later in the
cycle should be more hyperpolarized and/or possess lower intrinsic frequencies.
Summary
We have analyzed the dynamics of excitatory networks of
voltage-dependent pacemaker neurons as a model for the rhythm and inspiratory burst generating kernel in the pre-BötC of in vitro preparations. The general principles of operation of this pacemaker network have been investigated. We conclude that a heterogeneous network of the type of voltage-dependent burster cells modeled can
account for the synchronization, frequency control, patterns of
inspiratory cell activity, synaptic interactions, and network activity
found in vitro. The voltage-dependent properties of the bursting cells
combined with tonic drive and phasic excitatory interactions can
provide a functionally robust mechanism for frequency control over a
wide dynamic range. The model requires further experimental
verification, and we have presented predictions to facilitate
additional tests. At present the model is intended to account for
patterns of inspiratory activity generated in vitro. An important
extension of the model will be to embed the pacemaker kernel in a
pattern formation network (Smith 1997) that provides additional synaptic control, including phasic inhibition. Analyzing dynamic interactions between rhythm and pattern generating elements that produce the complete pattern of inspiratory and expiratory phase
activity in vitro and in vivo remains an important problem.
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ACKNOWLEDGMENTS |
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We thank N. Koshiya, S. Johnson, C. Wilson, and G. de Vries for helpful discussions and R. Burke and A. Sherman for critical readings of the manuscript. J. Rinzel thanks the Laboratory of Neural Control, NINDS, for hosting him in the summer of 1998.
This work was supported by the intramural research programs of National Institute of Neurological Disorders and Stroke and National Institute of Diabetes and Digestive and Kidney Diseases.
Present address of R. J. Butera, Jr.: School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250.
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FOOTNOTES |
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Address for reprint requests: J. C. Smith, Laboratory of Neural Control, NINDS, NIH, Bldg. 49, Room 3A50, 49 Convent Dr., Bethesda, MD 20892-4455.
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 23 September 1998; accepted in final form 9 February 1999.
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REFERENCES |
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