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. I. Bursting Pacemaker Neurons.
J. Neurophysiol. 82: 382-397, 1999.
A network of
oscillatory bursting neurons with excitatory coupling is hypothesized
to define the primary kernel for respiratory rhythm generation in the
pre-Bötzinger complex (pre-BötC) in mammals. Two minimal
models of these neurons are proposed. In model 1, bursting
arises via fast activation and slow inactivation of a persistent
Na+ current INaP-h. In model
2, bursting arises via a fast-activating persistent
Na+ current INaP and slow activation of a
K+ current IKS. In both models, action
potentials are generated via fast Na+ and K+
currents. The two models have few differences in parameters to facilitate a rigorous comparison of the two different burst-generating mechanisms. Both models are consistent with many of the dynamic features of electrophysiological recordings from pre-BötC
oscillatory bursting neurons in vitro, including voltage-dependent
activity modes (silence, bursting, and beating), a voltage-dependent
burst frequency that can vary from 0.05 to >1 Hz, and a decaying spike frequency during bursting. These results are robust and persist across
a wide range of parameter values for both models. However, the dynamics
of model 1 are more consistent with experimental data in
that the burst duration decreases as the baseline membrane potential is
depolarized and the model has a relatively flat membrane potential
trajectory during the interburst interval. We propose several
experimental tests to demonstrate the validity of either model and to
differentiate between the two mechanisms.
Breathing movements in mammals are generated by
networks of neurons in the lower brain stem that produce a rhythmic
pattern of neural activity. Recently the primary neuronal kernel for
rhythm generation has been located in the pre-Bötzinger complex
(pre-BötC), a subregion of the ventrolateral medulla
(Smith et al. 1991 The hybrid pacemaker-network model departs from previous network-based
models (Balis et al. 1994 There currently is limited experimental information on the ionic and
synaptic mechanisms generating and controlling the rhythmic bursting of
pre-BötC pacemaker cells. We therefore have formulated minimal
models for the pacemaker neurons with reduced parameter sets that
nevertheless retain the essence of what has been measured or
hypothesized for the cellular properties and synaptic interactions. Our
objective has been to formulate the models in a way that facilitates exploration of general principles and mechanisms for oscillatory burst
generation. In this paper, we present our pacemaker neuron models and
address questions about membrane conductance mechanisms that may
underlie the voltage-dependent oscillatory behavior of the candidate
pre-BötC pacemaker neurons found in vitro. We conducted a
systematic analysis of potential mechanisms regulating oscillatory bursting, burst frequency, and burst duration at the single neuron level. We also have derived tests that allow several general mechanisms to be distinguished that will guide experimental measurements. In the
succeeding paper (Butera et al. 1999 All simulations were performed on IBM RS/6000 or Pentium-based
UNIX/LINUX workstations. Most simulations were coded in the C
programming language using the numerical integration package CVODE
(Cohen and Hindmarsh 1996 Model development
The primary features of oscillatory bursting of the candidate
inspiratory pacemaker neurons in the pre-BötC (recorded from in
vitro transverse medullary slice preparations from neonatal rat) are
illustrated in Fig. 1 (see also
Koshiya and Smith 1999
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
). This discovery lead to the
development of rhythmic in vitro medullary slice preparations from
neonatal and juvenile rodents (Funk et al. 1994
;
Ramirez et al. 1996
; Smith et al. 1991
)
that capture this kernel and have become important experimental
preparations for analysis of cellular and network mechanisms of rhythm
generation. Current evidence (reviewed in Rekling and Feldman 1998
;
Smith 1997
; Smith et al. 1995
) indicates that rhythm generation in
these slice preparations, as well as in more en bloc in vitro
preparations, arises from a population of pre-BötC excitatory
interneurons with intrinsic oscillatory bursting or pacemaker-like
properties. It thus has become clear that to understand respiratory
rhythm generation, at least in vitro, mechanisms incorporating
intrinsic cellular pacemaker properties must be analyzed. Accordingly,
a new mechanistic model, the hybrid pacemaker-network model
(Smith 1997
; Smith et al. 1995
), has been
proposed in which rhythm arises from the dynamic interactions of both
intrinsic and synaptic properties within a bilaterally distributed
population of coupled bursting pacemaker neurons. In this and the
following paper (Butera et al. 1999
), we present
computational versions of this "hybrid" model that provide an
initial analytic framework for analyzing the potential roles of
cellular and synaptic processes in the generation and control of rhythm.
; Botros and Bruce
1990
; Duffin 1991
; Gottschalk et al.
1994
; Ogilvie et al. 1992
; Rybak et al. 1997
) in which respiratory rhythm has been postulated to arise mainly from network interactions, particularly inhibitory connections; synaptic interactions are proposed to operate cooperatively with intrinsic cellular properties of specific classes of interneurons to
produce the phase transitions required for a network-based respiratory
rhythm (see Ramirez and Richter 1996
; Richter et
al. 1992
). In these models, rhythmicity ceases when synaptic
inhibition is blocked. However, in the in vitro slice and en bloc
preparations, inspiratory phase respiratory activity persists when
inhibitory synaptic connections are blocked pharmacologically
(Feldman and Smith 1989
; Ramirez et al.
1996
; Shao and Feldman 1997
), and neurons with
intrinsic bursting oscillations have been identified (Johnson et
al. 1994
; Smith et al. 1991
). In the hybrid
model, inhibitory interactions are not essential; thus the fundamental
difference between this model and previous models is that a population
of synaptically coupled excitatory pacemaker-like neurons generates the
inspiratory phase of respiratory network activity. This rhythm and
inspiratory burst-generating kernel is embedded in a complex network,
however, that provides excitatory and inhibitory synaptic mechanisms
for control of oscillatory bursting as well as for generation of the
complete pattern of respiratory network activity including the phasic
firing of neurons during expiration (see discussion in Smith
1997
; Smith et al. 1995
). Our focus in this and
the following paper is on modeling the rhythm and inspiratory burst-generating kernel operating in vitro. These models serve as the
basis for the development of a more complete hybrid model of the
respiratory network incorporating both rhythm-generating kernel and
pattern-formation networks (Smith 1997
) that must be included to account for inspiratory and expiratory patterns of activity
in vitro and in vivo.
), we extend the
analysis to the cell population level and consider a population of
synaptically coupled bursting pacemaker neurons
a model of the
inspiratory rhythm-generating kernel in in vitro preparations. We
address a number of issues about the dynamics of this pacemaker network kernel, including how bursting of neurons within the population is
synchronized and how synaptic interactions and intrinsic cellular properties dynamically regulate inspiratory burst frequency and duration. Preliminary reports of these modeling results have been presented in condensed form (Butera et al. 1997b
,
1998a
,b
).
METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
) available at
http://netlib.cs.utk.edu/ode/cvode.tar.Z. For final simulations,
relative and absolute error tolerances were
10
6 for all
state variables. Some simulations also were performed using the
interactive differential equation simulation package XPP available at
ftp://ftp.math.pitt.edu/pub/bardware.
; Smith et al.
1991
). After block of synaptic transmission by
low-Ca2+ conditions in the slice bathing medium, the neuron
exhibits voltage-dependent oscillatory bursting behavior. As the cell
is depolarized by a steady applied current under whole cell current
clamp, the cell undergoes a transition from a state of rest to a state
of oscillatory bursting. As the cell is depolarized further, the burst
period, as well as the burst duration (Fig. 1B), decreases.
Additional depolarization maintains the cell above the action potential
threshold and causes a transition to a state of beating (i.e., tonic
spiking). Another salient feature is a steadily decreasing spike
frequency throughout the duration of the burst (see Fig. 1B)
(see also the spike-frequency histograms of Johnson et al.
1994
). Thus these neurons are presumed to have multiple
functional states (quiescence, oscillatory bursting, and beating); they
have been referred to as conditional pacemaker neurons (Smith
1997
; Smith et al. 1991
, 1995
) to indicate that
particular conditions must exist (ranges of depolarization level or
magnitudes of burst-generating conductances) for oscillatory
pacemaker-like bursting to occur.
View larger version (30K):
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Fig. 1.
Example of voltage-dependent properties of pre-Bötzinger complex
(pre-BötC) inspiratory bursting neurons. Traces show whole cell
patch-clamp recordings from a single candidate pacemaker neuron in the
pre-BötC of a 400-µm-thick neonatal rat transverse medullary
slice with rhythmically active respiratory network. Recordings in
A and B were obtained respectively before
and after block of synaptic transmission by low Ca2+
conditions identical to those described in Johnson et al.
(1994) (i.e., 0.2 mM Ca2+, 4 mM Mg2+, 9 mM K+ in slice bathing solution). Patch pipette solution
and procedure for whole cell recording were as described previously
(Smith et al. 1991
, 1992
). Before block of synaptic
transmission, the neuron bursts in synchrony with the inspiratory phase
of network activity as monitored by the inspiratory discharge recorded
on the hypoglossal (XII) nerve (Smith et al. 1991
).
After block of synaptic activity (30 min under low-Ca2+
conditions), the cell exhibits intrinsic voltage-dependent oscillatory
behavior. As the cell is depolarized by constant applied current, it
undergoes a transition from silence (baseline potential below
65 mV,
left) to oscillatory bursting to beating (baseline
potential above
45 mV, right). In the bursting regime,
the burst period and duration decreases (see expanded time-base traces
in B) as the baseline membrane potential is
depolarized.
In investigating the biophysical basis of these features, a fast
depolarizing mechanism for burst initiation and a slower opposing
mechanism for burst termination must be considered. In the two models
developed in the following text, initiation occurs by the activation of
a persistent Na+ current (INaP).
However, burst termination in the two models occurs by contrasting
mechanisms. In model 1, burst termination occurs by the
slow-inactivation of INaP-h, a persistent
Na+-current with slow inactivation. In model 2, a slowly activating K+-current (IKS)
is responsible for burst termination. Results for both models will be
presented in this paper. These models cover the two major mechanisms
(slow inactivation of inward current, slow activation of outward
current) for "type I bursting" (Bertram et al.
1995a; Rinzel and Lee 1987
) to occur in
oscillatory cells. Detailed justification for our choices of
conductance mechanisms is presented in the DISCUSSION.
FORMULATION OF MODEL 1.
Our minimal model is based on a single-compartment Hodgkin-Huxley (HH)
formalism. The model's dynamics are described completely by an
autonomous set of differential equations. The time course of the
membrane potential is obtained by applying Kirchoff's current law to a
single compartment neuron. In this case, the transmembrane current is
equal to the sum of the intrinsic and externally applied currents, as
follows:
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(2) |
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(3) |
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(4) |
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(5) |
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(6) |
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(7) |
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FORMULATION OF MODEL 2.
Because there are not sufficient quantitative data to fully constrain
our parameter choices, we formulated model 2 as similar as
possible to model 1, enabling a fair and complete assessment of the dynamic behavior of both models. In model 2, we
introduce an additional slow K+ current,
IKS. Thus the current balance equation is
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(8) |
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(9) |
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RESULTS |
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Voltage-dependent behavior
There are several methods through which the level of
depolarization, and thus the model of activity exhibited by the
bursting neurons, may be controlled. Some of these methods are
illustrated in Fig. 3. Note that changing
Iapp to control the neuron's level of
depolarization is equivalent to varying EL
(Iapp = L
EL).
|
Figure 4 illustrates the effect of
varying EL on the oscillatory activity of
model 1. The time course of membrane potential is
illustrated in Fig. 4A, 1-4, and the time course of
the INaP-h inactivation (h) and
instantaneous membrane conductance are illustrated in Fig. 4B,
1-4. As EL is increased, the model
makes a transition from silence (A1) to bursting (A,
2 and 3). One burst cycle consists of an active phase,
denoted by the repetitive firing of action potentials, and a silent
phase, where the membrane potential varies slowly at a hyperpolarized
value. The duration of the active phase is referred to as the burst
duration, and the duration of the silent phase is referred to as the
interburst interval. The burst period is the sum of the burst duration
and interburst interval. The burst period and burst duration decrease
and the cell becomes less hyperpolarized during the silent phase as
EL is increased. Vmin,
the minimum membrane potential encountered during the silent phase of
the burst cycle, varies from 58 mV at the onset of bursting activity
(EL =
60.5 mV) to
48 mV
(EL =
57 mV) at the transition between
bursting and beating activity. At all levels of depolarization the
spike frequency decreases throughout the burst (e.g., Fig. 4, C,
2 and 3, D, 2 and 3). As the cell
model becomes further depolarized, it undergoes a transition to tonic
beating behavior (A4). All of these features are similar to
those shown in the experimental recording of Fig. 1. During bursting,
INaP-h is inactivated progressively with the
firing of each action potential (Fig. 4D, 2 and
3). During the interburst phase, inactivation is removed and
the instantaneous membrane conductance gradually increases as
INaP-h recovers from inactivation. Similar
results to those in Fig. 4A are obtained if
EL is separated into K+ and
Na+ components and EK is varied.
|
Modulating the level of tonic drive can bias the neurons to different activity modes. This is analyzed in our model by varying gtonic-e, which represents the mean level of excitatory input to the bursting neuron population from a tonically beating cell population (Fig. 3). The depolarizing effects of gtonic-e on the mode of activity and pattern of bursting are similar to those of EL (quantified in the following text).
Figure 5 illustrates the effect of varying EL on the oscillatory activity of model 2. The time course of membrane potential is illustrated in Fig. 5A, 1-4, and the time course of the IKS activation and instantaneous membrane conductance are illustrated in B, 1-4. The dynamics of model 2 are similar in many respects to model 1. As EL is increased, the model progresses through regimes of silence, bursting, and beating activity. During oscillatory bursting, the burst is initiated by INaP and terminates when IKS has been activated sufficiently. The spike frequency decays throughout the active phase of the burst. The current IKS is activated slowly and progressively with the firing of each action potential and deactivates during the interburst interval. The instantaneous membrane conductance (gm) decreases throughout most of the silent phase and increases toward the end of the phase. This differs from model 1, where gm steadily increases throughout the silent phase. This difference in the time course of gm may provide a means for differentiating between the mechanisms of the two models in vitro (see DISCUSSION).
|
The dynamics of model 2 were investigated across a wide range of parameters for IKS and INaP. For all parameter ranges where bursting activity was supported, we found three key differences from the dynamics of model 1. The membrane potential trajectory was not as flat during the interburst interval, the burst duration did not decrease with depolarization (Fig. 8), and gm did not exhibit a monotonic trend during the silent phase.
Robustness of frequency control
Models 1 and 2 support burst frequencies
that vary over at least an order of magnitude with
EL or gtonic-e, and this
range of burst frequencies is robust across a range of parameter
values. Figure 6 quantifies these effects
for model 1 for various values of
NaP. The burst duration, burst period,
and frequency of bursting and beating are plotted as
EL (Fig. 6A) and
gtonic-e (Fig. 6B) are varied.
Equivalent values of Iapp with
EL fixed to
65 mV are also shown on
A.
|
In both panels, there is a minimum value of
NaP for which oscillatory bursting can
occur. If
NaP is too low, the only supported modes of activity are quiescence and beating as shown for
model 1 in Fig. 7. For all
values of gNaP where oscillatory bursting
occurs, the burst frequency may be varied by approximately an order of
magnitude, and the burst period and burst duration decrease as the
neuron is depolarized. This is consistent with the data illustrated in
Fig. 1. Increasing
NaP increases the dynamic range of values of EL where bursting is
supported. We also have observed (not shown) that increasing
NaP increases the range of values of
Vmin encountered as a function of
EL during oscillatory bursting. At a given level
of EL, increasing
NaP increases burst frequency.
|
Figure 8 quantifies the effects of
EL on model 2 bursting for a range of
values of the subthreshold conductances
NaP and
KS.
For all values of
NaP and
KS shown, the model possesses regimes of
silent, bursting, and beating activity. During bursting activity, the
burst period decreases as the neuron is depolarized. However, the
parameterization of the dynamics of model 2 by
EL differed from that of model 1 in
several ways. First, the burst duration increased slightly with
depolarization. Second, during tonic firing model 2 could
not achieve spike frequencies as low as those obtained in model
1. Third, model 2 supported bursting over a range of
EL (and Iapp)
approximately twice as large as the range of EL
where bursting occurred in model 1. These trends persisted
as the other parameters of the subthreshold currents (e.g.,
k and
k) were varied. Similar results
were obtained when gtonic-e was varied.
|
The large range of Iapp in model 2 may be attributed to the difference in whole cell conductance
(gm) between the two models. Model 2 has an additional conductance, KS, which
is not present in model 1. This conductance is activated
during bursting and beating. The increased gm
means that the "gain" of the cell is decreased. A larger
increment in Iapp is required to achieve a given
increment in depolarization. This is why the frequency-current curves
in Fig. 8 are flatter and extend over a larger
Iapp range than those in Fig. 6.
The critical input levels for the transitions between activity modes
(i.e., from silence to bursting and bursting to beating), show distinct
dependencies on the conductances NaP and
KS. In both models (1 and
2), the critical value of EL or
Iapp that changes the cell from quiescence to
bursting is quite sensitive to
NaP. The
persistent Na+ current is the primary voltage-dependent
current (in model 1, the only one) that begins to activate
in the subthreshold regime. At hyperpolarized voltages, these
Na+ channels have a low open probability (exponentially
small with decreasing V) and the input resistance is high.
Thus excitability and the critical level of input needed for bursting
is strongly influenced by how many channels there are per unit area,
i.e., by
NaP. In model 2,
KS affects this critical level only
modestly, partly because the driving force for
KS is much lower than that of
NaP and partly because it possesses slow
(compared with INaP) activation kinetics and is
relatively inactive at subthreshold voltages. At the
bursting-to-beating transition, the cell is depolarized and
conductances are well activated. Thus the extra sensitivity to
NaP due to high input resistance and low
channel activation that we see at the silence to bursting transition is
diminished here. The critical levels for EL and
Iapp are almost independent of
NaP. The transition is affected by
KS because this current controls burst
termination (Fig. 5B3). Although these sensitivities for the
silence to bursting transition can be understood qualitatively by
considering the subthreshold currents (Isub, see
following text) alone, the corresponding sensitivities for the bursting to beating transition are difficult to intuit because the components of
Isub interact with the spike generating currents.
Mechanism for mode transitions and frequency control
In this section, we offer a qualitative biophysical description of
how control of the baseline membrane potential, illustrated by varying
EL, alters the activity mode (silence, bursting,
and beating as shown in Figs. 4 and 5) of both models. The activity modes exhibited by the model are explained by using steady-state (SS)
and quasi-steady-state (QSS) I-V curves of the subthreshold currents (Isub IL + INaP-h for model 1,
Isub
IL
+INaP + IKS for
model 2). This explanation suffices because without the
action potential currents INa and
IK, reduced models having only the Isub currents display oscillatory and
nonoscillatory states (hyperpolarized silence, oscillatory activity,
depolarized silence) that are qualitatively similar to those in the
corresponding full models. Figure 9
illustrates I-V curves corresponding to
Isub in cases shown in Figs. 4 and 5. The SS
curves (thick lines) represent Isub versus
V when the slow negative feedback process (h
inactivation for model 1, k activation for
model 2) is set to steady state [(i.e., h = h
(V) in model 1, and
k = k
(V) in
model 2]). The QSS curves (thin lines) represent
Isub versus V when the slow process
(h or k) is fixed to a particular value. These
curves are generally N-shaped, revealing the regenerative nature of the
persistent sodium current. For the remainder of this section, we will
only discuss model 1 (Fig. 9A, 1-4).
An analogous explanation suffices for model 2.
|
For all values of EL the SS I-V curves have a single positive-sloped crossing of the 0 nA axis. This crossing is an equilibrium state that may be stable or unstable. If stable, the equilibrium state corresponds to the model's resting potential. However, stability is not governed only by the slope's sign at the zero crossing. If all voltage-dependent conductances vary on time scales faster than the membrane time constant (CM/gm), then a positive slope implies stability. When the time scales are significantly disparate (such as h, which has a time constant of seconds), SS I-V curves may not correctly predict stability, although stability may be inferred through the use of QSS I-V curves. When the QSS I-V curve has a positive crossing of the 0 nA axis, the model is stable at that membrane potential at that particular value of h (i.e., as if h was clamped at the value). Neither the SS I-V nor QSS I-V curves provide any information on the dynamics of h itself.
When EL = 65 mV, the model cell is silent (as
shown in Fig. 4A1). The rest potential, at approximately
62 mV where the SS I-V curve of
Isub crosses the 0 nA axis, is stable. The
equilibrium point is stable because INaP-h is
relatively deactivated and INaP-h cannot be
adequately recruited to sustain bursting activity. In this case the QSS
I-V curves (Fig. 9A1) show that for all values of
h, even when inactivation is completely removed
(h = 1), the hyperpolarized zero crossing is stable
(i.e., a positive slope).
For EL equal to 60 mV (Fig. 4A2) or
57.5 mV (Fig. 4A3), the model is in the bursting mode and
the QSS I-V curves change during the cycle (Gola
1974
). An animated QuickTime movie of several burst cycles,
illustrating V(t), h(t),
Isub(t), and QSS
I-V(V, t), is available on the
World-Wide Web at
http://intra.ninds.nih.gov/smith/movie/moviejnp-99.html. Figure
9A, 2 and 3, shows
Isub at the maximum (a) and minimum (b) values of h during one complete burst cycle.
The burst begins at a value of h (a) where
Isub is inward at all hyperpolarized potentials
below the action potential threshold (approximately
45 mV). The cell
depolarizes across the action potential threshold and repetitive firing
occurs. With each action potential, h inactivates an
additional small amount, and this inactivation decreases the inward
contribution of INaP-h to
Isub, shifting the QSS I-V curve outward. Firing persists until Isub is net
outward at Vmin, the minimum membrane potential
encountered during an action potential. At this point (b),
firing stops, and V falls to the hyperpolarized equilibrium
point (0 nA crossing with positive slope) of the QSS I-V
curve (b). As h gradually deinactivates, the QSS
I-V curve moves inward, depolarizing the cell as the
pseudosteady equilibrium point drifts rightward. Another burst begins
when Isub is net inward at all subthreshold
potentials (a), and the hyperpolarized equilibrium point of
the QSS I-V curve disappears.
As the cell is depolarized further (from A2 to
A3, Fig. 4), the values of h encountered during
one burst cycle are biased toward smaller values, i.e., less inward
current from INaP-h is necessary to
counterbalance the reduced outward current from shifting EL. In addition, the dynamic range of
h is reduced, from a h of ~0.1 in Fig.
4A2 to <0.02 in Fig. 4A3. This is the essence of
the voltage-dependent frequency control in the model: as the cell is
depolarized toward higher values of EL, the cell
does not hyperpolarize as far at the end of each burst cycle, and less time is necessary to recover from inactivation of
INaP-h to begin the following burst. At all
values of EL where bursting occurs, INaP-h is active throughout the burst cycle, and
it is the cyclical variation in h that controls the cycle timing.
With still further depolarization such that the equilibrium point of
the SS I-V is above the action potential threshold, the model changes to a state of tonic firing (Fig. 4A4,
EL = 54 mV). For this activity mode, h is
nearly constant, oscillating at low amplitude with the action potential
frequency about a mean value. The QSS I-V curve in Fig.
9A4, generated for this mean value of h (0.315),
illustrates that on average Isub is net inward
at all subthreshold potentials during tonic spiking.
In summary, the model may be thought of as having two voltage-dependent
thresholdsa threshold for burst initiation (approximately
60 mV)
and a threshold for action potential initiation (approximately
45
mV). When the equilibrium point of the SS I-V curve is below the burst threshold, the model is quiescent. When the equilibrium point
is above the action potential threshold, the model is firing tonically.
When the equilibrium point is between these two values, oscillatory
bursting is likely. This explanation is only qualitative, however, and
a more quantitative explanation may be provided using bifurcation
theory (e.g., Rinzel 1985
).
Model predictions
In this section, we consider several other functionally important features of the models' dynamic behaviors that we predict should be observed experimentally.
When action potentials are blocked, each model still exhibits activity
modes and transitions that are analogous to those of our full model.
Figure 10 demonstrates the models'
dynamic responses when Na = 0. This
predicts what we expect in experiments when INa
is blocked (e.g., with QX-314 or TTX-see
DISCUSSION), and cells are depolarized progressively. For
both models, a subthreshold oscillation remains, with the trends in
burst duration, burst period, and interburst depolarization consistent
with those of the more complete models (Figs. 4A, 1-4, and
5A, 1-4). Thus in spite of complex interactions between the
spike-generating currents and the subthreshold burst-generating
currents, especially at the transition between bursting and beating,
these figures demonstrate that the subthreshold currents generally
control the burst period and burst duration during repetitive bursting.
|
Cells in hyperpolarized silent mode may retain burst excitability if
NaP is not too small, showing single
burst responses to transient inputs. A brief depolarizing input (50 ms)
of sufficient magnitude is capable of triggering a single sustained
burst lasting several hundred milliseconds (Fig.
11A). Similar responses
occur across a wide range of values of EL and
NaP where the model is initially silent,
providing that
NaP is not too small. At rest, INaP-h is relatively deactivated
(m is low) and deinactivated (h is high). A brief
depolarization activates INaP-h. Because INaP-h inactivates slowly,
INaP-h remains relatively deinactivated and a
larger conductance for INaP-h is elicited,
generating sufficient inward current to trigger a single burst. As
analyzed in the companion paper (Butera et al. 1999
),
such burst triggering is important for recruiting inactive cells and
achieving synchronous bursting in a synaptically coupled population of
these pacemaker cells with heterogeneous properties where a substantial
fraction of the population is intrinsically in a silent mode.
|
A single burst also may be elicited as a form of posthyperpolarization
rebound (Fig. 11B). A sustained hyperpolarizing input removes some inactivation from INaP-h (i.e.,
h increases). On abrupt release of the hyperpolarizing
input, the model depolarizes. Because inactivation of
INaP-h develops slowly,
INaP-h has an increased conductance as
V nears its rest value, and this increased conductance may
trigger a single posthyperpolarization burst. This response may occur
across a wide range of parameters provided that the stimulus is of
sufficient duration (the inactivation of
INaP-h is a slow process), the
hyperpolarizing stimulus is of sufficient magnitude (so that
h increases sufficiently), and the resting neuron is not too
hyperpolarized (so that sufficient h remains to be
deinactivated). For example, the response of Fig. 11B was
generated for EL = 62 mV. When
EL =
65 mV, the resting value of h
is sufficiently large (0.92) that even a large hyperpolarization (to
bring h nearly to 1) does not produce an increase in the
conductance of INaP-h strong enough to elicit a
posthyperpolarization burst, unless an additional depolarizing input is
coincident with the release from hyperpolarization. Thus we predict
that this response can be elicited by hyperpolarization release only
for a window of membrane potentials that is not too hyperpolarized.
During repetitive bursting, transient inputs can reset the rhythm. A
brief (50 ms) hyperpolarizing input (10 pA) is capable of resetting the
active phase of a burst (Fig. 12). This
resetting reduces the duration of the current burst as well as the
following interburst interval. The duration of the following interburst interval varies with the duration of the preceding burst. The duration
of the subsequent burst is unaffected. For example, a transient current
pulse applied early in the burst (b1) terminates the burst, and the following interburst interval is significantly shortened, with the next burst firing at b2. A
transient current pulse applied late in the burst
(c1) also terminates the burst, with the next
burst firing at c2. Although the poststimulus
interburst interval is shortened in both B and C,
the poststimulus interburst interval in B is shorter. This
behavior would be important for controlling burst cycle timing under
conditions where there is dynamic resetting by inhibitory synaptic
mechanisms (see discussions in Smith 1997; Smith
et al. 1995
).
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DISCUSSION |
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Justification of models
We have developed minimal models for oscillatory bursting neurons
found in the pre-Bötzinger complex in vitro. Although there is
strong evidence that neurons with bursting pacemaker-like properties are generating the respiratory rhythm in vitro, it has not been proven
that the neurons on which our models are based, with the behavior
depicted in Fig. 1, actually generate the rhythm. Such causality
remains difficult to establish definitively experimentally. However,
these cells are the only inspiratory neurons with intrinsic pacemaker-like properties that have been identified in the
pre-Bötzinger complex from electrophysiological recording and
mapping of neuron activity in slice preparations (Johnson et al.
1994; Smith et al. 1991
; Koshiya and
Smith 1998
, 1999
). They are currently candidates for the
rhythm-generating cells in vitro (see Smith et al.
1995
).
Intrinsic conductances responsible for the oscillatory bursting
behavior of these cells have not yet been identified experimentally. Furthermore there is likely to be considerable heterogeneity in cellular parameters for different neurons in the rhythm-generating cell
population, e.g., maximal conductances of burst-generating currents
(NaP) and baseline membrane potentials
(EL); this heterogeneity is treated in the
companion paper (Butera et al. 1999
). Figure 1 only
provides one example of voltage-dependent behavior in a bursting
inspiratory neuron. Variations in properties such as the voltage range
for oscillatory bursting, oscillatory frequency range, and the extent
to which a given neuron in the population will exhibit oscillatory
bursting (see Functional states of pacemaker neurons),
remain to be quantified experimentally. Given the limited experimental
data, we therefore have postulated a minimum set of conductance
mechanisms and explored a range of parameter values to determine if
they can account for the observed behaviors as well as to investigate
the range of behaviors likely to be exhibited by cells with these types
of conductances. Besides fast Na+ and K+
currents responsible for the generation of action potentials, ionic
mechanisms must be postulated for both the initiation and termination
of bursting. Although we have assigned specific types of currents to
these functions, as justified in the following text, the models are
general and are designed to provide insights into plausible mechanisms.
BURST INITIATION.
In numerous invertebrate and mammalian bursting neurons, the onset of
bursting is caused by an inward cationic current that activates at
subthreshold potentials. This current is responsible for maintaining
the negative-slope region of the I-V curve. T-type and
persistent Ca2+ currents have been implicated in burst
generation in mammalian neurons (e.g., Llinás
1988). Several types of Ca2+ currents exist in
respiratory neurons (Onimaru et al. 1996
); however,
inspiratory neurons have been identified which continue to undergo
rhythmic bursting under low Ca2+ conditions (Johnson
et al. 1994
) (see also Fig. 1). This evidence suggests that
Ca2+ currents are not essential for the generation of
bursting behavior. Therefore although it is known that the
pre-BötC oscillatory bursting cells have Ca2+
currents (Koshiya and Smith 1998
) and it is likely that
Ca2+ currents play a role in modifying a neuron's firing
properties, they are not considered in our formulation of a minimal
model for bursting activity.
BURST TERMINATION.
Spike-frequency histograms of pre-BötC bursting neurons in
rhythmically active slices (Johnson et al. 1994) show
that the firing frequency decays throughout the burst phase (Fig. 1).
Also the initiation and termination of the burst are accompanied by a
rapid transition between the silent phase and the firing of action
potentials (Fig. 1) and vice versa. In light of theoretical studies
(Bertram et al. 1995a
; Rinzel 1987
) of
the mechanisms by which individual neurons may exhibit bursting
behavior, both of these lines of evidence suggest that a minimal
mechanism for bursting requires just one slow recovery process, such as
intracellular Ca2+-accumulation or a slow voltage-dependent
gating mechanism. Because rhythmic bursting pre-BötC neurons have
been identified in low-Ca2+ preparations (Johnson et
al. 1994
), we postulate that the recovery process is voltage
dependent. This process could be either a slowly inactivating inward
current or a slowly activating outward current (presumably
K+).
Comparison of models
Models 1 and 2 both demonstrate several
qualitative features of the oscillatory bursting behavior of
pre-BötC inspiratory neurons. These features include
voltage-dependent control of the model activity: as the neuron is
depolarized via Iapp or
gtonic-e, or varying EL
(e.g., by controlling [K+]o), the neuron can
be biased from silence to bursting to beating activity. During
bursting, the spike-frequency decays during the burst and the bursting
frequency is strongly dependent on the level of depolarization.
However, model 1 is more consistent with our experimental
data in two respects: the membrane potential trajectory during the
interburst interval is much flatter (in membrane potential) and the
burst duration decreases as the baseline membrane potential is biased
toward more depolarized levels. It is possible that a more complex
repertoire of ionic currents may allow model 2 to possess a
flatter interburst interval (e.g., see Smith et al.
1995). Also, it is possible that we have not identified a
possible parameter set for model 2 where the burst duration
decreases, rather than increases, with depolarization. Nevertheless, we
have searched parameter space for all of the parameters of the
subthreshold currents (specifically, maximal conductances and
parameters responsible for the dynamics of the gating variables:
KS,
Nap,
m,
m,
k,
k)
in model 2 and have not identified parameter regimes where
model 2 exhibits such behavior.
We found that by varying additional parameters of
IKS we can mathematically transform the ionic
current equations of model 1 into a "nearly dynamically
equivalent" model that possess the ionic current equations of
model 2. Although the nature of this transformation is
beyond the scope of this paper, the nearly equivalent model
2 possesses a relatively flat interburst interval, and a burst
duration that decreases with depolarization until near the threshold
for beating. However, to achieve these dynamics using the ionic
currents of model 2, we found it necessary to either make
the reversal potential of IKS 55 mV or
incorporate an additional fast-activating gating variable to
IKS such that IKS is only
active at membrane potentials above
55 mV. With either of these
manipulations to model 2, gm
decreases throughout almost the entire silent phase.
The relative flatness of model 1's interburst voltage time
course can be understood as follows. During the interburst phase, the
subthreshold currents are essentially balanced and they sum to zero:
Isub 0. Thus V is essentially at
a hyperpolarized pseudo-steady-state and drifts upwards as the gating
variable [h(t) or k(t)]
slowly evolves. By differentiating this pseudo-steady-state relation, one can estimate the expected variations in V as the slow
gating variable evolves:
V/
h or
V/
k. For model 1, we find that
V/
h is proportional to
INaP (i.e., INaP-h with
h = 1), while for model 2
V/
k is proportional to
KS(V
EK) (i.e., IKS with
k = 1). Because INaP-h is almost
totally deactivated (m
0) in the voltage range of
the interburst phase we see that
V/
h is
much less than
V/
k. This qualitative
prediction of model 1's flat interburst voltage trajectory
was partial motivation for us to develop this mechanism early in our
study. A similar argument shows that the interburst voltage is more
sensitive to Iapp in model 1 than in model 2 by a factor of
2.
Models with a voltage- and time-dependent burst inactivation process
will exhibit some depolarizing trajectory of the interburst membrane
potential, as shown by model 1 and to a greater extent by
model 2. It has been speculated that postburst
hyperpolarization and depolarizing drift of interburst membrane
potential may be a signature of pre-BötC pacemaker cells (see
Rekling and Feldman 1998). Recordings of the candidate
pre-BötC pacemaker neurons, however, can exhibit nearly flat
interburst interval trajectories (Fig. 1) (Smith et al.
1991
). This may indicate the presence of a more complex mix of
subthreshold currents (e.g., Smith et al. 1995
) and/or a
nonuniform spatial distribution of ion channels between the soma and
proximal dendrites (e.g., Li et al. 1996
).
Experiments to test models
In light of the differences discussed above and the robustness of
the dynamics of model 1, we are inclined to present
model 1 as a more feasible mechanism for oscillatory
bursting in pre-BötC neurons. However, due to the limitations
previously discussed, we must test for both mechanisms in vitro. One
test that could support or refute the ionic currents of model
1 would be to measure the membrane conductance at various points
throughout the silent phase of the cycle (Atwater and Rinzel
1986). In model 1, the membrane conductance
increases monotonically as INaP-h recovers from
inactivation (Fig. 4B, 1-4). The recovery of h
and the gradual depolarization of the membrane potential (via the fast
activation of INaP-h) both contribute to an
increase in
Nap-h, and thus gm. The presence of additional
subthreshold currents with voltage-dependent conductances (e.g.,
IR or IH), however, could
alter this trend.
In model 2 during the silent phase of the burst cycle, there are two competing effects with respect to gm. The deactivation of IKS contributes toward a reduction of gm, while the membrane's gradual depolarization activates INaP, contributing toward an increase in gm. Thus the overall change in gm in model 2 depends on the relative strength of INaP and IKS. In Fig. 5 it is shown that gm decreases then increases during the silent phase of the burst cycle.
Another test to differentiate between the two mechanisms would be to
apply a voltage-clamp prepulse to 0 mV for 10 s and then hyperpolarize to 75 mV. In model 2, the prepulse would
activate IKS, and on hyperpolarization (as long
as we hyperpolarize above EK) we would expect to
see a slowly decaying outward current. In model 1 we would
not expect to see the slow decay as long as the clamp potential was
below the activation threshold for INaP-h. Therefore the existence of a slow current following a long depolarized prepulse would demonstrate sufficiency for model 2 and would
be inconclusive for model 1.
Figure 6B quantifies the effect of a mean level of tonic
synaptic drive on the behavior of model 1. This cannot be
tested directly in vitro, since the act of decoupling excitatory
synaptic connections between bursting neurons (e.g., via a low
Ca2+ bathing medium) also blocks tonic synaptic input to
these cells as well. However, this mechanism could be demonstrated
through the use of a dynamic clamp (Sharp et al. 1993)
after synaptic connections have been blocked. Nevertheless these
results illustrate that the depolarizing inputs of extrinsic tonic
drive have similar effects on the activity of model 1 as
varying EL or Iapp.
Similar results to Fig. 6 are obtained as EK is
varied (with IL separated into Na+
and K+ components). We have found that
[K+]o regulates the oscillation frequency of
individual pacemaker-like cells under low-Ca2+ or
6-cyano-7-nitroquinoxaline-2,3-dione conditions (R. Butera, C. Del
Negro, and J. Smith, unpublished data). Similar results to
those shown in Fig. 8 for model 2 are obtained as
EK or gtonic-e is varied.
The subthreshold mechanisms responsible for oscillatory bursting in
pre-BötC neurons may be elucidated more easily experimentally if
the fast Na+ current is blocked, as in Fig. 10. Our models
predict that a subthreshold oscillation should remain under these
conditions, allowing differences in the trajectories and voltage
dependence of potential to be distinguished for the two models.
INaP, in some mammalian neurons, is
TTX-sensitive (Fleidervish and Gutnick 1996),
necessitating block of fast Na+ by intracellular QX-314,
although TTX-insensitive forms of INaP also have
been described (Hoehn et al. 1993
; Oka
1996
).
It also will be necessary for recordings from oscillatory bursting
pre-BötC neurons to possess transient responses similar to those
shown in Figs. 11 and 12. These tests will not provide evidence in
favor of model 1 or model 2 but will simply
demonstrate that either model is consistent with experimental data. The
responses are not unique, however, and a more complex bursting neuron
with additional slow variables and ionic currents may possess similar responses (e.g., Butera et al. 1995, 1997a
; Demir et al.
1997
). Rebound bursting has been proposed as an important mechanism for producing transitions from inactive to active phases in neural oscillators, particularly when there is phasic inhibition
hyperpolarizing burst-generating cells. IH and
IT have been identified as ionic mechanisms that
contribute to rebound bursting, and these currents may exist in
respiratory neurons (Ramirez and Richter 1996
;
Richter 1996
). Our models demonstrate that
INaP-h is another current mechanism promoting
posthyperpolarization bursting (Fig. 11B). This may be functionally important in the respiratory oscillator under conditions where the pacemaker cells are embedded in the respiratory pattern generation network and receive phasic inhibitory inputs (see
discussions in Smith 1997
).
Comparison with other bursting cells
Voltage-dependent frequency control of oscillatory bursting has
been described in a variety of neuronal preparations, including neuron
R15 in Aplysia (Mathieu and Roberge 1971;
Wilson 1982
), neuron AB in stomatogastric ganglion
(Abbott et al. 1991
), medial mammalian body (MMB)
neurons in the guinea pig hypothalamus (Alonso and Llinás
1992
), and magnocellular neurons in the rat hypothalamus (Li and Hatton 1996
). In all of these cases, the
frequency of bursting increases, and the baseline membrane potential is
depolarized as Iapp is increased. Similar
effects of Iapp on burst frequency are exhibited
by models of R15 (Canavier et al. 1991
) and AB
(Abbott et al. 1991
; Epstein and Marder
1990
).
The models presented in this paper often are described as square-wave,
or type I, bursters (Bertram et al. 1995a; Rinzel
1987
). Various models of this type have been used to describe
the electrical bursting activity of pancreatic
-cells
(Bertram et al. 1995b
; Chay and Keizer
1985
; Keizer and Smolen 1991
; Sherman and
Rinzel 1992
). In all of these models, burst termination occurs
by either a slowly activating K+ current or a slowly
inactivating Ca2+ current. Models of electrical bursting in
pancreatic-
cells that rely on either of these slow processes have
used time constants with maximal values of 30-50 s (Bertram et
al. 1995a
; Keizer and Smolen 1991
;
Sherman and Rinzel 1992
). The effect of membrane depolarization in these secretory cells has been studied by modifying a
glucose-dependent conductance and not with Iapp.
Thus we reexamined a few of these
-cell models (Sherman and
Rinzel 1992
; Sherman et al. 1990
; Smolen
and Keizer 1992
) to study the effects of
Iapp and found that all three demonstrate
voltage-dependent frequency control as described in the preceding text.
To our knowledge, the models that we developed here are among the first
examples of type I bursting (Bertram et al. 1995a
;
Rinzel and Lee 1987
) in a neuronal preparation. The
appearance of type-I like bursting behavior has also been reported in
trigeminal motoneurons (Del Negro et al. 1998
).
It is difficult to generalize any principles regarding
voltage-dependent control of burst duration. In some cases, the burst duration increases with Iapp, such as R15
(Canavier et al. 1991; Mathieu and Roberge
1971
; Wilson 1982
) and MMB neurons
(Alonso and Llinás 1992
), whereas in other cases
burst duration decreases (Sherman and Rinzel 1992
;
Sherman et al. 1990
) or has very little change
(Abbott et al. 1991
; Epstein and Marder
1990
). Models with similar general mechanisms for burst
initiation and termination (e.g., model 2 of this paper)
(Sherman and Rinzel 1992
) yield opposite results with
respect to the voltage-dependent control of burst duration. These
results may depend on the interaction of action potential currents with
subthreshold processes, a dynamically complex process (de Vries
and Miura 1998
; Pernarowski et al. 1992
; Terman 1991
) that warrants further theoretical investigation.
Intrinsic control mechanisms
Neuromodulatory afferent inputs to the pre-BötC pacemaker
cells that regulate maximum conductances or voltage-dependent
parameters of the intrinsic subthreshold currents would play a major
role in burst frequency/duration control (see discussions in
Smith 1997; Smith et al. 1995
).
Models 1 and 2 offer uniquely different possibilities with respect to the possible role of subthreshold currents as neuromodulatory targets. For model 1
NaP may be viewed as a tunable parameter
that scales the dynamic range of bursting (Figs. 6 and 7). As
NaP is increased, two effects are observed: the range of values of EL where
bursting is supported is increased and the range of burst periods
possible as EL is varied is increased. The burst
duration is primarily a function of EL and relatively
independent of values of
NaP.
Model 2 offers two distinct subthreshold conductances for
controlling the model's burst dynamics,
NaP and
KS
(Fig. 8). Decreasing
KS and/or increasing
NaP increases burst duration.
KS and
NaP
have complementary roles with respect to controlling the frequency of
bursting:
NaP controls the minimal value
of EL where bursting is supported and the
maximum burst period, whereas
KS controls
the maximal value of EL (or
gtonic-e or Iapp) where bursting is supported and the minimal burst period. Although
KS appears to set an upper bound on the
range of EL values that support bursting, Figure
8B2 illustrates that
KS has a
minimal effect on the overall burst period, which may be considered to
be primarily a function of EL. Thus
KS may serve as a mechanism for varying duty cycle with minimal effects on the overall burst frequency.
In both models, gL provides a mechanism for
controlling baseline membrane potential and burst frequency.
K+-dominated conductances that provide a leakage or
background current in the subthreshold voltage range, including inward
rectifying K+ conductances, have been proposed to play an
important role in control of pre-BötC pacemaker cell activity
(Smith et al. 1995) as a target for neuromodulation. Our
models show that high gL results in quiescence.
As gL is reduced, bursting occurs at a frequency
dependent on the value of gL. Sufficiently low
values of gL cause beating. Thus the
K+-dominated leak conductance can control the activity mode
(quiescence, bursting, beating) and provide frequency control in the
voltage range for bursting.
Functional states of pacemaker neurons
The model pacemaker neurons have several functional states as a
consequence of the voltage-dependent properties of
INaP: quiescence, oscillatory bursting, and
beating. Previously we have called these pacemaker cells conditional
pacemakers to signify that conditions (appropriate ranges for level of
depolarization or magnitude of burst-generating conductances) must
exist for oscillatory bursting to occur (Smith et al. 1991,
1995
). As indicated in Figs. 6 and 7 for model 1,
the magnitude of
NaP determines whether
the neuron will exhibit oscillatory bursting: at low values of
NaP the cells only exhibit quiescence or
beating (Fig. 7), but they still (but not necessarily) may be burst
capable so that a self-terminating burst can be triggered by a brief
transient depolarization. As
NaP is
increased, the cells exhibit a voltage regime where intrinsic oscillatory bursting occurs. In our pacemaker cell population models,
as analyzed in detail in the companion paper (Butera et al.
1999
), we presume that there is ordinarily heterogeneity in values of
NaP and
EL within the cell population such that there is
a distribution of cells in the different states; only a fraction of the
cells (when uncoupled) are in the oscillatory bursting mode. With
excitatory synaptic coupling, synchronous oscillatory bursting occurs
and this heterogeneity results in a functionally more robust oscillator
than if the cells were homogeneous (e.g., all in the oscillatory
bursting mode). Because of the inactivation properties of
INaP and synaptic interactions, synchronous
oscillations can occur in the coupled population even if none (low
INaP) or few of the cells exhibit oscillatory
bursting. As discussed earlier,
NaP and
gL are parameters potentially tunable by
neuromodulation. We therefore view the range of behaviors exhibited by
the pacemaker cells as a functional continuum, and the fraction of
cells in the population that are in any state could be tunable by
neuromodulatory afferent inputs.
Summary
The minimal models we have developed provide plausible mechanisms for generating the multistate, voltage-dependent behavior observed for the candidate pre-BötC pacemaker neurons. We conclude that a burst-generating mechanism dominated by the activation-inactivation properties of a single cationic conductance, postulated to be a persistent Na+ conductance, can account for the voltage-dependent bursting behavior. The dynamics of the burst cycle are controlled principally by the kinetics of inactivation and recovery from inactivation of this conductance. Although the actual burst-generating current in the pre-BötC pacemaker neurons remains to be identified, our model indicates the essential features that are required to produce the experimentally observed bursting behavior.
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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, National Institute of Neurological Disorders and Stroke, for hosting him in the summer of 1998.
This work was supported by the intramural research programs of the National Institutes of Health.
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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