INTRODUCTION

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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). 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.

The hybrid pacemaker-network model departs from previous network-based models (Balis et al. 1994; 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.

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), we extend the analysis to the cell population level and consider a population of synaptically coupled bursting pacemaker neuronsa 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

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) available at http://netlib.cs.utk.edu/ode/cvode.tar.Z. For final simulations, relative and absolute error tolerances were 10⁶ 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.

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; 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): [in this window] [in a new window]   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 (I_NaP). However, burst termination in the two models occurs by contrasting mechanisms. In model 1, burst termination occurs by the slow-inactivation of I_NaP-h, a persistent Na⁺-current with slow inactivation. In model 2, a slowly activating K⁺-current (I_KS) 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:

(1)

where C is the whole cell capacitance (pF), V is membrane potential (mV), and t is time (ms). The ionic currents on the right-hand side are described in the following text. C is set to 21 pF, consistent with whole cell capacitance measurements from inspiratory neurons in vitro (Smith et al. 1992).

(2)

(3)

(4)

(5)

(6)

(7)

View larger version (23K): [in this window] [in a new window]   Fig. 2. Gating and I-V characteristics of components of models 1 and 2. A: spike generating kinetics. m3(V) and h(V) of INa and n(V) and n(V) of IK: note that h = 1  n. B1: gating characteristics of INaP: m(V), h(V), and h(V) (bold). Left: y axis scale for steady-state gating functions. Right: y axis scale for h(V). B2: I-V plots of INaP for h = h(V) and h = 1. First case results in a small window current at subthreshold potentials. Second case corresponds to INaP-h with complete removal of inactivation. C1: gating characteristics of IKS: k(V) and k(V) (bold). Left: y axis scale for activation function; right: y axis scale for k(V). C2: I-V plots of INaP + IKS for k = k(V) and k = 0. First case results in a small current at subthreshold potentials. Second case corresponds to INaP with complete removal of the opposing IKS.

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, I_KS. Thus the current balance equation is

(8)

where C is the same as in model 1. The ionic currents I_Na, I_K, I_L, I_tonic-e, and I_app are identical to those in model 1. I_NaP is identical to I_NaP-h in model 1 except that it does not possess the inactivation term h.

(9)

where _KS = 5.6 nS, _k = 38 mV, _k = 6 mV, _k = 10,000 ms, and E_K = 85 mV. The voltage dependency of the gating kinetics for k was chosen so that I_KS was activated by the firing of action potentials and was also dynamically active in the subthreshold potential range from 60 to 45 mV. Figure 2C1 illustrates k(V) and _k(V). Figure 2C2 illustrates I_NaP + I_KS when k = k(V) (slow voltage ramp with other membrane currents eliminated) and k = 0 (removal of I_KS, similar to a fast voltage ramp). The shapes of these plots of I_NaP + I_KS are similar to those of I_NaP-h illustrated in Fig. 2B2.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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 I_app to control the neuron's level of depolarization is equivalent to varying E_L (I_app = _LE_L).

View larger version (37K): [in this window] [in a new window]   Fig. 3. Mechanisms for controlling voltage-dependent activity in a single bursting neuron. Baseline membrane potential of the model may be controlled by application of an extrinsic current pulse (Iapp), changing [K+]o (which in turn affects EK) or adjusting the level of tonic synaptic input, the mean level of which is represented by gtonic-e.

Figure 4 illustrates the effect of varying E_L 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 I_NaP-h inactivation (h) and instantaneous membrane conductance are illustrated in Fig. 4B, 1-4. As E_L 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 E_L is increased. V_min, the minimum membrane potential encountered during the silent phase of the burst cycle, varies from 58 mV at the onset of bursting activity (E_L = 60.5 mV) to 48 mV (E_L = 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, I_NaP-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 I_NaP-h recovers from inactivation. Similar results to those in Fig. 4A are obtained if E_L is separated into K⁺ and Na⁺ components and E_K is varied.

View larger version (45K): [in this window] [in a new window]   Fig. 4. Dynamic response of model 1 as a function of EL. Left: (A, 1-4) membrane potential. Middle: (B, 1-4) instantaneous membrane conductance (gm) and inactivation of INaP-h (h) as cell is depolarized (EL = 65, 60, 57.5, and 54 mV for 1-4, respectively). C2/D2 and C3/D3: expanded time course of V and h and the spike-frequency profile () during 1 burst in A2/B2 and A3/B3. Instantaneous conductance plots are truncated during the firing of action potentials.

Modulating the level of tonic drive can bias the neurons to different activity modes. This is analyzed in our model by varying g_tonic-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 g_tonic-e on the mode of activity and pattern of bursting are similar to those of E_L (quantified in the following text).

Figure 5 illustrates the effect of varying E_L 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 I_KS 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 E_L is increased, the model progresses through regimes of silence, bursting, and beating activity. During oscillatory bursting, the burst is initiated by I_NaP and terminates when I_KS has been activated sufficiently. The spike frequency decays throughout the active phase of the burst. The current I_KS is activated slowly and progressively with the firing of each action potential and deactivates during the interburst interval. The instantaneous membrane conductance (g_m) decreases throughout most of the silent phase and increases toward the end of the phase. This differs from model 1, where g_m steadily increases throughout the silent phase. This difference in the time course of g_m may provide a means for differentiating between the mechanisms of the two models in vitro (see DISCUSSION).

View larger version (43K): [in this window] [in a new window]   Fig. 5. Dynamic response of model 2 as a function of EL. Left: (A, 1-4) membrane potential. Right: (B, 1-4) instantaneous membrane conductance (g) and activation of IKS (k) as cell is depolarized (EL = 65, 59.5, 50, and 40 mV for 1-4, respectively). C2/D2 and C3/D3: expanded time course of V and k and the spike-frequency profile () during 1 burst in A2/B2 and A3/B3. Instantaneous conductance plots are truncated during the firing of action potentials.

The dynamics of model 2 were investigated across a wide range of parameters for I_KS and I_NaP. 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 g_m 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 E_L or g_tonic-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 E_L (Fig. 6A) and g_tonic-e (Fig. 6B) are varied. Equivalent values of I_app with E_L fixed to 65 mV are also shown on A. 

View larger version (26K): [in this window] [in a new window]   Fig. 6. Intrinsic and extrinsic mechanisms for frequency control in model 1. Panels show effect of varying EL (A) and gtonic-e (B) on burst duration, burst period, burst frequency, and beat frequency. These curves were calculated for multiple values of NaP. Bursting activity does not occur for NaP = 2.0 nS in A and for NaP = 2.0 and 2.4 nS in B. IAPP below EL scale shows equivalent variation in IAPP with EL fixed to 65 mV.

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 g_NaP 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 E_L where bursting is supported. We also have observed (not shown) that increasing _NaP increases the range of values of V_min encountered as a function of E_L during oscillatory bursting. At a given level of E_L, increasing _NaP increases burst frequency.

View larger version (41K): [in this window] [in a new window]   Fig. 7. Intrinsic modes of single-cell activity of model 1 as a function of NaP and EL. When NaP exceeds 2.2 nS, the model neuron exhibits 3 functional states: silent, oscillatory bursting, and beating. Below 2.2 nS, oscillatory bursting does not occur at any level of EL. Range of NaP used in the figure spans those values used in the simulations presented in this and the companion paper (Butera et al. 1999).

Figure 8 quantifies the effects of E_L 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 E_L 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 E_L (and I_app) approximately twice as large as the range of E_L 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 g_tonic-e was varied.

View larger version (28K): [in this window] [in a new window]   Fig. 8. Mechanisms for frequency control in model 2. Panels show effect of varying EL on burst duration, burst period, burst frequency, and beat frequency of model 2. These curves were calculated for multiple values of NaP (A) and KS (B). IAPP below EL scale shows equivalent variation in IAPP with EL fixed to 65 mV.

The large range of I_app in model 2 may be attributed to the difference in whole cell conductance (g_m) 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 g_m means that the "gain" of the cell is decreased. A larger increment in I_app 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 I_app 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 E_L or I_app 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 I_NaP) 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 E_L and I_app 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 (I_sub, see following text) alone, the corresponding sensitivities for the bursting to beating transition are difficult to intuit because the components of I_sub 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 E_L, 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 (I_sub  I_L + I_NaP-h for model 1, I_sub  I_L +I_NaP + I_KS for model 2). This explanation suffices because without the action potential currents I_Na and I_K, reduced models having only the I_sub 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 I_sub in cases shown in Figs. 4 and 5. The SS curves (thick lines) represent I_sub 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 I_sub 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.

View larger version (28K): [in this window] [in a new window]   Fig. 9. Subthreshold currents of model 1 (INaP-h + IL) (left) and model 2 (INaP + IKS + IL) (right) as EL is varied. Values of EL correspond to values used in Figs. 4 and 5 and are 65, 60, 57.5, and 54 mV (A, 1-4) and 65, 59.5, 50, and 40 mV (B, 1-4). Thick lines are steady-state I-V curves [h = h(V) in A, 1-4, k = k(V) in B, 1-4]. Thin I-V plots are quasi-steady-state (QSS) I-V curves, where the slow variable h (A, 1-4) or k (B, 1-4) is fixed to a particular value. In A1 and B1, these values are indicated on the figure. In A, 2 and 3, and B, 2 and 3, these values correspond to the values of h or k at the beginning (a) and end (b) of the active phase of the burst cycle (I and II, top). In A4 and B4, the QSS I-V curve was generated by setting h or k to the mean value during tonic firing (see Figs. 4B4 and 5B4).

For all values of E_L 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 (C_M/g_m), 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 E_L = 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 I_sub crosses the 0 nA axis, is stable. The equilibrium point is stable because I_NaP-h is relatively deactivated and I_NaP-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 E_L 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), I_sub(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 I_sub 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 I_sub 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 I_NaP-h to I_sub, shifting the QSS I-V curve outward. Firing persists until I_sub is net outward at V_min, 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 I_sub 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 I_NaP-h is necessary to counterbalance the reduced outward current from shifting E_L. 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 E_L, the cell does not hyperpolarize as far at the end of each burst cycle, and less time is necessary to recover from inactivation of I_NaP-h to begin the following burst. At all values of E_L where bursting occurs, I_NaP-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, E_L = 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 I_sub 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 I_Na 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.

View larger version (17K): [in this window] [in a new window]   Fig. 10. Oscillatory model behavior in the absence of fast Na+ currents. Panels show effects of setting Na to 0 in model 1 (A, 1-5) and in model 2 (B, 1-5). Parameter sets for A, 1-4, and B, 1-4, correspond to parameters used in Figs. 4A, 1-4, and 5 A, 1-4, respectively. Parameters for A5 and B5 are EL = 53 mV and EL = 32 mV, respectively. - - -, 60 and 30 mV references.

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 E_L and _NaP where the model is initially silent, providing that _NaP is not too small. At rest, I_NaP-h is relatively deactivated (m is low) and deinactivated (h is high). A brief depolarization activates I_NaP-h. Because I_NaP-h inactivates slowly, I_NaP-h remains relatively deinactivated and a larger conductance for I_NaP-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.

View larger version (19K): [in this window] [in a new window]   Fig. 11. Transient burst responses. A: brief but sufficiently large transient depolarizing input can elicit a single sustained burst. Model was at rest (EL = 65 mV), and depolarizing current pulses (50 ms, 10 and 15 pA) were applied. B: transient hyperpolarizing input of sufficient magnitude and duration may elicit a single posthyperpolarization rebound burst. Model was at rest (EL = 62 mV), and a hyperpolarizing current pulse (500 ms, 60 pA) was applied. In both cases, the parameters EL and NaP correspond to points in Fig. 7 to the left of the bursting regime.

A single burst also may be elicited as a form of posthyperpolarization rebound (Fig. 11B). A sustained hyperpolarizing input removes some inactivation from I_NaP-h (i.e., h increases). On abrupt release of the hyperpolarizing input, the model depolarizes. Because inactivation of I_NaP-h develops slowly, I_NaP-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 I_NaP-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 E_L = 62 mV. When E_L = 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 I_NaP-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).

View larger version (22K): [in this window] [in a new window]   Fig. 12. Resetting response. Brief but sufficiently strong transient hyperpolarizing input can reset a burst. Simulations performed with EL = 59 mV, which corresponds to a burst period of ~4 s. Control period shown in A, with bursts occurring at times 1 and 2. Simulations in B and C were started with identical initial conditions, and a 50-ms, 10-pA hyperpolarizing input was applied early (b1) or late (c1) in the burst cycle. In both cases, the brief input reset the burst with the following burst occurring earlier. Duration until the subsequent burst was proportional to how early or late in the burst the transient input was applied.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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 (E_L); 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