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Ionic Current Model of a Hypoglossal Motoneuron

Laboratory for Neuroengineering, Georgia Institute of Technology, Atlanta, Georgia

Submitted 9 July 2004; accepted in final form 28 September 2004

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX: MODEL EQUATIONS GRANTS ACKNOWLEDGMENTS REFERENCES

 
We have developed a single-compartment, electrophysiological, hypoglossal motoneuron (HM) model based primarily on experimental data from neonatal rat HMs. The model is able to reproduce the fine features of the HM action potential: the fast afterhyperpolarization, the afterdepolarization, and the medium-duration afterhyperpolarization (mAHP). The model also reproduces the repetitive firing properties seen in neonatal HMs and replicates the neuron's response to pharmacological experiments. The model was used to study the role of specific ionic currents in HM firing and how variations in the densities of these currents may account for age-dependent changes in excitability seen in HMs. By varying the density of a fast inactivating calcium current, the model alternates between accelerating and adapting firing patterns. Modeling the age-dependent increase in H current density accounts for the decrease in mAHP duration observed experimentally, but does not fully account for the decrease in input resistance. An increase in the density of the voltage-dependent potassium currents and the H current is required to account for the decrease in input resistance. These changes also account for the age-dependent decrease in action potential duration.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX: MODEL EQUATIONS GRANTS ACKNOWLEDGMENTS REFERENCES

 
The hypoglossal motoneuron (HM) innervates the tongue and along with its roles in mastication and swallowing, the tongue plays an important role in breathing (Bartlett et al. 1990; Lowe 1980; reviewed in Sawczuk and Mosier 2001). The muscle of the tongue contributes to effective breathing by maintaining a patent upper airway and problems with the control of HMs can cause the breathing disorder obstructive sleep apnea (Fogel et al. 2000; Remmers et al. 1978). For this reason, extensive experimental work has been done on these motoneurons. In this communication, we present a computational model of a HM that provides a framework for analyzing the role of ionic currents in these neurons.

The HM is well suited for modeling because the ionic currents found in rat HMs have been thoroughly studied for over a decade (reviewed in Berger 2000). Numerous ionic currents have been identified in neonatal rat HMs and characterized by voltage-clamp experiments. Besides the sodium and potassium channels that generate the action potential (Haddad et al. 1990; Lape and Nistri 1999, 2001; Powers and Binder 2003; Viana et al. 1993a, b), HMs have several different types of calcium channels (Powers and Binder 2003; Umemiya and Berger 1994, 1995; Viana et al. 1993a) as well as a calcium-dependent potassium channel (Lape and Nistri 2000; Sawczuk et al. 1997; Viana et al. 1993b). HMs also include a hyperpolarization-activated cation channel whose density changes with age (Bayliss et al. 1994). Other age-dependent changes that have been found are input resistance (R_N), action potential duration, and firing behavior (Viana et al. 1994, 1995). The firing behavior of neonatal HMs changes from a decrementing firing pattern during the 1st wk postnatal to an incrementing firing pattern during the 2nd wk postnatal (Viana et al. 1995). These changes are investigated in the model by varying the densities of the ionic currents.

Previous modeling work from our laboratory has focused on models of rhythmic activity in a transverse brain slice preparation containing the pre-Bötzinger complex (pBC) (Butera et al. 1999a, b; Del Negro et al. 2001). The pBC is a subregion of the ventrolateral medulla and contains a population of cells that is a critical component of the respiratory rhythm generating network in mammals (Gray et al. 2001; Smith et al. 1991). The hypoglossal nucleus is also included in this transverse slice preparation (Bou-Flores and Berger 2001; Smith et al. 1991), and receives excitatory synaptic input from the pBC (Funk et al. 1993). Recordings from the rootlets of this nucleus are commonly used experimentally as a measure of the output of the network (Del Negro et al. 2001; Smith et al. 1991). A model of the HM is required to model the transformation of the pattern generating rhythm in the pBC to effector motoneurons.

The model is a Hodgkin–Huxley (Hodgkin and Huxley 1952) style, single-compartment, electrophysiological model of a HM from a neonatal rat containing a repertoire of ionic currents. The currents found in our HM model include the fast sodium and delayed-rectifier potassium currents, high- and low-voltage-activated calcium currents, voltage- and calcium-dependent potassium currents, a hyperpolarization-activated cationic current, and a persistent sodium current. The model is composed of these currents with parameters determined from experimental data where possible, along with a simplified model of the dynamics of internal calcium concentration used by other investigators (Bertram 1993; Booth et al. 1997). The model reproduces several characteristics of HMs and replicates the neuron's response to pharmacological experiments. The model is used to explore the roles of each current in shaping action potential dynamics, and to explore the age-dependent changes seen in HMs.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX: MODEL EQUATIONS GRANTS ACKNOWLEDGMENTS REFERENCES

 
Simulations

Simulations were performed using the interactive differential equation simulation package XPP (Ermentrout 2002). The integration was done using the Dormand–Prince integrator built into XPP. MATLAB (The MathWorks, Natick, MA) was also used for model formulation and data analysis.

Formulation of model

The model is based on a single-compartment Hodgkin–Huxley formalism. The membrane potential is found using the differential equation

where V is the membrane potential (mV), C_m is the whole cell capacitance (nF), t is time (ms), I_stim is the applied stimulus current (nA), and I_ionic are the ionic currents listed in the APPENDIX and have the form

where E_ionic is the equilibrium reversal potential for the ionic species carried by the current and

where is the maximum conductance of each current and y is the product of one or more gating variables raised to integer powers, as described below.

The dynamics of the conductances of the ionic currents regulated by voltage-dependent activation or inactivation variables are described according to

where x(V) is the steady-state voltage-dependent (in)activation function of x and _x(V) is the voltage-dependent time constant. x(V) is a sigmoid with a half-(in)activation at V = _x and a slope factor _x. _x(V) is a bell-shaped curve. If no experimental data exist for time constant measurements, or no significant change in the model results from including a voltage-dependent time constant, it is represented by a constant B.

The dynamics of the calcium-dependent potassium (SK) conductance is described according to the same equation for dx/dt, but x(V) is replaced by a Michaelis–Menten binding curve with a Hill coefficient of 2

where [Ca²⁺]_i is the internal calcium concentration of the cell with a half-activation at [Ca²⁺]_i = _C. This SK current model was previously used by Engel et al. (1999).

The modeling of calcium dynamics is performed by considering the cell to contain a single compartment with intracellular calcium accumulation and degradation of the form

where I_Ca is the sum of the 3 calcium currents, K₁ is an accumulation scaling factor, and K₂ is a decay rate constant that represents calcium uptake and diffusion. A similar model has been used by other investigators in the past (Bertram 1993; Booth et al. 1997). In the model, the calcium-activated potassium (SK) current is the only current whose conductance depends on the internal calcium concentration.

All state variables and initial conditions are listed in Table 1. The maximum conductance and reversal potential for each current, along with various other parameters, are listed in Table 2.

	MODEL DEVELOPMENT

TOP ABSTRACT INTRODUCTION METHODS MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX: MODEL EQUATIONS GRANTS ACKNOWLEDGMENTS REFERENCES

HMs have a fast sodium current (I_Na) that when blocked by tetrodotoxin (TTX) removes the ability of the cell to generate an action potential (Haddad et al. 1990; Lape and Nistri 2001; Viana et al. 1993a). In addition, the voltage-ramp data of Powers and Binder (2003) suggest the existence of a persistent sodium current (I_NaP) in HMs. Those data show an increase in sodium current with increased ramp speed, and no decrease in sodium current at voltages above –30 mV after application of TTX, which indicates a slowly inactivating sodium current (herein, persistent will mean slowly inactivating). Lape and Nistri (2001) present voltage-clamp data for the sodium current, but do not distinguish between the fast sodium and persistent sodium currents as was done by Rybak et al. (2003). Therefore parameters for the fast sodium current in the model are more depolarized than the data of Lape and Nistri (2001). Because the persistent sodium current has not yet been fully described in HMs, the parameters for this current are based on data from other neurons (Kononenko et al. 2004; Rybak et al. 2003). Kononenko et al. (2004) report a time constant of inactivation of 50–250 ms at the peak of activation. The value for the time constant of inactivation of I_NaP chosen for the model is in this range [_NaP(V) = 150 ms].

Voltage-dependent potassium currents

Another current found in rat HMs is a tetraethylammonium (TEA)-sensitive potassium current (I_K), the delayed rectifier (Haddad et al. 1990; Lape and Nistri 1999; Viana et al. 1993b). HMs also possess a potassium current that is sensitive to 4-aminopyridine (4-AP). This current is thought to be the fast transient A-type current (Lape and Nistri 1999; Viana et al. 1993b). Viana et al. (1993b) show a dose-dependent increase in action potential duration with application of TEA and 4-AP in neonatal rat HMs, indicating the importance of both of these currents in shaping the action potential. Parameters for I_K and I_A in the model are based on the voltage-clamp data of Lape and Nistri (1999, 2000), although the time constants used are faster than the reported values to produce an appropriate action potential duration (<2 ms; Viana et al. 1994). The model does not distinguish between (in)activation and de(in)activation as reported for the time constant data of I_A (Lape and Nistri 1999).

Calcium currents

It has been shown that rat HMs have both low-voltage-activated (LVA) and high-voltage-activated (HVA) calcium currents (Powers and Binder 2003; Umemiya and Berger 1994, 1995; Viana et al. 1993a). The LVA calcium current is a T-type current (Umemiya and Berger 1994). The properties of the T-type current were found by characterizing the residual current after application of antagonists to the 3 HVA currents. The 3 HVA currents found in HMs are P-type, N-type, and L-type (Powers and Binder 2003; Umemiya and Berger 1995). Umemiya and Berger (1995) report half-(in)activation and time constant values for these currents found by single-channel recordings. The time constants used in the model are based on these recordings; however, the half-(in)activation values are shifted to a more depolarized value to provide a better fit to whole cell data (Umemiya and Berger 1994; Viana et al. 1993a). The half-activation value for I_N required a considerable hyperpolarizing shift (55 mV) from the reported value of 25 mV to provide a significant contribution to the total calcium current that is consistent with the whole cell data. The L-type HVA current in HMs has dynamics similar to that of the P-type channel (i.e., noninactivating, high-voltage-activated; Umemiya and Berger 1995). However, the L-type current makes up <10% of the total calcium current (Powers and Binder 2003; Umemiya and Berger 1994). For these reasons, the L-type current is not included in this model.

Calcium-activated potassium current

One important feature of rat HMs is the medium-duration afterhyperpolarization (mAHP) after the action potential (Fig. 1). Previous experimental work has shown that the mAHP in rat HMs is calcium dependent, has a reversal potential near that of potassium, and is blocked by apamin (Lape and Nistri 2000; Sawczuk et al. 1997; Viana et al. 1993b). This indicated the existence of the calcium-activated potassium current, or SK current (I_SK). Lape and Nistri (2000) made several important observations by subtracting outward currents measured from voltage-clamp data before and after application of apamin: 1) membrane depolarizations as short as 1 ms produce enough calcium entry to evoke measurable SK currents; 2) the membrane conductance underlying I_SK is voltage independent; 3) apamin does not affect resting membrane potential, suggesting I_SK is not activated at resting calcium concentrations; and 4) the I–V relation indicated an activation region between –40 and –10 mV. The model uses a noninactivating and voltage-independent conductance that activates with an increase in internal calcium concentration, similar to that used by other researchers (Engel et al. 1999). This current, along with the model of internal calcium, satisfies the observations listed above through: 1) fast activation, 2) no voltage-dependent gating variables, 3) an appropriate half-activation calcium level, and 4) dependency on activation of calcium currents in the –40 to –10 mV range.

View larger version (8K): [in this window] [in a new window]   FIG. 1. Action potential. Simulation results of the membrane potential during an action potential showing the afterpotentials seen in a neonatal rat hypoglossal motoneuron (HM). Action potential is evoked by a brief (1-ms) current pulse of 1 nA. Action potential is truncated to emphasize the fast afterhyperpolarization (fAHP), afterdepolarization (ADP), and medium-duration afterhyperpolarization (mAHP) after the spike.

HMs also have a hyperpolarization-activated cationic current (I_H) that shows a large increase during postnatal development (Bayliss et al. 1994; Haddad et al. 1990). In neonatal rat HMs, the H current is only about one-tenth the density of that in adult rat HMs (Bayliss et al. 1994). The H current activates much more slowly than the other currents in HMs, with a peak activation time constant >200 ms (compared with about 10 ms for the calcium currents).

Passive properties

The membrane capacitance of the model is set to match experimental data (Lape and Nistri 1999; Umemiya and Berger 1994). Resting potential and R_N were set to match experimentally determined values for neonatal rat HMs (Viana et al. 1994) by adjusting the maximum conductances of the currents active at rest. The model's resting membrane potential is –72 mV. This value is within the range measured by many different observers for neonatal HMs (Haddad et al. 1990; Lape and Nistri 2000; Viana et al. 1994). The R_N of neonatal rat HMs is reported to be about 35 M by Viana et al. (1994), about 120 M by Robinson and Cameron (2000), and about 400 M by Lape and Nistri (1999). The large difference in observed R_N is probably attributable in part to the different recording techniques and the use of sharp electrodes (Viana et al. 1994) versus whole cell patch-clamp (Lape and Nistri 1999; Robinson and Cameron 2000). The difference between the 2 values found using whole cell patch-clamp techniques could be explained by the time elapsed before recording R_N, as suggested by Robinson and Cameron (2000). The R_N of the model is around 40 M, closer to that reported by Viana et al. (1994).

Maximum conductances

As stated above, the maximum conductances of the currents active at rest were set to match experimental data for R_N and resting potential. Also, the maximum conductance of I_Na, I_NaP, I_K, and I_A were adjusted to provide an action potential of the appropriate amplitude and duration (Viana et al. 1994). The maximum conductance of the calcium currents were then constrained to match whole cell voltage-clamp data (Umemiya and Berger 1994; Viana et al. 1993a). The maximum conductance of I_SK was chosen to provide a mAHP of appropriate amplitude and to reproduce the behaviors seen during repetitive firing (Lape and Nistri 2000; Viana et al. 1993b, 1994, 1995). The maximum conductance of I_H was set to match the voltage-clamp data of Bayliss et al. (1994) for neonatal HMs. All of these parameters are listed in Table 2.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX: MODEL EQUATIONS GRANTS ACKNOWLEDGMENTS REFERENCES

 
Action potential

A typical action potential produced by the model is shown in Fig. 1. The action potential is elicited by a 1-ms depolarizing current pulse of 1 nA. Following the standard fast depolarization and repolarization, HM action potentials display 3 additional features: a fast afterhyperpolarization (fAHP), an afterdepolarization (ADP), and a medium-duration afterhyperpolarization (mAHP). HMs do not fire spontaneously and thus require a stimulus current to elicit an action potential. The stimulus current causes a sudden activation of the sodium currents, which leads to the rapid depolarization of the action potential. The repolarization is caused by a combination of sodium current inactivation, activation of the delayed rectifier I_K, and activation of the fast transient potassium current I_A. If the sodium currents are removed from the model, no action potential is generated. Removing I_K from the model extends the duration of the action potential from 1.4 to about 4 ms. If I_A is removed, the action potential duration is increased by about 0.5 ms and the resting potential is increased. Removing I_A also allows burst firing in the model, like that seen in some very young HMs (Viana et al. 1993a). These 4 currents, in combination with the slower activation of the calcium currents, cause the fAHP. A plot of the behavior of I_Na, I_NaP, I_K, I_A, and I_leak during an action potential is illustrated in Fig. 2, A and C. Because the conductance of I_H does not vary significantly over a single action potential, the voltage trace of I_H is similar in shape to that of I_leak during a single action potential and is not included in this figure. During repetitive firing, however, the magnitude of I_H does vary from spike to spike.

View larger version (18K): [in this window] [in a new window]   FIG. 2. Model currents during an action potential. Currents active during the action potential shown in Fig. 1. A 2-ms timescale is used in A and C, and a 20-ms timescale is used in B and D. A: Na and NaP currents are active during the first 2 ms of the action potential. B: calcium currents are active during the first 20 ms of the action potential. I-Ca is the sum of the 3 calcium currents. C: K, A, and leak currents are outward currents active during the first 2 ms. D: SK current activates after the calcium currents and stays on throughout the remainder of the mAHP.

View larger version (11K): [in this window] [in a new window]   FIG. 3. ADP. A: dependency of the ADP on the low-voltage-activated (LVA) calcium current, IT. A 50% increase in T results in a 1.8-mV increase in the height of the ADP. B: voltage dependency of the ADP. Holding potentials of –72, –75, and–78 mV produced by hyperpolarizing prepulses of 0, –0.1, and –0.2 nA, respectively. Voltage traces are superimposed and truncated to show the increase in ADP amplitude from baseline membrane potential.

Apamin simulation

The importance of the SK current in producing the mAHP is clearly identified by apamin, which selectively blocks SK channels (Blatz and Magleby 1986). Viana et al. (1993b) show that apamin reduces the peak amplitude of the mAHP by 90% in HMs, while having no effect on HM passive properties, the depolarizing phase of the action potential, or the time course of repolarization. Lape and Nistri (2000) also report a complete block of the mAHP in HMs on application of apamin. The model's response to apamin is simulated by setting the maximum conductance of I_SK to zero (_SK = 0), effectively removing the current from the model. Figure 4A illustrates the result of this simulated experiment where a single action potential is elicited by a brief pulse of current. With _SK = 0, the mAHP's dependency on the SK current in the model is clearly seen. As reported (Viana et al. 1993b), the mAHP is blocked without affecting the passive properties or the depolarizing/repolarizing phase of the action potential. There is also an increase in ADP height, consistent with experimental findings (Lape and Nistri 2000; Viana et al. 1993b).

View larger version (24K): [in this window] [in a new window]   FIG. 4. Apamin simulation. Effects of apamin are simulated by reducing the maximum conductance of the SK current. A: response of the model to a 1-ms current pulse stimulus, SK = 0 µS. ADP is enhanced and the mAHP is removed. B: response of the model to an extended current pulse with a reduced amplitude (Istim = 0.33 nA), SK = 0.03 µS. Arrow indicates stimulus onset.

Firing properties

NEAR THRESHOLD. When the stimulus is near the threshold for firing, some HMs display 2 features that can be reproduced by the model (Fig. 5). The first feature is spike frequency adaptation leading to a cessation of firing after several spikes (Fig. 5A). The number of spikes before cessation is determined by the stimulus amplitude. A slightly smaller (or larger) stimulus amplitude will cause fewer (or more) spikes before ceasing. The persistent sodium current's contribution to spike frequency adaptation and AP generation can be seen here. It is the decrease in I_NaP amplitude resulting from slow inactivation that slows down the firing rate and leads to cessation of firing. If I_NaP is removed from the model, there is no stimulus amplitude that will show adaptation or lead to a cessation of firing once firing has been initiated. The second feature reproduced by the model is delayed excitation (DE), which is a common phenomenon caused by I_A where the onset of the first spike is delayed as a result of the large initial potassium current (Fig. 5B). A small amount of DE can be seen in HM recordings (Lape and Nistri 2000; Viana et al. 1993b).

View larger version (19K): [in this window] [in a new window]   FIG. 5. Near threshold firing. Response of the model to a stimulus amplitude near the threshold for firing action potentials, Istim = 0.22 nA. A: model shows spike frequency adaptation followed by a cessation of firing. B, inset: * in A showing delayed excitation (DE) of the first spike. Arrow indicates stimulus onset.

View larger version (20K): [in this window] [in a new window]   FIG. 6. Spike frequency adaptation. Stimulus current is a 250-ms step of current. A: a train of action potentials generated by a 2.0-nA stimulus current. B: frequency–time measurements. Amount of adaptation increases with an increase in stimulus current amplitude. Adaptation occurs during the first 2 interspike intervals. Arrow indicates stimulus onset.

View larger version (18K): [in this window] [in a new window]   FIG. 7. Spike frequency acceleration. T current is increased by 10-fold (see Fig. 8) from that shown in Fig. 6. Stimulus current is a 250-ms-long step of current. A: a train of action potentials generated by a 2.0-nA stimulus current. B: frequency–time measurements. Amount of acceleration increases with an increase in stimulus current amplitude. Acceleration occurs during the first few interspike intervals. Arrow indicates stimulus onset.

ADAPTATION VERSUS ACCELERATION. To identify a possible mechanism that explains the variation in firing characteristics during postnatal development, we considered 2 relevant experimental results. First, the calcium-dependent potassium current, I_SK, plays an important role in the frequency control of HMs (Lape and Nistri 2000; Sawczuk et al. 1997; Viana et al. 1993b). Second, experiments have shown that the contribution of the T current to the total calcium current changes during neonatal development (Umemiya and Berger 1994). Because I_T affects I_SK through the buildup of internal calcium concentrations, and I_SK affects firing frequency, we hypothesized that the change in density of I_T plays a role in the age-dependent firing properties of HMs. To test this hypothesis, we varied the density of T channels in the model by varying _T over a 10-fold range and measured changes of the first 2 interspike intervals (ISIs). The ISIs are measured from the peak of one spike to the peak of the following spike. Because adaptation (or acceleration) occurs during the first 3 spikes, only the first and second ISIs are considered. Figure 8A illustrates the effect of varying _T on the first 2 ISIs. At low values of _T, the first ISI is smaller than the second ISI (i.e., the frequency decreases with time, or adaptation). An increase in _T increases the duration of the first ISI at a rate greater than that of the second ISI. At large values of _T, this leads to the first ISI being greater than the second ISI (i.e., the frequency increases with time, or acceleration). Where the 2 curves meet, the first ISI is equal to the second ISI and there is no change in frequency over time. This constant firing rate is also a feature seen in neonatal HMs (Viana et al. 1995). Because blocking the N current can eliminate the mAHP (Viana et al. 1993b), we also varied the density of the N current and measured its affects on the first 2 ISIs. The results of varying _N are similar to those of _T (Fig. 8B).

View larger version (20K): [in this window] [in a new window]   FIG. 8. Adaptation vs. acceleration. Effect of varying the T and N current densities on firing behavior. Interspike interval (ISI) is measured from the peak of one spike to the peak of the next spike. If the first ISI is smaller than the second ISI, adaptation occurs; if the first ISI is larger than the second ISI, acceleration occurs. A and B: ICa = IP + IN + IT. Increasing T (A) or N (B) changes the firing behavior from adaptation to acceleration. C: ICa = IP + IN. Adaptation is seen for all values of T. D: ICa = IP + IT. Adaptation is seen for all values of N. T is reduced to 0.02 µS for B and D.

MAHP DURATION AND THE H CURRENT. Along with changes in firing properties, other changes accompany the HMs transition from neonatal to adulthood. These changes include a decrease in the duration of the mAHP and a large increase in the density of I_H (Bayliss et al. 1994; Haddad et al. 1990; Viana et al. 1994). Bayliss et al. (1994) showed that the density of H channels increases by a factor of 10 from neonatal rats to adult rats. Therefore we increased _H and measured its effects on the model. The most significant effect was seen on the duration of the mAHP (measured from the repolarizing phase of the action potential to the point in the AHP that reaches the resting potential). Neonatal HMs have a mAHP duration of 110 ms, and adult HMs have a mAHP duration of 75 ms (Viana et al. 1994). Figure 9 is a plot of mAHP duration versus _H. The original model is of a neonatal HM and consequently has a mAHP duration of 109 ms. Increasing _H by a factor of 10 shortens the mAHP duration to 77 ms in the model, which is very similar to what is seen in the data (Viana et al. 1994). This model experiment indicates that the change in the density of the H current in the data can potentially completely account for the change in mAHP duration in the data. Thus our model supports this contribution of the H current suggested by Bayliss et al. (1994).

View larger version (13K): [in this window] [in a new window]   FIG. 9. H current vs. mAHP duration. Effect of varying H current density on mAHP duration. A 10-fold increase in H decreases the mAHP duration from that of neonatal HMs (about 110 ms) to that of adult HMs (about 75 ms).

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX: MODEL EQUATIONS GRANTS ACKNOWLEDGMENTS REFERENCES

 
Powers (1993) developed a variable-threshold model of a cat alpha-motoneuron that was able to account for several motoneuron behaviors. That model is a simplified motoneuron model and is well suited for large networks. Powers et al. (1999) also developed other simplified motoneuron models to explore the 3 phases of adaptation seen in adult HMs (Sawczuk et al. 1995). Our model is more complex and is derived mostly from data taken from neonatal HMs. The model is able to reproduce the fine details of a HM action potential including the fAHP, the ADP, and the mAHP. The model reproduces both types of firing behavior observed in neonatal HMs: adaptation and acceleration. We proposed possible explanations for the age-dependent differences in firing properties, mAHP duration, resting R_N, and action potential duration.

Apamin simulation

The model faithfully reproduces the response to a brief pulse of current when the SK current is removed. As in the experimental results, the passive properties and the depolarizing/repolarizing phase of the action potential are unchanged. This is accompanied by an enhancement of the ADP and the absence of the mAHP. Here, the critical role of the SK current in generating the mAHP can be clearly seen. The importance of I_SK for repetitive firing is revealed when the brief pulse of current is lengthened to a longer step of current. Reducing _SK from 0.3 to 0.03 µS leads to a large increase in firing frequency accompanied with spike frequency adaptation. A similar response is obtained by reducing the total calcium current in the model to zero (not shown). Under either of these conditions, the HM model is able to display calcium-independent spike frequency adaptation, which has been shown to occur experimentally (Lape and Nistri 2000; Sawczuk et al. 1997; Viana et al. 1993b). In the model, this adaptation is caused by the slow inactivation of I_NaP. However, under normal conditions, calcium and calcium-sensitive currents dominate the firing behavior of the model.

For the simulation in Fig. 4B, _SK was reduced to 0.03 µS and not 0 µS. When I_SK is completely removed (_SK = 0), the model does not display the initial spike frequency adaptation. Instead, the model displays acceleration caused by inactivation of the A current. This contradicts the role of the A current suggested by Lape and Nistri (1999) where they propose that the A current causes adaptation in HMs. The acceleration occurs regardless of whether the model parameters (before reduction of _SK attributed to apamin) are set to produce adaptation, as in Fig. 6, or acceleration, as in Fig. 7. As discussed in MODEL DEVELOPMENT, our implementation of the A current is simpler than the description given by Lape and Nistri (1999). Alternatively, a multicompartment model similar to that of Booth and Rinzel (1995) that physically separates the fast and slow currents may provide a more robust response.

Age-dependent changes

The model provides a possible explanation for the change in repetitive firing properties observed during development. If I_T or I_N is included in the calcium equation, then increasing the density of that current in the model changes the firing properties from adaptation to acceleration (Fig. 8, A and B). When the current density is low, the SK current is only partially activated during the first spike. The level of internal calcium increases during the second spike and causes summation of the SK current, which produces adaptation. When the T or N current density is large, the SK current is almost completely activated during the first spike. Because both the T and N currents quickly inactivate after the first spike, much less calcium enters the cell through those channels during subsequent spikes. This causes the level of internal calcium to reach its maximum during the first spike. As the level of internal calcium decreases, the activation of the SK current also decreases. The smaller SK current allows subsequent spikes to occur sooner (i.e., acceleration). This phenomenon is dynamic, and not merely a by-product of increasing the overall level of calcium entering the cell. If the density of the P current (a slowly activating and noninactivating calcium current) is increased, the model continues to show adaptation.

Experiments show that the density of the current resistant to the 3 HVA current antagonists (considered to be the T current) decreases during postnatal development. The decrease seen experimentally in T current density is measured between days 3 and 4 postnatal (Umemiya and Berger 1994). Experiments have shown that the majority of HMs show spike frequency adaptation during the 1st wk postnatal, and acceleration during the 2nd wk (Viana et al. 1995). Reducing the T current in the model changes the firing characteristics from acceleration to adaptation. If experiments confirm that the T current continues to monotonically decrease as the rat ages, then the model's results do not correlate (until the adult age). If a change in T current density as modeled is the sole cause of the change in repetitive firing properties during aging, then the T current density will have to increase during the 2nd wk postnatal and then decrease before adulthood. There have been no reports of an age-dependent change in N current density in HMs.

Figure 8 demonstrates that varying the density of a calcium current only affects the firing properties (adaptation vs. acceleration) if that calcium current has an influence on the SK current. Therefore colocalization of calcium channels and the SK channels could play a role in determining HM firing behavior. Colocalization between specific calcium channels and calcium-dependent potassium channels has been observed in other neurons (Bowden et al. 2001; Kobayashi et al. 1997; Marrion and Tavalin 1998) and was proposed in HMs by Viana et al. (1993b). Colocalization would allow the influx of calcium from only certain channel types to affect activation of the SK current. Thus a current such as the L-type calcium current (not modeled) that makes up <10% of the total calcium current in HMs could have a substantial effect on the shape of the action potential. The dynamics of a single calcium channel type could control the firing properties of HMs through the SK current. If the SK channels are localized to the P or L channels at birth, and then shift their localization to the T or N channels, this will change the firing properties from adaptation to acceleration. Another possibility is that the SK channels may change from a global channel to one that is localized to a certain calcium channel type during aging, again changing the firing properties with age. To verify this, experiments to determine the existence of colocalization between the SK current and any of the calcium currents in HMs would need to be performed. Viana et al. (1993a) show a neuron where pharmacologically blocking N channels removes the mAHP. Although it is not stated, this neuron appears to display acceleration. This could then be tested by blocking P and/or L channels in a HM that displays adaptation.

Mammalian motoneurons experience anatomical and physiological changes during development (reviewed in Cameron and Nunez-Abades 2000). Some of the age-dependent changes of neonatal rat HMs (reviewed in Berger et al. 1996) have been explored here. These changes include a decrease in mAHP duration, a decrease in R_N, a decrease in action potential duration, a change in repetitive firing properties, and changes in the densities of different ion channels. Factors that affect the duration of the mAHP in the model include 1) the magnitude and time course of the calcium currents, 2) the SK current, 3) the A current, 4) the H current, and 5) the depletion rate of calcium from the cell. We showed in Fig. 9 that, in the model, the 10-fold increase in density of the H current is enough to account for the decrease in mAHP duration observed between neonatal and adult rats. However, the change in H current density is not enough to account for all of the age-dependent changes seen in HMs. According to Viana et al. (1994), the resting membrane potential of HMs is the same for neonatal and adult rats. In the model, the resting membrane potential increases by about 5 mV as the H current density is increased. Also, increasing the density of the H current in the model cannot completely account for the decrease in R_N or produce the decrease in action potential duration observed in HMs (Viana et al. 1994). Concurrent changes in other currents are needed to account for these behaviors in the model. Cameron and Nunez-Abades (2000) agree that the decrease in R_N is partly caused by a proliferation of ion channels. However, they suggest that this decrease is also caused by an increase in the number of synaptic inputs, something that is not included in our model.

Adult HMs fire at a higher rate than neonatal HMs (Viana et al. 1995). The decrease in mAHP duration could allow for the higher firing rates observed in adult HMs. In addition to its possible role in modulating the steady-state firing rate, the H current could potentially play a role in spike frequency adaptation seen in adult HMs. In the model, the magnitude of the H current between spikes (during the mAHP) decreases over time with a time constant of about 300 ms during a train of action potentials. If the density of the H current is sufficiently large, this spike-to-spike decrease in H current magnitude could result in an increase in mAHP duration between spikes. However, increasing _H to 0.1 µS causes only a small amount of adaptation in the model (<1 Hz decrease over 1 s).

Limitations of the model

CURRENTS. The currents included in this model are only those known to exist in neonatal rat HMs (reviewed in Berger 2000). We did not include those currents that have been observed in other motoneurons or other animal species. Currents such as the voltage- and calcium-activated potassium current (BK) and the sodium-activated potassium current (K, NA) found in other motoneurons (reviewed in McLarnon 1995; Rekling et al. 2000) could also exist in neonatal rat HMs. These currents could certainly affect the model's behavior, but are not included because there are no data describing these currents in neonatal HMs. This is both a strength and a limitation of the model. A potential role of the BK current is the age- and calcium-dependent change in action potential shape during a train of action potentials observed in HMs (Viana et al. 1995). An increase in density of a BK current, or an increase in density of a calcium current that activates a BK current, could cause this age-dependent change.

MORPHOLOGY. Many motoneuron morphologies have been described previously. Some of these studies include a 3-dimensional analysis of cat alpha-motoneuron morphology (Cullheim et al. 1987a, b) and phrenic motoneuron morphology from a neonatal rat (Lindsay et al. 1991). HM morphologies have also been investigated in the cat (Withington-Wray et al. 1988), guinea-pig (Mosfeldt Laursen and Rekling 1989; Viana et al. 1990), and rat (Altschuler et al. 1994). The morphology of rat genioglossal motoneurons has been shown to change during postnatal development (Mazza et al. 1992; Nunez-Abades and Cameron 1995; Nunez-Abades et al. 1994). It is clear that morphology plays an important functional role in motoneurons (reviewed in Rekling et al. 2000). However, our single-compartment HM model is able to accurately reproduce several electrophysiological characteristics of HMs. Extension of this model will be required to explain the effects of synaptic integration, including modeling dendritic structures. Developmental changes in HM morphology may also partially account for the age-dependent changes in resting membrane potential and R_N.

	APPENDIX: MODEL EQUATIONS

TOP ABSTRACT INTRODUCTION METHODS MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX: MODEL EQUATIONS GRANTS ACKNOWLEDGMENTS REFERENCES

 
Sodium current: I_Na

Persistent sodium current: I_NaP

Delayed-rectifier current: I_K

Leak current: I_leak

LVA calcium current: I_T

HVA calcium current: I_N

HVA calcium current: I_P

Calcium-dependent potassium current: I_SK

Fast-transient potassium current: I_A

Hyperpolarization-activated current: I_H

	GRANTS

TOP ABSTRACT INTRODUCTION METHODS MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX: MODEL EQUATIONS GRANTS ACKNOWLEDGMENTS REFERENCES

 
This work was supported by National Institute of Mental Health Grant R01-MH-62057 to R. J. Butera. L. K. Purvis was supported in part by the Georgia Tech–Student and Teacher Enhancement Project, a GK-12 program National Science Foundation Award DGE-0086420.

	ACKNOWLEDGMENTS

TOP ABSTRACT INTRODUCTION METHODS MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX: MODEL EQUATIONS GRANTS ACKNOWLEDGMENTS REFERENCES

 
We thank Dr. Chris Wilson (Case Western Reserve University), Dr. Jeff Smith (National Institutes of Health), and Dr. Bob Lee (Emory University and Georgia Institute of Technology) for input while developing this model. We thank the reviewers for helpful comments that led to improvements in the model. We also thank all of the researchers that have studied HMs and thereby contributed to this work.

	FOOTNOTES

 
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.

Address for reprint requests and other correspondence: R. J. Butera, 313 Ferst Drive, Atlanta, GA 30332-0535 (E-mail: rbutera{at}ece.gatech.edu)

	REFERENCES

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