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The Journal of Neuroscience, December 15, 2000, 20(24):9290-9297

Modeling of Membrane Excitability in Gonadotropin-Releasing Hormone-Secreting Hypothalamic Neurons Regulated by Ca²⁺-Mobilizing and Adenylyl Cyclase-Coupled Receptors

Andrew P. LeBeau¹, Fredrick Van Goor², Stanko S. Stojilkovic², and Arthur Sherman¹

¹ Mathematical Research Branch, National Institute of Diabetes and Digestive and Kidney Diseases, and ² Endocrinology and Reproduction Research Branch, National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, Maryland 20892

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Gonadotropin-releasing hormone (GnRH) secretion from native and immortalized hypothalamic neurons is regulated by endogenous Ca²⁺-mobilizing and adenylyl cyclase (AC)-coupled receptors. Activation of both receptor types leads to an increase in action potential firing frequency and a rise in the intracellular Ca²⁺ concentration ([Ca²⁺]_i) and neuropeptide secretion. The stimulatory action of Ca²⁺-mobilizing agonists on voltage-gated Ca²⁺ influx is determined by depletion of the intracellular Ca²⁺ pool, whereas AC agonist-stimulated Ca²⁺ influx occurs independently of stored Ca²⁺ and is controlled by cAMP, possibly through cyclic nucleotide-gated channels. Here, experimental records from immortalized GnRH-secreting neurons are simulated with a mathematical model to determine the requirements for generating complex membrane potential (V_m) and [Ca²⁺]_i responses to Ca²⁺-mobilizing and AC agonists. Included in the model are three pacemaker currents: a store-operated Ca²⁺ current (I_SOC), an SK-type Ca²⁺-activated K⁺ current (I_SK), and an inward current that is modulated by cAMP and [Ca²⁺]_i (I_d). Spontaneous electrical activity and Ca²⁺ signaling in the model are predominantly controlled by I_d, which is activated by cAMP and inhibited by high [Ca²⁺]_i. Depletion of the intracellular Ca²⁺ pool mimics the receptor-induced activation of I_SOC and I_SK, leading to an increase in the firing frequency and Ca²⁺ influx after a transient cessation of electrical activity. However, increasing the activity of I_d simulates the experimental response to forskolin-induced activation of AC. Analysis of the behaviors of I_SOC, I_d, and I_SK in the model reveals the complexity in the interplay of these currents that is necessary to fully account for the experimental results.

Key words: GT1 neurons; mathematical modeling; voltage-gated calcium entry; calcium-mobilization; phospholipase C; adenylyl cyclase

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Embryonic and green fluorescent protein-tagged gonadotropin-releasing hormone (GnRH) neurons, as well as immortalized GnRH-secreting neurons (hereafter GT1 cells), display spontaneous action potential (AP) firing (Kusano et al., 1995) as well as fluctuations in intracellular Ca²⁺ concentration ([Ca²⁺]_i) (Constantin and Charles, 1999; Spergel et al., 1999; Van Goor et al., 1999a,b). GnRH secretion from native and immortalized neurons correlates with changes in the pattern of electrical activity (Knobil, 1980; Krsmanovic et al., 1992; Martínez de la Escalera 1992a; Wetsel et al., 1992) and with spontaneous [Ca²⁺]_i oscillations (Krsmanovic et al., 1992; Terasawa et al., 1999). Both GnRH secretion and oscillations in membrane potential (V_m) and [Ca²⁺]_i are abolished in Ca²⁺-deficient medium, demonstrating the dependence of neuropeptide secretion on spontaneous electrical activity. Conversely, AP-driven [Ca²⁺]_i oscillations and GnRH secretion are facilitated by activating adenylyl cyclase (AC)- and phospholipase C-coupled receptors (Krsmanovic et al., 1991; Martínez de la Escalera et al., 1992b,c; Van Goor et al., 1999a,b).

Experimental identification and characterization of the pacemaker currents underlying AP activity and their modulation by AC- and phospholipase C-coupled receptors in GnRH neurons is incomplete. AC-induced cAMP production occurs in response to dopamine and norepinephrine (Jarry et al., 1990; Martínez de la Escalera 1992b,c; Al-Damluji et al., 1993). In general, cAMP can act indirectly, by modulating various ionic currents in a protein kinase A-dependent manner, or directly, via modulation of cyclic nucleotide-gated (CNG) channels, which have been identified in GT1 neurons (Vitalis et al., 2000). The phospholipase C pathway in GT1 cells is activated by at least two receptors, GnRH and endothelin (Krsmanovic et al., 1991, 1993), leading to Ca²⁺ mobilization from endoplasmic reticulum (ER) stores. As is widely observed in nonexcitable cells (Parekh and Penner, 1997) and in some excitable cells (Bertram et al., 1995; Bennett et al., 1998; Fomina and Nowycky, 1999), including GT1 neurons (Van Goor et al., 1999a), store emptying activates a Ca²⁺-carrying current (I_SOC). When activating both Ca²⁺ mobilization and Ca²⁺ entry pathways in GnRH neurons, communication between the ER and plasma membranes is vital for coordination of responses, with a critical role of I_SOC and I_SK (Van Goor et al., 1999a). Thus, GT1 cells have a wide repertoire of potential membrane electrical activity and [Ca²⁺]_i response types, which can be invoked by distinct agonists, and therefore these cells may serve as an excellent experimental and theoretical model for neuroendocrine cells.

One of the major issues that has not been resolved in neuroendocrine cells is the role of multiple Ca²⁺ pools (e.g., cytosol, ER) feeding back to plasma membrane-regulated intracellular signaling. The complexity of such signaling means that determining quantitative consistency between interpretations of experimental records is difficult; although a coherent qualitative representation of the system may be formed, lack of interexperimental constraint means important information may be lost. To address this issue, we have developed a mathematical model of GT1 cell electrophysiology and [Ca²⁺]_i signaling, allowing simulation of responses to GnRH and other agonists or pharmacological agents.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

GT1 cell culture. All experiments were performed on the GT1-7 subtype of immortalized GnRH neurons (Mellon et al., 1990), which were originally provided by Richard I. Weiner (University of California, San Francisco, CA). The cells were grown in 75 ml culture flasks containing culture medium (DMEM and F-12; 1:1, with L-glutamate, pyridoxine hydrochloride, 2.5 gm/l sodium bicarbonate, 10% heat-inactivated fetal bovine serum, and 100 µg/ml gentamycin; Life Technologies, Grand Island, NY). At confluence, the cells were dispersed by trypsinization (0.05% trypsin) for 10 min, resuspended in culture medium, and plated (50,000 cells/ml) in 35 mm tissue culture dishes (Corning, Corning, NY) with poly-L-lysine-coated (0.01%) coverslips. After incubation for 48 hr, the culture medium was replaced with medium containing B-27 serum-free supplement (Life Technologies) to induce morphological differentiation of the cells. All experiments were performed 3-5 d after serum removal.

Simultaneous measurement of V_m and [Ca²⁺]_i. Changes in V_m were monitored using the perforated-patch recording technique, as previously described in Van Goor et al. (1999a). All current-clamp recordings were performed at room temperature using an Axopatch 200 B (Axon Instruments, Foster City, CA) patch-clamp amplifier. Patch pipette tips (3-5 M) were briefly immersed in amphotericin B-free solution containing (in mM): 70 KCl, 70 K-aspartate, 1 MgCl₂, and 10 HEPES, pH adjusted to 7.2 with KOH, and then backfilled with the same solution containing amphotericin B (240 µg/ml). To monitor changes in [Ca²⁺]_i, GT1 neurons were incubated for 30 min at 37°C in phenol red-free medium 199 containing Hank's salts, 20 mM sodium bicarbonate, 20 mM HEPES, and 0.5 µM indo-1 AM (Molecular Probes, Eugene, OR). The coverslips with cells were then washed twice with modified Krebs'-Ringer's solution containing (in mM): 120 NaCl, 4.7 KCl, 2.6 CaCl₂, 2 MgCl₂, 0.7 MgSO₄, 10 HEPES, and 10 glucose, pH adjusted to 7.4 with NaOH, and mounted on the stage of an inverted epifluorescence microscope (Nikon). A photon counter system (Nikon) was used to simultaneously measure the intensity of light emitted at 405 and at 480 nm after excitation at 340 nm. Background intensity at each emission wavelength was corrected. The data were digitized at 4 kHz using a personal computer equipped with the Clampex 8 software package in conjunction with a Digidata 1200 analog-to-digital converter (Axon Instruments).

[Ca²⁺]_i was calibrated in vivo according to Kao (1994). Briefly, R_min was determined by exposing the cells to 10 µM Br-A23187 in the presence of Krebs'-Ringer's solution with 2 mM EGTA and 0 Ca²⁺ for 60 min; 15 mM Ca²⁺ was then added to determine R_max. The values used for the [Ca²⁺]_i calibration parameters: R_min, R_max, S_f,480/S_b,480, and K_d were 0.472, 3.634, 3.187, and 230 nM, respectively (for details, see Kao 1994). All reported V_m were corrected for a liquid junction potential between the pipette and bath solution of +10 mV. The bath contained <500 µl of saline and was continuously perfused at a rate of 2 ml/min using a gravity-driven superfusion system. The outflow was placed near the cell, resulting in complete solution exchange around the cell within 2 sec. A solid Ag-AgCl reference electrode was connected to the bath via a 3 M KCl agar bridge. In some experiments, the cells were preloaded with the membrane-permeant Ca²⁺ chelator BAPTA-AM (1 µM) for 45 min at 37°C.

GT1 cell model. We have previously published a GT1 cell model (Van Goor et al., 2000) that included fast, voltage-dependent currents (I_Na, I_CaL, I_CaT, I_KDR, I_M, I_ir and a linear depolarizing leak current, see the online Appendix at http://www.jneurosci.org), but no [Ca²⁺]_i, [Ca²⁺]_er, or [Ca²⁺]-dependent currents. Except for the leak current, the extended model presented here keeps all the fast currents exactly as used in Van Goor et al. (2000). From Van Goor et al. (1999a) we have experimental evidence for I_SOC activity in GT1 neurons, and therefore such a current was added. Also, as described in Results, interpretation of the experimental data led us to hypothesize that the leak current (I_d) is a Ca²⁺-inactivated nonspecific cation current, which is described below. I_SK was also added, the parameters for which were derived from experimental data (Van Goor et al., 1999a).

Initially, the Ca²⁺ dynamics were modeled as a simple one-compartment cytosol with a spatially homogeneous [Ca²⁺]_i, interacting with a similarly defined ER compartment and a constant [Ca²⁺] extracellular compartment (see Appendix). The ER was 10% of the total cell volume (the rest being cytosolno allowance was made for the nucleus or other subcompartments), distributed evenly throughout the total volume. However, it became apparent that to accurately simulate the experimental results it was necessary to subdivide each of the cytosolic and ER pools to allow compartmentalization of Ca²⁺ dynamics in the region at the inner face of the plasma membrane. We refer to the subdivided regions as the "shell" and the "bulk" compartments. The shell represents the 80-nm-deep region (see below) of the cytosol immediately adjacent to the plasma membrane. The bulk represents the rest of the cell interior, ~97.8% of the total volume. There is no physical barrier separating the shell and bulk compartments, and Ca²⁺ is exchanged in a manner and at a rate representing simple diffusion. Each compartment contains the same ratio of cytosol and ER as described above. Thus, there are four distinct pools of intracellular Ca²⁺ with concentrations given by: shell cytosolic [Ca²⁺] ([Ca²⁺]_is), bulk cytosolic [Ca²⁺] ([Ca²⁺]_ib), shell ER [Ca²⁺] ([Ca²⁺]_ers), and bulk ER [Ca²⁺] ([Ca²⁺]_erb). Because the bulk compartment represents almost 98% of the total volume, [Ca²⁺]_ib corresponds to [Ca²⁺]_i reported by the fluorescent dye in the experimental traces.

The choice of 80 nm for the depth of the shell was based on two criteria: (1) that the volume of the shell be small enough to provide a clear separation in the [Ca²⁺]_i time scales in the shell and bulk compartments, and (2) that the depth of the shell be sufficient to encompass the outer regions of the ER, including the proposed IP₃ receptor-SOC channel complex (see below). The value of 80 nm fulfills both criteria. In early trial simulations, a value of ~400 nm was also used, and this gave similar results to those for 80 nm, indicating that the value for shell depth used here is not critical.

Within each pool, Ca²⁺ is 99% buffered, and the given Ca²⁺ concentrations represent free (i.e., unbuffered) Ca²⁺ (see Appendix). Ca²⁺ is pumped out of the cell via plasma membrane pumps (Ca-ATPase and Na⁺/Ca²⁺ exchanger) and into the ER by sarcoendoplasmic ATPase (SERCA) pumps. Ca²⁺ release from the ER into the cytosol is modeled by a simple term given as the product of a constant (though adjustable) efflux rate and the difference between the ER and cytosolic [Ca²⁺]. GnRH application is simulated by increasing the efflux rate above its low, resting value. We found that this simple system was sufficient to simulate the effects of IP₃-induced mobilization of stored Ca²⁺ and that a specific description of IP₃ receptor and its kinetics was not necessary. Our simple system does not distinguish whether basal release of stored Ca²⁺ is via IP₃ receptors activated by resting IP₃ levels or via a separate leak pathway.

It was also necessary to include mitochondrial Ca²⁺ handling. Again, uptake was modeled as being via a Ca²⁺ pump. Pivovarova et al. (1999) have provided evidence that excess Ca²⁺ taken into the ER is stored as a precipitate, which means that the free intramitochondrial [Ca²⁺], and therefore passive release of mitochondrial Ca²⁺, may be fairly constant. This assumption allows us to dispense with equations governing mitochondrial [Ca²⁺], retaining just the flux terms in the [Ca²⁺]_i equations (see Appendix). To accurately reproduce the experimental records, particularly the response to SERCA pump inhibition by thapsigargin (Tg), it was necessary to have higher mitochondrial activity in the shell compartment than in the bulk. This is consistent with reports of localization of mitochondria near the plasma membrane in neurons (Pivovarova et al., 1999) and with ER membranes (Rizzuto et al., 1993; Csordás et al., 1999).

I_SK was modeled in the usual way as a product of a macroscopic conductance, fractional activation by [Ca²⁺]_is, and a linear voltage driving force (see Appendix for details). I_SOC was similarly defined, but with a fractional activation that was inversely related to [Ca²⁺]_ers, consistent with a recent report suggesting that activation of I_SOC may be specifically regulated by subcompartments of the ER (Broad et al., 1999). Moreover, several other recent reports have suggested that I_SOC is activated via a direct coupling with IP₃ receptors in the ER membrane (Kiselyov et al., 1998, 1999; Boulay et al., 1999; Ma et al., 2000) (also see Putney, 1999). The details of such an interaction have not been determined, and so we do not specifically implement this effect. However, our description of I_SOC regulation is compatible with such a mechanism.

A schematic diagram of the key elements of the model is given in Figure 1. An animated version may be viewed at http://mrb.niddk.nih.gov/alebeau.

View larger version (31K): [in this window] [in a new window]   Figure 1.   Schematic diagram of the key elements of the model, showing the separation of both the cytosolic and ER pools into shell and bulk compartments and the three pacemaker currents, ISK, ISOC, and Id. ISOC is thought to be activated via a direct coupling to shell IP3 receptors (see Materials and Methods for details), whereas Id is activated by cAMP. A rise in shell [Ca2+]i activates ISK and inactivates Id. An animated version of this diagram may be viewed at http://mrb.niddk.nih.gov/alebeau/gt1.html.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

We start by describing five experimental records and reviewing published results, the interpretation of which provides a qualitative understanding of electrical activity and Ca²⁺ signaling in GT1 cells. Our goal is to characterize the roles of the previously described I_SK and I_SOC in regulating pacemaking in spontaneously and GnRH-induced activity and to determine whether these channels are sufficient to explain the basal and agonist-induced effects. We then test this understanding by simulating the experimental traces with the model. The experimental records and simulations are presented in five figures in which the layout and, as much as practical, the scaling were kept constant to allow direct comparison between the respective results.

Experimental results

As described previously (Van Goor et al., 1999a), unstimulated GT1 cells fired spontaneous APs at a frequency of ~0.5-1.0 Hz. GnRH application induced a transient hyperpolarization and cessation of AP firing (Fig. 2A) and a spike rise in [Ca²⁺]_i (Fig. 2B) caused by IP₃-mediated emptying of the ER Ca²⁺ store. Subsequently, [Ca²⁺]_i fell to a plateau level that was higher than the prestimulus level, the V_m depolarized, and AP activity resumed with an increased firing rate and decreased amplitude.

View larger version (21K): [in this window] [in a new window]   Figure 2.   Responses of GT1 neurons and model to GnRH stimulation. Simultaneous Vm (A) and [Ca2+]i (B) responses to 100 nM GnRH in GT1 neurons. C-G, Model GnRH response, simulated by increasing ER membrane permeability to Ca2+ 25-fold. In this and following figures, heavy and light lines denote bulk and shell compartments, respectively. C, Vm response. D, [Ca2+]i and [Ca2+]er (E) response. F, ISK and ISOC (G, light line) and Id (G, heavy line) membrane current responses. Calibration in A applies to A and B, and in C applies to C-G.

Application of apamin, an I_SK blocker, completely prevented GnRH-induced hyperpolarization (Fig. 3A). When added during the sustained phase of GnRH stimulation, apamin caused an increase in the AP firing space (Van Goor et al., 1999a), whereas in spontaneously active cells, apamin was ineffective. These results indicate that the spike and plateau elevations in [Ca²⁺]_i, but not basal [Ca²⁺]_i, are sufficient to activate SK channels. The results further indicate that the increase in firing frequency after GnRH occurs despite an increase in I_SK activity. Thus, activation of a depolarizing current or inhibition of a hyperpolarizing current is needed to overcome the negative effects of activated SK channels on pacemaking.

View larger version (24K): [in this window] [in a new window]   Figure 3.   Responses of GT1 neurons and model to addition of GnRH during blockade of SK channels by apamin. Simultaneous Vm (A) and [Ca2+]i (B) responses to 100 nM GnRH in GT1 neurons during constant perfusion with 100 nM apamin. C-G, Model simulations of GnRH plus apamin response. Apamin was simulated by setting the conductance of ISK to zero. GnRH was simulated as in Figure 2. Calibration in A applies to all traces.

Several lines of observations indicate that store emptying activates I_SOC in GT1 neurons, which may account for this agonist-induced increase in firing frequency (Van Goor et al., 1999a). As shown in Figure 4B, Tg, a blocker of SERCA pumps, caused a slow rise in [Ca²⁺]_i, presumably reflecting a slow depletion of the ER Ca²⁺ stores, in contrast to the rapid release of Ca²⁺ observed in GnRH-stimulated cells (Fig. 2B). This may account for the inability of Tg to induce a transient hyperpolarization and cessation of AP firing (Fig. 4A). Consistent with a role of I_SOC in control of pacemaker activity, Tg was also able to induce a sustained increase in the firing frequency. Together, the results in Figures 2-4 suggest that I_SOC and I_SK act coordinately to regulate AP activity during the response to GnRH, with I_SOC overcoming the hyperpolarizing effects of the I_SK to increase AP firing rate.

View larger version (34K): [in this window] [in a new window]   Figure 4.   Responses of GT1 neurons and model to addition of endoplasmic reticulum Ca2+-ATPase pump blocker thapsigargin (Tg). Simultaneous Vm (A) and [Ca2+]i (B) responses to 5 µM Tg in GT1 neurons. C-G, Model simulations of GnRH plus Tg response. Tg was simulated by setting the SERCA pump rates to zero. Calibration in A applies to A and B, and in C applies to C-G.

However, the cessation of AP activity, which persisted even when SK channels were blocked by apamin (Fig. 3A), cannot be explained by a change in I_SOC activity (unless I_SOC is sensitive to [Ca²⁺]_i; see Discussion). Activation of I_SOC may be delayed if total store emptying is required, but a delay in I_SOC activation should lead to a slow increase in firing rate, not a cessation of activity. Therefore, these results suggest that another current underlies the cessation of activity. Modulation of this current must occur before, or initially override, the increase in I_SOC activity, to stop the firing. I_SOC must eventually dominate, to produce the increase in firing frequency seen during the sustained response phase. We designate the new current I_d. Therefore, it appears that at least three individual currents act coordinately to regulate AP firing in GT1 cells: I_SOC, I_SK, and I_d.

In further experiments, the Ca²⁺ buffer BAPTA was injected into GT1 cells to prevent an increase in [Ca²⁺]_i during the GnRH response. This abolished the spike and plateau rise in [Ca²⁺]_i (Fig. 5B). BAPTA also prevented the GnRH-induced membrane hyperpolarization, confirming the dependence of SK channels on [Ca²⁺]_i (Fig. 5A). Moreover, the transient cessation of AP firing observed in GnRH-stimulated cells with blocked SK channels was also abolished by BAPTA. As shown in Figure 5A, there was an increase in AP firing frequency almost immediately after GnRH was applied, in contrast to the experiment shown in Figure 3A. This suggests that both I_d and I_SK are regulated by [Ca²⁺]_i. By inhibiting both currents, BAPTA reveals the effects of the (now clearly) rapidly activated I_SOC.

View larger version (25K): [in this window] [in a new window]   Figure 5.   Responses of GT1 neurons and model to GnRH stimulation during [Ca2+]i buffering with BAPTA. Simultaneous Vm (A) and [Ca2+]i (B) responses to 100 nM GnRH in GT1 neurons with [Ca2+]i clamped to ~200 nM. C-G, Model simulations of GnRH plus BAPTA response. BAPTA was simulated by setting the fraction of free cytosolic Ca2+ (fcyt) to 1 × 105. Calibration in A applies to A and B, and in C applies to C-G.

I_d could be either a Ca²⁺-inactivated current that is active under basal conditions or a Ca²⁺-activated, apamin-insensitive outward current. In accordance with the former possibility, we were unable to observe any effect of charybdotoxin or iberiotoxin, two specific blockers of BK-type I_KCa, on GnRH-induced electrical activity (Van Goor et al., 1999a). If present in GT1 cells, two other Ca²⁺-activated channels, chloride and nonselective, would depolarize cells under our recording conditions. At the present time, we do not have specific information regarding the identity of Ca²⁺-inactivated I_d. However, Vitalis et al. (2000) have reported recently that the olfactory subtype of CNG channels is expressed in GT1 cells. These channels have a preference for cAMP over cGMP (Wei et al., 1998) and are inhibited by [Ca²⁺]_i either directly (Finn et al., 1996; Wei et al., 1998) or indirectly via modulation of AC activity (Hurley, 1999). Consistent with these findings, forskolin, an AC activator, caused an increase in AP frequency with no cessation in firing, no hyperpolarization, and no change in AP amplitude (Fig. 6A), whereas [Ca²⁺]_i rose very slightly (Fig. 6B).

View larger version (28K): [in this window] [in a new window]   Figure 6.   Responses of GT1 neurons and model to activation of adenylyl cyclase by forskolin. Simultaneous Vm (A) and [Ca2+]i (B) responses to 10 µM forskolin in GT1 neurons. C-G, Model simulations of forskolin response. Forskolin was simulated by a threefold increase in the conductance of Id. Calibration in A applies to A and B, and in C applies to C-G.

Model simulations

Figure 2C shows that the model fires spontaneous AP at a frequency of ~0.7 Hz, similar to the average value observed in GT1 cells. During basal activity, bulk [Ca²⁺]_i ([Ca²⁺]_ib) is constant (on the scale shown; Fig. 2D) as in the experimental trace (Fig. 2B). However, shell [Ca²⁺]_i ([Ca²⁺]_is) shows excursions of ~0.5 µM, coincident with each AP, representing influx of Ca²⁺ during a single spike. The rise in [Ca²⁺]_is is brief because the compartment is small, and the effects of plasma membrane pumps and diffusion into the bulk compartment act rapidly to lower [Ca²⁺]_i.

The model also simulates GnRH-induced store emptying and the consequent effects on plasma membrane electrical activity (Fig. 2C,D), when the permeability (p in the equations for j_relx; see Appendix) of the ER membrane for Ca²⁺ flux is increased 25-fold. This change is applied equally in both the shell and bulk compartments. In response, [Ca²⁺]_ib rises rapidly to a peak slightly >2 µM, then declines more slowly to a very slowly decaying plateau (Fig. 2D). [Ca²⁺]_is also rises rapidly in response to GnRH, but the rise is limited because of plasma membrane pumping. The spike rise in model [Ca²⁺]_i causes membrane hyperpolarization (Fig. 2C) and a temporary cessation of AP firing, as in the experimental trace (Fig. 2A). We found that with the parameters required to simulate the pattern of AP firing (our primary focus), the model exhibited a relatively low interspike potential (although within the range observed experimentally; see Fig. 4A), which means that it tends to have a more modest GnRH-induced hyperpolarization than typically seen in experiments. However, the cause of the hyperpolarization is well characterized experimentally (activation of I_SK; see Fig. 3A and Van Goor et al., 1999a). The mechanism underlying the hyperpolarization in the model is the same (see below), so the difference is only quantitative and does not indicate an alternative process.

Finally, Figure 2E shows that the shell and bulk ER [Ca²⁺] ([Ca²⁺]_ers and [Ca²⁺]_erb, respectively) are essentially constant during the prestimulus period but that both fall rapidly after application of GnRH. [Ca²⁺]_ers falls slightly faster initially, but both compartments of the store are essentially empty within 20 sec of stimulation. This is consistent with the failure of ionomycin to elicit additional [Ca²⁺]_i rises when added shortly after GnRH in GT1 cells (data not shown).

Figure 2, F and G, shows the ionic currents that are principally responsible for controlling plasma membrane excitability and the pattern of AP firing in the GT1 cell model. During basal activity there is no significant activation of I_SK (Fig. 2F), which is consistent with the lack of effect of apamin on spontaneous AP activity (Van Goor et al., 1999a). During the spike phase of the [Ca²⁺]_i response, sustained I_SK is evoked, reaching a peak of ~12 pA and causing the transient membrane hyperpolarization shown in Figure 2C. The current declines as [Ca²⁺]_is falls, but when AP firing resumes, the residual activation of I_SK leads to large current deflections during each spike because of the change in K⁺ driving force.

In the model, I_SOC is partially activated under basal conditions (Fig. 2G, light trace). In response to GnRH, rapid emptying of the ER stores further activates I_SOC, reaching a peak of approximately 18 pA. The current then decreases slightly as the driving force for Ca²⁺ falls with V_m depolarization. However, although I_SOC is activated, AP firing does not resume until the I_SK decays sufficiently for I_SOC to dominate. The coordinate actions of I_SOC and I_SK alone are sufficient to explain some effects of GnRH on electrical activity and Ca²⁺ signaling. However, in the absence of I_d, AP firing frequency after the [Ca²⁺]_i spike is much greater than the observed experimental range (data not shown). Such rapid firing is further evidence that a third pacemaker current is required; this current must be modulated to compensate for the activation of I_SOC. Thus, in addition to the qualitative evidence for I_d provided by the experiments described above, quantitative evidence that such a current is necessary to explain the response to GnRH is provided by the model.

Analysis of the experimental records suggests that I_d is inactivated by [Ca²⁺]_i (see above), and model simulations support this hypothesis. GnRH-induced store emptying raises [Ca²⁺]_is and inactivates I_d, compensating for the concurrent (although slower) activation of I_SOC (Fig. 2G). These bidirectional effects result in AP firing frequencies within the experimental range (Fig. 2).

The model simulation of the response to GnRH, during exposure to apamin, is shown in Figure 3C-G. The effects of apamin were simulated by setting the conductance of I_SK to zero (Fig. 3F). This has no effect on the pre-GnRH AP firing frequency (compare Fig. 2C), consistent with experimental results discussed above. Also, under these conditions GnRH application causes no membrane hyperpolarization (Fig. 3C), demonstrating that inactivation of I_d cannot account for this effect, and no change in the rate of store depletion (Fig. 3D,E). Finally, the model is able to mimic the cessation of AP activity observed experimentally in cells with blocked SK channels.

Figure 3G suggests an explanation for why AP firing was briefly halted in the absence of any I_SK. When GnRH is applied, I_d quickly inactivates because of the rapid rise in [Ca²⁺]_is. On the other hand, I_SOC activates more gradually, as the ER depletes with a slower time course. The cessation of firing in the model is thus a result of differences in the relative time courses of the changes in [Ca²⁺]_i levels in the cytosolic and ER pools and hence, the currents they regulate.

The effects of Tg were simulated by setting the SERCA pump rates to zero in both the shell and bulk compartments. This causes an initial slowing of the firing frequency and hyperpolarizes the nadir of the spikes (Fig. 4C). Subsequently, the firing rate increases beyond the prestimulus level. [Ca²⁺]_i in both cytosolic compartments increases after Tg application (Fig. 4D). The value for [Ca²⁺]_ib is lower than that observed in the representative experimental trace (Fig. 4B), but it is within the range observed from all Tg experiments (data not shown). [Ca²⁺]_er decreases more slowly with Tg application than with GnRH (compare Figs. 2E, 4E), as expected from the slow leak of Ca²⁺ induced by Tg relative to the rapid store emptying induced by IP₃. The increase in [Ca²⁺]_is activates I_SK modestly (Fig. 4F), consistent with the effects of apamin on the firing frequency observed experimentally (Van Goor et al., 1999a). Figure 4G shows the pattern of I_SOC and I_d after Tg application. As the store (specifically the shell ER) empties, I_SOC is slowly activated. At the same time, the rise in [Ca²⁺]_i (specifically [Ca²⁺]_is), which is also slow, inactivates I_d. The two currents essentially switch activity levels, at about the same rate, so there is no cessation of AP firing. I_d decreases more rapidly than I_SOC increases, resulting in a slight initial reduction of AP firing frequency immediately after Tg application. After ~20 sec, I_SOC dominates I_d sufficiently to cause the firing frequency to increase beyond the prestimulus level.

The effects of BAPTA were simulated by decreasing f_cyt, the fraction of free Ca²⁺ in the cytosol, such that [Ca²⁺]_ib does not rise significantly when GnRH is applied (Fig. 5D, heavy line). We confirmed that this treatment prevents membrane hyperpolarization and cessation of AP activity in the model as in the experiments (Fig. 5C). There is a brief period of rapid firing that was not observed experimentally, suggesting that we have perhaps not fully captured the complexity of [Ca²⁺]_i-induced changes in pacemaker currents (see below). However, after this the firing frequency settles down to a rate moderately faster than before GnRH, consistent with the experimental record. Figure 5D shows that BAPTA simulation prevents the brief pulses of [Ca²⁺]_is before GnRH application, but that after GnRH, [Ca²⁺]_is rises slowly, reflecting an inability of the enhanced buffer to completely prevent a rise in [Ca²⁺]_i in the small-volume shell compartment.

This slow rise in [Ca²⁺]_is is important for the response of the model to GnRH in the presence of BAPTA. Figure 5G shows that with GnRH application, I_SOC is activated in response to store emptying, but at the same time I_d is slowly decreased by the slow rise in [Ca²⁺]_is. Both the rise and its slowness are importantif no rise occurred, I_d would not be inactivated, and the firing frequency would be too high, as in the initial rapid-firing phase. Moreover, because the rise in [Ca²⁺]_is is slow (on approximately the same time scale as I_SOC activation), I_d inactivates more slowly than during GnRH (Fig. 2) or GnRH plus apamin (Fig. 3), so there is no cessation of AP activity. Instead, there is a transfer of control of AP activity from I_d to I_SOC, with I_SK playing no role (Fig. 5F). The lack of a rapid firing phase in the experimental data suggests that transfer of control of firing frequency is even more tightly regulated in the cells than is captured by our relatively simple descriptions of the two currents.

As a final test, we asked whether I_d could represent a CNG channel. Specifically, we tested whether the experimental response to forskolin could be modeled by increasing the conductance of I_d. In fact, this causes an increase in firing frequency without a cessation of AP activity, membrane hyperpolarization, or a change in AP amplitude (Fig. 6C). [Ca²⁺]_i in the shell or bulk compartments does not rise significantly (Fig. 6D), and [Ca²⁺]_er is unaffected (Fig. 6E). I_SK and I_SOC (Fig. 6F,G, respectively) are unaffected by forskolin, and I_d is slightly increased (Fig. 6G). This latter effect is solely responsible for the increased AP firing frequency.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

We have presented here a mathematical model that provides a quantitative description of the regulation of AP pacemaking and the associated [Ca²⁺]_i signaling in GT1 neurons. The results indicate that a complex interplay of at least three pacemaker currents, modulated by at least two distinct Ca²⁺ pools (cytosol and ER), regulates AP firing in GT1 cells. The pacemaker currents included in the model are I_SOC, I_SK, and the cAMP-regulated I_d. Spontaneous firing and the associated Ca²⁺ signaling in the model are predominantly controlled by I_d. Increasing the activity of I_d also simulates the experimental response to forskolin-induced activation of AC. In contrast, activation of I_SOC by depletion of the intracellular Ca²⁺ pool and I_SK by the concomitant rise in [Ca²⁺]_i, as well as a decrease in I_d, is required to mimic the increase in the firing frequency and Ca²⁺ influx observed in cells stimulated with Ca²⁺-mobilizing agonists.

In general, the expression of I_SOC in excitable cells provides a potential mechanism to link AP-driven Ca²⁺ influx with agonist-induced store depletion (Berridge, 1998). In cells with a leaky ER Ca²⁺ pool (i.e., with relatively high basal levels of leak and uptake), the same channels may play a role in the control of pacemaking in spontaneously active cells. Coupled to I_SK, such a system should lead to controlled Ca²⁺ influx to replenish the depleted pool. However, from the analysis of the experimental data presented here and published earlier (Van Goor et al., 1999a), no evidence for a regulatory role of I_SOC and I_SK in spontaneously active cells was observed. The need for an additional pacemaker current also emerged from our simulations with an early-stage model, which contained only I_SK and I_SOC as pacemaker currents. In the absence of I_d, such a model was unable to mimic all aspects of spontaneous and agonist-induced AP firing. Furthermore, the integration of I_d helped to mimic the action of AC-coupled receptors in these cells (Martínez de la Escalera et al., 1992b,c).

In our model, control of basal electrical activity is predominantly achieved by I_d, a channel that is activated by cAMP and inhibited by [Ca²⁺]_i. Although we have not recorded I_d experimentally, the CNG channel recently described by Vitalis et al. (2000) in GT1 cells is an obvious candidate. Another, more complex, possibility is that I_d could actually represent [Ca²⁺]_i-mediated inactivation of I_SOC (Lewis, 1999). That is, the ER store and [Ca²⁺]_i would act antagonistically on I_SOC, and a separate population of CNG channels would mediate the effects of AC-coupled agonists. We have performed simulations to confirm that this is a viable mechanism. However the model cannot discriminate between the two possible systems because the pacemaker currents (I_SOC and I_d) are not well characterized. The model is able to tell us what the characteristics of the currents need to be to generate the observed firing patterns, and the single current (I_SOC with inactivation) can produce equivalent results provided it behaves more-or-less like the sum of the two original currents. Without specific experimental characterization of the currents in these cells, the model can only make predictions that can be tested experimentally. We favor separating the channels because: (1) "slow inactivation" of I_SOC (Zweifach and Lewis, 1995b; Liu et al., 1998) is too slow for the effects necessary in our model, and "fast inactivation" of the Ca²⁺-release activated Ca²⁺ current (CRAC) form of I_SOC (Zweifach and Lewis, 1995a) is only marginally active at the interspike potentials in GT1 neurons; and (2) there is experimental evidence for a CNG channel in GT1 neurons (Vitalis et al., 2000). Because we are not aware of any experimental evidence for cAMP activation of I_SOC, a second channel is still necessary. Therefore two currents, as described here, is the simplest form, and additional experimental evidence would be necessary to support a more complex system.

When integrated with I_d, two other channels, I_SK and I_SOC, play specific roles in control of pacemaking. In our model, I_SK is not significantly activated in spontaneously active cells, and I_SOC, although modestly activated, provides only a fairly constant background conductance. The background activity of I_SOC is consistent with findings by Bennett et al. (1998) in PC12 cells and also with our observation of the relatively rapid effects of Tg on AP firing frequency, which suggests that little store emptying is needed for an increase in I_SOC. This could easily be achieved if I_SOC is already partially activated.

The real importance of I_SOC is in mimicking the action of Ca²⁺-mobilizing agonists. In our model, an increase in AP firing frequency in response to GnRH is driven primarily by I_SOC, with I_d and I_SK playing regulatory roles. I_SK generates the membrane hyperpolarization, although it is not clear whether this has any physiological function, and keeps a partial brake on AP frequency once firing resumes. I_d is rapidly inactivated by the spike [Ca²⁺]_i rise, passing control of AP firing over to I_SOC. As the store refills slowly after removal of GnRH, I_SOC gradually relinquishes control back to I_d (data not shown).

Store-operated channels have been studied most intensively in nonexcitable cells with much less attention given to their presence and role in excitable cells. Here we have shown how I_SOC can integrate the state of store Ca²⁺ content into excitable membrane activity. During the response to GnRH, the cytosolic and ER Ca²⁺ stores act in opposite directions on membrane excitability. GnRH causes a rise in [Ca²⁺]_i which acts to depress membrane activity by increasing I_SK activation and inactivating I_d, whereas the depleted ER pool increases I_SOC activity to increase membrane activity. In other situations the two pools may work in concert. For example, in the model we allow a low level of I_SOC activation at rest, so that if a depolarizing stimulus were applied, there would be an increase in AP activity that would increase [Ca²⁺]_i, some of which would be taken up into the ER. The increase in [Ca²⁺]_er would cause I_SOC to be reduced and, together with a [Ca²⁺]_i-induced decrease in I_d, would counteract the depolarizing stimulus.

The two cases in which the cytosolic and ER Ca²⁺ pools act either in opposition or together in regulating membrane electrical activity suggest a general rule for Ca²⁺ handling: in the absence of a Ca²⁺-mobilizing agonist, the cytosolic and ER pools rise and fall together in response to changes in Ca²⁺ entry (regulated by membrane electrical activity in these cells). For both pools a rise in [Ca²⁺] has an inhibitory effect on electrical activity (Chay, 1997). However, store-emptying breaks the connection, and the pools work antagonistically until the store is refilled.

In the model, the Ca²⁺-mobilizing agonist (GnRH), transiently shuts off I_d and stimulates I_SOC, whereas activation of I_d simulates the experimental response to an AC activator, supporting I_d being the CNG channel found by Vitalis et al. (2000) in GT1 neurons. The model also suggests that in contrast to Ca²⁺-mobilizing agonists, which modulate all three pacemaker currents, AC-coupled agonists increase AP activity by modulating I_d alone and thus act independently of the ER. This suggests that AC-coupled agonists act to a greater degree on [Ca²⁺]_i local to the plasma membrane, compared with calcium-mobilizing agonists, which act globally through emptying of the ER store. These differences may reflect differential roles of the respective agonists on the patterns of secretion and protein synthesis (Dolmetsch et al., 1998).

In the model, it was necessary to compartmentalize the cytosolic and ER Ca²⁺ pools to get the appropriate changes in the pacemaker currents after agonist-drug addition. The model suggests that the region of the cytosol immediately adjacent to the plasma membrane has a degree of functional separation from the interior cytosol. This separation allows [Ca²⁺]_i in this region to be more dynamic than, and act somewhat independently of, bulk [Ca²⁺]_i. Similarly the ER in this region may also be more dynamic than the bulk ER, allowing rapid modulation of I_SOC and thus electrical activity. Broad et al. (1999) and Gregory et al. (1999) have also reported evidence supporting the concept of specialized Ca²⁺ signaling in the periphery of the cell. Moreover, a close functional connection between the shell ER and I_SOC is consistent with several recent reports indicating a physical connection between IP₃Rs and I_SOC channels (Kiselyov et al., 1998, 1999; Boulay et al., 1999; Ma et al., 2000) (also see Putney, 1999).

The shell Ca²⁺ pools cannot, however, function completely independently of their bulk counterparts. In the case of the ER this raises an intriguing possibility. The model shows that the activity of I_SOC can be modulated on the time scale of the shell ER. Because there is still slow communication between ER compartments, I_SOC activity can also be modulated on the time scale of the bulk ER, which is on the order of 10-20 min in the absence of elevated [IP₃]. Therefore, the ER can potentially regulate plasma membrane excitability on both fast (seconds) and slow (tens of minutes) time scales.

In conclusion, the experimental and theoretical results present here provide a model for spontaneous electrical activity of neuroendocrine cells that also accommodates the integration of two major receptor pathways, Ca²⁺-mobilizing and AC-coupled, in control of spontaneous electrical activity. The model suggests that in spontaneously active cells, basal AC activity, possibly activated by voltage-gated Ca²⁺ influx, generates cAMP and thus activation of CNG-like I_d channels. Control of spontaneous electrical activity and the [Ca²⁺]_i-dependent AC is achieved by an increase in localized [Ca²⁺]_i, which transiently inactivates I_d. Activation of Ca²⁺-mobilizing receptors also leads to inhibition of I_d but stimulates I_SOC, which is responsible for reinitiation of electrical activity, whereas agonist-induced activation of AC facilitates I_d independently of the status of I_SOC. At the same time, I_SK protects the cells from Ca²⁺ overload when the pacemaking is predominantly controlled by I_SOC.

	FOOTNOTES

Received June 12, 2000; revised Sept. 11, 2000; accepted Sept. 28, 2000.

Correspondence should be addressed to Dr. Andrew LeBeau, Mathematical Research Branch, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, BSA Building, Suite 350, 9190 Rockville Pike MSC 2690 Bethesda, MD 20892-2690. E-mail: lebeau{at}nih.gov.

	APPENDIX

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

The Appendix for this manuscript may be viewed at http://www.jneurosci.org.

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TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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