Computational modeling of anoctamin 1 calcium-activated chloride channels as pacemaker channels in interstitial cells of Cajal

Abstract

Interstitial cells of Cajal (ICC) act as pacemaker cells in the gastrointestinal tract by generating electrical slow waves to regulate rhythmic smooth muscle contractions. Intrinsic Ca²⁺ oscillations in ICC appear to produce the slow waves by activating pacemaker currents, currently thought to be carried by the Ca²⁺-activated Cl⁻ channel anoctamin 1 (Ano1). In this article we present a novel model of small intestinal ICC pacemaker activity that incorporates store-operated Ca²⁺ entry and a new model of Ano1 current. A series of simulations were carried out with the ICC model to investigate current controversies about the reversal potential of the Ano1 Cl⁻ current in ICC and to predict the characteristics of the other ion channels that are necessary to generate slow waves. The model results show that Ano1 is a plausible pacemaker channel when coupled to a store-operated Ca²⁺ channel but suggest that small cyclical depolarizations may still occur in ICC in Ano1 knockout mice. The results predict that voltage-dependent Ca²⁺ current is likely to be negligible during the slow wave plateau phase. The model shows that the Cl⁻ equilibrium potential is an important modulator of slow wave morphology, highlighting the need for a better understanding of Cl⁻ dynamics in ICC.

smooth muscle contractions in the gastrointestinal tract are partly controlled by spontaneous electrical events called slow waves, which originate in pacemaker cells known as interstitial cells of Cajal (ICC). In the stomach and small intestine of mice and guinea pigs, slow waves are generated by ICC located in the region of the myenteric plexus between the circular and longitudinal muscle layers (ICC-MY), while in the colon the primary pacemaker ICC are in the region of the submuscular plexus (ICC-SMP). Other classes of ICC regenerate slow waves within the smooth muscle layers and mediate neural signals (15, 33). Slow waves recorded from ICC-MY are often called pacemaker potentials to differentiate them from the smaller amplitude slow waves recorded from smooth muscle. In this article we refer to the electrical events in both ICC and smooth muscle cells (SMC) as slow waves, reflecting the origin of all slow wave activity in ICC (52).

Slow waves recorded from ICC-MY in the stomach and small intestine have a characteristic morphology: a rapid upstroke phase, in which the cell depolarizes from a resting membrane potential between −80 and −50 mV to a peak potential between −25 and 0 mV (33, 52); a plateau phase, in which the membrane potential is maintained close to the peak potential; and a repolarization phase, in which the cell returns to the resting membrane potential.

Cyclical changes in intracellular Ca²⁺ underlie the slow wave activity in ICC (51, 64). The Ca²⁺ cycle involves Ca²⁺ release from and uptake into inositol 1,4,5-trisphosphate (IP₃)-mediated stores in the endoplasmic reticulum (ER). Ca²⁺ oscillations occur at the same frequency as slow waves, but to our knowledge intracellular Ca²⁺ concentration ([Ca²⁺]_i) has not been quantitatively measured in ICC. However, the amplitude of global Ca²⁺ transients in ICC is likely to be similar to that seen in other cell types, such as cardiac myocytes, in which [Ca²⁺]_i typically increases from a resting level ∼100 nM to a peak ∼1 μM (54).

IP₃ receptors (IP₃Rs) are known to contribute to the pacemaker mechanism, because slow waves are absent in gastric smooth muscle tissue from mice lacking type 1 IP₃R (59), and pacemaker activity is inhibited when phospholipase C blockers are used to inhibit IP₃ production (38, 40). In previous pacemaker hypotheses, IP₃-mediated ER Ca²⁺ release was proposed to initiate the pacemaker cycle, either by Ca²⁺-dependent activation of pacemaker channels (33) or by stimulating mitochondrial Ca²⁺ uptake to relieve Ca²⁺-dependent inhibition of pacemaker channels (52).

Another possible pacemaker mechanism involves activation of pacemaker channels by store-operated Ca²⁺ entry (SOCE) (36, 64). In brief, SOCE is the activation of Ca²⁺-permeable ion channels, called store-operated Ca²⁺ (SOC) channels, in the plasma membrane in response to depletion of Ca²⁺ stores in the ER. SOCE is initiated when stromal interaction molecule (STIM) proteins in the ER membrane sense a depletion of ER Ca²⁺ stores and respond by translocating to sections of ER close to the cell membrane (35, 74). SOC channels are formed by the oligomerization of STIM with pore-forming proteins in the cell membrane. STIM can interact either with pore-forming proteins called Orai to form Ca²⁺ release-activated Ca²⁺ (CRAC) channels that are highly selective for Ca²⁺ or with canonical transient receptor potential (TRPC) channel proteins to form nonselective cation channels (17, 45, 46) (see refs. Refs. 56, 58 for further details).

Evidence for the importance of SOCE in ICC comes from the inhibitory effects of 2-aminoethoxydiphenyl borate (2-APB) on pacemaker activity (36, 38), although it should be noted that 2-APB has complex effects on Ca²⁺ influx and release. 2-APB has varying effects on Orai, but in general >10 μM 2-APB inhibits CRAC channels, while low micromolar concentrations of 2-APB potentiate CRAC channels (50, 56). At similar concentrations, 2-APB inhibits a variety of TRPC and TRPM channels (26, 71). 2-APB inhibits IP₃Rs with an IC₅₀ of 42 μM to 1 mM, depending on the IP₃ concentration (5, 41). The effects of 2-APB on ICC must be interpreted with caution, but 2-APB appears to be a more consistent and potent inhibitor of SOC channels than IP₃Rs, suggesting a role for SOCE in ICC pacemaking.

The exact mechanism of interaction between the intracellular Ca²⁺ dynamics and the ion currents contributing to the slow wave potential is a matter of ongoing discussion. Recent reviews have described the possible roles of ion channels that have been identified in ICC (2, 33). A crucial step in understanding pacemaker activity is identifying the pacemaker channel that initiates slow waves. Proposed candidates for the pacemaker channel include Ca²⁺-activated Cl⁻ channels (69); Ca²⁺-inhibited nonselective cation channels (30); Ca²⁺-activated nonselective cation channels (21); or TRPM7 nonselective cation channels (26). A likely candidate is anoctamin 1 (Ano1, previously called TMEM16A), which has been identified as a Ca²⁺-activated Cl⁻ channel (8, 53, 72). Ano1 is highly expressed in all classes of ICC in the human and murine gastrointestinal tracts (20) and appears to be essential for the electrical pacemaker activity of ICC, because Ano1 knockout mice do not exhibit slow waves (23).

Ano1 channels have a complex dependence on Ca²⁺ and voltage. They are slowly activating and outwardly rectifying at low [Ca²⁺]_i, but at high [Ca²⁺]_i they activate rapidly and have a linear voltage-current relationship (70). Ca²⁺-activated Cl⁻ currents were studied extensively before the discovery that the anoctamin family of proteins, particularly Ano1, is a major component of Ca²⁺-activated Cl⁻ channels (7, 62). Mathematical models of these Ca²⁺-activated Cl⁻ currents were developed (7, 32), but we cannot ascertain whether the experimental recordings on which these models were based were exclusively carried by Ano1 or whether other Ca²⁺-activated Cl⁻ channels were involved, requiring us to develop a novel Ano1 channel model.

Furthermore, it is important to understand how the Ano1 channel fits into a model of pacemaker activity. The existing biophysically based models of ICC slow wave activity were completed before the discovery of Ano1 in ICC, so none of these models incorporated the Ano1 channel (10, 16, 73). In addition, SOCE was not represented in the existing ICC models. Both the Faville small intestine ICC model (16) and the Corrias and Buist gastric ICC model (10) implemented a Ca²⁺-inhibited nonselective cation channel as the pacemaker channel, based on a pacemaker hypothesis that has since been called into question (23, 33).

Therefore, we have developed a novel ICC model that includes our Ano1 model as the pacemaker channel and incorporates SOCE as a key component of the Ca²⁺ dynamics. We chose to represent the pacemaker activity of ICC-MY from murine small intestine with this model, because mice are a common animal model for studying ICC electrophysiology. Slow waves in the murine small intestine typically occur at 16–30 min⁻¹ (21, 29), but frequencies of 10–50 min⁻¹ have been observed (61). The overall morphology of a slow wave, rapid upstroke, plateau phase, and repolarization, is determined by the summation of the individual ion currents active during the slow wave.

The primary aim of the modeling work presented here is to elucidate how Ano1 Ca²⁺-activated Cl⁻ channels contribute as pacemaker channels in ICC. The precise function of Ano1 current in ICC depends on the Cl⁻ equilibrium potential (E_Cl). Measurements of intracellular Cl⁻ concentration ([Cl⁻]_i) indicate that E_Cl in murine ICC-MY is 40 to −50 mV (76), but microelectrode techniques used to record slow waves may lead to Cl⁻ loading (4), and previous studies have suggested that Cl⁻ currents reverse positive to the slow wave plateau (29). Therefore, we also aimed to test the effect of varying E_Cl on slow waves in ICC-MY.

In addition, the identity of the ion channels active during the plateau phase remains unclear. We also aimed to predict the characteristics of the ion channels that are required to reproduce the expected slow wave morphology. Specifically, suitable ion channel candidates may be activated by voltage or Ca²⁺, remain active for the duration of the plateau phase, and carry an inward current at the plateau.

We present first the development of the ICC and Ano1 models, followed by verification that the Ano1 model exhibits the expected Ca²⁺ and voltage-dependent behavior of Ano1 channels. We then report the results of simulations testing the impact of altering E_Cl, showing that the ability to reproduce the characteristic slow wave morphology is highly dependent on E_Cl. Finally, we report the results of varying the ion currents in the model and show that the properties of Ca²⁺ transients and slow waves are particularly sensitive to the properties of Ca²⁺ currents included in the model.

METHODS

Overview of Baseline Cell Model

The baseline cell model presented in this study represents the pacemaker activity of ICC-MY from murine small intestine. Our model is predicated on the fundamental hypothesis that Ano1 channels activated by SOCE initiate slow waves in ICC-MY. A baseline model of this pacemaker hypothesis was developed and is adapted to investigate the effects of varying the ion currents and E_Cl in the model. Several key assumptions were made in the development of the baseline model, and the experimental evidence supporting each assumption is outlined below, followed by a summary of the pacemaker hypothesis.

The first assumption is that cyclical oscillations of Ca²⁺ in the ER and cytosol are an essential component of the ICC pacemaker mechanism. The importance of intracellular Ca²⁺ dynamics in ICC pacemaker activity is well established. Pacemaker activity is inhibited when Ca²⁺ uptake via the sarco-endoplasmic reticulum ATPase (SERCA) is blocked (40, 66). IP₃Rs and ongoing IP₃ generation are essential for pacemaker activity in ICC (38, 40, 59). Mitochondrial Ca²⁺ handling has also been proposed to play a role in slow wave generation (66), but more recent evidence from ICC in situ indicates that mitochondria may not be essential for Ca²⁺ cycling (38), so they were not included in our model.

The second assumption is that SOCE is a key part of ICC pacemaker activity. The occurrence of SOCE following Ca²⁺ store depletion was demonstrated in ICC by inhibiting SERCA pumps with thapsigargin (64). Furthermore, inhibition of Ca²⁺ oscillations by Ca²⁺-free external solution and the SOC channel inhibitors SK&F96365 and 2-APB also demonstrated the importance of SOCE (36, 64).

The third assumption is that Ano1 Ca²⁺-activated Cl⁻ currents are crucial for the initiation of slow waves (23).

The fourth assumption is that Ano1 channels in ICC are colocalized with SOC channels, such that Ca²⁺ release from the ER causes SOCE, resulting in Ca²⁺-dependent activation of nearby Ano1 channels. Ano1 gating occurs in a complex Ca²⁺ and voltage-dependent manner. The literature is not consistent, but in general the EC₅₀ for Ca²⁺ decreases from 5.9 ± 25 μM at +100 mV to 0.4 ± 01 μM at +100 mV (70). Global Ca²⁺ transients in ICC are likely to be similar in magnitude to those in other cell types, which typically peak around 1 μM (54), but at this level of Ca²⁺ the Ano1 channel variant we modeled is <10% activated at resting membrane potential (see Fig. 2). Ano1 channels can be exposed to sufficiently high Ca²⁺ concentrations by being in close proximity to a Ca²⁺ source. For example, Ca²⁺-activated Cl⁻ channels are activated by SOC channels in vascular SMC (1) and Xenopus oocytes (31, 32), and Ano1 channels were recently shown to be coupled to IP₃Rs in nociceptive sensory neurons (24). We assume that Ano1 channels are colocalized with SOC channels in ICC because of the evidence detailed above about the importance of SOCE.

Fig. 2.

Steady-state open probability of the Ano1 (O_Ano1) model as a function of membrane potential (V_m) and the intracellular Ca²⁺ concentration ([Ca²⁺]_i) seen by the Ano1 channel (Ca_Ano1). Maximal open probability is achieved with simultaneously high Ca²⁺ concentration and positive membrane potential.

The final assumption is that T-type Ca²⁺ channels contribute to the upstroke phase of slow waves in ICC. A number of voltage-dependent ion channels have been identified in ICC-MY, and the role of these ion channels remains a matter of ongoing discussion (2, 33), but experimental evidence shows that T-type Ca²⁺ currents are important for the upstroke phase (19, 29). In addition, Ca²⁺ influx through T-type Ca²⁺ channels can modulate the intrinsic Ca²⁺ cycle by activating quiescent IP₃Rs, enabling voltage-dependent coordination and propagation of slow waves (52).

In summary, the proposed pacemaker cycle we model begins when IP₃-mediated Ca²⁺ release from the ER leads to depletion of the ER Ca²⁺ stores, activating SOCE via SOC channels. Ano1 channels colocalized with SOC channels are activated by the local rise in Ca²⁺ in microdomains surrounding the open SOC channels, causing depolarization of the ICC. The depolarization initiates a slow wave by activating voltage-dependent ion channels. The morphology of the slow wave, including the plateau phase and repolarization, is determined by the balance of voltage- and Ca²⁺-dependent ion currents. The pacemaker cycle concludes when Ca²⁺ influx via SOC channels and uptake via SERCA pumps is sufficient to refill the ER stores, resulting in deactivation of the SOC channels and Ano1 channels.

This pacemaker hypothesis shares similarities with previous pacemaker hypotheses (33, 52), notably the importance of ER Ca²⁺ handling, Ca²⁺ microdomains, and the role of T-type Ca²⁺ channels.

The key components of this pacemaker hypothesis are incorporated in the baseline cell model, as illustrated in the schematic diagram of an ICC in Fig. 1. Each component is described in more detail below. The model supports the pacemaker hypothesis if it can quantitatively and qualitatively reproduce experimentally observed results. If the model cannot adequately simulate slow waves, then the pacemaker hypothesis needs to be revised accordingly.

Fig. 1.

Schematic diagram of the cell, including ion channels and transporters in the plasma membrane and endoplasmic reticulum (ER). The cell is divided into two compartments: the bulk cytosol and the ER. Ca²⁺ changes in the bulk cytosol are determined by the Ca²⁺ flux through the inositol 1,4,5-trisphosphate (IP₃) receptors (IP₃R) and sarco-endoplasmic reticulum ATPase (SERCA) pumps in the ER, and the T-type and store-operated Ca²⁺ (SOC) channels and plasma membrane Ca²⁺ ATPase (PMCA) in the cell membrane. The Ca²⁺ microdomain around the colocalized Ano1 and SOC channels is indicated by a dotted line. The microdomain is not modeled as a separate compartment but rather is directly dependent on the Ca²⁺ flux through the SOC channel. The membrane potential across the plasma membrane is governed by ion currents through the Ano1, SOC, and T-type Ca²⁺ channels along with additional inward and outward currents passed by other channels as specified later.

The baseline ICC model is a compartmental model in which the membrane potential within the cell and the ion concentrations within each subcellular compartment are homogeneous. The model was implemented using a Hodgkin-Huxley type formulation, in which the cell membrane lipid bilayer is represented as a capacitance (C_m), and the ion channels in the membrane are represented as conductances. The change in the transmembrane potential (V_m) over time depends on the total ionic current:

$$\frac{\text{d}{V}_{\text{m}}}{\text{d}t}=-\frac{I_{\text{ion}}}{C_{\text{m}}},$$ (1)

where I_ion is the sum of the individual ion currents through each class of ion channel in the cell. C_m was set to 25 pF, which is a representative value for a murine small intestinal ICC (21, 27).

Each class of ion channel is represented as a variable conductor, where the conductance depends on the channel open probability as a function of voltage or [Ca²⁺]_i. The whole cell current through each class of ion channel is proportional to the difference between V_m and the reversal potential of the channel. For an ion channel selective for a single ion, the reversal potential is equal to the Nernst or equilibrium potential of that ion. The equations for the open probability of each channel are given in the appendix. The model equations were solved in MATLAB (R2012a).

Calcium Model

Two intracellular Ca²⁺ compartments were included in the model: bulk cytosol Ca²⁺ ([Ca²⁺]_i) and an endoplasmic reticulum Ca²⁺ store ([Ca²⁺]_ER), as shown in Fig. 1. To model Ca²⁺ flux into and out of the ER, a simple model of intracellular Ca²⁺ dynamics was adapted from the oscillatory Ca²⁺ model developed by Wang et al. (65) for airway SMC. The specific components used were a sigmoidal model of the SERCA pump and an IP₃R model based on the De Young and Keizer model (12). The ryanodine receptor was excluded because it appears to be less important for ICC pacemaker activity (40). The equations describing the IP₃R and SERCA Ca²⁺ flux were unchanged from the model presented by Wang et al. (65) with parameter values modified to adjust the frequency of Ca²⁺ oscillations to an appropriate value for murine small intestine (see appendix for details).

SOC Channel Model

The type of SOC channel that might be present in ICC is unknown. Activation of SOC channels occurs when STIM1 senses depletion of the [Ca²⁺]_ER. The [Ca²⁺]_ER at which STIM1 activates is around 200–250 μM (6, 57). Measured Hill coefficients for activation of the STIM1-Orai1 complex by ER depletion range from 4 to 8 (6, 39). Therefore, a Hill function with a coefficient of 8 and K_d of 200 μM was used to model SOC channel open probability as a function of [Ca²⁺]_ER.

When STIM1 is activated it can oligomerize with either Orai or TRPC proteins to form CRAC channels or nonselective cation channels, respectively (45, 46). CRAC channels are highly selective for Ca²⁺, and native CRAC currents are very small (56). In this model the Ca²⁺ influx via the SOC channel was important for activating Ano1, but the SOC channel itself was expected to make a negligible contribution to depolarizing the membrane potential, as evidenced by the lack of slow waves in smooth muscle tissue from Ano1 knockout mice (23). Therefore, the SOC channel was implemented as a highly Ca²⁺-selective CRAC channel with a small conductance of 0.1 nS. The mechanism by which Ca²⁺ flux through the SOC channel acts on Ano1 is described below.

Ano1 Ca²⁺-Activated Cl⁻ Channel Model

The voltage- and Ca²⁺-dependent behavior of mouse Ano1 channels heterologously expressed in HEK 293 cells was recently characterized. Equations were reported for both the steady-state Ca²⁺-dependent activation as a function of voltage and voltage-dependent activation as a function of [Ca²⁺]_i. The time constants of activation and deactivation were also quantified (70).

Using this data we developed a novel Hodgkin-Huxley type model of coupled voltage- and Ca²⁺-dependent Ano1 activation. The steady-state activation for an ion channel is given by G/G_max (or I/I_max), whereas the previously published equations scaled G to the maximal conductance recorded at a set value of [Ca²⁺]_i or V_m; G_max,Ca or G_max,V, respectively (70). Thus, we model the steady-state open probability (O_Ano1) by scaling G/G_max,V by the G/G_max,Ca at a very high [Ca²⁺]_i, giving O_Ano1 as the product of a Boltzmann function for voltage-dependent activation and a Hill equation for Ca²⁺ binding:

$$O_{\text{Ano1}}=\frac{1}{\left[1+e^{(V_{\text{h}}-V_{\text{m}})k_{\text{v}}}\right]\left[1+{\left(\frac{{\text{EC}}_{50}}{{\text{Ca}}_{\text{Anol}}}\right)}^2\right]}$$ (2)

where V_h is the half-activation potential at high [Ca²⁺], k_v is the slope factor associated with the gating charge of the channel, EC₅₀ is a voltage-dependent expression for the half-concentration of Ca²⁺-dependent activation, and Ca_Ano1 is the [Ca²⁺]_i in the vicinity of the Ano1 channel. The steady-state open probability of the Ano1 channel model is shown in Fig. 2.

Calcium Microdomain Calculation

Ano1 channels are likely to be located within a Ca²⁺ microdomain near a Ca²⁺ channel. In previous models of ICC, microdomains were represented using a small compartment with separate differential equations for the microdomain Ca²⁺ concentration and the bulk cytosol [Ca²⁺]_i (10, 16) or using a spatial representation of Ca²⁺ concentration throughout the microdomain and bulk cytosol (44).

The compartmental method does not have a heavy computational load, although computational efficiency decreases as the number of microdomains represented increases (10, 16). However, the Ca²⁺ concentration calculated within the microdomain compartment is unlikely to be realistic. On the other hand, the spatial method provides greater accuracy in estimating Ca²⁺ concentration but is computationally expensive (44).

In the present study the [Ca²⁺]_i seen by the Ano1 channel due to SOC channel Ca²⁺ influx is represented using the analytical formulation for the steady-state solution to the rapid buffering approximation of [Ca²⁺]_i near the entrance of a point source (55). This enables colocalized channels to be modeled without the additional computational expense or uncertainties introduced by spatial or compartmental methods.

The [Ca²⁺] seen by Ano1 (Ca_Ano1) was calculated using the solution for the [Ca²⁺] near a channel mouth with one mobile buffer in the cytosol (55). The solution depends on the bulk Ca²⁺ concentration far away from the point source, i.e., [Ca²⁺]_i (Ca_i); the [Ca²⁺] at the mouth of the SOC channel (Ca_SOC); and the distance from the SOC channel to the Ano1 channel, which was set to 50 nm to provide sufficiently high Ca_Ano1. To ensure Ca_Ano1 represented the [Ca²⁺] seen by an individual Ano1 channel, Ca_SOC was calculated by dividing the total flux of Ca²⁺ into the cell through SOC channels by an estimate of the number of SOC channels or channel clusters in the cell, which was estimated to be 50 channel clusters, based on the estimation that there are approximately 50 separate Ca²⁺ microdomains in an ICC (16).

Voltage-Gated Calcium Channel Model

The final component implemented in the baseline model was a voltage-dependent T-type Ca²⁺ channel. This channel was modeled using the voltage-dependent dihydropyridine-resistant (VDDR) Ca²⁺ channel model from the Corrias and Buist ICC model (10), except where otherwise stated.

Total Calcium Fluxes in the Cytosol

Cytosolic Ca²⁺ in the model was determined by the Ca²⁺ fluxes across the ER and plasma membranes, and cytosolic Ca²⁺ buffers were assumed to be fast and linear. Thus the total change in [Ca²⁺]_i for each of the model variations presented below, except where otherwise stated, was given by

$$\frac{{\text{dCa}}_{\text{i}}}{\text{d}t}=b_{\text{c}}(J_{\text{IPR}}-J_{\text{SERCA}}+J_{\text{SOC}}+J_{\text{CaT}}-J_{\text{PMCA}})$$ (3)

where b_c is the buffering constant in the cytosol, J_IPR is the flux out of the ER through the IP₃R, J_SERCA is the uptake into the ER, J_SOC is the total flux through the SOC channels, J_CaT is the total flux through the voltage-gated T-type Ca²⁺ channels, and J_PMCA is the rate of extrusion of Ca²⁺ out of the cell via the plasma membrane Ca²⁺ ATPase (PMCA) pump.

Model Variations

The baseline cell model was used to develop six model variations to investigate the currents active during the plateau phase and the effects of changing E_Cl. E_Cl in ICC-MY is reported to be around −40 to −50 mV (76). However, previous estimates of E_Cl were often higher than this (63) and depolarizing Ca²⁺-activated Cl⁻ currents were suggested to produce the plateau phase (29). In contrast, Cl⁻ loading during microelectrode recordings may raise [Cl⁻]_i and E_Cl above endogenous levels, as discussed later (4). Therefore, we test two different levels of E_Cl: one in which E_Cl is close to the plateau potential; and one in which E_Cl is at the reported value of −50 mV (76).

Ano1 Cl⁻ currents can initiate the slow wave and T-type Ca²⁺ channels contribute to the upstroke, but the identity of the ion channels active during the plateau phase remains undetermined, so four hypothetical plateau currents are considered. These currents encompass a range of ion channel properties that could theoretically maintain a depolarizing current at the peak potential of a slow wave, including currents that have thus far not been identified in ICC. The four scenarios considered are as follows: 1) plateau current is carried by voltage-dependent Na⁺ channels; 2) plateau current is carried by voltage-dependent nonselective channels; 3) plateau current is carried by Ca²⁺-dependent nonselective channels that are activated by a voltage-dependent Ca²⁺ influx; or 4) plateau current is carried by voltage-dependent Ca²⁺ channels.

No long-lasting Na⁺ channels have been identified in ICC to date, but this type of channel is theoretically capable of generating the plateau phase. A candidate Ca²⁺ channel that may stay open long enough to generate the plateau current is the L-type Ca²⁺ channel (27). The voltage- and Ca²⁺-dependent nonselective channels may be either nonselective cation channels of maxi channels, both of which pass currents that reverse close to 0 mV. Nonselective cation channels in ICC are likely to be Ca²⁺ dependent (60). Maxi channels observed in ICC in situ are permeable to both Cl⁻ and Na⁺ ions and display voltage-dependent activation and inactivation (49).

Originally, a Cl⁻ current generated by the Ano1 channel was also considered as a fifth plateau current candidate. However, we found that a Cl⁻ current could not maintain the plateau in our model unless the conductance was unrealistically large and E_Cl was unphysiologically positive, so this possibility was not considered plausible.

The six model variations have different hypotheses regarding the ion currents involved in generating slow waves, such that the “additional currents” mentioned in Fig. 1 differ in each variation. The model variations are grouped into two categories based on [Cl⁻]_i and the resultant E_Cl.

High-Cl.

E_Cl is close to the peak of the slow wave at −20 mV. Ano1 current is responsible for initiating the slow wave, but an additional plateau current is included to maintain depolarization through the peak and plateau. There were four variations to the High-Cl hypothesis:

1) A voltage-dependent Na⁺ current (I_NaV) acts to maintain depolarization through the peak and plateau.

2) A voltage-dependent nonselective cation or maxi current (I_NSV) with a reversal potential of 0 mV maintains depolarization through the peak and plateau.

3) The voltage-dependent T-type Ca²⁺ current, activated by the initial depolarization caused by Ano1, contributes to the upstroke and produces a larger increase in [Ca²⁺]_i than that due to Ca²⁺ influx through SOC channels alone. The Ca²⁺ transient activates a Ca²⁺-dependent nonselective cation or maxi current (I_NSCa) with a reversal potential of 0 mV to maintain depolarization through the peak and plateau.

4) A long-lasting voltage-dependent Ca²⁺ current (I_CaV) acts to maintain depolarization through the peak and plateau.

Low-Cl.

E_Cl is −50 mV, as measured in ICC in situ (76). Depolarization to the peak potential and during the plateau phase is due to either a voltage-gated current or a Ca²⁺-activated current. Based on the High-Cl simulation results, there was little value in testing I_CaV at the lower E_Cl. The similarities between the High-Cl results with I_NaV and I_NSV and the effect of E_Cl on the simulation results, as discussed later, meant that a single voltage-gated current could be tested in the Low-Cl hypotheses without affecting the conclusions of the study. Therefore, the candidate plateau currents tested were as follows:

1) A voltage-activated Na⁺ current, as in High-Cl(NaV), or

2) A Ca²⁺-activated nonselective current that is activated by transient voltage-gated Ca²⁺ current, as in High-Cl(NSCa).

Repolarization at the end of the plateau phase may be initiated by a combination of a delayed activation or increased amplitude of a K⁺ current or Ano1 current, and a voltage-, Ca²⁺-, or time-dependent inactivation of the inward plateau current.

The full complement of ion currents included in each model variation is detailed in results. The equations and parameters for each of these ion channels are given in the appendix.

RESULTS

Ano1 Model

The first step was to verify our novel Ano1 model against published data at various Ca²⁺ and voltage levels. The steady-state open probability of the Ano1 channel model in Fig. 2 shows that open probability is increased by both Ca_Ano1 and membrane potential, resulting in strong outward rectification of steady-state Ano1 current at lower Ca²⁺ concentrations, as expected. The kinetics of the Ano1 model (Fig. 3) were compared with published voltage-clamp and fast Ca²⁺ perfusion experiments (see Figs. 1 and 3 in Ref. 70). For these simulations Ano1 currents were calculated by setting a reversal potential of 0 mV and a maximal conductance of 0.5 nS.

Fig. 3.

A–C: simulated voltage-clamp experiments to test the response of the Ano1 model to the voltage protocol used by Xiao et al. (see Fig. 1, A–E, in Ref. 70). From a holding potential of 0 mV, the Ano1 model was stepped for 800 ms to a V_m between −100 and +100 mV in 40-mV increments, followed by a hyperpolarizing step to −100 mV, as shown in the inset. Depolarizing steps activate Ano1 currents more strongly and rapidly at higher Ca²⁺ concentrations, as expected. However, above 1 μM Ca²⁺ the decay of the simulated tail currents following a hyperpolarizing step occurs more rapidly than expected based on experimental observations. D: simulated rapid Ca²⁺ perfusion experiments, to compare the response of the Ano1 model to published rapid perfusion experiments (see Fig. 3, A and B, in Ref. 70). V_m in the Ano1 model was set at a fixed holding potential ranging from −100 to +100 mV and Ca_Ano1 was stepped from 0.1 to 20 μM for 800 ms. Ano1 activation upon application of 20 μM Ca²⁺ occurs quickly and the rate of activation is only weakly voltage-dependent, while deactivation upon removal of 20 μM Ca²⁺ is slow and the deactivation rate is strongly voltage dependent.

In the voltage-clamp simulations the voltage protocol stepped the Ano1 model from a holding potential of 0 mV to a V_m between −100 and +100 mV in 40-mV increments for 800 ms, followed by a hyperpolarizing step to −100 mV (Fig. 3A, inset). Voltage clamp simulations were carried out at three different values of fixed [Ca²⁺]: 0.1 μM (Fig. 3A), 1 μM (Fig. 3B), and 10 μM (Fig. 3C). The simulated voltage-dependent activation qualitatively matches the experimentally observed behavior, with higher Ca²⁺ concentrations producing larger and more rapid activation. However, at higher Ca²⁺ concentrations the decay of the tail currents following a hyperpolarizing step occurs much more rapidly in the simulations than in published data (see Fig. 1, A–E, in Ref. 70), indicating that the model does not fully capture the time dependence of voltage-dependent deactivation (Fig. 3C).

Rapid perfusion experiments were performed in which [Ca²⁺] was changed within several milliseconds (70). These experiments were reproduced using the Ano1 model by setting V_m at a fixed holding potential (ranging from −100 to +100 mV in 40-mV increments) and applying a step change in Ca_Ano1 from 0.1 μM to 20 μM for 800 ms. The Ano1 model produces fast Ca²⁺-dependent activation at all voltages and slower deactivation upon Ca²⁺ removal with a strongly voltage-dependent deactivation rate (Fig. 3D), in qualitative agreement with published data (see Fig. 3, A and B, in Ref. 70).

ICC Model: Impact of Altering E_Cl

Following validation of the Ano1 channel model we tested the impact of varying E_Cl and the plateau current on the ICC model. Two values of E_Cl were tested: −20 and −50 mV, referred to as High-Cl and Low-Cl model variations, respectively. The results of each model variation are presented below.

Simulations were carried out using both the normal (“wild-type”) cell model and an “Ano1 knockout” scenario in which the maximal whole cell conductance of Ano1 (g_Ano1) is set to 0 nS to verify that Ano1 channels are essential for initiating the slow wave in each model variation.

The resting membrane potential, amplitude, frequency, and half-width (duration at half-maximal amplitude) were quantified, as summarized in Table 1. All the simulated slow waves have similar resting membrane potentials and amplitudes, and these values are within the expected range for ICC. The greatest variations in the quantified model results are in the frequency and half-width values, indicating that these temporal characteristics of the slow wave are more dependent than the membrane potential on the changes to E_Cl and the ion currents in the model.

Table 1.

Quantitative characteristics of slow waves simulated by the six model variations

	RMP, mV		Amplitude, mV	Half-Width, s	Frequency, min−1
Variation	Wild Type	Knockout	Wild Type	Wild Type	Wild Type	Knockout
High-Cl
NaV	−66.4	−67.2	44.4	1.2	18.8	17.7
NSV	−64.6	−65.1	44.4	1.5	19.2	17.8
NSCa	−65.3	−65.1	41.9	1.1	15.7	19.1
CaV	−65.1	−63.2	48.8	0.6	9.6	—
Low-Cl
NaV	−66.7	−66.0	42.9	0.8	18.3	17.8
NSCa	−67.2	−65.0	47.6	0.8	14.9	21.3

RMP is the resting membrane potential; amplitude is the difference between resting membrane potential and peak slow wave potential; half-width is the duration of slow wave as measured at half of the amplitude; wild-type denotes the simulations using normal model variations; knockout denotes anoctamin 1 (Ano1) knockout simulations; V is voltage; NS is nonselective.

High-Cl simulations.

The four High-Cl model variations (Fig. 4) set E_Cl at −20 mV, close to the peak potential of a slow wave. In addition to the three ion channels described in the baseline model, SOC channels (I_SOC), Ano1 channels (I_Ano1), and T-type Ca²⁺ channels (I_CaT), each High-Cl model variation incorporates a voltage-activated inward current to enable the plateau phase to be generated, as detailed below. Two “background” or “leak” channels are also included, a background K⁺ current (I_Kb) and a background Na⁺ current (I_Nab). I_Kb acts to maintain the resting membrane potential and to repolarize the cell following a slow wave, while I_Nab also contributes to maintenance of the resting membrane potential.

Fig. 4.

Membrane potential and Ca²⁺ oscillations produced from High-Cl model simulations with Cl⁻ equilibrium potential (E_Cl) set to −20 mV. The plateau current is generated by a voltage-activated Na⁺ channel (A and E), a voltage-activated nonselective channel (B and F), a Ca²⁺-activated nonselective channel (C and G), or a voltage-activated Ca²⁺ channel (D and H). A–D: membrane potentials simulated using the wild-type (WT) and Ano1 knockout (KO) scenarios (black and gray lines, respectively). E–H: oscillations in [Ca²⁺]_i simulated using the WT and Ano1 KO scenarios (black and gray lines, respectively).

Each of the High-Cl model variations were tested in both the wild-type scenario, with normal Ano1 current, and in the Ano1 knockout scenario, in which Ano1 conductance is set to zero (Fig. 4, A–D, black and gray lines, respectively). The background Na⁺ current plays an important role in inhibiting membrane potential oscillations in the Ano1 knockout simulations. When g_Ano1 is set to zero in test simulations without the background Na⁺ current, the depolarizations generated by the SOC current are tens of millivolts in amplitude (not shown), which runs counter to the key assumption that loss of Ano1 blocks slow waves.

As well as the Na⁺ background channel, the efficacy of two other channels at inhibiting membrane potential oscillations in Ano1 knockout simulations was trialed. A voltage-activated K⁺ channel with strong voltage dependence at resting membrane potential and a Ca²⁺- activated K⁺ channel colocalized with the SOC channel both act to reduce membrane potential oscillations in the Ano1 knockout scenario, but these channels are less effective than the Na⁺ leak channel (not shown).

Slow wave-like potentials are successfully produced by the High-Cl(NaV) model variation, which incorporates a voltage-activated Na⁺ channel (Fig. 4A). Qualitatively similar results are obtained using a voltage-activated nonselective channel [High-Cl(NSV), see Fig. 4B] or a Ca²⁺-activated nonselective channel in conjunction with a transient voltage-activated Ca²⁺ channel [High-Cl(NSCa), see Fig. 4C] but not with a longer lasting voltage-activated Ca²⁺ channel [High-Cl(CaV), see Fig. 4D], as described below.

HIGH-CL(NAV): SODIUM CHANNEL.

For the High-Cl(NaV) variation, a hypothetical voltage-activated, noninactivating Na⁺ current (I_NaV) is added to the model, so the total ionic current across the membrane is

$$I_{\text{ion}}=I_{\text{SOC}}+I_{\text{Ano}1}+I_{\text{CaT}}+I_{\text{Kb}}+I_{\text{Nab}}+I_{\text{NaV}}.$$ (4)

The model produces slow wave-like depolarizations that peak 2.0 mV negative to E_Cl (Fig. 4A; Table 1). Simulating Ano1 knockout results in a loss of slow waves, with residual oscillations <3 mV in amplitude (Fig. 4A, gray line) and little change in the resting membrane potential or frequency (Table 1).

Cytosolic Ca²⁺ oscillations occur at the same frequency as slow waves and persist in Ano1 knockout simulations (Fig. 4E). In the wild-type simulation, the bulk cytosolic Ca²⁺ concentration (Ca_i) peaks at 0.6 μM, whereas the [Ca²⁺] in the microdomain near the SOC channel mouth is ∼14 times greater than this (not shown), enabling a much higher level of Ano1 activation than would occur if Ano1 channels were not localized near SOC channels.

HIGH-CL(NSV): VOLTAGE-ACTIVATED NONSELECTIVE CHANNEL.

Currents with reversal potentials close to 0 mV have been observed in ICC, either carried by nonselective cation channels (21, 30) or maxi channels permeable to Cl⁻ and Na⁺ (49). Maxi channels in particular display activation and inactivation in response to voltage changes (49). Thus the High-Cl(NSV) model variation incorporates a voltage-dependent nonselective current (I_NSV) that reverses at 0 mV and displays voltage-dependent activation and partial inactivation. The total ionic current is

$$I_{\text{ion}}=I_{\text{SOC}}+I_{\text{Ano}1}+I_{\text{CaT}}+I_{\text{Kb}}+I_{\text{Nab}}+I_{\text{NSV}}.$$ (5)

The quantitative and qualitative characteristics of the membrane depolarizations generated by this model variation are comparable to those of slow waves generated by murine small intestine ICC. Membrane potential oscillations are reduced to 1.5 mV in amplitude in Ano1 knockout simulations, with little change in resting membrane potential or frequency (Fig. 4B, gray line; Table 1). Ca²⁺ oscillations occur at the same frequency as the slow waves, and are maintained in the Ano1 knockout simulation (Fig. 4F).

HIGH-CL(NSCA): CA²⁺-ACTIVATED NONSELECTIVE CHANNEL.

Ca²⁺ dependence has been identified in nonselective cation channels (30, 60). The Ca²⁺ dependence of the maxi channel identified in situ is unknown, although the maxi Cl⁻ channel observed in cultured ICC is Ca²⁺ dependent (69). The High-Cl(NSCa) model variation includes a Ca²⁺-activated nonselective channel 6 (I_NSCa) and the total ionic current is

$$I_{\text{ion}}=I_{\text{SOC}}+I_{\text{Ano}1}+I_{\text{CaT}}+I_{\text{Kb}}+I_{\text{Nab}}+I_{\text{NSCa}}.$$ (6)

The parameters for I_CaT in this model variation are modified to generate a larger voltage-dependent Ca²⁺ influx, and the K_d for the I_NSCa current is set higher than the peak voltage-independent Ca²⁺ amplitude. The resultant Ca²⁺-activated nonselective channel is activated by voltage-dependent Ca²⁺ influx in the wild-type scenario, generating slow waves that peak at −23.4 mV (Fig. 4C, black line; Table 1). However, the channel remains relatively inactive during Ano1 knockout simulations in which the voltage-dependent Ca²⁺ influx is negligible, so the amplitude of the residual membrane potential oscillations during Ano1 knockout simulations is reduced to 2.5 mV (Fig. 4C, gray line).

HIGH-CL(CAV): CALCIUM CHANNEL.

The reversal potentials of both Na⁺ and Ca²⁺-selective currents are much more positive than the peak potential of a slow wave, suggesting that a long-lasting voltage-activated Ca²⁺ current could be just as effective at generating the slow wave plateau phase as the voltage-activated Na⁺ current used in High-Cl(NaV). The High-Cl(CaV) model variation incorporates a long-lasting voltage-gated Ca²⁺ channel (I_CaV), so the total ionic current across the membrane is

$$I_{\text{ion}}=I_{\text{SOC}}+I_{\text{Ano}1}+I_{\text{CaT}}+I_{\text{Kb}}+I_{\text{Nab}}+I_{\text{CaV}}.$$ (7)

Correspondingly, the change in cytosolic [Ca²⁺]_i is

$$\frac{{\text{dCa}}_{\text{i}}}{\text{d}t}=b_{\text{c}}(J_{\text{IPR}}-J_{\text{SERCA}}+J_{\text{SOC}}+J_{\text{CaT}}+J_{\text{CaV}}-J_{\text{PMCA}}).$$ (8)

The peak potential of depolarizations generated by this model variation is within the expected range for an ICC-MY, and oscillations are abolished by Ano1 knockout (Fig. 4D). However, the model exhibits a much faster initial repolarization than that of a normal slow wave because of the impact of the long-lasting Ca²⁺ influx on the intrinsic Ca²⁺ dynamics (Fig. 4H). Consequently, the results of the High-Cl(CaV) model variation differ from those of the other High-Cl models and will be considered in more detail in ICC Model: Impact of Varying Ca²⁺ Currents.

In summary, the High-Cl model variations generate membrane potentials with amplitudes comparable to slow waves from the mouse small intestine, and this activity is inhibited when simulating Ano1 knockout (Fig. 4, A–D). Membrane potential oscillations occur at the same frequency as Ca²⁺ oscillations (Fig. 4, E–H). The Ano1 channel contributes negligible current at the slow wave peak in these model variations because the peak potential is close to E_Cl. The plateau phase is generated by a current with a more positive reversal potential, such as a nonselective or Na⁺ current.

Low-Cl simulations.

In the High-Cl model variation E_Cl is set at −20 mV, but this is higher than the E_Cl measured in ICC from the mouse small intestine (76). Furthermore, none of the model variations presented so far are able to depolarize the cell past E_Cl while retaining the characteristic plateau phase of the slow wave. The delayed repolarization in the High-Cl model variations is facilitated by deactivation of the Ano1 current at the end of the Ca²⁺ transient. The Low-Cl model variations set E_Cl at −50 mV, as measured in ICC-MY in situ (76). The slow wave depolarizes the ICC membrane potential past E_Cl, so a combination of time-, voltage-, and Ca²⁺-dependent currents is necessary to reproduce the plateau phase and initiate the repolarization phase.

When developing the Low-Cl model variations, many different combinations of inward and outward currents were tested in an effort to reproduce ICC slow wave activity. The model variations described below represent two of the most successful attempts. Both produce cyclical depolarizations with an amplitude and frequency comparable to those of slow waves recorded from murine small intestine ICC-MY and similar to the simulated slow waves in the High-Cl model variations. However, the morphology of the depolarizations produced by the Low-Cl model variations do not match the expected slow wave morphology (Fig. 5).

Fig. 5.

Membrane depolarizations and Ca²⁺ oscillations produced from model simulations with E_Cl set to −50 mV. The plateau current is carried by either a voltage-activated Na⁺ channel (A and C), or a Ca²⁺-activated nonselective channel (B and D). A and B: membrane potentials simulated using the WT and Ano1 KO scenarios (black and gray lines, respectively). C and D: oscillations in [Ca²⁺]_i simulated using the WT and Ano1 KO scenarios (black and gray lines, respectively).

LOW-CL(NAV): SODIUM CHANNEL.

The Low-Cl(NaV) model variation uses a voltage-dependent Na⁺ channel (I_NaV) to generate depolarization during the plateau phase, similar to High-Cl(NaV). In addition, two voltage-gated K⁺ currents are introduced, a transient K⁺ current (I_Kt), and an ERG-like K⁺ current (I_KERG) that activates upon repolarization (2). The total ion current is

$$I_{\text{ion}}=I_{\text{SOC}}+I_{\text{Ano}1}+I_{\text{CaT}}+I_{\text{Kb}}+I_{\text{Nab}}+I_{\text{NaV}}+I_{\text{Kt}}+I_{\text{KERG}}.$$ (9)

This model produces slow wave-like depolarizations with a similar resting membrane potential, amplitude, and frequency to the previous model variations (Table 1). However, the “plateau” phase is steeper than expected for an ICC, and the slow wave duration at half amplitude is 27–47% shorter than the High-Cl(NaV), High-Cl(NSV), and High-Cl(NSCa) model variations. In addition, repolarization slows down when the cell repolarizes past −50 mV as Ano1 current becomes inward again (Fig. 5A).

In the Ano1 knockout simulations, the membrane potential oscillations are reduced to 3.9 mV in amplitude, with little change in the resting membrane potential and frequency, while Ca²⁺ oscillations are retained (Fig. 5, A and C; Table 1).

LOW-CL(NSCA): CA²⁺-ACTIVATED NONSELECTIVE CHANNEL.

The final model variation tested is a Low- Cl variation in which a Ca²⁺-dependent nonselective channel is used to maintain depolarization during the plateau in response to a larger voltage-dependent Ca²⁺ influx, similar to High-Cl(NSCa).

Thus, the Low-Cl(NSCa) model variation incorporates a Ca²⁺-activated nonselective current (I_NSCa), as well as a modified T-type Ca²⁺ current to enable larger Ca²⁺ influx to activate the nonselective current (I_CaT), and a Na⁺/Ca²⁺ exchanger to balance the larger Ca²⁺ influx (I_NCX). The total ionic current is

$$I_{\text{ion}}=I_{\text{SOC}}+I_{\text{Ano}1}+I_{\text{CaT}}+I_{\text{Kb}}+I_{\text{Nab}}+I_{\text{NSCa}}+I_{\text{NCX}}.$$ (10)

The corresponding change in cytosolic [Ca²⁺]_i is

$$\frac{{\text{dCa}}_{\text{i}}}{\text{d}t}=b_{\text{c}}(J_{\text{IPR}}-J_{\text{SERCA}}+J_{\text{SOC}}+J_{\text{CaT}}-J_{\text{PMCA}}-J_{\text{NCX}}).$$ (11)

The Low-Cl(NSCa) model variation generates depolarizations with a similar resting membrane potential, amplitude and frequency to the other model variations (Table 1). As with the Low-Cl(NaV) variation, the plateau phase is less pronounced than in the High-Cl models. The fast upstroke phase is also not captured well by the Low-Cl(NSCa) variation, and the slow wave duration at half-amplitude is 27–47% shorter than the High-Cl model durations (Fig. 5B; Table 1).

During Ano1 knockout simulations the remaining membrane potential oscillations are 1.5 mV in amplitude and occur 43% faster than the wild-type oscillations, corresponding to the change in frequency in the Ca²⁺ oscillations (Fig. 5, B and D; Table 1).

In summary, the amplitude and frequency of the wild-type depolarizations in the Low-Cl simulations are within the established range for murine small intestine slow waves, and Ano1 knockout simulations inhibit membrane depolarization. However, the ability of the models to reproduce the characteristic plateau phase of a slow wave is less clear. Both the Low-Cl(NaV) and Low- Cl(NSCa) models rapidly repolarize immediately after the peak because repolarizing currents are greater in magnitude than depolarizing currents at the peak potential, resulting in a shorter slow wave duration than the High-Cl model variations. The Low-Cl(NSCa) model also does not adequately reproduce the fast upstroke phase of a slow wave. This reflects the altered balance of inward and outward currents compared with the High-Cl models as the Ano1 current reverses at −50 mV.

ICC Model: Impact of Varying Ca²⁺ Currents

In addition to testing the effect of altering E_Cl, this study also tests whether a variety of different ion currents are plausible candidates for the plateau current that maintains the plateau phase of the slow wave. When varying the plateau current, the impact of altering the Ca²⁺ influx has a notable impact on the quantitative and qualitative characteristics of the simulated pacemaker activity in both wild-type and Ano1 knockout scenarios.

Three of the six model variations, High-Cl(NaV), High-Cl(NSV), and Low-Cl(NaV), have a relatively small voltage-dependent Ca²⁺ influx carried by the T-type Ca²⁺ channel. The Ca²⁺ oscillations produced by these variations all have similar amplitudes and frequencies. In the wild-type simulations, the magnitude of Ca²⁺ oscillations is smaller and the frequency is faster than in the Ano1 knockout simulations (Figs. 4, E and F, and 5C; Table 1).

The three other model variations, High-Cl(NSCa), High-Cl(CaV), and Low-Cl(NSCa), have a comparatively larger voltage-dependent Ca²⁺ influx carried by either a modified version of the T-type Ca²⁺ channel [High-Cl(NSCa) and Low-Cl(NSCa)] or a longer lasting Ca²⁺ current designated ICaV [High-Cl(CaV)]. In these variations, larger and slower Ca²⁺ oscillations are generated in the wild-type simulations than in the Ano1 knockout scenario; for example, the High-Cl(NSCa) wild-type Ca²⁺ oscillations peak at 1.9 μM compared with 0.53 μM in Ano1 knockout simulations. The exception is the High-Cl(CaV) model variation, which does not oscillate in the Ano1 knockout scenario (Figs. 4, G and H, and 5D; Table 1).

Furthermore, the wild-type Ca²⁺ oscillations produced by the three models with large Ca²⁺ influx are larger in amplitude and slower in frequency than those from the four smaller Ca²⁺ influx models. For example, in wild-type simulations with High-Cl(NSCa), [Ca²⁺]_i peaks at 1.9 μM (Fig. 4G), whereas with High-Cl(NaV) and High-Cl(NSV) [Ca²⁺]_i peaks at 0.6 μM (Fig. 4, E and F).

In the Low-Cl(NSCa) model variation, the Ca²⁺ oscillations in the wild-type simulations are similar in magnitude and frequency to those from the High-Cl(NSCa) variation, due to the similarities in the Ca²⁺-activated nonselective plateau current in these two models. Conversely, the Ca²⁺ oscillations in the Low-Cl(NSCa) Ano1 knockout simulations are smaller and faster than in any of the other model variations because of the additional Ca²⁺ extrusion mechanism, the plasma membrane NCX, included in this model (Fig. 5D).

The High-Cl(CaV) model variation has a significantly larger Ca²⁺ influx than the other variations, with wild-type [Ca²⁺]_i peaking as high as 4.7 μM, compared with 1.9 μM in High-Cl(NSCa) or 0.6 μM in High-Cl(NaV) and High-Cl(NSV) (Fig. 4, E–H). As a result, the membrane potential and Ca²⁺ oscillations are significantly slower in the High-Cl(CaV) model variation, and in the Ano1 knockout simulation the lack of voltage-dependent Ca²⁺ entry results in complete loss of Ca²⁺ oscillations. Furthermore, while the resting membrane potential and amplitude of the depolarizations are within an appropriate range for slow waves recorded from ICC, the frequency and the morphology of the depolarizations are not. In particular, the characteristic plateau phase is not evident and instead a rapid repolarization follows immediately after the peak of the upstroke phase (Fig. 4, D and H; Table 1).

The main pattern that emerges from these results is that simulations with larger Ca²⁺ oscillations, corresponding to larger Ca²⁺ influx, tend to have slower frequencies; this holds true for both wild-type and Ano1 knockout simulations. Consequently, the model variations that have a larger voltage-dependent Ca²⁺ influx have more substantial changes in frequency in Ano1 knockout compared with wild-type simulations, associated with significant changes in the amplitude of Ca²⁺ oscillations in the absence of voltage-dependent Ca²⁺ influx (Figs. 4–5; Table 1).

DISCUSSION

To our knowledge, this article describes the first mathematical model of ICC to implement Ano1 as a pacemaker channel and to incorporate SOCE. We also present a novel mathematical model of the Ano1 Ca²⁺-activated Cl⁻ channel based on electrophysiological measurements of Ano1 currents.

Ano1 Model

Multiple isoforms of Ano1 are produced by alternative splicing of four different exons, a, b, c, and d. The Ano1 model presented here is based on the properties of mouse Ano1(a,c) expressed in HEK 293 cells (70). However, the expression of alternative splice variants is known to alter the kinetics and Ca²⁺ sensitivity of the channel (47).

It is possible that the Ano1(a,c) variant used to develop the Ano1 model presented in this study is not the most prevalent splice variant in ICC. Verifying which Ano1 splice variants are commonly expressed in human and mouse ICC and characterizing the electrophysiological properties of these variants would allow more precise modeling of the role of Ano1 in ICC. Different Ano1 isoforms were shown to be expressed at different rates in human diabetic gastroparesis compared with nondiabetic controls, but there was also variable expression of Ano1 isoforms in the control subjects (43). The expression of human Ano1 variants is controlled by a recently identified promoter, which may impact on the properties of Ano1 currents (42), and different Ano1 variants can interact, further modifying their behavior (47). These factors may complicate attempts to determine which isoform or isoforms should be included in an ICC model.

ICC Model Development

One of the general limitations of modeling biophysical processes is that models can be sensitive to minor parameter changes, and care must be taken to ensure this does not inadvertently alter the model results. In the process of developing this ICC model it became apparent that the most important parameter was the Cl⁻ reversal potential. Altering E_Cl has a significant impact on the ability of the model to generate slow waves in response to an initial depolarization by Ano1 channels. Furthermore, the channels responsible for maintaining depolarization during the plateau phase have not yet been identified. Consequently, instead of presenting a single model of ICC pacemaker activity, the dependence of the baseline model on E_Cl was explored in depth by developing six different model variations.

Voltage-gated Ca²⁺ influx is also shown to be a significant factor in the ability to generate slow waves. However, the qualitative results of the High-Cl model variations are relatively insensitive to model parameters other than E_Cl. As shown in Fig. 4, it is possible to simulate slow wave-like depolarizations when the plateau current is carried by a voltage-gated Na⁺ channel, a voltage-gated nonselective channel, or a Ca²⁺-activated nonselective channel. The exception is the High-Cl(CaV) model variation that includes a large voltage-gated Ca²⁺ current, which is unable to generate slow wave-like events because of the effect of the large Ca²⁺ influx on the intrinsic Ca²⁺ cycle.

Conversely, with the Low-Cl model variations the ability to reproduce slow wave-like events is highly dependent on the parameters used. Many of the parameter sets tested were unable to generate slow waves at all. However, those that were able to generate oscillatory depolarizations produced qualitatively and quantitatively similar results to those shown in Fig. 5, in which the plateau phase is not well reproduced.

Therefore, although a single pacemaker hypothesis underlies each of the model variations, this article does not present an exact replication of the pacemaker mechanism in ICC. Rather, we assess the viability of some of the components and processes that may contribute to ICC slow wave activity. We test the ability of four candidate ion channels to generate the plateau phase of the slow wave in response to an initial depolarization by Ano1 current and consider how E_Cl affects this mechanism.

Impact of Altering E_Cl

The High-Cl(NaV), High-Cl(NSV), and High-Cl(NSCa) model variations present mathematically plausible mechanisms for generating slow waves, thereby supporting the hypothesis that Ano1 acts as a pacemaker channel.

Conversely, the quantitative characteristics of the Low-Cl models are within the appropriate range for slow waves (Table 1) but the results do not qualitatively resemble ICC slow waves (Fig. 5, A and B). In particular, the prolonged plateau phase of slow waves is not adequately reproduced by the Low-Cl models, because the large outward current carried by Ano1 at the peak potential contributes to rapid repolarization.

The simulation results indicate that a typical slow wave morphology is more readily produced when E_Cl is near or above the peak slow wave potential, whereas the ability to generate slow waves is complicated by lowering E_Cl well below the peak potential. The short slow wave duration of the Low-Cl model variations restricts their use in other simulations, but the results can inform further experimental research.

Factors affecting Cl⁻ equilibrium potential.

Previous estimates of the equilibrium or reversal potentials of Cl⁻ currents in ICC varied widely, ranging from −10 (63) to −52 mV (48). Experimental (29) and computational (10) studies have suggested that Cl⁻ channels contribute an inward current to maintain the plateau phase of the slow wave.

Subsequent measurements of [Cl⁻]_i in ICC-MY from murine jejunum show that E_Cl is around −50 mV in situ or −41 mV in explant culture (76), suggesting that Cl⁻ channels generate outward current during the plateau phase and could contribute to repolarization of the slow wave. However, our model results suggest that the reversal potential of Ano1 Cl⁻ current is likely to be closer to the slow wave peak potential than to resting membrane potential.

It is possible that the [Cl⁻]_i measured in situ does not accurately reflect the [Cl⁻]_i seen by Ano1 channels during slow wave recordings. During microelectrode recordings, [Cl⁻]_i may be raised by the high KCl concentration within the microelectrode tip. Glass microelectrodes are typically filled with an aqueous solution of 3 M KCl, and this highly concentrated filling solution causes a leak of K⁺ and Cl⁻ ions into the impaled cell. The result is Cl⁻ loading, in which significant changes in [Cl⁻]_i and E_Cl can occur, thus altering the reversal potential of Cl⁻ currents (28). For example, Cl⁻ leak rates of 4.4 fmol/s and elevations in [Cl⁻]_i up to 72 μM/min were observed in nonexcitable cells when using a 1 M KCl electrode filling solution (4). In a cell ∼1 pl in volume, the estimated volume of a murine ICC (10, 73), a leak rate on the order of 1 fmol/s will cause up to a 1 mM/s rise in [Cl⁻]_i, which corresponds to a 4–8% increase per second in ICC-MY where [Cl⁻]_i is reported to be 12.8 to 26.1 mM (76). Even accounting for mechanisms to balance [Cl⁻]_i, such as diffusion out of the cell or the action of Cl⁻ transporters, a 1 mM/s Cl⁻ leak into the cell can have a dramatic effect on the reversal potential of Cl⁻ currents.

An important implication of E_Cl being raised by a Cl⁻ leak into the cell is that the slow wave morphology observed during microelectrode recordings may not accurately reflect the true morphology of slow waves in ICC. This could account for the difficulties we encountered in reproducing the typical slow wave morphology when E_Cl was set to −50 mV in our Low-Cl model variations. Physiological [Cl⁻]_i can also be altered in whole cell patch-clamp recordings of slow waves in cultured and isolated ICC, due to dialysis of the intracellular solution with the pipette solution (34). Manipulation of [Cl⁻]_i using whole cell patch clamp has been used in some studies to identify the Cl⁻ selectivity of ion currents (75), but in other studies dialysis may alter the physiological [Cl⁻]_i, making it difficult to determine the precise behavior of Cl⁻ currents and slow waves in ICC.

An alternative explanation for a discrepancy between the [Cl⁻]_i measured in situ (76) and the [Cl⁻]_i that facilitates simulation of slow wave-like depolarizations in this study is that Ano1 channels are located within Cl⁻ microdomains. Microdomains of Ca²⁺ are most common because of the highly buffered state of Ca²⁺ in the cytosol, but microdomains of other ions also occur, particularly close to ion transporters, such as Na⁺ microdomains observed near Na⁺/K⁺ ATPase and NCX (22). Cl⁻ microdomains in ICC could form due to accumulation of Cl⁻ by ion transport proteins in the cell membrane. ICC-MY are known to express several Cl⁻ transporters, including a Na-K-Cl cotransporter (NKCC1) and three varieties of K-Cl cotransporters (9, 68). NKCC1 can maintain [Cl⁻]_i above electrochemical equilibrium, and in murine small intestine ICC-MY this function is critical for the pacemaker mechanism (68).

Impact of Varying Ca²⁺ Currents

Other than E_Cl, the main determinant of the viability of the model variations presented in this study was the magnitude of Ca²⁺ influx associated with the plateau current.

The High-Cl(CaV) model variation, which includes a long-lasting voltage-activated Ca²⁺ current large enough to depolarize the cell above −20 mV, is not a valid model of slow wave activity. The large Ca²⁺ influx that accompanies depolarization alters the intrinsic Ca²⁺ cycle, resulting in low frequency spiking events instead of slow waves (Fig. 4D) and generating Ca²⁺ transients much larger than those observed in other cell types (54), but a smaller Ca²⁺ current would not cause sufficient depolarization for a slow wave. Thus Ca²⁺ currents are unlikely to contribute a major component of the plateau current.

However, it is plausible that a voltage-gated Ca²⁺ current is responsible for activating the plateau current, as in the High-Cl(NSCa) and Low-Cl(NSCa) model variations (Figs. 4C and 5B). The Ca²⁺ transients in these two model variations are somewhat larger than those observed in cardiac myocytes (54). If the plateau channel model was more sensitive to Ca²⁺, it could be activated by the smaller voltage-dependent Ca²⁺ influx used in the High-Cl(NaV) and High-Cl(NSV) model variations, but this would obviate the need for Ano1 to initiate slow waves. Alternatively, the Ca²⁺-activated nonselective channel could be colocalized with the T-type Ca²⁺ channel. This is consistent with previous pacemaker hypotheses in which the plateau phase occurs in response to voltage-dependent Ca²⁺ influx via T-type Ca²⁺ channels located in microdomains with IP₃Rs and Ca²⁺-dependent cation channels (33, 52). Modeling microdomains containing T-type Ca²⁺ channels and the plateau channel were outside the scope of the present study.

Mathematically plausible models of pacemaker activity in which voltage-dependent Ca²⁺ influx does not play a crucial role are also presented in our study. In the High-Cl(NaV) model variation, the plateau current is a voltage-dependent Na⁺ current, and in the High-Cl(NSV) model variation, the plateau current is a voltage-dependent nonselective current (Fig. 4, A and B). T-type Ca²⁺ channels are present but are not essential for producing the plateau phase, or for the Ca²⁺ cycle, as evidenced by the continuation of Ca²⁺ oscillations in the Ano1 knockout simulations. Nevertheless, Ca²⁺ influx via SOC channels remains an intrinsic component of the pacemaker mechanism in these model variations.

Although T-type Ca²⁺ channels may not be crucial for generating slow waves in isolated ICC, voltage-dependent Ca²⁺ currents appear to play an important role in the propagation of slow waves through ICC networks (52). We intend to use this ICC model to simulate the behavior of ICC networks in future studies.

Effects of Ca²⁺ influx on Ano1 knockout simulations.

All the model variations except for High-Cl(CaV) demonstrate continuation of Ca²⁺ oscillations in the Ano1 knockout simulations, suggesting that Ca²⁺ oscillations occur independently of slow waves. These oscillations result from cyclical release and uptake of ER Ca²⁺ combined with SOCE. The frequency and magnitude of the Ca²⁺ oscillations in Ano1 knockout simulations also depend on the properties of the voltage-gated Ca²⁺ channel and the plasma membrane Ca²⁺ extrusion mechanism included in each model variation.

In the High-Cl(NaV), High-Cl(NSV), and Low- Cl(NaV) model variations, the amplitude of Ca²⁺ oscillations in the wild-type simulation is smaller than in the Ano1 knockout simulation (Figs. 4, E and F, and 5C). This result is unexpected, because the Ca²⁺ cycle in the wild-type simulations is augmented by a voltage-dependent Ca²⁺ influx that is inactive in the Ano1 knockout simulations. However, the increased amplitude of Ca²⁺ oscillations in the Ano1 knockout scenario is due to an increased driving force for SOCE at the more negative membrane potentials.

Conversely, in the High-Cl(NSCa) and Low-Cl(NSCa) model variations, the amplitude of Ca²⁺ oscillations in the wild-type simulations is much larger than in the Ano1 knockout simulations because of the larger voltage-dependent Ca²⁺ influx in the wild-type simulations (Figs. 4G and 5D). These model variations also demonstrate increased frequencies in Ano1 knockout simulations resulting from the decreased amplitude of Ca²⁺ oscillations, likely due to the greater capacity of the plasma membrane Ca²⁺ pumps (see Table A1 in the appendix).

The 43% increase in frequency in the Low-Cl(NSCa) Ano1 knockout simulations appears to contradict experimental evidence showing that Ca²⁺-dependent currents recorded from ICC under voltage clamp oscillate at similar frequencies to slow waves (61, 63). However, the impact of Ca²⁺ influx on the simulated Ca²⁺ transient depends on the Ca²⁺ model used. The Low-Cl(NSCa) model variation includes simple phenomenological models of two Ca²⁺ extrusion mechanisms: a high-affinity plasma membrane Ca²⁺ ATPase (PMCA) and a low-affinity Na⁺/Ca²⁺ exchanger (NCX) (37). Introducing a more complex model of an adaptive Ca²⁺ extrusion mechanism may enable the slow wave model to adjust to the changes in Ca²⁺ influx observed in the wild-type and Ano1 knockout simulations to maintain similar frequencies of Ca²⁺ oscillations.

Impact of Ano1 Knockout on Slow Waves

One of the key assumptions of the baseline model is that slow waves should be absent in an Ano1 knockout scenario. In recordings from Ano1 knockout mice the membrane potential of SMC is quiescent (23). However, to our knowledge, there have been no membrane potential recordings from ICC in Ano1 knockout mice, so we cannot be certain that the absence of slow waves in SMC reflects a complete loss of slow waves in ICC.

In the Ano1 knockout simulations presented here, slow wave activity is inhibited but membrane potential oscillations are not completely abolished, except in the High- Cl(CaV) model in which both membrane potential and Ca²⁺ oscillations ceased. The residual membrane potential oscillations in the Ano1 knockout scenarios are generated primarily by the cyclical activation and deactivation of the SOC current in response to changes in ER Ca²⁺ (Figs. 4 and 5). Therefore, our results predict that Ano1 knockout mice retain small Ca²⁺-driven depolarizations that are not transmitted to the SMC.

Leak currents are required to minimize the depolarizing effect of SOC current during Ano1 knockout simulations. The Na⁺ leak current may be carried by a Na⁺ leak channel (NALCN) recently identified in murine small intestine ICC (25). The K⁺ leak current is likely to be carried by KCNK3 (also called TASK-1), which encodes a background K⁺ channel that is abundantly expressed in ICC-MY (9).

Role of SOCE

Another of the key assumptions made in the model development is that slow waves are initiated when Ca²⁺ influx via SOC channels activates Ano1 channels. Store depletion and refill cycles in nonexcitable cells typically occur on slow time scales, on the order of tens of seconds, due to the time it takes for STIM1 to translocate to the plasma membrane (13), whereas pacemaker activity in mouse small intestine ICC can have a period <2 s (61). However, rapid SOCE has been observed in skeletal muscle cells, in which a splice variant of STIM1 proteins is permanently localized to the plasma membrane, allowing SOC channels to activate within milliseconds of depletion occurring (11, 13). Therefore, SOCE can function at the frequencies of Ca²⁺ oscillations observed in ICC.

The inhibitory effects of cyclopiazonic acid, thapsigargin, and SK&F96365 on ICC pacemaker activity suggest that SOCE plays a key role in ICC (36, 64). 2-APB, which has been routinely described as an IP₃R inhibitor in the ICC field (38), is a more reliable inhibitor of SOCE than IP3Rs (5), lending further support to the hypothesis that SOC channels are important for initiating the pacemaker cycle. Similarly, inhibition of pacemaker activity by Ni²⁺ is commonly used to indicate a role for T-type Ca²⁺ channels in ICC (27, 29) but similar concentrations of Ni²⁺ also inhibit SOC channels (18). The application of Ni²⁺ or Ca²⁺-free external solution inhibited a Ca²⁺-activated Cl⁻ current in ICC (75), indicating that Ca²⁺ influx, rather than Ca²⁺ release, is important for activating Ano1 in ICC. Therefore, it is likely that Ano1 channels in ICC are colocalized with SOC channels.

Nevertheless, it is plausible that the Ca²⁺ pathway for Ano1 activation in ICC may instead be IP₃R channels in sections of ER proximal to the plasma membrane. The lack of slow waves in IP₃R1 knockout mice (59) and inhibition of slow wave activity when inhibiting IP₃ production with phospholipase C inhibitors (38, 40) show that IP₃Rs are a crucial component of the pacemaker Ca²⁺ cycle in ICC. Modeling the activation of Ano1 by IP₃Rs localized near the cell membrane could be implemented with the same method that was used to model localization of Ano1 to SOC channels in this study, with little impact on the rest of the model.

CONCLUSIONS

We have developed the first mathematical model of ICC that implements Ano1 as a pacemaker channel and incorporates SOCE. We present a series of model variations that enable us to draw several conclusions: Firstly, the models presented here support the theory that Ano1 can initiate slow waves in response to cyclical activation of SOC channels. The simulations reproduce the observation that blocking or knocking out Ano1 channels causes loss of slow wave activity. Secondly, E_Cl plays an important role in modulating the shape of the slow wave. If E_Cl is close to the potentials measured in ICC-MY (76), then Ano1 current may contribute to repolarizing the cell but the plateau phase is not well reproduced. Conversely, if E_Cl is at least as high as the slow wave peak, membrane potential oscillations that recreate experimentally recorded slow waves can be easily simulated. It is possible that E_Cl is artificially raised by the microelectrode recording techniques used to measure membrane potentials, such that recorded slow wave traces are different to those present in undisturbed ICC. Thirdly, voltage- or Ca²⁺-activated nonselective channels or Na⁺ channels are plausible candidates for generating the plateau current, but large or prolonged Ca²⁺ currents are unlikely to contribute a major component of the slow wave depolarization. Fourthly, membrane potential oscillations in ICC-MY from Ano1 knockout mice may not be completely abolished but are minimized by background currents. Further experiments are needed to confirm or refute these conclusions and to provide more evidence about the contributions of various ion channels to slow waves generated by pacemaker ICC in the gastrointestinal tract. In particular, it is important to confirm the report of [Cl⁻]_i in ICC (76) and to search for any evidence that the reversal potential of Ano1 currents differs from the expected value for E_Cl.

GRANTS

R. Lees-Green was supported by a University of Auckland Doctoral Scholarship. This work was funded, in part, by grants from the Health Research Council of New Zealand and National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-64775 and DK-57061.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: R.L.G., S.J.G., G.F., J.S., and L.K.C. conception and design of research; R.L.G. performed experiments; R.L.G. analyzed data; R.L.G., S.J.G., G.F., J.S., and L.K.C. interpreted results of experiments; R. L.G. prepared figures; R.L.G. drafted manuscript; R.L.G., S.J.G., G.F., J.S., and L.K.C. edited and revised manuscript; R.L.G., S.J.G., G.F., J.S., and L.K.C. approved final version of manuscript.

Appendix

The equations used in our model are detailed below. Parameter values common to all model variations are given in Table A1, while parameter values and gating variables that differ across the model variations are in Tables A2 and A3, respectively.

Table A1.

Parameter values common to all model variations

Parameter	Value	Reference
Cell constants
Cm	25 pF	21, 27
v	1 × 10−12 l	10
vcyto	0.7 × 10−12 l	10
ver	0.1 × 10−12 l	10
F	96 485 C/mol
R	8.314 J·mol−1·K−1
T	310.0 K
Plasma membrane current parameters
[Ca2+]o	2 mM	10, 16
[Cl−]o	166 mM	76
[K+]i	140 mM	67
[K+]o	5 mM	67
[Na+]i	30 mM	10
[Na+]o	140 mM	10
EK	−89.0 mV	C
ENa	41.2 mV	C
ENS	0 mV	30, 49
gSOC	0.1 nS
Cytosolic Ca2+ flux parameters
KSOC	200 μM	6, 57
KPMCA	0.1 μM
bc	0.01	14
IP3R parameters
P	0.5 μM	65
k1	4000 μM−1·s−1	65*
k2	2.0 μM−1·s−1	65*
k3	4000 μM−1·s−1	65*
k4	2.0 μM−1·s−1	65*
k5	200 μM−1·s−1	65*
k−1	520 s−1	65*
k−2	2.1 s−1	65*
k−3	3772 s−1	65*
k−4	0.29 s−1	65*
k−5	16.4 s−1	65*
kIPR	7.0 s−1
Jleak	0.01 s−1
SERCA pump parameters
Ve	80 μM/s
Ke	0.1 μM	65
Ano1 model parameters
EC50(0mV)	1.39 μM	70
kc	0.012 48 mV−1	70
Vh	−100 mV
kv	0.0156 mV−1	70
Ano1-SOC localization parameters
Dc	250 μm2/s	55
Dm	75 μm2/s	55
Km	1 μM	55
Bm	50 μM	55
r	0.05 μm
vscale	1015 μm3/l	C
nSOC	50	16

C denotes that the parameter value was calculated using other parameters; SERCA is sarco-endoplasmic reticulum ATPase; SOC is store-operated Ca²⁺ channel; see text for other definitions. References are given where available; other parameter values were chosen to reproduce correct slow wave activity.

^*Inositol 1,4,5-trisphosphate receptor (IP₃R) k_i values are double the values in Ref. 65 to reproduce correct Ca²⁺ oscillation frequency.

Table A2.

Parameter values that differ across model variations

Parameter	Value	Reference
Cl− parameters
[Cl−]i
H	78 mM	C
L	25.85 mM	76
ECl
H	−20.2 mV	C
L	−49.7 mV	C
Ion channel maximal conductances
gAno1
H(NSV,NSCa)	20 nS
H(NaV,CaV), L(NaV)	15 nS
L(NSCa)	10 nS
gCaT
H(NaV,NSV,CaV), L(NaV)	3 nS	10
H(NSCa), L(NSCa)	4 nS
gKb
H(NSV,NSCa,CaV)	9 nS
H(NaV), L(NaV,NSCa)	5 nS
gNab
H(NSV,NSCa,CaV)	2 nS
H(NaV), L(NaV,NSCa)	1 nS
gNaV
H(NaV)	6 nS
L(NaV)	10 nS
gNSV
H(NSV)	30 nS
gNSCa
H(NSCa), L(NSCa)	30 nS
gCaV
H(CaV)	6.2 nS
gKt
L(NaV)	6 nS
gKERG
L(NaV)	3 nS
Ca2+ pump maximal fluxes
VPMCA
H(NaV,NSV), L(NaV)	50 μM/s
H(NSCa)	100 μM/s
H(CaV)	1500 μM/s
L(NSCa)	110 μM/s
VNCX
L(NSCa)	225 μM/s

C denotes that the parameter value was calculated using other parameters; see text for other definitions. References are given where available; other parameter values were chosen to reproduce correct slow wave activity. Model variations in which each parameter value is used are denoted by H for High-Cl and L for Low-Cl.

Table A3.

Gating variable parameters

Variable	Vh, mV	K, mV	a	τx, s	Reference
dCaT
H(NaV,NSV,CaV), L(NaV)	−26	−6	1	0.006	10
H(NSCa)	−40	−3	1	0.006
L(NSCa)	−45	−3	1	0.003
fCaT
H(NaV,NSV,CaV), L(NaV)	−66	6	1	0.04	10
H(NSCa), L(NSCa)	−55	5	1	0.1
dNaV
H(NaV)	−25	−5.6	1	Eq. 40	3*
L(NaV)	−39	−5.6	1	Eq. 46	3
fNaV
L(NaV)	−55	−4.5	0.53	1.0
dNSV
H(NSV)	−30	−5	1	0.005
fNSV
H(NSV)	−50	5	0.3	0.8
dCaV
H(CaV)	−26	−6	1	0.006	10
dKt
L(NaV)	−20	−4.6	1.0	0.003
fKt
L(NaV)	−45	4.4	1.0	0.045
dKERG
L(NaV)	−35	−7.1	1.0	Eq. 49
fKERG
L(NaV)	−42	4	1.0	0.003

Parameters for voltage-dependent gating variables for x_∞ = a/[1 + e(^{V_m − V_h/k})] + (1 − a) and for τ_x, where x = d_CaT, f_CaT, d_NaV, f_NaV, d_CaV, d_NSV, f_NSV, d_Kt, f_Kt, d_KERG, or f_KERG. Model variations in which each parameter value is used are denoted by H for High-Cl and L for Low-Cl. References are given where available; other parameter values were chosen to reproduce correct slow wave activity; see text for other definitions.

^*V_h modified from values in Ref. 3.

State Variables

Membrane potential.

$$\frac{\text{d}{V}_{\text{m}}}{\text{d}t}=-\frac{I_{\text{ion}}}{C_{\text{m}}}$$ (12)

where I_ion for each model variation is defined in Eqs. 4–7, 9, and 10.

Ion channel gating variables.

$$\frac{\text{d}x}{\text{d}t}=\frac{x_{\infty }-x}{{\tau }_x}$$ (13)

for x = O_Ano1, d_CaT, f_CaT, d_NaV, f_NaV, d_CaV, d_NSV, f_NSV, d_Kt, f_Kt, d_KERG, or f_KERG. Parameter values for x_∞ and τ_x are listed in Table A3, except for O_Ano1∞ and τ_Ano1 given in Eqs. 24 and 29, respectively, τ_dNaV given in Eqs. 40 and 46, and τ_dKERG given in Eq. 49.

Calcium concentration.

The change in [Ca²⁺]_ER is given by:

$$\frac{{\text{dCa}}_{\text{ER}}}{\text{d}t}=(J_{\text{SERCA}}-J_{\text{IPR}})\frac{v_{\text{cyto}}}{v_{\text{ER}}}.$$ (14)

The change in [Ca²⁺]_i is given by:

$$\frac{{\text{dCa}}_{\text{i}}}{\text{d}t}=b_{\text{c}}(J_{\text{IPR}}-J_{\text{SERCA}}+J_{\text{SOC}}+J_{\text{CaT}}-J_{\text{PMCA}})$$ (15)

as in Eq. 3, except in High-Cl(CaV) and Low-Cl(NSCa) where it is replaced by Eqs. 8 and 11, respectively.

IP3R inactivation.

The proportion of IP₃R channels inactivated by Ca²⁺ (y), as a function of Ca_i and IP₃ concentration (P), from Ref. 65:

$$\frac{\text{d}\mathit{y}}{\text{d}t}={\varphi }_1(1-y)-{\varphi }_{2}y,$$ (16)

$${\varphi }_1=\frac{(k_{-4}{K}_2{K}_1+k_{-2}{K}_{4}P){\text{Ca}}_{\text{i}}}{K_4{K}_2(K_1+P)},$$ (17)

$${\varphi }_2=\frac{k_{-2}P+k_{-4}{K}_3}{K_3+P},$$ (18)

where K_i = k_−i/k_i.

Calcium Flux

Ca²⁺ flux across the plasma membrane.

Plasma membrane Ca²⁺ ATPase:

$$J_{\text{PMCA}}=\frac{V_{\text{PMCA}}{{\text{Ca}}_{\text{i}}}^2}{{K_{\text{PMCA}}}^2+{{\text{Ca}}_{\text{i}}}^2}$$ (19)

Na⁺/Ca²⁺ exchanger, from Ref. 37:

$$J_{\text{NCX}}=\frac{V_{\text{NCX}}{\text{Ca}}_{\text{i}}}{3.6+{\text{Ca}}_{\text{i}}}$$ (20)

Ca²⁺ flux across the ER membrane.

Total Ca²⁺ release from the ER, adapted from Ref. 65:

$$J_{\text{IPR}}=(k_{\text{IPR}}{O}_{\text{IPR}}+J_{\text{leak}})({\text{Ca}}_{\text{ER}}-{\text{Ca}}_{\text{i}}),$$ (21)

$$O_{\text{IPR}}={\left(\frac{{\text{Ca}}_{\text{i}}P(1-y)}{(P+K_1)({\text{Ca}}_{\text{i}}+K_5)}\right)}^3$$ (22)

Ca²⁺ uptake into the ER, from Ref. 65:

$$J_{\text{SERCA}}=\frac{V_{\text{e}}{\text{Ca}}_{\text{i}}^2}{K_{\text{e}}^2+{\text{Ca}}_{\text{i}}^{\text{2}}}$$ (23)

Ano1 Model

Steady-state open probability of the Ano1 channel model, developed using data from Ref. 70:

$$O_{\text{Ano1}\infty }=\frac{1}{[1+e^{(V_{\text{h}}-V_{\text{m}})k_{\text{v}}}]\left[1+{\left(\frac{{\text{EC}}_{50}}{{\text{Ca}}_{\text{Ano}1}}\right)}^2\right]},$$ (24)

$${\text{EC}}_{50}={\text{EC}}_{50(0\text{mV})}{e}^{-k_{\text{c}}{V}_{\text{m}}}.$$ (25)

Ca²⁺concentration near Ano1 channels as a function of Ca²⁺ flux through SOC channels from Ref. 55:

$${\text{Ca}}_{\text{Ano}1}=(-D_{\text{c}}{K}_{\text{m}}+\frac{\sigma }{2\pi \mathrm{r}}+C_2+\sqrt{{(D_{\text{c}}{K}_{\text{m}}+\frac{\sigma }{2\pi \mathrm{r}}+C_2)}^2+4D_{\text{c}}{D}_{\text{m}}{B}_{\text{m}}{K}_{\text{m}}})/2D_c,$$ (26)

$$C_2=D_{\text{c}}{\text{Ca}}_{\text{i}}-\frac{D_{\text{m}}{B}_{\text{m}}{K}_{\text{m}}}{K_{\text{m}}+{\text{Ca}}_{\text{i}}},$$ (27)

$$\sigma =\frac{v_{\text{scale}}{J}_{\text{SOC}}{v}_{\text{cyto}}}{n_{\text{SOC}}}$$ (28)

Time constant for the Ano1 channel model, developed using data from Ref. 70:

$${\tau }_{\text{Ano}1}=t_1+t_2{\text{e}}^{V_{\text{m}}/{\text{t}}_3}$$ (29)

$$t_1=0.08163{\text{e}}^{-0.57{\text{Ca}}_{\text{Ano}1}}$$ (30)

$$t_2=0.07617{\text{e}}^{-0.05374{\text{Ca}}_{\text{Ano}1}}$$ (31)

$$t_3=70.3{\text{e}}^{0.153{\text{Ca}}_{\text{Ano}1}}$$ (32)

Ion Currents

Ion currents common to all models.

SOC channel:

$$I_{\text{SOC}}=g_{\text{SOC}}\cdot O_{\text{SOC}}(V_{\text{m}}-E_{\text{Ca}})$$ (33)

$$O_{\text{SOC}}=\frac{{K_{\text{SOC}}}^8}{{K_{\text{SOC}}}^8+{{\text{C}\text{a}}_{\text{ER}}}^8}$$ (34)

Ano1 Ca²⁺-activated Cl⁻ channel:

$$I_{\text{Ano}1}=g_{\text{Ano}1}\cdot O_{\text{Ano}1}(V_{\text{m}}-E_{\text{Cl}})$$ (35)

Na⁺ background current:

$$I_{\text{Nab}}=g_{\text{Nab}}\cdot (V_{\text{m}}-E_{\text{Na}})$$ (36)

K⁺ background current:

$$I_{\text{Kb}}=g_{\text{Kb}}\cdot (V_{\text{m}}-E_{\text{K}})$$ (37)

T-type Ca²⁺ current:

$$I_{\text{CaT}}=g_{\text{CaT}}\cdot d_{\text{CaT}}\cdot f_{\text{CaT}}(V_{\text{m}}-E_{\text{Ca}})$$ (38)

Ion current specific to High-Cl(NaV).

Voltage-gated Na⁺ channel:

$$I_{\text{NaV}}=g_{\text{NaV}}\cdot d_{\text{NaV}}(V_{\text{m}}-E_{\text{Na}})$$ (39)

$${\tau }_{d\text{NaV}}={\left\{50+1000/\left[1+e^{-0.06(V_{\text{m}}+20)}\right]\right\}}^{-1}$$ (40)

Ion currents specific to High-Cl(NSV).

Voltage-gated nonselective channel:

$$I_{\text{NSV}}=g_{\text{NSV}}\cdot d_{\text{NSV}}\cdot f_{\text{NSV}}(V_{\text{m}}-E_{\text{NS}})$$ (41)

Ion currents specific to High-Cl(NSCa).

Ca²⁺-activated nonselective channel:

$$I_{\text{NSCa}}=g_{\text{NSCa}}\cdot O_{\text{NSCa}}(V_{\text{m}}-E_{\text{NS}})$$ (42)

$$O_{\text{NSCa}}=\frac{{\text{Ca}}_{\text{i}}^4}{\text{1}{\text{.8}}^{\text{4}}+{\text{Ca}}_{\text{i}}^{\text{4}}}$$ (43)

Ion current specific to High-Cl(CaV).

Voltage-gated Ca²⁺ channel:

$$I_{\text{CaV}}=g_{\text{CaV}}\cdot d_{\text{CaV}}(V_{\text{m}}-E_{\text{Ca}})$$ (44)

Ion currents specific to Low-Cl(NaV).

Voltage-gated Na⁺ channel:

$$I_{\text{NaV}}=g_{\text{NaV}}\cdot d_{\text{NaV}}\cdot f_{\text{NaV}}(V_{\text{m}}-{\text{E}}_{\text{Na}})$$ (45)

$${\tau }_{d\text{NaV}}={\left\{50+1000/\left[1+e^{-0.06(V_{\text{m}}+20)}\right]\right\}}^{-1}$$ (46)

Transient K⁺ channel:

$$I_{\text{Kt}}=g_{\text{Kt}}\cdot d_{\text{Kt}}\cdot f_{\text{Kt}}(V_{\text{m}}-E_{\text{K}})$$ (47)

ERG-like K⁺ channel:

$$I_{\text{KERG}}=g_{\text{KERG}}\cdot d_{\text{KERG}}\cdot f_{\text{KERG}}(V_{\text{m}}-E_{\text{K}})$$ (48)

$${\tau }_{d\text{KERG}}={\left\{1+1/\left[1+{\mathrm{e}}^{0.15({\mathrm{V}}_{\text{m}}+60)}\right]\right\}}^{-1}$$ (49)

Ion currents specific to Low-Cl(NSCa).

Ca²⁺-activated nonselective channel:

$$I_{\text{NSCa}}=g_{\text{NSCa}}\cdot O_{\text{NSCa}}(V_{\text{m}}-E_{\text{NS}})$$ (50)

$$O_{\text{NSCa}}=\frac{{\text{Ca}}_{\text{i}}^5}{\text{1}{\text{.3}}^{\text{5}}+{\text{Ca}}_{\text{i}}^5}$$ (51)

Na⁺/Ca²⁺ exchanger, from Ref. 37:

$$I_{\text{NCX}}=-2F\cdot J_{\text{NCX}}\cdot v_{\text{cyto}}$$ (52)

Nernst equilibrium potential.

$$E_{\text{Ca}}=\frac{RT}{2F}\hspace{0.17em}\ln \hspace{0.17em}\left(\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{o}}}{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}\right)$$ (53)

$$E_{\text{Cl}}=\frac{RT}{F}\hspace{0.17em}\ln \hspace{0.17em}\left(\frac{{\left[{\text{Cl}}^{-}\right]}_{\text{i}}}{{\left[{\text{Cl}}^{-}\right]}_{\text{o}}}\right)$$ (54)