Department of Zoology, University of Melbourne, Victoria 3010, Australia
Submitted 27 March 2003 ; accepted in final form 28 May 2003
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ABSTRACT |
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mathematical model; stomach; slow wave; interstitial cells of Cajal; inositol 1,4,5-trisphosphate
Regenerative potentials that share the characteristics of the secondary component of the slow wave can be initiated in isolated individual bundles of circular muscle (6, 28). Preparations were impaled with two intracellular electrodes, with one electrode to pass depolarizing current pulses and the other to record membrane potential (Em) changes. Depolarization evoked a regenerative response with a minimum latency of 1 s (28). The repolarizing step, ending a period of membrane hyperpolarization, also evoked a regenerative response but with a longer minimum latency of
2.5 s (28). Regenerative responses appear to depend on the release of Ca2+ from intracellular stores followed by the activation of anion-selective ion channels (13). They are abolished by depleting internal Ca2+ stores (28), by buffering the cytosolic free Ca2+ concentration, [Ca2+]i to low levels (6), and by 2-APB (9, 13, 14), an agent that prevents the release of Ca2+ from inositol 1,4,5-trisphosphate [Ins(1,4,5)P3]-dependent stores.
Em recordings from isolated bundles of circular muscle are dominated by membrane noise. Spectral analysis suggests that this noise results from an ongoing discharge of depolarizing unitary potentials that have a characteristic time course (6). Depolarizing current evokes regenerative responses with spectral profiles similar to those of the membrane noise but having higher levels of power. The spectral properties of regenerative responses were consistent with a cascade of unitary potentials rather than the activation of sets of voltage-dependent ion channels (6).
This model tests the possibility that regenerative responses are summations of many unitary membrane conductance modulations that arise at random times and that the mean time between unitary potentials becomes briefer after activation of a voltage-sensitive messenger pathway.
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MODEL FORMULATION |
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The long latencies seen in this tissue were alternatively modeled as a two-stage chemical reaction. Voltage sensitivity was applied to the forward rate constant (mhKF) for the first reaction step, presumably occurring at the membrane, which converts a precursor molecule to an intermediate reagent. The activation variable was designated m, and the inactivation variable, h; these designations were borrowed from Hodgkin and Huxley's (15) squid axon Na current description. The first reaction step was represented as a reversible reaction with the reverse rate constant (KB). The second step from the intermediate reagent to the messenger proceeded with net formation rate (KM), and the messenger was proposed to hydrolyze with the rate constant (KH) as shown.
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The equations describing the variations in intermediate concentration ([intermediate]) and messenger concentration ([messenger]) were
![]() | (1) |
The following equations were used to calculate m and h.
![]() | (2) |
These equations are illustrated in Fig. 1A, which shows steady-state activation and inactivation values (m and h
) and associated time constants (
m and
h) as functions of Em.
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Initial conditions were: Em = 65 mV; [messenger] = 340 nM; [intermediate] = 220 nM; m = 0.18; and h = 0.28.
Discharge rate of unitary potentials depends on [messenger]. The mean rate of unit discharge () was proposed to depend on the concentration of the intracellular messenger according to the following dose-effect relationship, which is shown graphically in Fig. 1B
![]() | (3) |
The maximum mean rate of unit discharge (max) was 140 Hz in control conditions, the Hill coefficient (H) was 6.8, and the affinity (KD) was 750 nM. Although the value chosen for KD was arbitrary because changes in
could be compensated by adjustment of KM (Eq. 1), 750 nM is a value reported for a mammalian Ins(1,4,5)P3 receptor (23) and is within the range (100 to 1,000 nM) offered by Marchant and Taylor (22) for Ins(1,4,5)P3 receptors in hepatocytes.
Regenerative potentials are composed of discrete unitary potentials. The spontaneous unitary membrane depolarizations seen in recordings of intracellular potential made in this tissue have been previously characterized (6). Individual unitary depolarizations were well fitted by the difference between two exponential functions raised to the third power. The intervals between successive unitary potentials were approximately Poisson distributed. Therefore, the spectral density in the region between 0.1 and 20 Hz can be described by a curve whose gradient approaches a theoretical extreme of 8 at high frequencies. See Edwards et al. (6) for full description.
Unitary potentials were assumed to reflect unitary membrane conductance increases having a similar time course. The equilibrium potential for the current they carry, obtained by extrapolation, is close to 20 mV (13). Clearly, although this may not be a measure of the true reversal potential, for example, if rectification occurs at more positive potentials, the extrapolated value of reversal potential is appropriate for potential changes occurring in the physiological ranges from which the extrapolated value was obtained. If a particular unitary conductance modulation, gj(t), begins at time tj and has amplitude Aj, it can be represented as follows (6)
![]() | (4) |
The time constants 0.434 and 0.077 s used in these calculations are the average values obtained from 17 experiments (6). Aj is an instance of a random variable whose frequency distribution is shown in Fig. 2A. This is an exponentially smoothed version of the frequency function shown in Fig. 9D of Edwards et al. (6).
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The total gu(t) is the sum of the N individual unitary modulations that begin before t
![]() | (5) |
The intervals between the starting times of unitary conductance modulations were drawn from a Poisson distribution (6). The mean frequency of unit discharge is by definition the inverse of the mean interval between units, and is given by
![]() | (6) |
At each integration step, was recalculated from the concurrent value of [messenger] (Eq. 3) and the intervals between forthcoming unitary conductance modulations were rescaled accordingly.
Electrical equivalent cell. A short segment of a single bundle of circular muscle from guinea pig antrum comprises 200 ICCIM distributed throughout a syncytium of
2,000 smooth muscle cells. The ICCIM are syncytially interconnected and make electrical connections with many surrounding muscle cells (26). When a segment of a single circular muscle bundle was impaled with two microelectrodes, the recordings were always very similar even when the electrodes were hundreds of micrometers apart. Within a bundle, voltage responses to injected current pulses never differed by >5% in amplitude, regardless of electrode position and separation. Moreover the entire time course of an electrotonic potential could be described by a single exponential function. These observations suggest that the intracellular compartment within a muscle bundle can be considered electrically isopotential and that a bundle can therefore be represented electrically as a single equivalent cell. The equivalent cell membrane was represented as the parallel combination of the total unitary conductance [gu(t)] at t with an equilibrium potential equal to the extrapolated reversal potential for unitary events (Eu), an aggregate linear background conductance (gb) with an equilibrium potential equal to the resting potential (Eb) and a capacitor representing membrane capacitance (Cm) (Fig. 2B).
Therefore, the resistive membrane current (im) was represented as the sum of two components. The first was the current carried by the total unitary conductance modulation with an equilibrium potential of 20 mV. The second was an equivalent net background current with an equilibrium potential of 65 mV and a conductance of 227 nS to agree with the mean input resistance of preparations of 4.4 M [Table 1, Edwards et al. (6)].
![]() | (7) |
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The variation in Em is given by
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METHODS |
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BAPTA-AM and nifedipine (each obtained from Sigma, St. Louis, MO) were used in these experiments.
Computational methods. Simulations involving integration were carried out by using MATLAB 6.0 (The MathWorks, Natick, MA). Stiff differential equation solvers ode15s and ode23s were used. Other simulations were performed by using Daos 7.0 (SciTech, Preston South, Victoria, Australia). Fast Fourier transforms were carried out by using Origin 5.0 (Microcal Software, Northampton, MA). Computations were done on an Intel Pentium 4-based desktop computer.
Seven model parameters (unitary potential time constants 0.434 and 0.077 s, unitary amplitude probability density function Aj, equilibrium potentials 65 and 20 mV, background conductance 227 nS, and the value of 36.8 nF for Cm) were individually estimated from the data. One parameter (750 nM for KD) was estimated with reference to the literature (23). Selection of suitable values for the 15 remaining parameters was done by hand optimization of candidate models against 33 experiments that correspond to previously published physiological results [29 experiments shown in Figs. 48 of Suzuki and Hirst (28); 4 experiments shown in Figs. 2Aa and 8 of Edwards et al. (6)].
To confirm that the intervals between simulated unitary potentials were in conformity with Poisson statistics, natural logarithm survivor (ln(survivor)) curves were constructed for periods of simulated membrane noise [method in Hashitani and Edwards (11)]. In each case examined, the ln(survivor) plot was well fitted by a straight line indicating that the distribution of intervals was well described by Poisson statistics. Spectral analysis allowed another test on the accuracy of simulated membrane noise characteristics. The spectral densities of simulated membrane noise were estimated by fast Fourier transform and compared with the theoretical curve obtained by direct Fourier transformation of the function defined in Eq. 4 (6). In each case, good agreement was obtained, as is the case for comparative physiologically acquired data. The figures of simulations presented in this paper were each selected from a set of simulated trials, each having the same initial conditions and stimulus but a different sample of the specified unitary conductance modulation distribution.
Model parameter uncertainties. Each of the 15 unknown parameters appearing in Eqs. 13 was individually increased in value until one or more of the experimental simulations yielded an unsatisfactory simulation. Candidate models were rejected because of failure to produce regenerative responses, spontaneous regenerative potentials occurring at unphysiologically rapid rates, or unacceptable estimates of latency and duration of regenerative potentials. This process was repeated by individually reducing the value of each parameter. Table 1 shows the minimum percent increase and reduction to each parameter that in isolation were sufficient to degrade the simulation capacity of the model to an unacceptable level. These limits indicate the relative uncertainties associated with each parameter. It can be seen that the smallest uncertainties are associated with Vm and Vh that define the voltage window for formation of the intermediate compound.
Because 15 parameters were estimated from fits to a set of 33 experiments, the system of equations was overdetermined. Such redundancy acts to reduce the range of parameter combinations that produce satisfactory simulations.
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RESULTS |
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Regenerative potentials can be initiated by depolarizing the muscle bundles. Characteristically, they begin after a variable latency when triggered by an invariant threshold depolarization. If the depolarization is increased, the latency and its variability are reduced until potentials are initiated with little latency variation after a minimum latency of 1 s (28). In the period after each regenerative potential, the discharge of membrane noise is suppressed (6). Figure 4A shows a family of such regenerative responses to depolarizing currents of increasing intensity. Figure 4B shows a family of simulated regenerative responses to applied depolarizing currents of 5-s duration and of different intensities (3, 6, 9, 12, and 15 nA). Note that, as with the physiological recordings, the latency shortened to
1 s for the highest stimulus current and that the ongoing membrane noise in the baseline regions was depressed after the regenerative potentials.
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Because regenerative responses are made up of populations of unitary potentials, successive responses show variation, even when triggered by identical stimuli. This is particularly apparent when threshold stimuli are applied (Fig. 5, A and C). Simulations of three regenerative potentials evoked by a near-threshold depolarizing current of 2 nA for 5 s are shown in Fig. 5B. They show latencies ranging between 3 and 5 s. Figure 5D shows simulated responses to 8-nA current pulses of the same duration. The mean latency is shorter, and the variability in latency is reduced. Physiologically acquired comparative data are shown in Fig. 5, A and C, respectively.
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Physiological experiments have shown that regenerative potentials can be evoked with a current pulse of short duration (28). In these cases, Em returns toward its resting value before the regenerative potential starts, suggesting that a period of membrane depolarization can activate a process of messenger formation. To trigger a regenerative potential with a briefer pulse, the intensity of the applied current must be increased (28). Such an inverse correlation between threshold current intensity and pulse duration is accommodated by the present model. Thus the model suggests that brief intense depolarization causes the formation of a threshold level of the intermediate reagent.
Regenerative potentials are also triggered at the break of a period of membrane hyperpolarization. After a period of mild hyperpolarization, the regenerative potential occurs with a long and variable latency. With more profound hyperpolarization, the latency and its variance are reduced until a minimum of 2.5 s is achieved (28). A family of regenerative potentials recorded from a circular muscle bundle and evoked by release of hyperpolarization is shown in Fig. 6A. The Hodgkin-Huxley formulation (15) used to describe the voltage-dependent nature of the first forward rate constant in the proposed reaction chain allows simulation of regenerative potentials that are evoked by release of hyperpolarizing current. In Fig. 6B, a family of simulated regenerative potentials evoked in this way is shown. The magnitudes of the applied hyperpolarizing currents were 3.5, 7, 10.5, 14, and 17.5 nA. For the largest currents, the latency shortened to
3 s.
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A more detailed description of a regenerative potential induced by the release of hyperpolarization is shown in Fig. 7. Figure 7A shows the time courses of the m and h in response to applied current shown in Fig. 7E. The onset of the hyperpolarizing pulse caused h to increase with a time constant of 6 s so that when m recovered from hyperpolarization, the product mh was greater than during the baseline region. Figure 7B shows the time course of [intermediate], which has an early peak. Also shown is the time course of [messenger]. The reduction in [messenger] during the period of hyperpolarization and its ensuing slow increase leads to the long latency seen before the regenerative response after the release of hyperpolarization [see Figs. 5C and 7C in Suzuki and Hirst (28)]. The time course of the mean rate of unitary potential discharge, which is a function of [messenger] (Eq. 3, Fig. 1B), is shown in Fig. 7C. The consequent Em time course, including the regenerative potential, is shown in Fig. 7D.
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Characteristically, after each regenerative response is initiated, the discharge of membrane noise falls to a low value. The period during which the discharge of membrane noise is at a low value is associated with a period of depressed excitability. Thus at short intervals, after a regenerative response, the preparation is refractory. As the interval between stimulating pulses is increased, the responses show partial refractory behavior, with full recovery occurring over some 15 to 25 s. Figure 8A shows physiologically acquired data illustrating the recovery of excitability after a conditioning regenerative response. This behavior is replicated by the model, because the voltage-sensitive rate mhKF in Eq. 1 recovers from inactivation with a time constant near 6 s at resting Em (h, Eq. 2 and Fig. 1A). Figure 8B shows that if simulated depolarizing pulses were delivered to the model 5 s apart, the second pulse failed to elicit a response. A 10-s interval allowed sufficient recovery from inactivation for a small regenerative potential to develop. Longer intervals allowed progressively more recovery, and a full amplitude regenerative potential could be evoked after 1520 s.
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When BAPTA-AM (1020 µM) was added to the physiological saline for 1015 min, membrane noise decreased and individual unitary potentials were detected (Fig. 9A) (6). Regenerative responses to applied depolarizing current (Fig. 9C) and release of hyperpolarizing current (Fig. 9E) could still be evoked, but their amplitudes were reduced to a few millivolts, and individual unitary potentials occurring at a frequency higher than the baseline rate were detected in each response (6). Presumably BAPTA had reduced [Ca2+]i. It has been shown in hepatocytes that the threshold Ins(1,4,5)P3 concentration for Ins(1,4,5)P3-induced Ca2+ release is inversely dependent on [Ca2+] within the store, which in turn depends on [Ca2+]i (24). Similarly, it was proposed that the transduction process that responds to changes in [messenger] by altering the rate of unitary potential discharge was inhibited in the presence of low [Ca2+]i. This inhibition was simulated by reducing the value of max in Eq. 3 from the control value of 140 to 18 Hz. Figure 9B shows a period of simulated Em in the absence of stimulation. Note the occurrence of unitary potentials in isolation and in sparse clusters comparable to the record shown in Fig. 9A. Figure 9D illustrates the consequent reduction in simulated regenerative potential amplitude and is comparable with the physiological record shown in Fig. 9C. Figure 9F shows a simulated response to a hyperpolarizing current pulse and is comparable with the physiological record shown in Fig. 9E.
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DISCUSSION |
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An isopotential group of cells can be modeled electrically as a single equivalent cell, even when the component cells have heterogeneous membrane properties. If the cells are sufficiently well coupled so that electrical potential differences between cell interiors are insignificant, then particular channels that arise on only one cell type contribute current that will influence every cell's Em. A segment of muscle bundle is then electrically equivalent to a single large cell whose membrane contains all of the membrane channels found on all the smooth muscle cells in the preparation to which are added all the membrane channels found on the ICCIM. The equivalent cell's surface area is the sum of the surface areas of all the cells, so its membrane capacitance is equal to that of the entire segment of muscle bundle.
When muscle bundles are depolarized, they respond by giving an increased rate of discharge of unitary potentials that together produce a regenerative potential (6). Although not specified by the model, it is assumed that unitary potentials are generated by ICCIM (4, 13). Because the physiological recordings have shown that changes in Em initiate a cascade of events that triggers a response after a minimum latency of 1 s, the first approach was to model the gating process in the way described by Hodgkin and Huxley (15). Assuming a single-step reaction, such delayed responses could only be achieved by dramatically increasing the exponent associated with the activation variable. An alternative approach, which was able to accommodate the physiological observations, was to examine the possibility that a gating mechanism, like that described by Hodgkin and Huxley (15), triggered formation of messenger via a two-step reaction chain. Presumably, this stands in place of a more complex reaction scheme that includes the gated formation of messenger, the release of Ca2+ from intracellular stores, and the activation of channels, perhaps calcium-activated chloride channels, in the membranes of ICCIM (13). Hence, the state variables precursor and intermediate have not been correlated with identified substances. The equations defining voltage-sensitive behavior are descriptive, in the style of Hodgkin and Huxley's (15) axonal Na channel activation and inactivation, with their physical parameters being chosen from fits to sets of experimental data.
Implicit in the construction of this schematic model of regenerative potentials are a number of simplifications. First, no attempt has been made to individually model the diffusional, binding, and Ca2+ release steps that lead from a change in [messenger] to a change in unit discharge rate. These actions presumably take place on a much more rapid time scale than the observed changes in Em. Therefore, the dose-effect relationship (Eq. 3) was implemented as an instantaneous transfer function. Second, no mechanistic basis is offered for the time course of a unitary conductance modulation. The decay phase proceeds more rapidly than simple exponential dissipation. This could result from Ca2+ inactivation of Ca2+ release, or partial inactivation of Ins(1,4,5)P3 receptors by Ins(1,4,5)P3 (22). Such self-inhibition would also be expected to affect the time courses of regenerative potentials. Adding a mechanism of this kind to the model would be expected to improve the fidelity of the decay phases of simulated regenerative potentials.
Frequently, simulated periods of low unitary release rate appear as unrealistically flat baselines (e.g., refractory periods in Fig. 8B). A partial explanation for this may be an inaccuracy in the unitary amplitude histogram shown in Fig. 2A. The first bar in Fig. 2A, which contains the smallest measurable unitary potentials (00.5 mV), has 11.5% of the observations. This low value might reflect analytical difficulty in resolving overlapping small events, some of which are lost in noise from other sources. Certainly, increasing the proportion of small events results in simulations that have noisier baseline regions. There is, however, no better empirical measure of small unit frequency in this preparation than the value used.
The linear background conductance used in the equivalent cell membrane model is clearly a simplification. In the experiments described in this paper, L-type Ca2+ channels were blocked by using nifedipine. However, smooth muscle cells have a number of other voltage and Ca2+-sensitive conductances that are presumably activated during a regenerative potential (7). ICCMY, which are not present in antral circular muscle bundles, are likewise reported to have voltage and Ca2+-dependent conductances in their membranes (12, 18, 19). Similar conductances may be present in ICCIM, but no definitive measurements have been made from ICCIM.
Many of the voltage-dependent membrane channels inactivate at the depolarized level that is sustained for several seconds during a regenerative potential. T-type Ca2+ channels generally inactivate rapidly at Em positive of 60 mV (7, 34). Similarly, Koh et al. (20) showed that a voltage-dependent nonselective cation current in murine colonic myocytes inactivated with a time constant of 86 ms at 45 mV. Delayed rectifier K+ channel activation in murine colonic smooth muscle cells is incomplete in the Em range of the regenerative potential. Moreover, the major component of this current inactivates with a time constant at 0 mV and room temperature of 35°C, possibly faster at 1.5 s (21). A-type K+ current, which has been described in gastrointestinal tissues, is largely inactivated above approximately 40 mV (1, 32). A window current via this conductance contributes to Em between slow waves in murine antral tissue, but when this current was blocked, changes in spontaneous slow-wave shape were not evident (1). Inward rectifier K+ channels are found in canine colonic circular smooth muscle and may be predominantly located in ICC (8), but little inward rectifier current passes at depolarized potentials. In rat fundus, a transient Na+ current inactivates within a few milliseconds (33). Furthermore, application of tetrodotoxin has no effect on the mean rate or the variability in rate of generation of slow waves in preparations of guinea pig antrum, so few transient Na+ channels may be present (14).
Some other channel types are likely to be present in numbers too small to make a major contribution to regenerative potential shape. Suzuki et al. (29) showed that specific block of large conductance Ca2+-activated K+ channels with charybdotoxin or of apamin-sensitive Ca2+-activated K+ channels led to increases of a few millivolts in slow-wave amplitude and increased the probability of spiking. In the present study, the contribution due to these conductances is likely to be even smaller, because of the reduction in the range of [Ca2+]i due to nifedipine block of L-type Ca2+ channels. Ito et al. (16) showed that glibenclamide caused no detectable change in spontaneous contractions in circular smooth muscle from guinea pig stomach, which suggests that ATP-activated K+ channels are not open under normal conditions in this tissue.
It is certain that voltage and Ca2+-dependent channels make a contribution to the Em recorded from circular smooth muscle bundles. However, when simulating slow changes within the Em range of the regenerative potential, a linear background conductance provides an adequate approximation for testing the proposed messenger mechanism.
Whereas there is evidence suggesting that the intracellular messenger acting in this process could be Ins(1,4,5)P3 (14), which furthermore displays voltage dependency in other guinea pig tissue (10), there is no clear idea of what the membrane precursor substance is, or how a change in Em could activate its transformation to an intermediate reagent. Certainly any polar sections in a precursor molecule situated in the cell membrane would be subject to the electrostrictive effects of an electrical charge gradient across the membrane. The molecule need not span the whole membrane for this to be true. Even molecules bound to one side of the membrane will be subject to an electric field due to charge imbalance between the inside and outside of the cell. What proportion of the electric field differential is maintained across the precursor molecule depends on its dielectric constant relative to that of the lipid substrate. The forces that an electric field imposed on polar sections of a precursor molecule or its coreagents could cause a conformal change that catalyzed production of an intermediate form.
It has been tentatively suggested that a G protein might act as part of the voltage-sensing mechanism involved in the initiation of regenerative potentials (13). It has been shown previously that in smooth muscle cells, if Ins(1,4,5)P3 formation has been initiated by applying the appropriate agonist, its formation can be increased or decreased by depolarizing or hyperpolarizing the preparations (17). In these tissues, whereas a G protein forms part of the stimulatory pathway (2), these observations do not indicate that it is necessarily the G protein that displays voltage sensitivity. However, in bundles of antral muscle, voltage gating can be selectively uncoupled by treating the tissues with N-ethylmaleimide without blocking the resting discharge of unitary potentials (13). Although this agent has widespread actions, one of its properties is its ability to alkylate and, hence, inactivate G proteins (27).
Despite these simplifications, the model developed in this paper allows the replication of many characteristics seen in physiologically acquired data. The responses have a long latency, which is inversely proportional to stimulus strength. Responses to repeated stimuli of constant magnitude show latency jitter, the range of which is related inversely to stimulus strength. Occasional failures of response are observed at low levels of stimulation. The threshold depolarization for evoking a regenerative response is a decreasing function of pulse duration. Regenerative responses can be evoked by the release of applied hyperpolarizing current-anode break excitation. Subsequent to a regenerative response, there is a period of absolute refraction followed by a relative refractory period. In control conditions, spontaneous regenerative depolarizations are observed. Such spontaneous transients always begin with a smoothly accelerating pacemaker depolarization that leads into the rising phase and terminate with a more abrupt repolarization. Antral tissue is seen to contract repetitively and fairly regularly. Such oscillations can be modeled by increasing the maximum discharge rate of unitary depolarizations. Responses to depolarizing and hyperpolarizing current injections in the presence of BAPTA-AM can be simulated by reducing the maximum discharge rate of unitary depolarizations. The intervals between unitary potentials can be shown by ln(survivor) plots to have approximately Poisson distributions in both the physiological and simulated cases. The spectral density profiles of membrane noise and its simulation share a characteristic profile, having a knee at a few Hertz and a steep decline, approaching but never exceeding a gradient of 8 at high frequencies.
In summary, the model reproduces a wide range of spontaneous and evoked Em changes characteristic of guinea pig antral circular muscle bundles. Therefore, this study is consistent with the hypothesis that an intracellular messenger is formed whose production can be modulated by changes in Em and that the rate of discharge of unitary potentials varies with the concentration of the messenger.
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DISCLOSURES |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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