Intercellular coupling mediated by potassium accumulation in peg-and ...

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Abstract—Coupling of smooth muscle cells is important for coordination of gastrointestinal motility. Small structures called peg-and-socket junctions (PSJs) have ...
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 47, NO. 12, DECEMBER 2000

Intercellular Coupling Mediated by Potassium Accumulation in Peg-and-Socket Junctions Edward J. Vigmond, Berj L. Bardakjian*, Lars Thuneberg, and Jan D. Huizinga

Abstract—Coupling of smooth muscle cells is important for coordination of gastrointestinal motility. Small structures called peg-and-socket junctions (PSJs) have been found between muscle cells and may play a role in electrical coupling due to extracellular potassium accumulation in the narrow cleft between the muscle cells. A model was developed in which an electrical boundary element model of the cell morphology is used in conjunction with a finite difference model which described ionic fluxes and diffusion of extracellular potassium in the PSJ. The boundary element model used a combination of triangular and cylindrical elements to reduce computational demand while ensuring accuracy. Barrier kinetics were used to model the underlying ionic transport mechanisms. Seven ionic transport mechanisms were used to create the transmembrane voltage waveform. Results indicate that PSJs may produce significant coupling between smooth muscle cells under appropriate conditions. Coupling increased exponentially with increasing length and with decreasing intercellular gap. Index Terms—Boundary element method, coupling, morphology, peg-and-socket junctions, potassium accumulation, smooth muscle.

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LECTRICAL pacemaker activity in the gastrointestinal tract is manifested by electrical rhythmic activity, called electrical control activity (ECA), generated in the musculature. The ECA originates in an interconnected network of interstitial cells of Cajal (or ICC) and smooth muscle cells [1]–[3]. Dissection of tissues from the stomach and small intestine reveal different frequencies than those recorded in situ with proximal sites having higher intrinsic frequencies [4]–[6]. In situ, muscle cells are electrically coupled and the ECA is synchronized with the proximal slow waves leading and creating an apparent propagation in aboral direction. The rhythmic nature of the pacemaker activity and the entrainment of intrinsic frequencies has led to modeling of the ECA as a population of coupled relaxation oscillators [7], [8], [5], [9]. Indeed, normal functioning of the stomach and proximal small intestine relies on entrainment of intrinsic ECA activity. Apparent propagation in aboral direction governs propulsive contractile activity when the musculature is appropriately stimulated by distension and/or excitatory neural activity [10]–[12]. Coupling between ICC and smooth muscle cells and coupling between smooth muscle cells is essential for entrainment of ECA to obtain coordinated contraction of muscle cells. If only Manuscript received June 10, 1999; revised July 18, 2000. This work was supported by the Medical Research Council of Canada. Asterisk indicates corresponding author. E. L. Vigmond, L. Thuneberg, and J. D. Huizinga are with the University of Toronto, Toronto, ON, M5S 3G9 Canada. *B. L. Bardakjian is with the University of Toronto, 4 Taddle Creek Rd., Toronto, ON, M5S 3G9 Canada (e-mail: [email protected]). Publisher Item Identifier S 0018-9294(00)10190-9.

coupling is altered between such oscillators then the mode of the oscillation can change from periodic to chaotic [13]. Gap junctions play an important role in intercellular communication since they provide direct electrical and metabolic coupling between the cytoplasm of neighboring cells [14], [15] but cells are coupled also by a variety of other structures [16], [17]. Pacemaker cells may be coupled to smooth muscle cells by gap junctions as in the submuscular plexus of the canine colon [18], but at other sites, such as between ICC and the muscle layers of the mouse small intestine, gap junctions are not to be found and coupling appears to be mediated by an abundance of close apposition contacts [19]. Within certain muscle layers no gap junctions can be recognized by electron microscopic techniques such as in the longitudinal muscle of the intestine and colon of a variety of species [20], [21]. In these muscle layers, muscle cells are seen to communicate in different ways. In such tissues, close apposition contacts are always observed. Whether such contacts provide direct cytoplasmic communication is debated. Such contacts may contain small groups of connexons which would allow electrical and metabolic communication although it could not be identified as a gap junction; only a large aggregate of connexons can be recognized as a gap junction by electron microscopy. Evidence for direct cytoplasmic communication comes from the observation that tracer molecules can be found to diffuse from one cell to the other [21], [22]. Nevertheless, electrical communication other than purely resistive is likely to occur. One type of conspicuous intimate contact is the peg-andsocket junction (PSJ), where an extrusion or short branch of one cell, forms a peg which penetrates a neighboring cell, and fits into an intrusion or socket with an intercellular gap of 20 nm. This extrusion can be up to 10 m long, undergo a 90 bend upon entering a cell, and be up to 1.5 m in diameter [23]. The abundance of these structures in areas of no demonstrable gap junctional coupling and their absence in areas shown to be well coupled by gap junctions [23], [24] suggest they may be important for indirect electrical coupling. These structures facilitate field coupling [25], but may also facilitate propagation of a dein the extensive polarization pulse through accumulation of narrow gap between the cells created by the PSJs. Such a mechanism for cell to cell coupling has been proposed in other systems [26]–[30] and its study in smooth muscle is the subject of the present study. This study modeled a PSJ to determine the conditions under which coupling becomes significant between the two cells comprising the PSJ. One cell was called a source cell and the voltage it induced in a second cell, called the receiving cell, was com-

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were much more detailed. Finally, a differential equation solver based on Gear’s method, LSODE [32], was used to integrate the system. The elements of the model are discussed in detail below. A. Ionic Current Equations The equations describing the ionic current are a modified form of the barrier kinetic (BK) model presented by Skinner and Bardakjian [33]. Seven ionic transport mechanisms were incorporated into the membrane to produce a waveform which resembled the ECA of canine smooth muscle. The total ionic current density, , was composed of the following components: (1)

Fig. 1. Schematic of PSJ cross section showing pertinent dimensions. The intercellular region (shaded gray) was discretized into toroidal regions (shown) concentration was modeled using a finite difference scheme. The and the concentration at the open end was kept constant at 6 mM since it was assumed in the PSJ would be much larger than that outside of that the change in the PSJ due to the larger membrane separation between source and receiving cells. Only the portion of the source cell comprising the intrusion (thick line) was modeled. The inset box on the right shows the source cell and the receiving cell with the PSJ (circled and not to scale).

K

[K]

puted. The PSJ was modeled as a series of compartments, each with active membranes for both cells and extracellular potassium concentrations were computed using a finite difference scheme. The compartmental PSJ model was coupled to a boundary element model (BEM) of the receiving cell. The BEM model used a mixture of triangular and cylindrical elements to reduce computation while providing a realistic geometry [31]. I. METHODS Two isolated cells in an infinite bath were considered, a source cell with an intrusion and a receiving cell with an extrusion (see Fig. 1). Together, the extrusion and intrusion and the gap between them comprised the PSJ. It was assumed an ECA waveform propagated over the source cell and, furthermore, that the transmembrane voltage was constant over the intrusion while the receiving cell was assumed to remain in a resting state. The only part of the source cell membrane that was modeled was the portion which flux through the channels and constituted the intrusion. pumps in the source and receiving cell membranes of the PSJ concentration in the gap which, in turn, altered altered the -dependent current. The effect of these currents on the the voltage of the receiving cell was computed. The complete model was divided into two parts, a finite difference scheme which was coupled to a BEM model. The finite difference scheme was used to compute the transmembrane currents for the portions of source and receiving cells’ membranes which formed the PSJ as well as tracking the extracelin the gap of the PSJ. The BEM model then used the lular transmembrane voltage and current information of the receiving cell to compute the derivatives of the receiving cell transmembrane voltages. The ionic currents of the cell body outside the PSJ were calculated using a passive network to reduce computation while the equations used for the membranes of the PSJ

where the subscripts of the individual current densities refer to the transport mechanism as follows: for the voltage-dependent channel; for the voltage-dependent channel; for the voltage-dependent channel; for the -dependent channel; for the pump; for the exchanger; for the pump. Barrier kinetic equations (Eyring rate theory) [34] were used to describe the voltage-dependent channels which have the generic form for ionic species (2) where transmembrane voltage; valence of ; membrane thickness; Faraday’s constant; is the gas constant; temperature (310.15K); rate constant; transmission coefficient represented by a periodic function of time which was used to describe the gating of the channel (see Fig. 3) while the other parameters describe the permeation through an open channel. were solely the result of the changing of the Changes in transmission coefficients with time; there was no current stimulation. Transmission coefficients were chosen to produce a physiological looking waveform, and, were further constrained to properly balance the voltage and ionic concentrations over one ECA cycle. For the source cell, the transmission coefficients changed with time since the cell was active, but, for the receiving cell, they remained at their basal values. The calcium pump was treated as a channel operating only in an outward direction with no transmission coefficient. The equations for the remaining ionic transport mechanisms used are described in [33] and [35] were

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(3)

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TABLE I MODEL PARAMETERS

(8) (9) where is the volume/area of the source cell and is a factor representing the effect of calcium buffering. Values used for the parameters are given in Table I. The intracellular concentrations were assumed spatially to be constant over the cell since the volume of the cell is much larger than the gap. Of the three conwas by far the most important. Changing of centrations, had very little effect since it was much larger than so that the ratio of the two was relatively unaffected by a change . is only significant during the iniof several mM in tial spike of the ECA and is less important than in neurons and myocytes for producing the upstroke. C. Diffusion in the PSJ The PSJ was divided into toroidal compartments of rectangular cross section with each compartment containing a portion of source cell membrane, receiving cell membrane and the intercellular volume between the two portions of membrane. The rate concentration within each comof change of extracellular partment was given by a finite difference equation of the form (10) where

(4)

(superscript) denotes the compartment; diffusion coefficient of ; compartmental length; compartmental volume; total current introduced into the compartment of extracellular space

(5) (11) where and

half maximal concentrations, normalization factor; , and rate constants. The values of the model parameters are given in Table I. B. Source Cell In addition to modeling the ionic current flow across the region of source cell membrane which formed the intrusion, the transmembrane voltage of the intrusion was determined from (6) where the superscript “ ” denotes the source cell. It was also necessary to model the intracellular concentrations of the ions in the source cell in order to produce the proper ECA because of ion concentration-dependent currents. The rates of change of concentrations in the source cell were given by (7)

where the superscript “ ” denotes receiving cell quantities and is the surface area of the source or receiving portion of membrane. and were assumed to The concentrations of the remain constant in the gap since the concentration of these two , and, a very large ionic species is much higher than that of change in extracellular concentration of these ions must occur for there to be a significant effect because of the transmembrane ion concentration ratios. D. BEM Model of the Receiving Cell The finite difference method considered both the source and receiving cells but only over the PSJ. To compute the induced voltage in the receiving cell, the entire cell had to be considered, so, a BEM representation was used for the receiving cell (see Fig. 2). The smooth muscle cell was modeled as a cylinder of length 100 m with a radius of 5 m, an extrusion of radius 0.31 m and extrusion length with a nominal value of 2 m. The surface of the extrusion was discretized into a set of cylinders

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TABLE II INITIAL CONDITIONS

where is the membrane capacitance. Given the linear portion , the individual of the membrane had a static conductance, entries of the ionic current density vector were computed from extrusion nonextrusion Fig. 2. The figure on the left shows the BEM model of the receiving cell with the extrusion circled while the right figure shows the extrusion in detail. The extrusion membrane was modeled by a series of cylinders while the rest of the cell membrane was discretized into a set of triangles, necessitating a special interfacial element between the two regions: a triangle with a circle cut into it.

of length 0.2 m while the rest of the cell body was discretized into a set of 204 triangles (see Fig. 2). For the BEM, there must be no gaps in the surface and, hence, the cylindrically discretized extrusion had to interface exactly with the triangularly discretized body. A special element was used to join the two sections, a triangle with a hole formed by a negative disk element centered on the triangle (Fig. 2). Had triangles been used for the entire surface, many more would have been required. This would be due to the small size of the triangles needed to discretize the extrusion. Although the body section does not require small triangles, it must merge with the extrusion and the transition zone between the two sizes of elements would have been appreciable. Assuming equal intra- and extracellular conductivities, , to reduce computational requirements, and by transforming Laplace’s equation to an integral form and discretizing it, the vector of transmembrane current densities, , can be related to as previously described the transmembrane voltage vector, [25]

(14)

Thus, a system of first-order differential equations was constructed. E. Resultant System of Equations The final number of equations solved depended on the length concentration in of the interdigitation. Convergence in the the gap was obtained when the toroidal compartments of the PSJ were less than 0.3 m long. Typically, compartments of length 0.2 m were used. For the nominal case of a PSJ of length 2 m, the BEM model solved a system of 214 equations (204 triangular elements and ten cylindrical elements) to determine the derivatives of the transmembrane voltages of the receiving cell. This meant that a system of 228 equations needed to be integrated which was broken down as follows: three equations to , monitor the intacellular concentrations of the source cell, , ; the source transmembrane voltage, ; and of each of the ten compartments of the PSJ; and, the transmembrane voltages of the 204 triangular elements of the receiving cell body. The number of equations to solve was, thus, only dependent on the PSJ length. LU decomposition was used to solve for the derivatives and the integrator used was capable of handling stiff systems [32]. The initial conditions are given in Table II. II. RESULTS A. ECA Cycle

(12) where and correspond to the monopole and dipole current sources on the membrane respectively. Analytic formulae were available for the double layer calculations involving triangles [36] [37], and discs and cylinders [31]. For disc and cylinders, single layer calculations had to be performed with one-dimensional Gaussian quadrature [31] while analytic expressions were available for triangular elements [37]. was determined, the derivative of the transmemOnce brane voltage could be found [38] (13)

Variation in several quantities during one cycle of the ECA are plotted in Fig. 3. There are two phases to the cycle: 1) the resting phase where the voltage in the source cell was constant, and 2) the active phase where the voltage in the source cell was a pulse of 30-mV amplitude with a small initial spike. Although the voltage was flat during the resting level portion of the ECA cycle, current flow through each transport mechanism as well as the ionic concentrations were changing. The system was in a steady state, i.e., currents cancelled out on the whole during the resting phase of the ECA. During the active phase, 1) the intracellular concentration of decreased, the concentration of increased and the concentration decreased slightly, and 2) the outward flow of

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Fig. 4. Effect of open end boundary condition on received voltage. The effect of the [K] boundary condition, fixed at 6 mM or no flux, on the received voltage was ascertained for the nominal case of a 2-m-long PSJ. The fixed condition was used for the rest of the simulations.

Fig. 3. Voltages, concentrations and current densities during ECA cycle. From top to bottom, the graphs display transmembrane voltages, source cell transmission coefficients, intracellular Ca concentration of source cell, intracellular Na concentration of source cell, intracellular K concentration of source cell, extracellular concentration of K in PSJ gap, total K currents (thick lines = source cell, thin lines = receiving cell) at different points along the PSJ, and total ionic current of receiving cell along the PSJ. The lower legend applies to the bottom three graphs where the point of measurement is indicated: Open, end where PSJ interfaces with bulk medium with constant K concentration; Midway, halfway along the PSJ; and Closed, end farthest into the PSJ. The PSJ length was 2 m with a gap of 30 nm and a receiving cell conductance of 0.01 S/m .

the cells is an order of magnitude larger, meaning that the surface area per unit volume, which determines the change in concentration, will be an order of magnitude lower. The extreme outside the PSJ following the case of the concentration of inside the PSJ, i.e., the no flux boundary concentration of condition, was simulated to determine the effect of this assumption. As can be seen in Fig. 4, a no flux boundary condition increases the amplitude of the received voltage oscillation by a factor of three. The average of the voltage was slightly increased from 71.3 mV for the fixed concentration case to 70.0 mV for the no flux case meaning the minimum was reduced by 2.2 mV while the maximum was increased by 2.6 mV. These changes occurred because the coupling mechanism depends on an in. The fixed concentration allows to escape crease in from the PSJ during the resting phase, thereby lowering the attained. maximum C. Length of Junction

increased the extracellular concentration which decreased and , but increased the rate of the pump. The net effect was to decrease the current, resulting in depolarreached ization of the receiving cell. While the increase in its peak amplitude quickly, within approximately 150 ms, the voltage change occurred much slower, appearing to act with a time constant of 680 ms. and currents along the length of the Note the gradient in PSJ in the bottom three graphs of Fig. 3. Even at low currents, clearance from the PSJ is slow enough that accumulation the results. Changing the location of the PSJ along the length of the receiving cell body did not affect results. B. Open End Concentration The model as presented assumed that the concentration of at the open end of the PSJ was fixed at 6 mM. Due to the much larger separation between the source and receiving cells outside the outside the PSJ, it was assumed the change in PSJ would be much less than that inside. The distance between

The effect of changing the length of the PSJ is shown in Fig. 5. As the length was increased, the peak voltage induced in the receiving cell increased and the voltage resting level decreased. The voltage resting level was tied to the resting level of which also decreased. During the resting phase, there was an into the source cell which must be supplied by influx of from the bulk medium, at the open end. Thus decreased was fixed at the open end, the only along the PSJ. Since at the closed end. way to achieve this gradient was to lower resting level continued to As the length was increased, the , decrease and the peak continued to increase, but the peak which occurred in the closed end compartment, did not increase (the much after a length of 1 m. The extent of the peak exceeded a certain length of the PSJ over which the peak value) continued to increase with length, however, even though the peak level did not. and the decrease in the Both the increase of the peak served to increase the amplitude of the resting level of oscillation. The oscillation started to become significant after

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K

Fig. 5. Effect of PSJ length on induced transmembrane voltage and concentration in the PSJ gap. (Top) The peak voltage induced in the receiving cell during the ECA and the resting level of the voltage in the receiving cell and the basal are plotted as functions of the PSJ length. (Bottom) Peak level of . The receiving cell had a conductance of 0.01 S/m and the PSJ gap was 30 nm.

[K]

[K]

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K

Fig. 7. Effect of PSJ gap on induced transmembrane voltage and concentration in PSJ gap. (Top) The peak voltage induced in the receiving cell during the ECA and the resting level of the voltage in the receiving cell are plotted as functions of the gap between the intrusion and extrusion. (Bottom) as a function of gap size. The PSJ had a length of 2 Peak and resting level m and a conductance of 0.01 S/m .

[K]

TABLE III EFFECT OF CELL LENGTH ON v , , AND NORMALIZED CURRENT

[K]

E. Intercellular Gap

Fig. 6. Effect of receiving cell membrane conductance on induced concentration in PSJ gap. (Top) The peak transmembrane voltage and voltage induced in the receiving cell during the active phase and the resting level of the voltage in the receiving cell are plotted as functions of the conductance of receiving cell nonextrusion portion of membrane. (Bottom) as a function of receiving cell membrane conductance. The PSJ had Peak a length of 2 m and a gap of 30 nm.

K

[K]

approximately 0.5 m and continued to grow as the length inoscillations did not increase in amcreased. Conversely, the plitude after 1 m but a larger region experienced a higher as the length increased. D. Receiving Cell Membrane Conductivity , The effects of changing the receiving cell conductivity, , is shown in Fig. 6. Increasing the conductivity decreased showing the PSJ acted like a current source. Above 0.1 S/m (or, equivalently, less than 10 k ), the PSJ did not affect the receiving cell. There was no effect on the resting level of and only a small effect on the peak .

The gap between the cells was increased from the nominal value of 30 nm to 1000 nm (Fig. 7). Increasing the gap decreased the efficacy of the coupling. At smaller gaps, the resting level of decreased since the concentration gradient had to become stronger to overcome the increased resistance to diffusion while maintaining the same membrane current flow. The peak was constant for gaps of less than 500 nm. The extent of the peak region was larger at smaller gaps. The induced voltage decreased with increasing gap as the resting level increased. F. Cell Size Two other cell sizes were simulated. Results are shown in Table III. As the cell size was increased, the voltage was affected . The peak voltage decreased with cell size. The more than also decreased with cell size but to a smaller extent peak than the voltage reduction. The total peak current induced in the receiving cell normalized with respect to that induced at nominal 0.01 S/m , PSJ length 2 m, gap 30 conditions ( nm) is also shown in Table III. As the receiving cell increased in size, the induced current increased slightly but had a much larger surface to depolarize. The effect was only noticeable for the 200- m-long cell.

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[K]

Fig. 8. distribution along the PSJ normalized with respect to the peak . The PSJ was discretized into ten compartments. The nominal conditions were PSJ length 2 m and gap 30 nm. Differences from these values are given in the legend. the receiving cell was 100 m long with a membrane conductance of 0.01 S/m.

[K]

III. DISCUSSION Under certain conditions, extracellular accumulation in a PSJ due to an active source cell can induce significant changes in the transmembrane voltage of the receiving cell. Such coupling is in addition to any electric field coupling, i.e., coupling not through gap junctions, that may exist. The two mechanisms may be operating simultaneously. Although the simulations used the cell with the intrusion as the source cell, results would have been the same using the other cell as the source cell. accumulation acts to inject current Coupling through through the membrane of the PSJ. One may think of the PSJ as a current source which is relatively independent of cell size. Current flowing out through the PSJ must flow in through the rest of the cell with the current spreading itself out evenly. Hence, smaller cells experience higher current densities which cause greater voltage changes. The position of the PSJ makes no difference. The current injected is slightly dependent on cell size since the current–voltage relation is nonlinear. Larger receiving cells experience a slightly larger current. Electric field coupling, in contrast, induces a voltage which is highly spatially dependent, depends on the propagation velocity of the transmembrane voltage waveform over the membrane and increases with increasing cell size [25]. The peak voltage induced was increased at the same time that the resting level of the cell was reduced. Both these factors contributed to an increase in voltage oscillation amplitude. Of the two factors, the resting level was always affected as much, if not more, than the peak induced voltage. Decoupling two cells connected by a PSJ should result in a small depolarization in the resting levels of the cells. reached During the active phase of the ECA, the peak a saturation level (10.4 mM) unlike the peak voltage in the resaturated, the region of the peak ceiving cell. After the peak would expand as the induced voltage increased. This is normalized with respect demonstrated in Fig. 8 where the is shown along the entire length of the PSJ for to the peak three different cases. The peak extracellular potassium concen-

trations for the three cases were not very different, about 10.3 mM, yet the peak induced voltages range from 67 to 65 mV. The model suggested that the peak induced voltage increased elevation. Furtherwith the increase of the extent of the more, it was assumed that the interstitial volume outside the PSJ efflux, resulting in a negliwas large enough to buffer the outside the PSJ. If this assumption does gible change in not hold, possibly due to some pathological condition, and does increase outside the PSJ, diffusion of from the PSJ will and, subbe diminished, resulting in larger oscillations of sequently, voltage. The conductance of the cell body was not matched to the conductance of the PSJ since channel densities may change in different regions of the cell. The densities chosen for PSJ are based on those for a membrane with uniform properties. Differing channel densities may lead to greater coupling if a greater channels are present along the PSJ. This inconcentration of creased density would only have to be local for increased coupling. The perpendicular bend observed in some PSJ [23], [24] may serve to increase the length of the PSJ which will greatly enhance any coupling effects. The cells are ellipsoidal and get narrower at the ends. A 2- m-long PSJ without a bend might actually go completely though the cell if the PSJ were to occur near an end. Coupling may also be enhanced by the presence of more than one PSJ between cells. It has been observed that small intestinal smooth muscle cells of approximately 400 m in length have about ten PSJ per cell [23]. In conclusion, coupling between smooth muscle cells via exaccumulation can reach significant levels with tracellular the presence of one or more PSJ between the cells. The coupling is increased with increased PSJ length, a decreased PSJ gap and a lower conductance in the receiving cell. REFERENCES [1] A. J. Ward, S. M. Burns, S. Torihashi, and K. M. Sanders, “Mutation of the proto-oncogene c-kit blocks development of interstitial cells and electrical rhythmicity in murine intestine,” J. Physiol. (Lond), vol. 480, pp. 91–97, 1994. [2] L. Huizinga, J. D. Thuneberg, M. Kluppel, J. Malysz, H. B. Mikkelsen, and A. Bernstein, “The W/kit gene required for interstitial cells of cajal and for intestinal pacemaker activity and for intestinal pacemaker activity,” Nature, vol. 373, pp. 347–349, 1995. [3] L. W. C. Liu and J. D. Huizinga, “Electrical coupling of circular muscle to longitudinal muscle and interstitial cells of Cajal in canine colon,” J. Physiol., vol. 470, pp. 445–461, 1993. [4] N. E. Diamant and A. Bortoff, “Nature of the intestinal slow-wave frequency gradient,” Amer. J. Physiol., vol. 216, pp. 301–307, 1969. [5] B. L. Bardakjian and S. K. Sarna, “Mathematical investigation of populations of coupled synthesized relaxation oscillators representing biological rhythms,” IEEE Trans. Biomed. Eng., vol. BME-28, pp. 10–15, Jan. 1981. [6] B. L. Bardakjian, “The gastrointestinal system,” in The Biomedical Engineering Handbook, J. D. Bronzino, Ed. Boca Raton, FL: CRC, 2000, vol. 1, ch. 6. [7] S. K. Sarna, E. E. Daniel, and Y. K. Kingma, “Simulation of electrical control activity of the stomach by an array of relaxation oscillators,” Amer. J. Physiol., vol. 17, pp. 299–310, 1972. [8] B. L. Bardakjian and S. K. Sarna, “A computer model of human colonic electrical activity,” IEEE Trans. Biomed. Eng., vol. BME-27, pp. 193–202, Apr. 1980. [9] E. E. Daniel, B. L. Bardakjian, J. D. Huizinga, and N. E. Diamant, “Relaxation oscillator and core conductor models are needed for understanding of GI electrical activities,” Amer. J. Physiol., vol. 266, pp. G339–349, 1994.

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VIGMOND et al.: INTERCELLULAR COUPLING MEDIATED BY POTASSIUM ACCUMULATION IN PEG-AND-SOCKET JUNCTIONS

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Edward J. Vigmond received the B.A.Sc. degree in electrical and computer engineering from the University of Toronto, Toronto, Canada, in 1988. He also received the M.A.Sc. and Ph.D. degrees from the University of Toronto, Institute of Biomedical Engineering, in 1991 and 1996, respectively. He is a Post-Doctoral Fellow at Tulane University, New Orleans, LA. His research interests include numerical computation of electrical fields and modeling of nonlinear biosystems.

Berj L. Bardakjian received the Ph.D. degree in electrical engineering (Biomedical Engineering Group) from McMaster University, Hamilton, ON, Canada. He is currently a Professor of Biomedical Engineering at the University of Toronto. His research interests include modeling of nonlinear biosystems, biological clocks, generation and coupling mechanisms of cellular bioelectricity.

Jan D. Huizinga received the Ph.D. degree in pharmacology from the University of Groningen, Groningen, the Netherlands, in 1981. He is currently Professor in Biomedical Sciences, and MRC Scientist, at McMaster University. His research interests include the study of intestinal pacemaker activity, the communication between pacemaker cells and smooth muscle cells.

Lars Thuneberg is a Professor in anatomy at the Institute of Medical Anatomy, University of Copenhagen, Copenhagen, Denmark. His main interest is in morphology of the gastrointestinal tract, with particular emphasis on smooth muscle structure and regulation.

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