Linear electrical properties of passive and active currents in spherical heart cell clusters

A Bidomain Model for Lens Microcirculation

There exists a large body of research on the lens of the mammalian eye over the past several decades. The objective of this work is to provide a link between the most recent computational models and some of the pioneering work in the 1970s and 80s. We introduce a general nonelectroneutral model to study the microcirculation in the lens of the eye. It describes the steady-state relationships among ion fluxes, between water flow and electric field inside cells, and in the narrow extracellular spaces between cells in the lens. Using asymptotic analysis, we derive a simplified model based on physiological data and compare our results with those in the literature. We show that our simplified model can be reduced further to the first-generation models, whereas our full model is consistent with the most recent computational models. In addition, our simplified model captures in its equations the main features of the full computational models. Our results serve as a useful link intermediate between the computational models and the first-generation analytical models. Simplified models of this sort may be particularly helpful as the roles of similar osmotic pumps of microcirculation are examined in other tissues with narrow extracellular spaces, such as cardiac and skeletal muscle, liver, kidney, epithelia in general, and the narrow extracellular spaces of the central nervous system, the "brain." Simplified models may reveal the general functional plan of these systems before full computational models become feasible and specific.

Field stimulation of isolated chick heart cells: comparison of experimental and theoretical activation thresholds

INTRODUCTION

This study examines the accuracy of using membrane models to predict activation thresholds for chick heart cells during field stimulation.

METHODS AND RESULTS

Activation thresholds were measured experimentally in ten embryonic chick heart cells at 37 degrees C for stimulus durations 0.2 to 40 msec. Activation was assessed by observing the mechanical twitch of the cell. The heart cells ranged in diameter from 15.0 to 26.7 microm. Since the electric field required for activation depends on diameter, the thresholds were expressed as the maximum field-induced transmembrane potential, Vth = 1.5 a Eth, where a is the cell radius and Eth is the strength of the electric field at threshold. A cell model was created using a singular perturbation method and membrane models describing the ionic currents of a heart cell. The study used membrane models of Ebihara and Johnson (1980), Luo and Rudy (1991), Shrier and Clay (1994), and their combinations. The results show that for stimuli longer than 1 msec, theoretical activation thresholds were within one standard deviation of experimental thresholds. For shorter stimuli, the models failed to predict thresholds because of a premature deactivation of the sodium current. The modification of the m gates dynamics, so that they closed with a time constant of 1.4 msec, allowed to predict thresholds for all durations. The root mean square error between experimental and theoretical thresholds was 6.14%.

CONCLUSIONS

The existing membrane models can predict thresholds for field stimulation only for stimuli longer than 1 msec. For shorter stimuli, the models need a more accurate representation of the sodium tail current.

A quasi-one-dimensional theory for anisotropic propagation of excitation in cardiac muscle

It has been shown that propagation of excitation in cardiac muscle is anisotropic. Compared to propagation at right angles to the long axes of the fibers, propagation along the long axis is faster, the extracellular action potential (AP) is larger in amplitude, and the intracellular AP has a lower maximum rate of depolarization, a larger time constant of the foot, and a lower peak amplitude. These observations are contrary to the predictions of classical one-dimensional (1-D) cable theory and, thus far, no satisfactory theory for them has been reported. As an alternative description of propagation in cardiac muscle, this study provides a quasi-1-D theory that includes a simplified description of the effects of action currents in extracellular space as well as resistive coupling between surface and deeper fibers in cardiac muscle. In terms of classical 1-D theory, this quasi-1-D theory reveals that the anisotropies in the wave form of the AP arise from modifications in the effective membrane ionic current and capacitance. The theory also shows that it is propagation in the longitudinal, not in the transverse direction that deviates from classical 1-D cable theory.

Na/K pump current in aggregates of cultured chick cardiac myocytes

Spontaneously beating aggregates of cultured embryonic chick cardiac myocytes, maintained at 37 degrees C, were voltage clamped using a single microelectrode switching clamp to measure the current generated by the Na/K pump (Ip). In resting, steady-state preparations an ouabain-sensitive current of 0.46 +/- 0.03 microA/cm2 (n = 22) was identified. This current was not affected by 1 mM Ba, which was used to reduce inward rectifier current (IK1) and linearize the current-voltage relationship. When K-free solution was used to block Ip, subsequent addition of Ko reactivated the Na/K pump, generating an outward reactivation current that was also ouabain sensitive. The reactivation current magnitude was a saturating function of Ko with a Hill coefficient of 1.7 and K0.5 of 1.9 mM in the presence of 144 mM Nao. The reactivation current was increased in magnitude when Nai was increased by lengthening the period of time that the preparation was exposed to K-free solution prior to reactivation. When Nai was raised by 3 microM monensin, steady-state Ip was increased more than threefold above the resting value to 1.74 +/- 0.09 microA/cm2 (n = 11). From these measurements and other published data we calculate that in a resting myocyte: (a) the steady-state Ip should hyperpolarize the membrane by 6.5 mV, (b) the turnover rate of the Na/K pump is 29 s-1, and (c) the Na influx is 14.3 pmol/cm2.s. We conclude that in cultured embryonic chick cardiac myocytes, the Na/K pump generates a measurable current which, under certain conditions, can be isolated from other membrane currents and has properties similar to those reported for adult cardiac cells.

Repolarization currents in embryonic chick atrial heart cell aggregates

Outward membrane currents in aggregates of atrial cells prepared from 7-12-d-old chick embryonic hearts were measured with the two microelectrode voltage-clamp technique. Two outward current components, Ix1 and Ix2, were found in the plateau potential range of the action potential. The Ix1 component is activated between -50 and -20 mV; the Ix2 component is activated between -15 and +20 mV. The Ix1 component inwardly rectifies, whereas Ix2 has an approximately linear current-voltage relation. These preparations lack a time-dependent pacemaker current component, even though they beat spontaneously with an interbeat interval of approximately 1 s. A mathematical model of electrical activity is described based on our measurements of time-dependent outward current, and measurements in the literature of inward current components.

Linear impedance studies of voltage-dependent conductances in tissue cultured chick heart cells

Plateau and pacemaker currents from tissue cultured clusters of embryonic chick heart cells were studied in the time domain, using voltage-clamp steps, and in the frequency domain, using a wide-band noise input superimposed on a steady holding voltage. In the presence of tetrodotoxin to block the sodium channel, a depolarizing voltage step into the plateau range elicited: (a) a rapid (approximately equal to 2 ms) activation of the slow inward current; (b) a subsequent slower (approximately equal to 25 ms) decline in the slow inward current; and (c) activation of a very slow (5 to 10 s) outward current. Impedance studies in this voltage range could clearly resolve two voltage-dependent processes, which appeared to correspond to points b and c above because of their voltage dependence, pharmacology, and time constants. A correlate of point a was also probably present but difficult to resolve owing to the fast time constant of activation for the slow inward channel. At voltages negative to -50 mV a new voltage-dependent process could be resolved, which, because of its voltage dependence and time constant, appeared to represent the pacemaker channel (also termed If or IK2). In the Appendix, linear models of voltage-dependent channels and ion accumulation/depletion are derived and these are compared with our data. Most of the above-mentioned processes could be attributed to voltage-dependent channels with kinetics similar to those observed in time domain, voltage-clamp studies. However, the frequency domain correlate of the decline of the slow inward current was incompatible with channel gating, rather, it appears accumulation/depletion of calcium may dominate the decline in this preparation.

Phase resetting of the rhythmic activity of embryonic heart cell aggregates. Experiment and theory

Injection of a current pulse of brief duration into an aggregate of spontaneously beating chick embryonic heart cells resets the phase of the activity by either advancing or delaying the time of occurrence of the spontaneous beat subsequent to current injection. This effect depends upon the polarity, amplitude, and duration of the current pulse, as well as on the time of injection of the pulse. The transition from prolongation to shortening of the interbeat interval appears experimentally to be discontinuous for some stimulus conditions. These observations are analyzed by numerical investigation of a model of the ionic currents that underlie spontaneous activity in these preparations. The model consists of: Ix, which underlies the repolarization phase of the action potential, IK2, a time-dependent potassium ion pacemaker current, Ibg, a background or time-independent current, and INa, an inward sodium ion current that underlies the upstroke of the action potential. The steady state amplitude of the sum of these currents is an N-shaped function of potential. Slight shifts in the position of this current-voltage relation along the current axis can produce either one, two, or three intersections with the voltage axis. The number of these equilibrium points and the voltage dependence of INa contribute to apparent discontinuities of phase resetting. A current-voltage relation with three equilibrium points has a saddle point in the pacemaker voltage range. Certain combinations of current-pulse parameters and timing of injection can shift the state point near this saddle point and lead to an interbeat interval that is unbounded . Activation of INa is steeply voltage dependent. This results in apparently discontinuous phase resetting behavior for sufficiently large pulse amplitudes regardless of the number of equilibrium points. However, phase resetting is fundamentally a continuous function of the time of pulse injection for these conditions. These results demonstrate the ionic basis of phase resetting and provide a framework for topological analysis of this phenomenon in chick embryonic heart cell aggregates.

Linear electrical properties of isolated cardiac cells

A frequency domain equivalent circuit analysis of isolated ventricular cells indicated the presence of an internal membrane structure which has a total capacitance four- to sixfold larger than the surface membrane. The internal membrane was mainly attributed to the sarcoplasmic reticulum since other morphological studies have shown that its area is many-fold larger than that of the surface membrane. Corresponding estimates from the transverse tubular system indicate an area less than that of the surface; thus this structure is not a likely candidate for the observed internal capacitance. Measurements in hypertonic solutions showed that the access resistance to the internal membrane reversibly increased as the tonicity was elevated. Freeze-fractured electron microscopic studies confirmed that hypertonic solutions increased the volume of transverse tubular system, which thus appears to have little relation to the access resistance. The most probable source of the access resistance is the diadic junction to the sarcoplasmic reticulum, which therefore would electrically couple it to the surface membrane.

Electrical properties of sheep Purkinje strands. Electrical and chemical potentials in the clefts

The impedence of sheep Purkinje strands, measured to 3-5 kHz, is interpreted with circuit models based on morphology. The strand is described as a one-dimensional electrical cable. Clefts between myocytes of the strand allow radial current to flow in parallel with current across the outer membrane. A lumped model of the clefts, in which all the cleft membrane is in series with 100 omega-cm2, fits only below 20 Hz. Two distributed models, pie and disk, fit at all frequencies with somewhat different (31%) luminal resistivities, but with similar membrane parameters. Series resistance representing the endothelial sheath is small. Simulations of voltage clamp experiments include measured linear parameters and nonlinear membrane channels, as well as radial variation of cleft concentration, membrane flux, voltage, and current. Cleft potential is drastically nonuniform when sodium current flows. Cleft potential is reasonably uniform when calcium and potassium currents flow, but the calcium and potassium concentrations change markedly, enough to turn off the calcium current, even if the calcium channel did not inactivate. We conclude that physiological current flows produce significant nonuniformities in electrochemical potentials in the clefts of this cardiac preparation.

Ion conductances of the surface and transverse tubular membranes of skeletal muscle

A combination voltage clamp and admittance analysis of single skeletal muscle fibers showed that moderate depolarizations activated a steady-state negative sodium conductance in both the surface and transverse tubular membranes. The density of the voltage-dependent channels was similar for the surface and tubular conductances. The relaxation times associated with the negative conductance were in the millisecond range and markedly potential dependent. The negative tubular conductance has the consequence of increasing the apparent steady-state radial space constant to large values. This occurs because the positive conductance is counterbalanced by the maintained inward-going sodium current. The enhancement of the space constant by a negative conductance provides a means for the nearly simultaneous activation of excitation-contraction coupling.

The generation of the cardiac action potential: after the first millisecond

After an initial, transient voltage- and time-dependent burst of sodium current (equivalent to that occurring in nerve), the membrane current of cardiac muscle reverses in sign to a maximum value that is orders of magnitude smaller than that seen in nerve. The membrane of cardiac muscle, rather than exchanging an increased permeability to sodium ions (Na+) for one to potassium ions (K+), appears to become relatively impermeable to a variety of ions. It is argued that in a tissue such as cardiac muscle where the time when the cell is active is comparable to that when it is quiescent, the current generated by the active electrogenic transport/exchange of Na+, K+, and Ca2+ must be comparable to the corresponding currents generated by the passive transport of these ions. Consequently, the complex voltage and time dependency of the membrane current on the time scale of repolarization and beyond is generated, at least in part, by the complex time and voltage dependency of these transport/exchange processes. Measurement of the electrochemical properties of such transport/exchange mechanisms must ultimately be made on the individual mechanisms in isolation, e.g., in artificial membrane systems, before their contribution to the generation of the cardiac action potential can be unequivocably determined.

Effect of tortuous extracellular pathways on resistance measurements

There are many instances in which we are limited to measuring macroscopic quantities such as a bulk flow or an average field. In biology, wer are frequently interested in using such macroscopic measurements, for example, the total current from a tissue, to determine the microscopic properties of the cells or tubules of the tissue. The microstructure of the tissue will generally increase the resistance to flow over what would be measured in an unstructured medium. This paper derives a fairly general expression for the relationship between effective resistance to macroscopic flow and the specific resistance of the medium conducting the microscopic flow. This expression, called a tortuosity factor, is defined entirely in terms of measurable morphometric and geometric parameters of the tissue.