A two-dimensional simulation of the effects of late stimulation on a cardiac action potential.

The response of cardiac tissue to a stimulus delivered late in the action potential (AP) is important in studies of defibrillation and certain models of ischemia in which reentrant trajectories produce such stimulation. Such stimuli are often strong and delivered in complex spatial arrangements. In this work a two-dimensional computer simulation of a thin, flat layer of cardiac tissue subjected to rather small sizes of late stimulation delivered at a simple point has been performed. The calculation covered a plane of 95 x 140 space units of 72 u each and the stimulus was near the center. The Drouhard and Roberge version of the physiological routine due to Beeler and Reuter was used in an anisotropic bidomain model of the tissue. Late stimuli of either polarity were applied from 150 to 250 msec after initiation of the AP. Depolarizing (positive) voltage stimuli produced small positive voltage peaks, narrowly localized in space, which decayed with a time constant of 8 to 10 msec. Repolarizing (negative) stimuli, if not applied too late in the AP, produced, after a variable delay, a depolarizing fast sodium peak, which was also positive and spatially quite localized. It decayed quickly so that little effect on the AP duration was found. Since stimuli were of limited size and spatial extent in this study the small effects found here are expected but they indicate that major effects from late stimulation require strong pulses and/or large electrode arrays.

Simulation calculations of cardiac virtual cathode effects.

The anisotropic bidomain model has been used to perform computer simulations of cardiac tissue stimulated externally. Virtual cathode distances, Rvc, are found to be affected by type of stimulation and the degree of conductance between the extracellular domain and the ground provided by, for example, a saline bath. For no such conductance, hyperpolarizing blocking regions develop in the longitudinal direction of propagation. These regions are much reduced by even a small conductance to ground. They do not occur for transmembrane stimulation. They result in much reduced Rvc sizes in the longitudinal direction even in the presence of some ground conductance, and negative Rvc values for no such conductance.

Magnetic fields from simulated cardiac action currents.

A digital simulation has been performed of an idealized, thin, 2-D cardiac slice in the chi-y plane. The slice is stimulated near the center and the resulting action potential propagates outward, developing a distribution of electrical current with nonzero curl. An anisotropic bidomain model is used for the calculation, with membrane physiology based upon either just fast sodium fluxes or the more complete Beeler-Reuter myocardial model. The electrical anisotropy, expressed as the ratio of longitudinal to transverse electrical conductivity, is much greater for the inner domain than for the outer one, and this results in current loops that develop ahead of and behind the wavefront and produce a Bz magnetic field of order 10(-9) T 1 mm above the tissue, similar to recent experimental observations on canine cardiac tissue slices. The fields exhibit a quatrefoil symmetry which can be distorted by nonuniformities in the tissue. The field from repolarization currents is larger by almost an order of magnitude than might be predicted from considerations of rate of change of voltage.

The biomagnetic signature of a crushed axon. A comparison of theory and experiment.

The response of a crayfish medial giant axon to a nerve crush is examined with a biomagnetic current probe. The experimental data is interpreted with a theoretical model that incorporates both radial and axial ionic transport and membrane kinetics similar to those in the Hodgkin/Huxley model. Our experiments show that the effects of the crush are manifested statically as an elevation of the resting potential and dynamically as a reduction in the amplitude of the action current and potential, and are observable up to 10 mm from the crush. In addition, the normally biphasic action current becomes monophasic near the crush. The model reflects these observations accurately, and based on the experimental data, it predicts that the crush seals with a time constant of 45 s. The injury current density entering the axon through the crush is calculated to be initially on the order of 0.1 mA/mm2 and may last until the crush seals or until the concentration gradients between the intra- and extracellular spaces equilibrate.

A simulation of cardiac action currents having curl.

A digital simulation of a two dimensional cardiac slice has been performed. It is stimulated at the center and an action potential propagates outward. An anisotropic bidomain model is used in which fast sodium physiology connects the intracellular and extracellular domains. For cases in which the inner asymmetry (expressed as longitudinal versus transverse electrical conductivity) is greater than the outer asymmetry, a current flow pattern is observed for which there is nonzero curl. Such a result explains recent observations of nonzero Bz magnetic field detected above a slab of tissue in the x-y plane. The current loop producing this field consists of outer domain current in the longitudinal direction flowing around in space to return at the AP location in the transverse direction in the outer domain and then completing the loop in the longitudinal direction by passing distally through the AP in the inner domain where resistance is extremely low.

A numerical reconstruction of the effects of late stimulation on a cardiac ventricular action potential.

Numerical simulations of a propagating cardiac action potential utilizing Beeler-Reuter and Drouhard-Roberge physiological routines for the membrane current have been performed. These action potentials show increases in action potential duration when subjected to strong late stimuli of either positive or negative polarity. The mechanism is the same as that reported in an earlier paper which utilized a different physiological approach: repolarizing stimuli can reset the fast sodium gates locally so that they can be retriggered by diffusive return of charge from surrounding tissue. This results in a large depolarizing transient that lengthens action potential duration.

The effect of action potential propagation on a numerical simulation of a cardiac fiber subjected to secondary external stimulus.

Computer simulations have been performed on a particular model of propagating cardiac action potentials (AP), over a large enough strand (15 mm) and a long enough time (500 msec) that the complete AP can develop free of end effects. External stimuli of either polarity of transmembrane current are applied in the plateau phase of the AP to a position midway in the strand. In our case, stimuli of either polarity can produce increases in action potential duration (APD), whereas repolarizing stimuli produce decreases in APD for space-clamped calculations. An analysis of the ion fluxes allows us to characterize these events within the parameters of the model. For repolarizing stimuli in our long fiber case, we obtain an APD increase, whereas a decrease might be expected. The APD increase is due to a local deep drop in potential, which resets the fast sodium channels so that subsequent diffusional return of charge from regions of positive potential fires JNa at the stimulus site.

Computer simulations of cardiac action potentials in two dimensions.

A novel method of calculation is presented which allows computer simulations of action potentials for a two-dimensional, planar reconstruction of the depolarization phase of a cardiac action potential to be followed through nonuniform tissue. The calculation is explicit in type and assumes an infinite, grounded extracellular region. A 70 by 70 point region is calculated swiftly on a PC/AT. Results show elliptical isochrones when slow and fast directions have different resistivities. The time derivative of Vm is, however, similar in both directions. Insertion of a test region of tissue with high resistance results in wave-like propagation of the A.P. around the 'bad' region.

The effect of ohmic return currents on biomagnetic fields.

To illustrate the calculation methods used for biomagnetic fields we present a detailed calculation of the B-field from a spherical cell in an infinite ohmic bath. The calculation is done from three approaches and the results are used to clarify some misinterpretations that may seem to be biophysical problems but are, in fact, creatures of the formalism used.

Computer simulation of action potential propagation in septated nerve fibers.

The nonlinear, core-conductor model of action potential propagation down axisymmetric nerve fibers is adapted for an implicit, numerical simulation by computer solution of the differential equations. The calculation allows a septum to be inserted in the model fiber; the thin, passive septum is characterized by series resistance Rsz and shunt resistance Rss to the grounded bath. If Rsz is too large or Rss too small, the signal fails to propagate through the septum. Plots of the action potential profiles for various axial positions are obtained and show distortions due to the presence of the septum. A simple linear model, developed from these simulations, relates propagation delay through the septum and the preseptal risetime to Rsz and Rss. This model agrees with the simulations for a wide range of parameters and allows estimation of Rsz and Rss from measured propagation delays at the septum. Plots of the axial current as a function of both time and position demonstrate how the presence of the septum can cause prominent local reversals of the current. This result, not previously described, suggests that extracellular magnetic measurements of cellular action currents could be useful in the biophysical study of septated fibers.

Magnetic field of a nerve impulse: first measurements.

The magnetic field of the action potential from an isolated frog sciatic nerve was measured by a SQUID magnetometer with a novel room-temperature pickup coil. The 1.2 x 10(-10) tesla field was measured 1.3 millimeters from the nerve with a signal-to-noise ratio of 40 to 1.