Analytical solution for time-dependent potentials in a fiber stimulated by an external electrode.

This study provides an analytical solution for time-dependent potentials in a 3D cylindrical fiber stimulated by an extracellular point electrode. The membrane is passive and represented by surface resistance and surface capacitance. Separation of variables solution expresses intracellular and extracellular potentials as sums involving modified Bessel functions; the coefficients ([Formula: see text] and [Formula: see text]) depend on time. In contrast to previous analytical solutions, where [Formula: see text] and [Formula: see text] had to be determined numerically, here [Formula: see text] and [Formula: see text] are given by explicit formulas that resemble the formulas for potentials in a fiber stimulated by a transverse electric field. The comparison of the 3D analytical solution with the 1D cable model shows that the cable model approximates transmembrane potential with the error below 5 % when the distance between the electrode and the fiber is 0.2-4 mm and when the stimulus is longer than 3.3 ms. For stimuli between 0.43 and 3.3 ms, the range of fiber-electrode distances with error below 5 % shrinks, and it disappears completely for stimuli shorter than 0.43 ms. Thus, our study shows that the applicability of the 1D cable model may be more limited than previously considered.

Asymptotic model of electrical stimulation of nerve fibers.

We present a novel theory and computational algorithm for modeling electrical stimulation of nerve fibers in three dimensions. Our approach uses singular perturbation to separate the full 3D boundary value problem into a set of 2D "transverse" problems coupled with a 1D "longitudinal" problem. The resulting asymptotic model contains not one but two activating functions (AF): the longitudinal AF that drives the slow development of the mean transmembrane potential and the transverse AF that drives the rapid polarization of the fiber in the transverse direction. The asymptotic model is implemented for a prototype 3D cylindrical fiber with a passive membrane in an isotropic extracellular region. The validity of this approach is tested by comparing the numerical solution of the asymptotic model to the analytical solutions. The results show that the asymptotic model predicts steady-state transmembrane potential directly under the electrodes with the root mean square error of 0.539 mV, i.e., 1.04% of the maximum transmembrane potential. Thus, this work has created a computationally efficient algorithm that facilitates studies of the complete spatiotemporal dynamics of nerve fibers in three dimensions.

Spatial heterogeneity of restitution properties and the onset of alternans.

Traditionally, it was believed that cardiac rhythm stability was governed by the slope of the restitution curve (RC), which relates the duration of an action potential to the preceding diastolic interval. However, a single RC does not exist; rate-dependence leads to multiple distinct RCs. We measure spatial differences in the steady-state action potential duration (APD), as well as in three different RCs: the S1-S2 (SRC), constant-basic-cycle-length (BRC), and dynamic (DRC), and correlate these differences with the tissue's propensity to develop alternans. The results show that spatial differences in APD, SRC slope, and DRC slope are correlated with the tissue's propensity to exhibit alternans. These results may lead to a new diagnostic approach to identifying patients with vulnerability to arrhythmias, which will involve pacing at slow rates and analyzing spatial differences in restitution properties.

Variability of action potential duration in pharmacologically induced long QT syndrome type 1.

Long QT Syndrome (LQTS) is a congenital disorder associated with life-threatening arrhythmias. LQT1, a type of LQTS affecting the slow delayed rectifier potassium current, shows a higher incidence of arrhythmia associated with sympathetic stimulation than other types of LQTS. LQT1 patients show increased variability of repolarization with epinephrine infusion, as measured from the 12-lead ECG. We investigate the variability of repolarization measured as action potential duration (APD) in the rabbit left ventricle: how APD variability is affected by pacing rate, transmural location, LQT1 induced by chromanol 293b, and epinephrine infusion. Chromanol preferentially changes APD variability in the midwall. Infusing epinephrine returns the variability to near-control levels. These results differ substantially from clinical studies and show the need for further study.

Activating function of needle electrodes in anisotropic tissue.

We present an analytical solution for the electrical potential and activating function (AF) established by cylindrical needle electrodes in anisotropic tissue. We compare this activating function to (1) AF computed assuming line-source electrodes and (2) AF computed using a finite element program. The results show that when the fiber is two needle diameters away from the electrodes, the maximum of the AF for needle electrodes is 1.43-times larger than for line-source electrodes, which results in lower thresholds for stimulation and electroporation. Therefore, for fibers that are close to the stimulating electrodes, one would benefit from using the formula that accounts for the electrodes' geometry.