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The purpose of this paper is to revise the parameters of the Hodgkin-Huxley formulation for the Na+ current in ventricular myocardial cells. To this end we have assembled much of the recent voltage clamp data on cardiac preparations obtained with modern voltage clamp and patch clamp techniques. The selected activation and inactivation characteristics of the Na+ channel and other membrane parameters represent a good compromise between available experimental measurements and lead to a reasonable average representation of the cardiac Na+ membrane current. The resulting Na+ conductance changes during the action potential upstroke are much larger than in earlier models, so that the upstroke is much faster and the peak depolarization is close to the Na+ equilibrium potential. The firing threshold level is nearly constant for resting potentials in the range of -70 and -90 mV. The maximum rate of rise of the action potential displayed by the new model is quite comparable to experimental observations.

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Relative changes in the sodium conductance of the resting cardiac cell membrane are often estimated from relative changes in the maximum rate of rise of the action potential (Vmax). This approach has given rise to some controversy and it has not been possible so far to test it directly on an experimental basis. We have examined here the validity of this estimation using three different Hodgkin-Huxley representations of the cardiac membrane sodium current. The two basic requirements are a constant membrane capacitance and a negligible relative value of the nonsodium membrane currents at the time of Vmax. It is shown further that the approach leads to a satisfactory estimation only when the latency of Vmax is kept constant and a correction factor for the sodium driving force is applied to Vmax measurements. This conclusion applies either to a nonpropagated action or to an action potential propagated at constant velocity, provided that the membrane is not too strongly depolarized. It is valid for a wide range of sodium equilibrium potentials and a range of maximum sodium conductances limited to about 50% of the nominal value.

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A comparison between traditional numerical integration methods and a new hybrid integration method for the reconstruction of action potential activity is presented, using a mathematical model of the cardiac Purkinje fiber (MNT model). It is shown that the hybrid integration method reduces importantly the overall computation time required for solving the Hodgkin-Huxley differential equations describing membrane electrical events. To accomplish this, the particular form of the gating variable equations is exploited to reformulate the step-by-step computation. In this way, the time increment can be made much larger compared with traditional methods when the membrane potential changes slowly. A mathematical analysis of the hybrid integration method is presented also, together with a numerical verification of its performance both for the propagated and nonpropagated membrane action potential. It is shown that the local error, that is the error arising at each integration step, and the cumulative integration error are strictly controlled by the membrane potential offset. Using the MNT model, the nonpropagated cardiac Purkinje action potential can be reconstructed in real time with an accuracy of 1% for the potential and 5% for the time of occurrence of its main features. In reconstructing propagated events, the hybrid integration method allows computation time savings by a factor of 10 or more compared to accurate Runge-Kutta schemes.

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A simulation study was performed to evaluate different recovery procedures for computing the multipole components of the cardiac electrical activity. A series of dipolar potential distributions was first generated on a realistic numerical model of the human torso. Then, different procedures based on surface integration (SI) and least-squares (LS) minimization were used to compute the multipole components. The parameters of a single moving dipole (SMD) computed from the estimated multipoles were compared with those of the original dipole source. For a finite and homogeneous simulation as well as recovery medium, the results showed that SI employing the potentials over all 1216 surface elements of the torso model was not affected by the various numerical approximations used to perform the integration (e.g., rms error for the SMD position, p = 0.7 mm). By integrating the potentials with truncated capping surfaces at the neck and the waist, the recovery errors increased (p = 2.1 mm). Sampling the potentials at 63 sites, followed by interpolation over the rest of the torso surface, severely affected the SI results for the SMD (p = 6.4 mm), as compared with LS minimization using also 63 values (p = 0.9 mm). With lungs and intraventricular blood masses in the simulation medium but a finite and homogeneous recovery medium, SI was less effective (p = 10.8 mm) than LS (p = 8.6 mm). Adequate compensation for the effects of lungs was obtained by including regions of lower electrical conductivity in the recovery medium for LS, and by a correction matrix for SI. In general, LS gave better results than SI, but with a higher initial computation time.

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1. The sodium and potassium conductances of the HODGKIN-HUXLEY model are simulated by a field effect transistor with a series resistor. This arrangement leads to a simple analog model of the excitable membrane (fig. 1 and 2). 2. Normally, the model is silent (fig. 3), but it becomes automatic (fig. 4) when the decay time (de-activation) of the potassium conductance is at least twice the recovery from inactivation time of the sodium conductance (taud greater than 2 tauri). 3. The effects of changes in sodium (fig. 5 and 6) and potassium (fig. 7, 8 and 9) concentration gradients upon the membrane potential and the ionic currents are easily studied when the model is silent or automatic. 4. When automatic, an increase in the potassium concentration gradient induces a lengthening of the period and ultimately, when the gradient is very high, spontaneous activity is blocked (fig. 9). On the other hand, increases of sodium gradient over 30% of normal value do not modify the period (fig 6). 5. The potassium concentration gradient modifies the excitability solely through membrane polarization (fig. 8), while sodium concentration has no effect on it (fig. 5). 6. Results with the model strengthen the hypothesis that tetraethylammonium (TEA) acts on both the maximum potassium conductance (gK) and the mechanism of sodium conductance inactivation (Tauh) to lengthen the action potential as observed on the Ranvier node (fig. 10). Effects of TEA on potassium conductance activation are also discussed. 7. Because of its simplicity and accuracy, this model lends itself easily to many other simulations.

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