Beeler Reuter Model for Ventricular Cells

One Cell

This model is used to simulate the activation of ventricular cells. As in the Hodgkin-Huxley applet, one can change the time for the second stimulus (S2) and conductances. Also, one can plot all 8 variables of the model.

The continuous discovery of new ionic channels in mammalian cardiac cells, as well as the improvements in voltage-clamp techniques and data acquisition over the past 30 years, has led to a parallel increase in the complexity of mathematical models used to describe cardiac cell electrical dynamics. In 1977, Beeler and Reuter developed a model using four of eight different ionic conductances known at the time in cardiac muscle. They implemented an initial fast inward sodium current INa, similar to the one used by Hodgkin and Huxley, but they added an inactivator slow gate variable denoted by j, a time-activated outward current Ix1, a time-independent potassium outward current IK1, and a secondary slow inward current Is carried primarily by calcium ions. Calcium is responsible for the contraction of a cardiac cells and produces a much larger AP plateau in cardiac cells than in nerve cells. The total ionic current in the BR model is given by four currents and uses eight variables: membrane potential, six ionic gates ( m, h, j, x1, f and j ) and the intracellular calcium concentration ([Ca] ). The model equations can be found in J. Physiol. Vol 268 pg 177-210, 1977 by G.W. Beeler and H. Reuter.

When the BR applet is started, it shows two APs generated by two consecutive stimuli in the same way as the HH model . The difference in plateau duration, shape and restitution response can be compared to the HH and FHN models described before. The duration of an AP in Human ventricular cardiac tissue is on the order of 200-300 ms. In the applet, varying the ionic currents can change the duration and APD shape given by the model. For instance, the calcium current is largely responsible for the plateau phase of the AP, so that decreasing the calcium conductance gs decreases the calcium current and thus the APD. Decreasing gs from 0.09 to 0.04 mmho/cm^2 shortens the APD by about 50 percent, and the AP shape becomes more triangular. A further decrease of gs to 0.01 yields an AP similar to the HH model. On the other hand, increasing gs to 0.2 mmho/cm^2 prolongs the AP and changes its shape yet again by making the secondary rise in potential more prominent. As in the HH model, the sodium conductance gNa is responsible for the rise of the AP, and changes to this value also can affect the AP shape and duration. Reducing gNa to 1.0 or to 0.9, for example, decreases the maximum depolarization, allowing less time for the calcium to activate and producing a smaller AP. Increasing gNa increases the excitability of the system and makes it easier to induce activations. By changing gNa to 20 mmho/cm^2 , for instance, the second activation becomes substantially longer because the increased sodium conductance allows a higher depolarization and therefore more time for the calcium current to activate. Under these conditions, it is easier to induce subsequent activations.

In this model the restitution relation is very strong at small diastolic intervals, that is, the smaller the difference in time between the end of the first activation and the application of the second stimulus, the smaller the second activation. Observe for example the small activation that is obtained when S1 is at 20 ms and S2 is at 320 ms. The timing of S2 must be increased much farther to about 500 ms, lengthening the DI to roughly the duration of the first AP, in order to produce a second activation whose duration is almost the same as the initial APD. Second stimuli below about 315 ms do not affect the membrane potential much since a new activation cannot be produced yet, and the system continues to return to the rest state. However, when gNa is larger, a second activation can be produced earlier (e.g. at 312 ms when gNa is increased to 20 mmho/cm^2 ).