The Calculation of Electrotonic Potential Half-Time and its Derivative in Respect to Distance in One- and Two-Dimensional RC Media

In this work the analytical expressions of half-time T1/2 and its derivatives in respect to distance ∂T1/2/∂R in a one-dimensional RC medium (a current electrode has a shape of the segment) and in a two-dimensional RC medium (a current electrode has a shape of the circle) were received. First, by using a well-known in electrostatics a superposition principle, the current’s electrodes were divided into elementary point sources by positioning them on the perimeter or the surface of the electrode. Second, with the help of the computer-simulation, the dependencies of T1/2 and ∂T1/2/∂R on the current electrode dimensions, the degree of electrotonic anisotropy, and the distance between the current electrode and the potential measurement place were calculated. Our calculations demonstrate that the slope of the function T1/2 = f(R) depends both on the distance between the potential measurement place and the current electrode, as well as the measurement direction in respect to the fibers’ direction. Furthermore, the slope value can be less or greater to 0.5. If we apply a linear dependency T1/2 = 0.5R + const for the analysis of the electrotonic potential measurement data in close vicinity to the current electrode in the direction of X-axis, we can receive 40% smaller values of τm. The analogical estimations of τm on the Y -axis would lead to the errors of up to +40%.