Discrete Model for Simple Heart Dynamics
by Mike Martin and Fred Adler

This page allows you to iterate a difference equation model, given explicitly below, for N steps using selected values of the parameters. N can also be thought of as the number of "regular" heartbeats, but some of these beats may be missed in pathological conditions. The Vt variable represents the potential (or voltage) at the atrioventricular (AV) node after it responds to a signal from the sinoatrial node (SN) at time t. τ is the time between signals from the two nodes and α is the exponential decay rate for the potential at the AV node. u is the amount the potential at the AV node is increased when the heart actually beats and Vc is a threshold (or critical) voltage for the AV node, below which the heart will beat and above it the heart will not beat. Finally, Vo is the initial potential (or voltage) at the AV node. Typical parameter values for healthy individuals are utilized when the page is first loaded. Note that the values of the parameters have been conveniently rescaled.

Parameter ranges for α should range between 0.4 and 2, with some very interesting behavior around Log[2] (Mathematica's syntax for the natural logarithm of 2). Both the critical voltage, Vc, and the amount we increase by, u, are set to 1. The time between signals, τ, is also set to 1. As a result of these settings the initial voltage, Vo, should be chosen close to 1, say between 0 and 2.

α =

N =
Vo =

  Vt+1 = e- α τ Vt      provided that      e- α τ Vt > Vc

  Vt+1 = e- α τ Vt + u      provided that      e- α τ VtVc

This takes a little while to render the graphs, so please be patient.







References:

Adler, F. R. Modeling the Dynamics of Life: Calculus and Probability for Life Scientists. Brooks/Cole, Pacific Grove, CA, 1998.

Glass, L. and M. C. Mackey From Clocks to Chaos: The Rhythms of Life. Princeton University Press, Princeton, N.J., 1988.

Keener, J. P. On Cardiac Arrhythmias: AV Conduction Block. Journal of Mathematical Biology 12:215-225, 1981.