Stability of entrainment conditions for a particular form of mutually coupled Van der Pol oscillators

A commonly accepted mathematical model for the electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected oscillators of the Van der Pol type. The method of harmonic balance is used to obtain analytical results for the entrainment conditions of two or more Van der Pol oscillators mutually coupled between their "
x
" outputs. Two stable limit cycles are shown to exist with frequency ratios which can be adjusted by changing the coupling factor. The limit cycles have different amplitudes and phase shift in accordance with physiological data obtained from the human colon. It is shown analytically that there is a minimum value of coupling factor below which, for a given waveshape factor and frequency, only one stable limit cycle exists. This latter condition matches the measured data from the human small intestine. For oscillators having different uncoupled frequencies a hill-climbing method is used on the algebraic equations to obtain entrainment conditions and in particular the phase shift between oscillators. The analytical and hill-climbing results are compared throughout with analog simulation studies.