Biological Receptor Scheme for the Robust Synchronization of Limit Cycle Oscillators

Using the conceptual similarity of biological receptors and limit cycle oscillators, the coupled equations dP j /dt = F j (P j , R j ) and dR j /dt = G j (P j + γΣ N l = 1 P l , R j ), where γ is an arbitrary positive number that is chosen to correspond to the sensitivity of a biological receptor, were derived as a system exhibiting robust global synchronization of limit cycle oscillators when individual limit cycles are expressed as dP j /dt = F j (P j , R j ) and dR j /dt = Gj (P j , R j ). Despite the broad distribution of natural frequencies, in a numerical study, 100 van der Pol oscillators became globally synchronized and two van der Pol oscillators became synchronized even when the ratio of natural frequencies was as large as 100. It was also shown that the amplitude of the individual van der Pol oscillator is proportional to the square of the corresponding natural frequency in the synchronized state, and oscillators with higher natural frequencies maintain larger amplitudes.