||Four types of cell-cell coupling related to models in biological systems are considered here: coupled noisy excitable-excitable elements, coupled oscillatory-oscillatory elements, coupled oscillatory-excitable elements and coupled oscillatory-passive elements. Systematic numerical simulations are carried out for these systems, analytic methods in nonlinear dynamics and statistical physics are also employed to gain deeper insights for the corresponding physical mechanisms. The oscillatory behavior of the coupled noisy excitable elements is investigated for systems with diﬀerent topologies and sizes. From the frequency distributions at diﬀerent coupling strengths, it is found that the distribution width, which characterizes the synchronization level, decreases with the coupling strength; while the mean frequency varies nonmonotonically. The key result is that the system synchronizes at a frequency higher than the uncoupled intrinsic frequencies, exhibiting the “frequency enhancement” phenomenon, which can further be explained by Kramer’s escape rate the-|
ory. For coupled-oscillators in the absence of noise, simulation results indicate that by coupling in diﬀerent components(fast or small variable), one can obtain diﬀerent synchronized frequencies, which can be further explained by the asymmetric coupling eﬀect in phase reduction method. Frequency enhancement is also observed in coupled noise-free
oscillatory-excitable elements, and its mechanism is understood in terms of shortening of the oscillator’s excursion period in phase space. Analytical calculations indicate a Hopf bifurcation at some coupling strength. Finally, oscillators coupled with passive element are
employed to model the quorum sensing phenomenon. Under mean-ﬁeld approximation, the onsets of the collective behavior with both saddle-node bifurcation in phase model and Hopf bifurcation in amplitude model are calculated analytically and veriﬁed by simulations. Non-mean-ﬁeld model with the chemical signals exchange locally by diﬀusion are also investigated. In this case, the sensing process only involves the local chemical concentration, and the onset of collective behavior happens only when the local chemical
concentration is suﬃciently high. The instability for the onset of oscillatory behavior by reducing the inter-cell separation and diﬀusion coeﬃcient of the signaling chemicals are studied both analytically and by simulations.
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