我們考慮擁有不同特性之元素間的四種耦合: 噪聲下可激發元素間之耦合、自發性震盪元素間之耦合、自發性震盪元素與可激發元素之耦合以及自發性震盪元素與被動元素間之耦合，並透過數值模擬、非線性動力學之解析方法以及統計力學等方法，來有系統的對其所對應之物理機制有更深的了解。在噪聲下可激發元素間之耦合系統的動力學中，我們考慮了不同的系統拓撲結構以及系統的大小對動力學行為的影響。由頻率的分佈，我們發現能代表系統同步性的分佈寬度隨著耦合強度增加而減少，在此同時，系統的平均頻率隨著耦合強度非單調性的增加，而當系統同步時，系統的頻率大於沒有耦合時的頻率，此為頻率增昇現象並可以 Kramer’s escape rate 理論來進行解釋。而在自發性震盪元素耦合系統中，數值模擬的結果顯示，在擁有不同的時間尺度的變數上耦合，會形成不同的同步頻率，並可進一步利用 phase reduction 的化簡方式來對此現象進行了解。而在自發性震盪元素與可激發元素之耦合系統中也發現到頻率增昇的的現象，而我們發現此頻率增昇是因為自發性震盪元素與可激發元素的穩定之不動點作用，使得自發性震盪元素相空間中軌跡縮短所造成。當耦合強度持續增加，我們的解析計算顯示系統會由震盪轉為安靜狀態是由於 Hopf bifurcation 的發生所造成。最後，自發性震盪元素與被動元素間之耦合系統可以被用來描述利用群眾感應機制所產生的集體行為。在平均場的近似中，藉由相位模型的 saddle-node bifurcation 或是震幅模型的 Hopf bifurcation 所產生之集體行為可被解析分析並藉由數值模擬的方式驗證。而在非平均場的模型中，化學信號分子僅能在局部的利用擴散方式來傳遞，當其濃度夠高，細胞間之集體行為便會被啟動。同時利用解析及數值模擬的方式，對因為減少細胞間距以及擴散常數而使得系統失去穩定性產生集體震盪之集體行為的基本啟動機制進行探討。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.
