Theoretical Study On Nonlinear Dynamics Of Calcium Oscillations With Time Delay And Coupling Effect

Calcium is regarded as“a life and death signal”because of its vital significance for the control of many physiological processes.Almost every cellular function such as neuronal activity,cell apoptosis,gene transcription,muscle contraction is regulated by Ca²⁺signaling,which can cause complex dynamical behaviours ranging from stochas-tic spiking to regular oscillations and from periodic waves to spiral waves.Although a solid experimental foundation is necessary,theoretical analysis can also provide a sig-nificant approach to establish the inherent relation between the experimental data and parameters,which greatly save time and cost.In this thesis,calcium oscillations from single cell to coupled cells system,from simple calcium oscillation models to complex delay models are mainly focused.From the perspective of nonlinear dynamics,com-plicated dynamical behaviors are studied by employing the bifurcation theory.Based on the previous researches,we make a further analysis on the dynamical properties of calcium oscillations.The contents of this thesis are mainly as follows:The entry rate of Ca²⁺into cell through plasma membrane is a major modulator of intracellular Ca²⁺dynamics,including the voltage-gated Ca²⁺channel(VGCC),the store-operated Ca²⁺channel(SOCC)and the receptor-operated Ca²⁺channel(ROC-C).In this thesis,we modify an established four-dimensional dynamical model,which contains the SOCC and ROCC,and carry out a bifurcation analysis to study dynamics of the model.In particular,Hopf bifurcation is identified with the maximum flow of the SOCC chosen as the bifurcation parameter,and normal form theory is applied to consider the stability of bifurcating limit cycles.Bifurcation of multiple limit cycles arising from generalized Hopf bifurcation is also discussed,which may yield complex dynamical behaviours.Further,it is shown that the variation of the maximum flows for different calcium channels determines the parameter range for stable oscillations,as well as for the frequency and amplitude of oscillations.The results indicate that Hopf bifurcation is the main source to generate oscillating behaviours,yielding a dif-ferent bistable phenomenon which involves stable limit cycle and stable equilibrium.Moreover,it is shown that partially blocking the SOCC or the ROCC can change the parameter region of stable calcium oscillations,and the ROCC has more impact than the SOCC on amplitude or frequency of calcium oscillations.To be more realistic,every cell is not single in any issue or organ,which mean-s there exists different pathways of cell communication to exchange signals between cells.In this thesis,the gap junction permeable to Ca²⁺and to IP₃have been con-sidered to establish a new mathematical model for describing calcium oscillations in coupled astrocytes.We focus on the effect of store-operated calcium channel(SOCC)and receptor-operated calcium channel(ROCC)on the intercellular synchronization,respectively.By employing bifurcation analysis on this model,the dynamic behaviors of the coupled system with different physiological state cells are obtained with changes in the maximum capacity of the SOCC and the ROCC.The synchronization boundaries for different conditions are gained in two-parameter plane of the channel parameter and the coupling strength.The results suggest that the variation of the maximum flows for different calcium channels determines stable oscillations of the coupled system,as well as for frequency and amplitude of oscillations.The SOCC has an expected effect on the change of oscillatory interval while the ROCC demonstrated the influence on the ampli-tude modulation(AM).Furthermore,coupling strength and channel parameters could induce 1:1 locking of intercellular Ca²⁺oscillations and the synchronization region like Arnol'd tongue is found.Apart from the Ca²⁺influxes come through calcium channels on the cell mem-brane from extracellular,another important pathway is IP₃receptor(IP₃R)which the Ca²⁺flux pass through calcium channels on the membrane of the endoplasmic reticu-lum(ER)from calcium stores into cytoplasm.In this thesis,a further and detailed study is given for the intracellular calcium oscillation model which mainly describes the in-flux from ER into cytoplasm.An established two-dimensional model is chosen and we demonstrate the existence and stability of equilibrium point and analysis the bifurcation properties by stability and bifurcation theory.According to linear analysis,two Hopf bifurcation points are identified,indicating that Hopf bifurcation is the source of oc-currence and disappearance of the oscillation behaviours.Furthermore,bifurcation of multiple limit cycles is investigated by normal form theory.It is shown that three limit cycles can bifurcate from a Hopf critical point,which involve one unstable equilibrium,two stable limit cycles and one unstable limit cycle.It implies that calcium oscillations can be dynamically balanced either on a small amplitude or a large amplitude depending on initial values.After intensive literature review,it is found that the open probability of IP₃R de-pends on the state of the receptors which is mainly determined by the binding of Ca²⁺concentration and IP₃concentration.However,Ca²⁺controls the IP₃R in a biphasic manner.In this thesis,we explore a further study on an established two-dimensional model.We consider a time delay?for the binding kinetics of Ca²⁺channel and s-tudy the dynamical behaviors of time delay system for a single cell and coupled cells,respectively.For single cell,we identify the critical values for stability switches,show-ing various bifurcations may occur when the parameters are in the reasonable range by the method of bifurcation analysis.In particular,a sequence of Hopf bifurcations occur as time delay crosses some critical values and the critical stability boundaries in two-parameter plane are identified.According to the theoretical results,the effects of different parameters on calcium responds are investigated by numerical simulation.For the coupled cells,we consider the effects of both delay and coupling on the dynamical behaviors of calcium oscillations and build a time-delay coupling model of calcium os-cillations.The influence of time delay on the bifurcation and vibration characteristics of a coupled system is analyzed.Based on two physiological states of cell,?is selected as bifurcation parameter.The results reveal that time delay has remarkable influence on the dynamical behaviors,causing the enlarged oscillation region and the change of oscillation mode.The introduction of?alters the continuous variation of Ca²⁺con-centration with some parameters to a discrete type,which may improve the accuracy of encoding.Frequency modulation(FM)and amplitude modulation(AM)can be af-fected directly by?,implying the possibility of the effect on the regulation of calcium signal transduction.Moreover,the oscillation region in time-delay coupling system is narrower than that of the single time-delayed system,implying that the condition of synchronous oscillations is more difficult in time-delay coupling system and only when two cells are in oscillating state,can synchronous oscillations occur in the time-delay coupling system...