Analysis And Control Of Delayed Neutral Dynamics Model

Epilepsy is an intractable chronic recurrent disease characterized by sudden brain dysrhythmia that leads to temporary disorders of brain function and structural network.In contemporary society,epilepsy is still a difficult problem in the medical community.It not only poses a threat to the physical and mental health of patients,but also brings heavy medical burden and increasingly serious social problems.Focal epilepsy is the most common type of epilepsy,and patients with focal epilepsy account for approximately 60%of the total population with epilepsy in China.Because the firing patterns of focal epilepsies have more complex temporal and spatial scales,it is difficult to study the generation,propagation and termination mechanisms of focal epilepsies.If the long-term seizure of focal epilepsy can not be effectively controlled,it will eventually become intractable epilepsy.Focal seizures begin in one area of the brain’s epileptogenic network and rapidly propagate to other brain areas to affect the whole brain network.There is a significant time delay in the spread of the epileptic oscillators in different brain regions,even up to more than ten seconds.In recent years,many scholars have been studying the seizure mechanism,but the specific effect of time delay on seizures has not yet been revealed.To solve this problem,we propose a model of epileptic oscillator with time delay between network nodes to study the effect of time delay on seizures.The main research work includes:1.Based on the existing model of epileptic oscillators and the fact that there is a time delay in the propagation between different epileptic oscillators,a dynamical system of epileptic oscillators with time delay coupling is established systematically.We analyze the existence and stability of the equilibrium point of the system,as well as the effect of time delay caused by the propagation between epileptic oscillators on the stability of epileptic oscillator subsystem.We give the bifurcation condition and the critical value of Hopf bifurcation for the system.The validity of the theoretical analysis is verified by numerical simulation.The results show that the steady state of the model can be transformed into periodic oscillation due to the time delay in the propagation between the epileptic oscillators.2.For the epileptic oscillator model with time delay,we discuss the properties of Hopf bifurcation by using the central manifold theorem and the normal form method.we obtain the direction,period and stability of the periodic solution of the bifurcation.In addition,in order to achieve the stability control of the epileptic oscillator model,we add a feedback control to the quiescent current of the model to further explore the relationship between the control intensity and the stability of the epileptic system,and verify the effectiveness of the theoretical analysis by numerical simulation.We find that the time delay between nodes could make the brain network enter the state of epileptic seizure(SLE)through Hopf bifurcation,and the time delay was one of the important reasons for inducing epileptic seizure.Meanwhile,the delay threshold τs depends on the characteristic time scale τ0 and increases with the increase of τ0.When the intensity of control is increased to a certain value,the system will change from an unstable state to a locally stable state,which indicates that the brain network can be changed from seizure-like event to a stable state by changing the intensity of control.In summary,using the epilepsy oscillator coupling dynamics model with time delay,we have found that with the increase of the time delay between the nodes of the brain network,the normal brain regions will enter seizure-like event through Hopf bifurcation.The time delay is one of the important reasons for the induction of epileptic seizure.These results help to explore the transmission mechanism of epilepsy,and provide a theoretical reference for clinical treatment of epilepsy.