Nonlinear Dynamic Analysis Of Neural Mass Model Based On Bifurcation Theory

Epilepsy is a common chronic neurological disorder characterized by the recurrence of seizures that strongly alter the patient's quality of life.Epileptic seizures are transient manifestations of abnormal brain activity characterized by excessive and highly synchronous firing in networks of neurons distributing over focal or extended brain regions.Understanding from a theoretical point of view the origin and the nature of this condition and predicting seizures are hot topics in different scientific domains.This investigation should be based on comprehensive understanding of the collective behavior of neurons in epileptic networks and which depends on various types of computational and analytical models.The mathematical models adopted iin this article are Jansen&Rit neural mass model,a new improved time-delay model based on Jansen&Rit model and Wendling neural mass model model.These neural mass models can be defined by ordinary differential equations or delay differential equations.By studying the bifurcations of these equations we establish an important relation between their dynamic characteristics and cortical patterns of activity,including normal and pathological,especially epileptic patters.(1)The bifurcation analysis of Jansen&Rit neural mass model using the extrinsic input,average number of synaptic contacts and average time constant as a variable show that the model undergoes supercritical Hopf bifurcation,subcritical Hopf bifurcation,saddle-node bifurcation and fold bifurcation of limit cycles when the parameters change slowly.These bifurcations lead to different types of steady state in the model which have a necessarily close connection with the output of the model.Hopf bifurcations lead to harmonic oscillation limit cycles.The frequency of harmonic oscillation is not sensitive about the extrinsic input of the model.Through bifurcation analysis using average time constant as a variable,we found that the model can produce harmonic oscillation with frequency ranging from 1 to 70Hz and the frequency is decided by membrane average time and dendritic tree average time constant.This pattern is compatible with typical brain rhythms,such as the ?-? rhythm.Saddle-node bifurcations lead to homoclinic limit cycles that appear suddenly at high amplitude and low frequency,which we call anharmonic oscillation spike limit cycles.In general,the spike limit cycles are not harmonic but they have a spike-like appearance(anharmonic oscillation).The model produces a spike-like output when it is under a state of spike limit cycles,which is related to the sustained discharge of spikes.Their frequency depends very closely on the extrinsic input levels and average number of synaptic contacts.Hence,if the extrinsic input is fluctuating,the intervals between the wave peaks(or spikes)are variable.These phenomena are compatible with the hallmark of epileptic seizures(i.e.,suddenly occurring,irregular spiking patterns).The fold bifurcations of limit cycles lead to the transition different oscillation state.Through the bifurcation analysis we also give the parameter space of the model under different states.This analytical results plays an important theoretical guiding role in understanding the oscillation mechanism of the neural mass model.(2)we modify Jansen&Rit neural mass model by adding a time delay in the inhibitory feedback loop to simulate signal transmission delay among subset of neurons,resulting in a neural mass model with time delay.The new model is able to reproduce several different types of EEG activities including ? rhythm wave,interictal EEG,and ictal EEG,with all parameters(excepting extrinsic inputs)being kept constant and equal to the standard values.According to Wendling et al.'s research,Jansen&Rit model with altered parameters(a certain ratio that is related to the balance between excitatory and inhibitory)can produce epileptiform signals.Nevertheless,a time delay in the inhibitory feedback loop can also make the model produce epileptiform signals,with all parameters(excepting extrinsic inputs)being kept constant and equal to standard values.In order to reveal the reasons behind this phenomenon,we present a detailed description of the model's behavior with bifurcation diagrams.Through bifurcation analysis considering extrinsic input as a bifurcation parameter when the time-delay take different values and time-delay as a bifurcation parameter respectively,we find anharmonic oscillations(spikes)caused by saddle-node bifurcation become really prominent with the increasing of the time delay.Thus,the time-delay model produces epileptiform signals.These findings indicate that a signal transmission delay among subset of neurons may cause seizure-like activity in the brain when the degree of the delay reaches a certain value.The results of the bifurcation analysis of the time delay model can be used to explain how the time delay in subset of neurons lead to EEG abnormalities(3)Wendling neural mass model is an extension of Jansen and Rit model with the specific aim of reproducing the fast activity observed at the onset of seizures.They added a new inhibitory feedback loop to the previous model to represent a subset of interneurons that provide somatic inhibition to pyramidal cells.Six different types of EEG activity were produced using the model which closely resemble the real depth-EEG activity recorded in interictal or ictal period.Through bifurcation analysis considering key parameters as a bifurcation parameter,including extrinsic inputs,excitatory average synaptic gain,slow inhibitory and fast inhibitory average synaptic gain.As a function of extrinsic inputs,the occurrence of and the transitions between different states of the model are encoded by the topology of the bifurcation diagrams.Through the bifurcation analysis considering average synaptic gains as a bifurcation parameter,we show how the different average synaptic gains influence the kinetics states of the model.The analysis results provides a mathematical explanation of how Wendling model can produce the different activity observed at the onset of seizures,and then,necessarily theory supports for diagnosis and treatment of epilepsy.In this thesis,we provide a detailed analysis of the dynamic properties of three neural mass model for a single cortical area and describe the rich dynamic behavior by relatively concise topology structure of the bifurcation diagrams of these model.We dissected the regulating mechanism of the nolinear dynamics of neural mass models underlined by the key parameters and extrinsic inputs,which enable us to draw general conclusions about the interesting phenomena(such as epilepsy)and the parameter space they exist.Moreover,we show transitions between different oscillatory regimes,which can be observed in EEG.These transitions can be explained by smooth changes in the extrinsic input or the key parameters of models and which may be used as models of sudden transitions between brain states,caused by slow changes in system parameters.In short,the study on bifurcation of neural mass models model provides theoretically further reference for grasping the pathogenesis of epilepsy and the regulating mechanism of neural activities of higher level brain network.