Dynamical Analysis and Bifurcations of Jansen-Rit's Neural-Mass Model and Its Application in Describing Epileptic Seizures

Nowadays, with technological advancements and increasing computing power, the use of mathematical models to describe the functioning of the brain in normal and abnormal manners, especially the study of the formation causes and methods of controlling and treating some nervous system diseases, such as epilepsy, have become widespread and many models have been developed to simulate patterns appearing in the brain signals of these patients. One of the most commonly used types of modeling is neural mass models such as the Jansen-Rit model that those can simulate some of the essential brain patterns and rhythms that appear in the brain recorded signals. Therefore, in this paper, we have tried to provide a complete dynamical analysis of the Jansen-Rit model. To analyze this model, first, the equations of the model have been changed so that the output of the model be one of the system states variables. Then, the new equations have been nondimensionalized by defining a biological parameter (proportion of inhibition to excitation in neural populations of the model). In the following, the bifurcation diagram of the dimensionless model has been plotted with respect to nondimensional input and inhibition to excitation proportion parameters (codimension-two bifurcation) and the dynamical behavior of the system, such as bifurcations, periods and frequency of the limit cycles and time responses, have been investigated. Further, we have discussed two significant behaviors in this model, spike-and-wave discharges (SWDs) and alpha rhythms. In the present paper, we have been shown how these models can describe complex disease such as epilepsy and have been mentioned dynamical mechanism underlying transition from a normal state (background activity) to an abnormal situation (epileptic seizures). The innovations of this study one can be the definition of the new meaningful and significant biological parameter in the dimensionless model that all dynamical analysis are based on it. Also, some bifurcations and, consequently, some of the behaviors observed in the model are for the first time reported. Moreover, this new parameter contains two primary model parameters and then the effect of three parameters simultaneously in the system behavior has been investigated.