Bifurcation of the neuronal population dynamics of the modified theta model: Transition to macroscopic gamma oscillation

Interactions of inhibitory neurons produce gamma oscillations (30–80 Hz) in the local field potential, which is known to be involved in functions such as cognition and attention. In this study, the modified theta model is considered to investigate the theoretical relationship between the microscopic structure of inhibitory neurons and their gamma oscillations under a wide class of distribution functions of tonic currents on individual neurons. The stability and bifurcation of gamma oscillations for the Vlasov equation of the model is investigated by the generalized spectral theory. It is shown that as a connection probability of neurons increases, a pair of generalized eigenvalues crosses the imaginary axis twice, which implies that a stable gamma oscillation exists only when the connection probability has a value within a suitable range. On the other hand, when the distribution of tonic currents on individual neurons is the Lorentzian distribution, the Vlasov equation is reduced to a finite dimensional dynamical system. The bifurcation analyses of the reduced equation exhibit equivalent results with the generalized spectral theory. It is also demonstrated that the numerical computations of neuronal population follow the analyses of the generalized spectral theory as well as the bifurcation analysis of the reduced equation.