  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.
