Detailed study of bifurcations in a rate model with excitatory and inhibitory neurons and adaptation

The dynamical behaviour of a population-based rate model with firing adaptation is studied. An excitatory and inhibitory population of neurons is recurrently coupled and a negative feedback term is added to the excitatory population as firing adaptation. In several studies of these models, the UP-DOWN transitions are in focus, which are also exhibited by the investigated model. In this paper we provide a full characterization of equilibrium points with the precise conditions for their existence and stability. Calculations can be performed analytically and explicit formulas can be provided due to the fact that the activation function is a threshold linear function. We use bifurcation analysis to detect significant changes in the phase space. Complete list of bifurcation diagrams is provided with respect to the number of steady states. Oscillatory dynamics of neurobiological relevance are examined through local bifurcation analysis. The study demonstrates that sharp wave-ripple oscillations may emerge in certain regions of the parameter space.