Numerical bifurcation computations are used to characterize traveling waves for a family of models of thalamic neurons in a network. These models consist of two layers of neurons: one made up of excitatory neurons, and the other of inhibitory ones. The interplay of these two different couplings gives rise to the propagation of activity waves. This article contains some preliminary work on the characterization of the observed waves in a one-dimensional lattice and explores the effects of varying key parameters of the model. The stability of these solutions, as well as the presence of hysteresis and the coexistence of up to three different waves, are most naturally explained in terms of the theory of bifurcations of dynamical systems.