In this article dynamical system based on the generalized Lotka-Volterra equation of three coupled identical excitable elements is studied. The system has three parameters: one of them sets the value of the excitation threshold of the elements, the other two set the strength of the couplings between the elements. It is shown that the system is multistable. Along with the trivial attractor (stable equilibrium in origin) in the phase space of the system, for some values of the system parameters, there can exist a stable heteroclinic cycle consisting of six saddle equilibria and heteroclinic trajectories connecting them. On the plane of the system parameters, a set is found in which this heteroclinic cycle exists. Sufficient conditions for the stability of this heteroclinic cycle have been found. Due to multistability, the system has two main types of behavior depending on the initial states: a) absence of neuron-like activity (mathematical image is stable equilibrium in origin); b) non-decaying sequential switching activity, when the activity of one element is replaced by the activity of a pair of elements, then by the activity of another single element, then a pair of other elements, etc. (mathematically, this is a stable heteroclinic cycle).