A phenomenological model of a neuron, which takes the form of a differential equation with two delays, has been considered. It has been proven that, for a rational ratio of delays, there exists a countable family of homothetic solutions coexisting with each periodic solution. In this case, the homothety coefficients are less than one and depend on the period of the original solution. This result was applied to the well-known bursting cycle for the considered equation. For solutions that are homothetic to the bursting cycle, the problem of their stability was investigated. During the analysis, an exponential substitution was made in the equation to reduce it to a simpler form for analysis. The resulting equation was linearized around the periodic solution, which is homothetic to the bursting cycle with the largest coefficient, and a special monodromy operator was constructed. It was proved that there exists a multiplier outside the unit circle, which indicates instability.