We study the periodic solutions of a discrete neuron model for period two or period three of the parameter (internal decay rate) unmapped: inline-formula unmapped: math unmapped: msubsup unmapped: mrow unmapped: mo (unmapped: msub unmapped: mrow unmapped: mi βunmapped: mrow unmapped: mi nunmapped: mo )unmapped: mrow unmapped: mi nunmapped: mo =unmapped: mn 0unmapped: mrow unmapped: mi ∞. The novelty of this research is finding a chaotic attractor for certain interval, outside the defined interval the solution goes to positive infinity or to negative infinity. The investigation can be useful in the design of chaos-based neural networks architecture.