Mid-infrared quantum cascade lasers operating under external optical feedback and external periodic bias forcing are shown to exhibit a deterministic chaotic pattern composed of frequencies which are linked to the one of the forcing. Results also show that both the amplitude and the frequency of the forcing play a key role in the number of retrieved spikes per modulation period. These findings are of paramount importance for chaotic operation of quantum cascade lasers in applications such as optical countermeasure systems and secure atmospheric transmission lines, as well as for simulating neuronal systems and the communication between neurons due to sudden bursts.