We propose here efficient mathematical tracking control algorithms to design the spiking or bursting behavior in the four dimensional dynamical system modeling biological neurons represented by the Hodgkin-Huxley (HH) differential equations. The stimulating external electrical current serves as a control signal, while the membrane action potential is the target output. We use two alternative feedback algorithms, Fradkov’s speed gradient and Kolesnikov’s ‘synergetic’ target attractor control, to produce arbitrary spiking or bursting regimes in the model and to track the action potential of the system. Both algorithms demonstrate high efficiency and robustness for the controlled HH dynamics. For virtually any initial condition we are able to form a single spike at the chosen moment of time, the train with any number of spikes, the arbitrary-shaped burst, and also to switch between regular and chaotic regimes of bursting. Two approaches developed here could be easily adopted for the networks of neural clusters and used effectively for the purposes of neuro-informatics and for modeling neural dysfunctions like epileptiform or other abnormal behavior in Hodgkin-Huxley neuron clusters. This work has been supported by the TÜBİTAK project 116F049 “Controlling Spiking and Bursting Dynamics in Hodgkin-Huxley Neurons”.