Computing the Vulnerable Phase in a 2D Discrete Model of the Hodgkin-Huxley Neuron

In several neurological diseases, like essential tremor, the functions of the brain are severely impaired by synchronized processes, in which the neurons fire in a synchronized periodical manner at a frequency closely related to that of the tremor. Stimulation techniques have been developed to desynchronize these neuronal populations. One such technique is the electrical Deep Brain Stimulation (DBS) (Luders, 2004), (Mayberg, 2005) which is performed by administering a permanent high frequency periodic pulse train to the brain by means of so-called depth electrodes. The DBS method was developed empirically, and its mechanism has not yet been understood. Another stimulation technique is the perturbation with brief stimuli. Clinical results for this technique (some of them are briefly presented in this chapter) prove that a carefully chosen brief pulse applied at a specific time, denoted by the term “vulnerable phase”, can annihilate the firing behaviour in the neuron. It is believed that by determining the vulnerable phase of a neuron, the result can be generalized to a population of neurons. In this context, the first neural model analytically investigated in great detail was the Hodgkin-Huxley (HH) neuron, which exhibits stable periodic solutions for a certain range of constantly applied depolarizing currents. To study the latter from a variety of perspectives, we shall first present, in Section 1.1, the dynamics of the HH neuron. Then, in Section 1.2, we informally describe its annihilation and stability properties and compare its characteristics with the properties of some of its close “relatives”. Finally, in Section 1.3, we shall describe the HH model from the context of the works of Winfree and Guckenheimer.