MATH 723 Mathematical Neuroscience Spring 2008

1.1.1. The current balance equation. Imagine a cylindrical membrane, which separates intracellular space from the extracellular. We assume that the properties of membrane are spatially uniform and consider a small patch of the membrane. The membrane is covered with sufficiently large number of ionic channels. These channels can be in one of the two states ”open” or ”closed”. The ions from extracellular space can enter the cell (and vice versa, the ions from intracellular space can leave the cell) through open channels (Fig. 1). The net currents through ionic channels are called ionic currents or conductances. The dynamics of the Hodgkin-Huxley (HH) model is generated by three ionic conductances: sodium (INa), potassium (IK ) and a passive linear conductance, (IL). The latter is called the leak current. In addition, to describe many experimental results, it is convenient to include an additional current, I , the applied current, which can be controlled during the experiment. The dynamical state of the system is determined from the current balance equation I = INa + IK + IL + I ′, (1.1)