Dynamical Systems in Neuroscience:

38 Electrophysiology of Neurons2.3 Hodgkin-Huxley ModelIn Sect. 2.1 we have studied how the membrane potential depends on the membranecurrents assuming that ionic conductances are fixed. In Sect. 2.2 we have used theHodgkin-Huxley gate model to study how the conductances and currents depend onthe membrane potential assuming that the potential is clamped at different values.In this section we put it all together and study how the potential ↔ current nonlinearinteractions lead to many interesting phenomena such as generation of actionpotentials.2.3.1 Hodgkin-Huxley equationsOne of the most important models in computational neuroscience is the Hodgkin-Huxley model of the squid giant axon. Using pioneering experimental techniques of thattime, Hodgkin and Huxley (1952) determined that the squid axon curries three majorcurrents: voltage-gated persistent K + current with four activation gates (resulting inthe term n 4 in the equation below, where n is the activation variable for K + ), voltagegatedtransient Na + current with three activation gates and one inactivation gate (termm 3 h below), and Ohmic leak current, I L , which is carried mostly by Cl − ions. Thecomplete set of space-clamped Hodgkin-Huxley equations isC ˙V = I −I K{ }} {ḡ K n 4 (V − E K ) −ṅ = α n (V )(1 − n) − β n (V )nṁ = α m (V )(1 − m) − β m (V )mḣ = α h (V )(1 − h) − β h (V )h ,I Na{ }} {ḡ Na m 3 h(V − E Na ) −I L{ }} {g L (V − E L )where10 − Vα n (V ) = 0.01exp( 10−V ) − 1 ,10( ) −Vβ n (V ) = 0.125 exp ,8025 − Vα m (V ) = 0.1exp( 25−V− 1 ,10( ) −Vβ m (V ) = 4 exp ,18( ) −Vα h (V ) = 0.07 exp ,201β h (V ) =exp( 30−V ) + 1 .10