Dynamical Systems in Neuroscience:

40 Electrophysiology of NeuronsFigure 2.14: Studies of spike-generation mechanism in “giant squid axons” won AlanHodgkin and Andrew Huxley the 1963 Nobel prize for physiology or medicine (sharedwith Sir John Eccles); see also Fig. 4.1 in Keener and Sneyd (1998).Since we are interested in geometrical and qualitative methods of analysis of neuronalmodels, we assume that all variables and parameters have appropriate scalesand dimensions, but we do not explicitly state them. An exception is the membranepotential V , whose mV scale is stated in every figure.2.3.2 Action PotentialRecall that when V = V rest , which is 0 mV in the Hodgkin-Huxley model, all inwardand outward currents balance each other so the net current is zero, as in Fig. 2.15.The rest state is stable: A small pulse of current applied via I(t) produces a smallpositive perturbation of the membrane potential (depolarization), which results in asmall net current that drives V back to rest (repolarization). However, an intermediatesize pulse of current produces a perturbation that is amplified significantly becausemembrane conductances depend on V . Such a non-linear amplification causes V todeviate considerably from V rest — a phenomenon referred to as an action potential orspike.In Fig. 2.15 we show a typical time course of an action potential in the Hodgkin-Huxley system. Strong depolarization increases activation variables m and n and decreasesinactivation variable h. Since τ m (V ) is relatively small, variable m is relativelyfast. Fast activation of Na + conductance drives V toward E Na , resulting in furtherdepolarization and further activation of g Na . This positive feedback loop, depicted inFig. 2.16 results in the upstroke of V . While V moves toward E Na , the slower gatingvariables catch up. Variable h → 0, causing inactivation of Na + current, and variablen → 1, causing slow activation of outward K + current. The latter and the leak current