Dynamical Systems in Neuroscience:

42 Electrophysiology of NeuronsDepolarizationIncrease ingNaIncreasein gNaIncreasein gKNa +InflowDepolarizationRepolarizationHyperpolarizationFigure 2.16: Positive and negative feedback loops resulting in excited (regenerative)behavior in neurons.repolarize the membrane potential toward V rest .When V is near V rest , the voltage-sensitive time constants τ n (V ) and τ h (V ) arerelatively large, as one can see in Fig. 2.13. Therefore, recovery of variables n and his slow. In particular, outward K + current continues to be activated (n is large) evenafter the action potential downstroke, thereby causing V to go below V rest toward E K— a phenomenon known as afterhyperpolarization.In addition, Na + current continues to be inactivated (h is small) and not availablefor any regenerative function. The Hodgkin-Huxley system cannot generate anotheraction potential during this absolute refractory period. While the current deinactivates,the system becomes able to generate an action potential provided that the stimulus isrelatively strong (relative refractory period).To study the relationship between these refractory periods, we stimulate the Hodgkin-Huxley model with 1-ms pulses of current having various amplitudes and latencies. Theminimal amplitude of the stimulation needed to evoke a second spike in the model isdepicted in Fig. 2.17, bottom. Notice that around 14 ms after the first spike, themodel is hyper-excitable, that is, the stimulation amplitude is less than the baselineamplitude A p ≈ 6 needed to evoke a spike from the resting state. This occurs becausethe Hodgkin-Huxley model exhibits damped oscillations of membrane potential, whichwe discuss in Chap. 7.2.3.3 Propagation of the action potentialsThe space-clamped Hodgkin-Huxley model of squid giant axon describes non-propagatingaction potentials since V (t) does not depend on the location, x, along the axon. To describepropagation of action potentials (pulses) along the axon having potential V (x, t),radius a (cm) and intracellular resistivity R (Ω·cm), the partial derivative V xx is addedto the voltage equation to account for axial currents along the membrane. The resultingnon-linear parabolic partial differential equationC V t = a2R V xx + I − I K − I Na − I Lis often referred to as the Hodgkin-Huxley cable or propagating equation. Its importanttype of solution, a traveling pulse, is depicted in Fig. 2.18. Studying this equation goes