Dynamical Systems in Neuroscience:

234 Excitability0.20.150.10.05I 1I 1I 1K + gating variable, nI 0I 00-70 -60 -50 -40 -70 -60 -50 -40 -70 -60 -50 -40membrane potential, V (mV) membrane potential, V (mV) membrane potential, V (mV)0-20-40-60-80V(t)I 1 I 1II(t)1I 0I 0slow ramp from I 0 to I 1step from I 0 to I 1 shock pulse at I 1(a) (b) (c)Figure 7.9: The difference between ramp, step, and shock stimulation is in the resettingof initial condition.ing pulses of current. The system goes through a bifurcation of the equilibrium in theformer, but may bypass it and jump somewhere else in the latter.7.1.5 Ramps, steps, and shocksIn Fig. 7.9 we elaborate the difference between injecting slow ramps, steps, and shocks(i.e., brief pulses) of current. In the first two cases the magnitude of the injected currentchanges from I 0 to I 1 , while in the third case the current is I 1 except the infinitesimallybrief moment when it has an infinitely large strength. In all three cases the dynamicsof the model can be understood via analysis of its phase portrait at I = I 1 . The keydifference among the stimulation protocols is how they reset the initial condition.At the beginning of the slow ramp in Fig. 7.9a, the state of the neuron is at thestable equilibrium. As the current slowly increases, the equilibrium slowly moves, andthe trajectory follows it. When the current reaches I = I 1 , the trajectory is at the newequilibrium, so no response is evoked because the equilibrium is stable. In contrast,when the current is stepped from I 0 to I 1 in Fig. 7.9b, the location of the equilibriumchanges instantaneously, but the membrane potential and the gating variables do nothave the time to catch up. To understand the response of the model to the step, we