Dynamical Systems in Neuroscience:

Excitability 235(a)(b)frequencypulsesrampfrequency-current (F-I) relation3.88subcriticalAndronov-Hopfbifurcationhomoclinic5.25Andronov-Hopfcurrent, I(c)(d)0.7I=5.25near subcriticalAndronov-Hopfbifurcation(e)0.7I=3.8866near saddlehomoclinic orbitbifurcationK+ gating variable, nK+ gating variable, n0.1-80 membrane potential, V (mV) 00.1-80 membrane potential, V (mV) 0Figure 7.10: The I Na,p +I K -model undergoes subcritical Andronov-Hopf bifurcation yetcan exhibit low-frequency firing when pulses (but not ramps) of current are injected.Parameters: C = 1, I = 0, E L = −66.2, g L = 2, g Na = 5, g K = 4.5, m ∞ (V ) hasV 1/2 = −30 and k = 10, n ∞ (V ) has V 1/2 = −34 and k = 13, and τ(V ) = 1, E Na = 60mV and E K = −90 mV. The shaded region denotes the attraction domain of the restingstate. The inset shows a distorted drawing of the phase portrait.need to consider its dynamics at I = I 1 with the initial condition set to the locationof the old equilibrium marked by the white square in the figure. Such a step evokes aspike response even though the new equilibrium is stable. Finally, shocking the neuronresults in an instantaneous increase of its membrane potential to a new value. As anexercise, prove that the magnitude of the increase equals the product of pulse widthand pulse height divided by the membrane capacitance. This shifts the initial conditionhorizontally to a new point, marked by the white square in Fig. 7.9c, and results in aspike response.Now, let us revisit the Hodgkin experiments and demonstrate the fundamental differencebetween the stimulation protocols. In Fig. 7.10a,b,c we simulate the I Na,p +I K -model and show that it is Class 2 excitable in response to ramps of current but Class1 excitable in response to steps of current. The apparent contradiction is resolved in