Dynamical Systems in Neuroscience:

Excitability 277Figure 7.55: Ex. 6: Zero frequency firing near subcritical Andronov-Hopf bifurcationin the I Na,p +I K -model with parameters as in Fig. 6.16 and a high-threshold slow K +current (g K,slow = 25, τ K,slow = 10 ms, n ∞,slow (V ) has V 1/2 = −10 mV and k = 5 mV.)3. (Noise-induced oscillations) Consider the systemż = (−ε + iω)z + εI(t) , z ∈ C (7.3)which has a stable focus equilibrium z = 0 and is subject to a weak noisy inputεI(t). Show that the system exhibits sustained noisy oscillations with an averageamplitude |I ∗ (ω)|, whereI ∗ 1(ω) = limT →∞ T∫ T0e −iωt I(t) dtis the Fourier coefficient of I(t) corresponding to the frequency ω.4. (Frequency preference) Show that a system exhibiting damped oscillation withfrequency ω is sensitive to an input having frequency ω in its power spectrum.(Hint: use Ex. 3.)5. (Rush and Rinzel 1996) Use the phase portrait of the reduced Hodgkin-Huxleymodel in Fig. 5.21 to explain some small but noticeable latencies in Fig. 7.26.6. The neuronal model in Fig. 7.55 has a high-threshold slow persistent K + current.Its resting state undergoes a subcritical Andronov-Hopf bifurcation, yet it canfire low-frequency spikes, and hence exhibits Class 1 excitability. Explain. (Hint:Show numerically that the model is near a certain codimention-2 bifurcationinvolving a homoclinic orbit).