Dynamical Systems in Neuroscience:

508 Synchronization (see www.izhikevich.com)8. Show that the PRC of the leaky integrate-and-fire neuron (Sect. 8.1.1)˙v = b − v ,if v ≥ 1 (threshold), then v ← 0 (reset)with b > 1 has the formPRC (ϑ) = min {ln(b/(b exp(−ϑ) − A)), T } − ϑ ,where T = ln(b/(b − 1)) is the period of free oscillations and A is the amplitudeof the pulse.9. Prove that the quadratic integrate-and-fire neuron˙v = 1 + v 2 ,if v = +∞ (peak of spike), then v ← −∞ (reset)has PTC (ϑ) = π/2 + atan (A − cot ϑ).10. Find PRC of the quadratic integrate-and-fire neuron (Sect. 8.1.3)˙v = b + v 2 ,if v ≥ 1 (peak of spike), then v ← v reset (reset)with b > 0.11. Consider two mutually pulsed-coupled oscillators with periods T 1 ≈ T 2 and type1 phase transition curves PTC 1 and PTC 2 , respectively. Show that the lockingbehavior of the system can be described by the Poincare phase mapχ n+1 = T 1 − PTC 1 (T 2 − PTC 2 (χ n )) ,where χ n is the phase difference between the oscillators, i.e., the phase of oscillator2 when oscillator 1 fires a spike.12. [MATLAB] Write a program that solves the adjoint equation (10.10) numerically(hint: integrate the equation backwards to achieve stability).13. [MATLAB] Write a program that finds the infinitesimal PRC using the relationship˙ϑ = 1 + PRC (ϑ) εp(t) ,the moments of firings of a neuron (zero crossings of ϑ(t)), and the injectedcurrent εp(t); see Sect. 10.2.4 and Fig. 10.24.14. Use the approaches of Winfree, Kuramoto, and Malkin to transform the integrateand-fireneuron ˙v = b − v + εp(t) in Ex. 8 to its phase modelwith T = ln(b/(b − 1)).˙ϑ = 1 + ε ( e ϑ /b ) p(t) ,