Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 10 511b 2a 2a 1b 1g(x, y)=0f(x, y)=0a 1a 1|g(a 1 )| |g(a 1 )|Figure 10.47: Left: Relaxation oscillator in the limit µ = 0 near the onset of oscillation.Middle and right: A magnification of a neighborhood of the jump point a 1 for variousg(a 1 ) and µ. Canard (French duck) solutions can appear when g(a 1 ) ≪ µ.where coth, acoth, and sinh are hyperbolic cotangent, hyperbolic inverse cotangentand hyperbolic sine, respectively.27. [M.S.] Derive the PRC for an oscillator near saddle homoclinic orbit bifurcationthat is valid during the spike downstroke. Take advantage of the observation inFig. 10.39 that the homoclinic orbit consists of two qualitatively different parts.28. [M.S.] Derive PRC for a generic oscillator near fold limit cycle bifurcation.29. [M.S.] Simplify the connection function H for coupled relaxation oscillators(Izhikevich 2000) when the slow nullcline approaches the left knee, as in Fig. 10.47.Explore the range of parameters ε, µ, and |g(a 1 )| where the analysis is valid.30. [Ph.D.] Use ideas outlined in Sect. 10.4.5 to develop the theory of reduction ofweakly coupled bursters to phase models. Do not assume that bursting trajectoryis periodic.Solutions to Chapter 101. In polar coordinates, z = re iϑ , the system has the form˙ϑ = 1 , ṙ = r − r 3 .Since the phase of oscillation does not depend on the amplitude, the isochrons have the radialstructure depicted in Fig. 10.3.2. In polar coordinates, the oscillator has the form˙ϑ = 1 + dr 2 , ṙ = r − r 3 .The second equation has an explicit solution r(t), such thatr(t) 2 =11 − (1 − 1/r(0) 2 )e −2t .