Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 467that the phase ϑ 3 cannot be measured directly from the voltage trace in Fig. 10.10abecause pulse 2 changes the phase, so it is not the time since the last spike when pulse3 arrives. The Poincare phase map (10.3) takes into account such multiple pulses.The orbit approaches a point (called fixed point, see below) that corresponds to asynchronized or phase-locked state.A word of caution is in order. Recall that PRCs are measured on the limit cycleattractor. However, each pulse displaces the trajectory away from the attractor, asin Fig. 10.5. To use the PRC formalism to describe the effect of the next pulse, theoscillator must be given enough time to relax back to the limit cycle attractor. Thus,if the period of stimulation T s is too small, or the attraction to the limit cycle is tooslow, or the stimulus amplitude is too large, then the Poincare phase map may be notan appropriate tool to describe the phase dynamics.10.1.6 Fixed pointsTo understand the structure of orbits of the Poincare phase map (10.3), or any othermapϑ n+1 = f(ϑ n ) , (10.4)we need to find its fixed pointsϑ = f(ϑ)(ϑ is a fixed point),which are analogues of equilibria of continuous dynamical systems. Geometrically, afixed point is the intersection of the graph of f(ϑ) with the diagonal line ϑ n+1 = ϑ n ;see Fig. 10.10d or Fig. 10.11. At such a point, the orbit ϑ n+1 = f(ϑ n ) = ϑ n is fixed.A fixed point ϑ is asymptotically stable if it attracts all nearby orbits, i.e., if ϑ 1 is ina sufficiently small neighborhood of ϑ, then ϑ n → ϑ as n → ∞, as in Fig. 10.11, left.The fixed point is unstable if any small neighborhood of the point contains an orbitdiverging from it, as in Fig. 10.11, right.The stability of the fixed point is determined by the slopem = f ′ (ϑ)of the graph of f at the point, which is called Floquet multiplier of the mapping. It playsthe same role as the eigenvalue λ of an equilibrium of a continuous dynamical system.Mnemonically, the relationship between them is µ = e λ , to that the fixed point is stablewhen |m| < 1 (λ < 0) and unstable when |m| > 1 (λ > 0). Fixed points bifurcatewhen |m| = 1 (λ is zero or purely imaginary). They lose stability via flip bifurcation (adiscrete analogue of Andronov-Hopf bifurcation) when m = −1 and disappear via foldbifurcation (a discrete analogue of saddle-node bifurcation) when m = 1. The formerplays an important role in the period-doubling phenomenon illustrated in Fig. 10.14,bottom trace. The latter plays an important role in the cycle slipping phenomenonillustrated in Fig. 10.16.