Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 491- +spike- + - +spike spikex(t)spikex(t)=-cot tspike-1 1000x'=-1+x2 x'=0+x2 x'=1+x2Figure 10.34: Phase portraits and typical oscillations of the quadratic integrate-and-fireneuron ẋ = I + x 2 with x ∈ R ∪ {±∞}. Parameter: I = −1, 0, +1.x'=1+x2A0xnormalized PRC (PRC( ,A)/A)100A=1A=2A=3A=0.1phase of oscillation,TFigure 10.35: The dependence of PRC of the quadratic integrate-and-fire model on thestrength of the pulse A.on invariant circle (SNIC) bifurcation studied in Sect. 6.1.2. Appropriate rescaling ofthe membrane potential and time converts the model into the normal formx ′ = 1 + x 2 , x ∈ R .Because of the quadratic term, x escapes to the infinity in a finite time, producinga spike depicted in Fig. 10.34. If we identify −∞ and +∞, then x exhibits periodicspiking of infinite amplitude. Such a spiking model is called quadratic integrate-and-fire(QIF) neuron; see also Sect. 8.1.3 for some generalizations of the model.Strong pulseThe solution of this system starting at the spike, i.e., at x(0) = ±∞, isx(t) = − cot t ,as the reader can check by differentiating. It is a periodic function with T = π,hence, we can introduce the phase of oscillation via the relation x = − cot ϑ. Thecorresponding PRC can be found explicitly (see Ex. 9) and it has the formPRC (ϑ, A) = π/2 + atan (A − cot ϑ) − ϑ ,