Dynamical Systems in Neuroscience:

186 Bifurcationsmembrane potential, V (mV)0-20-40-60(a)stable limitcycleunstable limit cyclestablesubcriticalAndronov-Hopfbifurcationunstabledelayedloss of stabilitymembrane potential, V (mV)0-20-40-60(b)stable limitnoise-inducedsustained oscillationscycleunstable limit cyclesubcriticalAndronov-Hopfbifurcationinjectedcurrent, I(t)60400 20 40 60 80 100 120 140 160 180 200time, tFigure 6.17: Delayed loss of stability (a) and noise-induced sustained oscillations (b)near subcritical Andronov-Hopf bifurcation. Shown are simulations of the I Na,p +I K -model with parameters as in Fig. 6.16 and the same initial conditions. Small conductancenoise is added in (b) to unmask oscillations.waning rhythmic activity seen in the figure (see also Ex. 3 in Chap. 7). In Chapters 8and 9 we present many examples of noise-induced sustained oscillations in biologicalneurons, and in Chap. 7 we study their neuro-computational properties.6.2 Limit Cycle (Spiking State)In the previous section we considered all codimension-1 bifurcations of equilibria, whichtypically correspond to transitions from resting to spiking states in neuronal models.Below we consider all codimension-1 bifurcations of limit cycle attractors on a phaseplane. These bifurcations typically correspond to transitions from repetitive spiking toresting behavior, as we illustrate in Fig. 6.18, and they will be important in Chap. 9where we consider bursting dynamics.