Dynamical Systems in Neuroscience:

Bifurcations 1831K+ activation variable, n0.5stablelimit cycleunstable limitcyclen-nullclineV-nullclinestable equilibrium (rest)0-80 -70 -60 -50 -40 -30 -20 -10 0 10membrane voltage, V (mV)Figure 6.14: Phase portrait of the I Na,p +I K -model: Unstable limit cycle (dashed circle)is often surrounded by a stable one (solid circle) in two-dimensional neuronal models.Finally, notice that there is always a bistability, i.e., co-existence, of the restingattractor and some other attractor near a subcritical Andronov-Hopf bifurcation in2-dimensional conductance-based models, as in Fig. 6.14 (in non-neural models, thetrajectories could go to infinity and there need not be bistability). The bistabilitymust also be present at the saddle-node bifurcation of an equilibrium, but may or maynot be present at the saddle-node on invariant circle or at a supercritical Andronov-Hopf bifurcation.Delayed loss of stabilityIn Fig. 6.17a we inject a ramp of current into the I Na,p +I K -model to drive it slowlythrough the subcritical Andronov-Hopf bifurcation point I ≈ 48.75 (see Fig. 6.16). Wechoose the ramp so that the bifurcation occurs exactly at t = 100. Even though thefocus equilibrium is unstable for t > 100, the membrane potential remains near -50mV as if the equilibrium were still stable. This phenomenon, discovered by Shishkova(1973), is called delayed loss of stability. It is ubiquitous in simulations of smoothdynamical systems near subcritical or supercritical Andronov-Hopf bifurcations.The mechanism of delayed loss of stability is quite simple. The state of the systemis attracted to the stable focus while t < 100. Even though the focus loses stabilityat t = 100, the state of the system is infinitesimally close to the equilibrium, so ittakes a long time to diverge from it. The longer the convergence to the equilibrium,the longer the divergence from it, hence the noticeable delay. The delay has an upperbound that depends on the smoothness of the dynamical system (Nejshtadt 1985).It can be shortened or even reversed (advanced loss of stability) by weak noise thatis always present in neurons. This may explain why the delay has never been seen