Dynamical Systems in Neuroscience:

Bifurcations 169fastBifurcation of an equilibrium subthreshold amplitude frequencyoscillations of spikes of spikessaddle-node no non-zero non-zerosaddle-node on invariant circle no non-zero A √ I−I b → 0supercritical Andronov-Hopf yes A √ I−I b → 0 non-zerosubcritical Andronov-Hopf yes non-zero non-zeroFigure 6.2: Summary of codimension-1 bifurcations of an equilibrium. Here, I denotesthe amplitude of the injected current, I b is the bifurcation value, A is a parameter thatdepends on the biophysical details.bifurcation: the equilibrium loses stability, but does not disappear.Thus, there are only two qualitative events that can happen with a stable equilibriumin a dynamical system of arbitrary dimension: It can either disappear or lose stability.Of course, there could be a third event: All eigenvalues continue to have negative realparts, in which case the equilibrium remains stable.Since any equilibrium of a neuronal model is the zero of the steady-state I-V curveI ∞ (V ) (the net current at the equilibrium must be zero), analysis of the shape of theI-V curve can provide an invaluable information about possible bifurcations of the reststate.Two typical steady-state I-V curves are depicted in Fig. 6.3. The I-V curve inFig. 6.3a has a region with a negative slope so that it may have 3 equilibria: the leftequilibrium is probably 1 stable, the middle is unstable, and the right equilibrium couldbe stable or unstable depending on the kinetics of the gating variables (it is stable inthe one-dimensional case, i.e., when gating variables have instantaneous kinetics). TheI-V curve in Fig. 6.3b is monotone. A positive (inward) injected dc-current I shifts theI-V curves down. This leads to the disappearance of the equilibrium in Fig. 6.3a, butnot in Fig. 6.3b. Therefore, Fig. 6.3a corresponds to the saddle-node bifurcation andFig. 6.3b to Andronov-Hopf bifurcation. When exactly the equilibrium loses stabilityin Fig. 6.3b cannot be inferred from the I-V relations (for this, we need to considerthe full neuronal model). But what we can infer is that the bifurcation cannot be ofthe saddle-node type. Surprisingly, non-monotonic I-V curves result in saddle-nodebifurcations but do not exclude Andronov-Hopf bifurcations, as the reader is asked todemonstrate in Ex. 8. This phenomenon is relevant to the cortical pyramidal neurons1 It may be unstable; see Ex. 8