Dynamical Systems in Neuroscience:

Bifurcations 177matches numerically the found frequency of oscillation (Fig. 6.8, top) in a fairly broadfrequency range. For comparison, we plot the curve 1000/T 2 to show that neglectingthe duration of the spike, T 1 , can be justified only when I is very close to the bifurcationpoint.6.1.3 Supercritical Andronov-HopfIf a neuronal model has a monotonic steady-state I-V relation, a saddle-node bifurcationcannot occur. The resting state in such a model does not disappear, but it losesstability, typically via an Andronov-Hopf (sometimes called Hopf) bifurcation. Thelost of stability is accompanied either by an appearance of a stable limit cycle (supercriticalAndronov-Hopf) or by a disappearance of an unstable limit cycle (subcriticalAndronov-Hopf).Let us consider a two-dimensional system˙v = F (v, u, b)˙u = G(v, u, b)(6.4)and suppose that (v, u) = (0, 0) is an equilibrium when the bifurcation parameterb = 0, that is, F (0, 0, 0) = G(0, 0, 0) = 0. This system undergoes an Andronov-Hopfbifurcation at the equilibrium if the following three conditions are satisfied:• (Non-hyperbolicity) The Jacobian 2 × 2 matrix of partial derivatives at the equilibrium(see Sect. 4.2.2), ( )Fv FL =u,G v G uhas a pair of purely imaginary eigenvalues, ±iω ∈ C with ω ≠ 0. That is,tr L = F v + G u = 0 and ω 2 = det L = F v G u − F u G v > 0 at v = u = b = 0.The linear change of variablesconverts (6.4) into the formwhere functionsv = x and F u u = −F v x − ωy (6.5)ẋ = −ωy + f(x, y)ẏ = ωx + g(x, y) ,f(x, y) = F (v, u) + ωy and g(x, y) = −(F v F (v, u) + F u G(v, u))/ω − ωxhave no linear terms in x and y. Now we are ready to state the other two conditions:• (Non-degeneracy) The parameter(6.6)a = 1 16 {f xxx + f xyy + g xxy + g yyy }+ 116ω {f xy(f xx + f yy ) − g xy (g xx + g yy ) − f xx g xx + f yy g yy }is non-zero.(6.7)