Dynamical Systems in Neuroscience:

Bifurcations 171I(V, b) = 0 (equilibrium condition) we find b = I ∞ (V ). Non-hyperbolicity conditionimplies I V (V, b) = 0, so that the bifurcation occur at the local maxima and minima ofI ∞ (V ). We considered all these properties in Chap. 3.• (Non-degeneracy) The second-order derivative of I(V, b sn ) with respect to V isnon-zero, that is,a = 1 ∂ 2 I(V, b sn )≠ 0 (at V = V2 ∂V 2 sn ) . (6.1)That is, the piece of the I-V curve, I ∞ (V ), at the bifurcation point, V sn , lookslike the square parabola.• (Transversality) Function I(V, b) is non-degenerate with respect to the bifurcationparameter b; that is,c = ∂I(V sn, b)∂b≠ 0 (at b = b sn ) .This condition is always satisfied when the injected dc-current I is the bifurcationparameter, because ∂I/∂b = ∂I/∂I = 1/C.The saddle-node bifurcation has codimension 1 because only one condition (non-hyperbolicity)involves strict equality (“=”), and the other two involve inequalities (“≠”). The dynamicsof multi-dimensional neuronal systems near a saddle-node bifurcation can bereduced to that of the topological normal form˙V = c(b − b sn ) + a(V − V sn ) 2 , (6.2)where V is the membrane voltage, and a and c are defined above. In the context ofneuronal models, this equation with an after-spike resetting is called the quadraticintegrate-and-fire neuron, which we discuss in Chapters 3 and 8.Example: The I Na,p +I K -modelLet us use the I Na,p +I K -model (4.1, 4.2) with high-threshold K + current to illustratethese conditions. The saddle-node bifurcation occurs when the V -nullcline touchesthe n-nullcline, as in Fig. 6.4. Solving the equations numerically, we find that thisoccurs when I sn = 4.51 and (V sn , n sn ) = (−61, 0.0007). The Jacobian matrix at theequilibrium,( )0.0435 −290L =,0.00015 −1has two eigenvalues λ 1 = 0 and λ 2 = −0.9565, with corresponding eigenvectors( )( )11v 1 =and v0.000152 =,0.0034