Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 9 437Saddle-NodeSeparatrix LoopSaddle-Nodeon InvariantCircleSpikeInvariantFoliationFigure 10.39: A small neighborhood of the saddle-node point can be invariantly foliated by stablesubmanifolds.for all µ and a > 0, which is stable when a > 1. When a decreases and passes an µ-neighborhoodof a = 1, the equilibrium loses stability via Andronov-Hopf bifurcation. When 0 < a < 1, thesystem has a limit cycle attractor. Therefore, the canonical model exhibits bursting behavior.The smaller the value of a, the longer the interburst period. When a → 0, the interburst periodbecomes infinite.15. Take w = I − u. Then (9.7) becomes˙v = v 2 + w ,ẇ = µ(I − w) ≈ µI .16. Let us sketch the derivation. Since the fast subsystem is near saddle-node homoclinic orbitbifurcation for some u = u 0 , a small neighborhood of the saddle-node point v 0 is invariantlyfoliated by stable submanifolds, as in Fig. 10.39. Because the contraction along the stablesubmanifolds is much stronger than the dynamics along the center manifold, the fast subsystemcan be mapped into the normal form ˙v = q(u) + p(v − v 0 ) 2 by a continuos change of variables.When v escapes the small neighborhood of v 0 , the neuron is said to fire a spike, and v is resetv ← v 0 + c(u). Such a stereotypical spike also resets u by a constant d. If g(v 0 , u 0 ) ≈ 0, thenall functions are small, and linearization and appropriate re-scaling yields the canonical model.If g(v 0 , u 0 ) ≠ 0, then the canonical model has the same form as in the previous exercise.17. The derivation proceeds as in the previous exercise, yielding˙v = I + v 2 + (a, u) ,˙u = µAu .where (a, u) is the scalar (dot) product of vectors a, u ∈ R 2 , and A is the Jacobian matrixat the equilibrium of the slow subsystem. If the equilibrium is a node, it has generically twodistinct eigenvalues, and two real eigenvectors. In this case, the slow subsystem uncouples intotwo equations, each along the corresponding eigenvector. Appropriate re-scaling gives the firstcanonical model. If the equilibrium is a focus, the linear part can be made triangular to getthe second canonical model.18. The solution of the fast subsystem˙v = u + v 2 , v(0) = −1 ,with fixed u > 0 isv(t) = √ ( )√ut 1u tan − atan √u