Dynamical Systems in Neuroscience:

428 Solutions to Exercises, Chap. 8fast K +activationslow K +activationVhomoclinic orbitAndronov-HopfFigure 10.28: Co-dimension-2 Shilnikov-Hopf bifurcation.Solutions to Chapter 81. Consider two mutually coupled neurons firing together.2. The equationcan be written in the formwithprovided that b < b sn .˙V = c(b − b sn ) + a(V − V sn ) 2 ,˙V = a(V − V rest )(V − V thresh ) .V rest = V sn − √ c(b sn − b)/a and V thresh = V sn + √ c(b sn − b)/a3. The system ˙v = b + v 2 with b > 0 and the initial condition v(0) = v reset has the solution (checkby differentiating)v(t) = √ b tan( √ b(t + t 0 ))wheret 0 =1 √batan v reset√b.From the condition v(T ) = v peak = 1, we findT = √ 1 atan √ 1 − t 0 = 1 (√ atan √ 1 − atan v )reset√ ,b b b b bwhich can be alternatively written asT =1 √batan( )√b v reset − 1.v reset + b4. The system ˙v = −|b| + v 2 with the initial condition v(0) = v reset > √ |b| has the solution (checkby differentiating)v(t) = √ |b| 1 + e2√ b(t+t 0 )1 − e ,2√ b(t+t 0)wheret 0 = 12 √ |b| ln v reset − √ |b|v reset + √ |b| .From the condition v(T ) = 1, we find(T = 12 √ ln 1 − √ |b||b| 1 + √ |b| − ln v reset − √ )|b|v reset + √ |b|.