Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 9 4295. The saddle-node bifurcation occurs when b = 0 regardless of the value of v reset , which is astraight vertical line in Fig. 8.3. If v reset < 0, then the saddle-node bifurcation is on aninvariant circle. When b < 0, the unstable node (saddle) equilibrium is at v = √ |b|. Hence,the saddle homoclinic orbit bifurcation occurs when v reset = √ |b|.6. The change of variables v = g/2 + V , b = g 2 /4 + B transforms ˙v = b − gv + v 2 to ˙V = B + V 2with V reset = −∞ and V peak = +∞. It has threshold V = √ B, rheobase B = 0 and thesame F-I curve as in the original model with g = 0. In v-coordinates, the threshold is v =g/2 + √ b − g 2 /4, which is greater than √ b, the new rheobase is b = g 2 /4, which is greater thanb = 0, and the new F-I curve is the same as the old one, just shifted to the right by g 2 /4.7. Let b = εr with ε ≪ 1 be a small parameter. The change of variablestransforms (8.2) into the theta-neuron formv = √ ε tan ϑ 2˙ϑ = √ ε{(1 − cos ϑ) + (1 + cos ϑ)r} .uniformly on the unit circle except the small interval |ϑ−π| < 2 4√ ε corresponding to the actionpotential (v > 1); see Hoppensteadt and Izhikevich (1997) for more details.8. Use the change of variablesv =√ εϑ1 − |ϑ| .To obtain other theta neuron models, use the change of variablesv = √ εh(ϑ) ,where the monotone function h maps (−π, π) to (−∞, ∞) and scales like 1/(ϑ ± π) whenϑ → ±π. The corresponding model has the formϑ ′ = h 2 (ϑ)/h ′ (ϑ) + r/h ′ (ϑ) .In particular, h 2 (ϑ)/h ′ (ϑ) exists and is bounded and 1/h ′ (ϑ) = 0 when ϑ → ±π. These implya uniform velocity independent from the input r when ϑ passes the value ±π corresponding tofiring a spike.9. The equilibrium v = I/(b + 1), u = bI/(b + 1) has the Jacobian matrix( ) −1 −1L =ab −awith trL = −(a + 1) and detL = a(b + 1). It is a stable node (integrator) when b < (a +1) 2 /(4a) − 1 and a stable focus (resonator) otherwise.10. The quadratic integrate-and-fire neuron with a dendritic compartment˙V = B + V 2 + g 1 (V d − V )˙V d = g leak (E leak − V d ) + g 2 (V − V d )can be written in the form (8.3, 8.4), with v = V − g 1 /2, u = −g 1 V d , I = B − g 2 1/4 − (g 2 1g 2 +g leak E leak )/(g leak + g 2 ), a = g leak + g 2 , and b = −g 1 g 2 /a.11. A MATLAB program generating the figure is provided on the author’s webpage.12. An example is in Fig. 10.29.13. An example is in Fig. 10.30.