Dynamical Systems in Neuroscience:

One-Dimensional Systems 9115. Prove that the upstroke of the spike in the quadratic integrate-and-fire neuron(3.9) has the asymptote 1/(c − t) for some c > 0.16. (Cusp bifurcation) Draw the bifurcation diagram and representative phase portraitsof the system ẋ = a + bx − x 3 , where a and b are bifurcation parameters.Plot the bifurcation diagram in the (a, b, x)-space and on the (a, b)-plane.17. (Gradient systems) An n-dimensional dynamical system ẋ = f(x), with x =(x 1 , . . . , x n ) ∈ R n is said to be gradient when there is a potential (energy) functionE(x) such thatẋ = − grad E(x) ,wheregrad E(x) = (E x1 , . . . , E xn )is the gradient of E(x). Show that all one-dimensional systems are gradient (Hint:see Fig. 3.11). Find potential (energy) functions for the following one-dimensionalsystemsa. ˙V = 0 , b. ˙V = 1 , c. ˙V = −V ,d. ˙V = −1 + V 2 , e. ˙V = V − V 3 , f. ˙V = − sin V .18. Consider a dynamical system ẋ = f(x) , x(0) = x 0 .(a) (Stability) An equilibrium y is stable if any solution x(t) with x 0 sufficientlyclose to y remains near y for all time. That is, for all ε > 0 there existsδ > 0 such that if |x 0 − y| < δ then |x(t) − y| < ε for all t ≥ 0.(b) (Asymptotic stability) A stable equilibrium y is asymptotically stable if allsolutions starting sufficiently close to y approach it as t → ∞. That is, ifδ > 0 from the definition above can be chosen so that lim t→∞ x(t) = y.(c) (Exponential stability) A stable equilibrium y is said to be exponentiallystable when there is a constant a > 0 such that |x(t) − y| < exp(−at) for allx 0 near y and all t ≥ 0.Prove that (c) implies (b), and (b) implies (a). Show that (a) does not imply(b) and (b) does not imply (c); That is, present a system having stable but notasymptotically stable equilibrium, and a system having asymptotically but notexponentially stable equilibrium.19. (I NMDA -model) Show that voltage-depended activation of NMDA synaptic receptorsin a passive dendritic tree with a constant concentration of glutamate ismathematically equivalent to the I Na,p -model.