Dynamical Systems in Neuroscience:

One-Dimensional Systems 87follow approach similar to that in Nonlinear Dynamics and Chaos by Strogatz (1994):Instead of going from linear to non-linear systems, we go from one-dimensional nonlinearsystems (this chapter) to two-dimensional non-linear systems (next chapter).Rather than burdening the theory with a lot of mathematics, we use the geometricalapproach to stimulate the reader’s intuition. (There is plenty of fun math in exercisesand in the later chapters.)Exercises1. Consider a neuron having Na + current with fast activation kinetics. Assume thatinactivation of this current, as well as (in)activations of the other currents in theneuron are much slower. Prove that the initial segment of action potential upstrokeof this neuron can be approximated by the I Na,p -model (3.5). Use Fig. 3.15to discuss the applicability of this approximation.2. Draw phase portraits of the systems in Fig. 3.36. Clearly mark all equilibria,their stability, attraction domains, and direction of trajectories. Determine thesigns of eigenvalues at each equilibrium.F(V) F(V) F(V)V V Va b cFigure 3.36: Draw phase portrait of the system ˙V = F (V ) with shown F (V ).3. Draw phase portraits of the following systems(a) ẋ = −1 + x 2 ,(b) ẋ = x − x 3 .Determine the eigenvalues at each equilibrium.4. Determine stability of the equilibrium x = 0 and draw phase portraits of thefollowing piece-wise continuous systems{ 2x, if x < 0(a) ẋ =x, if x ≥ 0⎧⎨ −1, if x < 0(b) ẋ = 0, if x = 0⎩1, if x > 0{ −2/x, if x ≠ 0(c) ẋ =0, if x = 0