Dynamical Systems in Neuroscience:

128 Two-Dimensional SystemsabcdFigure 4.38: Determine the approximate direction of the vector field in each regionbetween the nullclines. Continuous (dashed) curve is the x-nullcline (y-nullcline), andthe direction of vector field in one region is indicated by the arrow.trajectories, as well as other representative trajectories. Estimate the signs ofeigenvalues at each equilibrium.4. Prove the classification diagram in Fig. 4.15.5. (van der Pol oscillator) Determine nullclines and draw phase portrait of the vander Pol oscillator (given in the Liénard (1928) form)where b > 0 is a parameter.ẋ = x − x 3 /3 − y ,ẏ = bx ,6. (Bonhoeffer–van der Pol oscillator) Determine the nullclines and sketch representativephase portraits of the Bonhoeffer–van der Pol oscillatorẋ = x − x 3 /3 − y ,ẏ = b(x − a) − cy ,in the case of c = 0. Treat a and b > 0 as parameters.7. (Hindmarsh-Rose spiking neuron) The following system is a generalization of theFitzHugh-Nagumo model (Hindmarsh and Rose 1982)ẋ = f(x) − y + I ,ẏ = g(x) − y ,