Introduction to the Modeling and Analysis of Complex Systems

8.3. HOPF BIFURCATIONS IN 2-D CONTINUOUS-TIME MODELS 143Figure 8.7 shows the results where a clear transition from a stable spiral focus (for r < 0)to an unstable spiral focus surrounded by a limit cycle (for r > 0) is observed.3r = -13r = -0.13r = 03r = 0.13r = 1222221111100000111112222233 2 1 0 1 2 333 2 1 0 1 2 333 2 1 0 1 2 333 2 1 0 1 2 333 2 1 0 1 2 3Figure 8.7: Visual output of Code 8.2.Exercise 8.3 FitzHugh-Nagumo model The FitzHugh-Nagumo model [30, 31]is a simplified model of neuronal dynamics that can demonstrate both excited andresting behaviors of neurons. In a normal setting, this system’s state convergesand stays at a stable equilibrium point (resting), but when perturbed, the system’sstate moves through a large cyclic trajectory in the phase space before comingback to the resting state, which is observed as a big pulse when plotted over time(excitation). Moreover, under certain conditions, this system can show a nonlinearoscillatory behavior that continuously produces a sequence of pulses. The behavioralshift between convergence to the resting state and generation of a sequenceof pulses occurs as a Hopf bifurcation, where the external current is used as acontrol parameter. Here are the model equations:()dxdt = c x − x33 + y + z (8.30)dydt= −x− a + byc(8.31)z is the key parameter that represents the external current applied to the neuron.Other parameters are typically constrained as follows:1 − 2 3 b < a < 1 (8.32)0 < b < 1 (8.33)b < c 2 (8.34)