Dynamical Systems in Neuroscience:

Two-Dimensional Systems 109τ0eigenvaluessaddle(real eigenvalues, different signs)saddle-node bifurcation saddle-node bifurcation(real positive eigenvalues)unstable nodeunstable focus(complex eigenvalues,positive real part)Andronov-Hopf bifurcationstable focus(complex eigenvalues,negative real part)stable node(real negative eigenvalues)τ2 − 4∆ = 0τ2 − 4∆ = 0∆0Figure 4.15: Classification of equilibria according to the trace (τ) and the determinant(∆) of the Jacobian matrix L. The shaded region corresponds to stable equilibria.approach the saddle equilibrium along the eigenvector corresponding to the negative(stable) eigenvalue and then diverge from it along the eigenvector correspondingto the positive (unstable) eigenvalue.Focus (Fig. 4.18): The eigenvalues are complex-conjugate. Foci are stable when theeigenvalues have negative real parts, and unstable when the eigenvalues have positivereal parts. The imaginary part of the eigenvalues determines the frequencyof rotation of trajectories around the focus equilibrium.When the system undergoes a saddle-node bifurcation, one of the eigenvalues becomeszero and a mixed type of equilibrium occurs — saddle-node equilibrium, illustratedin Fig. 4.14b. There could be other types of mixed equilibria, such as saddle-focus,focus-node, etc., in dynamical systems having dimension three and higher.Depending upon the value of the injected current I, the I Na,p +I K -model (4.1, 4.2)with a low-threshold K + current has a stable focus (Fig. 4.8) or an unstable focus(Fig. 4.10) surrounded by a stable limit cycle. In Fig. 4.19 we depict the vector fieldand nullclines of the same model with a high-threshold K + current. As one expectsfrom the shape of the steady-state I-V curve in Fig. 4.1a, the model has three equilibria:a stable node, a saddle, and an unstable focus. Notice that the third equilibrium isunstable even though the I-V relation has a positive slope around it.Also notice that the y-axis starts at the negative value -0.1. However, the gatingvariable n represents the proportion (probability) of the K + channels in the open state,hence a value less than zero has no physical meaning. So while we can happily calculatethe nullclines for the negative n, and even start the trajectory with initial condition