Dynamical Systems in Neuroscience:

126 Two-Dimensional SystemsReview of Important Concepts• A two-dimensional system of differential equationsẋ = f(x, y)ẏ = g(x, y) ,describes joint evolution of state variables x and y, which often are the membranevoltage and a “recovery” variable.• Solutions of the system are trajectories on the phase plane R 2 that are tangentto the vector field (f, g).• The sets given by the equations f(x, y) = 0 and g(x, y) = 0 are the x- andy-nullclines, respectively, where trajectories change their x and y directions.• Intersections of the nullclines are equilibria of the system.• Periodic dynamics correspond to closed loop trajectories.• Some special trajectories, e.g., separatrices, define thresholds and separateattraction domains.• An equilibrium or a periodic trajectory is stable if all nearby trajectories areattracted to it.• To determine the stability of an equilibrium, one needs to consider the Jacobianmatrix of partial derivatives( )fx fL =y.g x g y• The equilibrium is stable when both eigenvalues of L are negative or havenegative real parts.• The equilibrium is a saddle, a node, or a focus, when L has real eigenvaluesof opposite signs, of the same signs, or complex-conjugate eigenvalues,respectively.• When the equilibrium undergoes a saddle-node bifurcation, one of the eigenvaluesbecomes zero.• When the equilibrium undergoes an Andronov-Hopf bifurcation (birth or deathof a small periodic trajectory) the complex-conjugate eigenvalues becomepurely imaginary.• The saddle-node and Andronov-Hopf bifurcations are ubiquitous in neuralmodels, and they result in different neuro-computational properties.