Dynamical Systems in Neuroscience:

98 Two-Dimensional Systems(f(x(t),y(t)),g(x(t),y(t)))y(x(t),y(t))(f(x 0 ,y 0 ),g(x 0 ,y 0 ))(x 0 ,y 0 )xFigure 4.5: Solutions are trajectories tangent to the vector field.(c) Both V and n decrease: Both Na + and K + currents deactivate while V is small,leading to a refractory period.(d) V increases but n still decreases: Partial activation of Na + current combinedwith further deactivation of the residual K + current lead to a relative refractoryperiod, then to an excitable period, and possibly to another action potential.The intersection of the V - and n-nullclines in Fig. 4.4 is an equilibrium correspondingto the rest state. The number and location of equilibria might be difficult to infervia analysis of equations (4.1, 4.2), but it is a trivial geometrical exercise once the nullclinesare determined. Because nullclines are so useful and important in geometricalanalysis of dynamical systems, few scientists bother to plot vector fields. Followingthis tradition, we will not show vector fields in the rest of the book (except for thischapter). Instead, we plot nullclines and representative trajectories, which we discussnext.4.1.2 TrajectoriesA vector-function (x(t), y(t)) is a solution of the two-dimensional systemẋ = f(x, y) ,ẏ = g(x, y) ,starting with an initial condition (x(0), y(0)) = (x 0 , y 0 ) when dx(t)/dt = f(x(t), y(t))and dy(t)/dt = g(x(t), y(t)) at each t ≥ 0. This requirement has a simple geometricalinterpretation: A solution is a curve (x(t), y(t)) on the phase plane R 2 which is tangentto the vector field, as we illustrate in Fig 4.5. Such a curve is often called a trajectoryor an orbit.One can think of the vector field as a stationary flow of a fluid. Then a solution isjust a trajectory of a small particle dropped at a certain (initial) point and carried bythe flow. To study the flow, it is useful to drop a few particles and see where they are