Dynamical Systems in Neuroscience:

60 One-Dimensional Systems40bistability (I=0)excited40monostability (I=60)excitedmembrane potential, V (mV)200-20-40V(t)restingmembrane potential, V (mV)200-20-40V(t)-600 1 2 3 4 5time (ms)a-600 1 2 3 4 5time (ms)bFigure 3.6: Typical voltage trajectories of the I Na,p -model (3.5) having different valuesof I.membrane potential, VV 0V(0)V(h)V(2h)V(3h)V(t)=E L+(V 0-E L)e -g L t/CE Ltime, tFigure 3.7: Explicit analytical solution (V (t) = E L + (V 0 − E L )e −g Lt/C ) of the linearequation (3.1) and corresponding numerical approximation (dots) using Euler’s method(3.7).solid curve, is an explicit analytical solution to the linear equation (3.1) (check bydifferentiating).Finding explicit solutions is often impossible even for such simple systems as (3.5),so quantitative analysis is carried out mostly via numerical simulations. The simplestprocedure to solve (3.6) numerically, known as first-order Euler method, replaces (3.6)by the discretized system[V (t + h) − V (t)]/h = F (V (t))where t = 0, h, 2h, 3h, . . . , is the discrete time and h is a small time step. Knowing thecurrent state V (t), we can find the next state point viaV (t + h) = V (t) + hF (V (t)) . (3.7)Iterating this difference equation starting with V (0) = V 0 , we can approximate the