Dynamical Systems in Neuroscience:

Chapter 4Two-Dimensional SystemsIn this chapter we introduce methods of phase plane analysis of two-dimensional systems.Most concepts will be illustrated using the I Na,p +I K -model in Fig. 4.1:leak I Linstantaneous I Na,p IC ˙V{ }} { { }} { { }} K{= I − g L (V −E L ) − g Na m ∞ (V ) (V −E Na ) − g K n (V −E K ) , (4.1)ṅ = (n ∞ (V ) − n)/τ(V ) , (4.2)having leak current I L , persistent Na + current I Na,p with instantaneous activation kineticand a relatively slower persistent K + current I K with either high (Fig. 4.1a) orlow (Fig. 4.1b) threshold (the two choices result in fundamentally different dynamics).The state of the I Na,p +I K -model is a two-dimensional vector (V, n) ∈ R 2 on the phaseplane R 2 . New types of equilibria, orbits, and bifurcations can exist on the phase planethat cannot exist on the phase line R. Many interesting features of single neuron dynamicscan be illustrated or explained using two-dimensional systems. Even neuronalbursting, which occurs in multi-dimensional systems, can be understood via bifurcationanalysis of two-dimensional systems.This model is equivalent in many respects to the well-known and widely usedI Ca +I K -model proposed by Morris and Lecar (1981) to describe voltage oscillationsin the barnacle giant muscle fiber.4.1 Planar Vector FieldsTwo-dimensional dynamical systems, also called planar systems, are often written inthe formẋ = f(x, y) ,ẏ = g(x, y) ,where the functions f and g describe the evolution of the two-dimensional state variable(x(t), y(t)). For any point (x 0 , y 0 ) on the phase plane the vector (f(x 0 , y 0 ), g(x 0 , y 0 ))93