Dynamical Systems in Neuroscience:

Chapter 3One-Dimensional SystemsIn this chapter we describe geometrical methods of analysis of one-dimensional dynamicalsystems, i.e., systems having only one variable. An example of such a system isthe space-clamped membrane having Ohmic leak current I LC ˙V = −g L (V − E L ) . (3.1)Here the membrane voltage V is a time-dependent variable, and the capacitance C, leakconductance g L and leak reverse potential E L are constant parameters described in theprevious chapter. We use this and other one-dimensional neural models to introduceand illustrate the most important concepts of dynamical system theory: equilibrium,stability, attractor, phase portrait, and bifurcation.3.1 Electrophysiological ExamplesThe Hodgkin-Huxley description of dynamics of membrane potential and voltage-gatedconductances can be reduced to a one-dimensional system when all transmembraneconductances have fast kinetics. For the sake of illustration, let us consider a spaceclampedmembrane having leak current and a fast voltage-gated current I fast havingonly one gating variable p,Leak I L IC ˙V{ }} { { }} fast{= − g L (V − E L ) − g p (V − E) (3.2)ṗ = (p ∞ (V ) − p)/τ(V ) (3.3)with dimensionless parameters C = 1, g L = 1, and g = 1. Suppose that the gatingkinetics (3.3) is much faster than the voltage kinetics (3.2), which means that thevoltage-sensitive time constant τ(V ) is very small, i.e. τ(V ) ≪ 1, in the entire biophysicalvoltage range. Then, the gating process may be treated as being instantaneous,and the asymptotic value p = p ∞ (V ) may be used in the voltage equation (3.2) to55