Dynamical Systems in Neuroscience:

Conductance-Based Models 1615.2.4 Reduction to simple modelAll models discussed in this chapter can be reduced to two-dimensional systems havinga fast voltage variable, V , and a slower “recovery” variable, u, with N-shaped andsigmoidal nullclines, respectively. The decision to fire or not to fire is made at theresting state, which is the intersection of the nullclines near the left knee, as we illustratein Fig. 5.23a. To model the subthreshold behavior of such neurons and the initialsegment of the up-stroke of an action potential, we need to consider only a smallneighborhood of the left knee confined to the shaded square in Fig. 5.23. The rest ofthe phase space is needed only to model the peak and the down-stroke of the actionpotential. If the shape of the action potential is less important than the subthresholddynamics leading to this action potential, then we can retain detailed informationabout the left knee and its neighborhood and simplify the vector field outside theneighborhood. This approach results in a simple model capable to exhibit quite realisticdynamics, as we see in Chap. 8.Derivation via nullclinesThe fast nullcline in Fig. 5.23b can be approximated by the quadratic parabolau = u min + p(V − V min ) 2 ,where (V min , u min ) is the location of the minimum on the left knee, and p ≥ 0 is ascaling coefficient. Similarly, the slow nullcline can be approximated by the straightlineu = s(V − V 0 ) ,where s is the slope and V 0 is the V -intercept. All these parameters can easily bedetermined geometrically or analytically.Using these nullclines, we approximate the dynamics in the shaded region in Fig. 5.23by the system˙V = τ f{p(V − Vmin ) 2 − (u − u min ) } ,˙u = τ s {s(V − V 0 ) − u} ,where the parameters τ f and τ s describe the fast and slow time scales. Because ofthe term (V − V min ) 2 , the variable V can escape to infinity in a finite time. Thiscorresponds to the firing of an action potential, more precisely, to its upstroke. Tomodel the downstroke, we assume that V max is the peak value of the action potential,and we reset the state of the system(V, u) ← (V reset , u + u reset ) , when V = V max ,as if the spiking trajectory disappears at the right edge and appears at the left edgein Fig. 5.23b. Here V reset and u reset are parameters. Appropriate re-scaling of variables