Dynamical Systems in Neuroscience:

One-Dimensional Systems 57currentsinwardoutwardgatingactivation, minactivation, hI Na,pnegativeconductanceI hI(V)I(V)VVI KI KirI(V)I(V)VnegativeconductanceVFigure 3.3: Typical currentvoltage(I-V) relations of the fourcurrents considered in this chapter.Shaded boxes correspond tonon-monotonic I-V relations havinga region of negative conductance(I ′ (V ) < 0) in the biophysicallyrelevant voltage range.using any dynamical systems theory. The models might also appear too simple tomathematicians, who can easily understand their dynamics just by looking at thegraphs of the right-hand side of (3.4) without using any electrophysiological intuition.In fact, the models provide an invaluable learning tool, since they establish a bridgebetween electrophysiology and dynamical systems.In Fig. 3.3 we plot typical steady-state current-voltage (I-V) relations of the fourcurrents considered above. Notice that the I-V curve is non-monotonic for I Na,p andI Kir but monotonic for I K and I h , at least in the biophysically relevant voltage range.This subtle difference is an indication of the fundamentally different roles these currentsplay in neuron dynamics: The I-V relation in the first group has a region of “negativeconductance”, i.e., I ′ (V ) < 0, which creates positive feedback between the voltage andthe gating variable (Fig. 3.4), and plays an amplifying role in neuron dynamics. Werefer to such currents as amplifying currents. In contrast, the currents in the secondgroup have negative feedback between voltage and gating variable, and they often resultin damped oscillation of the membrane potential, as we show in the next chapter. Werefer to such currents as resonant currents. Most neural models involve a combinationof at least one amplifying and one resonant current, as we discuss in Chap. 5. Theway these currents are combined determines whether the neuron is an integrator or aresonator.3.1.2 Leak + instantaneous I Na,pTo ease our introduction into dynamical systems, we will use the I Na,p -modelC ˙V = I − g L (V − E L ) −instantaneous I Na,p{ }} {g Na m ∞ (V ) (V − E Na ) (3.5)withm ∞ (V ) = 1/(1 + exp {(V 1/2 − V )/k})