Dynamical Systems in Neuroscience:

268 Excitability(a)150100I K(M)(b)300200I fast (V)=I L +I Na,p +I Kcurrent, I500-50I (V)current, I1000-100I (V)-100I fast (V)=I L +I Na,p +I K-200-150-100 -50 0membrane potential, V (mV)-300-100 -50 0 50membrane potential, V (mV)Figure 7.45: Slow conductances can mask the true I-V relation of the spike-generatingmechanism. (a) The I Na,p +I K -model with parameters as in Fig. 4.1a has a nonmonotonicI-V curve I fast (V ). Addition of the slow K + M-current with parametersas in Sect. 2.3.5 and g M = 5 (dashed curve) makes the asymptotic I-V relation, I ∞ (V ),of the full I Na,p +I K +I K(M) -model monotonic. (b) Addition of a slow inactivation gateto the K + current of the I Na,p +I K -model with parameters as in Fig. 4.1b results in anon-monotonic asymptotic I-V relation of the full I Na,p +I A -model.gate to the persistent K + current, effectively transforming it into transient A-current.If the inactivation kinetics is sufficiently slow, the I Na,p +I A -model retains resonatorproperties on the millisecond time scale, i.e., on the time scale of individual spikes.However, its asymptotic I-V relation, depicted in Fig. 7.45b, becomes non-monotonic.Besides spike-frequency acceleration, the model acquires another interesting property— bistability. A single spike does not inactivate I A significantly. A burst of spikescould inactivate the K + A-current to such a degree that repetitive spiking becomessustained. Slow inactivation of the A-current is believed to facilitate the transitionfrom down- to up-states in neocortical and neostriatal neurons.When a neuronal model consists of conductances operating on drastically differenttime scales, it has multiple I-V relations, one for each time scale. We illustrate thisphenomenon in Fig. 7.46 using the I Na,p +I K +I K(M) -model with activation time constantof 0.01 ms for I Na,p , 1 ms for I K , and 100 ms for I K(M) . The up-stroke of an actionpotential is described only by leak and persistent Na + currents, since the K + currentsdo not have enough time to activate during such a short event. During the up-stroke,the model can be reduced to a one-dimensional system (see Chap. 3) with instantaneousI-V relation I 0 (V ) = I leak + I Na,p (V ) depicted in Fig. 7.46a. The dynamics during andimmediately after the action potential is described by the fast I Na,p +I K -subsystemwith its I-V relation I fast (V ) = I 0 (V ) + I K (V ). Finally, the asymptotic I-V relation,I ∞ (V ) = I fast (V ) + I K(M) (V ), takes into account all currents in the model.The three I-V relations determine the fast, medium, and asymptotic behavior ofa neuron in a voltage-clamp experiment. If the time scales are well separated (theyare in Fig. 7.46), all three I-V relations can be measured from a simple voltage-clamp