Dynamical Systems in Neuroscience:

Excitability 269(a)current, I3002001000-100-200I (V)I fast (V)I 0 (V)-300-100 -50 0 50membrane potential, V (mV)(b)current, I2000150010005000-500current response to voltage-clamp stepsI 0 (V)I fast (V)I (V)-2 -1 0 1 2 3time, (logarithmic scale, log 10 ms)(c)current, I10005000instantaneous I-V (d) fast I-V relation (e) steady-state I-V15002000I 0 (V)1000current, I5000I fast (V)current, I150010005000I (V)-5000 0.1 0.2time (ms)-5000 10 20time (ms)-5000 500 1000time (ms)Figure 7.46: (a) The I Na,p +I K +I K(M) -model in Fig. 7.45a has three I-V relations: InstantaneousI 0 (V ) = I leak (V ) + I Na,p (V ) describes spike upstroke dynamics. The curveI fast (V ) = I 0 (V ) + I K (V ) is the I-V relation of the fast I Na,p +I K -subsystem responsiblefor spike-generating mechanism. The curve I ∞ (V ) = I fast (V ) + I K(M) (V ) is thesteady-state (asymptotic) I-V relation of the full model. Dots denote values obtainedfrom a simulated voltage-clamp experiment in (b); notice the logarithmic time scale.Magnifications of current responses are shown in (c,d,e). Simulated time constants:τ Na,p (V ) = 0.01 ms, τ K (V ) = 1 ms, τ M (V ) = 100 ms.experiment, depicted in Fig. 7.46b. We hold the model at V = −70 mV and stepthe command voltage to various values. The values of the current, taken at t = 0.05ms, t = 5 ms, and t = 500 ms in Fig. 7.46b, result in the instantaneous, fast, andsteady-state I-V curves, respectively. Notice that the data in Fig. 7.46b are plottedon the logarithmic time scale. Various magnifications using the linear time scale aredepicted in Fig. 7.46c,d, and e. Numerically obtained values of the three I-V relationsare depicted as dots in Fig. 7.46a. They approximate the theoretical values quite wellbecause there is a 100-fold separation of time scales in the model.