Dynamical Systems in Neuroscience:

372 Bursting(a)membrane potential, V (mV)(b)activation gate0-20-40-60-800 100 200 300 400time (ms)0.250.20.150.10.05n slow (t)m slow (t)00 100 200 300 400time (ms)(c)K + activation gate, nslow(d)membrane potential, V (mV)-20-40-60-800.120.10.080.060.040.0220000 0.05 0.1 0.15 0.2 0.25Na + activation gate, m slow0.1restingK + activation gate, n slow0.05restingspiking00spiking0.2Na + activationgate, m slowFigure 9.33: “Circle/circle” bursting in the I Na,p +I K +I Na,slow +I K(M) -model. Parametersof the fast I Na,p +I K -subsystem are the same as in Fig. 4.1a with I = 5. Slow Na +current has V 1/2 = −40 mV, k = 5 mV, g Na,slow = 3, τ Na,slow (V ) = 20 ms. Slow K +current has V 1/2 = −20 mV, k = 5 mV, g K(M) = 20, τ K(M) (V ) = 50 ms.having parameters as in Fig. 9.33. Notice that these equations are uncoupled and hencecannot oscillate by themselves without the feedback from variable V .Let us describe the bursting mechanism in the full I Na,p +I K +I Na,slow + I K(M) -modelwith I = 5. Since I > 4.51, the resting state of the fast subsystem does not exist,and the model generates action potentials, depicted in Fig. 9.33a. Each spike activatesI Na,slow , produces even more inward current and hence more spikes. This, however,activates a much slower K + current, see Fig. 9.33b, and produces a net outward currentthat moves the fast nullcline down and eventually terminates spiking. The transitionfrom spiking to resting occurs via saddle-node on invariant circle bifurcation. While atrest, both currents deactivate and the fast nullcline slowly moves up. The net inwardcurrent, consisting mostly of the injected dc-current I = 5, drives the fast subsystemvia the same saddle-node on invariant circle bifurcation and initiates another burst, asshown in Fig. 9.33a.