Dynamical Systems in Neuroscience:

Bursting 35300−10−10membrane potential, V (mV)−20−30−40−50−60membrane potential, V (mV)−20−30−40−50−60−70−70−80−0.0210 0.02 fast K + 0.5activation, n0.04 0.06 0.08 0−80−0.0210 0.50.02 fast K + activation, n0.04 0.06 0.08 0slow K + activation, n slowslow K + activation, n slowFigure 9.12: Stereoscopic image of a bursting trajectory of the I Na,p +I K +I K(M) -modelin the three-dimensional phase space (V, n, n slow ) (for cross-eye viewing).the value of n slow . Gluing the phase portraits together, we see that there is a manifoldof limit cycle attractors (shaded cylinder) that starts when n slow < 0 and ends in asaddle homoclinic orbit bifurcation when n slow = 0.066. There is also a locus of stableand unstable equilibria that appears via a saddle-node bifurcation when n slow = 0.0033.Once we understand the transitions from one phase portrait to another as the slowvariable changes, we can understand the geometry of the burster. Suppose µ > 0 (i.e.,τ slow (V ) = 20 ms) so that n slow can evolve according to its gating equation.Let us start with the membrane potential at the stable equilibrium correspondingto resting state. The parameters of the I Na,p +I K +I K(M) -model (see caption to Fig. 9.4)are such that slow K + M-current deactivates at rest, i.e., n slow slowly decreases, andthe bursting trajectory slides along the bold half-parabola corresponding to the locusof stable equilibria. After a while, the K + current becomes so small, that it cannot holdthe membrane potential at rest. This happens when n slow passes the value 0.0033, thestable equilibrium coalesces with an unstable equilibrium (saddle), and they annihilateeach other via saddle-node bifurcation. Since the resting state no longer exists (see thephase portrait at the top left of Fig. 9.13) the trajectory jumps up to the stable limitcycle corresponding to repetitive spiking. This jumping corresponds to the transitionfrom resting to spiking behavior.While the fast subsystem fires spikes, the K + M-current slowly activates, i.e., n slowslowly increases. The bursting trajectory winds up around the cylinder correspondingto the manifold of limit cycles. Each rotation corresponds to firing a spike. After the9th spike in the figure, the K + current becomes so large that repetitive spiking cannotbe sustained. This happens when n slow passes the value 0.066, the limit cycle becomes ahomoclinic orbit to a saddle, and then disappears. The bursting trajectory jumps down