Dynamical Systems in Neuroscience:

358 Bursting0membrane potential, V (mV)-10-20-30-40-50-60-70V equiv (n slow )n slow (V)V equiv (n slow )0 0.05 0.1K + activation gate, nslowlimit cycle (max)unstable equilblimit cycle (min)n slow (V)saddlenode0 0.05 0.1K + activation gate, nslowFigure 9.16: Projection of bursting trajectory of the I Na,p +I K +I K(M) -model onto the(n slow , V ) plane.to bizarre bursters having amplifying slow currents, such as the one in Ex. 10.We depict the equivalent voltage of the I Na,p +I K +I K(M) -model in Fig. 9.16, left(variable u corresponds to n slow ). In the same figure, we depict the steady-state activationfunction n = n ∞,slow (V ) (notice the flipped coordinate system). We interpretthe two curves as fast and slow nullclines of the reduced (V, n slow )-system. During theactive (spiking) phase of bursting, the reduced system slides along the upper branch ofV equiv (n slow ) to the right. When it reaches the end of the branch, it falls down to thelower branch corresponding to resting, and slides along this branch to the left. When itreaches the left end of the lower branch, it jumps up to the upper branch, and therebycloses the hysteresis loop. Fig. 9.16, right, summarizes all the information needed tounderstand the transitions between resting and spiking states in this model. It depictsthe bursting trajectory, loci of equilibria of the fast subsystem, and the voltage range ofspiking limit cycle as a function of the slow gate n slow . With some experience, one canread this complicated figure and visualize the three-dimensional geometry underlyingbursting dynamics.9.2.5 Hysteresis loops and slow wavesSustained bursting activity of the fast-slow system (9.1) corresponds to periodic (orchaotic) activity of the reduced slow subsystem (9.6). Depending on the dimensionof u, i.e., on the number of slow variables, there could be two fundamentally differentways the slow subsystem oscillates.If the slow variable u is one-dimensional, then there must be a bistability of restingand spiking states of the fast subsystem so that u oscillates via a hysteresis loop. Thatis, the reduced equation (9.6) consists of two parts, one for V equiv (u) correspondingto spiking, and one for V equiv (u) corresponding to resting of the fast subsystem, as