Classification of bursting patterns: A tale of two ducks
Fig 1
Ducks at the transition from rest to spikes in the FitzHugh–Nagumo model.
The dynamics are represented in the phase plane (upper panels) and as illustrative time courses (lower panels). When increasing the applied current I, the system’s dynamics transitions from a stable equilibrium (rest state, black dot in (a)), which is excitable, to a strongly attracting limit cycle (repetitive spiking state, in blue in (b)). The excitable structure in panel (a) is further illustrated with 2 trajectories whose initial conditions are at the stable equilibrium (rest state) and with a step current of slightly different amplitude: One trajectory (in red) stays below threshold, while the other one (in purple) crosses threshold, fires an action potential, and then returns back to the rest state. The transition from (a) to (b) is continuous but confined to a very small range of I values around IT. At I ∼ IT, special solutions called ducks emerge (c), which stay close to the unstable part of the V-nullcline, shown as a cubic curve whose attracting branches (resp. repulsive branch) are displayed as solid (resp. dashed) black lines. Two ducks shown in red stay below threshold and correspond to subthreshold oscillations, while one (in purple) crosses threshold and corresponds to a near-threshold spiking solution. Also shown in the top row is the w linear nullcline and the equilibrium point (filled circled when stable, open circle when unstable) at the intersection between the 2 nullclines. Single (double) arrows represent fast (slow) flow. Notice that the purple trajectory in panel (c) has the shape of a leftward-directed duck’s profile; see top-right inset. The bottom row shows the time courses of the membrane potential V for all trajectories displayed on the top row, keeping the same colors; for panel (c), only the time series of the largest red cycle and of the purple one are shown. Equations and parameter values are given in S1 Text.