Bursting behavior is very common in the brain, where neurons slowly oscillate between active spiking phases and states of quiescence. Today we examine a very common mechanism for bursting in neurons using the Morris-Lecar model. A calcium-dependent potassium current activates when the cell is spiking to cause an outward current that hyperpolarizes the membrane and ultimately causes spiking to cease. This current activates much slower than the spiking system for the model (which we study in the V, W phase plane). Therefore, several spikes can be generated before the calcium-dependent potassium current sufficiently activates and causes burst termination. Likeswise, in the quiescent state, the current deactivates because no more calcium is flowing in to the cell. However, the current again is subject to its own slow kinetics and thus takes a significant amount of time to turn itself off again. But as it does turn off slowly, the membrane gradually depolarizes and ultimately begins to spike again, renewing the bursting cycle.

Today we will study how the slow dynamics of the bursting system cause the Morris-Lecar model to trace out a hysteresis loop through its bifurcation diagram, when the system is bistable and a branch of stable limit cycles co-exists with the branch of stable nodes.