Dynamical Systems in Neuroscience:

344 Burstingmembranepotential (mV)-20-40-60I(t)I(t)slow K + activationgate, n slow0.060.040.0200 20 40time (ms)a00 50 100 150 200time (ms)bFigure 9.4: Intrinsic bursting in the I Na,p +I K +I K(M) -model (7.1) consisting of theI Na,p +I K -model with parameters as in Fig. 4.1a and fast K + current (g K = 9, τ(V ) =0.152) and a slow K + current with g slow = 5, V 1/2 = −20 mV, k = 5 mV andτ slow (V ) = 20 ms. (a) Burst excitability when I = 0. (b) Periodic bursting whenI = 5.there is a coexistence of the resting and spiking states. The brief pulse of currentexcites the neuron, i.e., moves its state into the attraction domain of the spiking limitcycle and initiates periodic activity. Without any other modification, the model wouldproduce an infinite spike train. To stop the train, we added a slower high-thresholdpersistent K + current similar to I K(M) that provides a negative feedback. This M-current is deactivated at rest. However, during the active (spiking) phase, the currentslowly activates, as indicated by the slow build-up of its gating variable n slow in thefigure. The neuron becomes less and less excitable, and eventually cannot sustainspiking activity. If, instead of a pulse of current, a constant current is applied, theneuron can burst periodically, as in Fig. 9.4b.This model presents only one of many possible examples of bursters, which we studyin this chapter. However, it illustrates a number of important issues common to allbursters. For instance, in contrast to the forced bursting in Fig. 9.3, this bursting isintrinsic or autonomous. This stereotypical bursting pattern results from the intrinsicvoltage-sensitive currents, and not from a time-dependent input. The behavior inFig. 9.4a is called burst excitability to emphasize that the model is an excitable systemwith the exception that superthreshold stimulation elicits a burst of spikes instead of asingle spike. Hippocampal pyramidal neurons of the type “grade III bursters” depictedin Fig. 8.34Eb exhibit burst excitability.Biologists sometimes refer to the bursting in Fig. 9.4b as being conditional, becauserepetitive bursting occurs when a certain condition is satisfied, e.g., positive I is in-