Dynamical Systems in Neuroscience:

320 Simple Modelsv r = −80 mV, and we set v t = −25 mV, p = 1, and b = −20 to get 30 MΩ inputresistance and 300 pA rheobase current. We take a = 0.01 to reflect the slow inactivationof the K + A-current in the subthreshold voltage range. The membrane potentialin the figure reaches the peak of +40 mV during the spike and then resets to −55 mVor lower, depending on the firing frequency. The value d = 150 provides a reasonablematch of the interspike frequencies for all magnitudes of injected current. Notice thatb < 0, so that u represents either slow inactivation of I A or slow charging of the passivedendritic compartment, or both. In any case, it is a slow amplifying variable, which isconsistent with the observation that spiny neurons do not “sag” in response to hyperpolarizingcurrent pulses, do not “peak” in response to depolarizing pulses (Nisenbaumet al. 1994), and do not generate rebound (post-inhibitory) spikes.Injection of a depolarizing current shifts the v-nullcline of the simple model up, andthe resting state disappears via saddle-node bifurcation. The trajectory slowly movesthrough the ghost of the bifurcation point (shaded rectangle in the figure), resultingin the long latency to the first spike. The spike resets the trajectory to a point (whitesquare) below the ghost, resulting in significantly smaller delays to subsequent spikes.Because the resetting point is so close to the saddle-node bifurcation point, the simplemodel, and probably the spiny projection neuron in the figure, is near the co-dimension-2 saddle-node homoclinic orbit bifurcation discussed in Sect. 6.3.6.8.4.3 Mesencephalic V neurons of brainstemThe best examples of resonators, with fast subthreshold oscillations, Class 2 excitability,rebound spikes, etc., are mesencephalic V (mesV) neurons of brainstem (Wu et al.2001) and primary sensory neurons of dorsal root ganglion (Amir et al. 2002, Jian etal. 2004). Mes V neurons of the brainstem have monotone I-V curves, whereas thesimple model with linear equation for u does not. In Fig. 8.38 we use a modificationof the simple model to simulate the responses of mesV neuron (data from Fig. 7.3) topulses of depolarizing current.The model’s phase portrait is qualitatively similar to that of the FS interneuronsin Fig. 8.27. The resting state is a stable focus, resulting in damped or noise-inducedsustained oscillations of the membrane potential. Their amplitude and frequency dependon I and could be larger than 5 mV and 100 Hz, respectively. The focus losesstability via subcritical Andronov-Hopf bifurcation. Because of the co-existence of theresting and spiking states, the mesV neuron can burst, and so can the simple model ifnoise or a slow resonant variable is added.8.4.4 Stellate cells of entorhinal cortexThe entorhinal cortex occupies a privileged anatomical position that allows it to gatethe main flow of information to and out of the hippocampus. In vitro studies show thatstellate cells, a major class of neurons in entorhinal cortex, exhibit intrinsic subthresholdoscillations with a slow dynamics of the kind shown in Fig. 8.39b (Alonso and Llinas