Dynamical Systems in Neuroscience:

314 Simple Models(1) fast oscillations resulting from the interplay of amplifying and resonant currents,and (2) slow ramp resulting from the slow kinetic of an amplifying variable, such asslow inactivation of an outward current (e.g., K + A-current) or slow activation of aninward current, or both. In addition, the ramp could result from the slow charging ofthe dendritic compartment of the neuron.The exact mechanism responsible for the slow ramp in LS neurons is not known atpresent. Fortunately, we do not need to know the mechanism to simulate LS neuronsusing the simple model approach. Indeed, simple models with passive dendrites areequivalent to simple models with linear amplifying currents. For example, the model inFig. 8.28 consists of a 2-dimensional system (v, u) responsible for the spike-generationmechanism at the soma and a linear equation for the passive dendritic compartmentv d .When stimulated with the threshold current, i.e., just above the neuronal rheobase,LS neurons often exhibit stuttering behavior seen in Fig. 8.28, middle. Subthresholdoscillations, voltage ramps, and stuttering are consistent with the following geometricalpicture: Abrupt onset of stimulation evokes a transient spike followed by brief hyperpolarizationand then sustained depolarization. While depolarized, the fast subsystemaffects the slow subsystem, e.g., slowly charges the dendritic tree or slowly inactivatesthe K + current. In any case, there is a slow decrease of the outward current, or equivalently,slow increase of the inward current that drives the fast subsystem throughthe subcritical Andronov-Hopf bifurcation. Because of the co-existence of resting andspiking states near the bifurcation, the neuron can be switched from one state to theother by the membrane noise. Once the bifurcation is passed, the neuron is in thetonic spiking mode. Overall, LS neurons can be thought of as being FS neurons with aslow subsystem that damps any abrupt changes, delays the onset of spiking, and slowsdown its frequency.8.2.8 Diversity of inhibitory interneuronsIn contrast to excitatory neocortical pyramidal neurons, which have stereotypical morphologicaland electrophysiological classes (RS, IB, CH), inhibitory neocortical interneuronshave wildly diverse classes with various firing patters that cannot be classifiedas FS, LTS, or LS. Markram et al. (2004) reviewed recent results on the relationshipbetween electrophysiology, pharmacology, immunohistochemistry and gene-expressionpatterns of inhibitory interneurons. An extreme interpretation of their findings is thatthere is a continuum of different classes of interneurons rather than a set of 3 classes.Figure 8.29 summarizes five of the most ubiquitous groups in the continuum:• (NAC) non-accommodating interneurons fire repetitively without frequency adaptationin response to a wide range of sustained somatic current injections. ManyFS and LS neurons are of this type.• (AC) accommodating interneurons fire repetitively with frequency adaptationand therefore do not reach high firing rates of NAC neurons. Some FS and LS