Dynamical Systems in Neuroscience:

Simple Models 309100080010 msrecovery variable, u600400200v-nullcline, I=550 pAv-nullcline, I=0 pAAHP5432112 3 4 5AHP0u-nullclinespike-200-70 -60 -50 -40 -30 -20 -10 0 10 20membrane potential, v (mV)Figure 8.24: Phase portrait of the simple model in Fig. 8.23 exhibiting CH firingpattern.current. There are some subtle differences: The response of an LTS cell to a weak depolarizingcurrent consists of a phasic spike or a doublet with a relatively short latencyfollowed by low-frequency (less than 10 Hz) subthreshold oscillation of membrane potential.Stronger pulses elicit tonic spikes with slow frequency adaptation, decreasingamplitudes and decreasing after-hyperpolarizations, as one can see in Fig. 8.11.LTS neurons have more depolarized resting potentials, lower threshold potentials,and lower input resistances than those of RS neurons. To match the in vitro firingpatterns of LTS interneuron of rat’s barrel cortex in Fig. 8.25, we take the simple modelof RS neuron and adjust the resting and instantaneous threshold potentials v r = −56mV and v t = −42 mV, and the values p = 1 and b = 8 resulting in the rheobasecurrent of 120 pA and the input resistance of 50 MΩ. To model the decreasing natureof the spike and AHP amplitudes, we assume that the peak of the spike and the afterspikeresetting point depend on the value of the recovery variable u. This completelyunnecessary cosmetic adjustment has a mild effect on the quantitative behavior of themodel but renders a more “realistic” look to the simulated voltage traces in Fig. 8.25.The class of excitability of LTS neurons has not been studied systematically, thoughthe neurons seem to be able to fire periodic spike trains with a frequency as low as thatof RS neurons (Beierlein et al. 2003, Tateno and Robinson, personal communication).The conjecture that they are near saddle-node on invariant circle bifurcation, andhence are Class 1 excitable integrators, seems to be at odds with the observation thattheir membrane potential exhibits slow damped oscillation and that they can fire postinhibitoryrebound spikes (Bacci et al. 2003), called low-threshold spikes (hence thename). They are better characterized as being at the transition from integrators to