Dynamical Systems in Neuroscience:

312 Simple ModelsFS neuron (in vitro)simple model2 31I=400 pA2 3121I=200 pAAHP1 2AHPrecovery variable, u4003002001000I=100 pA-60 -50 -40 -30 -20 -10membrane potenrial, v (mV)20 mV40 ms-55 mVresetrecovery variable, u4020spiking limitv-nullclinecycleu-nullclineI=73.2 pAspike0rest-55 -50 -45 -40membrane potenrial, v (mV)Figure 8.27: Comparison of in vitro recordings of a fast spiking (FS) interneuron of layer5 rat’s visual cortex with simulations of the simple model 20 ˙v = (v +55)(v +40)−u+I,˙u = 0.2{U(v) − u}, if v ≥ 25, then v ← −45 mV. Slow nonlinear nullcline U(v) = 0when v < v b and U(v) = 0.025(v − v b ) 3 when v ≥ v b with v b = −55 mV. Shaded areadenotes the attraction domain of the resting state.neurons have monotone relation.The absence of rebound responses in FS neurons means that the phenomenologicalrecovery variable (activation of fast K + current) does not decrease significantly belowthe resting value when the membrane potential is hyperpolarized. That is, the slowu-nullcline becomes horizontal in the hyperpolarized voltage range. Accordingly, wesimulate FS neuron in Fig. 8.27 using the simple model (8.5) with nonlinear u-nullcline.The phase portraits and bifurcation diagram of the FS neuron model are qualitativelysimilar to the fast subsystem of a “subHopf/fold cycle” burster: Injection ofdc-current I creates a stable and unstable limit cycles via fold limit cycle bifurcation.The frequency of the newborn stable cycle is around 20 Hz, hence the discontinuityof the F-I curve and Class 2 excitability. There is a bistability of resting and spikingstates, as in Fig. 8.27, bottom, so that noise can switch the state of the neuron back