Dynamical Systems in Neuroscience:

296 Simple Modelslayer 5 neuronsimple modelI=100 pAI=85 pAI=70 pAinput+35 mV200 ms-60 mVinputresetAHPrecovery, u100500-50restresetAHPI>0I=0I=60 pA-60 -40 -20 0 20membrane potential, v (mV)spike35Figure 8.12: Comparison of in vitro recordings of a regular spiking (RS) pyramidalneurons with simulations of the simple model 100 ˙v = 0.7(v + 60)(v + 40) − u + I,˙u = 0.03{−2(v + 60) − u}, if v ≥ +35, then v ← −50, u ← u + 100.responses to positive pulses, sags and rebound spikes to negative pulses (as in Fig. 7.48),relatively short latencies of the first spike, and other resonance phenomena. The evenmore extreme example in Fig. 7.42 shows a pyramidal neuron executing a subthresholdoscillation before switching to a tonic spiking mode.The difference between the types in Fig. 8.13 can be explained by the sign of theparameter b in the simple model (8.5, 8.6), which depends on the relative contributionsof amplifying and resonant slow currents and gating variables. When b < 0 (or b ≈ 0,e.g., b = 0.5 in Fig. 8.14), the neuron is a pure integrator near saddle-node on invariantcircle bifurcation. Greater values of b > 0 put the model near the transition from anintegrator to a resonator the via co-dimension-2 Bogdanov-Takens bifurcation studiedin Sect. 6.3.3 and 7.2.11.The sequence of bifurcations when b > 0 is depicted in Fig. 8.15. Injection ofdepolarizing current below the neuron’s rheobase transforms the resting state intoa stable focus and results in damped oscillations of the membrane potential. The