Dynamical Systems in Neuroscience:

Simple Models 317figure are v r = −60 mV and v t = −50 mV. The value p = 1.6 results in 40 pA rheobasecurrent and 60 MΩ input resistance, and the membrane capacitance C = 200 pF givesthe right current-frequency (F-I) relationship. Thus, in the tonic mode, our model isessentially the quadratic integrate-and-fire neuron 200 ˙v = 1.6(v + 60)(v + 50) + I withthe after-spike reset from +35 mV to −60 mV.To model slow Ca 2+ dynamics in the burst mode, we assume that hyperpolarizationsbelow the Ca 2+ inactivation threshold of −65 mV decrease u, thereby creating inwardcurrent. In the linear case, we take ˙u = 0.01{b(v + 65) − u} with b = 0 when v ≥ −65and b = 15 when v < −65, resulting in the piecewise linear u-nullcline depicted inFig. 8.31, bottom. Prolonged hyperpolarization below −65 mV decreases u and movesthe trajectory outside the attraction domain of the resting state (shaded region in thefigure). Upon release from the hyperpolarization, the model fires a rebound burst ofspikes, variable u → 0 reflecting inactivation of Ca 2+ , and the trajectory reenters theattraction domain of the resting state. Steps of depolarized current produce reboundbursts followed by tonic spiking with adapting frequency. A better quantitative agreementwith TC recordings can be achieved when two slow variables, u 1 and u 2 , areused.8.3.2 Reticular thalamic nucleus (RTN) neuronsReticular thalamic nucleus (RTN) neurons provide reciprocal inhibition to TC relayneurons. RTN and TC cells are similar in the sense that they have two firing modes,illustrated in Fig. 8.32: They fire trains of single spikes following stimulation fromresting or depolarized potentials in the tonic mode, and rebound bursts upon releasefrom hyperpolarized potentials in the burst mode.The parameters of the simple model in Fig. 8.32 are adjusted to match the in vitrorecording of the RTN cell in the figure, and they differ from the parameters of the TCmodel cell. Nevertheless, the mechanism of rebound bursting of RTN neuron is thesame as that of the TC neuron in Fig. 8.31, bottom. In contrast, the tonic mode of firingis different. Since b > 0, the model neuron is near the transition from an integratorto a resonator; It can fire transient spikes followed by slow subthreshold oscillationsof membrane potential; It has co-existence of stable resting and spiking states, withthe bifurcation diagram similar to the one in Fig. 8.15, and it can stutter and produceclustered spikes when stimulated with barely threshold current. Interestingly, similarbehavior of TC neurons was reported by Pirchio et al. (1997), Pedroarena and Llinas(1997) and Li et al. (2003). We will return to the issue of subthreshold oscillations andstuttering spiking when we consider stellate cells of entorhinal cortex in Sect. 8.4.4.8.3.3 Thalamic interneuronsIn contrast to TC and RTN neurons, thalamic interneurons do not have a prominentburst mode, though they can fire rebound spikes upon release from hyperpolarization(Pape and McCormick 1995). They have action potentials with short duration, and