Dynamical Systems in Neuroscience:

310 Simple ModelsLTS neuron (in vitro)simple modelI=300 pAspike peakresetdecreasing AHPsdecreasing amplitudesI=200 pAspike peak20 mV100 msresetv-nullclinev-nullcline, I=0u-nullclineI=125 pAspike peak-56 mVrestdampedoscillationrecovery, u3002001000attractiondomainfocusspikeI=100 pAspike peak-60 -40 -20 0 20 40membrane potential, v (mV)Figure 8.25: Comparison of in vitro recordings of a low threshold spiking (LTS) interneuron(rat’s barrel cortex, data provided by B. Connors) with simulations of thesimple model 100 ˙v = (v + 56)(v + 42) − u + I, ˙u = 0.03{8(v + 56) − u}, if v ≥ 40 − 0.1u,then v ← −53 + 0.04u, u ← u + 20.resonators, with phase portraits as in Fig. 8.15.A possible explanation for the subthreshold oscillations in LTS (and some RS)neurons is given in Fig. 8.13, case b > 0. The resting state is a stable node whenI = 0, but it becomes a stable focus when the magnitude of the injected current isnear the neuron’s rheobase. After firing a phasic spike, the trajectory spirals in tothe focus exhibiting damped oscillation. Its frequency is the imaginary part of thecomplex-conjugate eigenvalues of the equilibrium, and it is small because the systemis near Bogdanov-Takens bifurcation.A possible explanation for the rebound spike in LTS (or some RS) neurons is givenin Fig. 8.26. The shaded region is the attraction domain of the resting state (blackcircle), which is bounded by the stable manifold of the saddle (white circle). A sufficientlystrong hyperpolarized pulse moves the trajectory to the new, hyperpolarized