Dynamical Systems in Neuroscience:

Simple Models 311membrane potential, v (mV)40200-20-40resting-60-80I=0I=0I=-1 nAinjected currentAsagB0 100 200 300time (ms)CDhyperpolarized staterestingrecovery variable, u2001000-100-200v-nullcline, I=-1 nAsagBv-nullcline, I=0ADresetresting-80 -60 -40 -20membrane potential, v (mV)Chyperpolarized statesaddlereboundspikeFigure 8.26: The mechanism of sag and rebound spike of the model in Fig. 8.25.equilibrium (black square), which is outside the attraction domain. Upon release fromthe hyperpolarization, the trajectory fires a phasic spike and then returns to the restingstate. Some LTS interneurons fire bursts of spikes, and for that reason are calledburst-spiking non-pyramidal (BSNP) neurons.8.2.6 Fast spiking (FS) interneuronsFast spiking neurons fire “fast” tonic spike trains of relatively constant amplitudeand frequency in response to depolarized pulses of current. In a systematic study,Tateno et al. (2004) have shown that FS neurons have Class 2 excitability in the sensethat their frequency-current (F-I) relation has a discontinuity around 20 Hz. Whenstimulated with barely superthreshold current, such neurons exhibit irregular firingrandomly switching between spiking and fast subthreshold oscillatory mode (Kubotaand Kawaguchi 1999, Tateno et al. 2004).The absence of spike frequency adaptation in FS neurons is mostly due to the fastK + current that activates during the spike, produces deep AHP, completely deinactivatesNa + current, and thereby facilitates the generation of the next spike. Blockingthe K + current by TEA (Erisir et al. 1999) removes AHP, leaves residual inactivationof the Na + current and slows down the spiking, essentially transforming the FS firingpattern into LTS.The existence of fast subthreshold oscillations of membrane potential suggest thatthe resting state of the FS neurons is near Andronov-Hopf bifurcation. Stutteringbehavior at the threshold currents points to the co-existence of resting and spikingstates, as in Fig. 8.16, and suggests that the bifurcation is of the subcritical type.However, FS neurons do not fire post-inhibitory (rebound) spikes — the feature usedto distinguish them experimentally from LTS types. Thus, we cannot use the simplemodel (8.5, 8.6) in its present form to simulate FS neurons because the model withlinear slow nullcline would fire rebound spikes according to the mechanism depicted inFig. 8.26. In addition, the simple model has a non-monotone I-V relation, whereas FS