Dynamical Systems in Neuroscience:

242 ExcitabilityThe existence of fast subthreshold oscillatory potentials is a distinguishable featureof neurons near Andronov-Hopf bifurcation. Indeed, the resting state of such a neuronis a stable focus. When stimulated by a brief synaptic input or an injected pulse ofcurrent, the state of the system deviates from the focus equilibrium, and then returnsto the equilibrium along a spiral trajectory, as depicted in Fig. 7.16, top, therebyproducing a damped oscillation. The frequency of such an oscillation is the imaginarypart of the complex-conjugate eigenvalues at the equilibrium (see Sect. 6.1.3), and itcan be as large as 200 Hz in mammalian neurons.In Ex. 3 we prove that noise can make such oscillations sustained. While the stateof the system is perturbed and returns to the focus equilibrium, another strong randomperturbation may push it away from the equilibrium, thereby starting a new dampedoscillation. As a result, persistent noisy perturbations create a random sequence ofdamped oscillations and do not let the neuron rest. The membrane potential of sucha neuron exhibits noisy sustained oscillations of small amplitude depicted in Fig. 7.16and discussed in Sect. 6.1.4.Injected dc-current or background synaptic noise increase the rest potential, changeits eigenvalues and, hence change the frequency and amplitude of noisy oscillations.Fig. 7.16 depicts typical cases when the frequency and the amplitude increase as theresting state becomes more depolarized.One should be careful to distinguish fast and slow subthreshold oscillations of membranepotential. Fast oscillations, as in Fig. 7.16, are those having period comparablewith the membrane time constant or with the period of repetitive spiking. In contrast,some neurons found in entorhinal cortex, inferior olive, hippocampus, thalamus, andmany other brain regions can exhibit slow subthreshold oscillations with the periodof 100 ms and more. These oscillations reflect the interplay between fast and slowmembrane currents, e.g., I h or I T , and may be irrelevant to the bifurcation mechanismof excitability. We will discuss this issue in detail in Sect. 7.3.3 and in Chap. 9. Amazingly,such neurons still possess many neuro-computational properties of resonators,such as frequency preference and rebound spiking, but exhibit these properties on aslower time scale.7.2.2 Frequency preference and resonanceA standard experimental procedure to test the propensity of a neuron to subthresholdoscillations is to stimulate it with a sinusoidal current having slowly increasing frequency,called a zap current, as in Fig. 7.17. The amplitude of the evoked oscillationsof the membrane potential, normalized by the amplitude of stimulating oscillatory current,is called the neuronal impedance — a frequency-domain extension of the conceptof resistance. The impedance profile of integrators is decreasing while that of resonatorshas a peak corresponding to the frequency of subthreshold oscillations, around140 Hz in the mesV neuron in the figure. Thus, integrators act as low-pass filters whileresonators act as band-pass filters to periodic signals.Instead of sinusoidal stimulation, consider more biological stimulation with pulses