Dynamical Systems in Neuroscience:

Simple Models 3211989, Alonso and Klink 1993, Klink and Alonso 1993, Dickson et al. 2000). The oscillationsare generated by the interplay between persistent Na + current and h-current, andthey are believed to set the theta rhythmicity in the entorhinal-hippocampal network.The caption of Fig. 8.39 provides parameters of the simple model (8.5, 8.6) thatcaptures the slow oscillatory dynamics of an entorhinal stellate cell recorded in vitroin adult rat. The cell sags to injected hyperpolarizing current in Fig. 8.39a and thenfires a rebound spike upon release from hyperpolarization. From a neurophysiologicalpoint of view, the sag and rebound response are due to the opening of the h-current;From a mathematical point of view, they are caused by the resonant slow variable u,which could also describe deinactivation of a transient Na + current and deactivationof low-threshold K + current. The geometrical explanation of these responses is similarto the one provided for LTS interneurons in Fig. 8.26. Positive steps of current evoke atransient or sustained spiking activity. Notice that the first spike is actually a doubletin the recording and in the simulation in Fig. 8.39a (I = 200 pA).Stellate cells in the entorhinal cortex of adult animals can exhibit damped or sustainedsubthreshold oscillations in a frequency range from 5 to 15 Hz. The oscillationscan be clearly seen when the cell is depolarized by injected dc-current, as in Fig. 8.39b.The stronger the current, the higher the amplitude and frequency of oscillations, whichoccasionally result in spikes or even bursts of spikes (Alonso and Klink 1993). The simplemodel also exhibits slow damped oscillations because its resting state is a stablefocus. The focus loses stability via subcritical Andronov-Hopf bifurcation, and hence itcoexists with a spiking limit cycle. To enable sustained oscillations and random spikes,we add channel noise to the v-equation (White et al. 2000).In Fig. 8.39c we explain the mechanism of random transitions between subthresholdoscillations and spikes, which is similar to the mechanism of stuttering in RS andFS neurons. When weak dc-current is injected (left), the attraction domain of theresting state (shaded region) is separated from the rest of the phase space by thestable manifold to the saddle equilibrium (denoted separatrix). Noisy perturbationsevoke small sustained noisy oscillations around the resting state with an occasionalspike when the separatrix is crossed. Increasing the level of injected dc-current resultsin the series of bifurcations similar to those in Fig. 8.15. As a result, there is a coexistenceof a large amplitude (spiking) limit cycle attractor and a small unstable limitcycle, which encompasses the attraction domain of the resting state (right). Noisyperturbations can randomly switch the activity between these attractors, resulting inthe random bursting activity in Fig. 8.39b.8.4.5 Mitral neurons of olfactory bulbMitral cells recorded in slices of rat main olfactory bulb exhibit intrinsic bistability ofmembrane potentials (Heyward et al. 2001). They spontaneously alternate betweentwo membrane potentials separated by 10 mV: a relatively depolarized (up-state) andhyperpolarized (down-state) potentials. The membrane potential could be switchedbetween the states by a brief depolarizing or hyperpolarizing pulse of current, as we