Dynamical Systems in Neuroscience:

322 Simple Modelsshow in Fig. 7.36. In response to stimulation, the cells are more likely to fire in theup-state than in the down-state.Current-voltage (I-V) relations of such mitral cells have three zeros in the subthresholdvoltage range confirming that there are three equilibria, two stable correspondingto the up- and down-state, and one unstable — saddle. There are no subthresholdoscillations in the down-state, hence it is a node, and the cell is an integrator. Thereare small-amplitude 40 Hz oscillations in the up-state, hence it is a focus and the cellis a resonator.To model the bistability, we use the simple model with a piece-wise linear slownullcline that approximates non-linear activation functions n ∞ (v) near the “threshold”of the current and a passive dendritic compartment. In many respects, the model issimilar to the one for late spiking (LS) cortical interneurons. In Fig. 8.40 we fine-tunethe model to simulate responses of a rat mitral cell to pulses of current of variousamplitude. To prevent noise-induced spontaneous transitions between the up- anddown-states, the cell in the figure was hold at −75 mV by injection of a large negativecurrent. Its responses to weak positive pulses of current show a fast rising phasefollowed by an abrupt step (arrow in the figure) to a constant value correspondingto the up-state. Increasing the magnitude of stimulation elicits trains of spikes with aconsiderable latency, whose cause has yet to be determined experimentally. The latencycould be the result of slow activation of an inward current, slow inactivation of anoutward current, e.g. K + A-current, or just slow charging of the dendritic compartment.All three cases correspond to an additional slow variable in the simple model, whichwe interpret as a membrane potential of a passive dendritic compartment.To understand the dynamics of the simple model, and hopefully of the mitral cell,we simulate its responses in Fig. 8.41 to the activation of the olfactory nerve (ON).In the top of Fig. 8.41, the cell is held at I = 0 pA. Its phase portrait clearly showsthe co-existence of stable node and focus equilibria separated by a saddle. The shadedregion corresponds to the attraction domain of the focus equilibrium. To fire a spikefrom the up-state, noise or external stimulation must push the state of the systemfrom the shaded region over the threshold to the right. The cell returns to the downstateright after the spike. Much stronger stimulation is needed to fire the cell fromthe down-state. Typically, the cell is switched to the up-state first, spend some timeoscillating at 40 Hz, and then fire a spike (Heyward et al. 2001).In the bottom of Fig. 8.41, the cell is held at a slightly depolarizing current I =7 pA. The node equilibrium disappeared via saddle-node bifurcation, so there is nodown-state, but only its ghost. Stimulation at the up-state results in a spike, afterhyperpolarization,and slow transition through the ghost of the down-state back tothe up-state. Further increasing the holding current results in the stable manifoldto the upper saddle (marked “threshold” in the figure) to make a loop, and thenbecome a homoclinic trajectory to the saddle giving birth to an unstable limit cycle,which shrinks to the focus and makes it lose stability via subcritical Andronov-Hopfbifurcation. Notice that this phase portrait and the bifurcation scenario is differentfrom the one in Fig. 7.36. However, in both cases, the neuron is an integrator in