Dynamical Systems in Neuroscience:

One-Dimensional Systems 6915 mV-59 mV-75 mVmembranepotential100 pA2 sFigure 3.16: Membrane potential bistability in a cat TC neuron in the presence ofZD7288 (pharmacological blocker of I h ; modified from Fig. 6B of Hughes et al. 1999).(a)(d)(a)(b)(b)(b)(c)I+F(V)VIV(c)Figure 3.17: Bistability and hysteresis loop as I changes.in Fig. 3.16, or −50 mV and −60 mV in mitral cells of the olfactory bulb (Heyward etal. 2001), or −45 mV and −60 mV in Purkinje neurons. Brief inputs can switch suchneurons from one state to the other, as in Fig. 3.16. Though the ionic mechanisms ofbistability are different in the three neurons, the mathematical mechanism is the same.Consider a one-dimensional system ˙V = I+F (V ) with function F (V ) having a cubicN-shape. Injection of a dc-current I shifts the function I + F (V ) up or down. When Iis negative, the system has only one equilibrium, depicted in Fig. 3.17a. As we removethe injected current I, the system becomes bistable, as in Fig. 3.17b, but its state isstill at the left equilibrium. As we inject positive current, the left stable equilibriumdisappears via another saddle-node bifurcation, and the state of the system jumps tothe right equilibrium, as in Fig. 3.17c. But as we slowly remove the injected currentthat caused the jump and go back to Fig. 3.17b, the jump to the left equilibrium doesnot occur until a much lower value corresponding to Fig. 3.17a is reached. The failureof the system to return to the original value when the injected current is removed iscalled hysteresis. If I were a slow V -depended variable, then the system could exhibitrelaxation oscillations depicted in Fig. 3.17d and described in the next chapter.