Dynamical Systems in Neuroscience:

152 Conductance-Based Models1 sec100 nA50 mV20 mV(a) (b) 1 secFigure 5.15: Anomalous (upside-down) spikes in (a) lobster muscle fibers (modifiedfrom Fig.2 of Reuben et al. 1961) and in (b) Ascaris Esophageal cells (modified fromFig.16 of del Castillo and Morales 1967; the cell is depolarized by injected dc-current).The voltage axis is not inverted.consequences: The node corresponds to a resting state, and the saddle correspondsto the threshold state. The large-amplitude trajectory that starts at the saddle andterminates at the node corresponds to an action potential, though to a weird one.Thus, the behavior of this model is similar to the behavior of other models with theexception that the V -axis is reversed.The existence of “upside-down” K + spikes may (or better say does) look bizarre tomany researchers, even though “inverted” K + and Cl − spikes were reported in manypreparations, including frog and toad axons, squid axons, lobster muscle fibers, dogcardiac muscle, etc., as reviewed by Reuben et al. (1961) and Grundfest (1971). Twosuch cases are depicted in Fig. 5.15. Interestingly, Reuben et al. (1961) postulated,albeit reluctantly, that the inverted spikes are caused by the inactivation of K + current.The reluctance was due to the fact that transient K + I A was not known at that time.By now the reader must have convinced himself that quite different models canhave practically identical dynamics. Conversely, the same model could have quite differentbehavior if only one parameter, e.g., V 1/2 , is changed by as little as 10 mV.Such dramatic conclusions emphasize the importance of geometrical phase plane analysisof neuronal models, since the conclusions can hardly be drawn from mere worddescriptions of the spiking mechanisms.5.1.8 Ca 2+ -gated minimal modelsSo far, we considered minimal models consisting of voltage-gated currents only. However,there are many ionic currents that depend not only on the membrane potential,but also on the concentration of intracellular ions, mostly Ca 2+ . Such currents arereferred to as being Ca 2+ -gated, and they are summarized in Fig. 5.16. In addition,there are Cl − -gated, K + -gated, and Na + -gated currents, such as SLO gene family ofCl − -gated K + currents discovered in C. elegans, and related “slack and slick” familyof Na + -gated K + currents (Yuan et al. 2000, 2003). Considering minimal modelsinvolving these currents goes outside the scope of this book, but it could be a good