Dynamical Systems in Neuroscience:
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Dynamical Systems in Neuroscience:

Dynamical Systems in Neuroscience: Dynamical Systems in Neuroscience:

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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

Conductance-Based Models 153Voltage-GatedCa 2+ -GatedCurrentsActivationInactivationActivationInactivationI leakInwardI Na,pI CafastI Na,tI Ca(T)fastfastI Ca(P)fastslowI Ca(L)fastslowI Ca(N)fastmediummediumI hmediumI CANslowOutwardI KfastI K(M)slowI AfastfastI K(D)mediumslowI K(Ca)fastfastI AHPslowI KirfastFigure 5.16: Some representative voltage- and Ca 2+ -gated ionic currents (Johnston andWu 1995, Hille 2001, Shepherd 2004).intellectual exercise for an expert reader (see also Ex. 7 and 8).Ca 2+ -gated currents can also be divided into amplifying and resonant. Ca 2+ -activated inward currents, such as the cation non-selective I CAN , act as amplifyingcurrents. Indeed, activation of such a current leads to an influx of Ca 2+ ions and tomore activation. Similarly, a hypothetical outward current inactivated by Ca 2+ , notpresent in the figure, might also act as an amplifying current. Indeed, a depolarizationdue to the Ca 2+ influx inactivates such a hypothetical outward current, therebyproducing a net shift toward inward currents and leading to more depolarization.In contrast, Ca 2+ -inactivating inward currents and Ca 2+ -activating outward currents,such as I Ca(L) and I AHP , respectively, act as resonant currents. Indeed, a depolarizationdue to the Ca 2+ influx inactivates the inward current and activates theoutward current, and resists further depolarization.

Conductance-Based Models 153Voltage-GatedCa 2+ -GatedCurrentsActivationInactivationActivationInactivationI leakInwardI Na,pI CafastI Na,tI Ca(T)fastfastI Ca(P)fastslowI Ca(L)fastslowI Ca(N)fastmediummediumI hmediumI CANslowOutwardI KfastI K(M)slowI AfastfastI K(D)mediumslowI K(Ca)fastfastI AHPslowI KirfastFigure 5.16: Some representative voltage- and Ca 2+ -gated ionic currents (Johnston andWu 1995, Hille 2001, Shepherd 2004).<strong>in</strong>tellectual exercise for an expert reader (see also Ex. 7 and 8).Ca 2+ -gated currents can also be divided <strong>in</strong>to amplify<strong>in</strong>g and resonant. Ca 2+ -activated <strong>in</strong>ward currents, such as the cation non-selective I CAN , act as amplify<strong>in</strong>gcurrents. Indeed, activation of such a current leads to an <strong>in</strong>flux of Ca 2+ ions and tomore activation. Similarly, a hypothetical outward current <strong>in</strong>activated by Ca 2+ , notpresent <strong>in</strong> the figure, might also act as an amplify<strong>in</strong>g current. Indeed, a depolarizationdue to the Ca 2+ <strong>in</strong>flux <strong>in</strong>activates such a hypothetical outward current, therebyproduc<strong>in</strong>g a net shift toward <strong>in</strong>ward currents and lead<strong>in</strong>g to more depolarization.In contrast, Ca 2+ -<strong>in</strong>activat<strong>in</strong>g <strong>in</strong>ward currents and Ca 2+ -activat<strong>in</strong>g outward currents,such as I Ca(L) and I AHP , respectively, act as resonant currents. Indeed, a depolarizationdue to the Ca 2+ <strong>in</strong>flux <strong>in</strong>activates the <strong>in</strong>ward current and activates theoutward current, and resists further depolarization.

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