Dynamical Systems in Neuroscience:

154 Conductance-Based Modelsvoltage-gatedinactivationof inwardcurrentresonant gateactivationof outwardcurrentinactivationof inwardcurrentCa 2+ -gatedactivationof outwardcurrentamplifying gatevoltage-gatedCa 2+ -gatedactivationof inwardcurrentinactivationof outwardcurrentactivationof inwardcurrentinactivationof outwardcurrentINa,p+IhINa,tIKir+IhICAN+IhI1I4+I3INa,p+IKIKir+IKIAICAN+IKICa+I2ICa(L)IKir+I2ICAN+I2ICAN,tI4+I2ICa+IK(Ca)ICAN+IK(Ca)Figure 5.17: Voltage- and Ca 2+ -gated minimal models. I 1 , . . . , I 4 are four hypotheticalcurrents that could exist theoretically, but has never been recorded experimentally; seetext.Any combination of one voltage- or Ca 2+ -gated amplifying current and one voltageorCa 2+ -gated resonant current leads to a minimal model for spiking. All such combinationsare depicted in Fig. 5.17. Here, I 1 denotes a hypothetical Ca 2+ -activatedvoltage-inactivated transient inward current. Though such a current is not currentlyknown, one can easily write a conductance-gated model for it. A biologist would treatsuch a current as hyperpolarization- and Ca 2+ -activated. I 2 is a hypothetical Ca 2+ currentthat is inactivated by Ca 2+ . I 3 is a hypothetical voltage-inactivated Ca 2+ current.I 4 is an outward Ca 2+ -inactivated current.We see that there are many minimal models in Fig. 5.17. Six of them are purelyvoltage-gated, and they have been investigated above. The others are mixed-mode orpurely Ca 2+ -gated models. An interested reader can work out the details of their phaseportraits.