Dynamical Systems in Neuroscience:

274 ExcitabilityK + activation gate, n0.10.05V-nullclineseparatrixn-nullclinesaddleADP-65 -60 -55 -50 -45membrane potential, V (mV)membrane potential, V (mV)0-20-40-60ADP-800 10 20 30 40time (ms)Figure 7.53: A long afterdepolarization (ADP) in the I Na,p +I K -model without any slowcurrents. Parameters as in Fig. 6.52.a long-lasting effect appears because the trajectory follows the separatrix, comes closeto the saddle point, and spends some time there before returning to the stable restingstate.An example in Fig. 7.54 shows the membrane potential of a model neuron slowlypassing through a saddle-node on invariant circle bifurcation. Because the vector fieldis small at the bifurcation, which takes place around t = 70 ms, the membrane potentialis slowly increasing along the limit cycle and then slowly decreasing along the locus ofstable node equilibria, thereby producing a slow ADP. In Chap. 9 we will show that suchADPs exist in 4 out of 16 basic types of bursting neurons, including thalamocorticalrelay neurons and R 15 bursting cells in the abdominal ganglion of the mollusk Aplysia.