Dynamical Systems in Neuroscience:

Excitability 251K + activationK + activation0.060(a)-70 -60 -50 -40(c)integratorsmall EPSPsmallEPSPthreshold manifoldresonatorhalf-amplitudespikespikespike0.50.210.80.60.4(b)threshold manifoldsmallEPSP-70 -60 -50 -40(d)resonatorresonatorcanardtrajectoryPspikespikes0.1thresholdset0.20-60 -50 -40 -30membrane potential, mV-70 -60 -50 -40 -30 -20membrane potential, mVFigure 7.27: Threshold manifolds and sets in the I Na,p +I K -model. Parameters in (a)as in Fig. 4.1a, in (b), (c), and (d) as in Fig. 6.16 with I = 45 (b) and I = 42 (c andd).separating the resting and the spiking states, as in Fig. 7.27b. Such an unstable cycleacts as a threshold manifold. Any perturbation that leaves the state of the neuroninside the attraction domain of the resting state, which is the shaded region boundedby the unstable cycle, results in subthreshold potentials. Any perturbation that pushesthe state of the neuron outside the shaded region results in an action potential. In theextreme case, a perturbation may put the state right on the unstable limit cycle. Then,the neuron exhibits unstable ”threshold” oscillations, at least in theory. In practice,such oscillations cannot be sustained because of noise, and they will either subside orresult in spikes.The bistable regime near subcritical Andronov-Hopf bifurcation is the only casewhen a resonator can have a well-defined threshold manifold. In all other cases, includingthe supercritical Andronov-Hopf bifurcation, resonators do not have well-definedthresholds. We illustrate this in Fig. 7.27c. A small deviation from the resting stateproduces a trajectory corresponding to a “subthreshold” potential. A large deviationproduces a trajectory corresponding to an action potential. We refer to the shadedregion between the two trajectories as a threshold set. It consists of trajectories corre-