Dynamical Systems in Neuroscience:

Excitability 275membrane potential, V (mV)0-10-20-30-40-50-60saddle-node on invariant circlebifurcationinvariant circleADPsaddlenode-70-800 20 40 60 80 100 120 140 160 180 200time (ms)Figure 7.54: Afterdepolarization in the I Na,p +I K -model passing slowly through saddlenodeon invariant circle bifurcation, as the magnitude of the injected current rampsdown.Review of Important Concepts• A neuron is excitable because, as a dynamical system, it is near abifurcation from resting to spiking activity.• The type of bifurcation determines the neuron’s computational properties,summarized in Fig. 7.15.• Saddle-node on invariant circle bifurcation results in Class 1 excitability:the neuron can fire with arbitrary small frequency andencode the strength of input into the firing rate.• Saddle-node off invariant circle and Andronov-Hopf bifurcations resultin Class 2 excitability: the neuron can fire only within a certainfrequency range.• Neurons near saddle-node bifurcation are integrators: they preferhigh-frequency excitatory input, have well-defined thresholds, andfire all-or-none spikes with some latencies.• Neurons near Andronov-Hopf bifurcation are resonators: they haveoscillatory potentials, prefer resonant-frequency input, and can easilyfire post-inhibitory spikes.