Dynamical Systems in Neuroscience:

Simple Models 323the down-state and a resonator in the up-state! The same property is exhibited bycerebella Purkinje cells (see Fig. 7.37), and possibly by other neurons kept in theup-state (intrinsically or extrinsically).Review of Important Concepts• Integrate-and-fire neuron is a linear model having a stable node equilibrium,an artificial threshold, and a reset.• Resonate-and-fire neuron is a linear model having a stable focusequilibrium, an artificial threshold, and a rest.• Though technically not spiking neurons, these models are useful foranalytical studies, i.e., to prove theorems.• The quadratic integrate-and-fire model captures the non-linearityof the spike-generation mechanism of real neurons having Class 1excitability (saddle-node on invariant circle bifurcation).• Its simple extension, model (8.5, 8.6), reproduces quantitatively subthreshold,spiking, and bursting activity of all known types of corticaland thalamic neurons in response to pulses of dc-current.• The simple model makes testable hypotheses on the dynamic mechanismsof excitability in these neurons.• The model is especially suitable for simulations of large-scale modelsof the brain.