Dynamical Systems in Neuroscience:

282 Simple Models1thresholdyz resetx0-1x-1 0 1100 20 40 60 80 100time, msFigure 8.2: Resonate-and-fire model with b = −0.05, ω = 0.25 and z reset = i. The spikewas added manually.spike. Young did not specify what happens after the spike. The resonate-and-firemodel is the Young model with the resetting: if V ≥ V thresh , then V ← V reset andW ← W reset , where V rest and W reset are some parameters.When the resting state is a stable focus, the model can be recast in complex coordinatesasż = (b + iω)z + I ,where b + iω ∈ C is the complex eigenvalue of the resting state, z = x + iy ∈ C is thecomplex-valued variable describing damped oscillations with frequency ω around theresting state. The real part, x, is a current-like variable. It describes the dynamicsof the resonant current and synaptic currents. The imaginary part, y is a voltage-likevariable. The neuron is said to fire a spike when y reaches the threshold y = 1. Thus,the threshold is a horizontal line on the complex plane that passes through i ∈ C, asin Fig. 8.2, though other choices are also possible. After firing the spike, the variablez is reset to z reset .The resonate-and-fire model illustrates the most important features of resonators:damped oscillations, frequency preference, post-inhibitory (rebound) spikes, and Class2 excitability. It cannot have sustained subthreshold oscillations of membrane potential.Integrate-and-fire and resonate-and-fire neurons do not contradict, but complementeach other. Both are linear, and hence are useful when we prove theorems and deriveanalytical expressions. They have the same flaws limiting their applicability, which wediscussed earlier. In contrast, two simple models described below are difficult to treatanalytically, but because of their universality they should be the models of choice whenlarge-scale simulations are concerned.8.1.3 Quadratic integrate-and-fireSubstituting −v by +v 2 in (8.1) results in the quadratic integrate-and-fire model˙v = b + v 2 , if v = v peak , then v ← v reset , (8.2)