Dynamical Systems in Neuroscience:

286 Simple Modelsbe an integrator or a resonator, as we illustrate in Fig. 8.5. The parameters c and ddo not affect steady-state subthreshold behavior. Instead, they take into account theaction of high-threshold voltage-gated currents activated during the spike and affectonly the after-spike transient behavior. If there are many currents with diverse timescales, then u, a, b, and d are vectors, and the equation (8.3) contains ∑ u instead ofu.The simple model may be treated as quadratic integrate-and-fire neuron with adaptationin the simplest case b = 0. When b < 0, the model can be treated as quadraticintegrate-and-fire neuron with a passive dendritic compartment (see Ex. 10). Whenb > 0, the connection to the quadratic integrate-and-fire neuron is lost, and the simplemodel represents a novel class of spiking models.In the rest of this chapter we tune the simple model to reproduce spiking andbursting behavior of many known types of neurons. It is convenient to use it in theformC ˙v = k(v − v r )(v − v t ) − u + I if v ≥ v peak , then (8.5)˙u = a{b(v − v r ) − u} v ← c, u ← u + d (8.6)where v is the membrane potential, u is the recovery current, C is the membranecapacitance, v r is the resting membrane potential, and v t is the instantaneous thresholdpotential. Though it looks like the model has ten parameters, but it is equivalentto (8.3,8.4) and hence it has only four independent parameters. As we described inSect. 5.2.4, the parameters k and b can be found knowing the neuron’s rheobase andinput resistance. The sum of all slow currents that modulate the spike-generationmechanism are combined in the phenomenological variable u with outward currentstaken with the plus sign.The sign of b determines whether u is an amplifying (b < 0) or a resonant (b > 0)variable. In the latter case, the neuron sags in response to hyperpolarized pulses ofcurrent, peaks in response to depolarized subthreshold pulses, and produces rebound(post-inhibitory) responses. The recovery time constant is a. The spike cut-off value isv peak , and the voltage reset value is c. The parameter d describes the total amount ofoutward minus inward currents activated during the spike and affecting the after-spikebehavior. All these parameters can be easily fit to any particular neuron type, as weshow in subsequent sections.Implementation and phase portraitThe following MATLAB code simulates the model and produces Fig. 8.6a.C=100; vr=-60; vt=-40; k=0.7;a=0.03; b=-2; c=-50; d=100;vpeak=35;T=1000; tau=1;n=round(T/tau);% parameters used for RS% neocortical pyramidal neurons% spike cutoff% time span and step (ms)% number of simulation steps