Dynamical Systems in Neuroscience:

288 Simple Modelsintegrate-and-fire modelsimple modelspike cutoffspikesare drawnby handspikes aregeneratedthresholdrestingresetthreshold ?restingresetinputinputFigure 8.7: Voltage reset in the integrate-and-fire model and in the simple model.is voltage-dependent, e.g., k = 0.7 for v ≤ v t and k = 7 for v > v t , or by usingthe modification of the simple model presented in Ex. 13 and Ex. 17. The seconddiscrepancy results from the instantaneous after-spike resetting, and it is of a lesserimportance because it does not affect the decision whether or when to fire. However,the slope of the downstroke may become important in studies of gap-junction-coupledspiking neurons.The phase portrait of the simple model is depicted in Fig. 8.6c. Injection of thestep of dc-current I = 70 pA shifts the v-nullcline (square parabola) up and makesthe resting state, denoted by a black square, disappear. The trajectory approaches thespiking limit cycle attractor, and when it crosses the cutoff vertical line v peak = 35 mV,it is reset to the white square, resulting in periodic spiking behavior. Notice the slowafterhyperpolarization (AHP) following the reset that is due to the dynamics of therecovery variable u. Depending on the parameters, the model can have other types ofphase portraits, spiking, and bursting behavior, as we demonstrate in the rest of thechapter.In Fig. 8.7 we illustrate the difference between the integrate-and-fire neuron and thesimple model. The integrate-and-fire model is said to fire spikes when the membranepotential reaches a preset threshold value. The potential is reset to a new value, andthe spikes are drawn by hand. In contrast, the simple model generates the up-strokeof the spike due to the intrinsic (regenerative) properties of the voltage equation. Thevoltage reset occurs not at the threshold, but at the peak of the spike. In fact, thefiring threshold in the simple model is not a parameter, but a property of the bifurcationmechanism of excitability. Depending on the bifurcation of equilibrium, the model maynot even have a well-defined threshold, similarly to conductance-based models.When implementing numerically the voltage reset, whether at the threshold or thepeak of the spike, one needs to be aware of the numerical errors, which translate intothe errors of spike timing. These errors are inversely proportional to the slope of thevoltage variable at the reset value. The slope is small in the integrate-and-fire model, so