with parameters and ; cf. Fig. 5.8A. For and initial condition , the voltage decays to the resting potential . For it increases so that an action potential is triggered. The parameter can therefore be interpreted as the critical voltage for spike initiation by a short current pulse. We will see in the next subsection that the quadratic integrate-and-fire model is closely related to the so-called -neuron, a canonical type-I neuron model (141; 291).
For numerical implementations of the model, the integration of Eq. (5.16) is stopped if the voltage reaches a numerical threshold and restarted with a reset value as new initial condition (Fig. 5.9B). For a mathematical analysis of the model, however, the standard assumption is and .
We have seen in the previous section that experimental data suggests an exponential, rather than quadratic nonlinearity. However, close to the threshold for repetitive firing, the exponential integrate-and-fire model and the quadratic integrate-and-fire model become very similar (Fig. 5.8B). Therefore the question arises, whether the choice between the two models is a matter of personal preferences only.
For a mathematical analysis, the quadratic integrate-and-fire model is sometimes more handy than the exponential one. However, the fit to experimental data is much better with the exponential than with the quadratic integrate-and-fire model. For a prediction of spike times and voltage of real neurons (cf. Fig. 5.5), it is therefore advisable to work with the exponential rather than the quadratic integrate-and-fire model. Loosely speaking, the quadratic model is too nonlinear in the subthreshold regime and the upswing of a spike is not rapid enough once the voltage is above threshold. The approximation of the exponential integrate-and-fire model by a quadratic one only holds if the mean driving current is close to the rheobase current.
In this section, we show that there is a one-to-one relation between the quadratic integrate-and-fire model (5.16) and the canonical type I phase model,
Let us denote by the minimal current necessary for repetitive firing of the quadratic integrate-and-fire neuron. With a suitable shift of the voltage scale and constant current the equation of the quadratic neuron model can then be cast into the form
For the voltage increases until it reaches the firing threshold where it is reset to a value . Note that the firing times are insensitive to the actual values of firing threshold and reset value because the solution of Eq. (5.18) grows faster than exponentially and diverges for finite time (hyperbolic growth). The difference in the firing times for a finite threshold of, say, and is thus negligible.
Thus Eq. (5.19) with given by (5.17) is a solution to the differential equation of the quadratic integrate-and-fire neuron. The quadratic integrate-and-fire neuron is therefore (in the limit and ) equivalent to the generic type I neuron (5.17).
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