Nonlinear generalizations of the one-dimensional equation of the leaky integrate-and-fire model can be found in the paper of Abbott and van Vreeswijk (1993). While the form of the nonlinearity was left open at that time, analytical methods from bifurcation theory suggested that there was a canonical one-dimensional model which describes the saddle-node-onto-limit-cycle bifurcation of type I models. This is called the canonical type I model or Theta model or Ermentrout-Kopell model (140; 141). While the quadratic nonlinearity in the voltage appeared in many early papers on bifurcations and type I models (141; 140; 499; 228) the active use of the quadratic integrate-and-fire model for simulations seems to have been started by Latham et al. in 2000 and later popularized by Izhikevich (239).

Based on biophysical and mathematical arguments, the exponential integrate-and-fire model was introduced in 2003 by Fourcaud-Trocme, Hansel, Vreeswijk and Brunel. While the quadratic integrate-and-fire model is the canonically correct type I model close to the bifurcation point, i.e. at the transition to repetitive firing, there is no fundamental reason that the quadratic nonlinearity should also be correct further away from the threshold, and it was unclear at that time what the nonlinearity of real neurons would look like. The issue was settled by the work of Badel et al. (35, 34) who designed an experimental method to measure the nonlinearity directly in experiments. The nonlinearity found in experiments is extremely well matched by the exponential integrate-and-fire model.

The link between ion channel activation or inactivation in conductance-based neuron models and the parameters of an exponential integrate-and-fire model with refractoriness (35; 34) is discussed in the overview paper of Platkiewicz and Brette (401).

We will see in the next chapter that the single-variable exponential integrate-and-fire model as it stands is not sufficient to account for the wide variety of neuronal firing patterns, but needs to be complemented by a second variable to account for slow processes such as adaptation - and this leads eventually to the Adaptive Exponential Integrate-and-Fire Model

1. 1.

   Quadratic vs. exponential integrate-and-fire For a comparison of the two models, take a look at Fig. 5.8B and answer the following questions:

   (a) Show for the exponential integrate-and-fire model that the minimum of the nonlinearity $f(u)$ occurs at $u=\vartheta_{rh}$. Calculate the curvature $d^{2}/du^{2}f(u)$ at $u=\vartheta_{rh}$.

   (b) Find parameters of the quadratic integrate-and-fire model so that it matches the location and curvature of the exponential model in (a).

   (c) Suppose that the value of the numerical threshold is $\theta_{\rm reset}=\vartheta_{rh}+2\Delta_{T}$. When the threshold is reached the membrane potential is reset to $u_{r}=\vartheta_{rh}-2\Delta_{T}$. Sketch qualitatively the trajectory $u(t)$ for both neuron models in the repetitive firing regime. Pay particular attention to the following questions: (i) For $u>\vartheta_{rh}$ which of the two trajectories rises more rapidly towards the numerical threshold? (ii) For $u=\vartheta_{rh}$, do both trajectories have the same speed of rise $du/dt$ or a different one? (iii) For $u<\vartheta_{rh}$, which of the two trajectories rises more rapidly? (iv) Should your drawing of the trajectories follow any symmetry rules? Compare your results with Fig. 5.9.

2. 2.

   Zero-curvature and rheobase threshold . Experimentalists sometimes determine the threshold voltage for spike initiation by searching for the zero-curvature point in the voltage trajectory, just before the upswing of the action potential. Show for a general nonlinear integrate-and-fire model that the zero-curvature point is equivalent to the rheobase threshold $\vartheta_{rh}$. Hints: (i) Minimal curvature means that the voltage trajectory passes through a point $d^{2}u/dt^{2}=0$. (ii) Note that the slope of the voltage trajectory $du/dt$ is given by the function $f(u)$. (iii) Recall that the minimum of $f$ is taken at $u=\vartheta_{rh}$.

3. 3.

   Exponential integrate-and-fire and sodium activation. To derive the exponential integrate-and-fire model from a Hodgkin-Huxley model with multiple channels, follow the procedure indicated in Section 5.2.2 and perform the following steps.

   a) Show that $N$ linear currents can always be rewritten as one effective leak current $\sum_{k}g_{k}\,(u-E_{k})=g^{\rm eff}\,(u-E^{\rm eff})$. Determine the effective conductance $g^{\rm eff}$ and the effective reversal potential $E^{\rm eff}$.

   b) Assume that the sodium activation is rapid and approaches an equilibrium value $m_{0}(u)=1/\{1+\exp[-\beta\,(u-\theta_{act})]\}=0.5\,\{1+\tanh[-\beta\,(u-\theta_{act})/2]\}$ where $\theta_{act}$ is the activation threshold of the sodium channel and $\beta$ the sharpness of the threshold. Using a Taylor expansion, show that for $u<\theta_{act}-\beta^{-1}$ the activation function can be approximated by an exponential $m_{0}(u)=\exp[\beta\,(u-\theta_{act})]$.

   c) Assume that $u<\theta_{act}-\beta^{-1}<E_{\rm Na}$ and show that the sodium current

   $$I_{\rm Na}=g_{\rm Na}[m_{0}(u)]^{3}\,h_{\rm rest}\,(u-E_{\rm Na})$$

   (5.21)

   gives rise to the exponential term.

   d) Using the steps (a) - (c), map the equation (5.15) to the exponential integrate-and-fire model

   $${du\over dt}=\tilde{f}(u)=-{{u-u_{\rm rest}}\over\tau}+{\Delta_{T}\over\tau}\,\exp\left({u-\vartheta_{rh}\over\Delta_{T}}\right)$$

   (5.22)

   Determine the parameters $u_{\rm rest}$, $\tau$ and $\Delta_{T}$.

4. 4.

   Refractory exponential integrate-and-fire model and sodium inactivation . In the previous exercise, we have assumed that $h=h_{\rm rest}$ is constant throughout the spiking process. Repeat the steps of the previous calculation, but set $h(t)=h_{\rm rest}+x(t)$ and show that the sodium inactivation dynamics leads to an increase in the the firing threshold a few milliseconds after the spike.

   Hints. (i) Linearize the dynamics of gating variable $h$ in the Hodgkin-Huxley model around $h=h_{\rm rest}$ so as to derive $dx/dt=-x/\tau_{0}$. (ii) Assume that during each spike, the inactivation variable $h$ increases by a fixed amount $\Delta h$.

5. 5.

   Quadratic integrate-and-fire model .

   Consider the dimensionless quadratic integrate-and-fire model

   $${{\text{d}}\over{\text{d}}t}u=\,(u-u_{\rm rest})\,(u-u_{c})+I\,,$$

   (5.23)

   with $u_{\rm rest}=-1$ and $u_{c}=1$ and $I=0$.

   (a) Suppose that a trajectory starts at time $t=0$ at $-\infty$. How long does it take to get close to the resting state and reach the value $u=-1-\epsilon$ where $\epsilon\ll 1$?

   (b) Suppose that a trajectory is initialized at $u(0)=1+\epsilon$. where $\epsilon\ll 1$ how long does it take to reach $u=\infty?$

   (c) Initialize as in (b) but stop the integration at $u=\theta_{\rm reset}$. How long does the trajectory take to reach the reset threshold? Is the difference between (b) and (c) important?

   Hint: Use $\int dx/[1-x^{2}]={\rm arcoth}(x)=0.5[{\rm ln}(x+1)-{\rm ln}(x-1)]$ and ${\rm cothh}(x)=[e^{x}+e^{-x}]/[e^{x}-e^{-x}]$

6. 6.

   Gain function of the quadratic integrate-and-fire model

   We consider a quadratic integrate-and-fire model in the super-threshold regime

   $${{\text{d}}\over{\text{d}}t}u=u^{2}+I\,,$$

   (5.24)

   with constant current $I>0$.

   (a) Give a transformation of variables from Eq. (5.23) to (5.24).

   (b) Calculate the duration $T$ of a trajectory which starts at time $t=0$ at $u(0)=-\infty$ and ends at time $T$ at $u(T)=+\infty$.

   Hint: Use $\int dx/[1+x^{2}]={\rm artan}(x)$

   (c) Plot the frequency $\nu=1/T$ as a function of $I$.