Formal neuron models where spikes are triggered by a threshold process have been popular in the sixties (494; 495; 170; 547), but the ideas can be traced back much earlier (288; 219). It has been recognized early that these models lend themselves for hardware implementations (158) and mathematical analysis (494; 496), and can be fitted to experimental data (69; 70).

Dynamic thresholds that increase after each spike have been a standard feature of phenomenological neuron models for a long time (162; 170; 547) and so have the slow subthreshold processes of adaptation (451; 333; 151; 481) While the linear subthreshold coupling of voltage and adaptation currents via a coupling parameter $a$ is nicely presented and analyzed in Richardson et al. (431), the spike-triggered jump $b$ of the adaptation current has been mainly popularized by Izhikevich (237) – but can be found in earlier papers (e.g. Gerstner et al. (179); Liu and Wang (305)), and much earlier in the form of a spike-triggered increase in the threshold (162; 170; 547).

The phase plane analysis of the AdEx model presented in this chapter is based on Naud et al. (360). The main difference between the AdEx model (67) and the highly influential model of Izhikevich (237) is that the AdEx uses in the voltage equation an exponential nonlinearity (as suggested by experiments (35)) whereas the Izhikevich model uses a quadratic nonlinearity (as suggested by bifurcation analysis close to the bifurcation point (141)).

The book of Izhikevich (238) as well as the Scholarpedia articles on the Spike Response Model (SRM) and the Adaptive Exponential Integrate-and-Fire (AdEx) model (184; 175), present readable reviews of the model class discussed in this chapter.

The functions $\eta$, $\kappa$ and $\epsilon_{ij}$ are response kernels that describe the effect of spike emission and spike reception on the variable $u_{i}$. This interpretation has motivated the name ‘Spike Response Model’. While the name and the specific formulation of the model equations (6.27)–(6.30) has been used since 1995 (182; 179; 260), closely related models can be found in earlier works; see, e.g., Hill (219); Geisler and Goldberg (170).

1. 1.

   Timescale of firing rate decay . The characteristic feature of adaptation is that, after the onset of a superthreshold step current, interspike-intervals become successively longer, or, equivalently, that the momentary firing rate drops. The aim is to make a quantitative prediction of the decay of the firing rate of a leaky integrate-and-fire model with a single adaptation current.

   a) Show that the firing rate of Eq. ( 6.7 ) and ( 6.8 ) with constant $I$ , constant $w$ and $a=0$ is

   $$f(I,w)=-\left[\tau_{m}\log\left(1-\frac{\vartheta_{\rm rh}-u_{\rm reset}}{R(I-w)}\right)\right]^{-1}.$$

   (6.57)

   
   

   b) For each spike (i.e., once per interspike interval), $w$ jumps by an amount $b$ . Show that for $I$ constant and $w$ averaged over one interspike interval, Eq. 6.8 becomes:

   $$\tau_{w}\frac{{\text{d}}w}{{\text{d}}t}=-w+b\tau_{w}\,f(I,w).$$

   (6.58)

   
   

   c) At time $t_{0}$ , a strong current of amplitude $I_{0}$ is switched on that causes transiently a firing rate $f\gg\tau_{w}$ . Afterward the firing rate decays. Find the effective time constant of the firing rate for the case of strong input current.

   Hint: Start from Eq. ( 6.58 ) and consider a Taylor expansion of $f(I,w)$ .

2. 2.

   Subthreshold resonance . We study a leaky integrate-and-fire model with a single adaptation variable $w$ . a) Assume $E_{0}=u_{\rm rest}$ and cast equation Eq. ( 6.7 ) and ( 6.8 ) in the form of Eq. ( 6.27 ). Set $\epsilon=0$ and calculate $\eta$ and $\kappa$ . Show that $\kappa(t)$ can be written as a linear combination $\kappa(t)=k_{+}e^{\lambda_{+}t}+k_{-}e^{\lambda_{-}t}$ with

   $$\lambda_{\pm}=\frac{1}{2\tau_{m}\tau_{w}}\left(-(\tau_{m}+\tau_{w})\pm\sqrt{\tau_{m}+\tau_{w}-4\tau_{m}\tau_{w}(1+aR)}\right)$$

   (6.59)

   and

   $$k_{\pm}=\pm\frac{R(\lambda_{\pm}\tau_{w}+1)}{\tau_{m}\tau_{w}(\lambda_{+}-\lambda_{-})}.$$

   (6.60)

   
   

   b) What are the parameters of Eq. ( 6.7 ) - ( 6.8 ) that lead to oscillations in $\kappa(t)$ ?
   c) What is the frequency of the oscillation?

   Hint: Section 4.4.3 .
   d) Take the Fourier transform of Eq. ( 6.7 ) - ( 6.8 ) and find the function $\hat{R}(\omega)$ that relates the current $\hat{I}(\omega)$ at frequency $\omega$ to the voltage $\hat{u}(\omega)$ at the same frequency, i.e., $\hat{u}(\omega)=\hat{R}(\omega)\,\hat{I}(\omega)$ . Show that, in the case where $\kappa$ has oscillations, the function $\hat{R}(\omega)$ has a global maximum. What is the frequency where this happens?

3. 3.

   Integrate-and-fire model with slow adaptation
   The aim is to relate the leaky integrate-and-fire model with a single adaptation variable, defined in Eq. ( 6.7 ) and ( 6.8 ) to the Spike Response Model in the form of Eq. ( 6.27 ). Adaptation is slow so that $\tau_{m}/\tau_{w}=\delta\ll 1$ and all calculations can be done to first order in $\delta$ .
   a) Show that the spike after-potential is given by

   	$\displaystyle\eta(t)$	$\displaystyle=$	$\displaystyle\gamma_{1}e^{\lambda_{1}t}+\gamma_{2}e^{\lambda_{2}t}$		(6.61)
   	$\displaystyle\gamma_{1}$	$\displaystyle=$	$\displaystyle\Delta u\,(1-\delta-\delta a)-b\,(1+\delta)$		(6.62)
   	$\displaystyle\gamma_{2}$	$\displaystyle=$	$\displaystyle\Delta u-\gamma_{1}$		(6.63)

   
   

   b) Derive the input response kernel $\kappa(s)$ .

   Hint: use the result from a).

4. 4.

   Integrate-and-fire model with time-dependent time constant. Since many channels are open immediately after a spike, the effective membrane time constant after a spike is smaller than the time constant at rest. Consider an integrate-and-fire model with spike-time dependent time constant, i.e., with a membrane time constant $\tau$ that is a function of the time since the last postsynaptic spike,

   $${{\text{d}}u\over{\text{d}}t}=-{u\over\tau(t-\hat{t})}+{1\over C}\,I^{\rm ext}(t)\,;$$

   (6.64)

   cf. Wehmeier et al. (546); Stevens and Zador (498). As usual, $\hat{t}$ denotes the last firing time of the neuron. The neuron fires if $u(t)$ hits a fixed threshold $\vartheta$ and integration restarts with a reset value $u_{r}$.

   (a) Suppose that the time constant is $\tau(t-\hat{t})=2ms$ for $t-\hat{t}<10ms$ and $\tau(t-\hat{t})=20ms$ for $t-\hat{t}\geq 10ms$. Set $u_{r}=-10mV$. Sketch the time course of the membrane potential for an input current $I(t)=q\,\delta(t-t^{\prime})$ arriving at $t^{\prime}=5ms$ or $t^{\prime}=15ms$. What are the differences between the two cases?

   (b) Integrate Eq. (6.64) for arbitrary input with $u(\hat{t})=u_{r}$ as initial condition and interpret the result.

5. 5.

   Spike-triggered adaptation currents . Consider a leaky integrate-and-fire model. A spike at time $t^{(f)}$ generates several adaptation currents $dw_{k}/dt=-{w_{k}\over\tau_{k}}+b_{k}\delta(t-t^{(f)})$ with $k=1,\dots,K$ .

   a) Calculate the effect of the adaptation current on the voltage.

   b) Construct a combination of spike-triggered currents that could generate slow adaptation.

   c) Construct a combination of spike-triggered currents that could generate bursts.