6 Adaptation and Firing Patterns

6.5 Summary

By adding one or several adaptation variables to integrate-and-fire models emulate a large variety of firing patterns found in real neurons, such as adaptation, bursting or initial bursting can be explained. The dynamics of the adaptation variables has two components: (i) a coupling to the voltage uu via a parameter aa which provides subthreshold adaptation and, in nonlinear neuron models, also a contribution to spike-triggered adaptation; and (ii) an explicit spike-triggered adaptation via an increase of the adaptation current during each firing by an amount bb. While positive values for aa and bb induce a hyperpolarization of the membrane and therefore lead to spike-frequency adaptation, negative values induce a depolarization and lead to delayed onset of spiking and spike frequency facilitation. Bursting is most easily achieved by a suitable combination of the reset parameters uru_{r} and bb.

The phenomenological adaptation variables wkw_{k} can be derived from the ionic currents flowing through different ion channels. Coupling of an integrate-and-fire model to a passive dendrite also yields effective adaptation variables which have, however, a facilitating influence.

The adaptation variables can be combined with a quadratic integrate-and-fire model which leads to the Izhikevich model; with an exponential integrate-and-fire model which leads to the AdEx model; or with a leaky integrate-and-fire model. In the latter case, the differential equations can be analytically integrated in the presence of an arbitrary number of adaptation variable. Integration leads to the Spike Response Model (SRM) which presents a general linear model combined with a sharp firing threshold. The Spike Response Model is the starting point for the Generalized Linear Models in the presence of noise which we will introduce in Chapter 9.


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 aa is nicely presented and analyzed in Richardson et al. (431), the spike-triggered jump bb 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 η\eta, κ\kappa and ϵij\epsilon_{ij} are response kernels that describe the effect of spike emission and spike reception on the variable uiu_{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 II , constant ww and a=0a=0 is

    f(I,w)=-[τmlog(1-ϑrh-uresetR(I-w))]-1.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), ww jumps by an amount bb . Show that for II constant and ww averaged over one interspike interval, Eq. 6.8 becomes:

    τw   𝑑   w   𝑑   t=-w+bτwf(I,w).\tau_{w}\frac{{\text{d}}w}{{\text{d}}t}=-w+b\tau_{w}\,f(I,w). (6.58)

    c) At time t0t_{0} , a strong current of amplitude I0I_{0} is switched on that causes transiently a firing rate fτwf\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)f(I,w) .

  2. 2.

    Subthreshold resonance . We study a leaky integrate-and-fire model with a single adaptation variable ww . a) Assume E0=urestE_{0}=u_{\rm rest} and cast equation Eq. ( 6.7 ) and ( 6.8 ) in the form of Eq. ( 6.27 ). Set ϵ=0\epsilon=0 and calculate η\eta and κ\kappa . Show that κ(t)\kappa(t) can be written as a linear combination κ(t)=k+eλ+t+k-eλ-t\kappa(t)=k_{+}e^{\lambda_{+}t}+k_{-}e^{\lambda_{-}t} with

    λ±=12τmτw(-(τm+τw)±τm+τw-4τmτw(1+aR))\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)


    k±=±R(λ±τw+1)τmτw(λ+-λ-).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 κ(t)\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 R^(ω)\hat{R}(\omega) that relates the current I^(ω)\hat{I}(\omega) at frequency ω\omega to the voltage u^(ω)\hat{u}(\omega) at the same frequency, i.e., u^(ω)=R^(ω)I^(ω)\hat{u}(\omega)=\hat{R}(\omega)\,\hat{I}(\omega) . Show that, in the case where κ\kappa has oscillations, the function R^(ω)\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 τm/τw=δ1\tau_{m}/\tau_{w}=\delta\ll 1 and all calculations can be done to first order in δ\delta .
    a) Show that the spike after-potential is given by

    η(t)\displaystyle\eta(t) =\displaystyle= γ1eλ1t+γ2eλ2t\displaystyle\gamma_{1}e^{\lambda_{1}t}+\gamma_{2}e^{\lambda_{2}t} (6.61)
    γ1\displaystyle\gamma_{1} =\displaystyle= Δu(1-δ-δa)-b(1+δ)\displaystyle\Delta u\,(1-\delta-\delta a)-b\,(1+\delta) (6.62)
    γ2\displaystyle\gamma_{2} =\displaystyle= Δu-γ1\displaystyle\Delta u-\gamma_{1} (6.63)

    b) Derive the input response kernel κ(s)\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 τ\tau that is a function of the time since the last postsynaptic spike,

       𝑑   u   𝑑   t=-uτ(t-t^)+1CIext(t);{{\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, t^\hat{t} denotes the last firing time of the neuron. The neuron fires if u(t)u(t) hits a fixed threshold ϑ\vartheta and integration restarts with a reset value uru_{r}.

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

    (b) Integrate Eq. (6.64) for arbitrary input with u(t^)=uru(\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)t^{(f)} generates several adaptation currents dwk/dt=-wkτk+bkδ(t-t(f))dw_{k}/dt=-{w_{k}\over\tau_{k}}+b_{k}\delta(t-t^{(f)}) with k=1,,Kk=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.