6 Adaptation and Firing Patterns

6.1 Adaptive Exponential Integrate-and-Fire

In the previous chapter we have explored nonlinear integrate-and-fire neurons where the dynamics of the membrane voltage is characterized by a function f(u)f(u). A single equation is, however, not sufficient to describe the variety of firing patterns that neurons exhibit in response to a step current. We therefore couple the voltage equation to abstract current variables wkw_{k}, each described by a linear differential equation. The set of equations is

τmdudt\displaystyle\tau_{m}\,\frac{{\text{d}}u}{{\text{d}}t} =\displaystyle= f(u)-Rkwk+RI(t)\displaystyle f(u)-R\,\sum_{k}w_{k}+R\,I(t) (6.1)
τkdwkdt\displaystyle\tau_{k}\,\frac{{\text{d}}w_{k}}{{\text{d}}t} =\displaystyle= ak(u-urest)-wk+bkτkt(f)δ(t-t(f)).\displaystyle a_{k}\,(u-u_{\rm rest})-w_{k}+{b_{k}\tau_{k}}\sum_{t^{(f)}}% \delta(t-t^{(f)}). (6.2)

The coupling of voltage to the adaptation current wkw_{k} is implemented by the parameter aka_{k} and evolves with time constant τk\tau_{k}. The adaptation current is fed back to the voltage equation with resistance RR. Just as in other integrate-and-fire models, the voltage variable uu is reset if the membrane potential reaches the numerical threshold Θreset\Theta_{\rm reset}. The moment u(t)=Θresetu(t)=\Theta_{\rm reset} defines the firing time t(f)=tt^{(f)}=t. After firing, integration of the voltage restarts at u=uru=u_{r}. The δ\delta-function in the wkw_{k} equations indicates that, during firing, the adaptation currents wkw_{k} are increased by an amount bkb_{k}. For example, a value bk=10b_{k}=10pA means that the adaptation current wkw_{k} is 10pA stronger after a spike than it was just before the spike. The parameters bkb_{k} are the ‘jump’ of the spike-triggered adaptation. One possible biophysical interpretation of the increase is that during the action potential calcium enters into the cell so that the amplitude of a calcium-dependent potassium current is increased. The biophysical origins of adaptation currents will be discussed in Section 6.3. Here we are interested in the dynamics and neuronal firing patterns generated by such adaptation currents.

Fig. 6.1: Multiple firing patterns in cortical neurons. For each type, the neuron is stimulated with a step current with low or high amplitude. Modified from Markram et al. (329).

Various choices are possible for the nonlinearity f(u)f(u) in the voltage equation. We have seen in the previous chapter (Section 5.2) that the experimental data suggests a nonlinearity consisting of a linear leak combined with an exponential activation term, f(u)=-(u-urest)+ΔTexp(u-ϑrhΔT)f(u)=-(u-u_{\rm rest})+\Delta_{T}\,\exp\left({u-\vartheta_{rh}\over\Delta_{T}}\right) . The Adaptive Exponential Integrate-and-Fire model (AdEx) consists of such an exponential nonlinearity in the voltage equation coupled to a single adaptation variable ww

τmdudt\displaystyle\tau_{m}\frac{{\text{d}}u}{{\text{d}}t} =\displaystyle= -(u-urest)+ΔTexp(u-ϑrhΔT)-Rw+RI(t)\displaystyle-(u-u_{\rm rest})+{\Delta_{T}}\,\exp\left({u-\vartheta_{rh}\over% \Delta_{T}}\right)-R\,w+R\,I(t) (6.3)
τwdwdt\displaystyle\tau_{w}\frac{{\text{d}}w}{{\text{d}}t} =\displaystyle= a(u-urest)-w+bτwt(f)δ(t-t(f)).\displaystyle a\,(u-u_{\rm rest})-w+b\tau_{w}\,\sum_{t^{(f)}}\delta(t-t^{(f)})\,. (6.4)

At each threshold crossing the voltage is reset to u=uru=u_{r} and the adaptation variable ww is increased by an amount bb. Adaptation is characterized by two parameters: aa couples adaptation to the voltage and is the source of subthreshold adaptation. Spike-triggered adaptation is controlled by a combination of aa and bb. The choice of aa and bb largely determines the firing patterns of the neuron (Section 6.2) and can be related to the dynamics of ion channels (Section 6.3). Before exploring the AdEx model further, we discuss two other examples of adaptive integrate-and-fire models.

Example: Izhikevich model

While the AdEx model exhibits the nonlinearity of the exponential integrate-and-fire model, the Izhikevich model uses the quadratic integrate-and-fire model for the first equation

τm   d   u   d   t\displaystyle\tau_{m}\,\frac{{\text{d}}u}{{\text{d}}t} =\displaystyle= (u-urest)(u-ϑ)-Rw+RI(t)\displaystyle(u-u_{\rm rest})(u-\vartheta)-R\,w+R\,I(t) (6.5)
τw   d   w   d   t\displaystyle\tau_{w}\frac{{\text{d}}w}{{\text{d}}t} =\displaystyle= a(u-urest)-w+bτwt(f)δ(t-t(f))\displaystyle a\,(u-u_{\rm rest})-w+b\tau_{w}\,\sum_{t^{(f)}}\delta(t-t^{(f)}) (6.6)

If u=θresetu=\theta_{\rm reset}, the voltage is reset to u=uru=u_{r} and the adaptation variable ww is increased by an amount bb. Normally bb is positive, but b<0b<0 is also possible.

Fig. 6.2: Multiple firing patterns in the AdEx neuron model. For each set of parameters, the model is stimulated with a step current with low or high amplitude. The spiking response can be classified by the steady-state firing behavior (vertical axis: tonic, adapting, bursting) and by its transient initiation pattern as shown along the horizontal axis: tonic (i.e. no special transient behavior), initial burst, or delayed spike initiation.

Example: Leaky model with adaptation

Adaptation variables wkw_{k} can also be combined with a standard leaky integrate-and-fire model

τm   d   u   d   t\displaystyle\tau_{m}\frac{{\text{d}}u}{{\text{d}}t} =\displaystyle= -(u-urest)-Rkwk+RI(t)\displaystyle-(u-u_{\rm rest})-R\,\sum_{k}w_{k}+R\,I(t) (6.7)
τk   d   wk   d   t\displaystyle\tau_{k}\frac{{\text{d}}w_{k}}{{\text{d}}t} =\displaystyle= a(u-urest)-wk+bkτkt(f)δ(t-t(f))\displaystyle a\,(u-u_{\rm rest})-w_{k}+b_{k}\tau_{k}\,\sum_{t^{(f)}}\delta(t-% t^{(f)}) (6.8)

At the moment of firing, defined by the threshold condition u(t(f))=θresetu(t^{(f)})=\theta_{\rm reset}, the voltage is reset to u=uru=u_{r} and the adaptation variables wkw_{k} are increased by an amount bkb_{k}. Note that in the leaky integrate-and-fire model the numerical threshold θreset\theta_{\rm reset} coincides with the voltage threshold ϑ\vartheta that one would find with short input current pulses.