First we focus on one variable $w$ at a time and study its subthreshold coupling to the voltage. In other words, the aim is to give a biophysical interpretation of the parameters $a$, $\tau_{w}$, and the variable $w$ that show up in the adaptation equation

The biophysical components of spike-triggered adaptation (i.e., the interpretation of the reset parameter $b$) is deferred to Section 6.3.2. Here and in the following we write $E_{0}$ instead of $u_{\rm rest}$ in order to simplify notation and keep the treatment slightly more general.

As discussed in Chapter 2, neurons contain numerous ion channels (Section 2.3). Rapid activation of the sodium channels, important during the upswing of action potentials, is well approximated (Fig. 5.4) by the exponential nonlinearity in the voltage equation of the AdEx model, Eq. (6.3). We will see now that the subthreshold current $w$ is linked to the dynamics of other ion channels with a slower dynamics.

Let us focus on the model of a membrane with a leak current and a single, slow, ion channel, say a potassium channel of the Hodgkin-Huxley type

where $R_{L}$ and $E_{L}$ are the resistance and reversal potential of the leak current, $\tau_{m}=R_{L}C$ the membrane time constant, $g_{K}$ the maximal conductance of the open channel and $n$ the gating variable (which appears with arbitrary power $p$) with dynamics

As long as the membrane potential stays below threshold, we can linearize the equations (6.16) and (6.17) around the resting voltage $E_{0}$, given by the fixed point condition

The resting potential is shifted with respect to the leak reversal potential if the channel is partially open at rest, $n_{0}(E_{0})>0$. We introduce a parameter $\beta=g_{K}\,p\,n_{0}(E_{0})^{p-1}\,(E_{0}-E_{K})$ and expand $n_{0}(u)=n_{0}(E_{0})+n_{0}^{\prime}\,(u-E_{0})$ where $n_{0}^{\prime}$ is the derivative $dn_{0}/du$ evaluated at $E_{0}$.

The variable $w=\beta\,[n-n_{0}(E_{0})]$ then follows the linear equation

We emphasize that the time constant of the variable $w$ is given by the time constant of the channel at the resting potential. The parameter $a$ is proportional to the sensitivity of the channel to a change in the membrane voltage, as measured by the slope $dn_{0}/du$ at the equilibrium potential $E_{0}$.

The adaptation variable $w$ is coupled into the voltage equation in the standard form

Note that the membrane time constant and the resistance are rescaled by a factor $[1+(R_{L}\,g_{K})\,n_{0}(E_{0})^{p}\,]^{-1}$ with respect to their values in the passive membrane equation, Eq. ((6.16)). In fact, both are smaller because of partial opening of the channel at rest.

In summary, each channel with nonzero slope $dn_{0}/du$ at the equilibrium potential $E_{0}$ gives rise to an effective adaptation variable $w$. Since there are many channels, we can expect many variables $w_{k}$. Those with similar time constants can be summed and grouped into a single equation. But if time constants are different by an order of magnitude or more then several adaptation variables are needed, which leads to the model equations (6.1) and (6.2).

We have seen in Chapter 2 that some ion channels are partially open at the resting potential, while others react only when the membrane potential is well above the firing threshold. We now focus on the second group in order to give a biophysical interpretation of the jump amplitude $b$ of a spike-triggered adaptation current.

Let us return to the example of a single ion channel of the Hodgkin and Huxley type such as the potassium current in Eq. (6.16). In contrast to the treatment before, we now study the change in the state of the ion channel induced during the large-amplitude excursion of the voltage trajectory during a spike. During the spike, the target $n_{0}(u)$ of the gating variable is close to one; but since the time constant $\tau_{n}$ is long, the target is not reached during the short time that the voltage stays above the activation threshold. Nevertheless, the ion channel is partially activated by the spike. Unless the neuron is firing at a very large firing rate, each additional spike activate the channel further, always by the same amount $\Delta_{n}$, which depends on the duration of the spike and the activation threshold of the current (Table 6.2). The spike-triggered jump in the adapting current $w$ is then

where $\beta=g_{K}\,p\,n_{0}(E_{0})^{p-1}\,(E_{0}-E_{K})$ has been defined before.

Again, real neurons with their large quantity of ion channels have many adaptation currents $w_{k}$, each with its own time constant $\tau_{k}$, subthreshold coupling $a_{k}$ and spike-triggered jump $b_{k}$. The effective parameter values depend on the properties of the ion channels (Table 6.2).

While in the previous paragraph, we have focused on the role of ion channels, here we show that a passive dendrite can also give rise to a subthreshold coupling of the form of Eq. (6.15).
We focus on a simple neuron model with two compartments, representing the soma and the dendrite, superscripts $s$ and $d$ respectively. The two compartments are both passive with membrane potential $V^{s},\,V^{d}$, transversal resistance $R_{\text{T}}^{s},\,R_{\text{T}}^{d}$, capacity $C^{s},\,C^{d}$ and resting potential $u_{\rm r},\,E^{d}$. The two compartments are linked by a longitudinal resistance $R_{\text{L}}$ (see Chapter 3). If current is injected only in the soma, then the two-compartment model with passive dendrites corresponds to

Such a system of differential equations can be mapped to the form of Eq. (6.15) by considering that the variable $w$ represents the current flowing from the dendrite into the soma. In order to keep the treatment transparent, we assume that $E^{d}=u_{\rm rest}=E$. In this case the adaptation current is $w=-(V^{d}-u_{\rm rest})/R_{\text{L}}$ and the two equations above reduce to

with an effective input resistance $R^{\rm eff}=R^{s}_{\rm T}/[1+(R^{s}_{\rm T}/R_{\rm L})]$, an effective somatic time constant $\tau^{\rm eff}=C^{s}\,R^{\rm eff}$ an effective adaptation time constant $\tau_{w}=R_{\rm L}\,C^{d}/[1+(R_{\rm L}/R_{\rm D})]$ and a coupling between somatic voltage and adaptation current $a=-[R_{\rm L}+(R^{2}_{\rm L}/R_{\rm D})]^{-1}$.

There are three conclusions we should retain from this mapping. First, $a$ is always negative, which means that passive dendrites introduce a facilitating subthreshold coupling. Second, facilitation is particularly strong with a small longitudinal resistance. Third, the timescale of the facilitation $\tau_{w}$ is smaller than the dendritic time constant $R^{d}_{\text{T}}C^{d}$ - so that, compared to other ‘adaptation’ currents, the dendritic current is a relatively fast one.

In addition to the subthreshold coupling discussed here, dendritic coupling can also lead to a spike-triggered current as we will see in the next example.