  Many types of neurons exhibit spike rate adaptation, mediated by intrinsic
slow $\mathrm{K}^+$-currents, which effectively inhibit neuronal responses. How
these adaptation currents change the relationship between in-vivo like
fluctuating synaptic input, spike rate output and the spike train statistics,
however, is not well understood. In this computational study we show that an
adaptation current which primarily depends on the subthreshold membrane voltage
changes the neuronal input-output relationship (I-O curve) subtractively,
thereby increasing the response threshold. A spike-dependent adaptation current
alters the I-O curve divisively, thus reducing the response gain. Both types of
adaptation currents naturally increase the mean inter-spike interval (ISI), but
they can affect ISI variability in opposite ways. A subthreshold current always
causes an increase of variability while a spike-triggered current decreases
high variability caused by fluctuation-dominated inputs and increases low
variability when the average input is large. The effects on I-O curves match
those caused by synaptic inhibition in networks with asynchronous irregular
activity, for which we find subtractive and divisive changes caused by external
and recurrent inhibition, respectively. Synaptic inhibition, however, always
increases the ISI variability. We analytically derive expressions for the I-O
curve and ISI variability, which demonstrate the robustness of our results.
Furthermore, we show how the biophysical parameters of slow
$\mathrm{K}^+$-conductances contribute to the two different types of adaptation
currents and find that $\mathrm{Ca}^{2+}$-activated $\mathrm{K}^+$-currents are
effectively captured by a simple spike-dependent description, while
muscarine-sensitive or $\mathrm{Na}^+$-activated $\mathrm{K}^+$-currents show a
dominant subthreshold component.
