How adaptation currents change threshold, gain and variability of neuronal spiking

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.