Neural systems adapt to changes in stimulus statistics. However, it is not known how stimuli with complex temporal dynamics drive the dynamics of adaptation and the resulting firing rate. For single neurons, it has often been assumed that adaptation has a single time scale. Here, we show that single rat neocortical pyramidal neurons adapt with a time scale that depends on the time scale of changes in stimulus statistics. This multiple time scale adaptation is consistent with fractional order differentiation, such that the neuron’s firing rate is a fractional derivative of slowly varying stimulus parameters. Biophysically, even though neuronal fractional differentiation effectively yields adaptation with many time scales, we find that its implementation requires only a few, properly balanced known adaptive mechanisms. Fractional differentiation provides single neurons with a fundamental and general computation that can contribute to efficient information processing, stimulus anticipation, and frequency independent phase shifts of oscillatory neuronal firing.

Neuronal firing is known to depend on the variance of synaptic input as well as the mean input current. Several studies suggest that input variance, or "noise," has a divisive effect, reducing the slope or gain of the firing frequency-current ( f-I) relationship. We measured the effects of current noise on f-I relationships in pyramidal neurons and fast-spiking (FS) interneurons in slices of rat sensorimotor cortex. In most pyramidal neurons, noise had a multiplicative effect on the steady-state f-I relationship, increasing gain. In contrast, noise reduced gain in FS interneurons. Gain enhancement in pyramidal neurons increased with stimulus duration and was correlated with the amplitude of the slow afterhyperpolarization (sAHP), a major mechanism of spike-frequency adaptation. The 5-HT 2 receptor agonist ␣-methyl-5-HT reduced the sAHP and eliminated gain increases, whereas augmenting the sAHP conductance by spike-triggered dynamic-current clamp enhanced the gain increase. These results indicate that the effects of noise differ fundamentally between classes of neocortical neurons, depending on specific biophysical properties including the sAHP conductance. Thus, noise from background synaptic input may enhance network excitability by increasing gain in pyramidal neurons with large sAHPs and reducing gain in inhibitory FS interneurons.

The frequency response properties of neurons are critical for signal transmission and control of network oscillations. At subthreshold membrane potential, some neurons show resonance caused by voltage-gated channels. During action potential firing, resonance of the spike output may arise from subthreshold mechanisms and/or spike-dependent currents that cause afterhyperpolarizations (AHPs) and afterdepolarizations (ADPs). Layer 2-3 pyramidal neurons (L2-3 PNs) have a fast ADP that can trigger bursts. The present study investigated what stimuli elicit bursting in these cells and whether bursts transmit specific frequency components of the synaptic input, leading to resonance at particular frequencies. We found that two-spike bursts are triggered by step onsets, sine waves in two frequency bands, and noise. Using noise adjusted to elicit firing at ϳ10 Hz, we measured the gain for modulation of the time-varying firing rate as a function of stimulus frequency, finding a primary peak (7-16 Hz) and a high-frequency resonance (250 -450 Hz). Gain was also measured separately for single and burst spikes. For a given spike rate, bursts provided higher gain at the primary peak and lower gain at intermediate frequencies, sharpening the high-frequency resonance. Suppression of bursting using automated current feedback weakened the primary and high-frequency resonances. The primary resonance was also influenced by the SK channel-mediated medium AHP (mAHP), because the SK blocker apamin reduced the sharpness of the primary peak. Our results suggest that resonance in L2-3 PNs depends on burst firing and the mAHP. Bursting enhances resonance in two distinct frequency bands.