Fractional differentiation by neocortical pyramidal neurons

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. We found 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 found that its implementation required 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.