Capturing multiple timescales of adaptation to second-order statistics with generalized linear models: gain scaling and fractional differentiation

Single neurons can dynamically change the gain of their spiking responses to account for shifts in stimulus variance. Moreover, gain adaptation can occur across multiple timescales. Here, we examine the ability of a simple statistical model of spike trains, the generalized linear model (GLM), to account for these adaptive effects. The GLM describes spiking as a Poisson process whose rate depends on a linear combination of the stimulus and recent spike history. The GLM successfully replicates gain scaling observed in Hodgkin-Huxley simulations of cortical neurons that occurs when the ratio of spike-generating potassium and sodium conductances approaches one. Gain scaling in the GLM depends on the length and shape of the spike history filter. Additionally, the GLM captures adaptation that occurs over multiple timescales as a fractional derivative of the stimulus variance, which has been observed in neurons that include long timescale after hyperpolarization conductances. Fractional differentiation in GLMs requires long spike history that span several seconds. Together, these results demonstrate that the GLM provides a tractable statistical approach for examining single-neuron adaptive computations in response to changes in stimulus variance.