On the fractional Eulerian numbers and equivalence of maps with long term power-law memory (integral Volterra equations of the second kind) to Grünvald-Letnikov fractional difference (differential) equations

We examine the dynamics of a neural code in the context of stimuli whose statistical properties are themselves evolving dynamically. Adaptation to these statistics occurs over a wide range of timescales-from tens of milliseconds to minutes. Rapid components of adaptation serve to optimize the information that action potentials carry about rapid stimulus variations within the local statistical ensemble, while changes in the rate and statistics of action-potential firing encode information about the ensemble itself, thus resolving potential ambiguities. The speed with which information is optimized and ambiguities are resolved approaches the physical limit imposed by statistical sampling and noise.

We examined fluctuations in band-limited power (BLP) of local field potential (LFP) signals recorded from multiple electrodes in visual cortex of the monkey during different behavioral states. We asked whether such signals demonstrated coherent fluctuations over time-scales of seconds and minutes, and would thus serve as good candidates for direct comparison with data obtained from functional magnetic resonance imaging (fMRI). We obtained the following results. (i) The BLP of the local field displayed fluctuations at many time-scales, with particularly large amplitude at very low frequencies (<0.1 Hz). (ii) These fluctuations exhibited high coherence between electrode pairs, particularly for BLP signals derived from the gamma frequency range. (iii) Coherence in the BLP, unlike that in the raw LFP, did not fall off sharply as a function of cortical distance. (iv) The structure and coherence of BLP changes were highly similar under distinctly different behavioral states. These results demonstrate the existence of widespread coherent activity fluctuations in the brain of the awake monkey over very long time-scales. We propose that such signals may make a significant contribution to the high variability observed in the time course of physiological signals, including those measured with functional imaging techniques. The results are discussed in the context of combined fMRI/electrophysiological recordings.