Fractional Correlation Measure for Spatial Coherence

This paper presents a generalization of the correlation measure in space, which has been commonly used in the analysis of partial synchronization regimes, in particular chimera states, in systems of interacting oscillators. The chosen approach is based on a generalization of classical centered differences to their fractional analogue. The presented spectrum of fractional measures of spatial correlation makes it possible to classify partial synchronization modes developed at the level of total synchronization and at the level of phase waves of certain scales in discrete systems. The latter case is essential for neuroscience applications and mathematical study of the dynamic behavior of networks of interacting neurons, due to the importance of both partial synchronization states and large-scale cortical waves in sensory information processing and memory consolidation.