Vector strength after Goldberg, Brown, and von Mises: biological and mathematical perspectives

The vector strength, a number between 0 and 1, is a classical notion in biology. It was first used in neurobiology by Goldberg and Brown (J Neurophys 31:639-656, 1969) but dates back at least to von Mises (Phys Z 19:490-500, 1918). It is widely used as a means to measure the periodicity or lack of periodicity of a neuronal response to an outside periodic signal. Here, we provide a self-contained and simple treatment of a closely related notion, the synchrony vector, a complex number with the vector strength as its absolute value and with a definite phase that one can directly relate to a biophysical delay. The present analysis is essentially geometrical and based on convexity. As such it does two things. First, it maps a sequence of points, events such as spike times on the time axis, onto the unit circle in the complex plane so that for a perfectly periodic repetition, a single point on the unit circle appears. Second, events hardly ever occur periodically, so that we need a criterion of how to extract periodicity out of a set of real numbers. It is here where convex geometry comes in, and a geometrically intuitive picture results. We also quantify how the events cluster around a period as the vector strength goes to 1. A typical example from the auditory system is used to illustrate the general considerations. Furthermore, von Mises' seminal contribution to the notion of vector strength is explained in detail. Finally, we generalize the synchrony vector to a function of angular frequency, not fixed on the input frequency at hand and indicate its potential as a "resonating" vector strength.