A trade-off between error and synchrony when using temporal codes

Neural oscillations can improve the fidelity of neural coding by grouping action potentials into synchronous windows of activity but this same effect can interfere with coding when action potentials become "over-synchronized". Diseases ranging from Parkinsons to epilepsy suggest such over-synchronization can lead to pathological outcomes, but the precise boundary separating healthy from pathological synchrony remains an open theoretical problem. In this paper, we focus on measuring the costs of translating from an aperiodic code to a rhythmic one and use the errors introduced in this translation to predict the rise of pathological results. We study a simple model of entrainment featuring a pacemaker population coupled to biophysical neurons. This model shows that "error" in individual cells computations can be traded for population-level synchronization of spike-times. But in this model error and synchronization are not traded linearly, but nonlinearly. The bulk of synchronization happens early with relatively low error. To predict this phenomenon we conceive of "voltage budget analysis", where small time windows of membrane voltage in single cells can be partitioned into "oscillatory" and "computational" terms. By comparing these terms we discover a set of inequalities that align with an inflection point in the curve of measured errors. In particular, when the entrainment and computational voltage terms are equal, the error curve plateaus. We show this point serves as a reliable natural boundary to define pathological synchrony in neurons. We also derive optimal algorithms for exchanging computational error with population synchrony. New and Noteworthy. We establish exact conditions for when rhythmic entrainment of precise spike-times in a neural population will improve or harm its ability to communicate.