Hawkes process modeling quantifies complicated firing behaviors of cortical neurons during sleep and wakefulness

Despite the importance of sleep to the cerebral cortex, how much sleep changes cortical neuronal firing remains unclear due to complicated firing behaviors. Here we quantified firing of cortical neurons using Hawkes process modeling that can model sequential random events exhibiting temporal clusters. "Intensity" is a parameter of Hawkes process that defines the probability of an event occurring. We defined the appearance of repetitive firing as the firing intensity corresponding to "intensity" in Hawkes process. Firing patterns were quantified by the magnitude of firing intensity, the time constant of firing intensity, and the background firing intensity. The higher the magnitude of firing intensity, the higher the likelihood that the spike will continue. The larger the time constant of firing intensity, the longer the repetitive firing lasts. The higher the background firing intensity, the more likely neurons fire randomly. The magnitude of firing intensity was inversely proportional to the time constant of firing intensity, and non-REM sleep increased the magnitude of firing intensity and decreased the time constant of firing intensity. The background firing intensity was not affected by the sleep/wake state. Our findings suggest that the cortex is organized such that neurons with a higher probability of repetitive firing have shorter repetitive firing periods. In addition, our results suggest that repetitive firing is ordered to become high frequency and short term during non-REM sleep, while unregulated components of firing are independent of the sleep/wake state in the cortex. Hawkes process modeling of firing will reveal novel properties of the brain.