“…This noise can influence the transmission and integration of signals from other neurons and alter the firing activity of neurons in isolation [2][3][4], and there are some significant effects near bifurcation points [5,6], the weak neural noise, that seem to be less relevant when the neurons operate in spike generating regime for a suprathreshold; however, the situation is completely different in the neighborhood of threshold where noise can induce significant changes in the impulse patterns; furthermore, in the central neural system, the neurons often work in the neighborhood of threshold, but neurons are heterogeneous and noise is inevitable [7]. Sleep is essential for the maintenance of the brain and the body, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms; in addition, noise is an inevitable factor in real neuronal systems, which plays an important role in spatiotemporal dynamics of neuronal networks, for nearly a century of study; some regulation nonlinear sleep models about circadian [8], diversity-induced resonance [9], temporal dynamics [10], physiological substrates [11,12], and more have been proposed to investigate the neural regulatory mechanism for sleep-wake cycle; however, sleep and its underlying processes still hold many mysteries; it remains unclear how identified brain regions interact to bring about the different stages of sleep and wakefulness, how the timing of sleep depends on the length of time spent awake and work load, and how pathologies associated with sleep, such as narcolepsy, arise [13]. As we all know, there are homeostatic formation mechanism and biological function 2 Complexity between connection neurons [14,15]; in this work, we study the spatiotemporal behaviors of noise effects in the neighborhood of stimulus threshold for a mathematical model of homeostatic regulation of sleep-wake cycles proposed by Postnova et al [16].…”