The detailed molecular and cellular mechanisms underlying NREM sleep (slow-wave sleep) and REM sleep (paradoxical sleep) in mammals are still elusive. To address these challenges, we first constructed a mathematical model, Averaged Neuron Model (AN Model), which recapitulates the electrophysiological characteristics of the slow-wave sleep. Comprehensive bifurcation analysis predicted that a Ca²⁺-dependent hyperpolarization pathway may play a role in slow-wave sleep. To experimentally validate this prediction, we generate and analyze 26 KO mice, and found that impaired Ca²⁺-dependent K⁺ channels (Kcnn2 and Kcnn3), voltage-gated Ca²⁺ channels (Cacna1g and Cacna1h), or Ca²⁺/calmodulin-dependent kinases (Camk2a and Camk2b) decrease sleep duration, while impaired plasma membrane Ca²⁺ ATPase (Atp2b3) increases sleep duration. Genetical (Nr3a) and pharmacological intervention (PCP, MK-801 for Nr1/Nr2b) and whole-brain imaging validated that impaired NMDA receptors reduce sleep duration and directly increase the excitability of cells. Based on these results, we propose phoshporylation hypothesis of sleep that phosphorylation-dependent regulation of Ca²⁺-dependent hyperpolarization pathway underlies the regulation of sleep duration in mammals. We also recently developed a simplified mathematical model, Simplified Averaged Neuron Model (SAN Model), which uncover the important role of K⁺ leak channels in NREM sleep. In this talk, I will also describe how we identify essential genes (Chrm1 and Chrm3) in REM sleep regulation, and propose a plausible molecular definition of a paradoxical state of REM sleep.