**Mathematically modeling the
Neurochemistry of Human Sleep-wake Cycles**

Lisa Rogers, CIMS

Abstract:

In order to improve upon previously established mathematical models of human sleep/wake
system behavior, an analysis, computation and experimental data driven
construction of the complex system is presented. The correlations
between the chemical and mathematical aspects of specific neuron function is
used to determine the mechanics of the sleep/wake cycle. Aspects of circadian
biology theory are utilized, with an emphasis on the buildup of restorative
REM/NREM oscillations to demonstrate the unfailing circadian regulation
process. The construction of the model draws upon limit cycle behavior and
stability theory with references to an established background in the laws of
mass action. The initial assumptions and predictions are validated by a
combination of computational methods and previously established knowledge of
the function of human circadian processes. The consequences of qualitative and
quantitative analysis are utilized to establish various physiological
implications. We use neurotransmitter data from experimentalists to derive
equations. We use a linearized stability analysis as
well as eigenvalue-finding algorithms to help find
coefficients of the system and to determine a reasonable area in which
solutions exist. We then use experimental data from normal sleep- wake systems
and orexin knockout systems to verify the
physiological validity of the equations.