Abstract: Modeling Neuronal Activity using systems of ordinary differential Equations and the Hodgkin Huxley Equations

Within the brainstem, there are two groups of cells that control sleep-wake cycles: sleep-active and wake-active cells. In infancy, mammals such as rats randomly switch in between bouts of sleep and wakefulness. During sleep bouts, the sleep population is active and inhibits the wake population. Due to a random fluctuation, wake-active cells can become active and sleep cells are then inhibited by the wake population. Throughout rat infancy, periods of wakefulness and sleep become increasingly long. The length of these bouts can be measured and graphed in terms of the probability of a switch from one state to another. From postnatal days 0-10, when the probability of being in a sleep boat and being in a wake bout are each graphed versus time, an exponential distribution arises. However, this changes between postnatal days 10-12, when wake bouts become power rule distributed.

The hazard rate is defined as the probability of switching states, that is, the probability of switching from sleep to wakefulness and vice versa. A power rule distribution therefore means that, the longer a wake bout is, the smaller the probability of a switch to sleep population, the hazard rate, is. Theoretically, there are two possible causes of this: steadily increasing excitation to the wake population during a wake bout, or steadily increasing inhibition to the sleep population during a wake bout. Experimental results show that the Locus Coeruleus, another group of cells, is responsible for the switch from an exponential distribution to a power law distribution, although the biophysical reasoning is still unknown. Thus, the Locus Coeruleus is either engaging in a positive feedback loop with wake-active cells, causing increasing levels of excitation of the wake-active cells, or is progressively increasing inhibition to the sleep-active cells.

Furthermore, The Hodgkin Huxley Equations are a system of experimentally-developed ordinary differential equations that can be used to model action potentials in Neurons. Thus, these equations can be used to model the interactions between the Locus Coeruleus, the wake-active population, and the sleep-active population.

During the previous semester, we have worked to understand the Hodgkin Huxley Equations and to model neuronal activity. Over the summer, by expanding this code, we can thus create an interactive model of groups of neurons. That is, we can model the interactions between sleep-active cells, wake-active cells, and the Locus Coeruleus. One major hypothesis we hope to explore is that the LC works by inhibiting sleep-active cells directly. We can thus run computational simulations to explore how these groups of neurons are interrelated, and new knowledge about these interactions can emerge.