A design principle of slow-wave sleep firing pattern with Na+ dynamics

Non-rapid eye movement (NREM) sleep is characterized by electroencephalography (EEG) signals with high amplitude and low frequency. This signal is thought to originate from the synchronized activity of cortical neurons, showing the alternating bursting state (up state) and resting state (down state). This activity is termed as slow-wave sleep (SWS) firing pattern. We previously proposed the importance of Ca2+-dependent hyperpolarization pathway in generating this firing pattern by introducing the averaged-neuron (AN) model, which describes neuronal activity based on the Hodgkin-Huxley type model. In the AN model, Ca2+-dependent K+ channels are involved in the transition from the up to the down state. Here we focus on the intracellular Na+ dynamics which are not explicitly described in the AN model. A revised AN model, termed as Na+-centered AN (NAN) model, proposes that the activation of voltage-gated Na+ channels leads to intracellular Na+ accumulation, which in turn triggers the activation of Na+-dependent K+ (KNa) channels or Na+/K+ ATPases, resulting in the down state. Changes in the activation kinetics of voltage-gated Na+ channels are important in shaping SWS firing pattern as well as explaining the inter-spike interval changes between SWS and AWAKE firing pattern. Mathematically, transition from the up state to the down state occurs in accordance with the change in the number of the fixed point in the dynamical system with the changes in the intracellular Na+ concentration. The importance of Na+-dependent pathway is elucidated even with the coexistence of Ca2+-dependent pathway. Subsequent analysis with network model suggests that the result of averaged neuron model with Na+ pathway can be extended to the population of neurons. Therefore, our model proposes that voltage-gated Na+ channels and Na+-dependent K+ channels or Na+/K+ ATPases are also the candidate pathways for the generation of SWS firing pattern.