A Mathematical Model towards Understanding the Mechanism of Neuronal Regulation of Wake-NREMS-REMS States

<div><p>In this study we have constructed a mathematical model of a recently proposed functional model known to be responsible for inducing waking, NREMS and REMS. Simulation studies using this model reproduced sleep-wake patterns as reported in normal animals. The model helps to explain neural mechanism(s) that underlie the transitions between wake, NREMS and REMS as well as how both the homeostatic sleep-drive and the circadian rhythm shape the duration of each of these episodes. In particular, this mathematical model demonstrates and confirms that an underlying mechanism for REMS generation is pre-synaptic inhibition from substantia nigra onto the REM-off terminals that project on REM-on neurons, as has been recently proposed. The importance of orexinergic neurons in stabilizing the wake-sleep cycle is demonstrated by showing how even small changes in inputs to or from those neurons can have a large impact on the ensuing dynamics. The results from this model allow us to make predictions of the neural mechanisms of regulation and patho-physiology of REMS.</p> </div

The behavior of POAH and MRF in their respective phase planes corresponding to the voltage traces of Fig. 4.

<p>The dashed vertical line corresponds to the synaptic threshold <i>v_th</i>. In panel A. there are four different <i>v</i>-nullclines. The lowest one occurred during wake when the inhibition from MRF and ORX was present and the input from the homeostasis was at its smallest. The one above this corresponded to an increase in the homeostasis to just high enough to allow POAH to escape from the MRF and ORX inhibition. The third one corresponded to when POAH reached a local maximum of a <i>v</i>-nullcline indicating the end of the sleep state. The highest nullcline corresponded to when the system has just fallen asleep, the homeostasis was at its highest level and there was no inhibition from MRF and ORX. Note that the POAH trajectory was constantly changing the nullcline on which it lies. These nullclines were slowly shifting due to the changes in homeostatic and circadian input, so the POAH trajectory lay on a family of such nullclines bounded between the highest and the lowest.</p

Power spectral density of IMFs in wake, NREMS and REMS.

<p>The PSD of NREMS is larger for all of the modes than wake and REMS. In IMF2 and IMF7, the PSD is different for wake and REMS. These particular IMFs can be considered as characteristic of vigilance state that they represent. </p

Narcolepsy induced by removal of ORX inhibition.

<p>At time <i>t1</i>while the system was still asleep, the inhibition from ORX to POAH was removed. At the end of that sleep bout, the system was no longer able to consolidate wake and sleep into distinct episodes. The POAH trajectory was not sufficiently suppressed by the MRF inhibition alone to create a prolonged wake state.</p

The activity of neuronal groups during a sleep deprivation experiment.

<p>The parameters were tuned so that the activity of MRF was high and the activity of POAH was low during the second sleep wake cycle. The subsequent rebound sleep episode was longer in duration than the previous one. Depending on what phase of the circadian rhythm sleep deprivation was interrupted, brief awakenings appeared at the end of a sleep episode. Compare Panels (a) and (b).</p

The activity of various neuron groups during sleep-wake cycling.

<p>In the first sleep episode, the pre-synaptic inhibition from GABA-SNr was activated at two distinct times, each for different lengths. Pre-synaptic inhibition was triggered once during the second sleep episode. In each episode, this resulted in transient REM-on and REM-off oscillations. The overall transitions between wake and sleep were still governed by the homeostatic and circadian inputs as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042059#pone-0042059-g004" target="_blank">Fig. 4</a>.</p

The effect of feedback inhibition to ORX cells from REM-off neurons.

<p>The simulation begins with. This amount of feedback inhibition did not qualitatively change the basic sleep-wake cycling (compare with <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042059#pone-0042059-g003" target="_blank">Fig. 3</a>). At time <i>t1</i>, the feedback inhibition was changed to 0.5. The system was no longer able to stay in a prolonged state of sleep; sleep bouts were continually disrupted by brief awakenings. At time <i>t2,</i>was set to 0, which destroyed the cycling. At time <i>t3</i>was reset to 0.1 while <i>g_h</i> was increased to 7 which restored the normal cycling.</p

Fronto-frontal absolute EEG power spectral density (V2/Hz) in wake, NREMS and REMS of 10 sec epochs from n=12 rats.

<p>EEG power densities during NREMS were significantly higher than in REMS and wake. See text for details.</p

Brief awakenings induced by reducing the homeostatic sleep drive to POAH.

<p>At the time labeled <i>t1</i>, the maximal strength of the homeostatic drive to POAH was reduced resulting in longer subsequent wake duration and brief awakenings throughout the next sleep bout.</p

The putative REM homeostat.

<p>The strength of the CRF to REM-on excitatory synapse was allowed to depend on the length of the sleep and wake states. At the time denoted <i>t1</i>, the strengthis increased from 0 to 0.57, resulting in REM bouts in subsequent sleep cycles.</p