A mathematical model towards understanding the mechanism of neuronal regulation of wake-NREMS-REMS states

Abstract

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.

Figure 1. Schematic depiction of the model.

POAH- Pre-optic anterior hypothalamus, MRF- Midbrain reticular formation, CRF- Caudal reticular formation, R-ON- REM active neurons (LDT/PPT), R-OFF- REM inactive neurons (LC), GABA-LC- GABA-ergic interneuron in LC, ORX- Orexin-ergic neurons, CIR- Circadian pacemaker from SCN, GABA-SNr - GABA-ergic input from substantia nigra pars reticulate, Hom- Homeostatic sleep drive.

Figure 2. Nullclines and trajectories showing different intrinsic and synaptic behaviors.

In panels A, B and D, filled squares denote position of initial conditions that evolve toward particular fixed points (solid circles). Single arrows denote slow flows; double arrows denote fast transitions. A. Nullclines for an intrinsically active cell group intersect on the right branch of the v-nullcline. B. Nullclines for an intrinsically silent cell group intersect on the left branch of the v-nullcline. C. Nullclines for an intrinsically oscillatory cell group intersect on the middle branch of the v-nullcline (open circle). D. Inhibition to an intrinsically active group lowers its nullcline causing the trajectory to move from the solid square location, to the solid circle location.

Figure 3. The behavior of POAH and MRF in their respective phase planes corresponding to the voltage traces of Fig. 4.

The dashed vertical line corresponds to the synaptic threshold v_th. In panel A. there are four different v-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 v-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.

Figure 4. Homeostatic and circadian regulation of sleep-wake cycling.

Dark bars indicate the times during which the system was asleep. The wake-promoting neurons MRF were active during wake. During this time, the homeostatic drive increased preparing the system for sleep. When large enough, it allowed the sleep-promoting POAH neurons to become active. Reciprocal inhibitory connections between MRF and POAH and MRF and CRF helped in the transition between states. During sleep, the homeostatic drive decreased, but the length of the sleep cycle was determined by intrinsic properties of POAH as is shown more clearly in the phase plane in Fig. 3. The circadian rhythm modulated the onset of sleep as will be described in section 3.4.

Figure 5. The activity of neuronal groups during a sleep deprivation experiment.

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).

Figure 6. Nullclines and trajectories during REMS and NREMS.

Panel A shows REM-on activity while Panel B shows REM-off activity. Solid squares show that the trajectories in the (NREMS) REM-off controlled mode were stuck at fixed points. Trajectories with arrows depicted the oscillatory dynamics in the (REMS) REM-on controlled mode.

Figure 7. The activity of various neuron groups during sleep-wake cycling.

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 Fig. 4.

Figure 8. The role of MRF inhibition. MRF inhibition was necessary to prevent REM-on activity during wake.

At both the times t1 and t2 during wake, pre-synaptic inhibition from GABA-SNr was initiated. This resulted in REM-on activity only in the second case since MRF inhibition was not present. The re-application of MRF inhibition terminated REM-on activity at time t3.

Figure 9. The putative REM homeostat.

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 t1, the strengthis increased from 0 to 0.57, resulting in REM bouts in subsequent sleep cycles.

Figure 10. Brief awakenings induced by reducing ORX inhibition of POAH.

At the time labeled t1, the inhibition from ORX to POAH was reduced resulting in a brief excursion to sleep. The duration of the subsequent sleep cycle was longer, but was interrupted by several brief awakenings. Note the level of the homeostat at the onset of the second sleep cycle was smaller than at the onset of the first due to reduced ORX inhibition.

Figure 11. Brief awakenings induced by reducing the homeostatic sleep drive to POAH.

At the time labeled t1, 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.

Figure 12. Narcolepsy induced by removal of ORX inhibition.

At time t1while 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.

Figure 13. The effect of feedback inhibition to ORX cells from REM-off neurons.

The simulation begins with. This amount of feedback inhibition did not qualitatively change the basic sleep-wake cycling (compare with Fig. 3). At time t1, 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 t2,was set to 0, which destroyed the cycling. At time t3was reset to 0.1 while g_h was increased to 7 which restored the normal cycling.