A Physiologically Based Model of Orexinergic Stabilization of Sleep and Wake

The orexinergic neurons of the lateral hypothalamus (Orx) are essential for regulating sleep-wake dynamics, and their loss causes narcolepsy, a disorder characterized by severe instability of sleep and wake states. However, the mechanisms through which Orx stabilize sleep and wake are not well understood. In this work, an explanation of the stabilizing effects of Orx is presented using a quantitative model of important physiological connections between Orx and the sleep-wake switch. In addition to Orx and the sleep-wake switch, which is composed of mutually inhibitory wake-active monoaminergic neurons in brainstem and hypothalamus (MA) and the sleep-active ventrolateral preoptic neurons of the hypothalamus (VLPO), the model also includes the circadian and homeostatic sleep drives. It is shown that Orx stabilizes prolonged waking episodes via its excitatory input to MA and by relaying a circadian input to MA, thus sustaining MA firing activity during the circadian day. During sleep, both Orx and MA are inhibited by the VLPO, and the subsequent reduction in Orx input to the MA indirectly stabilizes sustained sleep episodes. Simulating a loss of Orx, the model produces dynamics resembling narcolepsy, including frequent transitions between states, reduced waking arousal levels, and a normal daily amount of total sleep. The model predicts a change in sleep timing with differences in orexin levels, with higher orexin levels delaying the normal sleep episode, suggesting that individual differences in Orx signaling may contribute to chronotype. Dynamics resembling sleep inertia also emerge from the model as a gradual sleep-to-wake transition on a timescale that varies with that of Orx dynamics. The quantitative, physiologically based model developed in this work thus provides a new explanation of how Orx stabilizes prolonged episodes of sleep and wake, and makes a range of experimentally testable predictions, including a role for Orx in chronotype and sleep inertia.


Nullclines and equilibriums
In this section we present equations that define the features of V v -V m plots, shown in Fig. 2 of the main text of the manuscript, and derived in previous work [1]. On timescales longer than τ v and τ m , D v and D m can be treated as control parameters of the fast model dynamics [1], which can be understood in terms of the nullclinesV v = 0: andV m = 0: Note that in this work, since V x contributes to D m , this analysis also assumes thatV x ≈ 0 on the timescale of τ m , τ v , which will be a good approximation for τ x τ v , τ m (as is the case here). Equilibriums occur at V v = V * v and V m = V * m , at the intersection of these two nullclines, and can be written implicitly in given where S (x) = dS(x)/dx. Equation (5) has solutions V v = V bif v , which allows us to then evaluate the corresponding drives at which bifurcations occur, D bif v , using The unstable manifold, W + , of a saddle point equilibrium, (V * v , V * m ), forms a separatrix between sleep and wake basins in V v -V m space. It is calculated numerically here by incrementally moving in the direction the negative of the vector field, (−V v , −V m ), after small positive and negative perturbations from the unstable equilibrium with the gradient , as derived using the Stable Manifold Theorem in previous work [1]. That is, (−V v , −V m ) is followed from initial points where is the small perturbation. A value = 0.01 mV is used here.

Parameter Constraints
The process through which the new structure and parameters of this model are constrained is explained in this section. Compared to the original Phillips-Robinson model, the current model includes several new parameters, as well as some adjustments of existing parameters, see Table 1  When simulating narcoleptic dynamics, we set ν mx = 0, so that the parameters controlling Orx dynamics have no effect on the sleep-wake switch. This allows a reduced number of parameters: A m , ν vc , and A v , to be constrained. The parameters A m (= 0.52 mV) and A v (= −8.5 mV), are set so that narcoleptic dynamics occur where thresholds for sleep-wake transitions are low, and could be modified to match clinical data for narcoleptics in future work. The 60% reduction in A m compared to previous work implies that approximately 60% of A m (the time-averaged drives to the MA modeled in previous work) can be attributed to Orx. In this model, the circadian input to the sleep-wake switch is now split between Orx and the VLPO, with a dominant pathway to Orx. The magnitude of the parameter ν vc is correspondingly lower than in previous work (i.e., ν vc = −2.9 mV s → ν vc = −0.29 mV s), motivated by the observation that circadian rhythmicity is strong in patients with VLPO legions [4]. The circadian phase dependence on sleep and wake for narcoleptics depends on A m and A v , and most importantly the oscillation magnitude ν vc .
Having fitted these parameters to reproduce key features of the narcoleptic phenotype, the remaining parameters are constrained by fitting to normal dynamics. Dynamical parameters for Orx are set to reproduce normal sleep-wake dynamics: τ x (= 2 min) is set to reproduce an approximate timescale for sleep inertia (as explained in the main text), ν xc (= 1.0 mV s) is set to provide an appropriate amount of circadian variation in waking arousal levels, ν mx (= 0.3 mV s) is set to relay an appropriate circadian variation to the MA during waking, and A x (= 1.0 mV) is set to a value that allowed the system to wake up in the early morning when the circadian drive is low. The parameters, µ h and η h are constrained by maintaining an approximately equal amount of time spent awake across the range of ν mx , that is, for both normals (that have Q m ≈ 6 s −1 during wake) and narcoleptics (that have Q m ≈ 2.4 s −1 during wake). The noise standard deviation, σ = 1 mV, is chosen to produce a realistic rate of arousal-state fragmentation in the narcoleptic phenotype.
Note that although many of the model parameters are set to values that produce sensible dynamics, they could be constrained more thoroughly, and also fitted to the sleep patterns of individuals, using clinical data. For example, the model could be fitted to state transition statistics and sleep-wake timings obtained from narcoleptic patients, as well as potentially constraining model parameters directly, using physiological data (e.g., of neuronal population firing rates in sleep and wake). We emphasize that our aim in this work is not to perform a thorough fitting of the model to real data, but to present a physiologically-plausible set of parameters that produces dynamics consistent with the known sleep-wake behavior, including how changes in Orx affect the dynamics.
The notation used in this work differs somewhat from previous work on the original Phillips-Robinson model [5]. In previous work, the only time-varying drive to the system was an input to the VLPO; in this work we have introduced a consistent labeling across the model equations for MA (m), VLPO (v), and Orx (x). For example, we label D (original model) as D v (current model), D 0 (original model) as A v (current model), and A (original model) as A m (current model).

Sleep deprivation
Because the model structure is different to that of previous work-with the addition of the Orx neuronal population and the dominant circadian input now acting through this population-we want to ensure that the behavior of the new model is consistent with previous results. Although it is beyond the scope of the current work to apply the new model to the full set of previously modeled phenomena, we have tested for consistency with both normal sleep-wake behavior and sleep deprivation. In this section, we characterize a simulated wake effort time series for sleep deprivation using the new model with Orx. The wake effort is defined as the minimal additional drive required to be applied to MA to maintain the system at a stable wake equilibrium [3]. As shown in Fig. 1, despite the presence of Orx in the model and that the dominant C drive is now afferent to MA rather than VLPO, the model produces the same pattern for the wake effort, with an oscillation superimposed on a increasing trend, but returning to baseline after the first night of total sleep deprivation [3]. The simulation is performed for nominal parameters without added noise by maintaining the system in the bistable region (and hence awake) whenever the system reaches the sleep bifurcation boundary. Although the Phillips-Robinson model only had a single time-varying drive, D v , sleep deprivation is simulated here in the same way: driving the system to wake using the minimal input to the MA [3].

Heuristic for labeling sleep and wake states
In this work, 'sleep' and 'wake' states sometimes need to be labeled from the output of our model, which produces time series for V v , V m , and V x . For this task, we use a simple heuristic that divides the model output into non-overlapping 20 s windows and performs a preliminary classification of each window as 'wake' if V m > V v more than half of the time, and 'sleep' otherwise. We then discounted bouts lasting less than 60 s as transient (indicative of the model not yet settling on a stable state), ignoring such brief interludes by labeling them as if they did not occur. The overall labeling produced by this procedure is not very sensitive to changes in these two parameters (i.e., the window length and minimum bout duration) across a range of reasonable choices.