Modeling the impact of impulsive stimuli on sleep-wake dynamics

A neuronal population model of the sleep-wake switch is extended to incorporate impulsive external stimuli. The model includes the mutual inhibition of the sleep-active neurons in the hypothalamic ventrolateral preoptic area (VLPO) and the wake-active monoaminergic brainstem populations (MA), as well as circadian and homeostatic drives. Arbitrary stimuli are described in terms of their relative effects on the VLPO and MA nuclei and represent perturbations on the normal sleep-wake dynamics. By separating the model’s intrinsic time scales, an analytic characterization of the dynamics in a reduced model space is developed. Using this representation, the model’s response to stimuli is studied, including the latency to return to wake or sleep, or to elicit a transition between the two states. Since sensory stimuli are known to excite the MA, we correspondingly investigate the model’s response to auditory tones during sleep, as in clinical sleep fragmentation studies. The arousal threshold is found to vary approximately linearly with the model’s total sleep drive, which includes circadian and homeostatic components. This relationship is used to reproduce the clinically observed variation of the arousal threshold across the night, which rises to a maximum near the middle of the night and decreases thereafter. In a further application of the model, time-of-night arousal threshold and body temperature variations in an experimental sleep fragmentation study are replicated. It is proposed that the shift of the extrema of these curves to a greater magnitude later in the night is due to the homeostatic impact of the frequent nocturnal disturbances. By modeling the underlying neuronal interactions, the methods presented here allow the prediction of arousal state responses to external stimuli. This methodology is fundamentally different to previous approaches that model the clinical data within a phenomenological framework. As a result, a broader understanding of how impulsive external stimuli modulate arousal is gained.

Figure 1

Schematic of the Phillips-Robinson model of the sleep-wake switch. The mutually inhibitory VLPO and MA populations form the core of the model, which also includes circadian $\left(C\right)$ and homeostatic $\left(H\right)$ drives to the VLPO and the constant drive $A$ to the MA. Excitatory (+) and inhibitory (−) interactions are shown with arrows and arousal state feedback from the MA to $H$ is shown dashed. A schematic of the typical time evolution of each drive is also displayed [cf., Figs. 2c, 2d].

Figure 2

Normal, unperturbed output from the Phillips-Robinson model. Time series are shown for (a) $V_v$ [Eq. (2)], (b) $V_m$ [Eq. (3)], (c) ${\nu }_{vh}H$ [cf., Eq. (8)], (d) ${\nu }_{vc}C$ [cf., Eq. (7)], and (e) $D_v=D_v^0={\nu }_{vc}C+{\nu }_{vh}H$ [Eq. (6)]. Sleep periods are shaded.

Figure 3

Hysteresis in the Phillips-Robinson model. The mean cell-body potentials $V_m$ (solid black line) and $V_v$ (dashed gray line) are plotted against $D_v$, showing hysteresis. Wake (low $D_v$), sleep (high $D_v$), and bistable (intermediate) regions are labeled. The arousal state alternates between sleep and wake as the $24\mathrm{h}$ periodic sleep drive $D_v$ oscillates with time. Note that this plot is equivalent to plotting the time series in Figs. 2a, 2b, for $V_v$ and $V_m$, respectively, against that shown in Fig. 2e for $D_v$.

Figure 4

The reduced manifold in $V_v\text{−}{V}_m\text{−}{D}_v$ space. Stable sleep (high $V_v$, low $V_m$) and wake (low $V_v$, high $V_m$) branches (black lines) are connected by the intermediate unstable branch (gray line). Projections of the manifold onto the two-dimensional subspaces are also plotted.

Figure 5

Eigenvalues ${\lambda }_{\pm }$ of the Jacobian as a function of the control parameter $D_v$ at a saddle point (dashed), wake node (thick), and sleep node (solid), as obtained from Eq. (24). The bistable region is shaded and the lines $\lambda =-{\tau }^{-1}=-0.1$, $\lambda =0$, and $\lambda =-0.2$ are dotted.

Figure 6

(Color online) Correspondence between (a) the $D_v\text{−}{V}_m$ hysteresis representation and (b)–(d) the $V_v\text{−}{V}_m$ nullcline representation of the model. (a) Bifurcation diagram for $V_m$ against the control parameter $D_v$, showing hysteresis. Arrows indicate the direction of ${\dot{V}}_m$ for $D_v=1,2,3\mathrm{mV}$. Regions of low $\dot{V}$ have been shaded, from $\dot{V}=0$ (black) to $\dot{V}>0.05\mathrm{mV}{\mathrm{s}}^{-1}$ (white), where the minimum $\dot{V}$ (across variation in $V_v$) has been collapsed onto this picture. (b)–(d) $V_v\text{−}{V}_m$ plots showing the $V_v$ nullcline [Eq. (9)] (thick solid line), $V_m$ nullcline [Eq. (10)] (thin solid line), stable nodes (circles), saddle points (squares), the $W^{+}$ manifold (dashed line), the $W^{-}$ manifold of stable nodes (dotted line), the separatrix formed by the stable invariant manifold $W^{-}$ of the saddle point (dot-dashed line), the velocity vector field $\stackrel{\mathrm{̇}}{\mathbf{V}}$ [Eq. (13)] (arrows), and regions over which $\dot{V}<0.1\mathrm{mV}{\mathrm{s}}^{-1}$ (shading) for (b) $D_v=1\mathrm{mV}$, a stable wake node, (c) $D_v=2\mathrm{mV}$, in the bistable region, and (d) $D_v=3\mathrm{mV}$, a stable sleep node. Near-stable wake and sleep ghosts (cf., Sec. 3D) are also labeled.

Figure 7

The model’s response discrete $\Delta D_m$ drives of different durations. (a) $\Delta D_m$ profiles for drives of duration (i) $0.5\mathrm{s}$, (ii) $2\mathrm{s}$, (iii) $5\mathrm{s}$, and (iv) $10\mathrm{s}$, with $\int \Delta D_{m}dt=180\mathrm{mV}\mathrm{s}$, a constant. (b) Corresponding trajectories through the $V_v\text{−}{V}_m$ plane during an impulse (black) and during relaxation (gray), at $D_v=3\mathrm{mV}$. As the impulse duration increases, the trajectories are increasingly distorted away from the $\delta$-function limit (a vertical line).

Figure 8

(Color online) Model responses to $\delta$-function $\Delta \mathbf{D}$ stimuli. $\Delta V_v\text{−}\Delta V_m$ and $V_v\text{−}{V}_m$ plots are shown for (a) and (b): $D_v^0=1\mathrm{mV}$ (initial condition: wake node), (c) and (d): $D_v^0=2\mathrm{mV}$ (initial condition: wake node), (e) and (f): $D_v^0=2\mathrm{mV}$ (initial condition: sleep node), and (g) and (h): $D_v^0=3\mathrm{mV}$ (initial condition: sleep node). In the $\Delta V_v\text{−}\Delta V_m$ plots (defined relative to the initial equilibrium), the sleep latency $t_{\mathrm{lat}}$ is represented using shading, from $0\min$ (black) to $>6\min$ (white). Sample $\delta$-function impulses are shown as white arrows, with the corresponding trajectories through $V_v\text{−}{V}_m$ space shown in the adjacent plots for the impulse (black) and during relaxation back to a stable node (gray). Equilibria (open white circles and black squares) and invariant manifolds (dotted, dashed, and dot-dashed lines) are labeled as per Fig. 6. Note that points in the $\Delta V_v\text{−}\Delta V_m$ plane should be interpreted in terms of the $\delta$-function drives $(\Delta D_v,\Delta D_m)$ that produce them (cf., Sec. 4A).

Figure 9

The model’s response to excitatory $\delta$-function sensory stimuli $\Delta D_m$ of varying magnitude applied during sleep $(D_v=3\mathrm{mV})$. (a) The time to return to sleep $t_{\mathrm{lat}}$ following the impulse is plotted as a function of its strength $\Delta V_m$. For the five points marked in (a): $\Delta V_m=\left(\mathrm{i}\right)$ $10\mathrm{mV}$, (ii) $16\mathrm{mV}$, (iii) $18.7\mathrm{mV}$ (the point of inflection), (iv) $20\mathrm{mV}$, and (v) $22\mathrm{mV}$, corresponding trajectories through the $V_v\text{−}{V}_m$ plane are shown in (b), where regions of $\dot{V}<0.1\mathrm{mV}{\mathrm{s}}^{-1}$ have been shaded. The point of inflection is shown as a larger, closed circle in (a), with the arousal threshold $\mathcal{A}$ taken from the $\Delta V_m$ at this point and the critical sleep latency $\mathcal{T}$ taken from the $t_{\mathrm{lat}}$ at this point, as labeled.

Figure 10

The model’s response to $\Delta V_m=18\mathrm{mV}$ $\delta$-function impulses at different values of the sleep drive $D_v$. Trajectories through the $V_v\text{−}{V}_m$ plane are plotted during excitation (black) and relaxation (gray) for $D_v=\left(\mathrm{a}\right)$ $2.5\mathrm{mV}$, (b) $3.0\mathrm{mV}$, (c) $3.5\mathrm{mV}$, and (d) $4.0\mathrm{mV}$. Regions of $\dot{V}<0.1\mathrm{mV}{\mathrm{s}}^{-1}$ are shaded and plots of $\dot{V}$ as a function of time along each trajectory are shown inset. Note that the vertical scale of the inset figures is chosen to show the main features of the return to equilibrium, which does not always include the initial maximum in $\dot{V}$.

Figure 11

Time-of-night variation in the arousal threshold $\mathcal{A}$ and critical sleep latency $\mathcal{T}$. (a) $\Delta V_m\text{−}{t}_{\mathrm{lat}}$ curves are plotted for $D_v=2.5$, 3.0, 3.5, and $4.0\mathrm{mV}$. Points of inflection are marked with circles. (b) $\mathcal{A}$ as a function of $D_v$ (circles) with a fitted dotted line [Eq. (30)] and $\mathcal{T}$ as a function of $D_v$ (squares) with a fitted dashed line [Eq. (31)]. (c) $\mathcal{T}$ and (d) $\mathcal{A}$ across the first $\sim 7\mathrm{h}$ of sleep (the portion for which $D_v>2.46\mathrm{mV}$), with the fits from (b) shown dashed and dotted, respectively. Clinical auditory arousal threshold $I_c$ data from Bonnet et al. [53] is displayed using open circles and a dot-dashed line in (d).

Figure 12

The model’s simulation of the sleep fragmentation protocol reported by Lammers et al. [55]. (a) $V_m$ and (b) $H$ as a function of time for fragmented (solid) and normal (dashed) sleep.

Figure 13

The model’s simulation of the sleep fragmentation protocol reported by Lammers et al. [55]. The auditory arousal threshold $I_c$ (closed circles) and body temperature $T_{\mathrm{bod}}$ (open circles) data are plotted alongside the model’s predictions for $I_c$ [as inferred from $D_v\left(t\right)$ using Eqs. (30, 32)] (solid line) and $T_{\mathrm{bod}}$ [as inferred from $D_v\left(t\right)$ using Eq. (33)] (dotted line). A $9\mathrm{dB}$ offset relative to the calibration to the Bonnet et al. data [53] is added to the model’s $I_c$ predictions. The model formulation emulates the clinical protocol: $2\mathrm{s}$ impulses of strength equal to the arousal threshold $\mathcal{A}$ are applied $4\min$ after every entry into Stage 2 sleep (defined as the $\dot{V}<0.1\mathrm{mV}{\mathrm{s}}^{-1}$ region surrounding the sleep node).