Mathematical model of network dynamics governing mouse sleep-wake behavior

Abstract

Recent work in experimental neurophysiology has identified distinct neuronal populations in the rodent brain stem and hypothalamus that selectively promote wake and sleep. Mutual inhibition between these cell groups has suggested the conceptual model of a sleep-wake switch that controls transitions between wake and sleep while minimizing time spent in intermediate states. By combining wake- and sleep-active populations with populations governing transitions between different stages of sleep, a "sleep-wake network" of neuronal populations may be defined. To better understand the dynamics inherent in this network, we created a model sleep-wake network composed of coupled relaxation oscillation equations. Mathematical analysis of the deterministic model provides insight into the dynamics underlying state transitions and predicts mechanisms for each transition type. With the addition of noise, the simulated sleep-wake behavior generated by the model reproduces many qualitative and quantitative features of mouse sleep-wake behavior. In particular, the existence of simulated brief awakenings is a unique feature of the model. In addition to capturing the experimentally observed qualitative difference between brief and sustained wake bouts, the model suggests distinct network mechanisms for the two types of wakefulness. Because circadian and other factors alter the fine architecture of sleep-wake behavior, this model provides a novel framework to explore dynamical principles that may underlie normal and pathologic sleep-wake physiology.

Figure 1

Hypnograms summarize 2 hours of typical experimentally observed alternations between wake (W), sleep (S), and REM sleep (R). If brief awakenings are neglected (left), the polyphasic sleep-wake structure is evident. Including brief awakenings (right), illustrates the qualitative differences between brief and sustained awakenings.

Figure 2

Nullclines of the form associated with Morris-Lecar-type equations have three distinct configurations determined by the branch of the cubic on which the intersection point is located: left branch (top, left), right branch (top, right), or middle branch (bottom).

Figure 3

Summary of model sleep-wake network architecture: filled circles denote inhibitory connections; arrows denote excitatory connections. The sign change on the projection from $\mathcal{S}$ to $\mathcal{R}$ represents the net effect of the projection from eVLPO to LDT-PPT: this projection is thought to synapse on inhibitory interneurons in LDT-PPT, thereby disinhibiting the (excitatory) cholinergic neurons in LDT-PPT and exerting a net excitatory effect.

Figure 4

Activation functions for coupling effects can be discrete (top) or continuous (bottom).

Figure 5

Hypnograms of typical simulated mouse sleep-wake behavior capture both the polyphasic nature of sustained wake bouts (left) and the qualitative differences between brief and sustained wake bouts (right).

Figure 6

There is close agreement between experimental and simulated data in the percentage of time spent in each behavioral state.

Figure 7

Experimental and simulated data show similar numbers of bouts of each behavioral state.

Figure 8

Mean bout durations for each behavioral state are similar between experimental and simulated data.

Figure 9

Distribution of wake bout durations in experimental and simulated data show close agreement.

Figure 10

Simulated data captures features of experimental distribution of NREM sleep bout durations.

Figure 11

Comparison of REM sleep bout durations between experimental and simulated data reveals an absence of long REM bouts in simulated data.

Figure 12

State transition probabilities in simulated behavior are similar to the probabilities calculated from experimental data.

Figure 13

Transition from wake to NREM sleep can be understood as intrinsic escape of $\mathcal{S}$: as inhibition from $\mathcal{W}$ to $\mathcal{S}$ decreases as a result of increasing NREM homeostatic sleep drive h, the intersection point of the nullclines, p_S, moves down the left branch of $N_v^S$. It is tracked by the trajectory. When the intersection point loses stability at the knee K_L, the trajectory jumps to the right branch.

Figure 14

Because activation of inhibition from $\mathcal{W}$ to $\mathcal{S}$ is associated with a time course, there is a regime in which brief awakenings occur without switching network behavior.