The authors have declared that no competing interests exist.
Wrote the paper: MP SP HAB EHG RT. Design of computer experiments: MP SP HAB EHG RT. Interpretation of numerical results: MP SP HAB EHG RT.
Recent advances in sleep neurobiology have allowed development of physiologically based mathematical models of sleep regulation that account for the neuronal dynamics responsible for the regulation of sleepwake cycles and allow detailed examination of the underlying mechanisms. Neuronal systems in general, and those involved in sleep regulation in particular, are noisy and heterogeneous by their nature. It has been shown in various systems that certain levels of noise and diversity can significantly improve signal encoding. However, these phenomena, especially the effects of diversity, are rarely considered in the models of sleep regulation. The present paper is focused on a neuronbased physiologically motivated model of sleepwake cycles that proposes a novel mechanism of the homeostatic regulation of sleep based on the dynamics of a wakepromoting neuropeptide orexin. Here this model is generalized by the introduction of intrinsic diversity and noise in the orexinproducing neurons, in order to study the effect of their presence on the sleepwake cycle. A simple quantitative measure of the quality of a sleepwake cycle is introduced and used to systematically study the generalized model for different levels of noise and diversity. The model is shown to exhibit a clear diversityinduced resonance: that is, the best wakesleep cycle turns out to correspond to an intermediate level of diversity at the synapses of the orexinproducing neurons. On the other hand, only a mild evidence of stochastic resonance is found, when the level of noise is varied. These results show that disorder, especially in the form of quenched diversity, can be a keyelement for an efficient or optimal functioning of the homeostatic regulation of the sleepwake cycle. Furthermore, this study provides an example of a constructive role of diversity in a neuronal system that can be extended beyond the system studied here.
All biological systems are inherently noisy and heterogeneous. Disorder is mostly expected to disturb proper functioning of a system, like it can be the case with noise in a radio signal. However, it has been demonstrated by numerous studies that noise can actually improve signal encoding – the socalled stochastic resonance phenomenon. Recently, it was discovered that quenched diversity (heterogeneity) can also enhance the response of a system to an external perturbation (diversityinduced resonance). In this study we investigate the role of noise and diversity in a neuronal model of sleepwake cycles based on the dynamics of the wakepromoting orexin neurons that is crucial for stability of wake and sleep states. We demonstrate that suitable levels of diversity introduced in the orexin neurons can significantly improve the quality of the sleepwake cycle, and may be essential for proper sleepwake periodicity. Noise, on the other hand, provides only a mild improvement.
Disorder, which originates from both noise and diversity, is naturally present in all biological systems. In neuronal systems some examples are the random opening and closing of ion channels, the multitude of stochastic input currents in the neurons, and the diversity of shapes, sizes, and electrophysiological properties of the neurons
In the present study we examine the effects of noise and diversity (heterogeneity) in a physiologically based neuronal model of sleepwake cycles
In the original model interaction between only two representative neurons is simulated: the orexin neuron and the local glutamate neuron that are reciprocally connected to each other according to the experimentally established physiological connections
In the present paper we extend the above described twoneuron model to a more realistic multiunit model with heterogeneous neurons. The aim of the study is to first of all investigate how the presence of diversity in the neuronal population affects sleepwake transitions, since it is wellknown that neurons are highly heterogeneous by their nature. In particular, within the orexin neurons population significant intrinsic diversity can be found: different electrophysiological properties, sizes in the diameter range
In this section the twoneuron model of sleepwake cycles
The original model of the homeostatic regulation of sleep has a minimal structure consisting of two representative interacting neurons A and B, as depicted in
The
Interaction between the neurons A and B takes place through glutamate and orexin neurotransmitters, as detailed below. The neuron A is acted upon by a stimulus in pace with the circadian rhythm, here treated as a periodic external signal — a simplification justified by its independence from the homeostatic process
Dynamics of the neurons A and B are based on a HodgkinHuxleytype model
Conductance ( 
Equilibrium Potential ( 
Slope Parameter ( 
Threshold Potential ( 
Time Scales ( 

L (Leakage current) 



Na (Sodium current) 




( 
K (Potassium current) 





gl (Glutamate current) 





ox (Orexin current) 










Periodic current 



In the following we give a detailed explanation of different parts of the model.
•
The
Main variables and interspike times
The
•
The
The depolarizing
The repolarizing
The activations of the glutamateinduced currents are modeled as:
The
The meaning of the product
The time constants
For numerical convenience, simulations are made over rescaled daily and orexin time scales: the daily period was assumed to be
All the parameter values for the currents are listed in
The system defined above is essentially an excitable feedback system, i.e. both the external input of sufficient strength and the AB coupling are essential elements for maintaining firing activity of the neurons. Orexinrelated dynamics, with the associated long time scales, are expected to direct the homeostatic sleep process, which regulates the sleepwake transitions. The healthy sleepwake cycles in this system are realized as follows:
Two examples of the twoneuron model dynamics without noise are illustrated in
As a step toward a more realistic model we generalize the twoneuron model into a heterogeneous multineuron model. For simplicity we first increase the number of orexin neurons only. To do this we replace the single neuron A by a set of
In reality a certain level of heterogeneity is observed in all neuronal parameters. However, given that our model neurons are simple pacemaking neurons such diversification of different model parameters (in a physiologically allowed range) would simply lead to slightly different firing rates of the neurons. This, in turn, will result in diversity in activations of synaptic currents, which can be mimicked by simply diversifying their activation thresholds. Thus, in the following we can limit ourselves to studying the effects of diversity in activation thresholds of synaptic currents without loss of generality. Furthermore, as a first step, the heterogeneity is only introduced in the glutamateinduced currents to avoid having a too complicated system, which would become difficult to understand.
With regard to the coupling topology among the orexin neurons, so far there is no detailed experimental data. Therefore, for simplicity, we chose an alltoall coupling via gap junctions, but other variations can be tested in the future. The intensity of the coupling has been chosen large enough to ensure that the neurons
•
For simplicity the same external current
Example of model system with
•
•
Note that diversity is again introduced in the activation thresholds of the glutamateinduced currents
The above described set of equations constitutes the multineuron heterogeneous model of the homeostatic regulation of sleep. Numerical results were obtained using a variation of the RungeKutta 2ndorder method, which is suitable for equations with stochastic terms, namely the Heun method
In this section a heuristic criterion is introduced in order to evaluate and compare the quality of the system responses obtained for different external signals or internal parameter values.
For this purpose, the period
For each period
The fractions
In this section we study how the presence of disorder affects the system response and discuss the main differences compared to the twoneuron model. The term “disorder” is used here to refer to either
In each of the examples considered, the initial configuration in the absence of noise and diversity is the same as the subthreshold state illustrated in
Here we investigate the effects of the noise currents in the equations for the membrane potentials. For clarity only the cases in which noise currents are present either in the neurons
To study the effects of the noise currents
(A). Ten periods of the raster plots of neuron B for different intensities
for small
the system's response becomes slightly more regular and periodic as
as
A sample of time dependence of the main variables in the interval
Sample of four periods of some relevant variables and interspike times
Here we consider the complementary case, in which
(A). Sample of ten periods of the raster plots of neuron B for different values of the intensity
The raster plot in
the smallest values of noise intensity
the response becomes periodic, and the length of the firing periods more regular for higher values of
at larger values of
A representative example of time dependence of selected variables of the neurons
Sample of four periods for some relevant variables and interspike times
The dependence of the coefficient
The effects introduced by a heterogeneity in the neurons are dramatic compared to the effects of noise. The corresponding improvement of the system response for suitable intermediate amounts of diversity can be detected very clearly. This is the main result of this paper and it is illustrated in this section. Noiseless neurons are assumed for easier estimation of the heterogeneity effects (
As in the study of noise described above, we carry out the study of diversity starting from the same configuration with a nonoptimal doubleperiodic response to the external periodic stimulus, corresponding to a zero diversity (homogeneous system). Heterogeneity is then introduced in the glutamateinduced currents, either in the thresholds
Diversifying the potential thresholds
The resulting raster plots of the activity of the neuron B are shown in
(A). Sample of ten periods of the raster plots of neuron B for different heterogeneity levels
Sample of four periods of some relevant variables and interspike times
The underlying mechanism leading the system from the double to the singleperiodic response as diversity is increased can be interpreted following the prototype mechanical model of diversityinduced resonance introduced in Ref.
To show that this is the actual mechanism in action,
Comparison between the responses of the heterogeneous system (left column) and homogeneous system (right column) in the first part of the time period
In order to study the effects of added heterogeneity in the glutamate synapses located at the neuron B, one has to diversify the potential threshold parameters
We now consider two limiting cases of the effective current given by
The system's response at different levels of heterogeneity,
(A). Sample of ten periods of raster plots of neuron B for different heterogeneity levels
Sample of four periods of the time dependence of some relevant system variables and interspike times
The main difference compared to the case in which noise intensity is varied, is that the response of the system remains more regular also at the highest levels of diversity considered, i.e. without random spikes appearing during the silent state and with a typical cycle well shared between a day and a night subperiod.
In the present work we have introduced a heterogeneous multineuron version of the previously developed physiologically motivated model of the homeostatic regulation of sleep. The multineuron model is composed of a population of conductancebased orexinproducing neurons and a single representative glutamatergic neuron. In this model the glutamatergic and orexinergic neurons are undergoing transitions between firing and silence depending on the external circadian input and internal homeostatic mechanisms. These transitions correspond to the transitions between wake (firing) and sleep (silence), with the homeostatic mechanism being dependent on the availability of orexin.
The specific aim of this study was to explore the effects of noise and diversity in the regulation of sleepwake cycles in such a model. It is clear that diversity and noise are integral parts of all biological systems, including the orexinergic neuronal population in the lateral hypothalamus. However, the role of disorder, and especially diversity, is rarely considered in the physiologically based mathematical models of sleeprelated systems
We have demonstrated the existence of a diversityinduced resonance, leading to a clear and strong improvement of the quality of the sleepwake cycles, at a physiologically justified intermediate level of diversity of the orexinproducing neurons. However, only a mild improvement was found with varying noise intensity (stochastic resonance phenomena).
We have considered the simplest system with only 20 heterogeneous orexin neurons and one local glutamate neuron. Also we have used a very simple alltoall network topology for the connections among orexinergic neurons. However, it can be expected that constructive effects of diversity will be found also in other model configurations. In the future, more realistic modifications of the model with a larger population of glutamatergic neurons and more sophisticated interpopulations connections should be considered. Furthermore, in the future studies interplay between noise and diversity should likewise be investigated, since in nature both types of disorder are normally present.
The validity of the result obtained within this model may be more general, since diversityinduced resonance is known to take place for suitable values of the parameters in general networks of interacting (nonlinear) oscillators. A question then naturally arises: whether the phenomena encountered here could also characterize other systems where there is a coupling between two very different time scales or, in other words, if homeostatically regulated biological systems may take advantage from a suitable level of heterogeneity of their components.
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