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Literature Review

Mathematical Models of Sleep and Circadian Rhythms: A Case for Using the 2-Process Model in Neuroscience Nursing

Mithani, Sara; Fink, Anne M.

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Journal of Neuroscience Nursing: February 2019 - Volume 51 - Issue 1 - p 48-53
doi: 10.1097/JNN.0000000000000408
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Patients with neurological conditions experience a variety of sleep disturbances. Neuroscience nurses are knowledgeable about patients’ sleep-related complaints—including insomnia and excessive sleepiness—and use interventions to improve sleep quality. There is no consensus in the field, however, about how to measure sleep and circadian rhythms in clinical practice or research. Considering the physiologic need for healthy sleep, it is important to ask: “Do neuroscience nurses need a framework to guide sleep practice and research?” The purpose of this article was to analyze the 2-process model of sleep regulation, a theory widely used in circadian biology and basic neuroscience.1 Nurse scientists cite the 2-process model in a small number of investigations, but not in studies of patients with neurological diseases.2–5 To determine whether this theory is suitable for studying sleep pathologies accompanying neurological conditions within a neuroscience nursing context, the 2-process model was analyzed using Fawcett and DeSanto-Madeya’s6 framework, a systematic approach for determining whether a theory is relevant to nursing.


Using Fawcett and DeSanto-Madeya’s6 method, theories are examined for their clarity, parsimony, philosophical significance, scope (level of abstraction), and testability. Testability indicates that a theory can be useful only if it demonstrates empirical and pragmatic adequacy for nursing. According to Fawcett, theoretical models that are incongruent with nursing’s metaparadigm and philosophies require modification before they can be implemented into practice or research.7


Overview of the 2-Process Model

Oscillations in 2 processes provide a predictive mathematical approach to defining sleep- and wake-related activities. Process S (sleep-dependent process) and process C (circadian-timing–dependent process) determine the times when an individual will wake.1 Process S represents the pressure to sleep, which depends on the time spent awake. While awake, process S increases exponentially, eventually reaching an upper threshold—the maximum propensity for sleep.8 Process S decreases during periods of sleep, reaching the lower threshold after approximately 8 hours of sleeping. Slow-wave activity in the cortical electroencephalogram, defined as electroencephalographic power in the 0.5- to 4-Hz range, is a commonly used variable representing process S.9 During wakefulness, brain activity increases (≥8 Hz).

Process C is a separate process, determined by the intrinsic circadian clock. For process C, wakefulness is promoted by activity of the suprachiasmatic nucleus (SCN) in the hypothalamus. Environmental light activates the SCN, and the activity of SCN neurons demonstrates circadian (approximately 24-hour) patterns.10 Considering that SCN activity regulates melatonin secretion, researchers often use hormonal patterns to measure process C.11 Core body temperature represents another marker of process C.12 Although processes S and C occur independently, if a subject’s sleep schedule (process S) changes in a manner altering light exposure, this can affect the phase of process C.13 Sine waves representing processes S and C are generated based on variables measured over 24 hours or longer. A common approach to characterizing the processes is to calculate acrophase (time when the peak of a rhythm occurs) and amplitude (difference between peak and trough, used to determine magnitude). Parametric statistics can be applied to determine whether interventions alter the amplitude or shift the phase of physiologic rhythms or whether groups differ in their process S or C curves.14

A healthy patient may demonstrate predictable rhythms, with processes S and C occurring in parallel.15 Sleeping for 8 hours during the night drives a propensity toward wakefulness the next morning; staying awake for 16 hours during the day increases the propensity for nighttime sleep. Circadian factors (eg, light-dark cues, SCN activity, melatonin levels) also drive a propensity for sleep occurring at predictable times (Figure 1A). The inability to sleep at night is predicted to result in an increased amplitude and a shift in phase for process S. The drive to sleep is increased because a rest period was interrupted. Because process C is time dependent, circadian rhythms are less likely to be disrupted (Figure 1B), unless patients have a prolonged hospitalization or if the timing of SCN exposure to light becomes altered.

2-Process Model of Sleep Regulation With Normal and Restricted Sleep

Theoretical Concepts and Propositions

Fawcett and DeSanto-Madeya6 stressed the importance of using the theorist’s words to define concepts (abstract representations of reality) and propositions (statements about the relationships between/among concepts). Unlike nursing theorists, Borbély does not explicitly identify concepts and propositions in the 2-process model, but 2 main concepts (processes S and C) can be inferred from Borbély’s original publication. Borbély also argued that the combined influence of processes S and C contributes to a third concept—total sleep propensity.1

Parsimony, Clarity, and Consistency

In nursing, theories are valued for having clear definitions and compatibility among concepts and their propositions.6 Parsimony—conciseness without oversimplification—is valuable when frameworks will be incorporated into complex patient care settings, such as ambulatory neurological disorder clinics or inpatient neurosurgery units. With 3 concepts (process S, process C, and total sleep propensity), parsimony is a strength of the 2-process model. Inconsistent use of terminology, however, hinders the ability to assign clear definitions to the 3 concepts. For example, the terms “process S,” “sleep-dependent process,” and “sleep homeostasis” are used interchangeably in the model. Process C is referred to as the circadian-timing–dependent process and also as the sleep-independent circadian process.1 Mathematical equations for calculating interactions between processes S and C and their contribution to calculating total sleep propensity received limited attention in the original publication,1 but subsequent articles expanded on the quantitative aspects and provided formulas for these calculations.8,16

Significance and Philosophical Claims

A theory is significant if the framework advances nursing and if the fundamental elements of the nursing discipline (person, health, environment, and nursing) are explicit.6 The 2-process model does not address any of these nursing-focused elements, but health and environment can be indirectly inferred. Sleep is a critical element of human health, and process C is directly influenced by environmental light-dark cues. Considering that the 2-process model has foundations in the biological sciences, research for developing and validating the model was conducted in laboratory animals and Drosophila.1,8,16,17 The purpose of these studies, however, was to advance sleep science to improve the health of humans. No philosophy is explicitly stated for the 2-process model, but the model is congruent with nursing’s reciprocal interaction philosophy, a holistic worldview where humans and their environments directly influence each other.18

Theory Scope, Antecedent Knowledge, and Testability

Nursing theories are classified according to their level of abstraction.6 The 2-process model meets the criteria of a middle-range theory because, unlike abstract grand nursing theories, it has measureable variables. Middle-range theories have direct relevance to practice because concepts, and relationships among concepts, can be tested empirically. The 2-process model is a predictive theory because it guides statistical predictions about sleep/wake activities in various environments. Antecedent knowledge for the model was derived from neurobiology, physiology, and mathematics.13,17 Middle-range theories require testing for evidence of their value to nursing practice. Although the 2-process model satisfies Fawcett and DeSanto-Madeya’s criteria as a theoretically sound theory, there is a lack of data demonstrating the use of the model in neuroscience nursing.


This analysis of the 2-process model using a nursing framework demonstrated that the model (1) is consistent with nursing’s metaparadigm because it is relevant to studying human subjects, health, and environmental factors; (2) meets the criteria for a middle-range theory; and (3) has concepts and propositions that nurses can test empirically. The next step is to apply the theory to neuroscience nursing.

Recent studies have focused on measuring the effects of patient care interventions on sleep, examining sleep/circadian characteristics associated with dementia and neurodegeneration, and defining the impact of caregiving responsibilities on the sleep of family members. In the neurological intensive care unit, patient care interventions, such as neurological assessments, monitoring of vital signs, and patient repositions, often disrupt nocturnal sleep patterns. Uğraş and colleagues19 demonstrated that sleep interruptions did not significantly impair patient satisfaction scores although interruptions were frequent. McLaughlin and colleagues20 prospectively evaluated outcomes in patients receiving hourly neurological examinations and found a high prevalence (75%) of delirium. Because these studies did not have control/comparison groups,19,20 questions remain about how fragmented sleep specifically contributes to outcomes in hospitalized patients. Frequent neurological assessments may be critical for the care of high-risk populations (eg, acute stroke, traumatic brain injury), but sleep deprivation/interruption may complicate outcomes in stable neurological patients. Neuroscience nurses could use the 2-process model to compare the impact of sleep loss and to tailor care plans and patient monitoring policies.

The inability to fall asleep, and to stay asleep, also afflicts patients after hospitalization. Chiu and colleagues examined an intervention for insomnia in patients with traumatic brain injury; patients randomized to undergo a warm-water footbath for 30 minutes 1 to 2 hours before bedtime demonstrated shorter sleep-onset times compared with controls. The researchers hypothesized that this intervention may influence cortisol rhythms, although they did not measure this variable.21 At home, disordered sleep afflicts family caregivers.22 Family members of patients who had a stroke experienced sleep loss and depressive symptoms in 1 investigation,22 underscoring the importance of assessing families when nurses are examining posthospitalization care plans.

The progression of neurodegenerative diseases has been associated with measurable changes in processes S and C. Chronic changes in sleep timing and architecture may predict future neurological disease and outcomes.23–25 For example, Alzheimer’s disease pathophysiology involves a phase delay, defined as a pattern where sleep onset and morning awakening occur later. This phenomenon is associated with neurodegeneration of the SCN. Alzheimer’s disease is also associated with frequent awakenings and reduced rapid eye movement (REM) sleep.23 Gehrman and colleagues24 found that a shift in circadian phase was associated with increased mortality in patients with Alzheimer’s disease. Abnormalities in circadian rhythms and REM sleep are also important to the pathophysiology of movement disorders. REM sleep disorders are associated with degeneration of neurons in the sublaterodorsal region of the pons. Patients with REM sleep behavior disorder have an elevated risk for developing Parkinson’s disease and demonstrate autonomic nervous system impairment.25 The latter increases the likelihood of hospitalizations for cardiovascular disorders. Videnovic and colleagues26 demonstrated that the amplitude of 24-hour melatonin rhythms was significantly lower in patients with Parkinson’s disease compared with controls; this finding could explain why excessive daytime sleepiness accompanies Parkinson’s disease. The researchers used questionnaires to measure sleepiness and sleep habits, but without objective sleep data, it is not possible to correlate the melatonin rhythm findings (process C) with patterns in process S. Collectively, the findings from these studies illustrate the importance of using objective sleep and circadian rhythm measurements when the purpose is to identify disease-specific targets for improving sleep.

Future Implications

Table 1 outlines recommendations for using the 2-process model. The first step is to identify measures of processes S and C relevant to neuroscience nursing. Quantitative, objective, and longitudinal data are required to model process S curves. Many studies rely exclusively on sleep questionnaires, but this approach does not permit researchers to determine timing of the upper and lower thresholds of process S or to determine sleep propensity in conjunction with circadian timing. In addition, poor correlations have been reported between self-reported sleep and objective measures of sleep, especially in patients with insomnia.27 In situations where polysomnography is not feasible (because of interference with patient care, lack of equipment, or cost), actigraphy may be an alternative. In hospitalized patients, devices should have sufficient sensitivity, considering findings that wrist actigraphy underestimated wake time (and overestimated sleep time) in hospitalized patients.28 Considering the role of the SCN in circadian timing, nurses might use light meters to quantify environmental light intensity/wavelength in the home or hospital environment.29 Many circulating biomarkers demonstrate circadian variability; this observation emphasizes the importance of obtaining serial measurements of metabolic, inflammatory, and hormonal laboratory values, when feasible, for research.

Recommendations for Applying the 2-Process Model in Nursing Practice and Research

In summary, Fawcett and DeSanto-Madeya’s theory critique method demonstrated the usefulness of the 2-process model and identified the limitations that nurses must consider. The use of consistent terminology to refer to processes S and C is important for improving semantic clarity of the model. The 2-process model can guide research measuring the impact of nursing interventions on processes S and C. In addition, comparisons among different neurological conditions could yield important data about risk factors and disease-specific characteristics, leading to an understanding of how disordered sleep alters neurobiological pathways. For example, investigating the alerting effects of light therapy or physical activity, as guided by the 2-process model, may provide important insights into prevention strategies targeting circadian dysfunction in neurodegenerative diseases.

Villarruel and colleagues30 argue that borrowed/shared theories have a place in nursing, stressing the need to critically examine theories to determine their required refinements. As a nonnursing theory, the 2-process model omits the last element in the nursing metaparadigm—nursing. Nurses, however, have the opportunity to test the 2-process model and, as a result, to contribute a unique perspective on sleep science.


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neuroscience; nursing interventions; sleep; theory; 2-process model of sleep regulation

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