Mathematical Models for Sleep-Wake Dynamics: Comparison of the Two-Process Model and a Mutual Inhibition Neuronal Model
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{"title"=>"Mathematical models for sleep-wake dynamics: Comparison of the two-process model and a mutual inhibition neuronal model", "type"=>"journal", "authors"=>[{"first_name"=>"Anne C.", "last_name"=>"Skeldon", "scopus_author_id"=>"6603272410"}, {"first_name"=>"Derk Jan", "last_name"=>"Dijk", "scopus_author_id"=>"7007018167"}, {"first_name"=>"Gianne", "last_name"=>"Derks", "scopus_author_id"=>"6602454668"}], "year"=>2014, "source"=>"PLoS ONE", "identifiers"=>{"scopus"=>"2-s2.0-84905369420", "doi"=>"10.1371/journal.pone.0103877", "sgr"=>"84905369420", "arxiv"=>"1311.1734", "isbn"=>"1932-6203", "pmid"=>"25084361", "issn"=>"19326203", "pui"=>"373686018"}, "id"=>"0c1bd757-abcf-35b7-910e-a090e5f8dd59", "abstract"=>"Sleep is essential for the maintenance of the brain and the body, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the mutual inhibition of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes occur at grazing bifurcations.This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the mutual inhibition model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is valuable because not only does it allow some features of the two-process model to be interpreted physiologically but it also means that knowledge gained from study of the two-process model can be used to inform understanding of the mutual inhibition model. This is important because the mutual inhibition model and its extensions are increasingly being used as a tool to understand a diverse range of sleep-wake phenomena such as the design of optimal shift-patterns, yet the values it uses for parameters associated with the circadian and homeostatic processes are very different from those that have been experimentally measured in the context of the two-process model.", "link"=>"http://www.mendeley.com/research/mathematical-models-sleepwake-dynamics-comparison-twoprocess-model-mutual-inhibition-neuronal-model", "reader_count"=>65, "reader_count_by_academic_status"=>{"Unspecified"=>6, "Professor > Associate Professor"=>3, "Researcher"=>12, "Student > Doctoral Student"=>4, "Student > Ph. D. Student"=>10, "Student > Postgraduate"=>5, "Student > Master"=>8, "Other"=>4, "Student > Bachelor"=>7, "Lecturer"=>2, "Lecturer > Senior Lecturer"=>1, "Professor"=>3}, "reader_count_by_user_role"=>{"Unspecified"=>6, "Professor > Associate Professor"=>3, "Researcher"=>12, "Student > Doctoral Student"=>4, "Student > Ph. D. Student"=>10, "Student > Postgraduate"=>5, "Student > Master"=>8, "Other"=>4, "Student > Bachelor"=>7, "Lecturer"=>2, "Lecturer > Senior Lecturer"=>1, "Professor"=>3}, "reader_count_by_subject_area"=>{"Unspecified"=>12, "Agricultural and Biological Sciences"=>11, "Business, Management and Accounting"=>1, "Computer Science"=>2, "Economics, Econometrics and Finance"=>1, "Engineering"=>7, "Environmental Science"=>1, "Biochemistry, Genetics and Molecular Biology"=>1, "Mathematics"=>6, "Medicine and Dentistry"=>5, "Neuroscience"=>8, "Physics and Astronomy"=>4, "Psychology"=>6}, "reader_count_by_subdiscipline"=>{"Medicine and Dentistry"=>{"Medicine and Dentistry"=>5}, "Physics and Astronomy"=>{"Physics and Astronomy"=>4}, "Psychology"=>{"Psychology"=>6}, "Mathematics"=>{"Mathematics"=>6}, "Unspecified"=>{"Unspecified"=>12}, "Environmental Science"=>{"Environmental Science"=>1}, "Engineering"=>{"Engineering"=>7}, "Neuroscience"=>{"Neuroscience"=>8}, "Economics, Econometrics and Finance"=>{"Economics, Econometrics and Finance"=>1}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>11}, "Computer Science"=>{"Computer Science"=>2}, "Business, Management and Accounting"=>{"Business, Management and Accounting"=>1}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>1}}, "reader_count_by_country"=>{"Colombia"=>1, "United Kingdom"=>1, "France"=>1, "Germany"=>1}, "group_count"=>3}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/1618344"], "description"=>"<p>Using the two-process model with parameters as indicated in (15). Figures (a)–(d) give sleep-wake cycles for different values of the homeostatic time constant ( illustrating that reducing results in more daily sleep episodes. (e) Sleep regions (shaded) as a function of . Note that the circadian maximum occurs at </p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "homeostatic"], "article_id"=>1124544, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g006", "stats"=>{"downloads"=>4, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Varying_the_homeostatic_constant_/1124544", "title"=>"Varying the homeostatic constant.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618345"], "description"=>"<p>Solutions of the two-process model showing periodicity on the period of two days. (a) (b) All other parameters are as in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#pone-0103877-g006\" target=\"_blank\">Figure 6</a> and can be found in (15).</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "cycles"], "article_id"=>1124545, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g007", "stats"=>{"downloads"=>0, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Sleep_wake_cycles_with_a_two_day_period_/1124545", "title"=>"Sleep-wake cycles with a two day period.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618346"], "description"=>"<p>The upper and lower thresholds are moved simultaneously via and with (a) (b) and all other parameters as in (15). Note that mean VLPO drive equals for consistency with <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#pone.0103877-Phillips2\" target=\"_blank\">[24]</a>. Sleep regions are shaded.</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "two-process"], "article_id"=>1124546, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g008", "stats"=>{"downloads"=>2, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Sleep_timing_in_the_two_process_model_/1124546", "title"=>"Sleep timing in the two-process model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618347"], "description"=>"<p>(a) Sleep-wake cycle showing the MA firing rate as a function of the drive to the VLPO . Over one cycle oscillates between high and low values. When is low, is high and the model is in a wake state. When is high, is low and the model is in a sleep state. The transitions from wake to sleep and sleep to wake occur at and respectively. The size of the hysteresis loop depends on , shrinking to nothing for and for (b) The path of and in the plane. and do not exist for values of that are either less than or greater than . Consequently for or increasing will result in a smooth change from high (wake) to low (sleep) instead of the jump from one state to the other shown in (a). (c) A blow up of (b), with the ‘normal’ sleep-wake cycle superimposed. (d) The plane showing the wake trajectory in a sleep deprivation experiment.</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "deprived", "pr"], "article_id"=>1124547, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g009", "stats"=>{"downloads"=>1, "page_views"=>14, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Normal_and_deprived_sleep_in_the_PR_model_/1124547", "title"=>"Normal and deprived sleep in the PR model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618348"], "description"=>"<p>(a) The two-process model, showing the typical trajectory of the homeostatic pressure during a sleep deprivation experiment. Using the wake effort concept of <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#pone.0103877-Fulcher2\" target=\"_blank\">[25]</a> suggests that the upper threshold moves simultaneously: the dashed line shows the position of the upper threshold after 4 days. (b) The difference between the homeostatic pressure and the value at the ‘normal’ threshold, . (c) The wake effort computed from the two-process model (10) (solid line), the PR model as in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#pone.0103877-Fulcher2\" target=\"_blank\">[25]</a> using (crosses), the PR model with (dashed line). (d) The dependence of , the upper asymptote, on time for the three different cases shown in (c). The downward spikes indicate that the model gets very close to falling asleep, hence gets very close to 0.</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "deprivation"], "article_id"=>1124548, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g010", "stats"=>{"downloads"=>1, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Sleep_deprivation_and_the_wake_effort_/1124548", "title"=>"Sleep deprivation and the wake effort.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618349"], "description"=>"<p>These values give the appropriate parameter values for the two-process model as in (15). The derivation of the values for , and for the PR switch model are given in the <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#s4\" target=\"_blank\">Methods</a> section. All parameters have been defined to be positive, consequently some of the signs in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#pone.0103877.e044\" target=\"_blank\">equations (6</a>) are opposite to their original definitions in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#pone.0103877-Phillips1\" target=\"_blank\">[17]</a>. The mean component of the circadian drive in the PR model has been incorporated in the definition of , , .</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "parameter", "pr"], "article_id"=>1124549, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.t001", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Typical_parameter_values_for_the_PR_model_and_the_equivalent_parameters_for_the_PR_model_with_a_hard_switch_/1124549", "title"=>"Typical parameter values for the PR model and the equivalent parameters for the PR model with a hard switch.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618350"], "description"=>"<div><p>Sleep is essential for the maintenance of the brain and the body, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the mutual inhibition of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes can be classified as grazing bifurcations. This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the mutual inhibition model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is valuable because not only does it allow some features of the two-process model to be interpreted physiologically but it also means that knowledge gained from study of the two-process model can be used to inform understanding of the behaviour of the mutual inhibition model. This is important because the mutual inhibition model and its extensions are increasingly being used as a tool to understand a diverse range of sleep-wake phenomena such as the design of optimal shift-patterns, yet the values it uses for parameters associated with the circadian and homeostatic processes are very different from those that have been experimentally measured in the context of the two-process model.</p></div>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "sleep-wake", "two-process", "inhibition", "neuronal"], "article_id"=>1124550, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877", "stats"=>{"downloads"=>23, "page_views"=>12, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Mathematical_Models_for_Sleep_Wake_Dynamics_Comparison_of_the_Two_Process_Model_and_a_Mutual_Inhibition_Neuronal_Model_/1124550", "title"=>"Mathematical Models for Sleep-Wake Dynamics: Comparison of the Two-Process Model and a Mutual Inhibition Neuronal Model", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618339"], "description"=>"<p>With the parameters as in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#pone.0103877-Daan1\" target=\"_blank\">[10]</a>, <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#pone-0103877-g003\" target=\"_blank\">Figure 3: </a> (a) (b) (c) The times when sleep occurs ( decreasing) are shaded.</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "cycles", "generated", "two-process"], "article_id"=>1124539, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g001", "stats"=>{"downloads"=>1, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Sleep_wake_cycles_generated_by_the_two_process_model_/1124539", "title"=>"Sleep-wake cycles generated by the two-process model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618340"], "description"=>"<p>(a) Diagrammatic description of the PR model showing the links between the VLPO, MA, the homeostatic and the circadian processes. (b), (c) and (d) show typical timeseries for the level of the homeostat, , and the firing rates of the MA and VLPO, and , respectively. The times where sleep occurs are shaded.</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "pr"], "article_id"=>1124540, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g002", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_PR_model_and_typical_solutions_/1124540", "title"=>"The PR model and typical solutions.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618341"], "description"=>"<p>(a) The dashed (black) line shows the firing function given by <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0103877#pone.0103877.e037\" target=\"_blank\">equation (5</a>); the thicker (red) line shows the portion that is used for the ‘normal’ PR cycle. (b) A magnified version of (a). The thin (blue) line shows the switch function (8). Panels (c),(d) and (e) show the behaviour of the homeostat, , and the firing rates and for the PR model (solid line) and the PR model with the hard switch (dashed line). The switch parameters are the mean firing rate of the neural population during wake; all other parameters are listed in the Tables section.</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "pr"], "article_id"=>1124541, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g003", "stats"=>{"downloads"=>1, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_PR_model_and_the_PR_switch_model_/1124541", "title"=>"The PR model and the PR switch model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618342"], "description"=>"<p>(a) Comparison of the PR switch model with the two-process model. (b) Comparison of the PR model with the two-process model. Crosses show the two-process model; solid line the PR model and (blue) dashed line the PR switch model.</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "two-process", "compared", "pr"], "article_id"=>1124542, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g004", "stats"=>{"downloads"=>1, "page_views"=>18, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_two_process_model_compared_to_the_two_PR_models_/1124542", "title"=>"The two-process model compared to the two PR models.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}
  • {"files"=>["https://ndownloader.figshare.com/files/1618343"], "description"=>"<p>(a) A single trajectory of the two-process model showing successive times of sleep onset. (b) Trajectories of the two-process model for different initial sleep onset times. Each different sleep onset time results in a different sequence, , but each sequence rapidly converges to the same sleep onset time, of modulo 1 day. (c) A zoom of (b), showing only the trajectories for and (d) First return map for the two-process model. The black line shows the return map, in other words for any value of sleep onset time on day , it shows the onset time of sleep on day , . The grey diagonal line is the line along which . One typical trajectory is plotted for showing the rapid convergence to the periodic cycle where modulo 1 day, the point at which the return map and the diagonal line intersect. The discontinuity is a result of the fact that neighbouring values of exist that lead to very different values for , as shown in (c). Parameter values for the two-process model are based on the PR model for the human sleep-wake cycle and can be found in (15).</p>", "links"=>[], "tags"=>["physiology", "Physiological processes", "sleep", "Theoretical biology", "one-dimensional", "two-process"], "article_id"=>1124543, "categories"=>["Biological Sciences"], "users"=>["Anne C. Skeldon", "Derk-Jan Dijk", "Gianne Derks"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0103877.g005", "stats"=>{"downloads"=>0, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_one_dimensional_map_for_the_two_process_model_/1124543", "title"=>"The one-dimensional map for the two-process model.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2014-08-01 02:58:45"}

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Relative Metric

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