Piecewise smooth maps for the circadian modulation of sleep-wake dynamics

Transcription

1 Piecewise smooth maps for the circadian modulation of sleep-wake dynamics Victoria Booth, Depts of Mathematics and Anesthesiology University of Michigan Cecilia Diniz Behn, Dept of Applied Mathematics & Statistics, Colorado School of Mines

2 Influences on sleep-wake dynamics 1. Internal circadian (24 hour) rhythms or circadian clock } Mediated by Suprachiasmatic Nucleus (SCN) Intracellular ~24 h transcription-translation cycle

3 Influences on sleep-wake dynamics 2. Homeostatic sleep drive } Unavoidable need for sleep after periods of waking and alertness } Perhaps caused by the build-up in the brain of adenosine, a by-product of energy consumption by cells } Caffeine blocks the effects of adenosine on brain cells

4 Sleep cycle dynamics } Human sleep: consolidated wake and sleep episodes Consists of rapid eye movement (REM) states and non-rem (NREM) states NREM-REM cycling during sleep (~ min cycle length)

5 Mathematical punchline } It s a system of coupled oscillators! } Circadian rhythm as a limit cycle oscillator } Sleep propensity as a: } Limit cycle oscillator } Hourglass oscillator or homeostatic process } Non-REM and REM cycling } Early mathematical models: Wever 1979; Kronauer et al 1982; Moore-Ede & Czeisler 1984; Strogatz 1987; McCarley & Hobson 1975

6 Circadian clock in the SCN } Intracellular transcription-translation loops of circadian clock have been identified and mathematically modeled (Forger and Peskin 2003; Leloup and Goldbeter 2003; Kim and Forger 2012) } Biology of projection of SCN cell signals to rest of brain and body is not completely determined

7 Anatomy and physiology of sleep-wake regulation } High activity in LC/DR/TMN and LDT/PPT promotes wake } Wake characterized by high expression of norepinephrine (NE), serotonin (5-HT) and acetylcholine (ACh) to thalamus and cortical regions Saper et al., 2001

8 Sleep states characterized by low activity in LC/DR/TMN. VLPO neurons have high activity during sleep and GABAergic projections to LC/DR/TMN In REM sleep, cholinergic populations reactivate. ACh in thalamus and brainstem induce REM characteristics.

9 Wake state Proposed rat sleep-wake network Sleep state

10 Physiological models of the sleep-wake regulatory network } Competing hypotheses for the structure of the network have led to multiple mathematical models Sleeppromoting Wakepromoting Sleeppromoting Wakepromoting Phillips and Robinson, 2007, 2008 Kumar et al 2012 Diniz Behn and B, 2010, 2011, 2013

11 Sleep-wake regulatory network model } Firing rates f i and neurotransmitter concentrations c j modeled for sleep- and wake-promoting neuronal populations df i dt F ( å g c )- f = t j j i f F ( ) dc C ( f )- c = dt t j j j j C ( ) Reduced form: c = C ( f ) j j Diniz Behn and B, 2010, 2011, 2013

12 Sleep-wake regulatory network model } Homeostatic sleep drive H promotes NREM population activity Light H ' ì( h - H)/ t max hw = í ý î-( H -hmin )/ t hs during wake during sleep states ü þ } 24h variation in SCN firing rate driven by circadian oscillator model C (Forger et al 1999) } Circadian oscillator C can be entrained to external light schedule Diniz Behn and B, 2010, 2011, 2013

13 Sleep-wake regulatory network model df dt LC df dt df dt VLPO R df dt df dt WR SCN (,,, ) F g c + g c - g c -F = t LC A LC A SCN LC SCN G LC G LC LC (,,,, ) F - g c - g c - g c h -F = t VLPO N VLPO N SCN VLPO SCN G VLPO G VLPO (,,,, ) (,, ) VLPO F g c - g c + g c - g c -F = t R A R A N R N SCN R SCN G R G R FWR ga WR ca - gg WR cg -FWR =, t Homeostatic sleep drive promotes NREM sleep WR (,, ) F C+ g c + g c -F = t SCN A SCN A N SCN N SCN h' SCN R ì( h - h)/ t max hw = í ý î-( h-hmin )/ t hs. during wake, during sleep states,, ü þ dc dt N dc dt G dc dt AR dc dt AWR dc dt SCN ( ) c F - c = t N LC N N ( ) c F - c = t G VLPO G G ( ) c F - c = t AR R AR AR ( ) c F - c = t AWR WR AWR AWR ( ) c F - c = t SCN SCN SCN cscn. Circadian rhythm C modeled by limit cycle oscillator model (Forger et al 1999)

14 Simulated typical human sleep-wake behavior Simulated behavioral state determined by active neuronal population Wake, NREM or REM sleep state determined by activity in state-promoting populations Sleep-wake cycle = 24 hrs (Wake ~ 16h, Sleep ~ 8h) REM cycling during sleep with ~90min cycle Sleep-wake cycle and SCN circadian rhythm are synchronized

15 Model as system of coupled oscillators } Circadian oscillator has limit cycle dynamics } Wake-NREM mutual inhibition with homeostat has hysteresis loop dynamics } Wake-REM reciprocal interaction has limit cycle dynamics Light

16 Under normal conditions, sleep and circadian rhythms are synchronized Experimental measures More interesting scenarios occur when sleep and circadian rhythms are desynchronized Dijk D, and Lockley S W J Appl Physiol 2002;92:

17 How to analyze desynchronized dynamics? } The entrained solution is the stable solution of the system } Desynchronized dynamics are transient solutions } Sleep deprivation: re-entrainment to stable entrained solution } Jet lag: re-entrainment to stable entrained solution } Construct a circle map } For a coupled oscillator system, it maps the timing of one oscillator relative to the cycle of the second oscillator

18 Circle map of circadian phases of sleep onsets 2.0 } For our sleep-wake regulatory network model, the map describes the timings of successive sleep onsets relative to the circadian cycle } F n = circadian phase of sleep onset } F n+1 = circadian phase of next sleep onset F n+1 (Day i) F n+1 (Day i+1) B, Xique, Diniz Behn SIAM J Appl Dyn Sys, 2017 F n (Day i)

19 Outline for remainder of talk 2.0 } Map construction } Analysis of map discontinuities } Applications of map: } Bifurcations of sleepwake patterns F n+1 (Day i) F n+1 (Day i+1) F n (Day i)

20 Map construction } To construct the map, we simulated the model from initial conditions corresponding to sleep onset occurring at different circadian phases } Need to define sleep onset in high dimensional model Sleep onset

21 H(t) (SWA %) Firing rate (Hz) Firing rate (Hz) Multiple timescales of sleep-wake behavior } Fast variables: } Firing rates of neuronal populations } Neurotransmitter concentration changes } Slow variables Wake NREM REM SCN } Homeostatic sleep drive } Circadian rhythm H } Consider both H and C as fixed parameters 0 8:30 20:30 8:30 Time 20:30 8:30

22 F W (Hz) Fast-slow decomposition: Bifurcation structure with respect to H } Circadian drive C fixed at its minimum value } Periodic solution is low amplitude in Wake population firing rate } But high amplitude in REM population firing rate Wake h Low circadian drive Stable Wake Unstable, NREM unstable Stable NREM-REM cycling NREM-REM cycling

23 F W (Hz) Circadian drive C modulates the hysteresis loop } Increases in circadian drive C shift bifurcation curve to higher H values } Promotes waking state } When C varies in time, bifurcation curve moves between these 2 cases Wake High C Low C NREM-REM cycling h

24 F W (Hz) Full model trajectory tracks slowly moving hysteresis loop } During wake, trajectory tracks steady states on upper stable branches } During sleep, trajectory moves along periodic branches generating NREM-REM cycling } Sleep onset is trajectory falling off upper saddle-node point Wake NREM-REM cycling H

25 Map construction } To compute the map: } for all circadian phases, we set initial conditions for the model at the upper saddle-node point } then simulated the model for many cycles } All trajectories traced out the map curve F n+1 (Day i) F n+1 (Day i+1) F n (Day i)

26 Understanding map discontinuities } Discontinuities reflect discontinuous jumps in the durations of wake or sleep episodes } Different branches represent different numbers of REM bouts during sleep episodes F n+1 (Day i) F n+1 (Day i+1) B, Xique, Diniz Behn SIAM J Appl Dyn Sys, 2017 F n (Day i)

27 Model with REM sleep suppressed } To simplify model dynamics, suppress activation of REM sleep population Light X

28 Firing rate (Hz) Map when REM sleep is suppressed 2.0 F n+1 (Day i) F n+1 (Day i+1) Long wake Long sleep F n = F n = F n = Wake NREM REM SCN Time (mins) F n (Day i)

29 Fast-slow decomposition with REM sleep suppressed Wake F W H Sleep C Thanks to Diya Sashidhar

30 Long wake discontinuity associated with a grazing bifurcation Wake F W H Sleep C Thanks to Diya Sashidhar

31 Map with REM sleep } Discontinuity due to long wake bout occurs similarly as in REM suppressed model } By grazing bifurcation at upper fold of bifurcation surface } Q: What causes cusps between branches with different numbers of REM bouts?? F n+1 (Day i) F n+1 (Day i+1) Long wake F n (Day i)

32 F W (Hz) Discontinuities due to changes in number of REM bouts } Path of trajectory across periodic solution surface determines number of REM bouts } Q: How does angle of approach to the homoclinic bifurcation at the lower fold dictate number of REM bouts? Wake NREM-REM cycling H

33 Understanding bifurcations using the map } Developmental changes in sleep patterns are thought to reflect changes in homeostatic sleep drive } We investigate bifurcations in model solutions as the time constants for H are decreased, leading to faster growth and decay } Bifurcation parameter c h %&'! c ) *+ during wake Homeostatic sleep drive! " =! h %,- c ) *. during sleep states Kalmbach, B, Diniz Behn, submitted 2018

34 1.0 Maps as c decreases } Fixed point for synchronized solution loses stability } Fixed point disappears in a border collision bifurcation c = 0.89 c = F n F n F n F n

35 Bifurcations in number of daily sleep episodes } As c decreases, increasing the rate of the homeostatic sleep drive, more sleep episodes occur per day 2 per day 1 per day Time (days) Scaling parameter c

36 Period-adding bifurcation in number of REM episodes } When fixed point loses stability, the number of REM bouts per night alternates between 4 and 5 Patterns of REM bouts add Scaling parameter c Number of REM bouts in periodic cycle {4,5,5} {4,5} {4,4,5,4,5} {4,4,5} {4} Red points = no stable periodic solution found quasi-periodic Scaling parameter c

37 Rotation number! Period-adding bifurcation in number of daily sleep episodes } Define rotation number! = # $ } M = number of sleep episodes and N = number of days in periodic solution Scaling parameter c Number of sleep episodes 2/1 Stable rotation numbers follow a Farey sequence 5/3 5/4 3/2 4/3 1/1 Red points = no stable periodic solution found- quasi-periodic Scaling parameter c

38 Bistability of 2 sleep cycle solutions } For c=0.58, 2 nd return map has 8 fixed points } 4 stable and 4 unstable } Pairs of stable fixed points correspond to 2 stable solutions A Φ n nd return map Φ n B Firing Rate (Hz) Firing Rate (Hz) Nap and nighttime sleep Time (days) Two-phase nighttime sleep Time (days) F W F N F R F SCN Kalmbach, B, Diniz Behn, submitted 2018

39 Conclusions & current work } We constructed a map representing a reduced dimensional manifold dictating model trajectories } Map describes the circadian modulation of sleepwake patterns } Using the framework of the map, we can } understand bifurcations of sleep-wake patterns as parameters vary } Current work is using the framework of the map to predict re-entrainment from jet lag

40 Acknowledgments Colorado School of Mines } Cecilia Diniz Behn } Kelsey Kalmbach (MS 2016) University of Michigan } Sofia Piltz } Ismael Xique (UM MS 2014) } Diya Sashidhar (REU 2016) Outstanding senior award Funding } NSF DMS , DMS

41 Thank you!