Coupled Flip-flops: Noise and Analysis for a Sleep-wake Cycle Model Justin Dunmyre and Victoria Booth Department of Mathematics, University of Michigan
Flip-flop model for sleep regulation Mutually inhibitory synaptic projections identified between wake- and sleep-promoting populations Saper et al, Neuron 2010
Flip-flop models for sleep regulation Multiple -on and -off populations identified with mutually inhibitory synaptic projections Saper et al, Neuron 2010 Luppi et al, Eur J Physiol 2012
Can a flip-flop replicate realistic sleep-wake patterning? Rat sleep over 4 hours of light period Variable transitions among wake, and sleep states 0 3600 7200 10800 14400 Time (s) Data courtesy of George Mashour Lab
Firing rate model formalism Neurotransmitter (NT) release depends on presynaptic firing rate Presynaptic Population NT Concentration Postsynaptic Population Postsynaptic firing rate depends on total NT concentration
Neurotransmitter/population firing rate model formalism dc j C ( f j) c j =, C ( f j) = tanh( f j) / γ j dt τ j C ( ) Presynaptic Population j Neurotransmitter j Postsynaptic Population X df F( ) X gc j j fx X max c β X =, F ( c) = 1 + tanh dt τ X 2 αx F ( ) Diniz Behn and Booth, J Neurophysiol, 2010
Mutual inhibition network = flip-flop Requires external drive to force transitions Homeostatic sleep drive Mediated by adenosine, modeled by h(t) sleep homeostatic drive Physiological mechanisms not determined, modeled by stp(t) stp(t) increases during and promotes termination of -off population to allow -on activation -on fr on (t) GABA cr on (t) GABA cr off (t) -off fr off (t) Franken, 2002
sleep flip-flop model -on: fr R ( g cr ) fr (tanh( fr ) / ) cr τ σ on off on on on on off, on on γ Ron ' =, cr ' = Ron Ron -off: fr off R ( g cr ) fr (tanh( fr ) / γ ) cr off on off off off on, off off Roff ' =, cr ' = τ Roff σ Roff stp ' stpmax stp on when fr < θron τ stp = stpmin stp on when fr θ Ron τ stp 8 R off (c) -2 0 2 4 off R off c β ( stp) R c = + off stp = f stp f 2 α off max off ( ) 1 tanh, β ( ) ( ) 6 4 2 0 2 1
Firing rate (Hz) stp Hysteresis loop cycling 5 4 3 2 1 0 Exploit slow time-scale of stp for Fast-Slow decomposition Can get asymmetric bout durations due to exponential stp dynamics fr on fr off stp max 1.0 0.8 0.6 0.4 0 500 1000 1500 2000 Time (s) stp min fr on (Hz) 5 4 3 2 1 0 0.8 0.9 1.0 1.1 stp
Can a flip-flop replicate realistic sleep-wake patterning? Rat sleep over 4 hours of light period Variable bout durations Extended and brief wake bouts 0 3600 7200 10800 14400 Time (s) Data courtesy of George Mashour Lab
Neurotransmitter variability Simulates variability of population-level neurotransmitter release due to stochasticity at single synapses Modeled as random time-varying and amplitude-varying multiplicative factor (mean=1.0) to neurotransmitter steady-state activation functions -on: cr on ' = ξ R on on (tanh( fr ) / γ ) σ Ron Ron cr on ξ C ( ) -off: cr off ' = ξ R off off (tanh( fr ) / γ ) σ Roff Roff cr off
Dynamic effects of neurotransmitter noise Neurotransmitter noise changes distance between knees and shape of S-shaped bifurcation curve Effect of ξ Roff only Effect of ξ Ron only
Dynamic effects of neurotransmitter noise Small amplitude ξ values make knees coalesce and hysteresis loop disappear Distance between knees of bifurcation curve
Dynamic effects of neurotransmitter noise Combined effects of noise in both populations on distance between knees ide hysteresis loop Narrow hysteresis loop
Dynamic effects of neurotransmitter noise Trajectory influenced by varying hysteresis loop 5 fr on fr off Firing rate (Hz) 4 3 2 1 fr on (Hz) 5 4 3 2 stp 0 1.0 0.8 0.6 0.4 stp max stp min 0 1000 2000 3000 4000 5000 Time (s) 1 0 0.7 0.8 0.9 1.0 1.1 stp
Dynamic effects of neurotransmitter noise Variability introduced in bout durations with mean similar to deterministic durations 0.10 -on bout durations 0.04 -off bout durations Fraction of bouts 0.08 0.06 0.04 0.02 Fraction of bouts 0.03 0.02 0.01 0.00 60 120 180 240 300 Bout duration (s) 0.00 240 480 720 960 1200 Bout duration (s)
Can a flip-flop replicate realistic sleep-wake patterning? Rat sleep over 4 hours of light period Variable bout durations Extended and brief wake bouts 0 3600 7200 10800 14400 Time (s) Data courtesy of George Mashour Lab
Variable external excitatory input Brief excitatory stimuli simulate external input/synaptic noise to population Modeled as additive randomly occurring, brief excitatory inputs to population on off on on R ( goff, on cr + S( t)) fr fr ' = -on: S(t) τ Ron -on fr on (t) GABA cr on (t) GABA cr off (t) -off fr off (t)
Firing rate (Hz) stp Dynamic effects of noisy external input Progress of trajectory around hysteresis loop is interrupted Transitions occur away from knees 5 4 3 2 1 0 fr on fr off 1.0 0.8 0.6 0.4 0 1000 2000 3000 4000 5000 Time (s) fr on (Hz) stp max stp min 5 4 3 2 1 0 0.7 0.8 0.9 1.0 1.1 stp
Dynamic effects of noisy external inputs Short -on bouts introduced -off bouts are fragmented 0.3 -on bout durations 0.08 -off bout durations Fraction of bouts 0.2 0.1 Fraction of bouts 0.06 0.04 0.02 0.0 20 40 60 80 100 120 140 160 Bout duration (s) 0.00 120 240 360 Bout duration (s)
How to couple Sleep/ake and on/-off flip-flops? Physiology not determined Consider transition dynamics in rat sleep recordings under different conditions (n=5) Control After 24h sleep deprivation 0 3600 7200 10800 14400 Time (s) 0 3600 7200 10800 14400 Time (s) Data courtesy of George Mashour Lab
State transition probabilities Control From ake From R N R From N From ake N R Post deprivation From R From N High / ake probability
State transition probabilities Control From ake From R N R From N From ake N R Post deprivation From R From N / transition should be robust
ake / transitions Occur after brief wake bouts Post deprivation 0 3600 7200 10800 14400 Time (s)
Coupled flip-flop model for sleep-wake regulation S(t) ake f(t) GABA c(t) GABA cs(t) Sleep fs(t) -on fr on (t) GABA cr on (t) GABA cr off (t) -off fr off (t)
ake effect on homeostatic drive During wake, STP shifted to low level forcing -off activation Sleep/wake flip-flop -on/-off flip-flop
Simulated rat sleep-wake behavior Control Post deprivation 0 3600 7200 10800 14400 Time (s)
Summary statistics for data and model 14 Mean number of bouts Mean bout duration (min) 12 10 8 6 4 2 0 40 30 20 10 Control - Data Control - Model Post Dep - Data Post Dep - Model Mean percent time in state 80 60 40 20 0 0
Conclusions & future directions e used the transition dynamics of experimental sleep recordings to propose network structure Matching number of bouts was key constraint in proposing coupling between -on and ake populations Identify/propose physiological substrates for population interactions Relate parameter differences between Control and SD cases to leading theories of sleep homeostasis Only three parameters were adjusted Model sleep deprivation and recovery as a dynamic process
Acknowledgements UM Dept of Anesthesiology George Mashour Dinesh Pal National Science Foundation DMS-1121361 Thanks!