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 REM sleep regulation Multiple REM-on and REM-off populations identified with

mutually inhibitory synaptic projections

Saper et al, Neuron 2010 Luppi et al, Eur J Physiol 2012

REM

REM

NREM

W

W

NREM

0 3600 7200 10800 14400Time (s)

Can a flip-flop replicate realistic sleep-wake patterning? Rat sleep over 4 hours of light period Variable transitions among wake, NREM and REM sleep states

Data courtesy of George Mashour Lab

Firing rate model formalism

Postsynaptic firing rate depends on total NT concentration

Postsynaptic Population NT Concentration

Presynaptic Population

Neurotransmitter (NT) release depends on presynaptic firing rate

Neurotransmitter/population firing rate model formalism

Postsynaptic Population X Neurotransmitter j Presynaptic

Population j

max( ), ( ) 1 tanh

2j j XX X

X X

F g c fdf X cF cdt

βτ α

∞∞

− −= = +

∑

( ), ( ) tanh( ) /j j j

j j jj

dc C f cC f f

dtγ

τ∞

∞

−= =

F∞(·)

Diniz Behn and Booth, J Neurophysiol, 2010

C∞(·)

Mutual inhibition network = flip-flop Requires external drive to force transitions Homeostatic sleep drive Mediated by adenosine, modeled by h(t)

REM sleep homeostatic drive Physiological mechanisms not determined, modeled by stp(t) stp(t) increases during NREM and promotes termination of REM-off

population to allow REM-on activation

REM-on fRon (t)

GABA cRon(t)

REM-off fRoff(t)

GABA cRoff(t)

Franken, 2002

max2 1

( )( ) 1 tanh , ( ) ( )

2

offoffoff

offoff

R c stpR c stp f stp f

ββ

α∞

−= + = −

,( ) (tanh( ) / )' , '

off on off off offon off Roffoff off

Roff Roff

R g cR fR fR cRfR cR

γτ σ

∞ − −= =

REM sleep flip-flop model

,( ) (tanh( ) / )' , 'on off on on on

off onon on Ron

Ron Ron

R g cR fR fR cRfR cR γτ σ

∞ − −= =REM-on:

REM-off:

onmax

onmin

when fR

'when fR

Ronstp

Ronstp

stp stp

stpstp stp

θτ

θτ

− < = − ≥

-2 0 2 40

2

4

6

8

Roff∞(c)

Hysteresis loop cycling Exploit “slow” time-scale of stp for Fast-Slow decomposition Can get asymmetric bout durations due to exponential stp dynamics

0.8 0.9 1.0 1.10

1

2

3

4

5

fRon

(Hz)

stpstpmax

stpmin

REM

NREM 0

1

2

3

4

5

Firin

g ra

te (H

z)

fRon

fRoff

0 500 1000 1500 20000.40.60.81.0

stp

Time (s)

REM

REM

NREM

W

W

NREM

0 3600 7200 10800 14400Time (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

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

ξ C∞(·)

(tanh( ) / )' off

off offRoffoff R

Roff

fR cRcR

ξ γσ

−=

(tanh( ) / )' on

on onRonon R

Ron

fR cRcR

ξ γσ

−=REM-on:

REM-off:

Dynamic effects of neurotransmitter noise Neurotransmitter noise changes distance between knees

and shape of S-shaped bifurcation curve

Effect of ξRoff only Effect of ξRon 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

Wide hysteresis loop

Narrow hysteresis loop

Dynamic effects of neurotransmitter noise Trajectory influenced by varying hysteresis loop

0

1

2

3

4

5

Firin

g ra

te (H

z)

fRon

fRoff

0 1000 2000 3000 4000 50000.40.60.81.0

stp

Time (s)

0.7 0.8 0.9 1.0 1.10

1

2

3

4

5

fRon

(Hz)

stp

stpmax

stpmin

REM

NREM

Dynamic effects of neurotransmitter noise Variability introduced in bout durations with mean similar

to deterministic durations

60 120 180 240 3000.00

0.02

0.04

0.06

0.08

0.10

Frac

tion

of b

outs

Bout duration (s)240 480 720 960 1200

0.00

0.01

0.02

0.03

0.04

Frac

tion

of b

outs

Bout duration (s)

REM-on bout durations REM-off bout durations

REM

REM

NREM

W

W

NREM

0 3600 7200 10800 14400Time (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

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

REM-on fRon (t)

GABA cRon(t)

REM-off fRoff(t)

GABA cRoff(t)

S(t)

,( ( ))'

on off onoff onon

Ron

R g cR S t fRfR

τ∞ + −

=REM-on:

Dynamic effects of noisy external input Progress of trajectory around hysteresis loop is

interrupted Transitions occur away from knees

0

1

2

3

4

5

Firin

g ra

te (H

z)

fRon

fRoff

0 1000 2000 3000 4000 50000.40.60.81.0

stp

Time (s)

0.7 0.8 0.9 1.0 1.10

1

2

3

4

5

fRon

(Hz)

stp

stpmax

stpmin

REM

NREM

Dynamic effects of noisy external inputs Short REM-on bouts introduced REM-off bouts are fragmented

20 40 60 80 100 120 140 1600.0

0.1

0.2

0.3

Frac

tion

of b

outs

Bout duration (s)120 240 360

0.00

0.02

0.04

0.06

0.08

Frac

tion

of b

outs

Bout duration (s)

REM-on bout durations REM-off bout durations

How to couple Sleep/Wake and REM-on/REM-off flip-flops? Physiology not determined Consider transition dynamics in rat sleep recordings

under different conditions (n=5)

REM

REM

NREM

W

W

NREM

0 3600 7200 10800 14400Time (s)

REM

REM

NREM

W

W

NREM

0 3600 7200 10800 14400Time (s)

Control After 24h REM sleep

deprivation

Data courtesy of George Mashour Lab

State transition probabilities

High REM / Wake probability

Control

Post REM deprivation

From Wake

From Wake

From NREM

From NREM

From REM

From REM

N

N

N

N

R

R

R

R

W

W

W

W

State transition probabilities

W / NREM transition should be robust

Control

Post REM deprivation

From Wake

From Wake

From NREM

From NREM

From REM

From REM

N

N

N

N

R

R

R

R

W

W

W

W

Wake / REM transitions Occur after brief wake bouts

REM

REM

NREM

W

W

NREM

0 3600 7200 10800 14400Time (s)

Post REM deprivation

Wake fW(t)

GABA cW(t)

Sleep fS(t)

GABA cS(t)

Coupled flip-flop model for sleep-wake regulation

REM-on fRon (t)

GABA cRon(t)

REM-off fRoff(t)

GABA cRoff(t)

S(t)

Wake effect on homeostatic REM drive During wake, STP shifted to low level forcing REM-off activation

Sleep/wake flip-flop REM-on/REM-off flip-flop

Simulated rat sleep-wake behavior

REM

REM

NREM

NREM

W

W

0 3600 7200 10800 14400Time (s)

Control

Post REM deprivation

Summary statistics for data and model

0

2

4

6

8

10

12

14

REMNREM

Mea

n bo

ut d

urat

ion

(min

)

Control - Data Control - Model Post REM Dep - Data Post REM Dep - Model

W

0

10

20

30

40

Mea

n nu

mbe

r of b

outs

W NREM REM

0

20

40

60

80

Mea

n pe

rcen

t tim

e in

sta

teW NREM REM

Conclusions & future directions We used the transition dynamics of experimental sleep

recordings to propose network structure Matching number of bouts was key constraint in proposing

coupling between REM-on and Wake populations Identify/propose physiological substrates for population

interactions Relate parameter differences between Control and

REMSD cases to leading theories of REM sleep homeostasis Only three parameters were adjusted

Model REM sleep deprivation and recovery as a dynamic process

Acknowledgements UM Dept of Anesthesiology George Mashour Dinesh Pal

National Science Foundation DMS-1121361

Thanks!