A COUPLED POISSON PROCESS MODEL FOR
WAKE-SLEEP CYCLING
Badal Joshi
Mathematics, Duke University
Stochastic Models in Neuroscience, CIRM
January 18, 2010
Outline

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
Biological motivation
Experimental background
Description of stochastic model
Analysis and results
Biological motivation
Behavioral/
Neurological
states
Wake
Sleep
(What is the
function/ properties?)
Some biology

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

All mammalian species show sleep behavior
e.g. rats, cats, mice, humans, bats, seals,
dolphins, platypus, birds etc. [Siegel (2005)]
Very few common properties.
Amount of sleep, number of bouts, duration,
timing, behavior during sleep, local/ global
sleep (in brain) all vary
Depend on species, age, gender, habitat,
season, time of day, foraging behavior etc.
Common features
Transition from :
sleep -> wake
wake -> sleep
are stochastic.
 Natural question: What is the
distribution of sleep and wake
durations?

Common features
Across mammalian species, in adults:
Sleep durations are exponentially distributed.
Wake durations are distributed as a power law.
 Sleep and wake durations are uncorrelated
(with each other and with itself).

Ref: C.C.Lo et al. (2002) EPL, (2004) PNAS
Blumberg et al. (2005) PNAS
Common features
Time
(seconds)
P (T > t)
Wake
Survivor
Function
P (T > t)
Survivor
Function
Sleep
Time
(seconds)
Different distributions
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Use development perspective to
understand neural mechanism.
For infants both sleep and wake are
exponentially distributed. [Blumberg et
al. (2005)]
As animal develops wake gradually
changes from exponential to power law.
Basic switch
Wake-Active
DLPT
Sleep-Active
PnO
DLPT = Dorsolateral Pontine Tegmentum, PnO = Nucleus Pontis Oralis
Two neuronal populations, with mutual
inhibition. [Blumberg et al. (2005)]
Outline of mathematical model
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Poisson process model whose rates are
stochastic processes
Take appropriate `average’ and get
deterministic dynamical system
Bifurcation analysis of deterministic system
Use information from bifurcation diagram to
make a prediction about stochastic system. In
particular, emergence of power law is related
to multiple fixed points.
Modeling approach
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
Model each population as a Poisson process
where the event is a spike occurring at
epoch ‘s’.
The population i is a Poisson process
N i (t ) with rate λ i (t )


Make the rate depend on the inputs i.e.
spikes in the input populations
Model described completely by giving
(stochastic) differential equations for rates.
Stochastic differential equations
for the rate processes
λ 'i = f i (λ i ) +
∑
g ij (λ i )∑ δ (T jk j − t )
∑
g ij (λ i ) N 'j
j
= f i (λ i ) +
kj
j
(where T jk j are firing epochs of pop. k)
The functions g ij (λ i ) can be read from
the wiring diagram
Proposed circuit diagram
Wake-Promoting
LC
LC = Locus Coeruleus
DLPT = Dorsolateral
Pontine Tegmentum
PnO = Nucleus Pontis
Oralis
Wake-Active
DLPT
Excitatory Connection
Inhibitory Connection
Sleep-Active
PnO
Population of neurons
Explicit model
Restoring autonomous term
fi (λi ) =
ki − λi
τi
, ki > 0, τi > 0
Inhibition j ⊣ i
gij (λi ) = −βij λi
,
0 ≤ βij ≤ 1
Excitation term j → i
λi
gij (λi ) = αij λi 1 −
si
where 0 ≤ αij ≤ 1, si > ki > 0, 0 ≤ λi ≤ si
Explicit model
k1 − λ1
′
− β12 λ1 N2 + α11 λ1 1 −
=
τ1
k2 − λ2
′
′
λ2 =
− β21 λ2 N1 + α22 λ2 1 −
τ2
k3 − λ3
′
′
− β32 λ3 N2 + α31 λ3 1 −
λ3 =
τ3
λ′1
λ1
λ1
′
N1 + α13 λ1 1 −
N3′
s1
s1
λ2
N2′
s2
λ3
N1′
s3
Solutions?
What are the solutions of the system of equations?
What are the class of behaviors?
More precisely, given parameter values
Solutions?
What are the solutions of the system of equations?
What are the class of behaviors?
More precisely, given parameter values
What are the distributions of the firing rates λ1 , λ2 and λ3 ?
Solutions?
What are the solutions of the system of equations?
What are the class of behaviors?
More precisely, given parameter values
What are the distributions of the firing rates λ1 , λ2 and λ3 ?
What is the distribution of the bout variable I(λ1 >λ2 ) ?
Solution
Solution: Study an appropriate ‘average’ system which is
deterministic.
Can use classical dynamical system theory to classify all
behaviors.
Solution
Solution: Study an appropriate ‘average’ system which is
deterministic.
Can use classical dynamical system theory to classify all
behaviors.
Find only two ‘regimes’
Solution
Solution: Study an appropriate ‘average’ system which is
deterministic.
Can use classical dynamical system theory to classify all
behaviors.
Find only two ‘regimes’
Either single stable fixed point or two stable fixed points.
Solution
Solution: Study an appropriate ‘average’ system which is
deterministic.
Can use classical dynamical system theory to classify all
behaviors.
Find only two ‘regimes’
Either single stable fixed point or two stable fixed points.
These correspond to exponential distribution or heavy tailed
distribution in bout durations.
Deterministic system
To identify the appropriate deterministic system to look at, we
use the following theorem.
Theorem: The expected change in firing rate in time h is given
by
X
E [λ(t + h) − λ(t)|Λt ] = hf (λ) +
hgj (λ(t))λj (t) + o(h)
j
Main Assumption in proof: The Poisson property of no two
spikes occurring simultaneously holds for the entire collection
of the populations of neurons.
Deterministic system
Theorem:
E [λ′ (t)|Λt ] = f (λ(t)) +
X
gj (λ(t))λj (t)
j∈input
In particular the zeros of the right hand side give values of λi for
which |λ′i | is smallest. This suggests studying the deterministic
system
λ̃′ (t) = f (λ̃(t)) +
X
gj (λ̃(t))λ̃j (t)
j∈input
Compare this with the original stochastic system
λ′ (t) = f (λ(t)) +
X
j∈input
gj (λ(t))Nj′ (t)
Classification of behaviors of deterministic system
First we rule out closed orbit solutions.
Classification of behaviors of deterministic system
First we rule out closed orbit solutions.
We show all solutions are bounded.
Classification of behaviors of deterministic system
First we rule out closed orbit solutions.
We show all solutions are bounded.
We show existence of fixed point solutions.
Classification of behaviors of deterministic system
First we rule out closed orbit solutions.
We show all solutions are bounded.
We show existence of fixed point solutions.
Conclusion: All trajectories converge to a fixed point solution.
Monotone dynamical systems
By changing sign of (2), we can make all the arrows positive.
So all bounded solutions converge to fixed points.
Simulation for two component system (Day 2)
9
8
7
6
5
4
3
2
1
100
105
110
115
120
125
130
135
140
Figure: Time course of λ1 (in black) and λ2 (in red)
Three component system - Effect of Development
C1
C2
20
10
8
15
6
10
4
5
2
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
mutualexc
0.6
0.8
1
mutualexc
C3
10
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
mutualexc
Figure: Steady state firing rates as a function of α := α13 = α31 for
β = 0.5, α11 = α22 = 0 and s1 = 20, s2 = 10, s3 = 10
Simulation of three component system (Day 21)
20
15
10
5
0
300
350
400
450
500
550
600
650
700
Figure: Time course of λ1 (in black), λ2 (in red) and λ3 (in orange)
Survivor plots (C.C.D.F.)
0
0
10
10
−1
−1
10
10
−2
−2
10
10
−3
−3
10
10
−4
10
−4
0
5
10
15
20
25
30
0
0
20
40
60
80
100
120
0
10
10
−1
−1
10
10
−2
−2
10
10
−3
−3
10
10
−4
10
10
−4
0
10
1
10
2
10
10
0
10
1
10
2
10
Figure: Survivor plots for sleep (left) and wake (right) on a semi-log
scale (top) and a log-log scale (bottom) for α13 = α31 = 0.6,
α11 = α22 = 0, β = 0.5
Thanks!
Reference: Badal Joshi, A doubly stochastic Poisson process
model for wake-sleep cycling, PhD Dissertation (2009), Ohio
State University.
Thanks to the organizers and faculty of ‘Stochastic Models
in Neuroscience’!