Badal Joshi Mathematics, Duke University Stochastic Models in - - PowerPoint PPT Presentation

A COUPLED POISSON PROCESS MODEL FOR WAKE-SLEEP CYCLING

Badal Joshi Mathematics, Duke University Stochastic Models in Neuroscience, CIRM January 18, 2010

Outline

 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

 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)

Survivor Function

P (T > t) Time (seconds) Sleep Wake

Survivor Function

P (T > t)

Different distributions

 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

DLPT = Dorsolateral Pontine Tegmentum, PnO = Nucleus Pontis Oralis

Two neuronal populations, with mutual

• inhibition. [Blumberg et al. (2005)]

Wake-Active DLPT Sleep-Active PnO

Outline of mathematical model

 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

 Model each population as a Poisson process

where the event is a spike occurring at epoch ‘s’.

 The population i is a Poisson process  Make the rate depend on the inputs i.e.

spikes in the input populations

 Model described completely by giving

(stochastic) differential equations for rates.

) ( rate with ) ( t t N

i i

λ

Stochastic differential equations for the rate processes

diagram wiring the from read be can ) ( functions The k) pop.

• f

epochs firing are (where ) ( ) ( ) ( ) ( ) (

' ' i ij jk j j i ij i i j k jk i ij i i i

g T N g f t T g f

j j j

λ λ λ δ λ λ λ

∑ ∑ ∑

+ = − + =

Proposed circuit diagram

Wake-Promoting LC Wake-Active DLPT Sleep-Active PnO Excitatory Connection Inhibitory Connection Population of neurons

LC = Locus Coeruleus DLPT = Dorsolateral Pontine Tegmentum PnO = Nucleus Pontis Oralis

Explicit model

Restoring autonomous term fi(λi) = ki − λi τi , ki > 0, τi > 0 Inhibition j ⊣ i gij(λi) = −βijλi , 0 ≤ βij ≤ 1 Excitation term j → i gij(λi) = αijλi

• 1 − λi

si

• where 0 ≤ αij ≤ 1, si > ki > 0, 0 ≤ λi ≤ si

Explicit model

λ′

1 = k1 − λ1

τ1 − β12λ1N′

2 + α11λ1

• 1 − λ1

s1

• N′

1 + α13λ1

• 1 − λ1

s1

• N′

3

λ′

2 = k2 − λ2

τ2 − β21λ2N′

1 + α22λ2

• 1 − λ2

s2

• N′

2

λ′

3 = k3 − λ3

τ3 − β32λ3N′

2 + α31λ3

• 1 − λ3

s3

• N′

1

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 E[λ(t + h) − λ(t)|Λt] = hf(λ) +

• j

hgj(λ(t))λj(t) + o(h) Main Assumption in proof: The Poisson property of no two spikes occurring simultaneously holds for the entire collection

• f the populations of neurons.

Deterministic system

Theorem: E[λ′(t)|Λt] = f(λ(t)) +

• j∈input

gj(λ(t))λj(t) 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 ˜ λ′(t) = f(˜ λ(t)) +

• j∈input

gj(˜ λ(t))˜ λj(t) Compare this with the original stochastic system λ′(t) = f(λ(t)) +

• j∈input

gj(λ(t))N′

j (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)

Figure: Time course of λ1 (in black) and λ2 (in red)

Three component system - Effect of Development

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)

Figure: Time course of λ1 (in black), λ2 (in red) and λ3 (in orange)

Survivor plots (C.C.D.F .)

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’!