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

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 Wake-Active Sleep-Active DLPT PnO DLPT = Dorsolateral Pontine Tegmentum, PnO = Nucleus Pontis Oralis Two neuronal populations, with mutual inhibition. [Blumberg et al. (2005)]

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 λ N ( t ) with rate ( t ) i i  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 = λ + λ δ − ' f ( ) g ( ) ( T t ) i i i ij i jk j j k j ∑ = λ + λ ' f ( ) g ( ) N i i ij i j j (where T are firing epochs of pop. k) jk j λ The functions g ( ) can be read from ij i the wiring diagram

Proposed circuit diagram Wake-Promoting LC = Locus Coeruleus DLPT = Dorsolateral LC Pontine Tegmentum PnO = Nucleus Pontis Oralis Wake-Active Sleep-Active DLPT PnO Excitatory Connection Population of neurons Inhibitory Connection

Explicit model Restoring autonomous term f i ( λ i ) = k i − λ i , k i > 0 , τ i > 0 τ i Inhibition j ⊣ i g ij ( λ i ) = − β ij λ i , 0 ≤ β ij ≤ 1 Excitation term j → i � � 1 − λ i g ij ( λ i ) = α ij λ i s i where 0 ≤ α ij ≤ 1 , s i > k i > 0 , 0 ≤ λ i ≤ s i

Explicit model 1 = k 1 − λ 1 � 1 − λ 1 � � 1 − λ 1 � − β 12 λ 1 N ′ N ′ N ′ λ ′ 2 + α 11 λ 1 1 + α 13 λ 1 s 1 s 1 3 τ 1 2 = k 2 − λ 2 � 1 − λ 2 � − β 21 λ 2 N ′ N ′ λ ′ 1 + α 22 λ 2 s 2 2 τ 2 3 = k 3 − λ 3 � � 1 − λ 3 − β 32 λ 3 N ′ N ′ λ ′ 2 + α 31 λ 3 s 3 1 τ 3

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 λ 1 , λ 2 and λ 3 ?

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 λ 1 , λ 2 and λ 3 ? What is the distribution of the bout variable I ( λ 1 >λ 2 ) ?

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 ) | Λ t ] = hf ( λ ) + hg j ( λ ( t )) λ j ( 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 ) | Λ t ] = f ( λ ( t )) + g j ( λ ( t )) λ j ( t ) � j ∈ input In particular the zeros of the right hand side give values of λ i for which | λ ′ i | is smallest. This suggests studying the deterministic system λ ′ ( t ) = f (˜ λ ( t )) + g j (˜ λ ( t ))˜ λ j ( t ) ˜ � j ∈ input Compare this with the original stochastic system λ ′ ( t ) = f ( λ ( t )) + � g j ( λ ( t )) N ′ j ( t ) j ∈ input

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 λ 1 (in black) and λ 2 (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 0.6 0.8 1 mutualexc 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 := α 13 = α 31 for β = 0 . 5, α 11 = α 22 = 0 and s 1 = 20 , s 2 = 10 , s 3 = 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 λ 1 (in black), λ 2 (in red) and λ 3 (in orange)

Survivor plots (C.C.D.F .) 0 0 10 10 −1 −1 10 10 10 −2 10 −2 −3 −3 10 10 −4 −4 10 10 0 5 10 15 20 25 30 0 20 40 60 80 100 120 0 0 10 10 −1 −1 10 10 10 −2 10 −2 −3 −3 10 10 −4 −4 10 10 10 0 10 1 10 2 10 0 10 1 10 2 Figure: Survivor plots for sleep (left) and wake (right) on a semi-log scale (top) and a log-log scale (bottom) for α 13 = α 31 = 0 . 6, α 11 = α 22 = 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’!