Feedback Particle Filter and its Applications to Neuroscience

Feedback Particle Filter and itsApplications to Neuroscience

3rd IFAC Workshop onDistributed Estimation and Control in Networked Systems

Santa Barbara, Sep 14-15, 2012

Prashant G. Mehta

Department of Mechanical Science and Engineeringand the Coordinated Science Laboratory

University of Illinois at Urbana-Champaign

Research supported by NSF and AFOSR

Background

Bayesian Inference/FilteringMathematics of prediction: Bayes’ rule

Signal (hidden): X X ∼ P(X ), (prior, known)

Solution

Bayes’ rule: P(X |Y )︸︷︷︸Posterior

∝ P(Y |X )P(X )︸︷︷︸Prior

This talk is about implementing Bayes’ rule indynamic, nonlinear, non-Gaussian settings!

2

Background



Observation: Y (known)

Solution




2

Background




Observation model: P(Y |X ) (known)

Solution




2

Background





Problem: What is X ?

Solution




2

Background






Solution




2

Background






Solution




2

Background

ApplicationsEngineering applications

Filtering is important to:

Air moving target indicator (AMTI) systems, Space situationalawareness

Remote sensing and surveillance: Air traffic management, weathersurveillance, geophysical surveys

Autonomous navigation & robotics: Simultaneous localization andmap building (SLAM)

3

Background






3

Background






3

Background






3

Background

Applications in BiologyBayesian model of sensory signal processing

4

Background

Applications in BiologyBayesian model of sensory signal processing

4

Part I

Theory: Nonlinear Filtering

Nonlinear Filtering

Nonlinear FilteringMathematical Problem

Signal model: dXt = a(Xt)dt + dBt , X0 ∼ p∗0(·)

Posterior is an information state

P(Xt ∈ A|Zt) =∫A

p∗(x , t)dx

E(Xt |Zt) =∫R

xp∗(x , t)dx

6

Nonlinear Filtering



Observation model: dZt = h(Xt)dt + dWt



p∗(x , t)dx

E(Xt |Zt) =∫R

xp∗(x , t)dx

6

Nonlinear Filtering




Problem: What is Xt ? given obs. till time t =: Zt



p∗(x , t)dx

E(Xt |Zt) =∫R

xp∗(x , t)dx

6

Nonlinear Filtering





Answer in terms of posterior: P(Xt |Zt) =: p∗(x , t).



p∗(x , t)dx

E(Xt |Zt) =∫R

xp∗(x , t)dx

6

Nonlinear Filtering








p∗(x , t)dx

E(Xt |Zt) =∫R

xp∗(x , t)dx

6

Nonlinear Filtering








p∗(x , t)dx

E(Xt |Zt) =∫R

xp∗(x , t)dx

6

Nonlinear Filtering

Pretty Formulae in MathematicsMore often than not, these are simply stated

Euler’s identity

e iπ =−1

Euler’s formula

v − e + f = 2

Pythagoras theorem

x2 + y 2 = z2

Kenneth Chang “What Makes an Equation Beautiful” in The New York Times on October 24, 2004

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Nonlinear Filtering

Kalman filterSolution in linear Gaussian settings

dXt = αXt dt + dBt (1)

dZt = γXt dt + dWt (2)

Kalman filter: p∗ = N(Xt ,Σt)

dXt = αXt dt + K (dZt − γXt dt)︸︷︷︸Update

Kalman Filter

Observation: dZt = γXt dt + dWt

Prediction: dZt = γXt dt

Innov. error: dIt = dZt − dZt

= dZt − γXt dt

Control: dUt = K dIt

Gain: Kalman gain

[?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8

Nonlinear Filtering






Kalman Filter




= dZt − γXt dt


Gain: Kalman gain


Nonlinear Filtering






Kalman Filter

-

+

Kalman Filter




= dZt − γXt dt


Gain: Kalman gain


Nonlinear Filtering






Kalman Filter

-

+

Kalman Filter




= dZt − γXt dt


Gain: Kalman gain


Nonlinear Filtering






Kalman Filter

-

+

Kalman Filter




= dZt − γXt dt


Gain: Kalman gain


Nonlinear Filtering






Kalman Filter

-

+

Kalman Filter




= dZt − γXt dt


Gain: Kalman gain


Nonlinear Filtering






Kalman Filter

-

+

Kalman Filter




= dZt − γXt dt


Gain: Kalman gain


Nonlinear Filtering






Kalman Filter

-

+

Kalman Filter




= dZt − γXt dt


Gain: Kalman gain[?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8

Nonlinear Filtering

Kalman filter

dXt = αXt dt︸︷︷︸Prediction

+ K (dZt − γXt dt)︸︷︷︸Update

This illustrates the key features of feedback control:

1 Use error to obtain control (dUt = K dIt)

2 Negative gain feedback serves to reduce error (K =γ

σ2W︸︷︷︸

SNR

Σt)

Simple enough to be included in the first undergraduate course on control

9

Nonlinear Filtering

Kalman filter



This illustrates the key features of feedback control:

1 Use error to obtain control (dUt = K dIt)


σ2W︸︷︷︸

SNR

Σt)


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Nonlinear Filtering

Kalman filter



Kalman Filter

-

+

This illustrates the key features of feedback control:1 Use error to obtain control (dUt = K dIt)


σ2W︸︷︷︸

SNR

Σt)


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Nonlinear Filtering

Filtering ProblemNonlinear Model: Kushner-Stratonovich PDE

Signal & Observations dXt = a(Xt)dt + σB dBt , (1)

dZt = h(Xt)dt + σW dWt (2)

Posterior distribution p∗ is a solution of a stochastic PDE:

dp∗ = L †(p∗)dt +1

σ2W

(h− h)(dZt − h dt)p∗

where h = E[h(Xt)|Zt ] =∫

h(x)p∗(x , t)dx

L †(p∗) =−∂ (p∗ ·a(x))

∂ x+

1

2σ

2B

∂ 2p∗

∂ x2

No closed-form solution in general. Closure problem.

[?] R. L. Stratonovich, SIAM Theory Probab. Appl., 1960. [?] H. J. Kushner, SIAM J. Control, 1964

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Nonlinear Filtering





dp∗ = L †(p∗)dt +1

σ2W



h(x)p∗(x , t)dx

L †(p∗) =−∂ (p∗ ·a(x))

∂ x+

1

2σ

2B

∂ 2p∗

∂ x2



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Nonlinear Filtering





dp∗ = L †(p∗)dt +1

σ2W



h(x)p∗(x , t)dx

L †(p∗) =−∂ (p∗ ·a(x))

∂ x+

1

2σ

2B

∂ 2p∗

∂ x2



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Nonlinear Filtering

Particle FilterAn algorithm to solve nonlinear filtering problem

Approximate posterior in terms of particles p∗(x , t) =1

N

N

∑i=1

δX it(x)

Algorithm outline

1 Initialization at time 0: X i0 ∼ p∗0(·)

2 At each discrete time step:

Importance sampling (Bayes update step)Resampling (for variance reduction)

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Nonlinear Filtering



N

N

∑i=1

δX it(x)

Algorithm outline




11

Nonlinear Filtering



N

N

∑i=1

δX it(x)

Algorithm outline




11

Nonlinear Filtering



N

N

∑i=1

δX it(x)

Algorithm outline




e.g. dZt = Xt dt + small noise11

Nonlinear Filtering



N

N

∑i=1

δX it(x)

Algorithm outline




e.g. dZt = Xt dt + small noise11

Nonlinear Filtering



N

N

∑i=1

δX it(x)

Algorithm outline




Innovation error, feedback?And most importantly, is this pretty?11

Control-Oriented Approach to Particle Filtering

Research goal: Bringing pretty back!

102 103

10−3

10−2

10−1

N (number of particles)

Bootstrap (BPF)

Feedback (FPF)

MSE


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Feedback Particle Filter

Signal & Observations dXt = a(Xt)dt + σB dBt (1)


Controlled system (N particles):

dX it = a(X i

t )dt + σB dB it + dU i

t︸︷︷︸mean field control

, i = 1, ...,N (3)

{B it}Ni=1 are ind. standard white noises.

Objective: Choose control U it , as a function of history

{Zs ,Xis : 0≤ s ≤ t}, such that the two posteriors coincide:∫

x∈Ap∗(x , t) dx = P{Xt ∈ A |Zt}∫

x∈Ap(x , t) dx = P{X i

t ∈ A |Zt}

Motivation: Work of Huang, Caines and Malhame on Mean-field games (IEEE TAC 2007)13






dX it = a(X i



, i = 1, ...,N (3)






t ∈ A |Zt}







dX it = a(X i



, i = 1, ...,N (3)






t ∈ A |Zt}



FPF SolutionLinear model

Controlled system: for i = 1, ...N:

dX it = αX i

t dt + σB dB it︸︷︷︸

Prediction

+ K[

dZt − γX it + µt

2dt]

︸︷︷︸Update (via mean field control)

(3)


-

+

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FPF SolutionLinear model

Controlled system: for i = 1, ...N:

dX it = αX i

t dt + σB dB it︸︷︷︸

Prediction

+ K[

dZt − γX it + µt

2dt]

︸︷︷︸Update (via mean field control)

(3)


-

+

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FPF Update StepsLinear model

Feedback particle filter Kalman filter

Observation: dZt = γXt dt + σW dWt dZt = γXt dt + σW dWt

Prediction: dZ it = 1

2

(γX i

t + γµt

)dt dZt = γXt dt

Innovation error: dI it = dZt − dZ it dIt = dZt − dZt

= dZt − γX it +µt

2 dt = dZt − γXt dt

Control: dU it = K dI it dUt = K dIt

Gain: K is the Kalman gain

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2

(γX i

t + γµt

)dt dZt = γXt dt






15






2

(γX i

t + γµt

)dt dZt = γXt dt






15






2

(γX i

t + γµt

)dt dZt = γXt dt






15






2

(γX i

t + γµt

)dt dZt = γXt dt






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Linear Feedback Particle FilterMean field model is the Kalman filter

Feedback particle filter:

dX it = αX i

t dt + σB dB it + K

[dZt −

γ

2

(X it +

1

N

N

∑j=1

X jt

)dt]

(3)

X i0 ∼ p∗(x ,0) = N(µ(0),Σ(0))

Mean-field model: Kalman filter! Let p denote cond. dist. of X it given

Zt . Then p = N(µt ,Σt) where

dµt = αµt dt +γΣt

σ2W

(dZt − γµt dt)

dΣt =

(2αΣt + σ

2B −

γ2Σ2t

σ2W

)dt

As N → ∞, the empirical distribution approximates the posterior p∗

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dX it = αX i


[dZt −

γ

2

(X it +

1

N

N

∑j=1

X jt

)dt]

(3)

X i0 ∼ p∗(x ,0) = N(µ(0),Σ(0))




σ2W

(dZt − γµt dt)

dΣt =

(2αΣt + σ

2B −

γ2Σ2t

σ2W

)dt


16




dX it = αX i


[dZt −

γ

2

(X it +

1

N

N

∑j=1

X jt

)dt]

(3)

X i0 ∼ p∗(x ,0) = N(µ(0),Σ(0))




σ2W

(dZt − γµt dt)

dΣt =

(2αΣt + σ

2B −

γ2Σ2t

σ2W

)dt


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Variance ReductionFiltering for simple linear model.

Mean-square error:1

T

∫ T

0

(Σ

(N)t −Σt

Σt

)2

dt

102 103

10−3

10−2

10−1

N (number of particles)

Bootstrap (BPF)

Feedback (FPF)

MSE

17


Methodology: Variational FormulationHow do we derive the feedback particle filter?

Time-stepping procedure:

Signal, observ. process:

dXt = a(Xt)dt + σB dBt

Ztn = h(Xtn) + Wtn

Feedback Particle filter

Filter: dX it = a(X i

t )dt + σB dB it

Control: X itn = X i

t−n+ u(X i

t−n)︸︷︷︸

controlConditional distributions:

p∗n(·): cond. pdf of Xt |Zt pn(·;u): cond. pdf of X it |Zt

Variational problem: minu

D (pn(u) ‖ p∗n)

As ∆t→ 0:

Optimal control, u = u◦, yields the feedback particle filter,Nonlinear filter is the gradient flow and u◦ is the optimal transport.

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Ztn = h(Xtn) + Wtn



t )dt + σB dB it


t−n+ u(X i

t−n)︸︷︷︸




D (pn(u) ‖ p∗n)

As ∆t→ 0:


18






Ztn = h(Xtn) + Wtn



t )dt + σB dB it


t−n+ u(X i

t−n)︸︷︷︸




D (pn(u) ‖ p∗n)

As ∆t→ 0:


18






Ztn = h(Xtn) + Wtn



t )dt + σB dB it


t−n+ u(X i

t−n)︸︷︷︸




D (pn(u) ‖ p∗n)

As ∆t→ 0:


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Feedback Particle FilterFiltering in nonlinear non-Gaussian settings

Signal model: dXt = a(Xt)dt + dBt , X0 ∼ p∗0(·)Observation model: dZt = h(Xt)dt + dWt

FPF: dX it = a(X i

t )dt + dB it + K(X i

t )◦ dI it︸︷︷︸Update

Innovations: dI it =:dZt −1

2(h(X i

t ) + h)dt, with cond. mean h = 〈p,h〉.

19


Feedback Particle FilterFiltering in nonlinear non-Gaussian settings

Signal model: dXt = a(Xt)dt + dBt , X0 ∼ p∗0(·)Observation model: dZt = h(Xt)dt + dWt

FPF: dX it = a(X i

t )dt + dB it + K(X i

t )◦ dI it︸︷︷︸Update

Innovations: dI it =:dZt −1

2(h(X i

t ) + h)dt, with cond. mean h = 〈p,h〉.

19


Update StepHow does feedback particle filter implement Bayes’ rule?

Feedback particle filter Linear case

Observation: dZt = h(Xt)dt + dWt dZt = γXt dt + dWt

Prediction: dZ it = h(X i

t )+h2 dt dZ i

t = γX it +γµt

2 dt

h = 1N ∑

Ni=1 h(X i

t )

Innov. error: dI it = dZt − dZ it dI it = dZt − dZ i

t

= dZt − h(X it )+h2 dt = dZt − γ

X it +µt

2 dt

Control: dU it = K (X i

t )◦ dI it dU it = K (X i

t )◦ dI it

Gain: K is a solution of a linear BVP K is the Kalman gain

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t )+h2 dt dZ i

t = γX it +γµt

2 dt

h = 1N ∑

Ni=1 h(X i

t )


t


X it +µt

2 dt



t )◦ dI it


20






t )+h2 dt dZ i

t = γX it +γµt

2 dt

h = 1N ∑

Ni=1 h(X i

t )


t


X it +µt

2 dt



t )◦ dI it


20






t )+h2 dt dZ i

t = γX it +γµt

2 dt

h = 1N ∑

Ni=1 h(X i

t )


t


X it +µt

2 dt



t )◦ dI it


20






t )+h2 dt dZ i

t = γX it +γµt

2 dt

h = 1N ∑

Ni=1 h(X i

t )


t


X it +µt

2 dt



t )◦ dI it


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Boundary Value ProblemEuler-Lagrange equation for the variational problem

Multi-dimensional boundary value problem

∇ · (Kp) =−(h− h)p

solved at each time-step.

Linear case: Nonlinear case:

21




∇ · (Kp) =−(h− h)p



21




∇ · (Kp) =−(h− h)p


Linear case:

Nonlinear case:

21




∇ · (Kp) =−(h− h)p


Linear case:

Nonlinear case:

21




∇ · (Kp) =−(h− h)p


Linear case:

Nonlinear case:

21




∇ · (Kp) =−(h− h)p



21




∇ · (Kp) =−(h− h)p



21




∇ · (Kp) =−(h− h)p



21




∇ · (Kp) =−(h− h)p



21


ConsistencyFeedback particle filter is exact

p∗ : conditional pdf of Xt given Zt ,

dp∗ = L †(p∗)dt + (h− h)(σ2W )−1(dZt − h dt)p∗

p : conditional pdf of X it given Zt ,

dp = L †(p)dt− ∂

∂ x(Kp) dZt −

∂

∂ x(up) dt +

σ2W

2

∂ 2

∂ x2

(pK2

)dt

Consistency Theorem

Consider the two evolution equations for p and p∗.Provided the FPF is initialized with p(x ,0) = p∗(x ,0), then

p(x , t) = p∗(x , t) for all t ≥ 0

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ConsistencyFeedback particle filter is exact

p∗ : conditional pdf of Xt given Zt ,

dp∗ = L †(p∗)dt + (h− h)(σ2W )−1(dZt − h dt)p∗

p : conditional pdf of X it given Zt ,

dp = L †(p)dt− ∂

∂ x(Kp) dZt −

∂

∂ x(up) dt +

σ2W

2

∂ 2

∂ x2

(pK2

)dt

Consistency Theorem

Consider the two evolution equations for p and p∗.Provided the FPF is initialized with p(x ,0) = p∗(x ,0), then

p(x , t) = p∗(x , t) for all t ≥ 0

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Kalman Filter

Kalman Filter

-

+

Innovation Error:

dIt = dZt −h(X )dt

Gain Function:

K = Kalman Gain



-

+

Innovation Error:

dI it = dZt−1

2

(h(X i

t ) + ht

)dt

Gain Function:

K is solution of a linear BVP.

23


Kalman Filter

Kalman Filter

-

+

Innovation Error:


Gain Function:

K = Kalman Gain



-

+

Innovation Error:

dI it = dZt−1

2

(h(X i

t ) + ht

)dt

Gain Function:


23


Kalman Filter

Kalman Filter

-

+

Innovation Error:


Gain Function:

K = Kalman Gain



-

+

Innovation Error:

dI it = dZt−1

2

(h(X i

t ) + ht

)dt

Gain Function:


23


Kalman Filter

Kalman Filter

-

+

Innovation Error:


Gain Function:

K = Kalman Gain



-

+

Innovation Error:

dI it = dZt−1

2

(h(X i

t ) + ht

)dt

Gain Function:


23


Kalman Filter

Kalman Filter

-

+

Innovation Error:


Gain Function:

K = Kalman Gain



-

+

Innovation Error:

dI it = dZt−1

2

(h(X i

t ) + ht

)dt

Gain Function:


23


Kalman Filter

Kalman Filter

-

+

Innovation Error:


Gain Function:

K = Kalman Gain



-

+

Innovation Error:

dI it = dZt−1

2

(h(X i

t ) + ht

)dt

Gain Function:


23

Part II

Neural Rhythms, Bayesian Inference

Oscillators in Biology

Normal Form ReductionDerivation of oscillator model

CdV

dt=−gT ·m2

∞(V ) ·h · (V −ET )

−gh · r · (V −Eh)− . . . . . .

dh

dt=

h∞(V )−h

τh(V )

dr

dt=

r∞(V )− r

τr (V )

[?] J. Guckenheimer, J. Math. Biol., 1975; [?] J. Moehlis et al., Neural Computation, 2004

25



CdV

dt=−gT ·m2

∞(V ) ·h · (V −ET )

−gh · r · (V −Eh)− . . . . . .

dh

dt=

h∞(V )−h

τh(V )

dr

dt=

r∞(V )− r

τr (V )




CdV

dt=−gT ·m2

∞(V ) ·h · (V −ET )

−gh · r · (V −Eh)− . . . . . .

dh

dt=

h∞(V )−h

τh(V )

dr

dt=

r∞(V )− r

τr (V )

Normal form reduction−−−−−−−−−−−−→

dθi (t) = ωi dt + ui (t) ·Φ(θi (t))dt



Collective Dynamics of a Large Number of OscillatorsSynchrony, Neural rhythms

26


Functional Role of Neural RhythmsIs synchronization useful? Does it have a functional role?

Books/review papers:

Buzsaki, Destexhe, Ermentrout, Izhikevich, Kopell, Trout and Whittington (2009), Llinas and

Ribary (2001), Pareti and Palma (2004), Sejnowski and Paulsen (2006), Singer (1993)...

Computations: Computing with intrinsic network states

Destexhe and Contreras (2006); Izhikevich (2006); Zhang and Ballard (2001).

Synaptic plasticity: Neurons that fire together wire together

And several other hypotheses:

Communication and information flow (Laughlin and Sejnowski); Binding by synchrony (Singer);

Memory formation (Jutras and Fries); Probabilistic decision making (Wang); Stimulus

competition and attention selection (Kopell); Sleep/wakefulness/disease (Steriade)

27


PredictionBrain as a reality emulator

“[Prediction] is the primary function of the neocortex,and the foundation of intelligence. If we want tounderstand how your brain works, and how to buildintelligent machines, we must understand the nature ofthese predictions and how the cortex makes them.”

“The capacity to predict the outcome of future events –critical to successful movement – is, most likely, theultimate and most common of all brain functions.”

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Filtering in Brain?Bayesian model of sensory signal processing

Theory:

Lee and Mumford, Hierarchical Bayesian inference Framework (2003)

Rao; Rao and Ballard; Rao and Sejnowski. Predictive coding framework (2002)

Dayan, Hinton, Neal and Zemel. The Helmholtz machine (1995)

Ma, Beck, Latham and Pouget. Probabilistic population codes (2006)

Kording and Wolpert. Bayesian decision theory (2006)

And others: See

Doya, Ishii, Pouget and Rao, Bayesian Brain, MIT Press (2007)

Rao, Olshausen & Lewicki, Probabilistic Models of Brain, MIT Press (2002)

29



Theory:

Lee and Mumford, Hierarchical Bayesian inference Framework (2003)

Rao; Rao and Ballard; Rao and Sejnowski. Predictive coding framework (2002)

Dayan, Hinton, Neal and Zemel. The Helmholtz machine (1995)

Ma, Beck, Latham and Pouget. Probabilistic population codes (2006)

Kording and Wolpert. Bayesian decision theory (2006)

And others: See

Doya, Ishii, Pouget and Rao, Bayesian Brain, MIT Press (2007)

Rao, Olshausen & Lewicki, Probabilistic Models of Brain, MIT Press (2002)

29



Experiments (see reviews):

Gold & Shadlen, The neural basis of decision making, Ann. Rev. of Neurosci. (2007)

R. T. Knight, Neural networks debunk phrenology, Science (2007)

Such theories naturally feed into computer vision & more generally on howto make computer “intelligent”

30



Experiments (see reviews):

Gold & Shadlen, The neural basis of decision making, Ann. Rev. of Neurosci. (2007)

R. T. Knight, Neural networks debunk phrenology, Science (2007)

Such theories naturally feed into computer vision & more generally on howto make computer “intelligent”

30


Bayesian Inference in NeuroscienceLee and Mumford’s hierarchical Bayesian inference framework

. . .Bayes’ rule Bayes’ rule Bayes’ rule

Similar ideas also appear in:

1 Dayan, Hinton, Neal and Zemel. The Helmholtz machine (1995)

2 Lewicki and Sejnowski. Bayesian unsupervised learning (1995)

3 Rao and Ballard; Rao and Sejnowski. Predictive coding framework (1999;2002)

31


Bayesian Inference in NeuroscienceLee and Mumford’s hierarchical Bayesian inference framework

. . .Bayes’ rule Bayes’ rule Bayes’ rule

. . .Part. Filter Part. Filter Part. Filter

Similar ideas also appear in:

1 Dayan, Hinton, Neal and Zemel. The Helmholtz machine (1995)

2 Lewicki and Sejnowski. Bayesian unsupervised learning (1995)

3 Rao and Ballard; Rao and Sejnowski. Predictive coding framework (1999;2002)

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Part III

Application: Filtering with Rhythms

Gait CycleBiological Rhythm

33


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Application: Ankle-foot OrthosesEstimation of gait cycle using sensor measurements

Ankle-foot orthoses (AFOs) : For lower-limb neuromuscular impairments.

Provides dorsiflexor (toe lift) and plantarflexor (toe push) torque assistance

Sensors: heel, toe, and ankle joint

Compressed CO2

Actuator

Solenoid valves:control the �ow of CO2 to the actuator

AFO system components: Power supply,

Valves, Actuator, Sensors.Professor Liz Hsiao-Wecksler

Acknowledgement: Professor Liz Hsiao-Wecksler for sharing the AFO device picture and sensor data.

34

Application: Ankle-foot OrthosesEstimation of gait cycle using sensor measurements

Ankle-foot orthoses (AFOs) : For lower-limb neuromuscular impairments.

Provides dorsiflexor (toe lift) and plantarflexor (toe push) torque assistance

Sensors: heel, toe, and ankle joint

Compressed CO2

Actuator

Solenoid valves:control the �ow of CO2 to the actuator

AFO system components: Power supply,

Valves, Actuator, Sensors.Professor Liz Hsiao-Wecksler

Acknowledgement: Professor Liz Hsiao-Wecksler for sharing the AFO device picture and sensor data.

34

Gait CycleSignal model

Stance phase Swing phase

Model (Noisy oscillator)

dθt = ω0 dt︸︷︷︸natural frequency

+ noise

35





+ noise

35





+ noise

35





+ noise

35





+ noise

35





+ noise

35





+ noise

35





+ noise

35

Problem: Estimate Gait Cycle θtSensor model

Observation model: dZt = h(θt)dt+ noise

Problem: What is θt given noisy observations?

36



Problem: What is θt given noisy observations?

36



Problem: What is θt given noisy observations?36







Solution: Particle FilterAlgorithm to approximate posterior distribution

“Large number of oscillators”

Posterior distribution:P(φ1 < θt < φ2|Sensor readings) = Fraction of θ

it in interval (φ1,φ2)

Circuit:dθ

it = ωi dt︸︷︷︸

natural freq. of ith oscillator

+ noisei + dU it︸︷︷︸

mean-field control

, i = 1, . . . ,N

Feedback Particle Filter: Design control law U it

37





Circuit:dθ




mean-field control

, i = 1, . . . ,N


37





Circuit:dθ




mean-field control

, i = 1, . . . ,N


37





Circuit:dθ




mean-field control

, i = 1, . . . ,N


37





Circuit:dθ




mean-field control

, i = 1, . . . ,N


37





Circuit:dθ




mean-field control

, i = 1, . . . ,N


37

Filtering for Oscillators

Signal & Observations dθt = ω dt + dBt mod 2π

dZt = h(θt)dt + dWt

− π 0 π

Particle evolution,

dθit = ωi dt + dB i

t + K(θit )◦ [dZt −

1

2(h(θ

it ) + h)dt] mod 2π, i = 1, ...,N.

where ωi is sampled from a distribution.

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− π 0 π

Particle evolution,



1

2(h(θ

it ) + h)dt] mod 2π, i = 1, ...,N.


38




− π 0 π

Particle evolution,



1

2(h(θ

it ) + h)dt] mod 2π, i = 1, ...,N.



-

+

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Simulation ResultsSolution of the Estimation of Gait Cycle Problem

[Click to play the movie]

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Filtering of Biological Rhythms with Brain RhythmsConnection to Lee and Mumford’s hierarchical Bayesian inference framework


Prior

Noisy input

. . .Part. Filter Part. Filter

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Prior

Noisy input


Noisy

measurements

Rhythmic

movement

Prior

Mumford’s

box with

neurons

Normal form

reduction

Normal form

reduction

Estimate

Mumford’s box with

oscillators

40



Prior

Noisy input


Noisy

measurements

Rhythmic

movement

Prior

Mumford’s

box with

neurons

Normal form

reduction

Normal form

reduction

Estimate


oscillators

40



Prior

Noisy input


Noisy

measurements

Rhythmic

movement

Prior

Mumford’s

box with

neurons

Normal form

reduction

Normal form

reduction

Estimate


oscillators

40

Acknowledgement

Adam Tilton Tao Yang Huibing Yin Liz Hsiao-Wecksler Sean Meyn

1 T. Yang, P. G. Mehta, and S. P. Meyn. Feedback particle filter with mean-field coupling. In Procs. of IEEE Conf. onDecision and Control, December 2011.

2 T. Yang, P. G. Mehta, and S. P. Meyn. A mean-field control-oriented approach to particle filtering. In Procs. ofAmerican Control Conference, June 2011.

3 A. Tilton, E. Hsiao-Wecksler, P. G. Mehta. Filtering with rhythms: Application to estimation of gait cycle. In Procs. ofAmerican Control Conference, 2012.

4 T. Yang, G. Huang and P. G. Mehta. Joint probabilistic data association-feedback particle filter with applications tomultiple target tracking. In Procs. of American Control Conference, 2012.

5 A. Tilton, T. Yang, H. Yin and P. G. Mehta. Feedback particle filter-based multiple target tracking using bearing-onlymeasurements. In Procs. of Information Fusion, 2012.

6 T. Yang, R. Laugesen, P. G. Mehta, and S. P. Meyn. Multivariable feedback particle filter. To appear in IEEE Conf. onDecision and Control, 2012.

7 T. Yang, P. G. Mehta, and S. P. Meyn. Feedback particle filter. Conditionally accepted to IEEE Transactions onAutomatic Control.

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Feedback Particle Filter and its Applications to Neuroscience

Technology