Jean LAURENS

Bayesian Modelling of Visuo-Vestibular Interactions

with Jacques DROULEZ

Laboratoire de Physiologie de la Perception et de l'Action,CNRS, Collège de France, Paris

Laurens, Droulez, Biol. Cyber. 2006

Bayesian model

Probabilisticcomputation

SemicircularCanals(noisy)

Otoliths(ambiguous)

PriorsP(motion)

P(sensory inputs | motion)Internal model of sensors

Motion estimatesP(motion | sensory inputs) = P(sensory inputs | motion).P(motion)

a

Plausible Improbable

GA

F

A priori

● VOR dynamic

● Somatogravic effect

Geometrical aspects

Angularacceleration

∫ ∫

Otolithsignal

Canalsignal

Linearacceleration

Linearvelocity

Headposition

∫ ∫

Angularvelocity

Headorientation

Double integration

Double integration

H-1

Noise issues

Angularacceleration

∫ ∫

Otolithsignal

Canalsignal

Linearacceleration

Linearvelocity

∫ ∫

Noise

Angularvelocity

Headorientation

Headposition

Double integration

Double integration

?

?

H-1

A priori

Angularacceleration

∫ ∫

Otolithsignal

Canalsignal

Linearacceleration

Linearvelocity

∫ ∫

A priori

Angularvelocity

Headorientation

A priori

Headposition

Double integration

Double integration

H-1

Noise

Visual informations

Angularacceleration

∫ ∫

Otolithsignal

Canalsignal

Linearacceleration

Linearvelocity

∫ ∫

A priori

Angularvelocity

Headorientation

A priori

Headposition

Double integration

Double integration

H-1

Noise

Visualsignal

Noise

Plan

● Introduction to Bayesian inference

● Visuo-vestibular interactions (Monkey)

● 3D Stimulations (Monkey, Human)

Bayesian inference

● Probability exam: normal and pronged dice (Gezinkter Würfel)

D1 : normal D2 : pronged6 in 50% throws

Bayesian inference

P(6 | D1) = 1/6

P(6 | D2) = 3/6

Likelihood :

P(D1 | 6) = 1/4

P(D2 | 6) = 3/4

?

Bayesian inference

● A priori:

P(D1) = 9/10

P(D2) = 1/10

Bayesian inference

● Bayes formula

P(6 | D2).P(D2)

P(6)P(D2 | 6) =

Likelihood A priori

Bayesian inference

Likelihood :

P(6 | D1) = 1/6

P(6 | D2) = 3/6

A priori :

P(D1) = 9/10

P(D2) = 1/10

A posteriori :

P(D1 | 6) = k * 1/6 * 9/10 = 3/4

P(D2 | 6) = k * 3/6 * 1/10 = 1/4?

Bayesian inference

● More observations

P(2 | D2).P(6 | D2).P(D2)

P(6,2)P(D2 | 6,2) =

Likelihood A priori

Bayesian inference and vestibular information

Likelihood :

P(Vest. Signal| Motion)

A priori :

P(Motion)

F

V = 0

P(Motion | Vest. Signal) = P(Vest. Signal| Motion).P(Motion)

Rotations around a vertical axis

Angularacceleration

∫Canalsignal

A priori40 °/s

Angularvelocity

Noise η10 °/s

Visualsignal

Noise η7 °/s

τ = 4 sH-1

Results: rotation in dark

0 50 100-1

-0.50

0.5

1

Rotation Stop

Rotation Vel.

Estimated vel. (τ = 20 s)

Vest. Signal (τ = 4 s)Velocity Storage

time (s)

Optokinetic stimulation

Light Dark

OKANOKN

0 20 40 60 80

0

0.5

1Stimulation velocity(Ω)

Estimatedvelocity (Ω)

Raphan,Matsuo,Cohen,1979

Optokinetic stimulation

Light Dark

0 20 40 60 80

0

0.5

1

No velocitystorage

Normal

No canals

Raphan, Cohen,Matsuo 1977

Stimulation velocity (°/s)

OK

N v

eloc

ity (

°/s)

Optokinetic stimulation

Light Dark

0 20 40 60 80

0

0.5

1

0 20 40 60 80

0

0.5

1

No canals

Normal

Canals plugging

Optokinetic stimulation

Rotation Stop

Light Dark

Rotation in dark

0 10 20-1

0

1

0 10 200

0.5

1

Angelaki & al. 1996

τ = 0.1 s

Velocity storage

0 50 100-1

-0.5

0

0.5

1

0 20 40 60 800

0.5

1

Rotation Stop Light Dark

Stimultation velocity

Estimated velocity (Ω)^

Vest. signal

'Velocity storage' (Ωt0)

^

Optokinetic stimulationRotation in Dark

Raphan, Cohen

Equivalence with Raphan-Cohen model

+

Conclusion

● Probabilistic modelling: noise on vestibular signal σ

= 10°/s

noise on visual signal σ = 7°/s A priori on velocity σ = 40°/s

Light Dark

0 20 40 60 80

0

0.5

1

3D model

Acceleration Tilt

F ≈ G

Gravito-inertial ambiguity

G

A

F

3D model

Angularacceleration

∫ ∫

Otolithsignal

Canalsignal

Linearacceleration

Linearvelocity

∫ ∫

A priori40 - 30 °/s

Angularvelocity

Headorientation

A priori3 - 5 m/s²

Headposition

Double integration

Double integration

Noise η10 °/s

Implementation: particle filter

Somatogravic effect

AccelerationA

Y (

m/s

²)

-2 0 2 4 6 8 10

0

2

4

Tilt

roll

(°)

-2 0 2 4 6 8 10

0

10

20

30

G

-A

F

Somatogravic: canals plugged

AY

(m

/s²)

-2 0 2 4 6 8 10

0

2

4ro

ll (°

)

-2 0 2 4 6 8 10

0

10

20

30

G

-A

F

Acceleration

Tilt

Tilt/translation discrimination

Acceleration (m/s²)

0 5 10 15-4-2024

tilt (°)

temps (s)0 5 10 15

-20

0

20

Acceleration (m/s²)

0 5 10 15-4-2024

tilt (°)

0 5 10 15

-20

0

20

Normal

Angelaki, 1999

Tilt/translation discrimination

0 5 10 15-4-2024

time (s)0 5 10 15

-20

0

20

0 5 10 15-4-2024

0 5 10 15

-20

0

20

Canals plugged

Angelaki, 1999

Acceleration (m/s²)

tilt (°)

Acceleration (m/s²)

tilt (°)

Post-rotatory tilt

Angular velocity

y (°/

s)

time (s)0 20 40 60 80 100 120

-50

0

50

Angelaki, 1994

Post-rotatory tilt

60 80 100 120-60-40-20

020

60 80 100 120

-20

0

20

time (s)60 80 100 120

-20

0

20

Angelaki, 1994

Centrifugation

G

A

F

Roll

r (

°)

0 50 100 150-20

020

4060

time (s)

OVAR

Benson, Bodin, 1965Guedry, 1974

after Guedry, 1974

60 °/s

180 °/s

OVAR

Head tilt (°)

0 50 100 150 2000

100

time (s)0 50 100 150 200

0

100

200

α

180 °/s

0 50 100 150 200

020

4060

60 °/s

Angular velocity (°/s)

Angular velocity (°/s)

OVAR

F

G

F

G

-A

Guedry, 1974

60 °/s

180 °/s

OVAR

Ang. vel. a priori: σΩ = 30°/s

Acceleration a priori : σA = 5 m/s²

60 °/s 78°/s 180 °/s

Rotation 2 σΩ

2.6 σΩ

6 σΩ

Acceleration (13 m/s²) 2.6 σA

2.6 σA

2.6 σA

Correia 1966,Lackner 1978,

Mittelstaedt 1989,Bos 2002 (90°/s)

Guedry 1965,Benson 1966,Correia 1966,

Wall 1990

OVAR

Denise, Darlot, Droulez, Berthoz 1989

Angular velocity (°/s)

0 50 100 150 200

0

20

40

60

Angular velocity (°/s)

time (s)0 50 100 150 200

0

20

40

60

Angelaki 2000, Kushiro 2002

OVAR

time (s)

Angelaki 2000, Kushiro 2002

0 20 40 60 80

0

20

40

60

Yaw velocity (°/s)

Motion sickness

0 20 40 6010

-2

100

102

104

accélération linéaire

rotation

inclinaison post-rotatoire

time (s)

k.P(Sensory Signal)

Conclusion

● 3 hypothesis Sensory signals uncertainty A priori Bayesian inference

● Lesion modelling (observer theory)● Bayesian approach● Extensions● Predictions Laurens, Droulez, Biol. Cyber. 2006

Thanks !