Bayesian Cognition Winter School Chamonix, January 6-11, 2008 Probabilistic interpretation of...

Bayesian Cognition Winter School

Chamonix, January 6-11, 2008

Probabilistic interpretation of physiological and psychophysical data

Jacques DroulezLaboratoire de Physiologie de la Perception et de l’ActionCNRS – Collège de Francejacques.droulez@college-de-france.fr

The agent …

Black box view of an agent

Sensory inputs: ot Motor outputs: at

The agent …

Black box view of a situated agent

otat

The world …

The agent …

Black box view of a situated agent(with an internal model of the world…)

otat

The world …

Stat ot

Question: P(St | a0→t-1 o0→t) and P(At | a0→t-1 o0→t)

Observation Action

a o

?

?

An evolutionary perspective

Eukaryotes

Opisthokonts Algae Plants Others...

Lebert, M. & Häder, D-P (2000) Photoperception and phototaxis in flagellated algae.Res. Adv. Photochem. Photobiol. 1: 201-226.Streb et al (2002) Sensory transduction of gravitaxis in Euglena gracilis. J. Plant. Physiol., 159: 855-862.

An evolutionary perspective

Eukaryotes

Opistokonts Algae Plants Others...

Adapted behaviors already exist in unicellular organisms, with:- specialized organelles for sensory signals (mechano / chemo / photoreceptors)- macromolecular assemblies and messengers for signal processing (ionic channels)- various effectors for locomotion / defense / predation

Bucci et al (2005) A role for GABAA receptors in the modulation of Paramecium swimming behaviour. Neuroscience Letters, 386:179-183

Evolutionary perspective

Eukaryotes

Opistokonts

Metazoan Fungi

Algae Plants

Sponges

Medusas

Bilateria

Cell specialization

Nervous system

Segmented brain

Others...

worms ... human

The space-time problem for multicellular organism ...

Diffusion law:

∂[C]/∂t = D ∂2[C]/∂x2

D ≈ 1 µm2/ms

Time Length2

⇒ Coordinated movements (in Sponges) are slow ...

Nickel, M. (2004) Kinetic and rhythm of body contractions in the sponge Tethya wilhelma, J. of Experimental Biology, 207:4515-4524

Evolutionary perspective

Eukaryotes

Opistokonts

Metazoan Fungi

Algae Plants

Sponges

Medusas

Bilateria

Cell specialization

Nervous system

Segmented brain

Others...

worms ... human

Schematic view of a cnidarian eyeCiliary type photoreceptor in redMelanin pigment cells in yellowBiconvex lens in blue

Nervous system of the cubomedusae jellyfishRN: ring nerveRh: rhopalia (eye + graviceptors)Garm et al (2006) Cell Tissue Res. 325: 333-343

Fast, single cell reaction (nematocysts)

Coordinated behaviour (nerve ring)

Intracellular recording of action potentials in motoneurons

Interpulse histogram for:1 (A) 2 (B) 3 (C ) 4(D) rhopalia(Bin width: 250 msec) Satterlie & Nolen (2001)J. of Exp. Biol. 204: 1413-1419

Evolutionary perspective

Eukaryotes

Opistokonts

Metazoan Fungi

Algae Plants

Sponges

Medusas

Bilateria

Cell specialization

Nervous system

Segmented brain

Others...

worms ... human

1: neurones2: cerebral ganglion3: medullae4: architecture en « échelle de corde »

A

B

C

D

E

A: flatworms B et C: molluscs D: annelids E: arthropods(from G. Roth & M.F. Wulliman, Brain evolution and cognition, 2001)

Evolution of the nervous system in bilaterian

Supra-spinal nervous system in some vertebrates

OB: olfactory bulb OT: optic tectum T: forebrain Cb: cerebellum D: diencephalon

frog

reptile

bird

rodent

horse

Three hierarchically organized levels

Organism Cell Macromolecule

Sensors

effectors

Chemical sensorsPhotosensors (eyespot), …

Neurotransmitter releaseMechanical effectors, …

Ligand binding siteStretching sensitivityVoltage sensor

Channel open/closedCatalytical action, …

Olfaction

Vision

Oculomotor (Eye/ Pupil)

Trochlear (Eye movement)

Abducens (Eye movement)

Trigeminal

Facial

Vestibulo-cochlear

…etc …

Observations Actions

The brain as a (segmented) agent

Example of cellular agent: the photoreceptor cell

cGMP

Ca2+/Na+

Glu

Photons

In darkness: high cGMP (2µM) Ca2+ & Na+ Inflow [Ca2+]i = 550 nM V = – 40 mV Voltage gated Ca2+ channels are open Continuous release of glutamate

In the light: hydrolysis of cGMP channels are closed [Cal+]i = 50 nM V = – 80 mV Voltage gated Ca2+ channels are closed Reduction of glutamate release

Depolarization

Hyperpolarization

Observation: Light intensityAction: chemical release

Example of molecular agent: a voltage gated Ca2+ channel

Action: Ca2+ Inflow

Observation: electric field across the membrane (~ 15 106 V/m !!)

10-12 s 10-9 s 10-6 s 10-3 s 1 s

Elementaryrelaxation(macromolecule)

Tertiary structuretransitions

ChannelsOpen/closed

Membrane timeconstant /spike propagation

Basic behaviours

103 s

Plastic adaptation

?

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The perception viewed as an (ill-posed) inverse problem...

Physical State S(self motion / object properties)

Observed sensory data O(inertial and visual sensors)

P(S) P(O | S)

P(S | O) P(S).P(O | S)

P ≠ P(S)● Blood pressure

● Attentional blindness

● neural activity under anesthesia

Media properties (absorption, diffusion, ...)

Object characteristics : position, orientation,

shape, texture, movement, ..

Primary & secondary light sources

Optic properties of the eye &Eye movements

The complete model is too complex for aBrain with limited resources...

Incidence of small ocular movements on visual perception

Akiyoshi Kitaoka, Out of focus (2001)

1. Luminance perception

~ 1 photon/µm2/s ~ 109 photon/µm2/s

Vincent Van Gogh, Nuit étoilée sur le Rhône (1888) Oliviers avec ciel jaune et soleil (1889)

Dawson (1990)

Girl reading a Letter at an Open Window (Jan Vermeer, 1657)

« Discrepancies between the real world and the world depicted by artists reveal as much about the brain within us as the artist reveals about the world around us. »P. Cavanagh, The Artist as Neuroscientist, Nature, 2005.

Deviation from Weber-Fechner’s laws: Perceived brightness versus luminance (Cd/m2)

A

B

Data from Nundy & Purves (2002)PNAS 99:14482-14487

The probabilistic explanation by Purves et al (2004) Psychol. Rev. 111:142-158

L < Background L > BackgroundL

Luminance ~ Illumination x Reflectance: I constant R LR constant I LI R R I L1/2

Basic functional schema of the retina

PhR

Bip

G

H HH

A AA

h

Glutamate

GABA

Hemi gap junction

Gap junction

GABA, Gly, Ser, ACh

Except for ganglion and some amacrine cells,information propagates without spikes.

Hypothesis of conservation of loal luminous intensity:

I(x+dx, y+dy, z+dt) ≈ I(x, y,z) or dI/dt = < Gt , V > = 0

Hypothesis not always nor eveywhere valid ! Ex.: apparition/disparition, occlusion, variation of luminous sources …

I(x,y,t) I(x+dx, y+dy,t+dt)

2. Visual motion perception

(dx,dy,dt)

The aspect is globally conserved, butthe local luminous intensity is not exactly the same in succesive images

P(It+dt | It V) or P(Gt | V)

Motion integration: numerous sources of uncertainty:

Nonhomogeneous distribution of contrasts

Low Contrast, Aperture problem, False correspondences

?? ?

G = 0 single oriented G multiple G

P(G V) = P(V). P(G | V) P(V | G) P(V).P(G | V)

P(V): prior favorable aux faibles vitesses

Weiss, Simoncelli & Adelson: Motion Illusions as Optimal Percepts. Nature (2002)

More recent works on the « low velocity prior » idea:

Carandini, M. (2006) Measuring the brain’s assumptions. Nat. Neuroscience, 4:469.Stocker, A. A. & Simoncelli, P. (2006) Noise characteristics and prior expectations in Human visual speed perception.

Nat. Neuroscience, 4: 578.Thomson et al (2006) Speed can go up as well as down at low contrast: implications for models of motion perception.

Vision Research, 46: 782-786.

2AFC speed discrimination

From Stocker & Simoncelli (2006)

DR

Scale ambiguity:

The depth & velocity scales cannot be estimated from the optic flow alone

V

dRv

Knowledge of self motion can be used (in principle) to solve thescale ambiguity problem.

V

D

d

Comparison SM (subject motion) versus OM (object motion)Subject’s Task: report whether or not the object is closer than 1 meter

DISTANCE [cm]

0 50 100 150 200

RE

SP

ON

SE

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

SMOM

DISTANCE [cm]

0 50 100 150 200

RE

SP

ON

SE

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

SMOM

Same relative velocity All trials

Panerai, Cornilleau-Pérès & Droulez, Perception & Psychophysics, 64: 717-731 (2002)

The convex/concave ambiguity:

● absent in large field stimulation● important in small field but strongly reduced in self-motion condition

Dijskra, Cornilleau-Pérès, Gielen & Droulez, Vision Research, 1995

Several examples of optic flow ambiguities

Perceptive inversion (Fronto-parallel plane symmetry for both object & motion)

≃

≃

Passive

Active

Wexler, Lamouret & Droulez, Vision Research, 41, 3023-3037 (2001)

Similar optic flows result from different combinations of rotation and translation

Results show a preference for the stationary object, even if it does not correspondto the most rigid solution

Wexler, Lamouret & Droulez, Vision Research, 41, 3023-3037 (2001)

Variability of perceptive responses (« shear effect »)

Van Boxtel, Wexler & Droulez, Journal of Vision 3(5) : 318-332. (2003)

Self-motion can change the interpretation of perspective cues

Wexler, Panerai, Lamouret & Droulez, Nature, 409, 85-88 (2001)

Perspective cues: « prior » knowledge on object shape & orientation.

The image of a regularlytextured plane ...

back-projected on another plane « trompe-l’œil » image

Shape from motion: combining prior, optic flow and self-motion

Optic Flow: rotation flow + translation flow (= rigidity hypothesis)Φ = W + p.T p = proximity map (depth-1)

The object shape & orientation (p) and its movement (W,T) determine the optic flow (Φ)Knowing Φ, how to compute the 3D shape & movement parameters?The direct problem P(Φ | p, W, T) is simpler than the inverse one …

The probabilistic model

P(Obj, Obs, Move, Flow) = P(Obj).P(Obs).P(Move | Obs).P(Flow | Move, Obj)

P(Obj) = “the fronto-parallel plane prior”

P(Obs) = “knowledge on self-motion”

P(Move | Obs) = “the stationarity hypothesis”

P(Flow | Move, Obj) = “the rigidity hypothesis”

Knowledge Formulation:

Exploitation (Question asked to the subject): P(Obj | Obs, Flow) ?

The variables of interest: Object orientation (Obj)Observer’s displacement (Obs)Relative Object movement (Move)Visual data (Flow)

F. Colas, J. Droulez, M. Wexler & P. Bessière Biol. Cybernetics, 2007.

Obj Obs

MoveFlow

Probabilistic model (results)

ImmobileSubject Shear 0°

ImmobileSubject Shear 90°

MovingSubjectShear 0°

MovingSubject Shear 90°

ImmobileSubject+TZ

MovingSubject+TZ

Self-motion perception

The vestibular sensor : 3 semi-circular canals (head angular acceleration) + 2 otolithic organs (head linear acceleration)

A first example of ambiguity: how to estimate the sustained angular velocity ?

Physical state (dH/dt)

Observation (from SCC)

Estimated velocity (from VOR)

While an exact integration (from filtered acceleration to velocity) is mathematicallystraightforward, it would yield error accumulation! The brain chooses to reduce as much as possible the estimated sustained velocity

Data from Büttner & Waespe (81)

Another well-known example of ambiguity:how to distinguish the inertial linear acceleration from gravity ?

A

GF

F = G - AThe physical state (A,G) cannot be inferred from the observed otolithic signal (F)without a priori on A or G.

G

A

F

The actual solution Another solution to the inverse problem

Both ambiguities can combine together !

g

-a

F

... during off-axis rotation (centrifugation)

Physical state

Perceived states

Decreasing the estimated angular velocity estimated gravity aligned with F estimated linear acceleration decreasing to 0

Denise, Darlot, Droulez, Cohen & Berthoz, EBR (1988)Data from Guedry, (1970)

... or during off-vertical axis rotation (OVAR / Barbecue)

Bayesian Filter with priors(Low angular velocity & linear acceleration)

Rotationalacceleration

∫ ∫

Otolithsignal

Canalsignal

Linearacceleration

Linearvelocity

∫ ∫

Noise ηN(0, 0.005) rad/s²

A prioriN(0, 0.3) rad/s

Double integration

Rotationalvelocity

Headorientation

A prioriN(0, 2) m/s²

Headposition

Double integration

a

J. Laurens & J. Droulez, Biol. Cybernetics, 2007

Centrifugation (Off Axis Rotation)

time (s)

Pitc

h (

rad

) Pitch

0 20 40 60 80

0

0.5

1

0

0.05

0.1

1 s 20 s 70 s50 s 60 s 62 s

g

-a

F

Vel

ocity

(ra

d/s) Yaw velocity

-2

0

2

00.10.20.30.4

time (s)

Vel

ocity

(ra

d/s)

Roll velocity

0 20 40 60 80-0.2

0

0.2

0.4

0.6

0

0.05

0.1

time (s)0 20 40 60 80

J. Laurens & J. Droulez, Biol. Cybernetics, 2007

?

The efficient coding hypthesis:

Attneave, F (1954) Informational aspects of visual perception. Psychol. Rev. 61, 183-193

Barlow, H.B. (1961) The coding of sensory messages. In Current Problems in Animal Behaviour (W.H. Thope & O.L. Zangwill, eds) Cambridge U. Press

van Hateren, J.H. & Ruderman, D.L. (1998) Independent component analysis of natural images yields spatio-temporal filters similar to simple cells in primary visual cortex. Proc. R. Soc. Lond. B 265: 2315-2320

Nirenberg et al (2001) Retinal ganglion cells act largely as independent encoders. Nature, 411: 698-701

Barlow, H.B. (2001) Redundancy revisited. Network, 12: 241-255

Simoncelli, E.P. (2003) Vision and the statistics of the visual environment. Current Opinion in Neurobiology, 13: 144-149

Non uniform distribution of natural stimuli: P(L)

Correlation in space or time: P(Lt+1 | Lt)

More complex dependencies (word …)

1) Sensory data compression (redundancy reduction)

2) Simplified representation of probability distribution

1. Redundancy reduction ? Yes, in the retina!

h12

0 M

pho

tore

cept

ors

wit

h gr

aded

out

put

Not from retina to LGN and from LGN to V1 (~ 1000 M neurones)

1.5

M g

angl

ion

cells

, di

scre

te o

utpu

t(s

pike

trai

ns)

From van Hateren, J.H. & Ruderman, D.L. (1998) Proc. R. Soc. Lond. B 265: 2315-2320

Linear Independent Component Analysis:I(x, y, t) = ∑i ai Ci(x, y, t) ai = ∑x, y Fi(x, y, t) I(x, y, t)

The component Ci are computed to maximize their statistical independence.The stimulus I(x, y, t) are 12x12x12 patches drawn from TV movies. The corresponding filters look like the spatio-temporal receptive field of V1 neurons.But not like ganglion cell of retina !

2. Correlations between cell can be ignoredNirenberg et al (2001) Retinal ganglion cells act largely as independent encoders. Nature, 411: 698-701

I = ∑s P(s).∑r1,r2 P(r1 r2 | s)log2P(r1 r2 |s) – ∑r1,r2 P(r1 r2)log2P(r1,r2)

Pind(r1 r2 | s) = P(r1 |s)P(r2 | s)

movie

Time var.of intensity

Is 10 % low? What about higher order correlations?

Population code

Single cell time code

Spatio-temporal analysis of optical imaging data (voltage sensitive dye) in the cat primary visual cortex (collaboration Chantal Milleret, Luc Foubert)

Instantaneous activity map according to themovement direction

Instantaneous activitymap according to thegrating orientation

Temporal profile (5ms per frame)

• Shift of the preferred orientation due to texture speed ... (not classic)

• Influence of texture element size (not classic)

Motion energy based model accounting for V1 optical imaging & unit recordings(Simon Capern, Daniel Bennequin, Jacques Droulez)

(Data from Basole, White & Fitzpatrick, Nature 03)

(classic)

Θ

M 0° 45° 90° 135°

Some combinations of movement (M) and orientation () elicit little or no activationin the population of V1 cells because the movement is nearly aligned to the contours:the distribution of their spatio-temporal characteristics is non uniform

Θ

M 0° 45° 90° 135°

For moving oriented textures, this non uniform distribution bias the population response

Ex: horizontal movement, texture oriented at 45° : biased toward 90° (67° in data)

Results

• Texture element size :

0 50 100 150 200 250 300 350010

20

30

40

50

60

70

80

90

Direction (°)

Rép

onse

2°

4°

10°

0 20 40 60 80 100 120 140 160 1800

10

20

30

40

50

60

70

80

90

100

pour

cent

age

Orientation (°)

2°

4°

10°

Uni

t re

cord

ing

Sim

ulat

ion

resu

lts

Sim

ulat

ion

resu

lts

PhR

Bip

G

H HH

A AA

h

Glutamate

GABA

Hemi gap junction

Gap junction

GABA, Gly, Ser, ACh

Ôlvevczky, Baccus & Meister (2003)Segregation of object and background motion in the retina. Nature, 423:402-408

+ Drift: 450 µm/s

Fixational eye movement(jitter) recorded & simulated

Proposed mecanism: inhibition by polyaxonal amacrine cells (Glycine)

Schwartz, Harris, Shrom, Berry (2007) Detection and prediction of periodic patternsby the retina. Nature Neuroscience, 10(5): 552-554

Response to the absence of a predicted stimulus:Could it simply result from an ON/OFF response to averaged intensity?

No, since: (1) the response timing depends on the stimulus frequency

(2) Still responds to stimulus absence with a constant mean intensity

Response to onset and offset of a movement (SupFig5)

Common point with data from Ôlvevczky et al: precise timing / temporal pattern

A common model ?

The space-time problem for a neuron

Diffusion equation of membrane potential : τ u/t = 2 2u/x2 2/τ ≈ 0.1 mm2/ms

Again: Time Length2

dq/dt = Cm u/t = gi 2u/x2 + gm(u,t) (Em – u)

Neurons can be extremely elongated !

Up to 1 m (motoneurons in the spinal chord) Tdiff ≈ 3 h

From 300 µm (bipolar neurons in the retina) Tdiff ≈ 1 ms

For l > 300 µm, it’s better to transmit discrete, regenerated signal (action potential)

∆t = 1 ms ∆x = 3 mm Tprop ≈ 300 ms

Non spiking code in neurons:

● Graded responses (ex. in retina)

● Calcium inflow (amacrine cells)

● Active chemical propagation in axons

Basic functional schema of the retina

PhR

Bip

G

H HH

A AA

h

Glutamate

GABA

Hemi gap junction

Gap junction

GABA, Gly, Ser, ACh

Except for ganglion and some amacrine cells,information propagates without spikes.

Rh

Rh*

h T

T*

PDE

PDE*

GTP

cGMP

GMP

GC

CNGC

The phototransduction results from a cascad of biochemical processes:● photoisomerisation of 11-cis retinal pigment into the (stable) all-trans isomere● conformational change of rhodopsine into the active (metaII) form (Rh*)● release by Rh* catalytic action of active transducine T*● activation of PDE by T* fixation● hydrolysis of cGMP, which is continuously produced by guanylate cyclase (GC)● cGMP diffuses in the cytoplasm and binds to a specific ionic channel (CNGC)● CNGC, permeable to Na+ and Ca2+, depolarizes the membrane in darkness ● In light, cGMP is removed, the channels close and the membrane repolarizes.

Na+, Ca2+

h

Rh*

PDE

J ~ V

Transduction without inactivation

h

Rh*

PDE

J ~ V

Transduction with inactivation

Rh* inactivated by RPK

PDE ~ Amp.t

cGMP cGMP

Inactivation of PDE

Rh* life time ~1-2 s (amphibian rods)

Rh

Rh*

RhP

h T

T*

PDE

PDE*

GTP

cGMP

GMP

RP

K

GC

CNGC

All the transduction chain is controlled by calcium feed-back

Ca2+

–

+? – –

–?

Calcium contribution to photoreceptor response (rods data from Matthiews, 1990)

Steady response

Incremental response (If = 0.8 ph.µm-2.s-1)

Ca2+ clamped to dark value

Ca2+ clamped to dark value

Unclamped Ca2+:S ~ Log Ib

Unclamped Ca2+ :R ~ I / I

If

Vo = -35mVR

If

Ib

-35mV

R

S

Flash in darkness

Flash above background

PhR

Bip

G

H HH

A AA

h

Glutamate

GABA

Hemi gap junction

Gap junction

GABA, Gly, Ser, ACh

Intracellular calcium and membrane potential signals recorded in starburst amacrine cells of the rabbit retina during moving gratings (left) and moving bars (right) in various directions (black arrows). The dendritic site of calcium recordings is shown by a cross on the cell image.

Euler, C., Detwiler, P.B. & Denk, W. (2001) Directionally selective calcium signals in dendrites of starburst amacrine cells. Nature,

418:845-851.

Fasano et al, (2007) Neuronal conduction of excitation without action potentials based on ceramide production. Plos One, 7: e612.

?

Who asks the question?

● One specification → P(A B C)

● Many questions (18):

P(A) ? …P(B C) ? …P(C |A) ? …P(B | A C) ? …P(B C | A) ? …

Each question requires the specific combination of 2 rules

Basic probabilistic reasoning:

1) Bayes’ rule: P(A B) = P(A).P(B | A) = P(B).P(A | B)

2) Marginalization rule: P(A) = B P(A B) P(B) = A P(A B)

An experimental protocol designed to directly test the marginalization rule:

Training phase with 2 color cues (A B) + 1 motion cue (C) Coherence = 30 %Learned distribution: P(C | A B train)

Test phase with either 2 color cues, 1 color cue (A or B) or no color cue + No coherent motion(9 cases)

Subject responses = priors P(C | A B test), P(C | A test), P(C | B test), P(C | test)

TrainTest

1) During test, without coherent motion, subjects responses are strongly biased2) The bias (prior on C) depends on the color cue configuration, even though subjects did

not at all remark any effect of color on their motion judgments3) According to the subject, this dependence reflects more or less the training dependency4) When one or both color cues are removed, the marginalization rules predicts subjects’

responses BUT TO A DIFFERENT QUESTION !

Examples of results

Who takes the decision?

Perceptive reports Behavioral choices

Are they simply randomly drawn from P(S) or P(A)?

Leopold et al (2003) Stable perception of visully ambigous patterns. Nat. Neuroscience, 5:605-609

Adaptation

K. Hokusai (1831)

Adaptation to visual metric distorsion

Two ways to consider parametric changes:

● P(X | θt) P(θt | θt-1)

● P(X | θ) with optimalization procedure for θ

Probability computation at different level(cell population / single cell / macromolecule)

Static models

Two diffusible messengers:[X1] P(O1 | S1=1) / P(O1 | S1=0) [X2] P(O2 | S2=1) / P(O2 | S2=0)

One macromolecule with (reversible) transitions between 4 conformation states.At equilibrium: P(M∞) = P(S1 S2 | O1 O2)

More complex static models …

A population of N macromolecules canrepresent any probability distribution over 2n states.

Ex: (rhodopsin) N=25000/µm2, n=12

S1 O1

S2 O2

S3

P(M open) = P(S3 | O1 O2)

Dynamic model: biochemical implementation of a Bayesian filter

Ot+∆t

Two diffusible messengers:x(t) = P(Ot | St=1) / P(Ot | St=0) y(t) = P(St=1 | O1→t) / P(St=0 | O1→t)

Two macromolecules:M1 (4 conformation states) computes the posterior distribution from the previous posterior ratio y(t) and the current likelihood ratio x(t).The net balance between release (from M1) and elimination (by M2) converts a posterior probability into a ratio: P(St=1 | O1→t) = y(t) / [1 + y(t)]

M1

M2

Examples of simulation results

At equilibrium, the 2nd messenger concentration(averaged on 20 ms) matches perfectly well theBayesian filter posterior ratio, when submitted to a constant observation, for a large range of likelihood ratio.

Dynamic evolution of the Bayesian filter(step 1 ms) as compared to the dynamicevolution of the biochemical system(1µm3, N=1000 macromolecules of each type)for staircase variation of the likelihood ratio.

Convergence speed of BBF and biochemical process

Fernandez et al (2006) DARPP-32 is a robust integrator of dopamine and glutamate signals.PloS computational biology, 2(12): e176.

DARPP-32 is a protein phosphatase. Itsactivity depends on the 4 phosphorylationsites (3 are shown). Each one is controlledby different messengers (cAMP, Ca2+).

It integrates dopamine and glutamate signalsin striatal GABAergic neurons.

Thank you for your attention,and for your patience …