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Bayesian Cognition Winter School Chamonix, January 6-11, 2008 Probabilistic interpretation of physiological and psychophysical data Jacques Droulez Laboratoire.

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Presentation on theme: "Bayesian Cognition Winter School Chamonix, January 6-11, 2008 Probabilistic interpretation of physiological and psychophysical data Jacques Droulez Laboratoire."— Presentation transcript:

1 Bayesian Cognition Winter School Chamonix, January 6-11, 2008 Probabilistic interpretation of physiological and psychophysical data Jacques Droulez Laboratoire de Physiologie de la Perception et de lAction CNRS – Collège de France


3 The agent … Black box view of an agent Sensory inputs: o t Motor outputs: a t

4 The agent … Black box view of a situated agent otot atat The world …

5 The agent … Black box view of a situated agent (with an internal model of the world…) otot atat The world … StSt atat otot Question: P(S t | a 0t-1 o 0t ) and P(A t | a 0t-1 o 0t )

6 Observation Action ao

7 ?

8 ?


10 An evolutionary perspective Eukaryotes OpisthokontsAlgaePlantsOthers... 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.

11 An evolutionary perspective Eukaryotes OpistokontsAlgaePlantsOthers...

12 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 GABA A receptors in the modulation of Paramecium swimming behaviour. Neuroscience Letters, 386:179-183

13 Evolutionary perspective Eukaryotes Opistokonts MetazoanFungi AlgaePlants Sponges Medusas Bilateria Cell specialization Nervous system Segmented brain Others... worms... human

14 The space-time problem for multicellular organism... Diffusion law: [C]/t = D 2 [C]/x 2 D 1 µm 2 /ms Time Length 2 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

15 Evolutionary perspective Eukaryotes Opistokonts MetazoanFungi AlgaePlants Sponges Medusas Bilateria Cell specialization Nervous system Segmented brain Others... worms... human

16 Schematic view of a cnidarian eye Ciliary type photoreceptor in red Melanin pigment cells in yellow Biconvex lens in blue Nervous system of the cubomedusae jellyfish RN: ring nerve Rh: rhopalia (eye + graviceptors) Garm et al (2006) Cell Tissue Res. 325: 333-343

17 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

18 Evolutionary perspective Eukaryotes Opistokonts MetazoanFungi AlgaePlants Sponges Medusas Bilateria Cell specialization Nervous system Segmented brain Others... worms... human

19 1: neurones 2: cerebral ganglion 3: medullae 4: 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


21 Supra-spinal nervous system in some vertebrates OB: olfactory bulb OT: optic tectum T: forebrain Cb: cerebellum D: diencephalon frog reptile bird rodent horse

22 Three hierarchically organized levels OrganismCell Macromolecule Sensors effectors Chemical sensors Photosensors (eyespot), … Neurotransmitter release Mechanical effectors, … Ligand binding site Stretching sensitivity Voltage sensor Channel open/closed Catalytical action, …

23 Olfaction Vision Oculomotor (Eye/ Pupil) Trochlear (Eye movement) Abducens (Eye movement) Trigeminal Facial Vestibulo-cochlear …etc … ObservationsActions The brain as a (segmented) agent

24 Example of cellular agent: the photoreceptor cell cGMP Ca 2+ /Na + Glu Photons In darkness: high cGMP (2µM) Ca 2+ & Na + Inflow [Ca 2+ ] i = 550 nM V = – 40 mV Voltage gated Ca 2+ 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 Ca 2+ channels are closed Reduction of glutamate release Depolarization Hyperpolarization Observation: Light intensity Action: chemical release

25 Example of molecular agent: a voltage gated Ca 2+ channel Action: Ca2+ Inflow Observation: electric field across the membrane (~ 15 10 6 V/m !!)

26 10 -12 s10 -9 s 10 -6 s 10 -3 s 1 s Elementary relaxation (macromolecule) Tertiary structure transitions Channels Open/ closed Membrane time constant / spike propagation Basic behaviours 10 3 s Plastic adaptation

27 ?

28 This page is intentionally black

29 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)

30 P P(S) Blood pressure Attentional blindness neural activity under anesthesia

31 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 a Brain with limited resources...

32 Incidence of small ocular movements on visual perception Akiyoshi Kitaoka, Out of focus (2001)

33 1. Luminance perception

34 ~ 1 photon/µm 2 /s ~ 10 9 photon/µm 2 /s Vincent Van Gogh, Nuit étoilée sur le Rhône (1888) Oliviers avec ciel jaune et soleil (1889) Dawson (1990)

35 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.

36 Deviation from Weber-Fechners laws: Perceived brightness versus luminance (Cd/m 2 ) A B Data from Nundy & Purves (2002) PNAS 99:14482-14487

37 The probabilistic explanation by Purves et al (2004) Psychol. Rev. 111:142-158 L < Background L > Background L Luminance ~ Illumination x Reflectance:I constant R L R constant I L I R R I L 1/2

38 Basic functional schema of the retina PhR Bip G HHH AAA h Glutamate GABA Hemi gap junction Gap junction GABA, Gly, Ser, ACh Except for ganglion and some amacrine cells, information propagates without spikes.

39 Hypothesis of conservation of loal luminous intensity: I(x+dx, y+dy, z+dt) I(x, y,z) or dI/dt = = 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)

40 The aspect is globally conserved, but the local luminous intensity is not exactly the same in succesive images P(I t+dt | I t V) or P(G t | V)

41 Motion integration: numerous sources of uncertainty: Nonhomogeneous distribution of contrasts Low Contrast, Aperture problem, False correspondences ? ? ? G = 0single oriented G multiple G

42 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)

43 More recent works on the « low velocity prior » idea: Carandini, M. (2006) Measuring the brains 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)

44 D R Scale ambiguity: The depth & velocity scales cannot be estimated from the optic flow alone V d R v

45 Knowledge of self motion can be used (in principle) to solve the scale ambiguity problem. V D d

46 Comparison SM (subject motion) versus OM (object motion) Subjects Task: report whether or not the object is closer than 1 meter Same relative velocityAll trials Panerai, Cornilleau-Pérès & Droulez, Perception & Psychophysics, 64: 717-731 (2002)

47 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

48 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)

49 Similar optic flows result from different combinations of rotation and translation Results show a preference for the stationary object, even if it does not correspond to the most rigid solution Wexler, Lamouret & Droulez, Vision Research, 41, 3023-3037 (2001)

50 Variability of perceptive responses (« shear effect ») Van Boxtel, Wexler & Droulez, Journal of Vision 3(5) : 318-332. (2003)

51 Self-motion can change the interpretation of perspective cues Wexler, Panerai, Lamouret & Droulez, Nature, 409, 85-88 (2001)

52 Perspective cues: « prior » knowledge on object shape & orientation. The image of a regularly textured plane... back-projected on another plane « trompe-lœil » image Shape from motion: combining prior, optic flow and self-motion

53 Optic Flow: rotation flow + translation flow (= rigidity hypothesis) Φ = W + p.Tp = 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 …

54 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) Observers displacement (Obs) Relative Object movement (Move) Visual data (Flow) F. Colas, J. Droulez, M. Wexler & P. Bessière Biol. Cybernetics, 2007. Obj Obs MoveFlow

55 Probabilistic model (results) Immobile Subject Shear 0° Immobile Subject Shear 90° Moving Subject Shear 0° Moving Subject Shear 90° Immobile Subject +TZ Moving Subject +TZ

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

57 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 mathematically straightforward, it would yield error accumulation! The brain chooses to reduce as much as possible the estimated sustained velocity Data from Büttner & Waespe (81)

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

59 A GF F = G - A The 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

60 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

61 Denise, Darlot, Droulez, Cohen & Berthoz, EBR (1988) Data from Guedry, (1970)... or during off-vertical axis rotation (OVAR / Barbecue)

62 Bayesian Filter with priors (Low angular velocity & linear acceleration) Rotational acceleration Otolith signal Canal signal Linear acceleration Linear velocity Noise η N(0, 0.005) rad/s² A priori N(0, 0.3) rad/s Double integration Rotational velocity Head orientation A priori N(0, 2) m/s² Head position Double integration a J. Laurens & J. Droulez, Biol. Cybernetics, 2007

63 Centrifugation (Off Axis Rotation) time (s) Pitch (rad) Pitch 020406080 0 0.5 1 0 0.05 0.1 1 s 20 s 70 s 50 s 60 s62 s g -a F Velocity (rad/s) Yaw velocity -2 0 2 0 0.1 0.2 0.3 0.4 time (s) Velocity (rad/s) Roll velocity 020406080 -0.2 0 0.2 0.4 0.6 0 0.05 0.1 time (s) 020406080 J. Laurens & J. Droulez, Biol. Cybernetics, 2007

64 ?

65 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

66 Non uniform distribution of natural stimuli: P(L) Correlation in space or time: P(L t+1 | L t ) More complex dependencies (word …) 1)Sensory data compression (redundancy reduction) 2)Simplified representation of probability distribution

67 1. Redundancy reduction ? Yes, in the retina! h 120 M photoreceptors with graded output Not from retina to LGN and from LGN to V1 (~ 1000 M neurones) 1.5 M ganglion cells, discrete output (spike trains)

68 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 a i C i (x, y, t) a i = x, y F i (x, y, t) I(x, y, t) The component C i 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 !

69 2. Correlations between cell can be ignored Nirenberg 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)log 2 P(r1 r2 |s) – r1,r2 P(r1 r2)log 2 P(r1,r2) P ind (r1 r2 | s) = P(r1 |s)P(r2 | s) movie Time var. of intensity Is 10 % low? What about higher order correlations?

70 Population code Single cell time code

71 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 the movement direction Instantaneous activity map according to the grating orientation Temporal profile (5ms per frame)

72 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)

73 Θ M 0° 45°90°135° Some combinations of movement (M) and orientation ( ) elicit little or no activation in the population of V1 cells because the movement is nearly aligned to the contours: the distribution of their spatio-temporal characteristics is non uniform

74 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)

75 Stimulus FFT for 2 velocities A « window of visibility » (Watson et al, 86; Mante & Carandini, 2005) is applied to the stimulus in the space-time domain Space-time frequency window (for a given preferred orientation)

76 Results Texture element size : 050100150200250300350 0 10 20 30 40 50 60 70 80 90 Direction (°) Réponse 2° 4° 10° 020406080100120140160180 0 10 20 30 40 50 60 70 80 90 100 pourcentage Orientation (°) 2° 4° 10° Unit recording Simulation results

77 Influence of the texture speed: 0102537.55062.57587.5100 80 90 100 110 120 130 140 Vitesse(°/s) Orientation préférée 020406080100120140160180 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Orientation(°) Réponse Normalisée Simulation results

78 PhR Bip G HHH AAA h Glutamate GABA Hemi gap junction Gap junction GABA, Gly, Ser, ACh

79 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

80 Proposed mecanism: inhibition by polyaxonal amacrine cells (Glycine)

81 Schwartz, Harris, Shrom, Berry (2007) Detection and prediction of periodic patterns by 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?

82 No, since: (1) the response timing depends on the stimulus frequency (2) Still responds to stimulus absence with a constant mean intensity

83 Response to onset and offset of a movement (SupFig5)

84 Common point with data from Ôlvevczky et al: precise timing / temporal pattern A common model ?

85 The space-time problem for a neuron Diffusion equation of membrane potential : τ u/ t = 2 2 u/ x 2 2 /τ 0.1 mm 2 /ms Again: Time Length 2 dq/dt = C m u/ t = g i 2 u/ x 2 + g m (u,t) (E m – u)

86 Neurons can be extremely elongated ! Up to 1 m (motoneurons in the spinal chord) T diff 3 h From 300 µ m (bipolar neurons in the retina) T diff 1 ms For l > 300 µm, its better to transmit discrete, regenerated signal (action potential) t = 1 ms x = 3 mm T prop 300 ms

87 Non spiking code in neurons: Graded responses (ex. in retina) Calcium inflow (amacrine cells) Active chemical propagation in axons

88 Basic functional schema of the retina PhR Bip G HHH AAA h Glutamate GABA Hemi gap junction Gap junction GABA, Gly, Ser, ACh Except for ganglion and some amacrine cells, information propagates without spikes.

89 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 +, Ca 2+

90 h Rh* PDE J ~ V Transduction without inactivation h Rh* PDE J ~ V Transduction with inactivation Rh* inactivated by RPK PDE ~ Amp.t cGMP Inactivation of PDE Rh* life time ~1-2 s (amphibian rods)

91 Rh Rh* RhP h T T * PDE PDE* GTP cGMP GMP RPK GC CNGC All the transduction chain is controlled by calcium feed-back Ca 2+ – +? – – –?

92 Calcium contribution to photoreceptor response (rods data from Matthiews, 1990) Steady response Incremental response (If = 0.8 ph.µm -2.s -1 ) Ca 2+ clamped to dark value Unclamped Ca 2+ : S ~ Log Ib Unclamped Ca 2+ : R ~ I / I If Vo = -35mV R If Ib -35mV R S Flash in darkness Flash above background

93 PhR Bip G HHH AAA h Glutamate GABA Hemi gap junction Gap junction GABA, Gly, Ser, ACh

94 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.

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

96 ?

97 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) ? …

98 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)

99 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) Train Test

100 1)During test, without coherent motion, subjects responses are strongly biased 2)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 judgments 3)According to the subject, this dependence reflects more or less the training dependency 4)When one or both color cues are removed, the marginalization rules predicts subjects responses BUT TO A DIFFERENT QUESTION ! Examples of results

101 Who takes the decision?

102 Perceptive reports Behavioral choices Are they simply randomly drawn from P(S) or P(A)?

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

104 Adaptation

105 K. Hokusai (1831)


107 Adaptation to visual metric distorsion

108 Two ways to consider parametric changes: P(X | θt) P(θt | θt-1) P(X | θ) with optimalization procedure for θ

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

110 Static models Two diffusible messengers: [X 1 ] P(O 1 | S 1 =1) / P(O 1 | S 1 =0) [X 2 ] P(O 2 | S 2 =1) / P(O 2 | S 2 =0) One macromolecule with (reversible) transitions between 4 conformation states. At equilibrium: P(M ) = P(S 1 S 2 | O 1 O 2 )

111 More complex static models … A population of N macromolecules can represent any probability distribution over 2 n states. Ex: (rhodopsin) N=25000/µm 2, n=12 S1S1 O1O1 S2S2 O2O2 S3S3 P(M open) = P(S 3 | O 1 O 2 )

112 Dynamic model: biochemical implementation of a Bayesian filter O t+t Two diffusible messengers: x(t) = P(O t | S t =1) / P(O t | S t =0)y(t) = P(S t =1 | O 1t ) / P(S t =0 | O 1t ) Two macromolecules: M 1 (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 M 1 ) and elimination (by M 2 ) converts a posterior probability into a ratio: P(S t =1 | O 1t ) = y(t) / [1 + y(t)] M1M1 M2M2

113 Examples of simulation results At equilibrium, the 2nd messenger concentration (averaged on 20 ms) matches perfectly well the Bayesian 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 dynamic evolution of the biochemical system (1µm 3, N=1000 macromolecules of each type) for staircase variation of the likelihood ratio. Convergence speed of BBF and biochemical process

114 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. Its activity depends on the 4 phosphorylation sites (3 are shown). Each one is controlled by different messengers (cAMP, Ca 2+ ). It integrates dopamine and glutamate signals in striatal GABAergic neurons.

115 Thank you for your attention, and for your patience …

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