Eye movements as a model for active inference - <a href="http://www.laconeu.cl">LACONEU2017: 4th Latin-American Summer School in Computational Neuroscience</a> January 20th, 2017 <BR> <a href="https://laurentperrinet.github.io/talk/2017-01-20-laconeu">https://laurentperrinet.github.io/talk/2017-01-20-laconeu</a>

Tutorial: Active inference for eye movements: Bayesian methods, neural inference, dynamics

Laurent U Perrinet, INT

Acknowledgements:

Mina Aliakbari Khoei, Guillaume Masson and Anna Montagnini - FACETS-ITN Marie Curie Training
Rick Adams and Karl Friston @ UCL - Wellcome Trust Centre for Neuroimaging
Wahiba Taouali, Giacomo Benvenuti, Frédéric Chavane - ANR BalaV1

LACONEU2017: 4th Latin-American Summer School in Computational Neuroscience January 20th, 2017
https://laurentperrinet.github.io/talk/2017-01-20-laconeu

(AUTHOR) Hi, I am Laurent Perrinet from (LOGO) the Institute de Neurosciences de la Timone in Marseille, a joint unit from the CNRS and AMU. Using computational models, I am investigating the link between the efficiency of behavioural responses in vision, their underlying neural code and their adaptation to the structure of the world.
(SHOW TITLE - THEME) = In particular, eye movements, that is, motion of the eyeballs targeted at orienting our gaze in the visual field, provide a large repertoire of behaviors at different time scales from fast, reflexive-like saccades to smooth movememnts and even to long-term adaptation.
(SHOW TITLE -OBJECTIVE) today, I will be focusing on the key challenges in modelizing visually guided eye movements and more generally to better understand decision making in biology (the living kingdom). in particular, we will focus on the dynamics of these mechanisms to unravel the sequence of processes which allow the efficient response of motor commands to the environment. As such, my goal here is to show that the solutions proposed by motion perception and EMs provide a generic (and rather unique) framework to finesse models of decision making in the generic theory championned by Karl Friston, active inference (AI)
(ACKNO) this endeavour involves different techniques, tools and... persons. From the head on, I wish to thank the people who collaborated to this project and in particular Rick Adams and Karl Friston and the Wellcome Trust Centre for Neuroimaging for providing the tools for a successful visit / but also Mina, Guillaume and Anna for the great oppportunity to link theories with psychophysical data; Wahiba, Giacomo and Frédéric for the challenging task of linking such models to the neural activity in NHP, and finally Jean-Bernard, Sohir and Laurent Madelain for their essential knowledge in adaptation and reinforcement.

more precisely... what are eye movements and how can they be useful for active inference?

... Eye movements can be of different nature, and I will propose to you now some gentle post-lunch gymnastics by trying to elicit such visually-guided movements with some short movies:

Most simply, saccades are an orienting behavior aiming at catching a specific position in visual space (or vergence movement in depth). if after fixating at the red fixation dot you try to catch the green dot, then you need to do such saccades, which can be very fast (up to 600 deg/s)

When the dot is now moving almong a trajectory, this visual motion can are also responsible for more complex movements such as the reflexive Ocular Following Response or the voluntary Smooth Pursuit EM which aims at stabilizing the visual image on the retina. INterestingly, in this latter case the position of the eye quickly centers on the present, physical position of the dot and we will later see why this is challenging

... Eye movements can be of different nature, and I will propose to you now some gentle post-lunch gymnastics by trying to elicit such visually-guided movements with some short movies:

The Behavioral receptive field

LP and Masson (2012) Bio Behav Reviews, https://laurentperrinet.github.io/publication/masson-12

Montagnini, and Masson (2015) in Biologically Inspired Computer Vision

... we have been busy in the last couple of years to make sense of this repertoire of behaviors, in particular in light with the architecture of the brain areas involved in producing these EMs, for instance for pursuit initiation, ...

the Left-side plot (extracted from this review paper) illustrates in primates a sketch of the hierarchy of neuronal population yielding to ocular following responses from the early visual areas (retina, thalamus, primary visual cortex), finessed by processing in motion specific information in downstream sensory areas (MT for speed or MST for global motion) to the brain stem and the oculomotor muscles which tranform this neural activity in a graded motor output. As such, moving stimuli covering the retina are processed by a cascade of neuronal populations, where information is processed through feedforward, lateral and feedback interactions.
in particular, we took advantage of a Bayesian, probabilistic framework to develop an unified theory, the "behavioral receptive field" which encompasses a large range of psychophysical experiments by modeling behavioral responses from population dynamics. Such cascade implements local, context-dependent extraction of motion information…
today, I would like today to focus on a particular problem which will help us unravel the dynamics of decision making: oculomotor delays. Indeed, one challenge for modelling is to understand EMs using AI as a problem of optimal motor control under axonal delays. The central nervous system has to contend with axonal delays, both at the sensory and the motor levels. For instance, in the human visuo-oculomotor system, it takes approximately $ au_s=50~ms$ for the retinal image to reach the visual areas implicated in motion detection, and a further $ au_m=40~ms $ to reach the oculomotor muscles.

The motion energy model

Weiss et al (2002) Nature Neuroscience

Standard Models of Motion detection are the well-known motion energy and Reichhardt models: these local motion detectors take the sequence as an input and give as an output different ``scores'' for a set of velocities.

(proba) in fact these models are equivalent at the functional level to a probabilistic measure. so if it's equivalent, why use probas? first, computation of a likelihood explicitly defines the generative models underlying the definition of motion and thus gives all underlying assumptions of a given model. second, we will see that probabilities are well adapted to manipulate bits of information,
(weiss) I show right the architecture of the model by Weiss02: top, the stimulus (here a rhombus), and in the middle what knowledge as represented in motion space (referenced here by the axis of horizontal and vertical translational velocities). we may extract internal knowledge about the motion in a receptive field and represent it by a subset in motion space, represented here by the "white" colours. In the case of the segment, most information is distributed along the line corresponding to the possible velocities. In the case of a dot or of an edge, motion is unambiguous, information lies on a small concentrated area of motion space.
(merge) information is then assumed to be conditionally independent and probabilities are multiplied to give a global readout. finally, the introduction of a bayesian prior was very successful in describing many behavioural and physiological data ... % however, the question is open concerning as how we may select and merge these measures

Eye movements and Bayes: motion integration

Outline

Eye movements

Motion-based predictive coding

Active inference, EMs & oculomotor delays

Perspectives: Saccadic adaptation

Motion-based predictive coding

Pack and Born (2001) Nature

(pack01) ... that the solution to the aperture problem seems to be dynamical. This data was recorded by Chris Pack and Rick Born in neurones from area MT in macaques: (2.a) the early response of neurones is preferentially tuned to the perpendicular to the line, (2.b) while it shifts to the physical direction (2.c) after approx 60 ms.
(OFR) this is also demonstrated at different levels of the system, and in particular for eye movements. (3) This is data recorded at their lab showing eye tracking to the slanted line a showing the same initial bias as with the neural data. These behavioural results replicate those obtained previously in humans by Guillaume Masson at our lab and show the close causality link from neural to behavioural response.
OBSOLETE (Pack03) as was shown physiologically or behaviorally, the transient response to line or line-ending detector detectors was similar, a difference emerged in their dynamics after few milliseconds. This observations suggest that these different cells have a common excitatory drive but that they integrate contextual information from neighboring neurons differently. It therefore suggests that the response to line-endings is not caused by specialized receptors but rather the consequence of a dynamical integration for which motion-based prediction gives a principled approach. (Figure 7 from Pack, 2003) Timing of V1 End-Stopping (A) The first row shows successive snapshots of the one-bar map for a strongly end-stopped V1 cell, for the preferred bar orientation. The cell initially responds well to a bar placed anywhere in its receptive field, but after $20-30 ms$, it responds only when the endpoints are in the receptive field. The second row shows the time course of the response when the analysis was limited to specific regions of the stimulus range. When the analysis was limited to instances when the bar was centered on the receptive field (green square), the neuron responded transiently (green line). When the analysis was limited to instances when the endpoint of the bar appeared in the receptive field (black square), the cell exhibited a strong maintained discharge (black line). (B) The time course of the response, averaged across the population of 29 neurons for the preferred bar orientation. Dotted lines indicate standard error of the mean.

Motion-based predictive coding

Pack and Born (2001) Nature

(pack01) ... that the solution to the aperture problem seems to be dynamical. This data was recorded by Chris Pack and Rick Born in neurones from area MT in macaques: (2.a) the early response of neurones is preferentially tuned to the perpendicular to the line, (2.b) while it shifts to the physical direction (2.c) after approx 60 ms.
(OFR) this is also demonstrated at different levels of the system, and in particular for eye movements. (3) This is data recorded at their lab showing eye tracking to the slanted line a showing the same initial bias as with the neural data. These behavioural results replicate those obtained previously in humans by Guillaume Masson at our lab and show the close causality link from neural to behavioural response.
OBSOLETE (Pack03) as was shown physiologically or behaviorally, the transient response to line or line-ending detector detectors was similar, a difference emerged in their dynamics after few milliseconds. This observations suggest that these different cells have a common excitatory drive but that they integrate contextual information from neighboring neurons differently. It therefore suggests that the response to line-endings is not caused by specialized receptors but rather the consequence of a dynamical integration for which motion-based prediction gives a principled approach. (Figure 7 from Pack, 2003) Timing of V1 End-Stopping (A) The first row shows successive snapshots of the one-bar map for a strongly end-stopped V1 cell, for the preferred bar orientation. The cell initially responds well to a bar placed anywhere in its receptive field, but after $20-30 ms$, it responds only when the endpoints are in the receptive field. The second row shows the time course of the response when the analysis was limited to specific regions of the stimulus range. When the analysis was limited to instances when the bar was centered on the receptive field (green square), the neuron responded transiently (green line). When the analysis was limited to instances when the endpoint of the bar appeared in the receptive field (black square), the cell exhibited a strong maintained discharge (black line). (B) The time course of the response, averaged across the population of 29 neurons for the preferred bar orientation. Dotted lines indicate standard error of the mean.

Motion-based predictive coding

LP and Masson (2012) Neural Computation, https://laurentperrinet.github.io/publication/perrinet-12-pred

We will now define a probabilistic framework for the representation of predictive motion information.

(prediction) First, we will define motion-based prediction which is adapted to smooth trajectories such as are observed in natural environments
(markov) Seeing this prediction as a prior on motion transition in the probabilistic domain, we then show how information may be pooled using a Hidden Markov Model, merging current information and measurement likelihood
(SMC) Differently to neural approximations that were used previously, we use a particle filtering method to implement this functional model. Prediction will now take the form of an anisotropic, context-dependent diffusion of local information.

But first, how do we define prediction?
In computer vision, one can define the prediction of the trajectory of some distinguished feature of the image (e.g., a local extremum of luminance) using its motion. here we define the feature to track as the motion at a given position and define Motion-based prediction by modelling smoothness in the trajectory of motion:
knowing the velocity $V_{t-dt}$ at one position $x_{t-dt}$, one expects that velocity will be transported to a new location, that is $x_{t} pprox x_{t-dt} + V_{t-dt} dt$ and approximately conserved in amplitude and direction $V_t pprox V_{t-dt} $. Nothing new: if we go to infinitesimal $dt$, this equation is the advection term in the navier-stokes equations.
This "approximately" may be included in our generative model by some random variable. different values for the variance of this state noise quantify the strength or precision of the prediction... it shapes the association fields in motion space similarly with what happens in the space domain with association field in the orientation domain of static images.
Note that contrary to classical models, position becomes an unknown variable instead of being a parameter as in OF. note also that it is auto-referent since prediction is based itself on motion %: neural = Series / models Grzywack, Burgi et al = "motion tracking"
then, we may use a classical Hidden Markov Model to recurrently update knowledge. This uses three steps knowing the probabilistic motion information at a current time and for simplicity I will just represent local probabilistic motion representations:
prediction of motion thanks to the generative model as a prior of motion transition and an integral over all possible motions. note that prediction is always diffusive
estimation of the likelihood of the predicted state,
merging both, we updated our knowledge from time $ t-dt$ to $ t$
repeating the operation, we define a dynamical system.
this theory seems easy on the paper, and there is nothing new here in the definition of the dynamical system . extit{BUT} ...... I have bad news: it is extremely hard to represent \& implement on classical machines: if we allow ourself a rather modest discretisation if the probability space (position and velocities), the very high dimensionality of the problem makes it tedious to represent ($2e9$ dimensions in the previously shown POF)...

Motion-based predictive coding

LP and Masson (2012) Neural Computation, https://laurentperrinet.github.io/publication/perrinet-12-pred

We will now define a probabilistic framework for the representation of predictive motion information.

(prediction) First, we will define motion-based prediction which is adapted to smooth trajectories such as are observed in natural environments
(markov) Seeing this prediction as a prior on motion transition in the probabilistic domain, we then show how information may be pooled using a Hidden Markov Model, merging current information and measurement likelihood
(SMC) Differently to neural approximations that were used previously, we use a particle filtering method to implement this functional model. Prediction will now take the form of an anisotropic, context-dependent diffusion of local information.

But first, how do we define prediction?
In computer vision, one can define the prediction of the trajectory of some distinguished feature of the image (e.g., a local extremum of luminance) using its motion. here we define the feature to track as the motion at a given position and define Motion-based prediction by modelling smoothness in the trajectory of motion:
knowing the velocity $V_{t-dt}$ at one position $x_{t-dt}$, one expects that velocity will be transported to a new location, that is $x_{t} pprox x_{t-dt} + V_{t-dt} dt$ and approximately conserved in amplitude and direction $V_t pprox V_{t-dt} $. Nothing new: if we go to infinitesimal $dt$, this equation is the advection term in the navier-stokes equations.
This "approximately" may be included in our generative model by some random variable. different values for the variance of this state noise quantify the strength or precision of the prediction... it shapes the association fields in motion space similarly with what happens in the space domain with association field in the orientation domain of static images.
Note that contrary to classical models, position becomes an unknown variable instead of being a parameter as in OF. note also that it is auto-referent since prediction is based itself on motion %: neural = Series / models Grzywack, Burgi et al = "motion tracking"
then, we may use a classical Hidden Markov Model to recurrently update knowledge. This uses three steps knowing the probabilistic motion information at a current time and for simplicity I will just represent local probabilistic motion representations:
prediction of motion thanks to the generative model as a prior of motion transition and an integral over all possible motions. note that prediction is always diffusive
estimation of the likelihood of the predicted state,
merging both, we updated our knowledge from time $ t-dt$ to $ t$
repeating the operation, we define a dynamical system.
this theory seems easy on the paper, and there is nothing new here in the definition of the dynamical system . extit{BUT} ...... I have bad news: it is extremely hard to represent \& implement on classical machines: if we allow ourself a rather modest discretisation if the probability space (position and velocities), the very high dimensionality of the problem makes it tedious to represent ($2e9$ dimensions in the previously shown POF)...

Particle filtering (Sequential Monte-Carlo)

Isard and Blake (1998) IJCV

... and at each point integration over all possible $x-Vdt$ (thus $2e9^2$ )...2 routes :
we may use Laplace's approximation for probabilities, and this Gaussian solution is a spatial Kalman filter: adapted to AP (one object), but lacks generality,
we may do an approximation, for instance by using the neuromimetic models inspired by Grossberg (Burgi, Watamaniuk) : but bad precision, bad prediction. this is why it has been discontinued in the literature,% but the novelty comes from the variables that are represented
instead, we use here a relatively old algorithm used in CV for tracking the contour of objects (a priori unrelated to tracking): CONDENSATION

put simply, it uses Particle filtering, that is implementing the Hidden Markov Model using a set of samples as "control points" (that's why it is also called SMC (Sequential Monte-Carlo) what is extit{NEW} compared to the previous route is that we consider a virtually continuous space with a controlled complexity (given by the number of particles) note that I do not claim that particles are implemented neurally, just that this algorithm provides a good approximation for the complex dynamical system (but see Maass et al) * I will not describe it here (it's not our point now, it's just a solution a method to an hardware problem and I will skip the details now) and we will rather focus on the results...

Prediction is sufficient to solve the aperture problem

LP and Masson (2012) Neural Computation, https://laurentperrinet.github.io/publication/perrinet-12-pred

we may evaluate an average cost on the tracking bias and precision, for each point in the state space here, we simply use as the cost the precision of the retrieved position x, y and velocity u,v of the dot % respectively on top and bottom plots, for left the horizontal component (x, u) on the left, and vertical component on the right (y, v) % \item averaging over 20 trials, one observes 3 types of behaviour: Tracking, FalseTracking, and NoTracking %there is a basin of parameters for which tracking emerges: its boundary is very steep% this will depend on the input showing that blur may be adapted to best fit natural scenes, %% \item this plot also shows that we reach precisions that show a super-resolution ($1e-2$ in position while we have XXX128XXX width images) % \item these properties define the extit{tracking} behaviour and explain why the system shows emergence of line-ending detectors. it would therefore be more correct to call them "smooth trajectory" detectors... in that basin, the system exhibits other properties like motion extrapolation ---that is that it will exhibit some kind of short-term memory%--- that we will detail later. \item[(Left)] Results show the emergence of different states for different prediction precisions: a smooth regime when prediction is lower (No Tracking - NT), a binary stage of motion tracking when it is set as previously (True Tracking - TT) and finally a regime corresponding to higher precisions with relatively efficient mean detection but high bias (False Tracking - FT). % : different stable tracking states. % TODO: As expected, replicating the results of (Weiss, 02) give a result similar to NT (dashed line). % (both axis are logarithmic)We also observe that edge-detector motion grabbers emerge as a property of the dynamical system. \item[(Right)] We give 3 representative examples of the emerging states at one contrast level ($C=0.1$) with starting (red) and end ending (blue) points and resp. NT, TT and FT by showing inferred trajectories for each trial. \item a second reason for which we solve the aperture problem is the fact that incoherent information is explained away by the competition implemented in the inference ...

Outline

Eye movements

Motion-based predictive coding

Active inference, EMs & oculomotor delays

Perspectives: Saccadic adaptation

* ... As a consequence, for a tennis player ---here (highly trained) Jo-Wilfried Tsonga at Wimbledon--- trying to intercept a passing-shot ball at a (conservative) speed of $20~m.s^{-1}$, the position sensed on the retinal space corresponds to the instant when its image formed on the photoreceptors of the retina and reaches our hypothetical motion perception area behind:

and at this instant, the sensed physical position is lagging behind (as represented here by $ au_s \cdot v 1~m$ ), that is, approximately at $45$ degrees of eccentricity (red dotted line),
while the position at the moment of emitting the motor command will be $.8~m$ ahead of its present physical position ($ au_m \cdot v$).
As a consequence, note that the player's gaze is directed to the ball at its present position (red line), in anticipatory fashion. Optimal control directs action (future motion of the eye) to the expected position (red dashed line) of the ball in the future --- and the racket (black dashed line) to the expected position of the ball when motor commands reach the periphery (muscles). This is obviously an interesting challenge for modelling an optimal control theory.

in a first study, we have proposed that PERCEPTION (following Helmhotz 1866) is an active process of hypothesis testing by which we seek to confirm our predictive models of the (hidden) world: Active inference: (cite TITLE). In theory, one could find any prior to fit any experimental data, but the beauty of the theory comes from the simplicity of the models chosen to model the data at hand, such as saccades...
... and even better if these models may find a possible correspondance into the neural anatomy and explain some deviation to a control behaviour, such as that we modelled for understanding some aspects of the EMs of schizophrenic patients
Today, I will again focus on the problem of sensorimotor delays in the optimal control of (smooth) eye movements under uncertainty. Specifically, we consider delays in the visuo-oculomotor loop and their implications for active inference and I will present the results presented in the following paper (show TITLE).

This schematic shows the dependencies among various quantities modelling exchanges of an agent with the environment. It shows the states of the environment and the system in terms of a probabilistic dependency graph, where connections denote directed (causal) dependencies. The quantities are described within the nodes of this graph -- with exemplar forms for their dependencies on other variables.
Hidden (external) and internal states of the agent are separated by action and sensory states. Both action and internal states -- encoding a conditional probability density function over hidden states -- minimise free energy. Note that hidden states in the real world and the form of their dynamics can be different from that assumed by the generative model; (this is why hidden states are in bold. )

Pursuit initiation

This figure reports the conditional estimates of hidden states and causes during the simulation of pursuit initiation, using a single rightward (positive) sweep of a visual target, while compensating for sensory motor delays. We will use the format of this figure in subsequent figures: the upper left panel shows the predicted sensory input (coloured lines) and sensory prediction errors (dotted red lines) along with the true values (broken black lines). Here, we see horizontal excursions of oculomotor angle (upper lines) and the angular position of the target in an intrinsic frame of reference (lower lines). This is effectively the distance of the target from the centre of gaze and reports the spatial lag of the target that is being followed (solid red line). One can see clearly the initial displacement of the target that is suppressed after a few hundred milliseconds. The sensory predictions are based upon the conditional expectations of hidden oculomotor (blue line) and target (red line) angular displacements shown on the upper right. The grey regions correspond to 90% Bayesian confidence intervals and the broken lines show the true values of these hidden states. One can see the motion that elicits following responses and the oculomotor excursion that follows with a short delay of about 64ms. The hidden cause of these displacements is shown with its conditional expectation on the lower left. The true cause and action are shown on the lower right. The action (blue line) is responsible for oculomotor displacements and is driven by the proprioceptive prediction errors.

This figure illustrates the effects of sensorimotor delays on pursuit initiation (red lines) in relation to compensated (optimal) active inference -- as shown in the previous figure (blue lines). The left panels show the true (solid lines) and estimated sensory input (dotted lines), while action is shown in the right panels. Under pure sensory delays (top row), one can see clearly the delay in sensory predictions, in relation to the true inputs. The thicker (solid and dotted) red lines correspond respectively to (true and predicted) proprioceptive input, reflecting oculomotor displacement. The middle row shows the equivalent results with pure motor delays and the lower row presents the results with combined sensorimotor delays. Of note here is the failure of optimal control with oscillatory fluctuations in oculomotor trajectories, which become unstable under combined sensorimotor delays.

This figure illustrates the effects of sensorimotor delays on pursuit initiation (red lines) in relation to compensated (optimal) active inference -- as shown in the previous figure (blue lines). The left panels show the true (solid lines) and estimated sensory input (dotted lines), while action is shown in the right panels. Under pure sensory delays (top row), one can see clearly the delay in sensory predictions, in relation to the true inputs. The thicker (solid and dotted) red lines correspond respectively to (true and predicted) proprioceptive input, reflecting oculomotor displacement. The middle row shows the equivalent results with pure motor delays and the lower row presents the results with combined sensorimotor delays. Of note here is the failure of optimal control with oscillatory fluctuations in oculomotor trajectories, which become unstable under combined sensorimotor delays.

This figure illustrates the effects of sensorimotor delays on pursuit initiation (red lines) in relation to compensated (optimal) active inference -- as shown in the previous figure (blue lines). The left panels show the true (solid lines) and estimated sensory input (dotted lines), while action is shown in the right panels. Under pure sensory delays (top row), one can see clearly the delay in sensory predictions, in relation to the true inputs. The thicker (solid and dotted) red lines correspond respectively to (true and predicted) proprioceptive input, reflecting oculomotor displacement. The middle row shows the equivalent results with pure motor delays and the lower row presents the results with combined sensorimotor delays. Of note here is the failure of optimal control with oscillatory fluctuations in oculomotor trajectories, which become unstable under combined sensorimotor delays.

This figure illustrates the effects of sensorimotor delays on pursuit initiation (red lines) in relation to compensated (optimal) active inference -- as shown in the previous figure (blue lines). The left panels show the true (solid lines) and estimated sensory input (dotted lines), while action is shown in the right panels. Under pure sensory delays (top row), one can see clearly the delay in sensory predictions, in relation to the true inputs. The thicker (solid and dotted) red lines correspond respectively to (true and predicted) proprioceptive input, reflecting oculomotor displacement. The middle row shows the equivalent results with pure motor delays and the lower row presents the results with combined sensorimotor delays. Of note here is the failure of optimal control with oscillatory fluctuations in oculomotor trajectories, which become unstable under combined sensorimotor delays.

Smooth Pursuit

We then consider an extension of the generative model to simulate smooth pursuit eye movements --- in which the visuo-oculomotor system believes both the target and its centre of gaze are attracted to a (hidden) point moving in the visual field.

This figure uses the same format as the previous figure -- the only difference is that the target motion has been rectified so that it is (approximately) hemi-sinusoidal. The thing to note here is that the improved accuracy of the pursuit previously apparent at the onset of the second cycle of motion has now disappeared -- because active inference does not have access to the immediately preceding trajectory. This failure of an anticipatory improvement in tracking is contrary to empirical predictions. \item \odot
the generative model has been equipped with a second hierarchical level that contains hidden states, modelling latent periodic behaviour of the (hidden) causes of target motion. With this addition, the improvement in pursuit accuracy apparent at the onset of the second cycle of motion is reinstated. This is because the model has an internal representation of latent causes of target motion that can be called upon even when these causes are not expressed explicitly in the target trajectory.

This figure uses the same format as the previous figure -- the only difference is that the target motion has been rectified so that it is (approximately) hemi-sinusoidal. The thing to note here is that the improved accuracy of the pursuit previously apparent at the onset of the second cycle of motion has now disappeared -- because active inference does not have access to the immediately preceding trajectory. This failure of an anticipatory improvement in tracking is contrary to empirical predictions. \item \odot
the generative model has been equipped with a second hierarchical level that contains hidden states, modelling latent periodic behaviour of the (hidden) causes of target motion. With this addition, the improvement in pursuit accuracy apparent at the onset of the second cycle of motion is reinstated. This is because the model has an internal representation of latent causes of target motion that can be called upon even when these causes are not expressed explicitly in the target trajectory.

Smooth Pursuit with oclusion

Finally, the generative model is equipped with a hierarchical structure, so that it can recognise and remember unseen (occluded) trajectories and emit anticipatory responses.

This figure uses the same format as the previous figure -- the only difference is that the target motion has been rectified so that it is (approximately) hemi-sinusoidal. The thing to note here is that the improved accuracy of the pursuit previously apparent at the onset of the second cycle of motion has now disappeared -- because active inference does not have access to the immediately preceding trajectory. This failure of an anticipatory improvement in tracking is contrary to empirical predictions. \item \odot
the generative model has been equipped with a second hierarchical level that contains hidden states, modelling latent periodic behaviour of the (hidden) causes of target motion. With this addition, the improvement in pursuit accuracy apparent at the onset of the second cycle of motion is reinstated. This is because the model has an internal representation of latent causes of target motion that can be called upon even when these causes are not expressed explicitly in the target trajectory.

This figure uses the same format as the previous figure -- the only difference is that the target motion has been rectified so that it is (approximately) hemi-sinusoidal. The thing to note here is that the improved accuracy of the pursuit previously apparent at the onset of the second cycle of motion has now disappeared -- because active inference does not have access to the immediately preceding trajectory. This failure of an anticipatory improvement in tracking is contrary to empirical predictions. \item \odot
the generative model has been equipped with a second hierarchical level that contains hidden states, modelling latent periodic behaviour of the (hidden) causes of target motion. With this addition, the improvement in pursuit accuracy apparent at the onset of the second cycle of motion is reinstated. This is because the model has an internal representation of latent causes of target motion that can be called upon even when these causes are not expressed explicitly in the target trajectory.

Outline

Eye movements

Motion-based predictive coding

Active inference, EMs & oculomotor delays

Perspectives: Saccadic adaptation

In the saccadic adaptation paradigm, ...

TODO: loop 4 times the standard and stop for a fifth one.

... Let us now look at the resulting eye movements

To study the recalibration process, one has to introduce a change in the sensory-motor mapping. This can be done by using a saccadic adaptation paradigm such as the McLaughlin, or double step, paradigm in which a saccade target is presented and then displaced before the saccade is completed. Prior to adaptation, the displacement introduces an error between the perceived location of the final target position of the double-step stimulus and the muscle command intended to move the eye to the initial target position (Hopp & Fuchs, 2004; McLaughlin, 1967). Within a few trials, the initial saccade amplitude is adjusted to minimize the visual error produced by the displacement. The perceived direction of a flashed saccadic stimulus is also affected by saccade adaptation (Bahcall & Kowler, 1999). We used the double-step paradigm to adapt saccades and a pointing task to investigate how such adaptation affects perceived direction.

In the saccadic adaptation paradigm, ...

TODO: loop 4 times the standard and stop for a fifth one.

... Let us now look at the resulting eye movements

Perspectives: Saccadic adaptation

* unfolding of a saccade

* one block : adaptation

* we are in the process of modeling such behavior with a hierarchical model (Mathys, 2011)

* synthetic saccades: non active vs active

Saccades (as I’ve described before in this very blog) are rapid point to point displacements of gaze. Unless you have a target which is moving that you can follow with your gaze, you make saccades. This is unique to eye movements

Courses of representative saccadic amplitude adaptations for monkeys (A) and humans (B). Note the difference in abscissa scales! (A) Top: Adaptation and recovery of saccades made to 10° horizontal target steps with a 30% backward adaptation step. Saccadic amplitude is plotted as a function of the number of the trial in each direction. Magenta points, data from preadaptation trials. Red points, data from trials in the adapted direction. Green points, data from interleaved trials in the non-adapted direction. Adaptation and recovery data are fit by exponential function blue. Arrow indicates initial rapid recovery phase. Bottom: Representative examples of target (gray dashed lines) and eye (black lines) position for different trials during preadaptation and adaptation. (B) Adaptation and recovery of saccades made to 15° horizontal target steps with a 33% backward adaptation step. (Note that little change occurs during recovery because the target was extinguished deliberately during the saccade.)

These simulations speak to a straightforward and neurobiologically plausible solution to the generic problem of integrating information from different sources with different temporal delays and the particular difficulties encountered when a system --- like the oculomotor system of JWT --- tries to control its environment with delayed signals.
this illustrates that this theory can be extended to more complex hierarchies : in the near future we will use such technology to "synchronize" the different sensors and effectors but also the internal modules of a drone and optimize its overall control.
more generally an interesting outlook are applications to machine learning : in big data, the challenge is to sort out the "good" data, to focus our gaze at the right chunk
to conclude, we have shown that EMs can be a great model at different time scales and that such models -and notablty the acute use of generalized coordinates- can be essential for theories to test predictive models such as the sensory system (as for the FLE), but also the oculomotor system as shown above
as such, an hypothesis that we will challenge in the context of our biomimetic drone is an interesting extension at a longer time scale how the system can adapt to smooth variability in the sensory motor contingencies [OR is to know why most mammals adopted a fovea and eye movements (catching preys) ]

Tutorial: Active inference for eye movements: Bayesian methods, neural inference, dynamics

Laurent U Perrinet, INT

Acknowledgements:

Mina Aliakbari Khoei, Guillaume Masson and Anna Montagnini - FACETS-ITN Marie Curie Training
Rick Adams and Karl Friston @ UCL - Wellcome Trust Centre for Neuroimaging
Wahiba Taouali, Giacomo Benvenuti, Frédéric Chavane - ANR BalaV1

LACONEU2017: 4th Latin-American Summer School in Computational Neuroscience January 20th, 2017
https://laurentperrinet.github.io/talk/2017-01-20-laconeu

Tutorial: Active inference for eye movements: Bayesian methods, neural inference, dynamics

Laurent U Perrinet, INT

Acknowledgements:

LACONEU2017: 4th Latin-American Summer School in Computational Neuroscience January 20th, 2017 https://laurentperrinet.github.io/talk/2017-01-20-laconeu

The Behavioral receptive field

The motion energy model

Eye movements and Bayes: motion integration

Outline

Eye movements

Motion-based predictive coding

Active inference, EMs & oculomotor delays

Perspectives: Saccadic adaptation

Motion-based predictive coding

Motion-based predictive coding

Motion-based predictive coding

Motion-based predictive coding

Particle filtering (Sequential Monte-Carlo)

Prediction is sufficient to solve the aperture problem

Outline

Eye movements

Motion-based predictive coding

Active inference, EMs & oculomotor delays

Perspectives: Saccadic adaptation

Outline

Eye movements

Motion-based predictive coding

Active inference, EMs & oculomotor delays

Perspectives: Saccadic adaptation

Perspectives: Saccadic adaptation

Tutorial: Active inference for eye movements: Bayesian methods, neural inference, dynamics

Laurent U Perrinet, INT

Acknowledgements:

LACONEU2017: 4th Latin-American Summer School in Computational Neuroscience January 20th, 2017 https://laurentperrinet.github.io/talk/2017-01-20-laconeu

LACONEU2017: 4th Latin-American Summer School in Computational Neuroscience January 20th, 2017
https://laurentperrinet.github.io/talk/2017-01-20-laconeu

LACONEU2017: 4th Latin-American Summer School in Computational Neuroscience January 20th, 2017
https://laurentperrinet.github.io/talk/2017-01-20-laconeu