# Decoding the population dynamics underlying ocular following response using a probabilistic framework

The machinery behind the visual perception of motion and the subsequent sensorimotor transformation, such as in Ocular Following Response (OFR), is confronted to uncertainties which are efficiently resolved in the primate's visual system. We may understand this response as an ideal observer in a probabilistic framework by using Bayesian theory~\citep{Weiss02} which we previously proved to be successfully adapted to model the OFR for different levels of noise with full field gratings or with disk of various sizes and the effect of a flickering surround (Perrinet, 07). More recent experiments of OFR have used disk gratings and bipartite stimuli which are optimized to study the dynamics of center-surround integration. We quantified two main characteristics of the global spatial integration of motion from an intermediate map of possible local translation velocities: (i) a finite optimal stimulus size for driving OFR, surrounded by an antagonistic modulation and (ii) a direction selective suppressive effect of the surround on the contrast gain control of the central stimuli~\citep{Barthelemy06,Barthelemy07}. Herein, we extended in the dynamical domain the ideal observer model to simulate the spatial integration of the different local motion cues within a probabilistic representation. We present analytical results which show that the hypothesis of independence of local measures can describe the initial segment of spatial integration of motion signal. Within this framework, we successfully accounted for the dynamical contrast gain control mechanisms observed in the behavioral data for center-surround stimuli. However, another inhibitory mechanism had to be added to account for suppressive effects of the surround. We explore here an hypothesis where this could be understood as the effect of a recurrent integration of information in the velocity map.

* F. Barthelemy, L. U. Perrinet, E. Castet, and G. S. Masson. Dynamics of distributed 1D and 2D motion representations for short-latency ocular following. Vision Research, 48(4):501–22, feb 2007. doi:10.1016/j.visres.2007.10.020.
* F. V. Barthelemy, I. Vanzetta, and G. S. Masson. Behavioral receptive field for ocular following in humans: Dynamics of spatial summation and center-surround interactions. Journal of Neurophysiology, (95):3712–26, Mar 2006. doi: 10.1152/jn.00112.2006.
* Y. Weiss, E. P. Simoncelli, and E. H. Adelson. Motion illusions as optimal percepts. Nature Neuroscience, 5(6):598–604, Jun 2002. doi:10.1038/nn858.

 Figure 2 To model spatial integration in the OFR in primates (humans and macaques), we use the tools of statistical inference with the hypothesis that information is represented in a probabilistic fashion. The architecture of the OFR system consists in this model of a stage extracting from the raw image the possible local translation velocities (V1) to represent the local probabilities of translational velocity (MT). These local bits of information are then pooled (MST) to give a single probabilistic representation of possible translational velocities to the oculomotor system, which then controls the eyes' motion as quickly and efficiently as possible to stabilize the image on the retina. The local probabilities may be often described as gaussian probabilities, and the gain response as a function to the signal to noise ratio is then given by a Naka-Rushton curve (Barthélemy et al., 2007).

 Figure 3 Analytical solution for the spatial integration: the ROG model. To model the integration over a grating limited to a disk, we may consider that the density (or weight) of neurons pooling responses for the OFR is a centered Gaussian with a width of $\omega$ degrees of visual space. However this class of models can only generate monotonously increasing response functions which are incompatible with the suppression observed in the response after a specific contrast (the so-called \emph{super saturation}). One may therefore add another integration term which accounts for a surround inhibition, pooling information toward the null velocity on a similar Gaussian distribution but with a larger size $\omega_i$. This framework gives a rationale for the Ratio-Of-Gaussian (ROG) model (Sceniak, 99; Cavanaugh,02) and explicitly states the underlying choice for the fitting formula. We plot the amplitude of the oculomotor response in the macaque to a central disk grating (dots) with fits to this model (continuous lines) and the original ROG (dashed lines), showing an improvement of the order of 2.5 in the $\chi^2$ score.

## reference

• Laurent Perrinet, Guillaume S. Masson. Decoding the population dynamics underlying ocular following responseusing a probabilistic framework, URL . In Proceedings of AREADNE, 2008 abstract.

 This work was supported by European integrated project FP6-015879, "FACETS".

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