Guessing unpredictable target displacements. *Vision Research*, 21(2), 191–203.

Krauzlis, R. J. (2004). Recasting the smooth pursuit eye movement system. *Journal of Neurophysiology*, 91(2), 591–603.

Krauzlis, R. J. (2005). The control of voluntary eye movements: new perspectives. *The Neuroscientist*, 11(2), 124–137.

Krauzlis, R. J., Goffart, L., & Hafed, Z. M. (2017). Neuronal control of fixation and fixational eye movements. *Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences*, 372(1718), 20160205.

Kuang, X., Poletti, M., Victor, J. D., & Rucci, M. (2012). Temporal encoding of spatial information during active visual fixation. *Current Biology*, 22(6), 510–514.

Langevin, P. (1908). On the theory of brownian motion. *Comptes Rendus*, 146, 530–533.

Madelain, L., & Krauzlis, R. J. (2003). Pursuit of the ineffable: Perceptual and motor reversals during the tracking of apparent motion. *Journal of Vision*, 3(11), 1, https://doi.org/10.1167/3.11.1.

Martinez-Conde, S., Macknik, S. L., & Hubel, D. H. (2004). The role of fixational eye movements in visual perception. *Nature Reviews Neuroscience*, 5(3), 229.

Martins, A. J., Kowler, E., & Palmer, C. (1985). Smooth pursuit of small-amplitude sinusoidal motion. *Journal of the Optical Society of America*, 2(2), 234–242.

Masson, G. S., & Stone, L. S. (2002). From following edges to pursuing objects. *Journal of Neurophysiology*, 88(5), 2869–2873.

Michalski, A., Kossut, M., & Żernicki, B. (1977). The ocular following reflex elicited from the retinal periphery in the cat. *Vision Research*, 17(6), 731–736.

Mihali, A., van Opheusden, B., & Ma, W. J. (2017). Bayesian microsaccade detection. *Journal of Vision*, 17(1), 13, https://doi.org/10.1167/17.1.13.

Miles, F., Kawano, K., & Optican, L. (1986). Short-latency ocular following responses of monkey: I. Dependence on temporospatial properties of visual input. *Journal of Neurophysiology*, 56(5), 1321–1354.

Moreno-Bote, R., & Drugowitsch, J. (2015). Causal inference and explaining away in a spiking network. *Scientific Reports*, 5, 17531.

Moreno-Bote, R., Knill, D. C., & Pouget, A. (2011). Bayesian sampling in visual perception. *Proceedings of the National Academy of Sciences*, 108(30), 12491–12496.

Moreno-Bote, R., Rinzel, J., & Rubin, N. (2007). Noise-induced alternations in an attractor network model of perceptual bistability. *Journal of Neurophysiology*, 98(3), 1125–1139.

Muirhead, R. J. (2010). Canonical correlation analysis. In *Aspects of multivariate statistical theory* (pp. 548–569). Hoboken, NJ: John Wiley & Sons.

Murphy, B. J., Kowler, E., & Steinman, R. M. (1975). Slow oculomotor control in the presence of moving backgrounds. *Vision Research*, 15(11), 1263–1268.

Nachmias, J. (1961). Determiners of the drift of the eye during monocular fixation. *Journal of the Optical Society of America*, 51(7), 761–766.

Nyström, M., & Holmqvist, K. (2010). An adaptive algorithm for fixation, saccade, and glissade detection in eyetracking data. *Behavior Research Methods*, 42(1), 188–204.

Orquin, J. L., & Loose, S. M. (2013). Attention and choice: A review on eye movements in decision making. *Acta Psychologica*, 144(1), 190–206.

Otero-Millan, J., Macknik, S. L., Langston, R. E., & Martinez-Conde, S. (2013). An oculomotor continuum from exploration to fixation. *Proceedings of the National Academy of Sciences*, 110(15), 6175–6180.

Palmer, S. E. (1999). Visual selection: Eye movements and attention. In *Vision science photons to phenomenology* (pp. 519–571). essay, MIT Press.

Patel, A., & Kosko, B. (2005). Stochastic resonance in noisy spiking retinal and sensory neuron models. *Neural Networks*, 18(5–6), 467–478.

Peel, T. R., Hafed, Z. M., Dash, S., Lomber, S. G., & Corneil, B. D. (2016). A causal role for the cortical frontal eye fields in microsaccade deployment. *PLoS Biology*, 14(8), e1002531.

Poletti, M., Fang, Y., & Rucci, M. (2017). Monocular microsaccades; do they really occur? *Journal of Vision*, 17(10), 893, https://doi.org/10.1167/17.10.893.

Poletti, M., Listorti, C., & Rucci, M. (2010). Stability of the visualworld during eye drift. *Journal of Neuroscience*, 30(33), 11143–11150.

Poletti, M., & Rucci, M. (2016). A compact field guide to the study of microsaccades: Challenges and functions. *Vision Research*, 118, 83–97.

Quaia, C., Sheliga, B. M., FitzGibbon, E. J., & Optican, L. M. (2012). Ocular following in humans:
Guessing unpredictable target displacements. *Vision Research*, 21(2), 191–203.

Krauzlis, R. J. (2004). Recasting the smooth pursuit eye movement system. *Journal of Neurophysiology*, 91(2), 591–603.

Krauzlis, R. J. (2005). The control of voluntary eye movements: new perspectives. *The Neuroscientist*, 11(2), 124–137.

Krauzlis, R. J., Goffart, L., & Hafed, Z. M. (2017). Neuronal control of fixation and fixational eye movements. *Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences*, 372(1718), 20160205.

Kuang, X., Poletti, M., Victor, J. D., & Rucci, M. (2012). Temporal encoding of spatial information during active visual fixation. *Current Biology*, 22(6), 510–514.

Langevin, P. (1908). On the theory of brownian motion. *Comptes Rendus*, 146, 530–533.

Madelain, L., & Krauzlis, R. J. (2003). Pursuit of the ineffable: Perceptual and motor reversals during the tracking of apparent motion. *Journal of Vision*, 3(11), 1, https://doi.org/10.1167/3.11.1.

Martinez-Conde, S., Macknik, S. L., & Hubel, D. H. (2004). The role of fixational eye movements in visual perception. *Nature Reviews Neuroscience*, 5(3), 229.

Martins, A. J., Kowler, E., & Palmer, C. (1985). Smooth pursuit of small-amplitude sinusoidal motion. *Journal of the Optical Society of America*, 2(2), 234–242.

Masson, G. S., & Stone, L. S. (2002). From following edges to pursuing objects. *Journal of Neurophysiology*, 88(5), 2869–2873.

Michalski, A., Kossut, M., & Żernicki, B. (1977). The ocular following reflex elicited from the retinal periphery in the cat. *Vision Research*, 17(6), 731–736.

Mihali, A., van Opheusden, B., & Ma, W. J. (2017). Bayesian microsaccade detection. *Journal of Vision*, 17(1), 13, https://doi.org/10.1167/17.1.13.

Miles, F., Kawano, K., & Optican, L. (1986). Short-latency ocular following responses of monkey: I. Dependence on temporospatial properties of visual input. *Journal of Neurophysiology*, 56(5), 1321–1354.

Moreno-Bote, R., & Drugowitsch, J. (2015). Causal inference and explaining away in a spiking network. *Scientific Reports*, 5, 17531.

Moreno-Bote, R., Knill, D. C., & Pouget, A. (2011). Bayesian sampling in visual perception. *Proceedings*

of the National Academy of Sciences, 108(30),
12491–12496.

Moreno-Bote, R., Rinzel, J., & Rubin, N. (2007). Noise-induced alternations in an attractor network model of perceptual bistability. *Journal of Neurophysiology*, 98(3), 1125–1139.

Muirhead, R. J. (2010). Canonical correlation analysis. In *Aspects of multivariate statistical theory* (pp. 548–569). Hoboken, NJ: John Wiley & Sons.

Murphy, B. J., Kowler, E., & Steinman, R. M. (1975). Slow oculomotor control in the presence of moving backgrounds. *Vision Research*, 15(11), 1263–1268.

Nachmias, J. (1961). Determiners of the drift of the eye during monocular fixation. *Journal of the Optical Society of America*, 51(7), 761–766.

Nyström, M., & Holmqvist, K. (2010). An adaptive algorithm for fixation, saccade, and glissade detection in eyetracking data. *Behavior Research Methods*, 42(1), 188–204.

Orquin, J. L., & Loose, S. M. (2013). Attention and choice: A review on eye movements in decision making. *Acta Psychologica*, 144(1), 190–206.

Otero-Millan, J., Macknik, S. L., Langston, R. E., & Martinez-Conde, S. (2013). An oculomotor continuum from exploration to fixation. *Proceedings of the National Academy of Sciences*, 110(15), 6175–6180.

Palmer, S. E. (1999). Visual selection: Eye movements and attention. In *Vision science photons to phenomenology* (pp. 519–571). essay, MIT Press.

Patel, A., & Kosko, B. (2005). Stochastic resonance in noisy spiking retinal and sensory neuron models. *Neural Networks*, 18(5–6), 467–478.

Peel, T. R., Hafed, Z. M., Dash, S., Lomber, S. G., & Corneil, B. D. (2016). A causal role for the cortical frontal eye fields in microsaccade deployment. *PLoS Biology*, 14(8), e1002531.

Poletti, M., Fang, Y., & Rucci, M. (2017). Monocular microsaccades; do they really occur? *Journal of Vision*, 17(10), 893, https://doi.org/10.1167/17.10.893.

Poletti, M., Listorti, C., & Rucci, M. (2010). Stability of the visualworld during eye drift. *Journal of Neuroscience*, 30(33), 11143–11150.

Poletti, M., & Rucci, M. (2016). A compact field guide to the study of microsaccades: Challenges and functions. *Vision Research*, 118, 83–97.

Quaia, C., Sheliga, B. M., FitzGibbon, E. J., & Optican, L. M. (2012). Ocular following in humans:
Spatial properties. *Journal of Vision*, 12(4), 13, https://doi.org/10.1167/12.4.13.

Ratliff, F., & Riggs, L. A. (1950). Involuntary motions of the eye during monocular fixation. *Journal of Experimental Psychology*, 40(6), 687.

Rolfs, M. (2009). Microsaccades: Small steps on a long way. *Vision Research*, 49(20), 2415–2441.

Rolfs, M., Kliegl, R., & Engbert, R. (2008). Toward a model of microsaccade generation: The case of microsaccadic inhibition. *Journal of Vision*, 8(11), 5, https://doi.org/10.1167/8.11.5.

Schwartz, J. L., Grimault, N., Hupé, J. M., Moore, B. C., & Pressnitzer, D. (2012). Multistability in perception: Binding sensory modalities, an overview. *Philosophical Transactions of the Royal Society B: Biological Sciences*, 367(1591), 896.

Shpiro, A., Moreno-Bote, R., Rubin, N., & Rinzel, J. (2009). Balance between noise and adaptation in competition models of perceptual bistability. *Journal of Computational Neuroscience*, 27(1), 37.

Sinn, P., & Engbert, R. (2016). Small saccades versus microsaccades: Experimental distinction and model-based unification. *Vision Research*, 118, 132–143.

Skinner, J., Buonocore, A., & Hafed, Z. M. (2018). Transfer function of the rhesus macaque oculomotor system for small-amplitude slow motion trajectories. *Journal of Neurophysiology*, 121(2), 513–529.

Spering, M., & Montagnini, A. (2011). Do we track what we see? Common versus independent processing for motion perception and smooth pursuit eyemovements: A review. *Vision Research*, 51(8), 836–852.

Stone, L. S., Beutter, B. R., & Lorenceau, J. (2000). Visual motion integration for perception and pursuit. *Perception*, 29(7), 771–787.

Taouali, W., Goffart, L., Alexandre, F., & Rougier, N. P. (2015). A parsimonious computational model of visual target position encoding in the superior colliculus. *Biological Cybernetics*, 109(4–5), 549–559.

Tatler, B. W., Hayhoe, M. M., Land, M. F., & Ballard, D. H. (2011). Eye guidance in natural vision: Reinterpreting salience. *Journal of Vision*, 11(5), 5, https://doi.org/10.1167/11.5.5.

Tomczak, M., & Tomczak, E. (2014). *The need to report effect size estimates revisited. An overview of some recommended measures of effect size*. *Trends in Sport Sciences*, 21(1).

Watanabe, M., Okada, K.-I., Hamasaki, Y., Funamoto, M., Kobayashi, Y., & MacAskill, M. (2019). Ocular drift reflects volitional action preparation. *European Journal of Neuroscience*, 50(2), 1892–1910.

Watanabe, T., Masuda, N., Megumi, F., Kanai, R., & Rees, G. (2014). Energy landscape and dynamics of brain activity during human bistable perception. *Nature Communications*, 5, 4765.

Wyatt, H. J. (2010). The human pupil and the use of video-based eyetrackers. *Vision Research*, 50(19), 1982–1988.

Yarbus, A. L. (1967). Eye movements during perception of complex objects. In *Eye movements and vision* (pp. 171–211). essay, Plenum Press.

## Appendix A: Notations used

In this work, we use $\boldsymbol{q}$ and $\boldsymbol{\dot{q}}$ for the two-dimensional position (in deg) and velocity vectors (in deg·s⁻¹). Subscripts $R$, $G$, and $S$ will respectively refer to the retinal image, the gaze, and the stimulus. Overlined notation will refer to the mean over a set of trials and a tilde to the median over a set of trials, for all metrics. Mean values will be reported with their standard deviation and median values with median absolute deviation (MAD), for instance, $\bar{\rho} = 2.93 \pm 0.01$ or $\tilde{\rho} = 3.01 \pm 0.02$.

## Appendix B: Metrics

We use $\cdot^\top$ for the transpose operator, and trace($\cdot$) will denote the trace operator (sum over the diagonal elements of a matrix). The identity matrix in dimension 2 will be denoted by $\text{Id}_2$.

### Variance-covariance and inertia

Let $\boldsymbol{q}_G(t) = [x_G(t), y_G(t)]^\top$, $\boldsymbol{q}_S(t) = (x_S(t), y_S(t))^\top$, and $\boldsymbol{q}_R = [x_R(t), y_R(t)]^\top$ be the screen Cartesian coordinates (column vectors) at time instant $t$ of the gaze, the stimulus, and the retinal image, respectively. Now, having samples at $n$ discrete times $\{t_i\}_{i=1}^n$, we estimate the center of gravity of a gaze trajectory $\{\boldsymbol{q}_G(t_i)\}_{i=1}^n$ by its empirical mean $\boldsymbol{m}_G = n^{-1} \sum_{i=1}^n \boldsymbol{q}_G(t_i)$. This estimate approaches the true center of gravity if we sample sufficiently regularly and beyond twice the Nyquist frequency, conditions that are met when working with the Eyelink 1000(+), sampling at about 1,000 Hz for each eye.

A second-order statistic of interest is the empirical variance-covariance matrix, which gives the inertia of the gaze trajectory defined as

$$ \boldsymbol{\Sigma}_G = n^{-1} \sum_{i=1}^{n} \boldsymbol{q}_G(t_i) \boldsymbol{q}_G^\top(t_i) - \boldsymbol{m}_G \boldsymbol{m}_G^\top $$

and analogously for the stimulus and retinal image empirical variance-covariance matrix. The
Spatial properties. *Journal of Vision*, 12(4), 13, https://doi.org/10.1167/12.4.13.

Ratliff, F., & Riggs, L. A. (1950). Involuntary motions of the eye during monocular fixation. *Journal of Experimental Psychology*, 40(6), 687.

Rolfs, M. (2009). Microsaccades: Small steps on a long way. *Vision Research*, 49(20), 2415–2441.

Rolfs, M., Kliegl, R., & Engbert, R. (2008). Toward a model of microsaccade generation: The case of microsaccadic inhibition. *Journal of Vision*, 8(11), 5, https://doi.org/10.1167/8.11.5.

Schwartz, J. L., Grimault, N., Hupé, J. M., Moore, B. C., & Pressnitzer, D. (2012). Multistability in perception: Binding sensory modalities, an overview. *Philosophical Transactions of the Royal Society B: Biological Sciences*, 367(1591), 896.

Shpiro, A., Moreno-Bote, R., Rubin, N., & Rinzel, J. (2009). Balance between noise and adaptation in competition models of perceptual bistability. *Journal of Computational Neuroscience*, 27(1), 37.

Sinn, P., & Engbert, R. (2016). Small saccades versus microsaccades: Experimental distinction and model-based unification. *Vision Research*, 118, 132–143.

Skinner, J., Buonocore, A., & Hafed, Z. M. (2018). Transfer function of the rhesus macaque oculomotor system for small-amplitude slow motion trajectories. *Journal of Neurophysiology*, 121(2), 513–529.

Spering, M., & Montagnini, A. (2011). Do we track what we see? Common versus independent processing for motion perception and smooth pursuit eyemovements: A review. *Vision Research*, 51(8), 836–852.

Stone, L. S., Beutter, B. R., & Lorenceau, J. (2000). Visual motion integration for perception and pursuit. *Perception*, 29(7), 771–787.

Taouali, W., Goffart, L., Alexandre, F., & Rougier, N. P. (2015). A parsimonious computational model of visual target position encoding in the superior colliculus. *Biological Cybernetics*, 109(4–5), 549–559.

Tatler, B. W., Hayhoe, M. M., Land, M. F., & Ballard, D. H. (2011). Eye guidance in natural vision: Reinterpreting salience. *Journal of Vision*, 11(5), 5, https://doi.org/10.1167/11.5.5.

Tomczak, M., & Tomczak, E. (2014). *The need to report effect size estimates revisited. An overview of some recommended measures of effect size*. *Trends in Sport Sciences*, 21(1).

Watanabe, M., Okada, K.-I., Hamasaki, Y., Funamoto, M., Kobayashi, Y., & MacAskill, M. (2019). Ocular drift reflects volitional action preparation. *European Journal of Neuroscience*, 50(2), 1892–1910.

Watanabe, T., Masuda, N., Megumi, F., Kanai, R., & Rees, G. (2014). Energy landscape and dynamics of brain activity during human bistable perception. *Nature Communications*, 5, 4765.

Wyatt, H. J. (2010). The human pupil and the use of video-based eyetrackers. *Vision Research*, 50(19), 1982–1988.

Yarbus, A. L. (1967). Eye movements during perception of complex objects. In *Eye movements and vision* (pp. 171–211). essay, Plenum Press.

## Appendix A: Notations used

In this work, we use $\mathbf{q}$ and $\dot{\mathbf{q}}$ for the two-dimensional position (in deg) and velocity vectors (in deg·s⁻¹). Subscripts $R$, $G$, and $S$ will respectively refer to the retinal image, the gaze, and the stimulus. Overlined notation will refer to the mean over a set of trials and a tilde to the median over a set of trials, for all metrics. Mean values will be reported with their standard deviation and median values with median absolute deviation (MAD), for instance, $\bar{\rho} = 2.93 \pm 0.01$ or $\tilde{\rho} = 3.01 \pm 0.02$.

## Appendix B: Metrics

We use $\cdot^\top$ for the transpose operator, and trace($\cdot$) will denote the trace operator (sum over the diagonal elements of a matrix). The identity matrix in dimension 2 will be denoted by $\text{Id}_2$.

### Variance-covariance and inertia

Let $\mathbf{q}_G(t) = [x_G(t), y_G(t)]^\top$, $\mathbf{q}_S(t) = (x_S(t), y_S(t))^\top$, and $\mathbf{q}_R = [x_R(t), y_R(t)]^\top$ be the screen Cartesian coordinates (column vectors) at time instant $t$ of the gaze, the stimulus, and the retinal image, respectively. Now, having samples at $n$ discrete times $\{t_i\}_{i=1}^n$, we estimate the center of gravity of a gaze trajectory $\{\mathbf{q}_G(t_i)\}_{i=1}^n$ by its empirical mean $\mathbf{m}_G = n^{-1} \sum_{i=1}^n \mathbf{q}_G(t_i)$. This estimate approaches the true center of gravity if we sample sufficiently regularly and beyond twice the Nyquist frequency, conditions that are met when working with the Eyelink 1000(+), sampling at about 1,000 Hz for each eye.

A second-order statistic of interest is the empirical variance-covariance matrix, which gives the inertia of the gaze trajectory defined as

$$ \Sigma_G = n^{-1} \sum_{i=1}^{n} \mathbf{q}_G(t_i) \mathbf{q}_G^\top(t_i) - \mathbf{m}_G \mathbf{m}_G^\top $$

and analogously for the stimulus and retinal image empirical variance-covariance matrix. The
<figure><img src="image_17.png" /><figcaption>Figure 6. Behavior of MPC scores over signal-to-noise ratios (SNRs) in simulated similarity computations with a Lissajous base signal from LJ, with varying signal sample sizes.</figcaption></figure>

inertia about its center of gravity is then given by
$I_{m_G} = n^{-1} \sum_{i=1}^{n} \|q_G(t_i) - m_G\|^2 = \text{trace}(\Sigma_G)$.

The inertia $I_r$ of the gaze trajectory $q_G$ with respect to any fixed point $r$ having screen coordinates $(x_r, y_r)$ is

$$I_r = \text{trace}(\Sigma_G + (m_G - r)^\top (m_G - r)).$$

## Maximally projected correlations

Taking now the simultaneously recorded gaze $\{q_G(t_i)\}_{i=1}^n$ and stimulus $\{q_S(t_i)\}_{i=1}^n$ signals, and their respective empirical variance-covariance matrices $G$ and $S$, we denote the intercovariance matrix by

$$\Sigma_{GS} \doteq n^{-1} \sum_{i=1}^{n} (q_G(t_i) - m_G)(q_S(t_i) - m_S)^\top = \Sigma_{SG}^\top.$$

This matrix is particularly useful when considering the inertia of gaze with respect to the time-changing coordinates of the stimulus. Indeed, after some manipulations, we obtain

$$I_{GS} = n^{-1} \sum_{i=1}^{n} \|q_G(t_i) - q_S(t_i)\|^2$$
$$I_{GS} = \text{trace}(\Sigma_G + \Sigma_S - \Sigma_{GS} - \Sigma_{SG}) + \|m_G - m_S\|^2.$$

Unfortunately, the inertia does not account for differences in scale or for coordinate translation, two characteristics that are typical aspects for pursuits and for which we require an invariance.⁶

## Noise robustness and signal size dependency

Figure 6 shows results of simulations operated on a Lissajous signal degraded by noise on the position of the stimulus at different signal-to-noise ratios (SNRs) and for different signal sizes. For signals with more than 167 samples, the behavior of MPC scores over SNR remains stable and shows quasi-unchanged dynamics.

## Appendix C: Model

Models come in a variety of forms, depending on the mathematical framework used to formalize and compute their mechanics. Two main families can be differentiated: descriptive statistical and generative mechanistic models. Here, we focus on the latter. The motivation is the following: Generative models can produce simulated and synthetic results that can be
<figure><img src="image_18.png" /><figcaption>Figure 6. Behavior of MPC scores over signal-to-noise ratios (SNRs) in simulated similarity computations with a Lissajous base signal from LJ, with varying signal sample sizes.</figcaption></figure>

inertia about its center of gravity is then given by
$I_{m_G} = n^{-1} \sum_{i=1}^{n} \|q_G(t_i) - m_G\|^2 = \text{trace}(\Sigma_G)$.

The inertia $I_r$ of the gaze trajectory $q_G$ with respect to any fixed point $r$ having screen coordinates $(x_r, y_r)$ is

$$I_r = \text{trace}(\Sigma_G + (m_G - r)^\top (m_G - r)).$$

## Maximally projected correlations

Taking now the simultaneously recorded gaze $\{q_G(t_i)\}_{i=1}^n$ and stimulus $\{q_S(t_i)\}_{i=1}^n$ signals, and their respective empirical variance-covariance matrices $G$ and $S$, we denote the intercovariance matrix by

$$\Sigma_{GS} \doteq n^{-1} \sum_{i=1}^{n} (q_G(t_i) - m_G)(q_S(t_i) - m_S)^\top = \Sigma_{SG}^\top.$$

This matrix is particularly useful when considering the inertia of gaze with respect to the time-changing coordinates of the stimulus. Indeed, after some manipulations, we obtain

$$I_{GS} = n^{-1} \sum_{i=1}^{n} \|q_G(t_i) - q_S(t_i)\|^2 \\ I_{GS} = \text{trace}(\Sigma_G + \Sigma_S - \Sigma_{GS} - \Sigma_{SG}) + \|m_G - m_S\|^2.$$

Unfortunately, the inertia does not account for differences in scale or for coordinate translation, two characteristics that are typical aspects for pursuits and for which we require an invariance.⁶

## Noise robustness and signal size dependency

Figure 6 shows results of simulations operated on a Lissajous signal degraded by noise on the position of the stimulus at different signal-to-noise ratios (SNRs) and for different signal sizes. For signals with more than 167 samples, the behavior of MPC scores over SNR remains stable and shows quasi-unchanged dynamics.

## Appendix C: Model

Models come in a variety of forms, depending on the mathematical framework used to formalize and compute their mechanics. Two main families can be differentiated: descriptive statistical and generative mechanistic models. Here, we focus on the latter. The motivation is the following: Generative models can produce simulated and synthetic results that can be
compared to observed empirical data. The model can then be studied and decomposed such that each internal force can be characterized, and their functional role in creating the analogous behavior can be investigated. Altogether, models remain key to understanding a phenomenon and making predictions for empirical and experimental work. We focused here on fixational eye movements in an attempt to explain and understand the data observed and reported in the article. Generative eye movement models use different approaches, including, for instance, probabilistic models (Tatler, Hayhoe, Land, & Ballard, 2011; Gide & Karam, 2017), accumulation process models (Orquin & Loose, 2013), or energy potential models (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011). Here we focus on the latter approach.

Recently, Engbert and colleagues (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011) proposed a generative model that could reproduce the statistical properties of fixational eye movements, stationary displacement, namely, the short-term persistence and long-term antipersistence of drift and tremors. They used a self-avoiding walk (Freund & Grassberger, 1992) in a discretized quadratic energy potential: At each iteration, the gaze, represented by a particle in the energy potential landscape, can go left, right, up, or down. The walker will choose the slot with the lowest energy. Once a step is made, the slot of the previous iteration is set to a high-energy value, and the entire energy landscape follows a linear relaxation law. Hence, fixational eye movements, bottom-up dynamics can be reproduced. Furthermore, the model also proposed to integrate micro-saccade generation by a threshold rule: When the particle is surrounded by options with energy higher than the threshold, it jumps to the global minimum of the energy landscape. Here, the authors provide an accumulation process linked to a global integration of the oculomotor field.

The integrated fixational eye movements model described above is a key foundation to bridge the oculomotor modeling communities and accounts for multiple fixational eye movement phenomena (e.g., drift displacement, micro-saccade, spatial orientation biases). However, it did not possess a mechanism to account for micro-pursuit, as these are hardly studied and reported. The observation of micro-pursuits presented in the article implies that the dynamics of the gaze within a fixation can be affected and attracted by motion of a perceptual object in or nearby the foveal field. Therefore, we propose a modeling approach, gravitational fixational eye movements (GraFEM), inspired by gravitational energy field theory to model motion of eye movements and derived from the work on integrating fixational eye movements in energy potential models (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011).

## Gravitational potential energy field modeling

Integrated and generative fixational eye movement models make use of energy potentials to generate self-avoiding walks, constrain the walks, and replicate oculomotor biases (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011). In fact, the latter is used to constrain the pseudo-random walk's spatial horizon. Furthermore, it can be considered an attractor of the energy landscape. Thus, the use of the particle in an energy potential framework can be adjusted to provide biases of the stimulus on the fixational eye movement generation. Combining attractors in the energy fields, which increase the probabilities of having the gaze at some spatial coordinates, and adding stochasticity to the movement of the particle can provide a simple mechanism for fixational eye movement generation.

The attractors' properties can be manipulated over time to affect the energy field and thus dynamics of the fixational eye movements generated. The energy field that is mapped to the visual field can be populated by an arbitrary number of n attractors of varying strength (see Figure 7A). Inspired by the formalism of gravitational fields, one can generate fields with the following equations. Let $\Phi_i$ represent the field generated by the ith attractor given by

$$ \Phi_i(\mathbf{q}, t) = -\frac{1}{\|\mathbf{q}(t) - \mathbf{a}_i(t)\|^2^{2\beta_i(t)} + \delta_i(t)} \quad (4) $$

with $\mathbf{q}$ and $\mathbf{a}_i$ corresponding to the spatial x-y coordinates (at time $t$) of the observer's gaze position and the ith attractor, respectively. The potential landscape can be fine-tuned according to assumptions on attractive attributes of the stimulus and the tasks. First, it is necessary to set how many attractors are present and give them spatial coordinates in the plane over time. Second, it is possible to handle the mass of those attractors and their subsequent force of attraction and distortion of the field by tuning two parameters: $\delta$ for the depth of the well and $\beta$ for the concavity of its slope. Summation and normalization of the field allow for the fusion of the multiple attractors.

$$ \Phi(\mathbf{q}, t) = \sum_{i=1}^{n} \Phi_i(\mathbf{q}, t). \quad (5) $$

A logarithmic attenuation is added to allow the possibilities of exploring high-energy areas of the visual/foveal field, giving the energy $E$:

$$ E = -\ln(-\Phi). \quad (6) $$

Memory of attractor motion (Figure 7B) is modeled by
compared to observed empirical data. The model can then be studied and decomposed such that each internal force can be characterized, and their functional role in creating the analogous behavior can be investigated. Altogether, models remain key to understanding a phenomenon and making predictions for empirical and experimental work. We focused here on fixational eye movements in an attempt to explain and understand the data observed and reported in the article. Generative eye movement models use different approaches, including, for instance, probabilistic models (Tatler, Hayhoe, Land, & Ballard, 2011; Gide & Karam, 2017), accumulation process models (Orquin & Loose, 2013), or energy potential models (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011). Here we focus on the latter approach.

Recently, Engbert and colleagues (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011) proposed a generative model that could reproduce the statistical properties of fixational eye movements, stationary displacement, namely, the short-term persistence and long-term antipersistence of drift and tremors. They used a self-avoiding walk (Freund & Grassberger, 1992) in a discretized quadratic energy potential: At each iteration, the gaze, represented by a particle in the energy potential landscape, can go left, right, up, or down. The walker will choose the slot with the lowest energy. Once a step is made, the slot of the previous iteration is set to a high-energy value, and the entire energy landscape follows a linear relaxation law. Hence, fixational eye movements, bottom-up dynamics can be reproduced. Furthermore, the model also proposed to integrate micro-saccade generation by a threshold rule: When the particle is surrounded by options with energy higher than the threshold, it jumps to the global minimum of the energy landscape. Here, the authors provide an accumulation process linked to a global integration of the oculomotor field.

The integrated fixational eye movements model described above is a key foundation to bridge the oculomotor modeling communities and accounts for multiple fixational eye movement phenomena (e.g., drift displacement, micro-saccade, spatial orientation biases). However, it did not possess a mechanism to account for micro-pursuit, as these are hardly studied and reported. The observation of micro-pursuits presented in the article implies that the dynamics of the gaze within a fixation can be affected and attracted by motion of a perceptual object in or nearby the foveal field. Therefore, we propose a modeling approach, gravitational fixational eye movements (GraFEM), inspired by gravitational energy field theory to model motion of eye movements and derived from the work on integrating fixational eye movements in energy potential models (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011).

## Gravitational potential energy field modeling

Integrated and generative fixational eye movement models make use of energy potentials to generate self-avoiding walks, constrain the walks, and replicate oculomotor biases (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011). In fact, the latter is used to constrain the pseudo-random walk's spatial horizon. Furthermore, it can be considered an attractor of the energy landscape. Thus, the use of the particle in an energy potential framework can be adjusted to provide biases of the stimulus on the fixational eye movement generation. Combining attractors in the energy fields, which increase the probabilities of having the gaze at some spatial coordinates, and adding stochasticity to the movement of the particle can provide a simple mechanism for fixational eye movement generation.

The attractors' properties can be manipulated over time to affect the energy field and thus dynamics of the fixational eye movements generated. The energy field that is mapped to the visual field can be populated by an arbitrary number of n attractors of varying strength (see Figure 7A). Inspired by the formalism of gravitational fields, one can generate fields with the following equations. Let $\Phi_i$ represent the field generated by the ith attractor given by

$$ \Phi_i(\mathbf{q}, t) = -\frac{1}{\|\mathbf{q}(t) - \mathbf{a}_i(t)\|^2^{2\beta_i(t)} + \delta_i(t)} \quad (4) $$

with $\mathbf{q}$ and $\mathbf{a}_i$ corresponding to the spatial x-y coordinates (at time $t$) of the observer's gaze position and the ith attractor, respectively. The potential landscape can be fine-tuned according to assumptions on attractive attributes of the stimulus and the tasks. First, it is necessary to set how many attractors are present and give them spatial coordinates in the plane over time. Second, it is possible to handle the mass of those attractors and their subsequent force of attraction and distortion of the field by tuning two parameters: $\delta$ for the depth of the well and $\beta$ for the concavity of its slope. Summation and normalization of the field allow for the fusion of the multiple attractors.

$$ \Phi(\mathbf{q}, t) = \sum_{i=1}^{n} \Phi_i(\mathbf{q}, t). \quad (5) $$

A logarithmic attenuation is added to allow the possibilities of exploring high-energy areas of the visual/foveal field, giving the energy $E$:

$$ E = -\ln(-\Phi). \quad (6) $$

Memory of attractor motion (Figure 7B) is modeled by
<figure><img src="image_18.png" /><figcaption>Figure 7. Examples of energy landscape surface plots setup using the gravitational fixational eye movements (GraFEM) model for the following: (A) shows three attractors (n = 3) with all attractors i having no motion and the following parameters:</figcaption></figure>

$\beta = 2$; $\delta = 1$; $\tau = 5$; $\lambda = 0.9$, and (B) shows two attractors (n = 2) with all attractors i having the following parameters:

$\beta_1 = 2$; $\beta_2 = 4$; $\delta_1 = \delta_2 = 1$; $\tau = 15$; $\lambda = 0.9$, and attractor motion computed with the following arbitrary sinusoidal motion:

$a_1(t) = [0, 0]$; $a_2(t) = a_2(t = 0) + [-5 \sin(2t), 5 \sin(3t)]$ on the 75th iterations. The motion of $a_2$ is shown in white. Though the model has many parameters, those manipulated in this work's results are exclusively the depth $\delta$ (or mass) of the attractors and the slope $\beta$ by affecting the concavity of the attractors' field.

<figure><img src="image_19.png" /><figcaption>Figure 8. Simulation examples generated with the GraFEM model. Simulations of fixations of 3.5 s with Euler-Muruyama time steps of $\Delta t = 13$ ms, with variable fixation dynamics generated by manipulating $\delta_{stim}$ and $\beta_{stim}$ parameters. Constant parameters of the model were number of attractors $n = 2$, with one for the fixation cross $a_{cross} = (0, 0)$ and another for the motion of the stimulus following Lissajous trajectories with the same parameters as in the three experiments (Necker, Square, Cross); $a_{stim}(t) = (\sin(2t), \sin(3t))$. The relaxation rate parameter $\lambda = 0.9$, memory temporal limit parameter $K = 5$, and noise $\xi \sim U[-0.5; 0.5]$ were used. (A) A simulated fixation with stable fixation dynamics with $\delta_{cross} = 100$; $\delta_{stim} = 100$; $\beta_{cross} = 1$; $\beta_{stim} = 1$. (B) A simulated fixation with micro-pursuit dynamics with $\delta_{cross} = 100$; $\delta_{stim} = 25$; $\beta_{cross} = 1$; $\beta_{stim} = 1$. (C) A simulated fixation with micro-saccade dynamics with $\delta_{cross} = 100$; $\delta_{stim} = 25$; $\beta_{cross} = 1$; $\beta_{stim} = 12$ and detected using the EK algorithm for micro-saccade detection.</figcaption></figure>

adding a moving average (MA) process (Hannan, 2009) on the field at a given time $t$:

$$E_{FEM}(q, t) = E(q, t) + \sum_{k=1}^{K} \frac{\lambda}{k+1} E(q, t - k\Delta t), \quad (7)$$

where $K$ is the temporal parameter limiting how far in time will the fields be summed over and with $\lambda$ the relaxation rate parameter, and $\Delta t$ is the temporal step size. It is also possible to set the impact of memory and anticipation through parameters that define the iteration window over which the field is deformed using

traces of the attractor in the past of a given current iteration and the rate $\lambda$ at which the deformation affects for a given lag.

A particle of position $(q)$ with negligible mass (or with very high friction) is dropped in the field and is disturbed by noisy force, in order to generate and simulate gaze dynamics. Therefore, given the fundamental relation for dynamics, where the accelerating second-order component is neglected, the gaze particle's motion is derived by the Langevin equation (Langevin, 1908), in which $m\ddot{q}$ is equal to the sum of forces applied to the particle, and can be
<figure><img src="image_19.png" /><figcaption>Figure 7. Examples of energy landscape surface plots setup using the gravitational fixational eye movements (GraFEM) model for the following: (A) shows three attractors (n = 3) with all attractors i having no motion and the following parameters:</figcaption></figure>

$\beta = 2$; $\delta = 1$; $\tau = 5$; $\lambda = 0.9$, and (B) shows two attractors (n = 2) with all attractors i having the following parameters:

$\beta_1 = 2$; $\beta_2 = 4$; $\delta_1 = \delta_2 = 1$; $\tau = 15$; $\lambda = 0.9$, and attractor motion computed with the following arbitrary sinusoidal motion:

$a_1(t) = [0, 0]$; $a_2(t) = a_2(t = 0) + [-5 \sin(2t), 5 \sin(3t)]$ on the 75th iterations. The motion of $a_2$ is shown in white. Though the model has many parameters, those manipulated in this work's results are exclusively the depth $\delta$ (or mass) of the attractors and the slope $\beta$ by affecting the concavity of the attractors' field.

<figure><img src="image_20.png" /><figcaption>Figure 8. Simulation examples generated with the GraFEM model. Simulations of fixations of 3.5 s with Euler-Muruyama time steps of $\Delta t = 13$ ms, with variable fixation dynamics generated by manipulating $\delta_{stim}$ and $\beta_{stim}$ parameters. Constant parameters of the model were number of attractors n = 2, with one for the fixation cross $a_{cross} = (0, 0)$ and another for the motion of the stimulus following Lissajous trajectories with the same parameters as in the three experiments (Necker, Square, Cross); $a_{stim}(t) = (\sin(2t), \sin(3t))$. The relaxation rate parameter $\lambda = 0.9$, memory temporal limit parameter K = 5, and noise $\xi \sim U[-0.5; 0.5]$ were used. (A) A simulated fixation with stable fixation dynamics with $\delta_{cross} = 100$; $\delta_{stim} = 100$; $\beta_{cross} = 1$; $\beta_{stim} = 1$. (B) A simulated fixation with micro-pursuit dynamics with $\delta_{cross} = 100$; $\delta_{stim} = 25$; $\beta_{cross} = 1$; $\beta_{stim} = 1$. (C) A simulated fixation with micro-saccade dynamics with $\delta_{cross} = 100$; $\delta_{stim} = 25$; $\beta_{cross} = 1$; $\beta_{stim} = 12$ and detected using the EK algorithm for micro-saccade detection.</figcaption></figure>

adding a moving average (MA) process (Hannan, 2009) on the field at a given time t:

$$E_{FEM}(q, t) = E(q, t) + \sum_{k=1}^{K} \frac{\lambda}{k+1} E(q, t - k\Delta t), \quad (7)$$

where K is the temporal parameter limiting how far in time will the fields be summed over and with $\lambda$ the relaxation rate parameter, and $\Delta t$ is the temporal step size. It is also possible to set the impact of memory and anticipation through parameters that define the iteration window over which the field is deformed using

traces of the attractor in the past of a given current iteration and the rate $\lambda$ at which the deformation affects for a given lag.

A particle of position (q) with negligible mass (or with very high friction) is dropped in the field and is disturbed by noisy force, in order to generate and simulate gaze dynamics. Therefore, given the fundamental relation for dynamics, where the accelerating second-order component is neglected, the gaze particle's motion is derived by the Langevin equation (Langevin, 1908), in which $m\ddot{q}$ is equal to the sum of forces applied to the particle, and can be
<figure><img src="image_20.png" /><figcaption>Figure 9. Simulations and analyses of the GraFEM model. Simulations of fixations of 1.5 s with Euler-Muruyama time steps of $\Delta t = 1$ ms, with variable fixation dynamics generated through the variation of $\delta_{stim} \in [0; 1200]$ and $\beta_{stim} \in [0; 50]$ parameters. Constant parameters of the model were number of attractors $n = 2$, with one for the fixation cross ($a_{cross} = [0, 0]$) and another for the motion of the stimulus following Lissajous trajectories with the same parameters as in the Necker cube experiment: $a_{stim}(t) = (\sin(2t), \sin(3t))$. The relaxation rate parameter $\lambda = 0.9$, memory temporal limit parameter $K = 5$, and noise $\xi \sim U[-0.5; 0.5]$ were used. (A) The behavior of inertia over the parameter space of the GraFEM model. (B) The behavior of the similarity between stimulus and simulated fixation motion using the MPC $\rho_1$. (C) The number of micro-saccades detected by the EK micro-saccade detection algorithm.</figcaption></figure>

rewritten as follows:

$$m\ddot{q} = -\gamma\dot{q} - \nabla E_{FEM}(q, t) + \xi(t) \quad (8)$$

with $m$ the negligible mass, $\gamma$ the friction, and $\xi$ an external force, here an oculomotor noise ($\eta$) applied to the gaze, such that $\eta(t) = \frac{\xi(t)}{\gamma}$. With the assumption of low mass and after normalization,⁷ such that $E_{FEM} = \frac{E_{FEM}}{\gamma}$, the dynamics can be expressed as

$$\dot{q} = -\nabla E_{FEM}(q, t) + \eta(t). \quad (9)$$

The evolution of the gaze particle’s dynamics can be computed by making the problem a discrete one using the Euler-Maruyama method (Kloeden & Platen, 2013), for instance.

## Model simulations: What are the parameters corresponding to ocular events and interpretation?

Fixations of 1.5 s, with a discretization Euler-Maruyama step $\Delta t = 1$ ms equal to the time step, were simulated using the GraFEM model with two attractors: $a_{cross}$ corresponding to the attractor of a fixation cross at the center and $a_{stim}$, the attractor representing the stimulus, with a Lissajous motion: $a_{stim} = (\sin(2t), \sin(3t))$. Only the slope and depth parameters were manipulated: $\beta_{stim} \in [0; 50]$ and $\delta_{stim} \in [0; 1200]$. All other parameters were kept constant with the other attractor position at $a_{cross} = (0, 0)$ with $\beta_{cross} = 1$ and $\delta_{cross} = 100$, the

relaxation rate parameter $\lambda = 0.9$, the memory temporal limit $K = 5$, and noise $\xi \sim U[-0.5; 0.5]$. These simulated fixations were then analyzed using the measures presented in this article, namely, inertia, MPC, and micro-saccade detection using the Engbert-Kliegl (EK) algorithm based on relative velocity thresholds (Engbert & Kliegl, 2003). Figure 9A shows that higher inertia follows a diagonal region along the $\{\beta_{stim}, \delta_{stim}\}$ space. When looking at Figure 9B, one can see that the same area in the parameter space has systematically high MPC. Finally, the EK algorithm was applied (without the binocularity criterion) to measure detected micro-saccades and summed over the time of a fixation. The results (Figure 9C) show that micro-saccades are detected when concavity is high due to a larger $\beta_{stim}$ parameter.

## Discussion and perspectives: Attractor, oculomotor, and perceptual multistability

The simulation results presented above show the following three points. First, fixations’ dynamics can be modeled, including a variety of fixational eye movements such as drift, tremors, micro-saccades, and micro-pursuits. Second, attractor dynamics can be intuitively manipulated by two parameters that control their slope and depth, hence imposing, by gravity, faster or slower dynamics on the gaze particle. Third, generalization to more complex stimuli or tasks can be maneuvered by such a model as attractors can be multiplied, if necessary. However, this work remains preliminary and calls for further investigation.
<figure><img src="image_21.png" /><figcaption>Figure 9. Simulations and analyses of the GraFEM model. Simulations of fixations of 1.5 s with Euler-Muruyama time steps of Δt = 1 ms, with variable fixation dynamics generated through the variation of δ<sub>stim</sub> ∈ [0; 1200] and β<sub>stim</sub> ∈ [0; 50] parameters. Constant parameters of the model were number of attractors n = 2, with one for the fixation cross (a<sub>cross</sub> = [0, 0]) and another for the motion of the stimulus following Lissajous trajectories with the same parameters as in the Necker cube experiment: a<sub>stim</sub>(t) = (sin(2t), sin(3t)). The relaxation rate parameter λ = 0.9, memory temporal limit parameter K = 5, and noise ξ ~ U[-0.5; 0.5] were used. (A) The behavior of inertia over the parameter space of the GraFEM model. (B) The behavior of the similarity between stimulus and simulated fixation motion using the MPC ρ<sub>1</sub>. (C) The number of micro-saccades detected by the EK micro-saccade detection algorithm.</figcaption></figure>

rewritten as follows:

$$m\ddot{q} = -\gamma\dot{q} - \nabla E_{FEM}(q, t) + \xi(t) \quad (8)$$

with m the negligible mass, γ the friction, and ξ an external force, here an oculomotor noise (η) applied to the gaze, such that η(t) = ξ(t)/γ. With the assumption of low mass and after normalization,⁷ such that E<sub>FEM</sub> = E<sub>FEM</sub>/γ, the dynamics can be expressed as

$$\dot{q} = -\nabla E_{FEM}(q, t) + \eta(t). \quad (9)$$

The evolution of the gaze particle's dynamics can be computed by making the problem a discrete one using the Euler-Maruyama method (Kloeden & Platen, 2013), for instance.

## Model simulations: What are the parameters corresponding to ocular events and interpretation?

Fixations of 1.5 s, with a discretization Euler-Maruyama step Δt = 1 ms equal to the time step, were simulated using the GraFEM model with two attractors: a<sub>cross</sub> corresponding to the attractor of a fixation cross at the center and a<sub>stim</sub>, the attractor representing the stimulus, with a Lissajous motion: a<sub>stim</sub> = (sin(2t), sin(3t)). Only the slope and depth parameters were manipulated: β<sub>stim</sub> ∈ [0; 50] and δ<sub>stim</sub> ∈ [0; 1200]. All other parameters were kept constant with the other attractor position at a<sub>cross</sub> = (0, 0) with β<sub>cross</sub> = 1 and δ<sub>cross</sub> = 100, the

relaxation rate parameter λ = 0.9, the memory temporal limit K = 5, and noise ξ ~ U[-0.5; 0.5]. These simulated fixations were then analyzed using the measures presented in this article, namely, inertia, MPC, and micro-saccade detection using the Engbert-Kliegl (EK) algorithm based on relative velocity thresholds (Engbert & Kliegl, 2003). Figure 9A shows that higher inertia follows a diagonal region along the {β<sub>stim</sub>, δ<sub>stim</sub>} space. When looking at Figure 9B, one can see that the same area in the parameter space has systematically high MPC. Finally, the EK algorithm was applied (without the binocularity criterion) to measure detected micro-saccades and summed over the time of a fixation. The results (Figure 9C) show that micro-saccades are detected when concavity is high due to a larger β<sub>stim</sub> parameter.

## Discussion and perspectives: Attractor, oculomotor, and perceptual multistability

The simulation results presented above show the following three points. First, fixations' dynamics can be modeled, including a variety of fixational eye movements such as drift, tremors, micro-saccades, and micro-pursuits. Second, attractor dynamics can be intuitively manipulated by two parameters that control their slope and depth, hence imposing, by gravity, faster or slower dynamics on the gaze particle. Third, generalization to more complex stimuli or tasks can be maneuvered by such a model as attractors can be multiplied, if necessary. However, this work remains preliminary and calls for further investigation.
<figure><img src="image_21.png" /><figcaption>Figure 10. GraFEM oculomotor interpretation. (A) A merger of measures applied on GraFEM synthetic data. Simulated fixations' inertia, MPC, and detected micro-saccade results are assigned to the red, green, and blue components of this RGB matrix, respectively. The yellow space provides $\beta$ and $\delta$ values that generate micro-pursuit, the blue pixels show a micro-saccade generation transition space, the dark area shows stable fixation parameters, and the red pixels could be interpreted as slow control. (B) A schematic interpretation of oculomotor dynamics generated in the parameter space of the GraFEM model with manipulation of the $\delta_{stim}$ and $\beta_{stim}$ parameters of the stimulus attractor, while keeping all other parameters constant. Micro-pursuit can be generate in a restrained subspace (yellow) while micro-saccades are detected in the surrounding space (dark blue). When $\delta_{stim}$ values are low, a transition area (light blue) exists where micro-saccades and similarity are high, but inertia is not. Finally, stable fixation subspace occupies the rest (gray).</figcaption></figure>

Such perspectives are discussed in the following paragraphs.

## Model interpretation for eye movements

The GraFEM model proposed in this article is capable of generating micro-saccade, drift, and tremor fixational eye movements (see Figure 8) as classified in the literature (Martinez-Conde, Macknik, & Hubel, 2004), as well as the micro-pursuits presented and detected in the article, as reported in Figure 9 and Figure 10. By using classified data (observations), the parameters of the model that allow the generation of these fixational eye movements could be inferred, and insights on the mechanics of micro-saccades, micro-pursuit, drift, and tremor generation and their interaction can be studied. The diagrams in Figure 10 already give a useful and overall understanding of the model, with respect to the manipulated parameters, but the work on parameter inference should be addressed in the near future in more details.

Given the observed data and the proposed model to account for it, questions and perspectives can be redefined with a novel angle for interpretation of fixational eye movements. Inversion and a full

analysis of a model, like GraFEM, with multiple free parameters is a complex task out of the scope of this thesis but should be tackled and reported in the near future.

The model presented here gives a mathematical framework in which eye movement phenomena can be generated and interpreted. Attractors are interesting as tools to explain and interpret cognitive and physiological behaviors as they allow an intuitive understanding of the evolution of dynamical systems (Watanabe, Masuda, Megumi, Kanai, & Rees, 2014; Kelso, 2012). Furthermore, complex learning systems—that is, neural networks—are known to develop such properties as the parameters of their processes tend to learn the statistics of the environment by creating attractors in the parameter space (Moreno-Bote, Rinzel, & Rubin, 2007; Shpiro, Moreno-Bote, Rubin, & Rinzel, 2009; Moreno-Bote, Knill, & Pouget, 2011; Moreno-Bote & Drugowitsch, 2015).

With this modeling framework, the fixational eye movements classification of the literature can be described and interpreted in terms of attractor spatiotemporal dynamics (Figures 9 and 10).

A stable fixation (Figure 8A) in the GraFEM model corresponds to a stabilization of an attractor
<figure><img src="image_22.png" /><figcaption>Figure 10. GraFEM oculomotor interpretation. (A) A merger of measures applied on GraFEM synthetic data. Simulated fixations' inertia, MPC, and detected micro-saccade results are assigned to the red, green, and blue components of this RGB matrix, respectively. The yellow space provides $\beta$ and $\delta$ values that generate micro-pursuit, the blue pixels show a micro-saccade generation transition space, the dark area shows stable fixation parameters, and the red pixels could be interpreted as slow control. (B) A schematic interpretation of oculomotor dynamics generated in the parameter space of the GraFEM model with manipulation of the $\delta_{stim}$ and $\beta_{stim}$ parameters of the stimulus attractor, while keeping all other parameters constant. Micro-pursuit can be generate in a restrained subspace (yellow) while micro-saccades are detected in the surrounding space (dark blue). When $\delta_{stim}$ values are low, a transition area (light blue) exists where micro-saccades and similarity are high, but inertia is not. Finally, stable fixation subspace occupies the rest (gray).</figcaption></figure>

Such perspectives are discussed in the following paragraphs.

## Model interpretation for eye movements

The GraFEM model proposed in this article is capable of generating micro-saccade, drift, and tremor fixational eye movements (see Figure 8) as classified in the literature (Martinez-Conde, Macknik, & Hubel, 2004), as well as the micro-pursuits presented and detected in the article, as reported in Figure 9 and Figure 10. By using classified data (observations), the parameters of the model that allow the generation of these fixational eye movements could be inferred, and insights on the mechanics of micro-saccades, micro-pursuit, drift, and tremor generation and their interaction can be studied. The diagrams in Figure 10 already give a useful and overall understanding of the model, with respect to the manipulated parameters, but the work on parameter inference should be addressed in the near future in more details.

Given the observed data and the proposed model to account for it, questions and perspectives can be redefined with a novel angle for interpretation of fixational eye movements. Inversion and a full

analysis of a model, like GraFEM, with multiple free parameters is a complex task out of the scope of this thesis but should be tackled and reported in the near future.

The model presented here gives a mathematical framework in which eye movement phenomena can be generated and interpreted. Attractors are interesting as tools to explain and interpret cognitive and physiological behaviors as they allow an intuitive understanding of the evolution of dynamical systems (Watanabe, Masuda, Megumi, Kanai, & Rees, 2014; Kelso, 2012). Furthermore, complex learning systems—that is, neural networks—are known to develop such properties as the parameters of their processes tend to learn the statistics of the environment by creating attractors in the parameter space (Moreno-Bote, Rinzel, & Rubin, 2007; Shpiro, Moreno-Bote, Rubin, & Rinzel, 2009; Moreno-Bote, Knill, & Pouget, 2011; Moreno-Bote & Drugowitsch, 2015).

With this modeling framework, the fixational eye movements classification of the literature can be described and interpreted in terms of attractor spatiotemporal dynamics (Figures 9 and 10).

A stable fixation (Figure 8A) in the GraFEM model corresponds to a stabilization of an attractor
with the energy landscape having little change. The gaze particle is stuck, and only the noise affecting its position may lead to small random movements of the eyes, as in other generative fixational eye movement models (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011; Herrmann, Metzler, & Engbert, 2017). In these models, constraints to the energy field of the fixation are used in an analogous fashion to reflect the higher probabilities of having fixational eye movements in horizontal and vertical directions. A fixation attractor can thus be predicted by the task or the stimulus controlled by the experiments, and its parameters can be inferred by a priori information and data. Hence, the model gives predictive capabilities that can be tested and requires assimilation of data to constrain its range of possibilities.

Micro-saccades (Figure 8C) correspond to sudden changes in the energy depth of attractors, with a new one emerging or deepening while the attractor of fixation has suddenly disappeared. They are likely to emerge as the gaze particle rushes down a gradient to the center of an attractor, giving it sufficient velocity. The depth and slope of the attractor can be manipulated (following the dynamics described in Figure 10), thus making it possible to infer, based on observed velocities and amplitudes, the saliency of that attractor. The GraFEM model does not use an explicit and separated mechanism for micro-saccade generation—as the model presented in Engbert, Mergenthaler, Sinn, and Pikovsky (2011)—though it is not incompatible.

Drifts correspond to a stability of the gaze particle with respect to the attractor by which it is transported. However, the attractor might itself slowly drift away in the visual space (independently from the target motion), or alternatively, the shape of the well might get larger (by manipulating the parameter $\beta$), allowing for the noisy gaze particle to explore further. These are two hypotheses that could be tested, in future work, by inferring the model parameters given sufficient data. These fixational eye movements are known to help reduce visual redundancy and extract features in complex visual stimuli (Kuang, Poletti, Victor, & Rucci, 2012) but are mostly considered consequences of the eye muscles and their neural control properties. Therefore, they have mostly been considered independent processes from the visual stimulus presented.

The micro-pursuits detected and described in the article could be interpreted as a form of stimulus-related drift, as its signal dynamics place it in similar ranges, and is captured by the proposed metric, namely, MPC. Consequently, this argues in favor of our proposition that drifts are composed of two categories stimulus independent and dependent and micro-pursuits logically fall within visually dependent ocular drifts. This dependency can be interpreted as the interference of bottom-up salient elements interrupting the top-down task of fixation. Micro-pursuits (Figure 8B)

are therefore close to drifts in the energy landscape dynamics.

## Model mechanics

The model sets the gaze as a particle in an evolving gravitational energy potential field. When the system has no dynamics added to the potential landscapes, the particle will fall into its nearest local minimum. In this implementation, at each iteration—here a discrete time step using Equation 9—the first derivative is computed to update the position of the particle in the plane, corresponding to the screen. Noise is then added to the deterministic dynamics and can drive fixational oculomotor decision-making with respect to attractors if its amplitude is sufficiently large (Shpiro, Moreno-Bote, Rubin, & Rinzel, 2009; Moreno-Bote, Rinzel, & Rubin, 2007). This mechanism is similar to bistable energy potential models, though it extends on the dimensions of the system. In a set of simulations reported in Figure 9, we show that through two continuous parameters applied to a target attractor, it is possible to generate and interpret oculomotor dynamics observed in fixational eye movements. However, here, there is no prior requiring the existence of different systems for each class of movements observed (Barnes, 2011). Fixational eye movement dynamics can be reproduced through a unique mechanism, as shown by the simulated examples in Figure 8.

Top-down intention processes can be tested and simulated, given the context of a task, by applying changes in the model's $\beta$ and $\delta$ parameters. Figure 10 can be used as a road map of the oculomotor dynamics and regimes expected, depending on parameter values. Moreover, bottom-up saliency or attentional effects can also be taken into account. This can be done with simpler assumptions, such as the ones presented here for the task used in the article, but can be more complex if using natural scene tasks, for instance. An interesting and practical perspective in this context lies in investigating how salience models, which derive probability distribution based on the statistics of images or videos, can be integrated such that only attractors are fed into a GraFEM oculomotor execution system.

## How would this be implemented in the brain?

Anatomically, oculomotor programming has been shown to be highly correlated and linked to a network of areas involving neural activity in the superior colliculus, the frontal eye field, and the lateral intraparietal cortex (Hafed, Goffart, & Krauzlis, 2009; Krauzlis, 2004, 2005; Krauzlis, Goffart, & Hafed, 2017; Astrand, Ibos, Duhamel, & Hamed, 2015; Peel, Hafed, Dash, Lomber, & Corneil, 2016; Taouali, Goffart, Alexandre, & Rougier, 2015). There are interindividual differences in anatomy and behavior for fixational eye movements
with the energy landscape having little change. The gaze particle is stuck, and only the noise affecting its position may lead to small random movements of the eyes, as in other generative fixational eye movement models (Engbert, Mergenthaler, Sinn, & Pikovsky, 2011; Herrmann, Metzler, & Engbert, 2017). In these models, constraints to the energy field of the fixation are used in an analogous fashion to reflect the higher probabilities of having fixational eye movements in horizontal and vertical directions. A fixation attractor can thus be predicted by the task or the stimulus controlled by the experiments, and its parameters can be inferred by a priori information and data. Hence, the model gives predictive capabilities that can be tested and requires assimilation of data to constrain its range of possibilities.

Micro-saccades (Figure 8C) correspond to sudden changes in the energy depth of attractors, with a new one emerging or deepening while the attractor of fixation has suddenly disappeared. They are likely to emerge as the gaze particle rushes down a gradient to the center of an attractor, giving it sufficient velocity. The depth and slope of the attractor can be manipulated (following the dynamics described in Figure 10), thus making it possible to infer, based on observed velocities and amplitudes, the saliency of that attractor. The GraFEM model does not use an explicit and separated mechanism for micro-saccade generation—as the model presented in Engbert, Mergenthaler, Sinn, and Pikovsky (2011)—though it is not incompatible.

Drifts correspond to a stability of the gaze particle with respect to the attractor by which it is transported. However, the attractor might itself slowly drift away in the visual space (independently from the target motion), or alternatively, the shape of the well might get larger (by manipulating the parameter $\beta$), allowing for the noisy gaze particle to explore further. These are two hypotheses that could be tested, in future work, by inferring the model parameters given sufficient data. These fixational eye movements are known to help reduce visual redundancy and extract features in complex visual stimuli (Kuang, Poletti, Victor, & Rucci, 2012) but are mostly considered consequences of the eye muscles and their neural control properties. Therefore, they have mostly been considered independent processes from the visual stimulus presented.

The micro-pursuits detected and described in the article could be interpreted as a form of stimulus-related drift, as its signal dynamics place it in similar ranges, and is captured by the proposed metric, namely, MPC. Consequently, this argues in favor of our proposition that drifts are composed of two categories stimulus independent and dependent and micro-pursuits logically fall within visually dependent ocular drifts. This dependency can be interpreted as the interference of bottom-up salient elements interrupting the top-down task of fixation. Micro-pursuits (Figure 8B)

are therefore close to drifts in the energy landscape dynamics.

## Model mechanics

The model sets the gaze as a particle in an evolving gravitational energy potential field. When the system has no dynamics added to the potential landscapes, the particle will fall into its nearest local minimum. In this implementation, at each iteration—here a discrete time step using Equation 9—the first derivative is computed to update the position of the particle in the plane, corresponding to the screen. Noise is then added to the deterministic dynamics and can drive fixational oculomotor decision-making with respect to attractors if its amplitude is sufficiently large (Shpiro, Moreno-Bote, Rubin, & Rinzel, 2009; Moreno-Bote, Rinzel, & Rubin, 2007). This mechanism is similar to bistable energy potential models, though it extends on the dimensions of the system. In a set of simulations reported in Figure 9, we show that through two continuous parameters applied to a target attractor, it is possible to generate and interpret oculomotor dynamics observed in fixational eye movements. However, here, there is no prior requiring the existence of different systems for each class of movements observed (Barnes, 2011). Fixational eye movement dynamics can be reproduced through a unique mechanism, as shown by the simulated examples in Figure 8.

Top-down intention processes can be tested and simulated, given the context of a task, by applying changes in the model's $\beta$ and $\delta$ parameters. Figure 10 can be used as a road map of the oculomotor dynamics and regimes expected, depending on parameter values. Moreover, bottom-up saliency or attentional effects can also be taken into account. This can be done with simpler assumptions, such as the ones presented here for the task used in the article, but can be more complex if using natural scene tasks, for instance. An interesting and practical perspective in this context lies in investigating how salience models, which derive probability distribution based on the statistics of images or videos, can be integrated such that only attractors are fed into a GraFEM oculomotor execution system.

## How would this be implemented in the brain?

Anatomically, oculomotor programming has been shown to be highly correlated and linked to a network of areas involving neural activity in the superior colliculus, the frontal eye field, and the lateral intraparietal cortex (Hafed, Goffart, & Krauzlis, 2009; Krauzlis, 2004, 2005; Krauzlis, Goffart, & Hafed, 2017; Astrand, Ibos, Duhamel, & Hamed, 2015; Peel, Hafed, Dash, Lomber, & Corneil, 2016; Taouali, Goffart, Alexandre, & Rougier, 2015). There are interindividual differences in anatomy and behavior for fixational eye movements
measuring and observed dynamics. For instance, it has
been shown that not only oculomotor behavior between
trained and untrained participants varies a lot, but
drift also accounts for more fixation correction motion
than micro-saccades (Cherici, Kuang, Poletti, & Rucci,
2012). The observations of micro-pursuits presented in
the article suggest that the dynamics of the gaze within
a fixation can be affected and attracted by motion of an
object in or nearby the foveal field.

However, rather than having an attractor with a
pseudo-random displacement, its motion follows a
deterministic and predictable trajectory, which can be
computed and estimated by the oculomotor system.
Moreover, that attractor is, given our observations so
far, only related to a target motion. This could, for
instance, be implemented in the brain by means of an
efference copy (Astrand, Ibos, Duhamel, & Hamed,
2015), though this idea remains speculative, and further
modeling and neurophysiological research are needed.
The low energy attributed to a decoded and perceived
object moving across space encourages the oculomotor
system to track it as it tries to minimize the energy of
the gaze particle. Finally, tremors are generated and
explained by the noise given to the particle over all
fixational eye movement events.

This model complements the eye movement field of
research with the possibility to program intentions,
salience, and their effects on the gaze dynamics by
simply using attractors and setting out their dynamics
in terms of motion on the visual field, depth, and
memory. For instance, the model can predict the
different dynamics reported based on the eccentricity
of an attractor corresponding to an afterimage, as
observed in Heywood and Churcher (1972). Thus, one
can use the model to generate statistical predictions of
eye movement dynamics. Given an understanding of
the visual attention or saliency effects of their stimulus
and taking into account all the associated intentions

to the tasks that participants are required to operate
during a trial, it is possible to use this modeling to
generate quantitative predictions of the oculomotor
dynamics. Moreover, the generative properties make
it possible to work on simulated data and extract
dynamics' statistics in terms of eye movements, and
this is possible using the traditional algorithms for
eye movements classification. Inversely, obtaining the
parameters of the model that replicate the dynamics of
observations could help understand better the internal
processes that drive eye movements.

Perspectives: Toward oculomotor multistability

A key aspect of this family of models is that it
showcases *multistability* regarding their attractors. This
phenomenon can emerge in many complex biological
systems and is present in many cognitive processes
(Schwartz, Grimault, Hup, Moore, & Pressnitzer,
2012). It is linked to coordination dynamics between
subsystems that have varying levels of coupling, leading
to monostable, multistable, or metastable dynamics
(Kelso, 2012). The consequent interpretation is that the
oculomotor system could have multistable dynamics
with respect to visual attractors. In this case, the
oculomotor dynamics are likely driven by noisy signals
(Braun & Mattia, 2010) representing other interfering
systems, such as perception, attention, intention, and
other cognitive systems. This framework connects
to the growing body of studies linking perceptual
decisions and multistable system dynamics. It also
creates a link for motor systems to studies of noise as
a component that helps a perceptual system operate
through stochastic resonance⁸ (Gammaitoni, Hänggi,
Jung, & Marchesoni, 1998; Patel & Kosko, 2005; Kim,
Grabowewcky, & Suzuki, 2006).
measuring and observed dynamics. For instance, it has
been shown that not only oculomotor behavior between
trained and untrained participants varies a lot, but
drift also accounts for more fixation correction motion
than micro-saccades (Cherici, Kuang, Poletti, & Rucci,
2012). The observations of micro-pursuits presented in
the article suggest that the dynamics of the gaze within
a fixation can be affected and attracted by motion of an
object in or nearby the foveal field.

However, rather than having an attractor with a
pseudo-random displacement, its motion follows a
deterministic and predictable trajectory, which can be
computed and estimated by the oculomotor system.
Moreover, that attractor is, given our observations so
far, only related to a target motion. This could, for
instance, be implemented in the brain by means of an
efference copy (Astrand, Ibos, Duhamel, & Hamed,
2015), though this idea remains speculative, and further
modeling and neurophysiological research are needed.
The low energy attributed to a decoded and perceived
object moving across space encourages the oculomotor
system to track it as it tries to minimize the energy of
the gaze particle. Finally, tremors are generated and
explained by the noise given to the particle over all
fixational eye movement events.

This model complements the eye movement field of
research with the possibility to program intentions,
salience, and their effects on the gaze dynamics by
simply using attractors and setting out their dynamics
in terms of motion on the visual field, depth, and
memory. For instance, the model can predict the
different dynamics reported based on the eccentricity
of an attractor corresponding to an afterimage, as
observed in Heywood and Churcher (1972). Thus, one
can use the model to generate statistical predictions of
eye movement dynamics. Given an understanding of
the visual attention or saliency effects of their stimulus
and taking into account all the associated intentions

to the tasks that participants are required to operate
during a trial, it is possible to use this modeling to
generate quantitative predictions of the oculomotor
dynamics. Moreover, the generative properties make
it possible to work on simulated data and extract
dynamics' statistics in terms of eye movements, and
this is possible using the traditional algorithms for
eye movements classification. Inversely, obtaining the
parameters of the model that replicate the dynamics of
observations could help understand better the internal
processes that drive eye movements.

Perspectives: Toward oculomotor multistability

A key aspect of this family of models is that it
showcases *multistability* regarding their attractors. This
phenomenon can emerge in many complex biological
systems and is present in many cognitive processes
(Schwartz, Grimault, Hup, Moore, & Pressnitzer,
2012). It is linked to coordination dynamics between
subsystems that have varying levels of coupling, leading
to monostable, multistable, or metastable dynamics
(Kelso, 2012). The consequent interpretation is that the
oculomotor system could have multistable dynamics
with respect to visual attractors. In this case, the
oculomotor dynamics are likely driven by noisy signals
(Braun & Mattia, 2010) representing other interfering
systems, such as perception, attention, intention, and
other cognitive systems. This framework connects
to the growing body of studies linking perceptual
decisions and multistable system dynamics. It also
creates a link for motor systems to studies of noise as
a component that helps a perceptual system operate
through stochastic resonance⁸ (Gammaitoni, Hänggi,
Jung, & Marchesoni, 1998; Patel & Kosko, 2005; Kim,
Grabowewcky, & Suzuki, 2006).