# Information Rate in Humans during Visuomotor Tracking

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## Abstract

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## 1. Introduction

#### 1.1. Information Processing Rate in Humans

#### 1.2. Pursuit-Tracking Task and Its Feedforward Component

## 2. Results

#### 2.1. Background

#### 2.2. Definition of Measures

**Basic assumptions.**Adopting a model-free approach, we did not make any specific assumptions on the mechanism involved in producing the observed tracking performance. For our analysis, we rely on two key properties of information sharing and transmission in the system.

**Feedback component information content.**Using the above properties, we were able to fix the free parameters in the transfer entropy formula in Equation (4), thus tailoring it to the quantification of the information rate of the feedback component, as desired. Given that the expected delay of information transfer from signal X to tracking Y is the VMD and that the target is a second-order autocorrelated signal, we set the delay d between the two processes to be VMD and the depths l and k to be 2. To ensure independence between successive samples, we further modified the transfer entropy term by conditioning it on ${Y}_{t-1}$, thus obtaining the following feedback component measure:

**Total information and feedforward component.**The total information shared between signal X and tracking Y has either of two origins: it arises via feedback information transfer with a non-reducible time delay $({X}_{t-VMD},{X}_{t-VMD-1}\to {Y}_{t})$, or it is due to prediction $({X}_{t},{X}_{t-1}\to {Y}_{t})$. This allowed us to compute it as the joint mutual information:

#### 2.3. Validation through Model Simulations

**Validation of ${I}_{FB}$ and ${I}_{FF}$**. To establish the ground truth for the feedback measure, ${\mathrm{T}}_{\mathrm{FB}}$, we took advantage of the linearity of the Kalman filter to directly quantify the information transfer through the feedback pathway in the model by computing the mutual information between observation x and state estimates s at the Kalman filter level. Given the Gaussian distribution of both the observation x and state estimates s, the mutual information can be expressed as:

**Validation of the Estimate for VMD.**While VMD can be directly extracted from the model for simulation data, there is no way to access it directly in real experimental data. We therefore needed to establish an estimate of VMD that could be applied to experimental data. A candidate for such an estimate was the peak latency of the transfer entropy from signal X to tracking Y, $T{E}_{X\to Y}$. Since the feedback component is delayed by VMD, the transfer entropy should peak at t-VMD. To evaluate the correspondence of this candidate measure to the true VMD, we generated simulated data corresponding to true VMD values from 9 to 19 frames while all other parameters were kept constant. Notably, the effective VMD of the simulation data was determined by the sum of the visual and motor delay parameters with an additional delay that was inherent to the joystick mechanism and which depended on the parameters of its state space representation (i.e., spring, mass and damping coefficients). Therefore, here again, we were seeking a correlation rather than a strict equality between inferred and reference values. The comparison showed perfect correlation between the peak latency of ${\mathrm{I}}_{\mathrm{FB}}$ and the VMD actually implemented in the model (${R}^{2}=1$), validating this estimate of VMD from data.

**Relationship between feedback component and performance lag.**The effect of prediction on tracking performance is two-fold: first, it provides a cognitively efficient way to encode the target signal, thus saving cognitive resources; second, it compensates for VMD by allowing subjects to act in advance, which contributes to a reduction in performance lag (that is, the lag corresponding to maximum cross-correlation between target and tracking). When prediction fails, we would expect the feedback component to take up more information load to maintain performance level. Due to the irreducible VMD of the feedback component, the more it is involved, the more performance lag will tend to VMD. We looked at our simulation data to confirm this effect by observing the relationship between the ratio of performance lag to VMD and the feedback component measure (Figure 2C). We found a strong exponential relationship between the two variables.

#### 2.4. Experimental Results

**Identifying VMD from experimental tracking data.**To be able to compute ${\mathrm{I}}_{\mathrm{FB}}$ and ${\mathrm{I}}_{\mathrm{FF}}$, one must first know the VMD of the system. Having confirmed that the peak latency of transfer entropy $T{E}_{X\to Y}$ corresponds perfectly with VMD in simulation data, we computed the trial-averaged peak latencies of $T{E}_{X\to Y}$ for each subject from their performance in the most complex condition to obtain an estimate for each subject’s VMD (Figure 3A). Our results showed that VMD lay between 14 and 16 frames (about 230 to 270 ms) for 10 out of the 11 subjects, while one subject showed a VMD of around 380 ms.

**Feedback information rate.**Using subject-specific VMD, we computed the feedback component ${\mathrm{I}}_{\mathrm{FB}}$ using Equation (5). Results showed that feedback information transfer increased with the complexity of the signal (F(3,40) = 34.9, p < 0.0001). Post hoc Tukey HSD test indicated that condition 1 and condition 4 were significantly different from all other conditions while the difference between condition 2 and 3 did not reach significance.

**Feedforward component and predictability of signal.**The feedforward component measure ${\mathrm{I}}_{\mathrm{FF}}$ cannot be interpreted as an information transfer rate per unit of time because, unlike ${\mathrm{I}}_{\mathrm{FB}}$, it is not an independent measure between successive samples. However, it can still be compared across conditions to help us gain insight about the role of prediction with regards to signals of different predictability. We observed a clearly opposite trend relative to that of the feedback component. As predictability of signals decreased, ${\mathrm{I}}_{\mathrm{FF}}$ also decreased, F(3,40) = 30.7, p < 0.0001. Post hoc Tukey HSD test once again indicated only condition 1 and condition 4 were significantly different from all other conditions (Figure 3C).

## 3. Discussion

#### 3.1. Information Processing Rate in Humans

#### 3.2. Information-Theoretic Approach to Evaluating Tracking Performance

#### 3.3. Limitations

## 4. Conclusions

## 5. Materials and Methods

#### 5.1. Participants

#### 5.2. Experimental Design

#### 5.3. Mutual Information Estimation Using Gaussian Copula

#### 5.4. Linear-Quadratic Regulator Model

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Cohen, M.R.; Maunsell, J.H. Attention improves performance primarily by reducing interneuronal correlations. Nat. Neurosci.
**2009**, 12, 1594. [Google Scholar] [CrossRef] - Smith, E.C.; Lewicki, M.S. Efficient auditory coding. Nature
**2006**, 439, 978–982. [Google Scholar] [CrossRef] [PubMed] - Zenon, A.; Solopchuk, O.; Pezzulo, G. An information-theoretic perspective on the costs of cognition. Neuropsychologia
**2019**, 123, 5–18. [Google Scholar] [CrossRef] [PubMed] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Hick, W.E. On the rate of gain of information. Q. J. Exp. Psychol.
**1952**, 4, 11–26. [Google Scholar] [CrossRef] - Fitts, P.M. The information capacity of the human motor system in controlling the amplitude of movement. J. Exp. Psychol.
**1954**, 47, 381. [Google Scholar] [CrossRef] [Green Version] - Gori, J.; Rioul, O.; Guiard, Y. Speed-accuracy tradeoff: A formal information-theoretic transmission scheme (fitts). ACM Trans. Comput.-Hum. Interact. (TOCHI)
**2018**, 25, 1–33. [Google Scholar] [CrossRef] - Grossman, E. The information-capacity of the human motor-system in pursuit tracking. Q. J. Exp. Psychol.
**1960**, 12, 1–16. [Google Scholar] [CrossRef] - Poulton, E. On prediction in skilled movements. Psychol. Bull.
**1957**, 54, 467. [Google Scholar] [CrossRef] - Yeo, S.H.; Franklin, D.W.; Wolpert, D.M. When optimal feedback control is not enough: Feedforward strategies are required for optimal control with active sensing. PLoS Comput. Biol.
**2016**, 12, e1005190. [Google Scholar] [CrossRef] - Maeda, R.S.; Cluff, T.; Gribble, P.L.; Pruszynski, J.A. Feedforward and feedback control share an internal model of the arm’s dynamics. J. Neurosci.
**2018**, 38, 10505–10514. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Drop, F.M.; Pool, D.M.; Damveld, H.J.; van Paassen, M.M.; Mulder, M. Identification of the feedforward component in manual control with predictable target signals. IEEE Trans. Cybern.
**2013**, 43, 1936–1949. [Google Scholar] [CrossRef] [PubMed] - Drop, F.M.; de Vries, R.; Mulder, M.; Bülthoff, H.H. The predictability of a target signal affects manual feedforward control. IFAC-PapersOnLine
**2016**, 49, 177–182. [Google Scholar] [CrossRef] - Trujillo, L.T. Mental Effort and Information-Processing Costs Are Inversely Related to Global Brain Free Energy During Visual Categorization. Front. Neurosci.
**2019**, 13, 1292. [Google Scholar] [CrossRef] [Green Version] - Schreiber, T. Measuring information transfer. Phys. Rev. Lett.
**2000**, 85, 461. [Google Scholar] [CrossRef] [Green Version] - Mulder, M.; Pool, D.M.; Abbink, D.A.; Boer, E.R.; Zaal, P.M.; Drop, F.M.; van der El, K.; van Paassen, M.M. Manual control cybernetics: State-of-the-art and current trends. IEEE Trans. Hum. Mach. Syst.
**2017**, 48, 468–485. [Google Scholar] [CrossRef] - Cover, T.M.; Thomas, J.A. Elements of Information Theory, 2nd ed.; Wiley-Interscience: Hoboken, NJ, USA, 2006. [Google Scholar]
- Rao, R.P.; Ballard, D.H. Predictive coding in the visual cortex: A functional interpretation of some extra-classical receptive-field effects. Nat. Neurosci.
**1999**, 2, 79–87. [Google Scholar] [CrossRef] - Kool, W.; McGuire, J.T.; Rosen, Z.B.; Botvinick, M.M. Decision making and the avoidance of cognitive demand. J. Exp. Psychol. Genl.
**2010**, 139, 665. [Google Scholar] [CrossRef] [Green Version] - Westbrook, A.; Braver, T.S. Cognitive effort: A neuroeconomic approach. Cogn. Affect. Behav. Neurosci.
**2015**, 15, 395–415. [Google Scholar] [CrossRef] - Shenhav, A.; Musslick, S.; Lieder, F.; Kool, W.; Griffiths, T.L.; Cohen, J.D.; Botvinick, M.M. Toward a rational and mechanistic account of mental effort. Annu. Rev. Neurosci.
**2017**, 40, 99–124. [Google Scholar] [CrossRef] [Green Version] - Hülsdünker, T.; Ostermann, M.; Mierau, A. The speed of neural visual motion perception and processing determines the visuomotor reaction time of young elite table tennis athletes. Front. Behav. Neurosci.
**2019**, 13, 165. [Google Scholar] [CrossRef] [Green Version] - Miall, R.; Weir, D.; Stein, J. Visuomotor tracking with delayed visual feedback. Neuroscience
**1985**, 16, 511–520. [Google Scholar] [CrossRef] - Foulkes, A.J.M.; Miall, R.C. Adaptation to visual feedback delays in a human manual tracking task. Exp. Brain Res.
**2000**, 131, 101–110. [Google Scholar] [CrossRef] - Ballard, K.J.; Robin, D.A.; Woodworth, G.; Zimba, L.D. Age-related changes in motor control during articulator visuomotor tracking. J. Speech Lang. Hear. Res.
**2001**, 44, 763–777. [Google Scholar] [CrossRef] - Bormann, R.; Cabrera, J.L.; Milton, J.G.; Eurich, C.W. Visuomotor tracking on a computer screen—An experimental paradigm to study the dynamics of motor control. Neurocomputing
**2004**, 58, 517–523. [Google Scholar] [CrossRef] - Lee, G.; Choi, W.; Jo, H.; Park, W.; Kim, J. Analysis of motor control strategy for frontal and sagittal planes of circular tracking movements using visual feedback noise from velocity change and depth information. PloS ONE
**2020**, 15, e0241138. [Google Scholar] [CrossRef] - Takagi, A.; Furuta, R.; Saetia, S.; Yoshimura, N.; Koike, Y.; Minati, L. Behavioral and physiological correlates of kinetically tracking a chaotic target. PLoS ONE
**2020**, 15, e0239471. [Google Scholar] [CrossRef] [PubMed] - Hamilton, A.F.d.C.; Jones, K.E.; Wolpert, D.M. The scaling of motor noise with muscle strength and motor unit number in humans. Exp. Brain Res.
**2004**, 157, 417–430. [Google Scholar] [CrossRef] [PubMed] - Ince, R.A.; Giordano, B.L.; Kayser, C.; Rousselet, G.A.; Gross, J.; Schyns, P.G. A statistical framework for neuroimaging data analysis based on mutual information estimated via a gaussian copula. Hum. Brain Mapp.
**2017**, 38, 1541–1573. [Google Scholar] [CrossRef] [PubMed] - Kraskov, A.; Stögbauer, H.; Grassberger, P. Estimating mutual information. Phys. Rev. E
**2004**, 69, 066138. [Google Scholar] [CrossRef] [Green Version] - Casella, G.; Berger, R.L. Statistical Inference; Duxbury: Pacific Grove, CA, USA, 2002; Volume 2. [Google Scholar]
- Sklar, A. Fonction de répartition dont les marges sont données. Inst. Stat. Univ. Paris
**1959**, 8, 229–231. [Google Scholar] - Jenison, R.L.; Reale, R.A. The shape of neural dependence. Neural. Comput.
**2004**, 16, 665–672. [Google Scholar] [CrossRef] - Misra, N.; Singh, H.; Demchuk, E. Estimation of the entropy of a multivariate normal distribution. J. Multivar. Anal.
**2005**, 92, 324–342. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**

**Example tracking data.**Example experimental data showing the x-coordinates of target signal (blue) and tracking response (orange) for condition 1 (top; most predictable condition) and condition 4 (bottom; least predictable condition).

**Figure 2.**

**Simulation design and results.**(

**A**) Schematic of Linear Quadratic Regulator model of the visuomotor tracking task. (

**B**) Correlation between true feedback measure ${\mathrm{T}}_{\mathrm{FB}}$ and proposed measure ${\mathrm{I}}_{\mathrm{FB}}$ from model data, R = 0.999. (left) Correlation between true feedforward measure ${\mathrm{T}}_{\mathrm{FF}}$ and proposed measure ${\mathrm{I}}_{\mathrm{FF}}$ from model data, R = 0.999. (right) Color code indicates the value of the noise parameter used to generate the signal (see Methods). Larger values correspond to higher complexity in signals, thus less predictable. (

**C**) Relationship between ${\mathrm{I}}_{\mathrm{FB}}$ and performance lag/visuomotor delay (VMD) ratio. An exponential function $\frac{\mathrm{PL}}{\mathrm{VMD}}=aexp\left(b{I}_{\mathrm{FB}}\right)$ was fitted on the data, with $\mathrm{PL}$ the performance lag and $\mathrm{VMD}$ the visuo-motor delay. The R squared of the fit was 0.98. a = 1.172 (95% confidence interval: 0.815–1.53), b = 4.282 (3.779–4.785).

**Figure 3.**

**Experimental results.**(

**A**) VMD of individual subjects (sorted in increasing order). Error bars represent the standard deviation across trials. (

**B**) Average real time information processing rate per second across subjects (

**C**) Average ${I}_{FF}$ across subjects for different conditions.

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Lam, S.-Y.; Zénon, A.
Information Rate in Humans during Visuomotor Tracking. *Entropy* **2021**, *23*, 228.
https://doi.org/10.3390/e23020228

**AMA Style**

Lam S-Y, Zénon A.
Information Rate in Humans during Visuomotor Tracking. *Entropy*. 2021; 23(2):228.
https://doi.org/10.3390/e23020228

**Chicago/Turabian Style**

Lam, Sze-Ying, and Alexandre Zénon.
2021. "Information Rate in Humans during Visuomotor Tracking" *Entropy* 23, no. 2: 228.
https://doi.org/10.3390/e23020228