# Towards formal models of inhibitory mechanisms involved in motor imagery

Ladislas Nalborczyk, Marieke Longcamp, Thibault Gajdos, Mathieu Servant, F.-Xavier Alario

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Ladislas Nalborczyk, Marieke Longcamp, Thibault Gajdos, Mathieu Servant, F.-Xavier Alario. Towards formal models of inhibitory mechanisms involved in motor imagery: A commentary on Bach, Frank, & Kunde (2022). Psychological Research, 2024, 10.1007/s00426-023-01915-8. http://dx.doi.org/10.1007/s00426-023-01915-8

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Distributed under a Creative Commons Attribution 4.0 International License
1 Towards formal models of inhibitory mechanisms involved in motor imagery: A

2 commentary on Bach, Frank, & Kunde (2022)

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4 Ladislas Nalborczyk¹,², Marieke Longcamp², Thibault Gajdos¹, Mathieu Servant³,⁴,⁵, & F.-Xavier

5 Alario¹

6 ¹ Aix Marseille Univ, CNRS, LPC, Marseille, France

7 ² Aix Marseille Univ, CNRS, LNC, Marseille, France

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9 ³ Laboratoire de Recherches Intégratives en Neurosciences et Psychologie Cognitive UR 481,
Université de Franche-Comté, France

10 ⁴ MSHE Ledoux USR 3124, Université de Franche-Comté, France

11 ⁵ Institut Universitaire de France

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13 Author note

14 Correspondence concerning this article should be addressed to Ladislas Nalborczyk,
15 Laboratoire de Psychologie Cognitive, AMU & CNRS (UMR 7290). E-mail:

16 ladislas.nalborczyk@gmail.com
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Abstract

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A vast body of research suggests that the primary motor cortex is involved during motor imagery. This raises the issue of inhibition: Given the role of the motor system in providing the multisensory content of motor imagery, how is it possible for motor imagery not to lead to motor execution? Bach et al. (2022, this issue) suggest that the motor execution threshold may be “upregulated” during motor imagery to prevent execution. Alternatively, it has been proposed that, in parallel to excitatory mechanisms, inhibitory mechanisms may be actively suppressing motor output during motor imagery. These theories are verbal in nature, with well-known limitations. Here, we describe a toy-model of the inhibitory mechanisms thought to be at play during motor imagery to start disentangling predictions from competing hypotheses.

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*Keywords:* motor imagery, motor inhibition, response inhibition, computational modelling, cognitive modelling

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A large body of behavioural, electrophysiological, and neuroimaging empirical evidence suggests that the motor system is involved during motor imagery (for review, see Guillot et al., 2012). This raises the "problem of inhibition of execution" (Jeannerod, 2001): Given the role of the motor system in providing the multisensory content of motor imagery, how is it possible for motor imagery not to lead to motor execution? It has been proposed that this may be achieved by modulating (e.g., upregulating) the execution threshold (e.g., Bach et al., 2022, this issue). Alternatively, parallel inhibitory processes may prevent execution during motor imagery (Berthoz, 1996; Guillot et al., 2012). These proposals are formulated as verbal theories, with well-known limitations (Smaldino, 2020; van Rooij & Blokpoel, 2020). Notably, these theories are insufficiently specified at the algorithmic level, and can be implemented in several formal models whose predictions may concur or conflict. Here, we describe an algorithmic toy-model of inhibitory mechanisms presumably at play during motor imagery and use it to clarify the predictions from competing hypotheses.

The toy model provides a simplified overarching description of how the motor system is involved over time during motor imagery, roughly corresponding to the activity of populations of excitatory and inhibitory neurons (MacDonald et al., 2014, 2017). In its current formulation, this toy model is not equipped to distinguish between different forms of inhibition occurring at the cortical, subcortical, or spinal levels (Guillot et al., 2012). The overall model structure is adapted from the activation threshold model of response inhibition (MacDonald et al., 2014, 2017). One important difference with the activation threshold model, however, is that here the overall level of *activation* is modelled throughout the entire trial to account for both reaction times (i.e., the time it takes to prepare and initiate execution/imagery) and movement times (i.e., the time it takes to execute/imagine an action) (Figure 1). In line with the “threshold upregulation hypothesis” of Bach et al. (2022, this issue), we expect *inhibition* to modulate the motor execution threshold during motor imagery.
<figure><img src="image_2.png" /><figcaption>Figure 1. Schematic overview of the model and its predictions. Left: the black line represents the average value of the activation function over the time course of a trial, grey lines represent the value of the activation function in each single trial. Reaction time is defined as the time at which the activation function crosses the threshold for motor execution (in executed trials) or for motor imagery (in imagined trials). Imagined and executed movement times are defined as the time “spent” above this threshold (i.e., the difference between the offset and the onset). Right: Distributions of reaction times and imagined movement times generated by the model's architecture. Between-trial variability in reaction and movement times is caused by adding a slight amount of Gaussian noise in the model's parameters.</figcaption></figure>

The overall level of activation is modelled as a rescaled lognormal function defined as:

$$f(t; A, \mu, \sigma) = A \cdot \exp\left[-\frac{(\ln t - \mu)^2}{2\sigma^2}\right], \quad t > 0 \qquad (1)$$

where A and μ respectively denote the peak amplitude and peak latency of the function; σ corresponds to its width. From a neurophysiological perspective, A is thought to result from
the temporal and spatial summation of excitatory inputs onto the alpha motoneuron pool, and $\sigma$ is thought to reflect the speed of neuronal firing (lower values of $\sigma$ imply more time to attain the peak, ceteris paribus) (Figure 1, left panel). From a psychological perspective, during motor imagery, $A$ may be related to the vividness of motor imagery percepts, $\mu$ may be related to the speed at which these mental percepts are established, and $\sigma$ may be related to their duration. The model also assumes two threshold parameters, for motor imagery (parameter $m$) and for motor execution (parameter $r$) respectively, to account for the onset and duration of imagined or executed movements. The value of the motor imagery threshold is expressed as a fraction of the motor execution threshold. We adapt the idea of a dual threshold from recent models integrating action and decision-making (Dendauw et al., 2022; Servant et al., 2021).

Analytic solutions for both predicted response time (RT) and movement time (MT) can be derived from our toy model. If the peak amplitude $A$ of the activation function is larger than the motor imagery threshold $m$, which is a necessary condition of our model to produce imagery, RT and MT for imagined trials are given by:

$$RT = \exp\left(\mu - \sigma\sqrt{2\ln(A/m)}\right) \qquad (2)$$

$$MT = 2 \exp(\mu) \cdot \sinh\left(\sigma\sqrt{2\ln(A/m)}\right) \qquad (3)$$

This preliminary formulation allows assessing the influence of modulating the motor execution threshold, and therefore confronting more explicitly the mechanism proposed by Bach et al. (2022, this issue) with the data. Consider Figure 2 and the interactive application linked in the “Code availability” statement. Upregulating (downregulating) the motor execution threshold will necessarily increase (decrease) the RT and decrease (increase) the MT. Therefore, longer RTs and MTs of executed or imagined responses following imagined responses observed in the action mode switching experiments (e.g., Bart et al., 2021c, 2021a,
2021b; Rieger et al., 2017) cannot be explained by an inter-trial modulation of the motor execution threshold. Similarly, modulating the amplitude (i.e., the height of the activation function) alone cannot account for such effects. However, joint modulations of the amplitude and the curvature, or more parsimoniously, modulations of the peak time, can account for these effects. Indeed, all other things being equal, increasing the peak time will increase both the predicted RT and MT (Figure 2). Therefore, this model tells us that the aftereffects observed in action-mode switching experiments cannot be accounted for by a modulation of the motor execution threshold but that they are compatible with a “shift” in the peak time.<fnref n="1" /> In other words, inhibition in the previous trial may slow down the accumulation of excitatory input in the next trial, rather than “modulating the execution threshold”, as suggested by Bach et al. (2022, this issue). It should be noted that a full parameter recovery study and a more extensive application of this model to empirical data is ongoing.

<footnote n="1">¹ This argument holds for any activation function that increases until a certain point in time and then decreases, and therefore is not specific to the lognormal activation function.</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 2. Impact of varying the peak time (left) or the motor thresholds (right) on the reaction time and movement time. Left: the impact of having a later peak time (represented by the dark blue density) as compared to a smaller peak time (represented by the light blue density) will increase both the reaction time (represented by the length of the leftmost horizontal arrow), and the movement time (represented by the length of the rightmost horizontal arrow). Right: the impact of having a higher motor threshold (here the dark blue horizontal dotted line), as compared to a lower motor threshold (here the light blue horizontal dotted line), will increase the reaction time (represented by the length of the leftmost horizontal arrow) but decrease the movement time (represented by the length of the rightmost horizontal arrow).</figcaption></figure>

Beyond providing an explanation of extant data, this model can be used to generate novel predictions, for instance about motor imagery strength or vividness. In the same way that the evidence accumulated during a decision-making task has been suggested to reflect sensory vividness (Pereira et al., 2022), the maximum value (or the integral of the surface above the threshold) of the activation function may be predictive of the vividness of subjective percepts associated with motor imagery. In addition, the value of this function throughout the
trial could also be related to modulations of EMG activity or cortical excitability recorded during motor imagery. This could be assessed in future studies combining the action-mode switching paradigm with EMG measurements and introspective scales assessing the vividness of motor imagery percepts.

In summary, our goal with this proposal is to help disambiguating the description of the mechanisms that prevent motor activation to trigger execution during motor imagery. What do we mean by stating that motor imagery is accompanied by a "subliminal activation of the motor system"? Is it subliminal because multiple processes compete with one another, and the result of this computation is a sub-threshold activation? Or is it because the motor execution threshold is modulated? If, so, how (i.e., by which processes) is it modulated? How is this implemented (e.g., at the neural or spinal level)? Much remains to be discovered; as a step in these directions, we provided a simple framework for clarifying some of these verbal descriptions, with the hope of stimulating future discussion and a detailed characterisation of the cognitive and neural mechanisms involved in preventing motor execution during motor imagery.

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Code availability

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The code underlying a preliminary version of the model and the Shiny application is available at: https://github.com/lnalborczyk/motor_imagery_inhibition_model. The interactive Shiny application is available at: https://barelysignificant.shinyapps.io/motor_imagery_inhibition_model/.

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Acknowledgements

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We want to thank Camille Grasso for insightful comments at various stages of the present research. This work, carried out within the Institute of Convergence ILCB (ANR-16-CONV-0002), has benefited from support from the French government (France 2030), managed by the French National Agency for Research (ANR) and the Excellence Initiative of Aix-Marseille University (A*MIDEX). A CC-BY 4.0 public copyright license (https://creativecommons.org/licenses/by/4.0/) has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant's open access conditions.
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