Volition is a crucial aspect of consciousness. How and when do we decide to act? There is no doubt that neural activities in our brain generate and initiate voluntary intentions. However, the gap between physical neural activities and abstract volitions remains largely unresolved [1,2,3,4]. Computational approaches have been used to understand human behavior and volition. For example, the free energy theory suggests that human behaviors are driven to minimize ambiguity in the environment [5, 6]. Learning the causal relationships in the environment, especially between one’s own actions and events in the external world, is essential for humans to establish the sense of “self” and optimize their behaviors when interacting with the environment. The subjective feeling of being able to voluntarily control one’s own actions, and through them, control external events, is called the sense of agency. There is evidence that humans start to learn the causal relationship between their actions and sensory consequences from a very early stage of development. For example, as early as 10 weeks old, infants show an increased frequency of foot thrusts when a cord is looped around their ankle and hooked to an overhead mobile, compared to when the mobile is moved similarly by the experimenter instead of the cord [7]. This well-known phenomenon has been widely used to study how infants acquire the sense of self through actions [8,9,10,11].

The sense of agency is largely examined in human adults because they possess well-established introspective abilities to provide subjective reports, as well as perform implicit measures tasks such as intentional binding and sensory attenuation [12]. It is more challenging to study the sense of agency in younger populations, such as infants and children. Many previous studies attempted to observe changes in infants’ behaviors when their movements produce conjugate sensory feedback to infer if they can learn about their control (i.e., the extent to which they can cause changes) in the environment [7, 13, 14]. However, there is a lack of solid evidence linking these simple behavioral pattern changes with the sense of agency [15]. Actions contain rich information regarding one’s intentions and cognitive processes. However, traditional methods of action analysis fail to abstract such information efficiently. For example, a previous study showed that movement speed increased when people had a low level of control over the stimuli [16]. Another study found that action selection, including movement speed, acceleration, and turns, varies depending on the uncertainty of control in the environment [17]. However, the action analyses used in previous studies—such as movement speed, number of turns, and acceleration—are not sufficient to predict the sense of agency.

A recent study reveals a new understanding of the sense of agency, emphasizing that this sense arises not merely from sensory feedback but from actively formed informative action policies [18]. Notably, individuals form high-level, low-dimensional “action plans” that represent sequences of actions to infer their degree of control over environmental changes. Utilizing deep learning techniques, particularly transformer-LSTM-based autoencoders, researchers have successfully captured these action plans. This innovative approach reveals the geometrical and dynamical properties of these plans, which accurately predict individuals’ behavioral responses in agency tasks. Specifically, by calculating the Euclidean distance and the dynamics (i.e., the difference between two adjacent time points) between two action plans (e.g., one for hand movement and another for the displayed movement), it is possible to predict which stimulus participants are most likely to feel control over. Importantly, predictions based on action plans are more accurate than those based on micro-movements [18]. Moreover, Change et al. (2024) also showed that the dimensionality of the action plan (i.e., action plan diversity) in each trial varied depending on the actual level of control over the target and the participant group (i.e., healthy controls or patients with schizophrenia), indicating that the abstracted action plan may be useful for measuring the sense of agency through actions. In summary, in contrast to previous research focusing on oversimplified aspects of behavior such as movement frequency, speed, and repetition ratios, the study based on action plans offers a more sophisticated method by capturing an internal representation of action policies. These policies non-linearly generate the complex high-dimensional motor sequences that inform our sense of agency. Therefore, this method is a promising approach to uncovering the link between actions and the sense of agency.

In the original study conducted by Chang et al. (2024), the dimensionality of action plans was calculated to quantify action plan diversity. Specifically, the mouse trajectories of each participant were used to train a model that could capture patterns in the trajectories. The hidden layer of the model was sufficient to recover each trajectory (see Methods), indicating that the movement patterns of each individual were compressed into the hidden layer space without loss. The index of action plan diversity was then calculated using principal component analysis to measure the dimensionality of each movement in the hidden layer space for each trial. The greater the action plan diversity, the more frequently participants changed their movement patterns in each trial. The study by Chang et al. (2024) reported that patients with schizophrenia exhibited lower action plan diversity and, more importantly, their action plan diversity did not vary with the actual level of control, compared to healthy controls [18]. This finding aligns with reports that patients with schizophrenia show an impaired sense of agency [19,20,21,22,23,24,25,26]. Therefore, action plan diversity may be a valuable indicator for revealing the relationship between actions and the sense of agency.

In the present study, we acquired a dataset of 167 children aged 6–16 years, focusing on their finger movements on a touchpad and their task performance in detecting control over multiple moving objects from a previous study [27]. In the experimental task, participants were asked to move their index finger on a touchpad to trigger the motion of three dots on the screen and to identify the target dot they felt was (partially) controlled by their finger movements. The detection accuracy of the target dot was used as a measure of the sense of agency [12]. The original study reported that the detection accuracy in the 5–6-year-old group was significantly worse compared to older groups, and there was a negative correlation between detection threshold and age among all participants (r = − 0.358, p < 0.001) [27]. This study was the first to examine the sense of agency during continuous movements in school-age children. However, a limitation of this task is that it requires attentional control over the three dots: either paying attention to all of them simultaneously or switching attention among them. Although this limitation can be mitigated by allowing sufficient time for exploration, it is still difficult to eliminate completely. We suggest that by conducting action analyses, it is possible to observe the sense of agency through actions while avoiding the potential influence of other cognitive functions, such as attention.

The current study aims to examine whether action diversity in children changes depending on their actual level of control and whether this diversity is linked to individual sensitivity of the sense of agency. We trained a model employing a transformer-LSTM-based autoencoder for each child using their finger movements, ensuring that the model could accurately restore the trajectories from a low-dimensional action space (see Methods and Chang et al., 2024 for details of the model). In the experimental task described above, participants typically generated patterns characterized by long trajectories. Movement-to-movement raw trajectory data alone cannot effectively capture these behavioral profiles. Converting these long trajectories into time series embeddings is necessary to represent the relationships between action plans. The autoencoder approach is useful for such a purpose. Furthermore, the action plan diversity was calculated for each trial to measure the dimensionality of the action plan space (see below). Chang et al. (2024) demonstrated that action plan diversity is sensitive to the level of control. In the present study, we aim to examine its relationship with the sense of agency both within and between participants.