The ability to perceive and remember the world changes radically throughout the first twenty years of life (Aslin and Smith, 1988; Schneider and Pressley, 2013). Older children and adolescents are better able to understand and interpret the world around them, and anticipate upcoming situations (Carpendale and Lewis, 2006; Casillas and Frank, 2017; Richardson and Saxe, 2020). Much of this increase in ability can be attributed to greater familiarity with the events that they are likely to encounter (Schneider and Pressley, 2013). For instance, children have better memory for situations that they have expertise in (e.g. Chess; Chi, 1978). Here, we characterize changes that occur in the brain’s response to complex naturalistic stimuli during this period of the acquisition of structured knowledge about the world.

Previous research has examined developmental changes in neural activity when learning principled relationships between items in a laboratory setting (Brod et al., 2017). Children, however, normally acquire and organize novel information about their world over the course of weeks, if not years (Nelson, 1986). Naturalistic narrative stimuli provide a tool for probing this complex, real-world knowledge that children have acquired across repeated experiences and can deploy automatically (Cantlon and Li, 2013; Lerner et al., 2019; Petroni et al., 2018; Moraczewski et al., 2018; Moraczewski et al., 2020; Richardson et al., 2019; Richardson and Saxe, 2020; Vanderwal et al., 2020).

We utilized functional magnetic resonance imaging (fMRI) data acquired while children and adolescents between the ages of 5 and 19 watched a short narrative cartoon that contained both social and emotional content (Alexander et al., 2017; Petroni et al., 2018). Although cartoons do not physically replicate the characteristics of everyday perception, there is evidence to suggest that children respond similarly to the social cues in cartoons and live action videos (Han et al., 2007). Analyzing brain responses to complex stories or movies is challenging, since we do not yet have models that can predict brain-wide responses to these types of stimuli, especially given the fact that the representation of meaning likely changes with age (Clark, 1973). A model-free approach to assess brain responses to this kind of stimuli is inter-subject correlation (ISC), which measures the similarity of brain responses in a brain region across movie viewers (Cantlon and Li, 2013; Lerner et al., 2019; Petroni et al., 2018; Moraczewski et al., 2018; Moraczewski et al., 2020). ISC within an age group has generally been found to increase with age, possibly due to more mature engagement with the content, greater shared knowledge about the world, or more exposure to widely used cinematic conventions (Cantlon and Li, 2013; Lerner et al., 2019; Moraczewski et al., 2018; Moraczewski et al., 2020). We therefore measured ISC in parcels throughout the cortex (Schaefer et al., 2018) to assess if processing becomes more similar with age. In line with previous studies, we hypothesized that the magnitude of ISC would increase with age, although decreases have been found in studies in other modalities (Petroni et al., 2018).

ISC has also been measured between age groups to assess the similarity between the responses of children and adults (Cantlon and Li, 2013; Lerner et al., 2019; Moraczewski et al., 2018). If a child is more similar to adults, they are considered more mature, and therefore likely to have better academic or social abilities (Cantlon and Li, 2013; Moraczewski et al., 2018). This previous work is consistent with the idea that children are simply noisy versions of adults, and the noise level decreases with age. We test an alternative hypothesis: that children and adolescents have different average response timecourses, corresponding to different age-related interpretations of the movie. We therefore employed a new across-group ISC measure that allows for comparisons across age groups while controlling for the degree of similarity within each age group. This allowed us to determine whether there was a consistent shift in the group-level temporal patterns of activity, independent of changes in within-group consistency. We hypothesized that in default mode network regions responsible for story interpretation and self-referential thought, even where within-age ISC magnitude does not change, the pattern of activity representing the movie will change across development, just as the semantic interpretation of movies changes with age (Nelson, 1986; Raichle et al., 2001).

How can we characterize the changes that are occurring in response timecourses? Although some knowledge of the hierarchical structure of the events that compose a narrative develops in infancy, the ability to reliably notice these events does not mature until at least the teenage years (Yates et al., 2021; Zacks and Tversky, 2001; Zheng et al., 2020). It is likely that the characterization of events changes with age. We therefore asked both children and adults to subjectively report where they believed meaningful scene changes occurred in the narrative. We hypothesized that although there would be no change in the behaviorally reported location of these coarse-grained narrative segments, the neural representation of events along the cortical hierarchy would change with age.

Recent work in adults has shown that the neural activity evoked by narratives are characterized by stable patterns of responses, and that the moments of transition between stable patterns in association areas correspond to these meaningful boundaries between ‘events’' in a continuous perceptual stream (Baldassano et al., 2020; Baldassano et al., 2018; Chen et al., 2017). These event patterns are driven in part by learned schematic scripts about common experiences in the world, such as the sequence of events that characterize a visit to a restaurant (Baldassano et al., 2018).

We hypothesized that with age, the strength of schematic event representations, that are able to generalize across different instances of similar events, will increase due to more experience with different exemplars. These kinds of representations should be stored in default mode regions such as the medial prefrontal cortex and posterior medial cortex (PMC). Following previous research in adults, we assessed the ability of children of different ages to segment their world into discrete chunks with a Hidden Markov model (HMM; Baldassano et al., 2020; Baldassano et al., 2018; Lee et al., 2021). We predicted that activity patterns will be more consistent among older subjects and become more stable across timepoints within events, providing a better fit to the HMM event segmentation model. If we do not find a general improvement in event model fits with age, this will refute the hypothesis that the schematic event representations that support event models are strengthened with age.

It is also likely that the timing of event transitions changes as a function of age. Activity in regions responsible for representing the viewpoints of others occurs earlier during the second presentation of a movie in children between six and seven years then in children between three and four years (Richardson and Saxe, 2020). Here we test whether, on an initial viewing of a naturalistic video, the timing of event transitions varies with age. Unlike previous work that considered anticipation at only a fixed offset of two seconds (Richardson and Saxe, 2020), the HMM-derived transition timings provide a flexible approach for detecting timing shifts between age groups that can vary across brain regions with different processing timescales (Baldassano et al., 2020; Hasson et al., 2015) and can vary across timepoints throughout the movie. In line with the idea that schematic event representations improve with age, older adolescents should be able to anticipate events further into the future due to their increased experience with the world.

Another characteristic of adult responses to narrative movies is the robust hippocampal activity evoked by boundaries between events (Baldassano et al., 2020; Ben-Yakov and Henson, 2018). Age-related changes in this signal have been previously observed in older adults, with response decreases between ages 18 and 88 in the posterior hippocampus (Reagh et al., 2020). It is therefore possible that development of event-structured responses in the cortex is mirrored by changes in the hippocampus’s ability to respond to event transitions. We predicted that the magnitude of the hippocampal response to event boundaries would increase with age, in line with a model of maturation wherein the ability to encode the unique episodes of daily experience increase into middle age and then decreases with senescence (Grady, 2012; Reagh et al., 2020). However, should we find a decrease in the hippocampal response to event boundaries with age, this would provide support for the idea that younger children may focus on the episodic encoding processes responsible for encoding the specific details of events as they work towards creating more stable schematic event representations (Keresztes et al., 2018; Maril et al., 2010).

We sought to uncover whether and how the neural representation of naturalistic stimuli change with age between 5 and 19 years. To accomplish this, we used a large, publicly available dataset (Alexander et al., 2017) of functional magnetic resonance imaging (fMRI) data recorded while children watch a short video animation with both social and emotional themes. As the data were not equally distributed with age, the results are derived from equally sized subsamples from each age group (each with an age span of approximately 3 years). To maximize the power and reproducibility of the results, our analyses were averaged across five random subsamples from each age group (See Methods; Poldrack et al., 2017).

We first measured inter-subject correlation (ISC) across all subjects, as well as within the youngest (5–8 years) and oldest (16–19 years) age groups, as a general measure of story comprehension and engagement (Figure 1—figure supplement 1 and Figure 1—figure supplement 2; Nastase et al., 2019). To statistically examine changes in this measure due to age, we compared ISCs for the youngest and oldest age groups, calculated within parcels (Schaefer et al., 2018). We found (Figure 1) that ISC increases with age in low level sensory regions such as the auditory cortex, and decreases with age in some higher level association regions such as the retrosplenial cortex (RSC) portion of the PMC. To ensure that the increases in ISC with age are not due to differences in the level of noise between the groups, we measured the relationship between the framewise displacement of each child in the Youngest group and their ISC with the other subjects. There was no relationship between framewise displacement and ISC in any of the parcels where ISC increased with age, indicating that motion did not drive the result in these parcels (all q’s>0.05).

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In most of the cortex, there is no difference in ISC magnitude as a function of age, which could indicate that children of all ages engage similarly with the movie (Petroni et al., 2018). It is also possible, however, that responses are highly consistent within each age group but differ systematically across age groups. For instance, it is possible that the most salient features of the video change with age, eliciting responses that are predictable within an age group but different across groups. To get a pure measure of across-group similarity that is insensitive to within-group variation, we use a novel measure called between-group ISC (ISC_b; see Appendix 1) (Figure 2a). To calculate the correlation between groups, ISC_b is computed by dividing the between-group (Youngest to Oldest) correlation by the geometric mean of the within-group correlation (Figure 2b).

Figure 2

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Eighty-one of the 100 (Schaefer et al., 2018) parcels tested show significant age-specific response timecourses including default mode regions, such as the temporoparietal junction (TPJ) and PMC (Raichle et al., 2001), and language and concept sensitive regions, such as the dorsolateral prefrontal cortex (dlPFC) and inferior temporal cortices, and orbitofrontal cortex (Binder and Desai, 2011; Ralph et al., 2017; Figure 2c).

This result demonstrates that responses are changing substantially with age in many brain regions, but does not indicate how these responses are changing. One possibility is that the interpretation of the events and scenes in the narrative changes with age. We therefore assessed when adults and children, age-matched to the fMRI sample, report that an event in the narrative has finished, and a new event has begun. There are small but significant differences in the timing of the event boundaries marked by adults and children (ISC_b = 0.97, p=0.009). This discrepancy was true for both the younger and older children, separated by a median split based on age (adult-older children ISC_b = 0.95, adult-younger children ISC_b = 0.91, both p’s<1 × 10^–5). The timing of event boundaries was also slightly, but significantly different between the younger half and older half of the children (ISC_b = 0.96, p=0.01, Figure 3). Given this overall similarity in behavior, we next ask whether there are differences in how these events are represented and tracked in the brain.

Figure 3

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In adults, movie responses are often characterized by rapid transitions between stable periods of brain activity, corresponding to meaningful events in the narrative (Baldassano et al., 2020; Chen et al., 2017). To identify the extent of this temporal clustering in our data, and the timescale of this clustering, we use a Hidden Markov Model (HMM) (Baldassano et al., 2020). Fitting the HMM separately to the Youngest and Oldest groups, we found that the HMM was able to identify event structure in at least one of the groups (Youngest and/or Oldest) for 70 out of our 100 parcels (Figure 5—figure supplement 1). In these 71 parcels, we found that the optimal timescale (optimal number of events) was highly correlated between the Youngest and Oldest ages (r=0.78, RMS difference between groups = 12.3 s), showing increasing timescales from sensory regions to higher level association regions (Figure 5—figure supplement 2). This replicates previous research showing increasing event timescales across the cortical hierarchy (Baldassano et al., 2020; Geerligs et al., 2021; Hasson et al., 2015), and shows that this timescale hierarchy is present even in our Youngest (5–8 years) age group.

As all 71 parcels had similar timescales in both groups, we jointly fit an HMM (with the parcel’s optimal number of events) simultaneously to the Youngest and Oldest groups. As expected based on previous research in adults, the event boundaries from this model correspond to the boundaries subjectively placed by human raters in association regions, such as PMC, TPJ, and precuneus (Figure 5—figure supplement 3; Baldassano et al., 2020). The jointly fit HMM produced an ordered set of event patterns which were shared across both groups (Figure 4a). We then tested this event model in held-out subjects, allowing us to measure in each group: (1) the model fit (log likelihood on held-out data), indicating the extent to which brain responses could be explained as an ordered sequence of these stable event patterns (Figure 4b–c), and (2) the timing of event transitions. We hypothesized that the Oldest subjects would have a better model fit than the Youngest subjects, indicating increased stability and reliability of event representations. We also hypothesized that the Oldest group would exhibit earlier event transitions than the Youngest group, due to predictive, anticipatory, or preparatory processes (Carpendale and Lewis, 2006; Lee et al., 2021; Richardson and Saxe, 2020).

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As hypothesized, the neural activity of the Oldest group can be better represented by an event model in most of the parcels that significantly differ with age. These regions include sensory areas, such as the auditory and visual cortex, as well as the left parietal operculum and dlPFC (Figure 5). In sensory regions, the largest change in event model-fit occurs between children 5–8 years and those 8–11 years (Figure 5, side panels). The left parietal operculum increases in model-fit through approximately age 13. These developmental trajectories hold not just for these regions' optimal event timescale, but also across a wide range of event timescales (Figure 5, side panels). Surprisingly, the HMM exhibits better model fits in the PMC, including both the posterior cingulate cortex (PCC) and RSC, for the Youngest group than the Oldest group. These results partially reflect changes in across-subject consistency as identified by the model-free ISC analysis (Figure 1), but also show effects in additional regions including the dlPFC.

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We also expected that event boundaries would occur earlier in time with age, because adolescents are more experienced with real-world schemas and can thus predictively represent upcoming situations. This hypothesis was borne out in sensory regions including auditory and visual cortex, as well as left lateralized associative regions including the left precuneus and left dlPFC (Figure 6a). Sensory regions are predictive at shorter time scales, consistent with their quicker processing speed (Baldassano et al., 2020; Hasson et al., 2015; Lee et al., 2021), while the higher level association regions anticipate upcoming events on average up to tens of seconds into the future. This anticipation increases rapidly across ages 5–11 in the left dlPFC and left precuneus, with a more variable trajectory in left postcentral sulcus (side panels). However, as was seen for model-fit, this trend reverses in the right ventral temporal cortex where the Youngest children anticipate upcoming events slightly sooner, and anticipation decreases almost linearly with age.

Figure 6

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To determine whether developmental changes in the timing of event transitions reflect an increased level of anticipation in the Oldest group, or, alternatively, a delayed response in the Youngest group, we compared the HMM-derived event boundaries to event transitions behaviorally identified by an age-matched group of children. Across the cortex we find that the Oldest ages generally anticipate event shifts, while the Youngest ages lag behind them to some degree (Figure 6b).

Finally, we asked whether hippocampal responses differ as a function of age at the event boundaries reported by our age-matched sample of children. We found a robust response to event boundaries in all ages (all p’s<0.0001 for each age group’s response at time 0, see Figure 7a, right side). This response decreases with age between 5 and 19 years old (r=0.–16, p=1 × 10^–3, N=414; Figure 7a). As previous research has indicated that the anterior and posterior HPC may potentially respond differently to event boundaries (Reagh et al., 2020), we examined whether the anterior or posterior HPC were driving this change in boundary-driven response with age. We found that there is a significant decrease in the anterior hippocampal (aHPC) response to event boundaries with age (r=−0.18, p=2 × 10^–4, Figure 7b). We did not find a significant decrease in the posterior HPC (pHPC; r=−0.09, p=0.06, Figure 7c), nor a significant interaction between HPC subregions (p=0.7). In none of these regions was the change in boundary-related response explained by anatomical volume changes (see Methods).

Figure 7

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Using a novel measure of response similarity, we found that the processes used to perceive a dynamic narrative change from ages 5–19 throughout the majority of the neocortex and in the hippocampus. These developmental changes are characterized by increasingly event-structured dynamics in the neocortex, with periods of stability punctuated by rapid transitions, and anticipatory shifts in event timing. In parallel, hippocampal responses at event boundaries decrease in magnitude with age. These changes suggest that the brain’s strategy for narrative processing shifts from encoding each event as a novel episode in the hippocampus to activating and maintaining generalized event knowledge in the cortex, storing only episode-specific information into long-term memory.

Older ages exhibit more across-subject synchrony in auditory cortex (Figure 1), consistent with prior work showing developmental increases in ISC evoked by other similarly highly produced animated movies (Moraczewski et al., 2018). These results are likely related to more strongly correlated semantic processing in adolescents, resulting in top-down increases in auditory understanding (Franchak et al., 2016; Kirkorian et al., 2012). Unfortunately, as eye movements were not recorded, we cannot establish the relationship between eye movements and synchrony in sensory cortex (Alexander et al., 2017). The growth of coordinated responses with age may also drive the stronger event models in these regions (Figure 5). The ISC_b analysis (Figure 2) found relatively small but significant shifts in the group-level responses in sensory regions, and auditory responses shifted slightly earlier in time with age (Figure 6). This indicates that the age-related changes in sensory regions are not only largely characterized by increasing synchrony across subjects, but also (to some extent) a shift from a more child-like to more adult-like timecourse of responses.

Theory of mind regions responsible for representing the mental states of others (Gallagher et al., 2000; Saxe and Kanwisher, 2003), including TPJ (Samson et al., 2004), develop immensely in the age range studied (Sebastian et al., 2012). We found very large age-related changes in the functional responses of TPJ (Figure 2). Older children, due to stronger schematic event representations, are likely able to construct better internal models of social and emotional structure (Dumontheil et al., 2010; Pavias et al., 2016), allowing them to form more cohesive models of the world. Previous work found that responses in a network of regions, including TPJ, shift approximately 2 s earlier when 6–7 year-olds watch a movie for a second time (Richardson et al., 2019). In the right posterior-superior temporal sulcus (part of TPJ as defined by Dufour et al., 2013) and in the TPJ-adjacent left postcentral sulcus, we found that neural representations in 16–19 year-olds can be as far as 20 s ahead of those for 5–8 year-olds on the first viewing of a movie clip. This reveals a protracted development in this region, likely due to continued growth in the ability to proactively represent the minds of others (Dumontheil et al., 2010; Pavias et al., 2016), or possibly due to a growth in attentional abilities in this region more generally (Corbetta and Shulman, 2002; Downar et al., 2001).

We also found age-related shifts in dlPFC (Figure 2) and increasing anticipation from ages 5–11 (Figure 6). This region, along with other regions showing age-related shifts such as the inferior temporal cortex and orbitofrontal cortex, are a part of a network responsible for processing conceptual knowledge about the world (Binder et al., 2009). The age-driven differences in story responses may therefore be partially due to changes in the understanding of the people, actions, and culture presented in the movie (Binder et al., 2009). The dlPFC is also critical for working memory, or the ability to keep recently learned information readily accessible (Curtis and D’Esposito, 2003). The dlPFC exhibits an improvement in event structure with age (Figure 5), which may be due to the ongoing development of working memory (Kwon et al., 2002), a cognitive function necessary to chunk incoming information into discrete events (Clewett et al., 2019). The changes in dlPFC may also be related to development of theory of mind as discussed above, as it overlaps with the theory of mind network identified by Richardson et al., 2019.

In the PMC event model fits worsened with age (Figure 5), and in the more posterior RSC portion, reliability also decreased with age (Figure 1). This finding contrasts with our hypothesis that schematic event representations should increase with age in regions such as the PMC. One possible reason for this discrepancy is that PMC performs additional processing due to its lack of strong schemas about observed events; Keidel et al., 2018 found that the PMC (RSC) had significantly more activity in response to movies when the narrative topic was new than when it was a continuation of themes from the previous clip. Younger children may therefore more consistently rely on this region to understand a relatively more novel environment. RSC is also known to be responsible for processing visual scenes and is fully developed by age 7 (Epstein and Baker, 2019; Meissner et al., 2019). It is therefore possible that young children are more consistently reliant on spatial scene representations as they process an ongoing narrative. PMC has an intrinsically long timescale for information processing, one that emerges in infancy, which may thus support strong event models in children as young as five (Hasson et al., 2015; Stephens et al., 2013; Yates et al., 2021). However, we did find both age-related shifts in the pattern by which the narrative was represented bilaterally (Figure 2) and increased anticipation in the left PMC (Figure 6). These findings in the more dorsal portion of PMC are notable because there were no significant age-related differences in this region, suggesting that these shifts are not due to overall changes in attention or executive function. This result is in line with our hypothesis that increasing age is associated with more anticipation of higher level themes, and different ages likely process these themes with a different overall pattern of activity.

We found small, but significant changes in the timecourse of narrative processing in the medial prefrontal cortex (Figure 2). This region has been previously implicated in tracking schematic knowledge during movie-watching (Baldassano et al., 2018; van Kesteren et al., 2010), suggesting that schematic responses evolve across this age range. However, we surprisingly did not find a change in the strength of event representations with age. This may be due to an overall low level of ISC in this region, either due to idiosyncratic cognition, inconsistent neuroanatomy across subjects, or noisy signals in this region (Figure 1—figure supplement 1), making it challenging to study this region using the group-level approaches in this study. This result is not entirely unexpected given that previous studies have found low levels of ISC in this region (Baldassano et al., 2020), especially in developmental populations (Lerner et al., 2019; Moraczewski et al., 2018). Furthermore, as has been found previously, it may be that only a subregion of the medial prefrontal cortex represents events (Baldassano et al., 2020; Brod et al., 2017; Masís-Obando et al., 2022; van Kesteren et al., 2010).

The HPC was found to reliably respond to the boundaries between events (Figure 7). This result extends evidence that the HPC tracks event changes in adults to children as young as five (Ben-Yakov and Henson, 2018; Baldassano et al., 2020; Reagh et al., 2020). This boundary-triggered response decreased with age across our sample, which is surprising given that adolescents have stronger event models and event anticipation in most of the cortical regions with significant effects. The decrease in HPC responsiveness with age may be a signature of age-related increases in the reliance on schematic representations of the environment which are stored in the cortex, thus decreasing the reliance on the HPC to encode newly encountered events (Sekeres et al., 2018). The HPC may thus have a more robust signal in young children who need to focus on encoding the specific details of events as they use them to extract commonalities across experiences (Keresztes et al., 2018; Maril et al., 2010). Likewise, the HPC has been shown to have increased activity during memory consolidation, and in response to novel events (Nadel and Moscovitch, 1997; Stern et al., 1996). It is possible that younger children found the events in the movie generally more novel, or needed to spend more effort consolidating them due to their impoverished schematic representations. However, if this was truly a novelty-related response we would have expected to see a greater age-related change in the pHPC which has been shown to be more responsive to novelty (Stern et al., 1996).

Interestingly, previous research in older adults also found a decrease in the HPC response to event boundaries (Reagh et al., 2020). However, in this previous research the decrease with age was seen in the pHPC (Reagh et al., 2020). Here, we find that only the response in the aHPC significantly decreases with age. This shift in age-related decline may be due to the developmental changes in the mnemonic roles along the long axis of the HPC (Demaster and Ghetti, 2013; Langnes et al., 2019). The decline in the response of the aHPC may also be related to the process of semanticization (Sekeres et al., 2018), whereas older ages may be experiencing age-related mnemonic, and thus perceptual declines (Reagh et al., 2020).

Jointly fitting an HMM across children of different ages was sensible because we found that the optimal event timescales in all regions with high model fits were similar between the Youngest and Oldest groups. This was not a foregone conclusion, because infants under one year of age segment ongoing narratives into far fewer events than adults (Yates et al., 2021) and behaviorally, we found small, but significant differences in segmentation behavior between adults and children, and children of different ages. However, at least neurally, children therefore develop enough narrative comprehension to segment the story in an adult-like manner by age five.

Although we largely attribute the better segmentation ability and prediction to adolescents’ greater experience with the world in general, it is possible that our Oldest cohort also had more previous experience watching the specific stimulus used (a clip from "Despicable Me"), as it was first released when they were young children themselves. Alternatively, younger children may have had more recent exposure to the film because it was more age appropriate for them, and this may have elicited a retrieval induced increase in HPC activity. Future research should collect data on prior stimulus exposure or aim to use stimulus-naive subjects for better control. Furthermore, we cannot distinguish whether the improvements that we see with age are due to the overall maturation of the brain or the increased life experience that underlies schematic knowledge. Additionally, although we chose to use this dataset because it is one of the few large databases of movie-watching in a developmental population, the subjects recruited for this study had a high prevalence of psychopathology and limited socioeconomic diversity. Future work should generalize our findings to more diverse and representative subject samples. Finally, our analyses require stimuli that exhibit multiple transitions between events that are at least 30 s. We were therefore unable to make use of the other movie available in this dataset or data from the short video clips used in other developmental datasets (Alexander et al., 2017; Richardson et al., 2019). Future studies should seek to generalize these results to stimuli with greater content diversity, and duration.

Our results reveal that brain responses to a narrative movie do not simply become more synchronized across children as they age, but in fact change their dynamics and timing to become more adult-like. Brain activity patterns in many regions become more cleanly structured into discrete, consistent events, and the timing of these events shifts to anticipate upcoming content in the movie. We also found that hippocampal responses to event boundaries are present in children as young as 5 years old, and that these event-boundary responses are in fact larger in children than in adolescents. This study provides the groundwork for assessing how children acquire schematic knowledge about the world and ultimately deploy that knowledge at the appropriate time.

For across-group analyses, we want to calculate the ISC between two groups of subjects, with two different sets of shared responses, g₁ and g₂, and different levels of variance, ${ϵ}_i$ and ${ϵ}_j$. To simplify the notation, we define $g_1^T{g}_1=a_1$, $g_2^T{g}_2=a_2$, $g_1^T{g}_2=a_B$, $E[{ϵ}_i^T{ϵ}_i]=b_1$ and $E[{ϵ}_j^T{ϵ}_j]=b_2$. We would like to estimate the correlation between g₁ and g₂, regardless of the difference in signal-to-noise levels between the two groups. That is, our goal is to compute:

If we know the ISCs for each group, from equation 1:

and

where $N$ is the number of subjects in each group, and the ISC between the two groups is:

$\begin{array}{cc}\hfill shIS{C}_B& =\frac{N^2{a}_B}{\sqrt{N^2{a}_1+4N{b}_1}\sqrt{N^2{a}_2+4N{b}_2}}.\hfill \end{array}$

Dividing $shIS{C}_B$ by the geometric mean of $shIS{C}_1$ and $shIS{C}_2$ allows us to calculate the between-group ISC without the $b$’s:

We start by assuming that the response timecourse for every subject consists of a response that is shared across subjects, $g$, and a subject-specific response, $ϵ$. The response timecourse for subject $i$ can therefore be represented as $s_i=g+{ϵ}_i$. For simplicity, all subject timecourses, $s$, have zero mean, we define $a$ as the norm of $g$ ($g^{T}g=a$), and we define $E[{ϵ}_i^T{ϵ}_i]=b$. If we have N subjects in total, the split half ISC ($shISC$) is:

where $f=a/b$, or the signal to noise ratio (the ratio of norms between $g$ and ${ϵ}_i$).

If we rearrange the variables in Equation 1, we find that:

We can thus calculate $pwISC$ and $looISC$ from $shISC$ as they relate to $f$ as follows. For $pwISC$ between subjects $i$ and $j$:

and for $looISC$, between one left-out subject, and all others:

Thus, for example, if there are 40 subjects, and $shISC$ is 0.5, the estimated $pwISC=0.05$ and the estimated $looISC=0.18$.

All neuroimaging data is available at: http://fcon_1000.projects.nitrc.org/indi/cmi_healthy_brain_network/sharing_neuro.html, and all behavioral data is available at: https://github.com/samsydco/HBN, (copy archived at swh:1:rev:278127d07b721c73679c11d0d1836631df778323).

1. 1. Cohen S

   (2022) GitHub

   ID 278127d. HBN.

   https://github.com/samsydco/HBN

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Author details

1. Samantha S Cohen

   Department of Psychology, Columbia University, New York, United States

   Contribution

   Conceptualization, Data curation, Formal analysis, Methodology, Software, Visualization, Writing - original draft, Writing - review and editing

   For correspondence

   samantha.s.cohen@gmail.com

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-3007-5372

2. Nim Tottenham

   Department of Psychology, Columbia University, New York, United States

   Contribution

   Conceptualization, Investigation, Methodology, Project administration, Resources, Supervision, Writing - review and editing

   Competing interests

   No competing interests declared

3. Christopher Baldassano

   Department of Psychology, Columbia University, New York, United States

   Contribution

   Conceptualization, Funding acquisition, Methodology, Resources, Supervision, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-3540-5019

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