Humans parsimoniously represent auditory sequences by pruning and completing the underlying network structure

“The fact then that many complex systems have a nearly decomposable, hierarchic structure is a major facilitating factor enabling us to understand, describe, and even ‘see’ such system and their parts” – H. Simon, The architecture of complexity (1962).

To interact efficiently with their environment, humans have to learn how to structure its complexity. In fact, far from being random, the sensory inputs we face are highly interdependent and often follow an underlying hidden structure that the brain tries to capture from the incomplete or noisy input it receives. For instance, Tenenbaum et al., 2011, proposed that learning implies building the simpler underlying relational model which can explain the data. Indeed, evidence suggests that humans can infer structures from data at different scales, ranging from local statistics on consecutive items (Saffran et al., 1996) to local and global statistical dependencies across sequences of notes (Basirat et al., 2014; Bekinschtein et al., 2009) or more high-order and abstract relationships such as pattern repetitions (Barascud et al., 2016), hierarchical patterns, and nested structures (Dehaene et al., 2015), networks (Garvert et al., 2017; Schapiro et al., 2013), and rules (Maheu et al., 2020).

At first, the extraction of local regularities in auditory streams was proposed as a major mechanism to structure the input, available from an early age since Saffran et al., 1996, showed that 8-month-old infants can use transition probabilities (TPs) - ${\displaystyle P\left(E_t|E_{t-1}\right)}$ - between syllables to extract words from a monotonous stream with no other available cues. Since then, the sensitivity of humans to local dependencies has been robustly demonstrated in the auditory and visual domain (Fiser and Aslin, 2002) without the focus of attention (Batterink and Choi, 2021; Batterink and Paller, 2019; Benjamin et al., 2021) and even in asleep neonates (Benjamin et al., 2023; Fló et al., 2022). Moreover, it is not limited to adjacent elements but can be extended to non-adjacent syllables - ${\displaystyle \mathrm{P}\left({\mathrm{E}}_{\mathrm{t}}|\mathrm{X}{\mathrm{E}}_{\mathrm{t}-2}\right)}$ - that could account for non-adjacent dependencies in language (Peña et al., 2002).

However, the computation of TPs between adjacent - $P\left(E_t|E_{t-1}\right)$ - and non-adjacent elements - $P\left(E_t|{XE}_{t-2}\right)$ - seems too limited to allow the extraction of higher-order properties without an infinite memory that the human brain does not have. Network science - an emerging interdisciplinary field - thus proposed a different description to characterize more complex streams (Lynn et al., 2020). In this framework, a stream of stimuli corresponds to a random walk in the associated probabilistic network. Several studies used this network approach to investigate how humans encode visual sequential information (Garvert et al., 2017; Mark et al., 2020). Schapiro et al., 2013, tested human adults with a network consisting of three communities (i.e. sets of nodes densely connected with each other and poorly connected with the rest of the graph; Newman, 2003) where transitions between all elements were equiprobable (each node had the same degree). This community structure is an extreme version of the communities and clustering properties that are often found in real-life networks, whether social, biological, or phonological (Girvan and Newman, 2002; Karuza et al., 2016; Siew, 2013). The authors reported that subjects discriminated transitions between communities from those within communities. Since local properties (TP) were not informative, this result revealed participants’ sensitivity to higher-order properties not covered by local probabilistic models. This sensitivity seems already to be in place at 6y-o (Pudhiyidath et al., 2020). Recently, Lynn et al., 2020, replicated a similar effect with a probabilistic sequential response task. They presented subjects with sequences of visual stimuli that followed a random walk into a network composed of three communities. After each stimulus, subjects were asked to press one or two computer keys, and their reaction time was measured as a proxy of the predictability of the stimulus. To explain the response pattern, the authors proposed an analytical model that optimizes the trade-off between accuracy and computational complexity by minimizing the free energy function. This model allows taking into account the probability of memory errors in the computation of the TPs between the elements of the stream. From now on, we will refer to this model as the free energy minimization model (FEMM: model D, explained below).

In this paper, we aim to merge these two lines of research and validate a model that can explain how humans learn local and high-order relations simultaneously present in sequences generated from noisy or incomplete structures. Moreover, we propose that adults do not encode the exact input but a parsimonious version based on the generalization of the underlying structure. To this end, we leveraged the community network framework and adapted it to expose adult participants to rapid sequences of sounds that followed a random walk through a network, building on the studies described above (Lynn et al., 2020; Schapiro et al., 2013), but using sparse communities with missing transitions between elements of the same community (see Figure 1). This design allows investigating whether participants are able to complete the network according to the high-order structure or whether, on the contrary, they rely on local transitions and reject impossible transitions ignoring the high-order structure. In other words, after training with an incomplete network, if new (‘unheard’) transitions are presented, are participants more willing to accept them if they belong to the community (i.e. within community transitions) than if they occur between communities? Moreover, while several papers have studied network learning in the visual domain (Karuza et al., 2019; Lynn et al., 2020; Schapiro et al., 2013), to our knowledge, it has never been tested in the auditory domain despite the better statistical learning capacities in the auditory modality (Conway and Christiansen, 2005), the sophisticated auditory sequence processing abilities observed in humans compared to other primates (Dehaene et al., 2015), and their potential importance in language acquisition. In addition, the original design was at a very slow rate, allowing for possible conscious decision to take place on the adequation of each element of the sequence to the structure. Here, we used a 4 Hz presentation, typically used in auditory sequence learning tasks, in order to force rapid processing of each element of the sequence and to be more comparable to the sequence learning literature. Finally, we compared how the different models proposed in the literature might fit our data and proposed a unified hypothesis of how any structure (local or global) might be extracted from a sequence.

Figure 1

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For this purpose, we tested three different experimental paradigms in an online task, using sequences of pure tones or of syllables (~240 adult participants tested in each paradigm). The first paradigm - full community - tested a network composed of two communities of six elements each, with all nodes within a community connected with each other (except two nodes at the border of the community to keep an equal degree for each node). In the second and third paradigms, the communities were incomplete, some connections being never presented during the exposure to the continuous sequence: In the sparse and high sparse community paradigms, respectively one and two possible edges for each node were removed. The performances in these two ‘sparse’ designs, relative to the full community design, are crucial to investigate the participants’ underlying representations of the sequences.

In each paradigm, participants were first asked to carefully listen to a continuous sequence for about 4 mn and then to press a key when they felt there was a natural break in the sequence (~2 mn). This task allowed measuring participants’ ability to parse the sequence and to compare their performances in the auditory domain with those published in the visual domain. In a following test phase, they were asked to choose between two isolated quadruplets, the most congruent with what they had heard before, during the familiarization sequence. With this test phase, we could present previously unheard transitions (‘new transitions’) and study whether participants were able to generalize the network structure (Figure 1), notably in the two incomplete networks (sparse and high sparse paradigms). These two tasks were done twice.

In the forced-choice task between the isolated quadruplets, we tested each other conditions against the familiar within community transitions (condition considered as the reference; Figure 1C). If participants did not learn the graph structure of the sequence, they had to be random in their familiarity choice between familiar within and between community transitions because all quadruplets have been presented and had the same local TPs between their elements. By contrast, if they had indeed learned the graph, their familiarity score should be below 50% denoting their preference for the familiar within community transitions (i.e. reference). The performances for the unheard transitions, which can be either within or between community transitions (i.e. new within community condition and new between community condition) relative to the reference should allow to separate the different models proposed in the literature to explain how structures are perceived. Therefore, we compared the participants’ behavior (i.e. their familiarity rating for the presented transition relative to the reference) to the predictions of different theoretical models proposed in the stream processing and graph learning literature (Figure 2).

Figure 2

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• Model A: TPs and Ngrams: Local transitions between consecutive elements - $P\left(E_t|E_{t-1}\right)$ - have been proposed as an efficient learning mechanism to structure streams of input. We tested the limits of this simple local learning computation in the presence of a high-order structure. Ngrams are similar to TP but take into account n previous items in the computation of the transition. For example, for trigrams, $P\left(E_t|E_{t-1}{E}_{t-2}{E}_{t-3}\right).$ Note that because our designs are random walks into Markovian networks, the TPs and Ngram models are identical, $P\left(E_t|E_{t-1}{E}_{t-2}{E}_{t-3}\right)=P\left(E_t|E_{t-1}\right)$. Chunking-based models, such as PARSER (Perruchet and Vinter, 1998), rely on the repetition of chunks of consecutive elements and, as TP and Ngrams, would reject any chunk with new transitions as they never occurred during familiarization.

• Model B: Non-adjacent TP: This metric is similar to the TPs but on non-consecutive items P (E_t |XE_t−2 ). We included it in our analysis because several studies have shown human sensitivity to such properties in streams (Peña et al., 2002).

• Model C: Graph communicability: This model comes from the network science literature and computes the relative proximity between nodes in the network, making it sensitive to cluster- like structures like communities. Interestingly, a recent study shows that this measure correlates with fMRI data (Garvert et al., 2017), suggesting a potential relevance in human cognition.

• Model D: FEMM: This model, recently proposed by Lynn et al., 2020 to account for community sensitivity by humans, is a trade-off between accuracy and computational complexity. It can be explained by memory errors while computing TP between elements in a stream. Participants exposed to a stream of elements reinforce the association between element i and i-1. However, errors in this process may lead participants to sometimes bind element i with element i-2, i-3, i-4.... with a decreasing probability (for a full description of the model, see Lynn et al., 2020). Mathematically, the distribution of the error size that minimizes the free energy function is a decreasing exponential (Boltzmann distribution). Therefore, the estimated mental model of TP is biased compared to the streams' objective TPs enabling participants to encode high-order structure. In more detail, the mental model is a linear combination of the TP matrix (A) and non-adjacent TPs of every order (A^∆t) with a weight of P (∆t) where ∆t is the order of non-adjacency (or size of the memory error, i.e. ∆t = n corresponds to P (E_t|X.....XE_t−n)). The estimated model can then be written as:

• ${\displaystyle \hat{A}=\sum _{\mathrm{\Delta }t=0}^{+\mathrm{\infty }}P(\mathrm{\Delta }t)A^{\mathrm{\Delta }t+1}}$

• with

• ${\displaystyle P\left(\mathrm{\Delta }t\right)=\frac{1}{Z}{e}^{-\beta \mathrm{\Delta }t}}$

• where A is the TP matrix of the graph. $\beta$ was previously estimated to 0.06 in a comparable task with human adults (Lynn et al., 2020). We therefore first used this value to test this model on our behavioral data and later confirmed this estimation with our data (see SI). In the reinforcement learning literature, the hippocampal place cells have been proposed to represent maps of probabilistic future states and reward by encoding successor representation instead of positional cognitive maps (Dayan, 1993; Stachenfeld et al., 2017). Successor representation has been formally defined as the sum of probabilistic future state and can be written SR = $\sum _{∆t}{\gamma }^{\Delta t}{A}^{\Delta t}$. This approach is very similar to FEMM with an infinite sum of all power of the transition matrix, pondered by an exponentially decreasing factor. Here, the factor is ${\gamma }^{∆t}$ with ${\displaystyle 0<\gamma <1}$ and generally $\gamma =\frac{0.85}{\lambda max}$ with $\lambda max$ the largest eigenvalue of the transition matrix (Garvert et al., 2017). This approach has been proposed to account for community perception (Pudhiyidath et al., 2022) but here we only included FEMM in our study, as the two models are identical with $\gamma =e^{-\beta }$ (with a different constant).

Another metric computing the same property but from a sequence point of view is the hitting time.

• Model E: Hitting time: This metric, also coming from network science, estimates the distance between two nodes in a graph as the average number of edges needed (path length) to move from one node to another during a random walk. Similar to communicability (model C) and FEMM (model D), it measures a ‘proximity’ between nodes in a network. To make it more comparable with the other models, we computed its inverse value.

Although the different models are partially correlated with each other, they give different predictions about participants’ familiarity responses. First, they were two kinds of local transitions: familiar transitions and new transitions (TP = 0). Since the TP calculation does not consider the community structure (model A), participants should equally reject new transitions regardless of their relation with respect to communities (new within communities = new between communities). Second, concerning the new transitions, FEMM and hitting time models predict that participants should better detect new between community than new within community transitions (completion effect). It is also partly the case for the communicability model, but not for the TP and non-adjacent TP models (models A and B). The similarity of the predictions of FEMM, hitting time, and communicability models is not surprising as they all describe the same property of the network: proximity between nodes. Intuitively, items from the same community will appear closer together than items from different communities, even if the two nodes are not connected. In fact, FEMM and communicability are mathematically very close but with a different decay (exponential vs. factorial). However, they can still be differentiated thanks to the high sparse paradigm were the relative predicted familiarity of new within and familiar between transitions are different between the two models.

In addition to those theoretical models, we considered two putative brain implementations using biologically realistic neural networks:

• Model F: Hippocampus CA1 similarity: This neural network aims to reproduce the hippocampus structure (Norman and O’Reilly, 2003), which is often described as a key structure in statistical and structure learning (Henin et al., 2021; Schapiro et al., 2017; Schapiro et al., 2016). We compute here the similarity in CA1 layer as it has been proposed to capture community-like structures in previous studies (Schapiro et al., 2017). Indeed, thanks to its ability to have overlapping representations of the input and direct connection with the entorhinal cortex through the monosynaptic pathway, CA1 structure is also sensitive to long-distance dependencies allowing high-order structure learning.

• Model G: Hebbian learning with decay: Hebbian learning is a biologically plausible implementation of associative learning. Some neurons fire specifically to some objects in the environment. When two of those neurons co-fire, the pair is reinforced. It has been suggested that learning TPs is based on such a mechanism in the cortex. Here, we adapted this idea to implement the FEMM computation instead of TP, specifically by adding a temporal exponential decay in the probability of a neuron firing after a stimulus’s presentation. When the exponential decay has the same β parameter as the FEMM, the results of the FEMM and the Hebbian learning with decay are mostly similar.

All Data and analysis are publicly available at https://osf.io/e8u7f/.

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Author details

1. Lucas Benjamin

   Cognitive Neuroimaging Unit, CNRS ERL 9003, INSERM U992, Université Paris-Saclay, NeuroSpin center, Gif/Yvette, France

   Contribution

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

   For correspondence

   lucas.benjamin@cea.fr

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-9578-6039

2. Ana Fló

   Cognitive Neuroimaging Unit, CNRS ERL 9003, INSERM U992, Université Paris-Saclay, NeuroSpin center, Gif/Yvette, France

   Contribution

   Conceptualization, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-3260-0559

3. Fosca Al Roumi

   Cognitive Neuroimaging Unit, CNRS ERL 9003, INSERM U992, Université Paris-Saclay, NeuroSpin center, Gif/Yvette, France

   Contribution

   Methodology, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-9590-080X

4. Ghislaine Dehaene-Lambertz

   Cognitive Neuroimaging Unit, CNRS ERL 9003, INSERM U992, Université Paris-Saclay, NeuroSpin center, Gif/Yvette, France

   Contribution

   Conceptualization, Supervision, Funding acquisition, Methodology, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-2221-9081

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