Phase Transitions in Development: From Dimensional to Topological Learning

H2: Introduction: The Shape of Growth

How does a child’s mind become an adult’s? For a century, developmental science has wrestled with this question, caught between two competing metaphors: growth as smooth accumulation versus growth as sudden, stage-like transformation. The former, a dimensional model, sees the child as a “small adult” who simply needs more time, data, and processing power. The latter, a stage model, sees the child as a qualitatively different thinker, a caterpillar awaiting metamorphosis (Piaget, 1952).

Both metaphors capture a partial truth, yet both fail to provide a precise, mathematically grounded mechanism for how genuine novelty emerges. Dimensional models struggle to explain the “aha!” moment—the discontinuous leap from non-understanding to understanding, such as the acquisition of object permanence or theory of mind. Stage models provide a compelling description of these leaps but offer few testable, mechanistic explanations for how the cognitive machinery reorganizes itself (Beilin, 1992).

This post proposes a new theory that synthesizes these views. The thesis is that developmental change is best modeled not as dimensional growth (more features in a static space) but as topological rewriting—the emergence of new connectivity invariants in the brain’s representational space.

I call this the Developmental Topological Transition (DTT) model. It defines a cognitive “stage” or milestone as a discrete topological surgery: the creation, annihilation, or reconnection of homology groups that encode the very accessibility of conceptual states. In this view, biological maturation provides the energy for these transitions, while environmental novelty supplies the boundary conditions that shape the resulting structure.

H2: The Problem with Dimensional Models

The dominant paradigm in cognitive science and machine learning implicitly favors dimensional models. We measure cognitive growth by tracking changes in vector representations, increases in synaptic weights, or the expansion of a feature set (Saxe et al., 2021). In these models, “learning” is a process of optimization within a high-dimensional but topologically static space. A concept is a point or region, and learning moves that point or sharpens that region’s boundaries.

Let us formally define “Dimensional Growth” as a quantitative increase in a feature set or a change in metric properties within a static representational space. For example, a child’s concept of “bird” might gain features (from “flies” to “has-feathers,” “has-beak”), or the cluster of “bird” exemplars in their semantic space might become tighter.

The fundamental limitation of this view is that it cannot explain emergence. A dimensional model can explain how a child gets better at a task, but it cannot easily explain how a child becomes capable of a new kind of task altogether. How does a system that only understands “here” and “now” suddenly develop a representation of “elsewhere” (object permanence) or “elsewhen” (episodic memory)? This is not just more data; it is a new axis, a new hole, a new structure in the representatonal manifold itself. These are qualitative, not quantitative, changes.

H2: Theory: The Developmental Topological Transition (DTT) Model

The DTT model posits that the substrate of cognition—the space of possible thoughts—is not a simple vector space but a dynamic topological manifold. The properties of this space dictate the very nature of the concepts available to the organism.

H3: Representational Space as a Manifold

First, we must define our terms. A “Representational Space” is the high-dimensional set of all accessible cognitive or neural states. We can model this space as a “Manifold,” a geometric object that locally resembles simple Euclidean space but can have a complex global structure (like the 2D surface of a sphere, which is locally flat but globally curved) (Sizemore et al., 2019).

The true power of this approach comes from computational topology, a field that provides tools for measuring the shape of data. The key concept is “Homology,” which algorithmically identifies the fundamental connectivity of a shape. Homology groups are algebraic descriptors of this connectivity, and their ranks are called “Betti Numbers” (Bk).

In simple terms, Betti numbers count the “holes” of different dimensions:

B0 (Betti zero) counts the number of disconnected components (pieces).
B1 (Betti one) counts the number of 1-dimensional “loops” or “tunnels.”
B2 (Betti two) counts the number of 2-dimensional “voids” or “cavities.”

H3: The DTT Model: Milestones as Topological Surgery

We can now state the core thesis precisely. The “Developmental Topological Transition (DTT)” is a cognitive milestone that occurs when the Betti numbers of the conceptual manifold change.

This is a topological surgery. For example:

Concept Formation: A child learns to distinguish “dogs” from “cats.” Two previously overlapping, indistinct clouds of representations (B0=1) bifurcate into two separate, stable components (B0=2).
Object Permanence: The concept of an object “out of sight” requires a new kind of representation. This could be modeled as the creation of a B1 loop—a cyclical path in state space that represents the object’s continued existence and potential return, a path that was previously impossible (a “hole” in the manifold was filled).
Theory of Mind: Representing another’s mind—a set of beliefs separate from one’s own—is a profound topological leap, perhaps creating a new high-dimensional void (Bk where k>2) that represents the “space” of another’s perspective.

In this model, cognitive development is a sequence of such transitions. The manifold of a pre-permanence infant is topologically different from that of a post-permanence toddler. They are not just running different software; they are running it on different hardware, where the very wiring of their conceptual space has been rewritten.

H3: Mechanisms: Energy and Boundary Conditions

The DTT model is not purely abstract; it proposes a mechanism grounded in biology and environment.

Hypothesis 1: Biological maturation provides the energy for topological transitions. Neural systems, like other physical systems, tend toward stability. A topological rewrite is a high-energy, non-linear event—a phase transition. We propose that maturational processes (e.g., myelination, synaptogenesis) “fuel” the system, increasing its energy or plasticity to a critical point where a topological jump becomes possible. This aligns with work on self-organized criticality in neural systems, where networks poise themselves at the “edge of chaos,” enabling rapid state transitions (Beggs & Plenz, 2003).

Hypothesis 2: Environmental novelty supplies the boundary conditions. The specific new topology that emerges is not random. It is constrained by the problems the environment poses. A structured, novel environment (like a parent playing peek-a-boo) provides consistent boundary conditions that “sculpt” the manifold, guiding the phase transition toward an adaptive new stable state. This is a formalization of Vygotsky’s (1978) concept of scaffolding.

H2: Supporting Evidence and Theoretical Precedents

The DTT model, while novel, stands on the shoulders of giants. It integrates insights from dynamic systems, network neuroscience, and computational topology.

H3: Precedent 1: Dynamic Systems Theory (Thelen & Smith, 1994)

Esther Thelen and Linda Smith’s pioneering work on development as a dynamic system provides the conceptual foundation for DTT. They argued that behavior is “softly assembled” from the real-time interaction of multiple components (body, task, environment) and that development is characterized by non-linear phase transitions. (Thelen & Smith, 1994).

The famous “A-not-B” error, in their view, is not the failure of a static “object permanence” module. It is a stable attractor state in a dynamic system. To “pass” the test, the system (infant + task) must undergo a phase transition to a new attractor landscape. DTT provides a formal language for this: the transition is a topological rewrite of the state space manifold.

H3: Precedent 2: Connectomics and Network Topology (Sporns, 2013)

Modern neuroscience has embraced topology through connectomics. The brain is a network, and its function is critically dependent on its topological structure—a balance of segregation and integration managed by hubs (Sporns, 2013; Sporns et al., 2007).

Sporns and others have shown that brain states—from sleep to task execution—can be described as different topological configurations of a functional network. The DTT model scales this principle: if transient brain states have unique topologies, then permanent developmental milestones are the result of permanent, large-scale rewrites of the underlying structural or functional connectivity manifold.

H3: Precedent 3: Computational Topology (Edelsbrunner & Harer, 2010)

The DTT model would be pure speculation without the tools to test it. (Edelsbrunner & Harer, 2010). This work, and the entire field of Topological Data Analysis (TDA), provides the precise algorithms (e.g., persistent homology) needed to reconstruct and measure the Betti numbers of high-dimensional point clouds.

These tools are already being applied to neural data. Researchers have used TDA to find topological signatures in fMRI data corresponding to stimulus processing and disease states (Saggar et al., 2018), to classify neuron firing patterns (Adcock et al., 2016), and to understand the intrinsic structure of neural correlations (Giusti et al., 2015). The DTT model simply proposes we point this “topological lens” at developmental data across time.

H2: Anticipated Objections and Responses

A novel theory must withstand scrutiny. We anticipate three primary objections.

H3: Is the DTT model testable?

Yes. The DTT model is not just descriptive; it is predictive. As stated in the core thesis, it makes a clear, falsifiable empirical prediction: EEG or fMRI manifold reconstructions during concept acquisition will exhibit discrete homological jumps (changes in Bk) preceding performance leaps. If, over the course of learning, the Betti numbers of a child’s neural manifold remain stable (i.e., Bk(t1)=Bk(t2)) while their performance improves, the DTT model would be falsified. Performance should only leap after a topological transition.

H3: Is this an unoperationalized metaphor?

This is a common and fair critique of “stage” theories. The DTT model avoids this trap by rejecting vague metaphor in favor of mathematical operationalization. We are not using “topology” as a loose analogy for “structure.” We are making a specific claim about Betti numbers—integers that can be algorithmically computed from data using persistent homology. The model is as operationalized as any statistical model that relies on p-values or variance, it simply uses a different, more powerful mathematical toolkit.

H3: Does this model violate parsimony?

One might argue that a dimensional model (“the child’s brain just got better”) is simpler. But this simplicity is illusory. A simple model that fails to explain the data—discontinuous leaps, emergent properties, and “all-or-nothing” conceptual understanding—is not parsimonious, it is simply inadequate. In the philosophy of science, explanatory power is prized over simplicity if that simplicity fails to account for the data (Popper, 1959). The DTT model, while mathematically more complex, offers superior explanatory power for the kind of change we observe in development.

H2: Synthesis: Reconciling Stage and Dimensional Theories

The DTT model does not invalidate dimensional models; it contextualizes them. It provides a synthesis that resolves the century-old conflict.

“Dimensional Growth” (quantitative improvement) is what happens within a stable topological regime. The child gets faster, more accurate, and more efficient as their system optimizes its state within a given manifold. This is the “accretion” phase.
“Stage Transitions” (qualitative change) are the DTTs themselves. The manifold rewrites. The system is destabilized and then settles into a new, topologically distinct landscape. This is the “re-organization” phase.

Development is therefore a process of punctuated equilibrium: long periods of stable dimensional learning (accretion) punctuated by rapid, non-linear topological phase transitions (re-organization).

H2: Implications for Developmental Science

If the DTT model is correct, its implications are significant.

H3: New Research Paradigms

This model demands a new empirical paradigm. The goal of developmental cognitive neuroscience should shift from just “where” (localization) or “how much” (activation) to “what shape” (topology) (Spreng & Turner, 2019). The primary tool for developmental data analysis should be TDA. Longitudinal studies are needed to capture neural data before, during, and after a predicted cognitive milestone (e.g., learning to read, acquiring theory of mind) to hunt for the predicted topological signatures of that transition.

H3: Clinical and Intervention Models

The DTT model reframes developmental disorders. Atypical development (e.g., in autism spectrum disorder or dyscalculia) may not be a “deficit” (less of a skill) but a “topological difference” (Lee et al., 2022). The representational space for “self/other” or “number” might be fundamentally structured differently—possessing different Betti numbers.

This has profound implications for intervention. An intervention for a “deficit” model aims to “fill up” a missing skill. An intervention for a “topological” model would aim to scaffold the environment to provide the precise boundary conditions and energetic inputs needed to induce a topological phase transition to a more adaptive state (Stiso & Bassett, 2018).

H2: Conclusion: The New Shape of Development

For too long, developmental science has been armed with linear tools to describe a non-linear process. The DTT model offers a new language, borrowed from mathematics, to describe the beautiful complexity of human growth. It reframes development as a series of topological phase transitions, where the very shape of thought itself is rewritten.

This model unifies biology (the energy for transitions) and environment (the boundaries that shape them). It resolves the stage-versus-accumulation debate by framing them as two parts of the same process: stable optimization within a manifold, followed by a rapid rewrite of the manifold itself.

The empirical work ahead is challenging but clear. We must stop just measuring the size of the mind and begin, in earnest, to measure its shape. The tools are ready.

H2: End Matter

H3: Assumptions

Representational Fidelity: The model assumes that cognitive/conceptual states can be meaningfully mapped to a high-dimensional geometric space (a manifold) whose topological structure is non-trivial.
Proxy Data: It assumes that data from fMRI, EEG, or other neural recordings (e.g., spike trains) serve as a sufficiently high-fidelity, high-dimensional proxy for this underlying representational space, such that its topological features (Betti numbers) can be reliably reconstructed using TDA methods.
Causality: The model assumes that the observed topological transitions are not mere epiphenomena but are causally linked to the observed functional, behavioral leaps.

H3: Limits

Mechanism Underspecified: The DTT model describes the what (the topological change) and when (preceding performance leaps) but the precise how—the specific neurobiological mechanism of “topological surgery” at the synaptic or circuit level—remains underspecified and is a key area for future research.
Methodological Challenges: Current TDA methods face challenges with noise sensitivity, parameter selection (e.g., the “radius” in persistent homology), and computational scalability for extremely large, complex neural datasets.
Directionality: The model does not yet fully explain why development proceeds in a specific, often universal, sequence of transitions. It posits that energy and environment guide the transitions, but the constraints on this sequence are not fully defined.

H3: Testable Predictions

Primary Prediction: As stated in the text: Longitudinal fMRI or EEG data of a subject acquiring a discrete concept (e.g., theory of mind, conservation of number) will show a discrete, statistically significant change in the Betti numbers (Bk) of the reconstructed neural manifold. This topological change will temporally precede the behavioral leap in performance.
Complexity Correlation: The topological complexity (e.g., the sum of Betti numbers, or the “persistence” of topological features) of a domain-specific representational manifold (e.g., social cognition) will correlate positively with the sophistication and flexibility of the corresponding skill set.
Disorder-Specific Topologies: Atypical developmental trajectories (e.g., ASD) will be characterized by measurably distinct topological signatures (different Bk values or different feature persistence) in relevant representational manifolds compared to neurotypical controls, rather than simple hypo- or hyper-activation.

H2: References

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Beggs, J. M., & Plenz, D. (2003). Neuronal avalanches in neocortical circuits. Journal of Neuroscience, 23(35), 11167-11177.

Beilin, H. (1992). Piaget’s enduring contribution to developmental psychology. Developmental Psychology, 28(2), 191–204.

Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society.

Giusti, C., Pastalkova, E., Curto, C., & Itskov, V. (2015). Clique topology reveals intrinsic geometric structure in neural correlations. Proceedings of the National Academy of Sciences, 112(44), 13455–13460.

Lee, H., Eom, H., & Chung, M. K. (2022). Topological analysis of brain dynamics in autism based on graph and persistent homology. bioRxiv. doi:10.1101/2022.05.14.491959

Piaget, J. (1952). The origins of intelligence in children. International Universities Press.

Popper, K. R. (1959). The logic of scientific discovery. Basic Books.

Saggar, M., et al. (2018). Changes in brain network topology as a function of treatment response in depression. Neuropsychopharmacology, 43(11), 2235–2244.

Saxe, A. M., Nelli, S., & Sommers, J. P. (2021). If deep learning is the answer, what is the question? Annual Review of Developmental Psychology, 3, 205-229.

Sizemore, A. E., Phillips-Cremins, J. E., Ghrist, R., & Bassett, D. S. (2019). The power of topology in neuroscience. Current Opinion in Neurobiology, 55, 105-112.

Sporns, O. (2013). Network analysis, topological states, and generative models of human brain connectomics. Nature Reviews Neuroscience, 14(2), 140–152. doi:10.1038/nrn3412

Sporns, O., Honey, C. J., & Kötter, R. (2007). Identification and classification of hubs in brain networks. PloS one, 2(10), e1049.

Spreng, R. N., & Turner, G. R. (2019). The future of developmental cognitive neuroscience. In J. Rubenstein & P. Rakic (Eds.), Comprehensive developmental neuroscience (2nd ed.). Elsevier.

Stiso, J., & Bassett, D. S. (2018). Topological data analysis in neuroimaging: A survey. arXiv preprint arXiv:1809.08182.

Thelen, E., & Smith, L. B. (1994). A dynamic systems approach to the development of cognition and action. MIT Press.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Harvard University Press.