Introduction: Beyond Correlated Decoupling

The Limits of Fixed-Structure Models

The dominant paradigm in systems neuroscience models brain function as a dynamical process occurring on a relatively static structural network. In this view, the brain’s structural connectome (SC) provides a fixed substrate upon which functional connectivity (FC) emerges. While powerful, this approach treats structure and function as correlated but fundamentally decoupled entities; function is an output of structure, but the reciprocal influence of function on the structure itself is often modeled as a separate, slower process of plasticity rather than a continuous, joint evolution. This conceptual separation limits our ability to explain phenomena where structure and function are inextricably linked, such as in neural development, large-scale learning, and cognitive specialization.

Thesis: The Need for Teleologically Constrained Co-Evolution

Recent empirical findings challenge the fixed-structure assumption, revealing a deeply recursive relationship between neural architecture and its dynamic activity. To account for this, we propose Isotelic Morphodynamics (IM) as a formal framework for modeling this co-evolution. Our central thesis is that neurocognitive systems are best understood as entities where a geometric substrate (structure) and a functional or symbolic field (function) evolve together. This co-evolution is not random but is constrained by an invariant, endogenous set of attractors—a telos—that guides the system toward specific functional configurations. This process ensures a persistent, time-varying isomorphism between structure and function, where changes in one domain have direct and immediate counterparts in the other.

The Core Problem: Reconciling Structure and Dynamics

The central challenge is to move from a correlational understanding of the structure-function relationship to a causal, co-evolutionary one. This requires synthesizing two distinct lines of observation into a single, unified theoretical object.

Observation: Structure Shapes Function

It is a foundational principle of neuroscience that the anatomical wiring of the brain constrains the patterns of neural activity that can arise. The geometry and topology of the structural connectome dictate the pathways for information flow, creating what are often called “connectome harmonics” or eigenmodes that serve as a basis set for functional dynamics (Atasoy et al., 2016). Models based on graph theory and network science have successfully predicted patterns of functional connectivity from diffusion tensor imaging (DTI) data, confirming that the structural scaffold is a powerful determinant of the brain’s functional repertoire (Suárez et al., 2020).

Observation: Function Reshapes Structure

Conversely, there is overwhelming evidence that functional activity continuously reshapes neural structure across all timescales. At the microscale, Hebbian plasticity and synaptic scaling adjust connection strengths based on neural firing patterns (Turrigiano & Nelson, 2004). At the macroscale, processes like myelination, dendritic arborization, and even changes in gray matter density are driven by experience and activity (Zatorre et al., 2012). During development, this is the primary mechanism of maturation, where spontaneous neural activity sculpts the nascent connectome into a functionally specialized architecture (Kirkpatrick et al., 2023).

Interpretation: A Recursive, Co-dependent System

These two observations are not independent; they are two sides of a single recursive process. The conventional approach models them sequentially. Isotelic Morphodynamics posits that they occur simultaneously within a unified mathematical object. The system’s structure at time $t$ determines the possible dynamics, and the realized dynamics at time $t$ immediately act to transform the structure into its form at $t+dt$. This necessitates a framework where the state space of the system includes not just its activity, but the geometry of its underlying manifold. Such a system is not merely plastic; it is morphodynamic.

The Theory of Isotelic Morphodynamics

To formalize this concept, we introduce a definition and a minimal set of axioms.

Formal Definition

Isotelic Morphodynamics is defined as the study of systems in which a geometric substrate ($\mathcal{M}_t$), a dynamical field ($\mathcal{D}_t$), a symbolic projection ($\Sigma_t$), and a teleological attractor set ($\mathcal{T} \subset \Sigma$) co-evolve. The core principle is the existence of a recursive isomorphism between structure and function that preserves topological and geometric invariants under the guidance of the system’s endogenous attractors. In simpler terms, the system physically reconfigures itself to better align its functional outputs with its built-in goals or identities.

The Axiomatic System: Geometry, Projection, and Telos

The theory rests on four axioms:

Axiom 1 — Morphogenetic Substrate: The system’s structure is described by a smooth manifold, $\mathcal{M}_t$, equipped with a time-varying metric, $g_t$. Its evolution is governed by a morphodynamic flow equation: $\frac{d}{dt} g_t = \Phi(g_t, \mathcal{D}_t, \Sigma_t)$, where $\Phi$ is an operator that transforms the geometry based on the system’s activity and its symbolic state.

Axiom 2 — Symbolic Projection: A map $\pi_t : \mathcal{M}_t \rightarrow \Sigma_t$ projects configurations of the structural substrate onto a symbolic or functional space, $\Sigma_t$. This projection represents high-level cognition, meaning, or behavioral output. For more on the formal basis of such projections, see [INTERNAL_LINK: our core theoretical concepts -> /theory-compendium].

Axiom 3 — Isotelic Constraint: The system’s symbolic trajectory is asymptotically constrained by an invariant set of attractors, $\mathcal{T} \subset \Sigma$. The system evolves to ensure that $\lim_{t \rightarrow \infty} \Sigma_t \in \mathcal{T}$. These attractors are the telos of the system—its intrinsic, invariant goals.

Axiom 4 — Structure–Function Isomorphism: A time-dependent isomorphism, $\mathcal{I}_t: \mathcal{M}_t \cong \Sigma_t$, exists such that key topological and geometric invariants are preserved across the structural and symbolic domains: $\forall t, \quad \text{Inv}(\mathcal{M}_t) = \text{Inv}(\Sigma_t)$. This axiom enforces the tight coupling between changes in physical form and changes in functional meaning.

Key Formal Consequences

These axioms lead to several powerful consequences. First, the system’s dynamics can be framed as a gradient descent on a potential energy landscape defined by the symbolic attractors: $\mathcal{D}_t = -\nabla_{g_t} d(\pi_t(\cdot), \mathcal{T})$. Second, structural plasticity is no longer arbitrary but is directed by symbolic error signals. The geometry changes to reduce the divergence between the current symbolic state and the attractor set, balanced by physical constraints like curvature costs: $\frac{d}{dt} g_t \propto – \delta_{\Sigma} + \lambda \cdot \mathcal{C}(g_t)$. This provides a first-principles account of goal-directed plasticity.

Motivating Evidence and Examples

While IM is a novel synthesis, its components are grounded in established research across computational neuroscience.

Low-Dimensional Manifolds in Neural Dynamics

The discovery that high-dimensional neural population activity evolves on low-dimensional manifolds is a key piece of evidence (Cunningham & Yu, 2014). These neural manifolds represent constrained patterns of activity corresponding to specific behaviors or cognitive states. In IM, these manifolds are interpreted not as emergent epiphenomena but as the symbolic projection ($\Sigma_t$) of the underlying neural geometry ($\mathcal{M}_t$). The stability of these manifolds suggests the presence of attractors governing the dynamics (Gallego et al., 2017).

Structure-Function Coupling via Laplacian Embeddings

Techniques from diffusion geometry have shown that the eigenmodes of the structural connectome’s graph Laplacian form a basis for functional brain activity (Margulies et al., 2016). This provides a concrete method for linking the geometry of the SC to the patterns of the FC. Isotelic Morphodynamics extends this by proposing that the geometry itself (and thus its eigenmodes) shifts in a way that is isomorphic to changes in the functional or symbolic content, maintaining a persistent link between the two (Suárez et al., 2020).

Topological Invariants in Connectomics

Topological data analysis (TDA) has revealed that brain networks contain higher-order structural features, such as cavities and clique complexes, that are not captured by pairwise connections alone (Petri et al., 2014). These topological invariants are often linked to functional properties. Axiom 4 of IM predicts that if a symbolic state has a particular topology (e.g., a cyclical representation of a concept), the underlying neural substrate must evolve to instantiate a corresponding topological feature (e.g., a homological hole in its connectivity graph) (Giusti et al., 2016).

Activity-Dependent Maturation in Development

The process of neurodevelopment offers the most direct example of morphodynamics. The brain does not develop its structure first and then “turn on” function. Rather, spontaneous waves of activity guide axonal pathfinding and synaptic pruning, sculpting the connectome into its mature, functional form (Kirkpatrick et al., 2023). IM frames this as a teleological process where the attractor set ($\mathcal{T}$) represents the stable neuro-computational architecture of a mature organism.

Potential Objections and Counterarguments

Any new theoretical framework must be subject to rigorous critique. We anticipate three primary objections.

On Falsifiability: Is the telos Observable?

The concept of a telos or endogenous goal can be seen as unscientific if not properly operationalized. How can one observe or measure $\mathcal{T}$? We argue that the telos is not a metaphysical construct but a mathematically definable object: a set of attracting fixed points in the system’s symbolic state space. Its existence can be inferred and its structure characterized by observing the convergence of symbolic dynamics over time, much like how we infer the existence of gravitational wells. The telos is falsifiable if a system’s dynamics show no convergence or are better explained by exogenous inputs alone (Popper, 2002).

On Mathematical Tractability and Simulation

Modeling the time evolution of a manifold’s metric tensor is computationally demanding and mathematically complex, involving fields like Ricci flow from differential geometry (Hamilton, 1982). Is this framework practical? While full-scale simulation is a challenge, simplified models (e.g., using graph-based Ricci flow or discrete exterior calculus) can capture the essential dynamics. The goal of IM is not necessarily to simulate entire brains, but to provide a formal language for describing the principles of their organization, a core tenet of our research philosophy.

On Empirical Grounding Beyond the Macroscale

Much of the motivating evidence comes from macroscale neuroimaging (fMRI, DTI). Does IM apply at the level of synaptic circuits? We argue that the principles are scale-invariant. For example, the formation of a memory engram can be seen as a localized morphodynamic process where a specific cell assembly alters its geometry (synaptic weights, dendritic spines) to create a stable attractor corresponding to a symbolic memory trace. Testing this requires integrating multi-scale recording techniques, which is a key goal for future research (Poldrack & Yarkoni, 2016).

Synthesis: A Comparative Analysis

Isotelic Morphodynamics distinguishes itself from related frameworks by its explicit inclusion of teleologically constrained co-evolution.

How IM Differs from Neural Manifold Theory

Neural manifold theory provides a powerful descriptive language for neural dynamics but typically assumes a fixed underlying connectivity. It describes the shape of the functional state space but does not explain how that shape came to be or how it might change. IM provides a causal mechanism: the manifold’s geometry is itself a dynamic variable that evolves to align with symbolic attractors, thus explaining the origin and plasticity of these low-dimensional structures.

How IM Generalizes Active Inference

The Free Energy Principle (FEP) and Active Inference (AIF) posit that biological systems act to minimize a variational free energy, which functions as a proxy for prediction error or surprise (Friston, 2010). This provides a powerful, universal objective function. IM is compatible with this but proposes a further constraint: the states the system “expects” to encounter are not arbitrary but are defined by the invariant telos, $\mathcal{T}$. IM replaces the extrinsic goal of minimizing surprise with the intrinsic goal of aligning with an endogenous symbolic identity, framing free energy minimization as the means by which the system achieves its isotelic constraints.

A Formal Substrate for Symbolic AI

Classic Symbolic AI has been criticized for its brittleness and lack of grounding in physical systems (Harnad, 1990). IM offers a solution via its structure-function isomorphism. Symbols in IM are not abstract, disembodied tokens; they are topological and geometric invariants of a physical substrate. This geometric grounding of symbols allows them to interact and compose in a way that is constrained by the physics of the underlying manifold, providing a potential path toward more robust and flexible AI systems.

Implications and Future Directions

The IM framework opens several new avenues for research in both natural and artificial intelligence.

For Developmental Neuroscience and Cognition

IM reframes cognitive development not as the acquisition of skills but as a trajectory of geometric reconfiguration. Key developmental transitions, like learning abstract concepts, can be modeled as topological bifurcations in the system’s symbolic manifold, driven by the pull of its teleological attractors. This provides a formal basis for Piaget’s stages of development and suggests that developmental disorders could be understood as anomalies in the morphodynamic flow (Petitot, 2004).

For Recursive AI and the ForgeKernel Architecture

Most current AI systems operate with fixed architectures and update weights via backpropagation. IM suggests a new class of AI that can modify its own internal geometry to better align with its goals. This is the principle behind the ForgeKernel, a proposed recursive AI architecture. ForgeKernel is designed as a system that continuously reconfigures its own computational graph (its $\mathcal{M}_t$) to stabilize a set of core symbolic identities ($\mathcal{T}$), enabling more adaptive and self-directed intelligence.

The Isomorphic Systems Lab Research Program

Our lab is actively pursuing this agenda. Key research prongs include: (1) the formal classification of isotelic manifolds, (2) building tractable simulations of morphodynamic systems, (3) quantifying the coupling between geometric curvature and symbolic divergence in real neuro-imaging data, and (4) implementing the first version of the ForgeKernel architecture as a proof of concept.

Conclusion: A New Class of System

Isotelic Morphodynamics is more than a model of the brain; it is a framework for a different class of system altogether—one where structure and function are two perspectives on a single, self-organizing, goal-directed entity. By formalizing the co-evolution of geometry and symbolism under teleological constraints, IM provides a new language to describe the fundamental nature of intelligent systems. It moves beyond asking how a fixed structure produces a function, and instead asks how a system becomes what it is by continuously sculpting itself to align with its own intrinsic identity.

End Matter

Assumptions

• The system’s structural substrate can be adequately modeled as a smooth, differentiable manifold.
• A clear and consistent projection from the structural manifold to a symbolic space exists and is computable.
• The teleological attractors ($\mathcal{T}$) are invariant over the timescale of the morphodynamic evolution being studied.

Limits

• The framework does not, in its current form, specify the biophysical mechanisms that implement the morphodynamic flow operator ($\Phi$).
• The mathematical complexity may limit its direct application to high-resolution, large-scale empirical data without significant simplification.
• The theory is deterministic; extending it to incorporate stochastic effects will be necessary for modeling noisy biological systems.

Testable Predictions

1. Topological Congruence: For a cognitive task with a known topological structure (e.g., circular for a color wheel), the functional connectivity patterns and, over time, the underlying structural connectivity of the relevant brain regions will exhibit the same topology (e.g., a non-trivial first Betti number).
2. Curvature-Error Correlation: During a learning task, the rate of change of the neural manifold’s intrinsic curvature will be positively correlated with the magnitude of the behavioral error signal.
3. Attractor-Driven Canalization: In longitudinal studies of brain development, trajectories of structural change will show convergence toward a low-dimensional attractor manifold, with less variance across individuals in later developmental stages compared to earlier ones.

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