Field closure, ice neurons, and when a dendrite is a motif

Dendritic silhouettes are cheap, but when a dendrite possesses an internal state and writes into its surrounding field, it becomes a biological neuron. This work develops a continuum system where ice neurons serve as the zero-gain limit of an interfacial system, demonstrating how finite source gain—modeled by field closure—transforms the purely morphological structure into a dynamical motif that exhibits persistence beyond what is achievable in passive growth models.

The core concept introduces the ice neuron as a structural analogue to a biological neuron.

The paper posits that dendritic silhouettes share a branching statistic because they share a growth instability, not functional similarity. The ice neuron is constructed as the zero-gain limit of an interfacial system where a FitzHugh–Nagumo state lives on the moving front and writes back into the field that drives it. This construction allows researchers to study how geometry, dynamics, and field co-sculpt one another.

Field closure is introduced as the mechanism that distinguishes ice neurons from passive growth structures.

At zero gain (γ = 0), morphology and internal dynamics decouple into a Mullins–Sekerka growth band and a damped oscillator. However, at finite γ, the interfacial state contributes directly to the flux balance: the motion of the front is no longer determined solely by the externally established gradient of ϕ: the interfacial state also modifies the flux that drives the front. This reciprocal coupling is termed field closure and corresponds to biological ephaptic coupling.

The model employs a continuum system combining morphological instability and interfacial dynamics.

The exterior field is governed by diffusion, while the interface motion is described by two boundary relations: the Gibbs–Thomson relation (setting the local equilibrium) and the Stefan flux balance. The interfacial state itself is modeled using a minimal internal state capable of an intrinsic Hopf instability: a FitzHugh–Nagumo pair. This system includes a fast variable ψ and a slower recovery variable w, coupled to the exterior field ϕ.

Linear theory reveals how field closure reshapes the spectral characteristics of the growth instability.

The linear analysis shows that the dispersion relation is modified by field closure. The classical Mullins–Sekerka contribution, which describes morphological instability, is scaled by p(k)k. Field closure introduces a correction proportional to p(k)γ, which warps the dispersion curve, giving greater relative weight to the long-wavelength part of the spectrum and shifting the selected instability mode toward smaller wavenumbers (smaller k).

The motif assay demonstrates that field closure enhances persistence in a way that is localized.

The study investigates persistence on a frozen tree by comparing growth at γ = 0 with finite γ. The linear response to an impulse is characterized by the DC gain Q(γ), which follows the first-order law: Q(γ) − Q(0) = γQ1 + O(γ2). This excess gain is localized to a small set of non-cable field shortcuts, which are identified by mapping the kernel Q1 back onto the frozen tree, showing that biological function is implemented through these specific field interactions rather than the overall silhouette.

The results show that field closure opens an oscillatory region where none existed before.

By analyzing the coupled cubic characteristic polynomial, it is found that finite γ can introduce a complex-dominant instability even when the isolated interfacial oscillator remains subcritical (below its own Hopf threshold). This γ-driven Hopf signifies a dynamical motif arising from the feedback loop, distinguishing the ice neuron from a passive structure. Furthermore, persistence enhancement occurs below the linear instability threshold at γ ≈ 0.20, suggesting that field closure is adding a systematic contribution to transmission rather than merely moving the system toward its instability.

The ensemble analysis confirms that these effects are generic across different growth realizations.

Testing over eighty independent seeds of the same growth rule confirms robustness: "Q1 > 0 on all eighty trees. The onset threshold γ∗ has a median of 0.14, and the first-order law holds for each tree within its own stable window. This indicates that while the magnitude is geometry-dependent, the direction of the first-order closure effect is not" dependent on the specific growth history.

The conclusion establishes a hierarchy between structure and computation.

The paper concludes by establishing a categorical account: growth supplies structure; an internal state supplies autonomous dynamics; γ supplies the physical loop; and computation, in the strong sense, is a further question about what the loop implements. Field closure provides the necessary physical loop that ice neurons lack, enabling them to function as dynamical motifs. The distinction between ordinary physical feedback and autonomous physical computation hinges on whether internally realized readout states must participate in selecting subsequent operations.

Key findings enumerated:

1. Ice neurons are the zero-gain limit of an interfacial system where a FitzHugh–Nagumo state writes back into the driving field.

Here are specific improvements to AI systems based on the concepts presented in this paper, categorized by the mechanism they enable:

)AI System Improvements Enabled by Field Closure and Dynamical Motifs

The core contribution of this work is moving from purely geometric motifs (like dendritic silhouettes) to dynamical motifs where structure and function co-sculpt one another via a bidirectional field interaction. Improving AI systems requires incorporating this reciprocal feedback loop into their architecture.

AI System Capability:

An AI system designed with a field-closed architecture will exhibit enhanced long-term memory, persistence of learned information, and the ability to dynamically modify its internal processing pathways based on the resulting state of its external environment (or internal field).

2. Mechanism for Improvement:

Implement a mechanism analogous to the FitzHugh–Nagumo pair (the fast interfacial variable and slower recovery variable) as an intrinsic state within every computational node or module. This is not just a static parameter but an active dynamical degree of freedom that evolves based on the field it generates and receives.

3. Specific AI Function:

When processing complex, noisy, or temporally extended data (e.g., long-term reinforcement learning, sequential decision-making), the system can use its internal state to modulate its susceptibility to input signals. In high-dimensional spaces, this allows the system to selectively amplify or suppress specific features of an external field that are coherent with its current internal state, leading to superior filtering and pattern recognition compared to purely feed-forward networks.

4. Mechanism for Improvement:

Incorporate the concept of field closure (the feedback loop where the interfacial state writes back into the driving field) as a primary control mechanism rather than just a passive input/output connection. This could be modeled via an ephaptic coupling term that links internal states across different processing layers or neurons.

5. Specific AI Function:

This enables true autonomous physical computation, where the system doesn't just react to its environment but actively shapes the environment in a way that reinforces its own operation. For example, in robotic control or autonomous agents navigating complex physical spaces, this means the agent's internal decision state dynamically generates local forces or field perturbations that guide its subsequent movements toward a goal, rather than relying solely on pre-programmed trajectories.

AI System Capability:

The system can transition from being purely reactive to being computationally active, capable of maintaining sustained activity (persistence) even when the external input signal weakens, provided the internal field structure remains stable within its operational window.

7. Mechanism for Improvement:

Design learning algorithms that explicitly seek out and stabilize regimes that reside in the stable linear window of their dynamics (i.e., operating below the threshold where instability occurs). The system should be trained to find parameters where the leading-order correction term, proportional to field closure gain, is positive and dominant over noise.

8. Specific AI Function:

This leads to robust long-term memory retention in neural networks or recurrent architectures. Instead of forgetting transient inputs (which happens when the linear operator has a pole), the system learns to maintain a persistent state that acts as a stable filter, effectively retaining relevant information over long time horizons.

AI System Capability:

The system can exhibit context-dependent computational modes, where different internal states correspond to different motifs or operational strategies (e.g., Planar vs. Real-Dominant spectral regimes).

10. Mechanism for Improvement:

Develop a meta-learning framework that allows the AI to switch between different functional modes based on the complexity or stability of its current input environment, analogous to how the system switches between a real-dominant and complex-dominant root in its linear spectrum.

11. Specific AI Function:

This allows for adaptive reasoning: when faced with a highly structured, predictable problem (e.g., simple pattern recognition), the system operates in an efficient, real-dominant mode; when faced with novel, chaotic, or high-uncertainty data (where the uncoupled system is unstable), it dynamically shifts to a more complex mode capable of exploring oscillatory solutions that might yield a breakthrough.

Dendritic silhouettes are cheap. Pond ice neurons, diffusion-limited aggregates, wiring-minimizing arbors, and biological neurons can share a branching statistic because they share a growth instability, not because they are functionally similar. We take that observation as a modeling constraint. The ice neuron only structurally resembles a biological neuron. A biological neuron carries an internal state and a field that it helps generate, letting geometry, dynamics, and field co-sculpt one another. In our construction, the ice neuron is the zero-gain limit of an interfacial system in which a FitzHugh--Nagumo state lives on the moving front and writes back into the field that drives it, with dimensionless gain γ. At γ=0, morphology and internal dynamics decouple into a Mullins--Sekerka growth band and a damped oscillator. Finite γ deforms the spectrum, shifts the selected instability, and opens an oscillatory region below the isolated Hopf threshold. On a dendrite grown at γ=0 and then frozen, field closure increases distal persistence without changing the geometry; its first-order effect remains positive across repeated realizations of the same growth rule and localizes to a small set of non-cable field shortcuts. Nonlinear dynamics remain finite beyond the linear instability: below threshold, closure is almost the transmission; at spike amplitude, it is a correction to a regenerative cable event. A dendritic tree therefore is not intrinsically a structural spandrel. Shape alone is morphology without an internal state or a source term. That is the dendritic silhouette of the ice neuron. Excitability adds an active cable. Field closure is the biophysical mélange in which the neuron helps write the field that acts back on it, sculpting its dynamics and, during growth, its structure. That is a biological neuron.