Neural: Systems, Computation & Codes

• Neural is a broad term defining biological brains, artificial networks, and formal algebraic models characterized by dynamic evolution and distributed processing.
• Research methodologies include synaptic learning, dynamical mode decomposition, and neural ring theory to analyze coding, representation, and computational potential.
• Applications span neuromorphic hardware, perception-informed networks, and sensor interfaces that integrate multimodal data for efficient processing.

“Neural” denotes a broad class of structures, processes, and models associated with biological nervous systems, artificial neural systems, and formal representations of population activity. In contemporary research, the term spans synaptically adaptive and self-organizing brain networks, artificial neural networks treated as depth-wise dynamical systems, algebraic encodings of neural codes, neural fields used as native geometric representations, and neuromorphic devices that transduce or process neural signals in hardware. Across these usages, recurring technical themes include distributed representation, dynamical evolution, multimodal integration, sparsity-aware computation, and biologically or physically anchored constraints (Yan et al., 28 Feb 2026).

1. Neural systems as biological and artificial information processors

In biological neuroscience, neural systems are described as learning statistical input regularities through synaptic learning, integrating functionally into increasingly larger interconnected neural assemblies, and self-organizing under multisensory input from vision, sound, smell, touch, and proprioception (Dresp-Langley, 2022). The same paper emphasizes that all sensory information is in a first instance topologically represented in the biological brain and thereafter integrated in somatosensory neural networks for multimodal and multifunctional control of complex behaviors, with the somatosensory cortex serving as a hub for integration and control (Dresp-Langley, 2022). In this sense, “neural” refers simultaneously to synaptic plasticity, modularity, recurrent sensorimotor loops, and embodied control.

A more abstract formulation treats both natural and artificial systems as “neural machines,” in which sensing, internal processing, and actuation are organized within a single informational framework (Petrila, 2024). That framework defines five basic informational characteristics—Function, Memory, Nondeterminism, Fragmentation, and Aggregation—and expresses the machine generically as

$$\text{Output} = \text{Function}(\text{Input}; \text{History}, \text{Environment}).$$

It further proposes “Neural Absolute Power” and “Neural Relative Power” as global connectivity-based measures of neural computing potential:

$$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$

$$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$

These definitions are intended to apply equally to brains, artificial neural systems, neuromorphic hardware, and integrated sensor–actuator controllers (Petrila, 2024).

Within machine learning, “neural” also names a family of architectures and meta-models rather than a specific biological analogy. A “neural predictor” for neural architecture search learns a regression model from architecture–validation-accuracy pairs and uses it to rank large numbers of candidate architectures, yielding a search procedure that is more than 20 times as sample efficient as Regularized Evolution on NASBench-101 and competitive with ProxylessNAS on ImageNet (Wen et al., 2019). “Perception-Informed Neural Networks” generalize physics-informed models by incorporating perception-based information—singular, probability distribution, possibility distribution, interval, fuzzy graph, and Z-number constraints—directly into loss functions, thereby extending informed neural modeling beyond crisp physics laws (Mazandarani et al., 2 May 2025). A plausible implication is that “neural” has become a unifying label for learnable systems that can absorb data, structure, and prior knowledge in multiple formal modes.

2. Neural representations, tuning functions, and dynamical alignment

A central contemporary usage of “neural” concerns representation: how activity patterns in brains or neural networks encode stimuli, tasks, or latent variables. The “Neural Functional Alignment Space” (NFAS) frames this question by asking how artificial neural networks can be characterized in a way that is meaningfully anchored to the brain (Yan et al., 28 Feb 2026). NFAS treats a deep network as a dynamical system over depth, interprets layer embeddings $\{\mathbf{x}_\ell\}_{\ell=1}^{L}$ as a depth-wise trajectory, and applies Dynamic Mode Decomposition (DMD) to extract a stable mode whose eigenvalue magnitude is closest to 1. The resulting stable dynamical representation

$$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$

is then convolved with a canonical hemodynamic response function and fitted to ROI-level fMRI time series through linear encoding models (Yan et al., 28 Feb 2026).

NFAS defines a biologically anchored coordinate system whose axes are Schaefer-200 functional ROIs normalized to MNI152 space, and each model is represented by an alignment vector of squared Pearson correlations with ROI responses (Yan et al., 28 Feb 2026). Across 45 pretrained models spanning vision, audio, and language, PCA of these vectors yields clear modality-specific clustering; PERMANOVA with 999 permutations gives a significant modality effect with $p < 0.001$, inter-modality cosine distances are larger than intra-modality distances (mean 0.060 vs. 0.0004), and the silhouette score reaches $S = 0.995$ with permutation $p < 0.001$ (Yan et al., 28 Feb 2026). NFAS also introduces the Signal-to-Noise Consistency Index,

$$\mathrm{SNCI}_r = \mathrm{sigmoid}\!\left( \frac{\mu_r}{\sigma_r + \epsilon} \right),$$

to summarize modality-level consistency across models (Yan et al., 28 Feb 2026). Visual models show strong SNCI in posterior occipital cortex, audio models in lateral temporal regions, and LLMs in prefrontal control regions; all modalities show elevation in the Limbic network when SNCI is aggregated into Yeo-7 systems (Yan et al., 28 Feb 2026). This suggests that “neural” representation can be compared across architectures and modalities only after specifying the dynamical and biological reference frame.

At the single-neuron level, “Factorized Neural Processes for Neural Processes” recasts neural system identification as $K$-shot prediction of a neuron's tuning function (Cotton et al., 2020). Each neuron is treated as a sampled function $$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$0, observed through a small set of stimulus–response pairs $$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$1, and modeled via a Neural Process:

$$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$2

The proposed Factorized Neural Process partitions the latent space into receptive field location $$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$3 and tuning properties $$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$4, yielding

$$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$5

The latent representation is inferred in a quick, single forward pass and achieves predictive accuracy comparable to, and for small $$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$6 even greater than, optimization-based approaches on real neural data from visual cortex (Cotton et al., 2020). Here, “neural” refers not only to the recorded neuron but also to the function-space model used to infer its tuning.

3. Neural codes, neural rings, and algebraic structure

A more formal and combinatorial sense of “neural” appears in the study of neural codes. A neural code on $$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$7 neurons is a subset $$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$8, where each codeword records which neurons are active and which are silent (Curto et al., 2015). When neurons have receptive fields $$\text{Neural Absolute Power} = \log_2(\text{Connections}),$$9, the associated code is

$$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$0

This formulation abstracts away timing and rate, retaining only combinatorial co-activity patterns (Curto et al., 2015).

The neural ring and neural ideal provide an algebraic encoding of this combinatorial data. For a code $$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$1, the neural ideal is

$$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$2

and the neural ring is

$$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$3

The ring is naturally isomorphic to the ring of functions $$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$4, and its canonical basis is given by characteristic functions

$$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$5

These constructions encode the full combinatorial data of the code and can be placed in canonical form to obtain a minimal description of the receptive field structure intrinsic to the code (Youngs, 2014). Type 1, Type 2, and Type 3 pseudo-monomials correspond, respectively, to empty intersections, containments of intersections in unions, and coverings of the stimulus space by receptive fields (Youngs, 2014).

Maps between neural codes are in bijection with ring homomorphisms between their neural rings, but unrestricted ring homomorphisms are too permissive to preserve neuron structure (Curto et al., 2015). This motivates neural ring homomorphisms, defined by the condition that each target neuron variable maps to either a single source neuron variable or a constant:

$$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$6

The associated code maps are characterized as compositions of five elementary operations: permutation, adding or deleting a trivial neuron, duplication or deletion of a duplicate neuron, neuron projection, and inclusion (Curto et al., 2015). Under surjective neural ring homomorphisms, convexity behaves monotonically: if $$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$7 is convex and $$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$8 corresponds to such a homomorphism, then $$\text{Neural Relative Power} = 100 \times \frac{\log_2(\text{Connections})}{\log_2(\text{Connections}_\text{Human})}.$$9 is convex and

$\{\mathbf{x}_\ell\}_{\ell=1}^{L}$0

This shows that structured code maps cannot increase minimal embedding dimension (Curto et al., 2015).

“Polarization of Neural Rings” extends this algebraic program by introducing a polarization operation that maps neural ideals, which are pseudomonomial ideals, to squarefree monomial ideals in a larger ring (Gunturkun et al., 2017). A pseudomonomial

$\{\mathbf{x}_\ell\}_{\ell=1}^{L}$1

is polarized to

$\{\mathbf{x}_\ell\}_{\ell=1}^{L}$2

and divisibility is preserved:

$\{\mathbf{x}_\ell\}_{\ell=1}^{L}$3

For a neural ideal $\{\mathbf{x}_\ell\}_{\ell=1}^{L}$4, the polarized ideal $\{\mathbf{x}_\ell\}_{\ell=1}^{L}$5 is squarefree and therefore amenable to Stanley–Reisner theory, free resolutions, and Cohen–Macaulay criteria (Gunturkun et al., 2017). The depolarization sequence

$\{\mathbf{x}_\ell\}_{\ell=1}^{L}$6

is regular on $\{\mathbf{x}_\ell\}_{\ell=1}^{L}$7, allowing free resolutions of $\{\mathbf{x}_\ell\}_{\ell=1}^{L}$8 to transfer back to $\{\mathbf{x}_\ell\}_{\ell=1}^{L}$9 (Gunturkun et al., 2017). In this algebraic setting, “neural” names a class of codes and morphisms whose structure can be studied with commutative algebra.

4. Neural computation, control, and informed learning frameworks

The term “neural” also names computational principles and architectures that need not resemble standard deep feedforward networks. In the “functional neural network” for decision processing, the basic unit is an Artificial Mirror Neuron composed of an intention wheel, a motor core, and a sensory core, with dynamics governed by cycloid, hypocycloid, and epicycloid parameterizations (Jumelle et al., 2021). Decision output is produced by racing dynamics and fuzzy truth values:

$$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$0

with analogous expressions for the sensory core (Jumelle et al., 2021). Group decisions are formed by averaging these fuzzy responses across neurons, and learning is framed through performance memorial, velocity adjustments, and expert sub-network selection rather than conventional backpropagation (Jumelle et al., 2021). This usage of “neural” is neuromorphic and decision-centric rather than layer-centric.

In the analysis of artificial computation more generally, “Neural Information Organizing and Processing – Neural Machines” argues that deep neural information processing should be understood as repeated alternations of fragmentation and aggregation (Petrila, 2024). The conventional neuron equation,

$$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$1

is said to linearize what are in natural systems non-linear fragmentation and aggregation stages. The proposed neural machine model therefore incorporates input interfaces, central graph- or hypergraph-like networks, output interfaces, internal feedbacks, and non-deterministic memorization within a single architecture (Petrila, 2024). This suggests a broader interpretation of “neural” as an informational regime characterized by distributed fragmentation, recombination, and state-dependent memory.

Perception-Informed Neural Networks extend informed neural computation into uncertain and linguistically specified domains (Mazandarani et al., 2 May 2025). They express knowledge as generalized constraints

$$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$2

and incorporate those constraints into training losses alongside data and, when available, physics residuals. In singular mode, this recovers PINN-style objectives such as

$$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$3

while probability, possibility, interval, fuzzy graph, and Z-number modes produce new classes such as MOEINNs, TKINNs, and FINNs (Mazandarani et al., 2 May 2025). FINNs, for example, constrain outputs through fuzzy graph losses

$$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$4

with $$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$5 built from fuzzy if–then rules (Mazandarani et al., 2 May 2025). A plausible implication is that “neural” now frequently designates hybrid models that absorb crisp equations, distributions, and linguistic rules within a single differentiable framework.

5. Neural hardware, interfaces, and native geometric representations

A distinct but related usage of “neural” appears in hardware and interfaces designed to sense, stimulate, or accelerate neural computation. NEMO introduces neural electro-mechano-optic sensors that convert electrophysiology signals into optical modulation through a NEMS electrostatic transducer coupled to a silicon photonic microdisk resonator (Cochran et al., 20 Apr 2026). The electrostatic force is modeled as

$$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$6

and the microdisk transmission near resonance by

$$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$7

The device achieves a limit of detection down to $$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$8 microvolts, sufficient to detect neural signals, while its tiny input capacitance reduces stimulation artifact decay from $$\boldsymbol{z} = \left( \frac{\tilde{\boldsymbol{\phi}}_s^\top \bar{\boldsymbol{x}}}{\|\tilde{\boldsymbol{\phi}}_s\|^2} \right)\tilde{\boldsymbol{\phi}}_s + \boldsymbol{\mu}$$9 in a conventional Intan front-end to $p < 0.001$0 in NEMO (Cochran et al., 20 Apr 2026). Because multiplexing is wavelength-based, the design also has the stated potential for massive multiplexing of neural recordings (Cochran et al., 20 Apr 2026).

At the tissue interface level, photoacoustic silk scaffolds integrate PEG-functionalized carbon nanotubes into silk fibroin to create a biocompatible neural scaffold that is also a light-addressable stimulation interface (Zheng et al., 2021). The photoacoustic pressure is modeled as

$p < 0.001$1

and the scaffold enables non-genetic activation of neurons with spatial precision defined by the area of light illumination (Zheng et al., 2021). Under reported conditions, 94% of photoacoustic stimulated neurons exhibit a fluorescence change larger than 10% in calcium imaging in the light illuminated area, neurite outgrowth increases by 1.74-fold in a dorsal root ganglion model compared to the unstimulated group, and BDNF levels rise by 1.96-fold relative to baseline control (Zheng et al., 2021). Here, “neural” refers simultaneously to the target tissue, the stimulation modality, and the scaffold’s regenerative function.

In computing hardware, the neuromorphic accelerator “NEURAL” implements spiking neural networks through hybrid data-event execution, elastic FIFOs, and on-the-fly QKFormer attention (Chen et al., 18 Sep 2025). Its Elastic Processing Element Array, Pipelined Sparse Detection Array, and W2TTFS-based Fully Connected Core execute single-timestep SNNs while keeping the entire pipeline in spike domain (Chen et al., 18 Sep 2025). At the algorithm level, KD-trained VGG-11 improves accuracy by 3.20% on CIFAR-10 and 5.13% on CIFAR-100; at the architecture level, compared to existing SNN accelerators, NEURAL achieves a 50% reduction in resource utilization and a 1.97x improvement in energy efficiency (Chen et al., 18 Sep 2025). This hardware-specific use of “NEURAL” emphasizes event sparsity, low-latency neuromorphic execution, and attention-compatible spike dataflow.

The term also appears in geometry processing, where spherical neural surfaces represent genus-0 surfaces directly as neural maps $p < 0.001$2 restricted to $p < 0.001$3 (Williamson et al., 2024). Differential-geometric operators are computed without meshing: the first fundamental form,

$p < 0.001$4

the unit normal,

$p < 0.001$5

and the Laplace–Beltrami operator,

$p < 0.001$6

The framework supports spectral analysis, heat flow, and mean curvature flow directly on neural representations, and is fully seamless for genus-0 surfaces (Williamson et al., 2024). In this context, “neural” refers not to neurons at all, but to the fact that the surface itself is encoded as a neural network.

6. Conceptual tensions and open directions

Across these literatures, “neural” does not denote a single ontology. In some settings it names biological substrates shaped by synaptic learning, self-organization, and multisensory integration (Dresp-Langley, 2022). In others it names artificial models whose functional identity lies in depth-wise representation dynamics rather than static layers (Yan et al., 28 Feb 2026), function-space priors over neuronal tuning (Cotton et al., 2020), algebraic encodings of co-firing patterns (Curto et al., 2015), or event-driven hardware that exploits spike sparsity (Chen et al., 18 Sep 2025). This breadth creates productive ambiguity but also methodological tension.

One tension concerns the level of abstraction at which biological and artificial systems should be compared. NFAS argues for stable dynamical modes anchored to distributed fMRI responses rather than best-layer correlations (Yan et al., 28 Feb 2026). Neural ring theory instead treats binary code structure as the invariant substrate from which convexity and embedding dimension may be inferred (Curto et al., 2015). Neural machines emphasize fragmentation, aggregation, and nondeterminism as scale-invariant informational primitives (Petrila, 2024). These frameworks are not incompatible, but they privilege different invariants: geometry of activity codes, dynamics across depth, or graph-level information flow. This suggests that “neural similarity” is representation-dependent.

A second tension concerns nonlinearity and approximation. Several frameworks rely on linearized or simplified structure: DMD and linear encoding models in NFAS (Yan et al., 28 Feb 2026), graph-based regression in neural architecture search (Wen et al., 2019), linear pullback maps in neural ring homomorphisms (Curto et al., 2015), or high-Q linearized optomechanical transduction in NEMO (Cochran et al., 20 Apr 2026). At the same time, the surveyed work repeatedly emphasizes irreducible nonlinearity: non-deterministic memorization and nonlinear fragmentation/aggregation in neural machines (Petrila, 2024), fuzzy or probabilistic generalized constraints in PrINNs (Mazandarani et al., 2 May 2025), and nonlinear decision geometry in functional neural networks (Jumelle et al., 2021). A plausible implication is that future uses of “neural” will continue to oscillate between tractable linear surrogates and richer nonlinear descriptions.

A third tension concerns embodiment and deployment. Some research pushes “neural” toward increasingly abstract formal objects—neural rings, neural surfaces, neural predictors—while other work pushes toward direct interfaces with tissue and devices, such as photoacoustic silk scaffolds, electro-mechano-optic recording probes, and neuromorphic FPGA accelerators (Zheng et al., 2021). The coexistence of these directions indicates that the term is now infrastructural: it can refer to codes, models, sensors, actuators, and substrates so long as they participate in the representation, transmission, or manipulation of neural information. In that sense, “neural” functions less as a narrow disciplinary adjective than as a cross-domain category linking brain science, machine learning, algebra, geometry, and hardware.