Neurophysiology-Grounded Axes

• Neurophysiology-grounded axes are defined as coordinate directions in neural state-spaces derived from observable physiological and behavioral features.
• They are identified using methods like sub-Riemannian metrics, spectral decomposition, and clustering to align geometric models with empirical neural selectivity.
• These axes underpin research across motor, hippocampal, and artificial domains, enabling robust, interpretable mappings between neural activity and computational models.

Neurophysiology-grounded axes refer to coordinate systems, directions, or subspaces in neural, behavioral, or artificial neural system state-spaces that are directly definable, interpretable, or derivable from measurable physiological properties. These axes form the structural backbone for understanding coding, transformation, and manipulation of neural activity by leveraging empirically validated physiological features and constraints. Research across motor, spatial, cognitive, and artificial domains demonstrates the construction, identification, and experimental utility of such axes, aligning geometric, statistical, and functional approaches with grounded neurophysiological observations.

1. Theoretical Motivation and Definitions

The core motivation for neurophysiology-grounded axes is to establish reference frames and dimensions within neural or artificial systems that correspond directly to observable or manipulable quantities in physiology or behavior. Unlike arbitrary mathematical or model-derived directions, these axes are anchored to biological measurements (e.g., preferred direction, speed, phase-locking in MEG) or experimentally defined transformations (e.g., sensory or motor maps).

Henri Poincaré articulated that geometry is a convention, not a fixed property of the world—a view directly relevant to neural systems, which lack a global, Euclidean, or simultaneity-enforced coordinate frame. Instead, neural axes or coordinates must be inferred from the relational structure of observed activity and physiological interactions (Goffart, 14 May 2024).

Empirically, axes are grounded by matching directions of state-space variation to (i) known tuning properties of neural units (e.g., direction selectivity in M1, head-direction cells in hippocampus), (ii) orthogonality constraints born from topological and anatomical organization, or (iii) projections that preserve or optimize the structure of synaptic or representational connectivity with measurable behavioral correlates.

2. Neurogeometric Models in Motor Cortex

Neurophysiological axes in primary motor cortex (M1) are derived from cellular, anatomical, and functional properties:

• Each "feature-tuning point" (motor cell) is parameterized by $\eta = (x, y, t, \theta, v, a)$, with $x, y$ (planar position), $t$ (time), $\theta$ (preferred direction), $v$ (speed), $a$ (acceleration) (Mazzetti et al., 22 Dec 2024, Mazzetti et al., 2022).
• The manifold $$\mathcal{M} = \mathbb{R}^2 \times \mathbb{R}_t \times S^1_\theta \times \mathbb{R}_v \times \mathbb{R}_a$$ integrates these physiological features.
• Three nonholonomic constraints (kinematic consistency via vanishing 1-forms) restrict physically plausible transitions and determine allowable local signal propagation directions:

$$\omega_1 = \cos\theta\,dx + \sin\theta\,dy - v\,dt = 0,\quad \omega_2 = -\sin\theta\,dx + \cos\theta\,dy = 0,\quad \omega_3 = dv - a\,dt = 0$$

• The system is equipped with three orthonormal horizontal vectors, $X_1$ (direction+time), $X_2$ (rotation in $\theta$), and $X_3$ (acceleration), which define principal directions in tangent space at each neural state.
• The resulting sub-Riemannian metric treats the physiological cost of changing each variable symmetrically, defining a geometrically principled distance $d_{\mathcal{M}}$ on $\mathcal{M}$.
• Spectral decomposition of the kernel $$K_{\mathcal{M}}(\eta_0, \eta) = \exp(-d_{\mathcal{M}}^2(\eta_0,\eta))$$ produces global coordinate axes (eigenfunctions) whose level sets or gradients align with known tuning curves and physiological features.
• Hierarchical clustering in the space of neural activity fragments organizes observed spatiotemporal patterns (e.g., "neural states") found experimentally in M1 (Mazzetti et al., 22 Dec 2024).

This approach confirms that the axes for time, direction, speed, and acceleration, as used in sub-Riemannian modeling, are not arbitrary but are dictated by the underlying physiological selectivity and connectivity.

3. Grounded Spatial Reference Frames in the Hippocampus

Spatial axes in hippocampal navigation circuitry emerge from the interaction of place and head-direction cell assemblies (Dabaghian, 2021):

• Place-cell activity is modeled as a simplicial complex $T_\sigma(t)$, stabilizing to the nerve of the place-field cover of environment $E$.
• Head-direction assemblies $T_\eta(t)$ represent the topology of $S^1$ in head-orientation space.
• Coactivity of place and direction cells defines "pose" simplices $T_\zeta(t)$, converging on the topology of $O = E \times S^1$.
• Synthetic affine-geometric rules (G1–G3) ensure that alignments, axes, and collinearity arise purely from co-firing statistics. Orthogonal axes (e.g., "north" and "east") are constructed by selecting two nonparallel, frequently observed head directions.
• The coordinate frame is formalized by mapping ensemble spike counts to vector directions using weighted centroids and integer lattice operations.

A summary table of biologically grounded axes in the hippocampal–head-direction system:

Axis Type	Construction Principle	Physiological Basis
Place–cell (location) axes	Simplicial coactivity + centroiding	Place field tuning
Head–direction (orientation) axes	Repeated direction selectivity (S¹ fibers)	Head-direction selectivity
Synthetic affine orthogonality	Frequent parallel/non-parallel "runs"	Statistical co-firing

This approach demonstrates that a coordinate system can emerge without presupposed Euclidean or physical axes, with all structural axes derivable from bottom-up statistics of spike trains and combinatorial rules (Dabaghian, 2021).

4. Brain-Grounded Axes in LLMs

Neurophysiology-grounded axes extend to artificial neural systems, notably in interpretability methods for LLMs (Andric, 22 Dec 2025):

• MEG signals from the SMN4Lang dataset are used to construct a word-level brain atlas of phase-locking value (PLV) patterns during language comprehension.
• Independent Component Analysis (ICA) on the 128-dimensional PLV-PCA state produces latent axes, interpretable via correlation with semantic and lexical properties (e.g., frequency, function/content, animacy).
• Brain-derived axes are mapped to LLM hidden states via linear ridge adapters, without altering pretrained weights.
• Manipulating LLM activations along these axes effects interpretable behavioral shifts in generated text (e.g., systematic shifts in lexical frequency or function/content word ratio), validated via text-level metrics and controlled for perplexity.
• The extracted axes are robust across ablation experiments (e.g., omitting text-based features or using word2vec), alleviating concerns about circularity or model-specific bias.

These results establish neural measurements as an independent, externally valid criterion for constructing and manipulating axes in model state-space, offering a mechanistically interpretable interface for behavior control.

5. Local Coding Geometry in Artificial Neural Networks

Neurophysiology-grounded axes are crucial for resolving ambiguities in model evaluation, revealing the limitations of purely performance-based metrics (McNeal et al., 27 Sep 2025):

• The "local coding axis" for a predicted neuron or voxel is defined as the input perturbation $\delta$ (constrained in norm) that maximally modulates the model's output, computeable via gradients or Jacobian singular value decomposition.
• Robust (adversarially trained) ANN-based brain models possess stable, transferable local coding axes that show semantic alignment with true neural selectivity (e.g., face structure in FFA, scene geometry in PPA).
• Standard models may achieve high predictive accuracy but lack reliability and do not generalize coding directions across architectures, as revealed by low subspace overlap and poor axis transferability.
• Adversarial sensitivity functions, transfer matrices, and subspace alignment metrics collectively support the conclusion that local representational geometry, not pointwise prediction accuracy, reflects mechanistic fidelity to brain-like coding axes.

Thus, neurophysiology-grounded axes serve both as diagnostic tools and as bases for formulating and testing causal hypotheses in neural system modeling.

6. Anatomical and Evolutionary Basis for Grounded Axes

The physical axes in vertebrate neuroanatomy are determined not simply by function but by developmental and evolutionary events:

• The ancestral axial twist hypothesis explains the origin of contralateral organization in the forebrain, optic chiasm, and sensory/motor mapping by a 90° rotation (and opposed compensatory migrations) during embryogenesis (Lussanet et al., 2010).
• Consequently, forebrain axes (e.g., dorsal/ventral orientation) must be interpreted as inverted with respect to trunk and spinal axes, grounding all anatomical and functional axes in evolutionary morphogenetic events rather than teleological function alone.
• Comparative, embryological, and molecular data corroborate this framework, explaining the systematic organization of crossed vs. uncrossed tracts and the conservation of axes in vertebrate neuroanatomy.

7. Debates and Methodological Considerations

There is ongoing debate about the interpretation and implementational status of physical or mathematical quantities in neural systems (Goffart, 14 May 2024):

• Quantities such as position, velocity, and acceleration are often recoverable from activity patterns, but their "representation" may be an analytic artifact rather than a mechanistic feature.
• No global, brain-wide "now" exists; all measurements and computations are realized via distributed, asynchronous, and convergent/divergent pathways.
• Firing-rate and population vector models can overstate the groundedness of particular axes due to analytic choices (e.g., smoothing kernels, averaging), measurement outcomes (which reflect net effect rather than process), and lack of direct evidence that these codes are inverted downstream.
• Axes in neural state-space are best interpreted as local, contextual constructs—poly-equilibria sensitive to task constraints and neuroanatomical boundaries—not absolute, intrinsic, or static coordinate grids.

A plausible implication is that neurophysiology-grounded axes must always be interpreted within the measurement, transformation, and reference-frame context in which they are constructed and validated.

• (Mazzetti et al., 22 Dec 2024) A sub-Riemannian model of neural states in the primary motor cortex
• (Mazzetti et al., 2022) Functional architecture of M1 cells encoding movement direction
• (Dabaghian, 2021) Learning Orientations: a Discrete Geometry Model
• (McNeal et al., 27 Sep 2025) Targeted perturbations reveal brain-like local coding axes in robustified, but not standard, ANN-based brain models
• (Andric, 22 Dec 2025) Brain-Grounded Axes for Reading and Steering LLM States
• (Lussanet et al., 2010) An ancestral axial twist explains the contralateral forebrain and the optic chiasm in vertebrates
• (Goffart, 14 May 2024) Cerebralization of mathematical quantities and physical features in neural science: a critical evaluation

These works anchor the contemporary understanding of neurophysiology-grounded axes in a multi-scale, multidomain, and empirically rigorous framework, connecting geometric, statistical, dynamical, and evolutionary perspectives to the structural organization of neural coding and computation.