Tensor-Network Ancestry and Basis-Transform Doctrine¶

Status: v0.2.29 conceptual documentation
Version: v0.2.28-aligned
run_status: tutorial_scaffold
Scope: Terminology, historical context, architectural parallels; no implementation notes

Purpose¶

This document situates TFNE within two distinct meanings of tensor network:

Pellionisz/Llinás (1980s–2000s): Tensor network as a geometric framework for sensorimotor transforms and cerebellar learning in neuroscience
Modern ML/Physics (2010s–present): Tensor network as a factorized state compression and contraction algorithm (tensor trains, MPS, PEPS, etc.)

jaxfne adopts the basis-transform architecture from the first tradition (multi-scale coordinate projections: emitter basis → source basis → field basis → readout basis) while remaining distinct from both traditions' full scope.

This document clarifies what TFNE does, what it does not implement, and why the basis-transform concept matters for modularity and extensibility.

Part 1: Two Meanings of "Tensor Network"¶

Pellionisz/Llinás Neuroscience Meaning¶

Era: Pellionisz & Llinás (1980–2010), continued by Porrill, Dean, and others

Core idea: Neurons solve coordinate-transform problems. A cerebellar Purkinje cell receives: - Sensory (retinotopic, somatosensory) input in one coordinate frame (retinal, skin) - Motor efference copy in another frame (joint angles, velocity) - Output command in a third frame (muscle activation)

The cerebellum learns to map between frames via a metric tensor.

Mathematical core:

\[V^{motor} = g^{motor,sensory}_{ij} S^{sensory}_{j} + \text{memory terms}\]

where \(g\) is a learned tensor (metric) that transforms sensory coordinates into motor coordinates.

Why "tensor network"? Multiple sensory and motor dimensions are contracted via a learned coupling tensor.

Modern ML/Physics Meaning¶

Era: Vidal (2003–present), developed in quantum information and condensed-matter physics

Core idea: A large high-dimensional tensor can be factorized into a network of smaller tensors connected along bonds (virtual indices).

Example: Tensor Train (TT) or Matrix Product State (MPS)

\[T(i_1, i_2, \ldots, i_n) = C_1(i_1) \times C_2(i_2) \times \cdots \times C_n(i_n)\]

(where \(\times\) denotes contraction along virtual indices)

Why "tensor network"? The high-dimensional tensor is built by contracting a network of lower-rank local tensors.

Applications: - Quantum state compression - Efficient PDE solvers - Variational autoencoders (VAE) with factorized latents

Part 2: TFNE Basis-Transform Architecture¶

TFNE adopts the coordinate-transformation philosophy (Pellionisz/Llinás) but does NOT implement: - Learned metric-tensor coefficients - Cerebellar learning dynamics - Full sensorimotor circuit models

TFNE Basis Contract¶

TFNE organizes computation as cascaded basis transforms:

Formal expression:

\[ \boxed{ Y^{readout}_a = P_{a\alpha c} Z^{field}_{\alpha c} } \]

Expanded (four basis transforms):

\[ X^{probe}_a = P_{a, basis(source \to probe)} \left( F_{basis(field \to source)} \left( Q_{basis(emitter \to field)} V^{state}_{state\_index} \right) \right) \]

Worded (English):

Emitter basis: Neural state (membrane voltage, gating variables, synaptic conductances) in a neuron-indexed frame
Source basis: Currents or source densities projected into a spatial frame (contact positions, voxels)
Field basis: Extracellular potentials or proxy potentials at measurement locations
Readout basis: Multiple simultaneous readouts (spikes, raw voltage, LFP proxy, CSD proxy, etc.)

Bridge terms (which basis transforms are computed):

Transform	Implemented	Status
Emitter → Source	✓ (always)	Mandatory; state projection via current/conductance emission
Source → Field	◑ (optional)	Proxy-only or PDE-based (reserved)
Field → Readout	✓ (selective)	User chooses probe operators (8 available: SPK, Vm, source, LFP, CSD, EEG, MEG, EMM)

Why Basis Transforms Matter¶

Modularity: Each stage is independent. A source can exist without field solve; a field without EEG readout.
Run boundary: Each basis transform carries its own status status. Source projection is deterministic; field solve is (currently) proxy-only.
Extensibility: New bases (ionic current frame, spectral frame, etc.) fit the same architecture without breaking the pipeline.
Tensor structure: The cascade is naturally a tensor contraction chain: state → source density → field potential → readout metrics.

Part 3: Probe-Basis Tensor Example¶

TFNE probe readouts implement the final basis transform:

Formal:

\[Y_a = P_{a,i,j} \, Z_{i,j}\]

where: - \(a\): readout index (spike, voltage, LFP, CSD, etc.) - \(P_{a,i,j}\): probe operator (includes geometry, filtering, etc.) - \(Z_{i,j}\): field basis (spatial location \(i\), time step \(j\)) - \(Y_a\): scalar or vector readout

Example (LFP proxy):

\[\text{LFP\_proxy}(t) = \sum_{contacts} w_{contact} \, \phi_e(contact, t)\]

(where \(\phi_e\) is extracellular potential and \(w_{contact}\) are weights from contact geometry)

Proxy note: LFP-proxy is a computational readout derived from simulated sources in relative units. See Limitations and future plans for calibration scope.

Part 4: What TFNE Does¶

TFNE Does¶

✓ Organize emitter → source → field → readout as a modular tensor-contraction pipeline
✓ Support multi-basis workflows (e.g., Izhikevich emitter in mV, source in nA, LFP proxy in arbitrary units)
✓ Provide 8 probe operators for simultaneous multimodal readouts
✓ Validate status checks: proxy-only readouts under the package truth gates
✓ Support extensions: new emitters, field operators, and probes within the same basis-transform architecture

The proxy-readout scope, reserved field-solver regimes, and calibration boundaries are catalogued in Limitations and future plans.

Part 5: Relationship to BasisSpec Contract¶

jaxfne's BasisSpec (introduced in v0.2.25) formalizes the basis-transform idea:

from jaxfne.core import BasisSpec

# Example: declare basis transforms
emitter_basis = BasisSpec(name="izhikevich_state", units="mV", n_dims=4)
source_basis = BasisSpec(name="synaptic_current", units="nA", n_dims=50)
field_basis = BasisSpec(name="laminar_potential", units="proxy_mV", n_dims=16)
readout_basis = BasisSpec(name="multimodal", units="mixed", n_dims=8)

# Pipeline preserves basis contracts through transformations

Doctrine: BasisSpec makes the tensor-coordinate structure explicit and testable. It is the operational instantiation of "basis transform" as a software contract.

Part 6: Reserved Optional Path (Not Implemented)¶

A reserved cerebellar/sensorimotor tutorial could use jaxfne's basis-transform architecture as a teaching tool:

Hypothetical example (NOT IMPLEMENTED):

# Hypothetical cerebellar learning model
# Would require:
# 1. A cerebellar-circuit emitter (Purkinje/Granule dynamics)
# 2. A metric-tensor learning rule in the source basis
# 3. Validation against empirical cerebellar response

# Such a tutorial would be a NEW module with its own:
# - Status checks
# - Validation tests
# - Scope Note that this is exploratory, not a biological proof

This path is deferred and not promised. If pursued, it would: - Require separate validation evidence - Use separate status checks (distinct from computational_scaffold) - Be a distinct research module, not a core jaxfne feature

Part 7: Distinction from Other Tensor-Network Meanings¶

Term	Meaning	jaxfne Role	Statement?
Tensor network (Pellionisz)	Sensorimotor coordinate transforms, metric-tensor learning	✓ Architectural inspiration	✗ No implementation
Tensor network (ML/Physics)	Factorized state compression (MPS, PEPS, TT)	✗ Not used	✗ Status note
Basis transform	Cascade of coordinate changes (emitter → source → field → readout)	✓ Core TFNE principle	✓ Implemented
Tensor contraction	Algebraic operation summing over shared indices	✓ Mathematical formalism	✓ Implicit in readouts

Summary: Why Basis-Transform Doctrine Matters¶

Architecture: Basis transforms make TFNE's modularity explicit. Users understand why sources can exist without fields.
Extensibility: New bases (ionic channels, spectral, population-level) fit the same framework.
Statement clarity: Each basis transform has its own status status and status check.
Teaching: The basis-coordinate idea connects TFNE to classical computational neuroscience (Pellionisz, Koch, Arleo) while remaining distinct.
Humility: By referencing Pellionisz/Llinás and NOT stating their results, we honor the intellectual history while respecting scope boundaries.