Skip to content

Mathematical Glossary Flow

Purpose

Mathematical glossary flow is the documentation format used to teach TFNE/jaxfne equations by connecting formal mathematics to physical intuition, computational implementation, and statement boundaries.

Each important equation is presented in: 1. Simple/familiar form 2. Tensor/general form (when useful) 3. Complete term glossary 4. Worded-equation (plain-language translation) 5. Critical bridge term (how this equation connects the pipeline) 6. Run boundary (what scientific status this equation carries) 7. Implementation location (where in code/report this shows up)

This structure ensures that equations remain teachable, grounded, and appropriately stated.

Core TFNE Equations

1. Emitter Dynamics

Formal equation:

\[\frac{dz}{dt} = F_\theta(z, u, t)\]

Tensor form: component-wise, same as above.

Term glossary: - \(z(t) \in \mathbb{R}^{n_\theta}\): neural state vector (membrane voltages, gating variables, synaptic conductances, etc.) - \(u(t)\): input (stimulus, noise, recurrent current) - \(F_\theta\): declared neural dynamics function (parametrized emitter) - \(\theta\): emitter parameters (time constants, thresholds, conductance scales, etc.) - \(t\): time

Worded-equation:

\[\boxed{\text{state change} = \text{declared neural dynamics applied to state, input, parameters, and time}}\]

Critical bridge term:

\(F_\theta\) is the Emitter → Source bridge. It transforms parameter choices and input currents into state evolution that later becomes membrane current and spike signals.

Run boundary:

  • Computational scaffold unless \(F_\theta\) is calibrated to empirical neural response
  • Not biological proof without comparison to experimental data
  • Izhikevich default in jaxfne; other emitters supported via custom models

Implementation: - jaxfne.emitters.IzhikevichParams — parameter container - jaxfne.emitters.simulate_eig_izhikevich() — forward simulation - jaxfne/core.py — Model interface

2. Source Projection

Formal equation:

\[q(x,t) = P_s[z(t), I(t), \chi(x)]\]

Tensor form:

\[q_\alpha(x,t) = \sum_k w_{k\alpha}(x) \cdot (a_k \cdot z_k(t) + b_k \cdot I_k(t))\]

where \(\alpha\) indexes spatial contact points, \(k\) indexes neurons, and \(w\) are spatial projection weights.

Term glossary: - \(q(x,t)\): field source density (current per unit volume or contact area) - \(P_s\): source projection operator (maps neural state/current to spatial domain) - \(z(t)\): neural state (from emitter) - \(I(t)\): input/drive current - \(\chi(x)\): spatial contact basis (probe geometry, source locations) - \(w_{k\alpha}(x)\): spatial projection weight from neuron \(k\) to contact \(\alpha\) (anatomical distance, contact coupling) - \(a_k\): state-to-source projection scalar for unit \(k\) (scales how much of the emitter state contributes to the source; proxy coefficient, not a calibrated physical constant) - \(b_k\): input-to-source projection scalar for unit \(k\) (scales how much of the drive/input current contributes to the source; proxy coefficient, not a calibrated physical constant)

Worded-equation:

\[\boxed{\text{field source density} = \text{projection of neural state and current into spatial tissue coordinates}}\]

Critical bridge term:

\(P_s\) is the Source → Field bridge. It transforms time-domain neural currents into spatially-localized sources that can be convolved with field operators.

Run boundary:

  • Physical only when source_calibration_status includes calibrated or empirical current evidence
  • Proxy in default linear_solver path (no physical current units stated)
  • Model current (Izhikevich model current + spike impulse proxy) by default
  • No double-counting of synaptic current (forbidden pattern: using both synaptic conductance-based current and spike-based source in the same field computation)

Implementation: - jaxfne.fields.project_laminar_sources() — source projection kernel - jaxfne/core.py → Signals — output structure - Manifest field: source_calibration_status

3. Ohmic Extracellular Current

Formal equation:

\[\mathbf{J}_e = -\sigma_e \nabla \phi_e\]

Tensor/index form:

\[J_e^i = -\sigma_e^{ij} \partial_j \phi_e\]

Term glossary: - \(\mathbf{J}_e(x,t)\): extracellular current density (A/m²) - \(\sigma_e(x)\): extracellular tissue conductivity tensor (S/m) - \(\phi_e(x,t)\): extracellular voltage (mV) - \(\nabla \phi_e\): voltage gradient

Worded-equation:

\[\boxed{\text{extracellular current density} = \text{passive conductive tissue response to voltage gradient}}\]

Critical bridge term:

\(\sigma_e\) is the Field → Current bridge. It encodes how tissue converts electrostatic voltage gradients into current flow.

Run boundary:

  • Proxy in current v0.2.24–v0.2.27 linear_solver path (assumed isotropic, no PDE solve)
  • Physical only when:
  • Field solver is active and verified (field_solver_status != linear_solver)
  • Conductivity is calibrated (conductivity_status == calibrated_physical)
  • Solution satisfies conservation constraints
  • Isotropy assumed in current proxy mode (conductivity is scalar)

Implementation: - jaxfne.fields.project_laminar_sources() — applies conductivity implicitly via kernel - Manifest: field_solver_status, boundary_condition, gauge

4. Field Compatibility Equation (Poisson-like)

Formal equation:

\[\nabla \cdot (-\sigma_e \nabla \phi_e) = q\]

Tensor/index form:

\[\partial_i (-\sigma_e^{ij} \partial_j \phi_e) = q\]

Term glossary: - \(\nabla \cdot\): divergence operator - \(q(x)\): source density (from source projection) - \(\phi_e\): extracellular potential

Worded-equation:

\[\boxed{\text{divergence of conductive extracellular current} = \text{declared source density}}\]

Critical bridge term:

\(q\) is the Source → Field boundary condition. It ensures the field solution (when computed) is compatible with declared sources.

Run boundary:

  • Current default: linear_solver — equation is declared but NOT solved
  • Reserved path: specified_future_solver for a calibrated field solver (see Limitations and future plans)
  • Not a physical status unless solver evidence exists

Implementation: - jaxfne.fields.project_laminar_sources() — declares but doesn't solve - Manifest: field_solver_status == "linear_solver" means this equation is metadata only

5. Current-Source Density (CSD)

Formal equation:

\[\mathrm{CSD}(x,t) = \nabla \cdot \mathbf{J}_e(x,t) = -\nabla \cdot (\sigma_e \nabla \phi_e)\]

Tensor form:

\[\mathrm{CSD} = \partial_i J_e^i\]

Term glossary: - \(\mathrm{CSD}\): current-source density (A/m³) - Positive CSD: extracellular current divergence (current flowing outward, suggesting membrane sink) - Negative CSD: current convergence (current flowing inward, suggesting membrane source)

Worded-equation:

\[\boxed{\text{current-source density} = \text{local divergence of extracellular current density}}\]

Critical bridge term:

CSD turns a field-current pattern into a source/sink-like readout. It is the primary laminar-field observable.

Run boundary:

  • CSD-proxy unless source AND field are calibrated/solved
  • Sign convention in jaxfne: positive_equals_extracellular_source (positive CSD = current flowing outward)
  • Not a direct biological measurement in v0.2.24–v0.2.27 (no physical conductivity or solved field)

Implementation: - jaxfne.fields.project_laminar_sources() — computes CSD proxy directly - Manifest: csd_sign_convention, field_model_status == "proxy_readout"

6. Probe Operator (General)

Formal equation:

\[Y_c(t) = P_c[\phi_e(t), \mathbf{J}_e(t), \mathrm{CSD}(t), \ldots]\]

Tensor form: operator-dependent (contact-wise, depth-wise, or spatially integrated).

Term glossary: - \(Y_c(t)\): channel readout (voltage, current, field quantity at contact \(c\)) - \(P_c\): probe operator (e.g., voltage sampling, CSD extraction, LFP bandpass) - \(c\): contact index (spatial location on probe)

Worded-equation:

\[\boxed{\text{channel readout} = \text{declared probe operator applied to field and source-derived quantities}}\]

Critical bridge term:

\(P_c\) is the Field → Probe bridge. It extracts spatially localized measurements from the field solution.

Run boundary:

  • Simulated/proxy readout unless:
  • Geometry (lead field) is calibrated to real electrodes
  • Field is solved (not proxy)
  • Validation against empirical data exists
  • Eight multimodal operators in jaxfne: SPK, Vm, source, LFP-proxy, CSD-proxy, EEG-proxy, MEG-proxy, EMM-proxy
  • See Probe Operators for operator-specific statement boundaries

Implementation: - jaxfne.fields.probe_laminar_modes() — operator definitions - jaxfne/core.py → Model.compute_readout() — readout computation - Manifest: operator_status for each probe operator

7. EMM-Proxy (Electro-Mechanical Metabolic-like Proxy)

Formal equation:

\[\mathrm{EMM}_\text{proxy}(t) = w_s \|\mathrm{source}(t)\|_2^2 + w_E \|\nabla\phi_e(t)\|_2^2 + w_J \|\mathbf{J}_e(t)\|_2^2\]

Term glossary: - \(\mathrm{EMM}_\text{proxy}\): dimensionless activity cost - \(w_s, w_E, w_J\): scalar weights (default: normalized by signal variance) - \(\|\cdot\|_2\): Euclidean (L2) norm

Worded-equation:

\[\boxed{\mathrm{EMM\text{-proxy}} = \text{weighted normalized source activity + field-gradient activity + current-density activity}}\]

Critical bridge term:

EMM-proxy is a relative within-run cost/activity index, not biological metabolism. It combines source, field-gradient, and current-density norms to measure "computational cost" or "activity intensity."

Run boundary:

  • Proxy-only in v0.2.24–v0.2.27
  • Computational cost metric — combines source and field norms, not ATP consumption or metabolic rate
  • Valid for relative within-run comparison (e.g., which timestep is most active)
  • Requires empirical calibration for any biological interpretation statements

Implementation: - jaxfne.fields.probe_laminar_modes() → EMM-proxy operator - Manifest: source_model includes EMM-proxy definition - See Probe Operators for full details

Conservation-Law Doctrine (Reserved Reference)

Poynting's Theorem

Formal equation:

\[\frac{\partial u_{em}}{\partial t} + \nabla \cdot \mathbf{S} + \mathbf{J} \cdot \mathbf{E} = 0\]

where

\[u_{em} = \frac{1}{2}\left(\epsilon_0 |\mathbf{E}|^2 + \frac{1}{\mu_0}|\mathbf{B}|^2\right)\]
\[\mathbf{S} = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B})\]

Term glossary: - \(u_{em}\): electromagnetic field energy density (J/m³) - \(\mathbf{S}\): Poynting vector (energy flux density, W/m²) - \(\mathbf{J} \cdot \mathbf{E}\): power density transferred to charges (W/m³) - \(\epsilon_0\): permittivity of free space - \(\mu_0\): permeability of free space

Worded-equation:

\[\boxed{\text{field energy change} + \text{energy flux leaving region} + \text{work done on charges} = 0}\]

Critical bridge term:

\(\mathbf{J} \cdot \mathbf{E}\) is the field-to-matter power density bridge. It quantifies electromagnetic power transferred to moving charges.

Run boundary:

  • Recorded here as conservation-principle motivation for the field diagnostics
  • Full Maxwell/stress-energy tensor dynamics belong to the reserved regimes in Limitations and future plans

Why included: Poynting's theorem provides a conservation-principle foundation for the field diagnostics and the reserved electrodynamic regimes catalogued in Limitations and future plans.

Implementation Checklist

When adding a new equation to jaxfne documentation:

  • [ ] Include formal equation (LaTeX)
  • [ ] Include tensor/index form (if useful for generalization)
  • [ ] Define every term in glossary
  • [ ] Provide worded-equation in plain language
  • [ ] Identify critical bridge term and pipeline location
  • [ ] State run boundary (computational scaffold? proxy? calibrated physical?)
  • [ ] Link to code location and manifest fields
  • [ ] Verify no forbidden phrasing (real EEG, biological metabolism, mechanism proof)

See Also