Conductivity and Tissue Properties

Conductivity & Tissue Properties

The accuracy of the extracellular field (and therefore of every threshold and selectivity number) rests on the tissue conductivities. golgi ships a physically grounded, frequency-aware, anisotropic material model with literature-backed defaults that are fully editable in the Materials drawer.

Source: golgi/conductivity/ — materials.py (canonical σ table), cole_cole.py (frequency dispersion), itis_db.py (IT'IS database), perineurium.py (perineurium morphology).

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Tissue regions

The model uses tagged regions in the tetrahedral mesh. The default conductivities (S/m), drawn from the peripheral-nerve modeling literature, are:

Tag	Region	σ (transverse)	Anisotropy	Source
1	Endoneurium	≈0.167 (1/6)	longitudinal ≈0.571 (1/1.75)	fascicle interior — strongly anisotropic (Ranck & BeMent 1965; Pelot et al. 2019)
2	Saline	1.76	isotropic	cuff fill (Pelot et al. 2019)
3	Silicone	≈1e-12	isotropic	cuff body — near-insulator (Pelot et al. 2019)
4	Muscle	0.086	longitudinal 0.35	far-field tissue — anisotropic (Gielen et al. 1984; Pelot et al. 2017)
5	Epineurium	≈0.159 (1/6.3)	isotropic	outer nerve sheath (Grill & Mortimer 1994; Pelot et al. 2019)
6 / 8	Contact metal	≈9.43e6	isotropic	platinum bulk, equipotential (De Podesta 2001)
7	Encapsulation / scar	≈0.159 (1/6.3)	isotropic	fibrotic shell around the implant (Grill & Mortimer 1994)
—	Perineurium	≈1/1149 (8.7e-4)	sheet	contact-impedance sheet, not a meshed volume — see below (Weerasuriya et al. 1984; Pelot et al. 2019)

These follow the finite-element conductivity conventions compiled for peripheral nerve by Pelot et al. (2019) and used in the ASCENT pipeline (Musselman et al. 2021); full citations are in the References below. Every value is overridable per project (Materials drawer or MeshConfig/state). Anisotropy is represented as a per-cell diagonal tensor σ = diag(σ_T, σ_T, σ_L) with the longitudinal axis along the nerve, so the endoneurium and muscle conduct preferentially along the fiber direction.

Available electrode-material presets in the GUI include Au, Pt, Pt-Ir, stainless steel, Ti, TiN, IrOx, and a perfect-conductor option.

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Frequency dependence: the Cole–Cole model

Tissue conductivity is frequency dependent. golgi evaluates a Cole–Cole dispersion model (Gabriel et al. 1996):

$$ \sigma(\omega) = \sigma_{\text{ionic}} + \omega,\varepsilon_0,\varepsilon''(\omega), \qquad \varepsilon^{*}(\omega) = \varepsilon_\infty + \sum_i \frac{\Delta\varepsilon_i}{1+(j\omega\tau_i)^{1-\alpha_i}} $$

The evaluator (cole_cole_sigma) accepts an arbitrary list of dispersion terms. The IT'IS database supplies up to four dispersions per tissue (the full Gabriel et al. 1996 parametric fit); the in-app Cole–Cole dialog uses a three-term fit as the default working model. The Cole–Cole dialog (opened per tissue in the Materials drawer) lets you pick a tissue, evaluate σ(f) across the spectrum, and commit a value at your stimulation frequency.

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IT'IS material database

golgi uses the IT'IS Foundation tissue database, version 4.2 (Hasgall et al.) — Gabriel-model parameters (Δε₁–₄, τ₁–₄, α₁–₄, σ_ionic) for hundreds of tissues. A curated subset (nerve, muscle, fat, blood, CNS tissues, bone, skin, …) is surfaced in the UI dropdowns, and itis_sigma_at(tissue, freq_hz) returns the Cole–Cole σ at any frequency. Standard preset frequencies are 100 Hz, 1 kHz, 10 kHz, 100 kHz, and 1 MHz. The Nerve entry is included as a peripheral-nerve-specific addition over the generic Gabriel et al. 1996 fits.

The IT'IS database is not redistributed with golgi. It is freely available from the IT'IS Foundation under their own license terms. Fetch it in one step — python -m golgi.conductivity.fetch_itis (or golgi fetch-tissue-db) — which downloads it directly from itis.swiss and installs it under resources/tissue_db/; or install it manually (see that folder's README.md). Without it golgi runs normally, the Cole–Cole dialog falls back to the Custom preset, and a one-time hint at startup points you to the download.

Attribution. golgi downloads the IT'IS Foundation Tissue Properties Database V4-1 (IT'IS Foundation, 2022-02-22; DOI 10.13099/VIP21000-04-1). Please cite it and observe the IT'IS license/terms when using golgi's tissue properties.

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Perineurium contact impedance

The perineurium is thin (microns) but highly resistive, and its trans-sheath voltage drop dominates how much of the applied field reaches the axons inside a fascicle. Meshing it as a volume would force µm-scale refinement, so golgi models it as a contact-impedance (Robin) sheet — ASCENT's thin-layer approximation (Musselman et al. 2021):

• Area-specific sheet resistance Rs = peri_thickness / σ_peri, with σ_peri ≈ 1/1149 S/m (Weerasuriya et al. 1984; adopted by Pelot et al. 2019).
• Thickness as a function of fascicle diameter, peri_thk(d) = slope·d + intercept, with species-specific coefficients (rat / pig / human) after Pelot et al. 2019, with vagus morphology refined in Pelot et al. 2020.
• A two-field formulation (solve_nerve_ci.py) couples the endoneurium potential and the surrounding potential across the endo↔epi interface through the sheet conductance g = 1/Rs.

Enable it with GOLGI_RECIP_CI=1 plus perineurium_ci=True and a positive thickness. It is off by default (single-field solve). Turning CI on attenuates the field reaching the axons and therefore raises thresholds and reshapes selectivity — it is the physically correct choice when fascicular voltage gradients matter, at ~10–20% extra solve cost. See the Finite-Element Solver page for the block-form details.

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References

The conductivity and perineurium references below are those used in the golgi methods paper (and recorded in golgi/conductivity/materials.py and perineurium.py):

• De Podesta M (2001). Understanding the Properties of Matter. Taylor & Francis. — platinum bulk conductivity.
• Gabriel S, Lau RW, Gabriel C (1996). The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues. Phys Med Biol 41:2271. — Cole–Cole / IT'IS dispersion model.
• Gielen FLH, Wallinga-de Jonge W, Boon KL (1984). Electrical conductivity of skeletal muscle tissue: experimental results from different muscles in vivo. Med Biol Eng Comput 22:569. — muscle σ.
• Grill WM, Mortimer JT (1994). Electrical properties of implant encapsulation tissue. Ann Biomed Eng 22:23. — encapsulation/scar (and epineurium).
• Hasgall PA, Di Gennaro F, Baumgartner C, Neufeld E, Lloyd B, Gosselin MC, Payne D, Klingenböck A, Kuster N. IT'IS Database for thermal and electromagnetic parameters of biological tissues, Version 4.2. IT'IS Foundation. https://itis.swiss/database
• Musselman ED, Cariello JE, Grill WM, Pelot NA (2021). ASCENT (Automated Simulations to Characterize Electrical Nerve Thresholds): a pipeline for sample-specific computational modeling of electrical stimulation of peripheral nerves. PLOS Comput Biol. — thin-layer perineurium approximation.
• Pelot NA, Behrend CE, Grill WM (2017). Modeling the response of small myelinated axons in a compound nerve to kilohertz frequency signals. J Neural Eng 14:046022. — muscle anisotropy.
• Pelot NA, Thio BJ, Grill WM (2019). On the parameters used in finite element modeling of compound peripheral nerves. J Neural Eng 16:016007. — adopted FE conductivities (endo/epi/saline/silicone/perineurium).
• Pelot NA, Goldhagen GB, Cariello JE, Musselman ED, Clissold KA, Ezzell JA, Grill WM (2020). Quantified morphology of the cervical and subdiaphragmatic vagus nerves of human, pig, and rat. Front Neurosci 14:1148. — perineurium thickness vs fascicle diameter.
• Ranck JB, BeMent SL (1965). The specific impedance of the dorsal columns of cat: an anisotropic medium. Exp Neurol 11:451. — endoneurium anisotropy.
• Weerasuriya A, Spangler RA, Rapoport SI, Taylor RE (1984). AC impedance of the perineurium of the frog sciatic nerve. Biophys J 46:167. — perineurium conductivity.

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See also

Finite-Element Solver · Meshing · Configuration Reference · Validation