                    IDNE 701 · Fall 2026 · Week 4 Lecture
                

Biophysical Foundations of Neural Contrast

From Quantum Spins to Photons, Volume Conduction & Acoustic Waves

                    Mark Bolding, PhD · Department of Biomedical Engineering
                

The 6-Order-of-Magnitude Gap

Non-invasive neuroengineering bridges microscopic physiology to macroscopic external sensors:

1 µm

Axon & Dendrite Caliber

1 pA

Single Channel Current

1 ms

Action Potential Duration

15 mm

Distance Through Cranium

Microscopic Reality

Individual ion channel gating and action potentials produce quadrupole current fields that decay as $1/r^3$, undetectable outside the dura mater.

Macroscopic Observation

We measure coherent spatial summations: synchronized postsynaptic dipoles, bulk water proton magnetization, or scattered diffuse photon flux.

The Four Physical Modalities

Every non-invasive brain technology operates via one of four physical carrier channels:

MRI Nuclear Spins

Water proton precession ($^1\text{H}$) in field $B_0$. BOLD detects paramagnetic deoxyhemoglobin ($\Delta\chi$) dephasing.

Law: Bloch Equations

fNIRS Photons

NIR light ($650 - 950\,\text{nm}$) in optical window. Mie scattering ($\mu_s' \gg \mu_a$) and chromophore absorption.

Law: Modified Beer-Lambert

EEG Ionic Currents

Synchronous EPSPs in Layer V pyramidal neurons. Quasi-static volume conduction through resistive skull.

Law: Poisson & Biot-Savart

fUS Acoustic Waves

High-frequency plane-wave ultrasound ($1 - 15\,\text{MHz}$). Doppler tracking of red blood cell backscatter.

Law: Wave & Doppler

Quantum Spins & Zeeman Splitting

Hydrogen nuclei ($^1\text{H}$, protons) possess spin $I = 1/2$. An external field $B_0$ breaks degeneracy into two energy levels:

Zeeman Energy Splitting

$$\Delta E = \gamma \hbar B_0$$

$\gamma / 2\pi = 42.58\,\text{MHz/T}$ for $^1\text{H}$

Thermal equilibrium produces a slight excess of parallel spins governed by the Boltzmann distribution.

Curie's Law (Net Magnetization)

$$M_0 = \frac{N_s \gamma^2 \hbar^2}{4 k_B T} B_0$$

Field Strength Scaling

At $3.0\,\text{T}$, only $\sim 10$ spins per million contribute to net magnetization $M_0$.

Because $M_0 \propto B_0$ and induction $\propto \omega_0$, raw SNR scales superlinearly: $\text{SNR} \propto B_0^{1.5 - 2.0}$.

The Vector Bloch Equations

Governing the relaxation and precession of macroscopic magnetization $\vec{M}(t)$:

$$\frac{d\vec{M}}{dt} = \gamma \vec{M} \times \vec{B} - \frac{M_x \hat{i} + M_y \hat{j}}{T_2} - \frac{(M_z - M_0)\hat{k}}{T_1}$$

Longitudinal Relaxation ($T_1$)

$$M_z(t) = M_0 - [M_0 - M_z(0)] e^{-t / T_1}$$

Spin-lattice energy dissipation to molecular lattice. $T_1 \approx 800 - 1800\,\text{ms}$ in brain tissue.

Transverse Relaxation ($T_2, T_2^*$)

$$M_{xy}(t) = M_{xy}(0) e^{-t / T_2^*} e^{-i \omega_0 t}$$

Spin-spin phase decoherence from molecular dipolar interactions + field gradients. $T_2^* \approx 30 - 60\,\text{ms}$.

The BOLD Effect in fMRI

Diamagnetic $\text{HbO}_2$: All electrons paired ($S = 0$). Volumetric susceptibility matches tissue ($\Delta\chi \approx 0$).
Paramagnetic $\text{HbR}$: 4 unpaired $d$-electrons ($S = 2$). Creates microscopic field gradients ($\Delta\chi \approx +0.2\,\text{ppm}$).
Dephasing: Water protons diffusing through gradients dephase faster, shortening $T_2^*$.

$$\Delta R_2^* = \frac{1}{T_2^*} - \frac{1}{T_2} \propto [\text{HbR}] \cdot B_0^\beta$$

The Hyperemic Paradox

Neural activation triggers local vasodilation via nitric oxide and adenosine.

Cerebral blood flow surges $30 - 60\%$, vastly outpacing oxygen consumption ($5 - 15\%$).

Result: Venous $[\text{HbR}]$ drops, $T_2^*$ lengthens, and MR signal increases ($1 - 5\%$).

Diffuse Optical Physics (fNIRS)

Optical interrogation relies on the diagnostic optical window ($650 - 950\,\text{nm}$):

Turbid Tissue Regime

Scattering completely dominates absorption:

$\mu_a \approx 0.01\,\text{mm}^{-1}$ (absorption path $\sim 100\,\text{mm}$)
$\mu_s \approx 10\,\text{mm}^{-1}$ (scattering path $\sim 0.1\,\text{mm}$)
Anisotropy $g \approx 0.9$ (forward Mie scattering)

$$\mu_s' = \mu_s(1 - g) \approx 1.0\,\text{mm}^{-1}$$

Photon Diffusion Equation

$$\frac{1}{c} \frac{\partial \Phi}{\partial t} - D \nabla^2 \Phi + \mu_a \Phi = S(\vec{r}, t)$$

Banana-Shaped Sensitivity

Surface optodes separated by $d_{\text{SD}} = 30\,\text{mm}$ yield a curvilinear migration profile:

$z_{\text{max}} \approx \frac{1}{2} d_{\text{SD}} \approx 12 - 15\,\text{mm}$

Interrogates only the superficial gyral crowns.

Modified Beer-Lambert Law (MBLL)

Unmixing oxy- and deoxyhemoglobin concentrations from multi-wavelength optical density changes:

$$\Delta\text{OD}(\lambda) = \left[ \varepsilon_{\text{HbO}_2}(\lambda) \Delta[\text{HbO}_2] + \varepsilon_{\text{HbR}}(\lambda) \Delta[\text{HbR}] \right] \cdot d_{\text{SD}} \cdot \text{DPF}(\lambda)$$

Dual-Wavelength Solution

Measuring at $\lambda_1 = 760\,\text{nm}$ (HbR dominant) and $\lambda_2 = 850\,\text{nm}$ ($\text{HbO}_2$ dominant):

$$\begin{bmatrix} \Delta[\text{HbO}_2] \\ \Delta[\text{HbR}] \end{bmatrix} = \frac{1}{d_{\text{SD}}} \mathbf{E}^{-1} \begin{bmatrix} \Delta\text{OD}_1 \\ \Delta\text{OD}_2 \end{bmatrix}$$

Scalp Contamination Challenge

Up to $80\%$ of the raw optical signal originates from scalp/skull blood flow.

Solution: Short-separation channels ($8\,\text{mm}$) + regression/Kalman filtering.

Electrodynamics: Pyramidal Dipoles

Why Postsynaptic Potentials?

Action potentials are quadrupole traveling fields with duration $\sim 1\,\text{ms}$; they cancel destructively.
EPSPs last $10 - 50\,\text{ms}$ and create stationary dipoles along apical dendrites ($\approx 1\,\text{mm}$ separation).
Layer V pyramidal cells form an open-field architecture perpendicular to cortex.

Elementary Current Dipole

$$\vec{p} = I \cdot \vec{d} \approx 20\,\text{fA}\cdot\text{m}$$

Population Coherence

To register a measurable scalp signal ($10 - 50\,\mu\text{V}$), approximately $10^4 - 10^5$ contiguous pyramidal neurons must fire synchronously.

$$\vec{Q}_{\text{net}} = \sum_{k} \vec{p}_k \sim 10 - 100\,\text{nA}\cdot\text{m}$$

Volume Conduction: Poisson's Law

In biological tissue at neurophysiological frequencies ($< 1000\,\text{Hz}$), quasi-static conditions hold:

$$\nabla \cdot (\sigma(\vec{r}) \nabla \Phi) = \nabla \cdot \vec{J}_p = I_s(\vec{r})$$

Poisson's Equation of Volume Conduction

Brain Tissue

$\sigma \approx 0.33\,\text{S/m}$. High conductivity electrolyte bath permitting easy current flow.

Skull Bone

$\sigma \approx 0.005\,\text{S/m}$. Resistivity ratio $\sim 60:1$. Acts as severe spatial low-pass filter!

Scalp Tissue

$\sigma \approx 0.33\,\text{S/m}$. Spreads current horizontally, smearing localized cortical spikes over $20 - 30\,\text{mm}$.

EEG vs. MEG: The Physical Distinction

Parameter	Electroencephalography (EEG)	Magnetoencephalography (MEG)
Signal Measured	Scalar electric potential $\Phi(\vec{r})$	Vector magnetic flux density $\vec{B}(\vec{r})$
Primary Source	Both radial (gyral) & tangential (sulcal)	Strictly tangential dipoles in sulcal walls
Skull Effect	Severe spatial blurring & attenuation	Completely transparent ($\mu_r \approx 1.0$)
Spatial Resolution	$20 - 30\,\text{mm}$ (without high-density montage)	$3 - 5\,\text{mm}$ (with anatomical MRI prior)
Sensor Physics	Ag/AgCl conductive contact electrodes	SQUID magnetometers / OPM quantum vapors
Environmental Noise	50/60 Hz mains, motion artifacts	Earth's magnetic field ($10^6 \times$ signal magnitude!)

Acoustic Waves & Functional Ultrasound

Transcranial Bone Challenges

Acoustic Mismatch: Bone impedance $Z \approx 6.0 \times 10^6\,\text{Rayl}$ vs brain $1.54 \times 10^6\,\text{Rayl}$.
Severe Attenuation: $\approx 10 - 20\,\text{dB/cm/MHz}$ causes cranial heating.
Phase Aberration: Variable skull thickness defocuses acoustic focal spot.

Ultrafast Plane-Wave Doppler (fUS)

Transmits unfocused plane waves at $5 - 20\,\text{kHz}$ PRF. Compound angles yield frame rates $> 1\,\text{kHz}$.

$$\text{PD} \propto \text{Cerebral Blood Volume (CBV)}$$

SVD filtering isolates capillary blood backscatter from tissue clutter.

Spatial resolution: $100\,\mu\text{m}$!

The Unified Trade-Off Landscape

fMRI

0.5 mm · 1-2 s · Whole Brain

fNIRS

15 mm · 10 ms · Cortex (15 mm)

EEG

25 mm · <1 ms · Surface Gyri

MEG

4 mm · <1 ms · Sulcal Cortex

Spatial Uncertainty Principle

Penetration depth through bone or turbid tissue inevitably compromises high spatial frequency information unless invasive windows are used.

Temporal vs. Metabolic Decoupling

Electrophysiology tracks instantaneous thought ($< 1\,\text{ms}$); hemodynamics tracks sluggish vascular infrastructure ($1 - 5\,\text{s}$).

Hands-On: The Simulation Sandbox

This week's workshop puts these mathematical derivations directly into code:

Bloch Vector Dynamics

Tune $B_0$ from $0.05\,\text{T}$ to $7.0\,\text{T}$. Watch longitudinal recovery and $T_2^*$ decay curves evolve with flip angle and TR.

fNIRS Photon Migration

Adjust $\mu_s', \mu_a$, and separation $d_{\text{SD}}$. Observe the banana profile and dual-wavelength hemodynamic unmixing.

Volume Conduction Dipole

Translate a cortical current dipole in depth and orientation. See the skull low-pass filter spread the scalp potential.

🔬 Open Biophysical Simulation Sandbox →

Required Foundational Literature

fMRI Ogawa et al. (1990)

Magn. Reson. Med., 14(1): 68–78
The landmark discovery of BOLD contrast and deoxyhemoglobin magnetic susceptibility gradients.

DOI: 10.1002/mrm.1910140108 →

fNIRS Villringer et al. (1993)

Neurosci. Lett., 154(1-2): 101–104
First demonstration of functional near-infrared spectroscopy through intact human skull.

DOI: 10.1016/0304-3940(93)90181-J →

EEG Nunez & Srinivasan (2006)

Electric Fields of the Brain (2nd Ed.)
The definitive treatise on postsynaptic pyramidal dipoles and head volume conduction.

Oxford University Press →

Looking Ahead to Week 5

Next: Sensor Hardware, Transduction Physics & Array Architecture

📝 Start Worksheet 4.1 🔬 Practice in Sandbox 📄 Read Full Lecture Notes

                    IDNE 701: Introduction to Neuroengineering · Department of Biomedical Engineering
                

Parameter	Electroencephalography (EEG)	Magnetoencephalography (MEG)
Signal Measured	Scalar electric potential \(\Phi(\vec{r})\)	Vector magnetic flux density \(\vec{B}(\vec{r})\)
Primary Source	Both radial (gyral) & tangential (sulcal)	Strictly tangential dipoles in sulcal walls
Skull Effect	Severe spatial blurring & attenuation	Completely transparent (\(\mu_r \approx 1.0\))
Spatial Resolution	\(20 - 30\,\text{mm}\) (without high-density montage)	\(3 - 5\,\text{mm}\) (with anatomical MRI prior)
Sensor Physics	Ag/AgCl conductive contact electrodes	SQUID magnetometers / OPM quantum vapors
Environmental Noise	50/60 Hz mains, motion artifacts	Earth's magnetic field (\(10^6 \times\) signal magnitude!)