Phase 2 · Biophysics & Sensors

Worksheet 4.1: Biophysical Forward Modeling & Signal Transduction

Derive the first-principles forward physics linking microscopic cellular pathology to external non-invasive sensor signals. Establish the differential governing equations, quantify depth-dependent signal attenuation, model theoretical Signal-to-Noise Ratio (SNR) boundaries, and formulate your in silico simulation test bench.

The Role of Forward Modeling in Neuroengineering: Before designing physical sensor arrays or writing reconstruction algorithms, engineers must build an accurate mathematical forward model: $$\mathbf{y} = \mathcal{F}(\mathbf{x}) + \mathbf{n}$$ where $\mathbf{x}$ represents the underlying neural source state, $\mathcal{F}$ represents the biophysical forward operator (Maxwell's equations, Bloch equations, diffusion equations), $\mathbf{n}$ is physical noise, and $\mathbf{y}$ is the raw recorded sensor vector. If your forward model is unphysical or violates conservation laws, your inverse reconstruction will yield nonsensical artifacts.

1 Sensing Modality & Biological Source Event

Specify your selected neuroimaging modality, physical energy form, and the microscopic physiological event acting as your signal generator.

Team / Project Identifier

Engineering Modeling Leads

Primary Biophysical Sensing Modality

Physical Energy Domain & Spectrum

Microscopic Neural Event (Source Generator)

2 Mathematical Forward Equations & Boundary Conditions

Formalize the differential or integral equations governing the propagation of your signal from the brain source through skull and scalp to the detector surface.

Canonical Governing Law (Selected Modality Reference)

$$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}') \times (\mathbf{r} - \mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|^3} \, d^3\mathbf{r}'$$

Biot-Savart Law for quasi-static magnetic field propagation from primary neuronal current density $\mathbf{J}_p$ and volume return currents $\sigma \nabla V$.

Your Explicit Analytical or Numerical Formulation

Volume Conductor Geometry & Boundary Conditions

Key Biophysical Constants & Tissue Parameters

3 Interactive Signal Attenuation & SNR Budget Calculator

Adjust the biophysical source depth, generator amplitude, and detector noise floor to verify that your neural signal exceeds the physical noise floor at the sensor plane.

Source Depth from Sensor Plane ($r$) 20 mm

Scalp thickness + skull + CSF + cortical depth

Source Current Dipole Moment ($Q$) 25 nA·m

Synchronized neural patch activity

Sensor White Noise Floor ($S_n^{1/2}$) 15 fT/√Hz

Transducer intrinsic noise limit

Recording Bandwidth ($\Delta f$) 250 Hz

Nyquist acquisition frequency range

Predicted Magnetic Field at Sensor ($B$): 312.5 fT

Total Integrated RMS Noise ($\sigma_n$): 237.2 fT

Single-Trial Signal-to-Noise Ratio (SNR): 1.32 (2.4 dB)

Required Averaging Epochs for SNR ≥ 5: 15 epochs

Detection Feasibility: Feasible with minimal averaging. Signal amplitude at 20 mm exceeds the single-trial sensor noise floor.

4 Physical Spatiotemporal Resolution Bounds

Identify the fundamental physical limits that prevent your system from achieving infinitely fine spatial or temporal resolution (diffraction, volume conduction, acoustic attenuation).

Theoretical Spatial Resolution Limit

Theoretical Temporal Resolution Limit

The Fundamental Physics Trade-off Curve

5 In Silico Forward Simulation Architecture

Outline the computational pipeline and simulation tools your team will deploy to generate synthetic ground-truth test data.

Computational Simulation Platform / Framework

Anatomical Head Model / Numerical Phantom

Synthetic Benchmark & Validation Protocol

6 Phase 2 Dossier Synthesis (Section 3: Biophysical Forward Model)

Synthesize your forward modeling derivation into formal technical prose for Section 3 of your Preliminary Design Concept (PDC) Dossier.

Executive Biophysical Forward Model Summary