Research

What can sparse, noisy measurements tell us about a biological mechanism — and when can a neural network be trusted to tell us?

Overview

My PhD research sits at the intersection of mathematical biology, scientific computing and numerical analysis. I develop physics-informed neural networks (PINNs) and numerical methods for forward and inverse problems in partial differential equations, with a focus on neuroscience: estimating dopamine diffusion and uptake from sparse electrode recordings, with applications to Parkinson’s disease, and modelling glioblastoma (GBM).

The mathematical setting

The dopamine work starts from a two-dimensional reaction–diffusion model of a single release event, with diffusion coefficient D and transporter-mediated uptake rate k:

\frac{\partial c}{\partial t} \;=\; D\, \Delta c \;-\; k\, c \;+\; s(\mathbf{x}, t),

where c(\mathbf{x}, t) is the dopamine concentration and s models the release. The inverse problem is to recover D and k from recordings at only one or a few electrode sites.

Research threads

1. Measurement-aware PINNs for dopamine transport

Electrodes record dopamine at only one or a few sites, with noise and with distortions from the sensor itself. I estimate the diffusion coefficient D and the transporter-mediated uptake rate k of a two-dimensional reaction–diffusion model of a single release event with an inverse PINN whose data term passes the network output through a model of the electrode: tip averaging, a first-order response lag, whitening of correlated noise and removal of an unknown baseline offset. Each estimate is compared with the Cramér–Rao bound of the recording design and with classical least squares on the true model, which separates the error of the network from the error the data allow.

Main findings: the network is about as accurate as the classical fit and close to the bound; the uptake rate is identifiable from a single electrode, while the diffusion coefficient needs a site near the release site or two sites; ignoring a 2-ms electrode lag more than doubles the estimated diffusion coefficient; and modelling the sensor with a short pre-stimulus baseline restores unbiased estimates with close to nominal uncertainty intervals.

Status: submitted to Biomedical Signal Processing and Control (Elsevier), October 2026. Code and data: doi.org/10.5281/zenodo.23518612.

Next: counterfactual (“what-if”) inference for dopamine transport, predicting how the concentration field would change under an intervention such as transporter blockade, first for the linear model and then for nonlinear Michaelis–Menten uptake.

2. Mathematical modelling of glioblastoma (GBM)

PDE models and numerical methods for glioblastoma, the most common malignant primary brain tumour.

3. Next: medical imaging

Physics-informed methods for medical imaging, possibly in 3D.

Tooling

  • Dopamine project: JAX, Flax NNX, Optax, NumPy/SciPy, Google Colab GPUs.
  • Compute: NVIDIA A100 / H100; multi-GPU via jax.pmap or torch.distributed.
  • Stack: JAX + Equinox, PyTorch, Hydra (config), Weights & Biases (logging), DeepXDE / NVIDIA Modulus.
  • Numerical solvers: FEniCSx, Dedalus (spectral), MATLAB.

Current focus

Extending the dopamine work to counterfactual inference and nonlinear Michaelis–Menten uptake, and modelling glioblastoma (GBM).