Category Spotlight #11 — Neuroscience & Medicine: Brain, Heart & Neural Models

Neural Electrophysiology

The brain is an electrical organ. Every thought, perception, and movement is encoded as patterns of voltage spikes — action potentials — propagating through networks of neurons. The Hodgkin-Huxley model, published in 1952 and awarded the Nobel Prize in Physiology, gives us the equations from which every modern neural model descends.

❤️ Cardiac Action Potential Simulate the Hodgkin-Huxley sodium/potassium channel dynamics in a ventricular cardiomyocyte. See depolarisation, plateau phase, and repolarisation driven by gated ion currents. Hodgkin-Huxley ODE system · adaptive RK4 · stiff solver 🧠 Spiking Neural Network A leaky integrate-and-fire network with 100 neurons. Adjust connection weights and external drive to observe synchrony, bursting, and irregular asynchronous firing. Leaky integrate-and-fire · STDP plasticity · sparse connectivity 🌊 Brainwave Oscillations Wilson-Cowan neural mass model with thalamocortical coupling. Adjust coupling strength and time constants to reproduce delta, theta, alpha, beta, and gamma bands. Wilson-Cowan equations · thalamocortical loop · spectral analysis 📈 ECG Simulator Generate realistic 12-lead ECG waveforms from a phenomenological cardiac dipole model. Introduce arrhythmias — atrial fibrillation, PVC, LBBB — and see how each changes the trace. Cardiac dipole model · lead projection · van der Pol oscillator

Hodgkin-Huxley Model (simplified)

C_m dV/dt = I_ext − g_Na · m³h · (V − E_Na) (sodium current)
− g_K · n⁴ · (V − E_K) (potassium current)
− g_L · (V − E_L) (leak current)

Gating variables m, h, n each satisfy: dx/dt = α_x(V)(1−x) − β_x(V)x
The voltage-dependent rate functions α, β are the model's empirical core.

Medical Imaging & Pharmacology

🔬 MRI Physics & k-space Visualise nuclear magnetic resonance: spin precession, T1/T2 relaxation, RF pulse excitation, and Fourier reconstruction of k-space data into a MR image. Bloch equations · 2D DFT · spin-echo sequence 💊 Drug Diffusion & Pharmacokinetics Two-compartment pharmacokinetic model with diffusion through a membrane. Compare first-order absorption, distribution, and elimination across dosing regimens. Two-compartment ODE model · Fick diffusion · half-life curves 🩸 Blood Flow (Navier-Stokes) Incompressible viscous flow through a branching vessel network. Poiseuille's law in straight segments, full Navier-Stokes in the bifurcation region. SIMPLE algorithm · pressure Poisson · vessel mesh FEM 🔵 Membrane Diffusion Fick's laws of diffusion across a semi-permeable biological membrane. Vary concentration gradients, membrane permeability, and temperature to observe flux rates and osmotic pressure. Fick's first and second laws · explicit FDM · osmotic pressure

Why are neural ODEs so hard to simulate? The Hodgkin-Huxley system is stiff — the sodium activation variable m has a time constant ~0.1 ms while the full action potential lasts ~2 ms. A naive Euler integrator needs Δt < 0.01 ms for stability, costing 200+ steps per spike. Adaptive RK4 (or the implicit Crank-Nicolson method) reduces this to ~10–15 function evaluations per spike.

Algorithms at a Glance

Hodgkin-Huxley ODE Adaptive RK4 Leaky integrate-and-fire STDP learning Wilson-Cowan equations Bloch equations 2D DFT k-space Two-compartment PK SIMPLE Navier-Stokes Fick diffusion FDM

Suggested Learning Paths

📘 Biology / Pre-Med Students

Cardiac Action Potential — ion channel basics
ECG Simulator — from cell to clinical trace
Drug Diffusion — pharmacokinetics concepts
Membrane Diffusion — Fick's laws in biology
Blood Flow — circulatory system physics

🎓 Biomedical / Computational Neuroscience

Spiking Neural Network — LIF and STDP
Brainwave Oscillations — neural mass models
MRI Physics — Bloch equations and k-space
Blood Flow — Navier-Stokes CFD in vessels
Cardiac AP — Hodgkin-Huxley in depth