Treating the brain as a dynamical system governed by (discoverable) differential equations
The brain is a physical system. Neural activity follows differential equations — reaction-diffusion of neurotransmitters, wave propagation of electrical signals, fluid dynamics of cerebrospinal flow. Physics-informed ML provides the tools to learn these equations from data.
Neural ODEs (Chen et al., 2018) parameterize the derivative dz/dt = f_θ(z, t) with a neural network. Instead of discrete layers, the network is a continuous dynamical system solved with ODE integrators.
Neural CDEs (Kidger et al., 2020) extend this to controlled differential equations — the input signal drives the dynamics. For brain time series: the observed fMRI/EEG signal controls the latent neural state evolution.
Counterfactual Neural CDEs answer: "How would this patient's brain trajectory have changed under a different treatment?" This connects physics-informed dynamics to causal inference in neuroscience.
Stochastic Differential VAE (Hess 2024): Models patient brain trajectories as stochastic differential equations in latent space. The drift term captures disease progression; the diffusion term captures individual variability.
Normal aging trajectory: z(t) = z₀ + ∫ f_healthy(z,s)ds + σW(t)
Disease trajectory: z(t) = z₀ + ∫ f_disease(z,s)ds + σW(t)
Deviation = disease - normal: captures pathological acceleration
The Virtual Brain (TVB): Simulates whole-brain dynamics using coupled neural mass models at each cortical region. TVB represents the "first principles" approach — physics-informed ML represents the "data-driven" approach. Their convergence is where neuroscience is heading.
Sparse Identification of Nonlinear Dynamical Systems (Brunton et al., 2016) discovers governing equations from time series data using sparse regression over a library of candidate terms.
Applied to neuroscience: Given fMRI or EEG time series, SINDy can discover:
- The effective connectivity equations between brain regions
- Nonlinear coupling terms (e.g., inhibition, facilitation)
- How these equations change with disease
Example: From resting-state fMRI of the default mode network:
dx₁/dt = -0.3x₁ + 0.2x₂ - 0.1x₁x₃
dx₂/dt = 0.15x₁ - 0.4x₂ + 0.05x₃²
These discovered equations are interpretable — each term has a neurobiological meaning (self-decay, excitatory/inhibitory coupling, nonlinear interaction).
PySR (Cranmer, 2023) provides an alternative approach — symbolic regression that evolves expressions rather than selecting from a fixed library. It might discover coupling terms that SINDy's library doesn't contain.
Physics-Informed Neural Networks model brain processes governed by PDEs:
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Tumor growth (Cho 2024): The Fisher-KPP equation describes tumor cell proliferation and diffusion through brain tissue. PINNs solve this PDE with MRI-derived brain geometry, predicting tumor progression.
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Neurotransmitter diffusion: Reaction-diffusion equations for dopamine, serotonin spreading through synaptic and extrasynaptic space.
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CSF flow: Navier-Stokes equations for cerebrospinal fluid dynamics — relevant for hydrocephalus and intrathecal drug delivery.
DIMON (Yin 2024): Learns the solution operator of PDEs — mapping initial conditions to solutions directly, without solving the PDE each time. Orders of magnitude faster than numerical solvers.
AFNO (Guibas et al., 2022): Fourier Neural Operator with attention — efficient for high-resolution 3D brain volumes where standard operators struggle with memory.
Fractal dimension (Wang 2024 review): The brain's cortical surface has fractal properties — self-similar structure across scales. Fractal dimension decreases with aging and neurodegeneration.
The physics connection: Scale-free dynamics (1/f noise, power-law distributions) in neural activity arise from criticality — the brain operates near a phase transition between order and chaos.
Jarzynski equality and nanoscale thermodynamics (Jarzynski 2011) connect to the brain through:
- Information thermodynamics: Neural computation has thermodynamic costs
- Free energy principle (Friston): The brain minimizes variational free energy — a concept directly from statistical physics
- Protein folding (prions, amyloids): The thermodynamics of misfolding underlies neurodegeneration
Bringing it all together: neurodegeneration can be modeled as a dynamical system where:
- Prion-like protein spreading (Prusiner, Kandel) follows a reaction-diffusion PDE on the brain's structural connectome
- Biomarker cascades (Jack et al., 2013) represent the temporal ordering of the system's state variables
- Normative deviations (Marquand et al., 2019) measure how far the system has departed from its healthy attractor
- Neural ODEs can model the continuous trajectory from health to disease
Structural Connectome (graph)
│
▼
Protein Spreading PDE ← Physics-Informed Neural Network
(reaction-diffusion
on connectome)
│
▼
Regional Atrophy Trajectory ← Neural ODE / SDE-VAE
│
▼
Normative Z-Scores ← GP / VAE Normative Model
│
▼
Clinical Staging ← Biomarker Cascade Model
The vision: A physics-informed digital twin of the brain that:
- Uses the structural connectome as its spatial domain
- Evolves protein pathology via reaction-diffusion PDEs
- Predicts atrophy trajectories via neural SDEs
- Estimates disease stage from normative deviations
- Is personalized to each patient using their own MRI
This is exactly what The Virtual Brain project aims to achieve — and physics-informed ML provides the computational tools to make it feasible.