NeuroPINNs: Neuroscience-Informed Neural Networks

Updated 11 November 2025

NeuroPINNs are machine learning models that embed biophysical neuron models (e.g., Hodgkin–Huxley, FitzHugh–Nagumo) into neural architectures to solve forward and inverse problems in neuroscience.
They leverage composite loss functions combining data fidelity with PDE/ODE constraints to achieve simultaneous state and parameter estimation from sparse, noisy data.
Recent advances integrate brain-inspired modularity and spiking neuron models, enhancing energy efficiency and deployment in clinical neuroimaging and neuroengineering applications.

NeuroPINNs—Neuroscience-Informed Physics-Informed Neural Networks—comprise a class of machine-learning models that embed neural, biophysical, or physiological constraints within deep neural architectures for the solution of forward and inverse problems in neuroscience and neurophysiology. By fusing principled neuronal models (e.g., Hodgkin-Huxley, FitzHugh-Nagumo), network physiology constraints, or brain-inspired architectural features with data-driven universal function approximation, these frameworks enable robust state and parameter estimation from sparse, noisy, or ill-posed datasets in computational neuroscience, neuroengineering, and clinical neuroimaging. Modern variants also introduce architectural motifs from biological neural systems (spiking, modularity, event-driven communication) to address efficiency and deployability challenges.

1. Biophysical Model Foundation and NeuroPINN Loss Construction

NeuroPINNs generalize PINNs by explicitly encoding neuron or network biophysics into the loss function and architecture. At their core, these models enforce PDE or ODE residuals derived from canonical neuron models. Notable examples include:

Hodgkin–Huxley (HH) model: Membrane potential $V(t)$ and gating variables evolve according to

$C\frac{dV}{dt} = -\Bigl(g_{\mathrm{Na}}\,m^3 h (V - E_{\mathrm{Na}}) + g_{\mathrm{K}}\,n^4 (V - E_{\mathrm{K}}) + g_L(V-E_L)\Bigr) + I_{\mathrm{ext}}(t)$

and

$\frac{dx}{dt} = \alpha_x(V)(1-x) - \beta_x(V)x, \quad x \in \{m, n, h\}.$

FitzHugh–Nagumo (FHN) reduction: Encodes essential excitability with variables $u$ and $v$ satisfying

$\frac{du}{dt} = u - \frac{u^3}{3} - v + I, \qquad \frac{dv}{dt} = \epsilon(u + a - b v).$

Physics-informed loss functions ensure neural network outputs not only fit measurements (e.g. membrane voltage trace) but also satisfy underlying neuron dynamics: $L_{\mathrm{total}} = L_{\mathrm{data}} + \lambda_{\mathrm{phys}} L_{\mathrm{phys}},$ where the data term matches observed $V(t)$ , and the physics term penalizes ODE/PDE violations at collocation points via residuals for both voltage and hidden variables.

2. Inverse Problem and Biophysical Parameter Inference

NeuroPINNs are optimized not only over neural network weights but jointly over biophysical parameters (e.g., conductances $g_\star$ , reversal potentials $E_\star$ , capacitance $C$ ) and possibly network couplings or spatial parameters. For instance, parameters such as $\phi = \{C, g_{\mathrm{Na}}, g_{\mathrm{K}}, g_L, E_{\mathrm{Na}}, E_{\mathrm{K}}, E_L\}$ are treated as trainable variables, subject to soft bounds or regularization.

Parameter inference leverages sparse or noisy data to reconstruct both observable (voltage) and latent (gating variable) trajectories with inferred parameter sets compatible with established biophysical ranges. Simultaneous state-trajectory and parameter identification is achieved by minimizing the composite loss, with regularization strategies to ensure biological plausibility.

3. Network Architectures and Brain-Inspired Modifications

Standard NeuroPINNs adopt deep feed-forward networks with tanh or sinusoidal activations; for multi-state systems, distinct subnetworks can be used for each state variable or subproblem. Recent advances introduce architectural modifications grounded in neurobiological motifs:

Brain-Inspired Modular Training (BIMT): Imposes sparsity, locality, and modularity via L1 penalties, distance-based connection costs, and unit-swapping (Markidis, 28 Jan 2024). This process yields compact, energy-efficient architectures with minimal necessary connections—a “bare-minimum” PINN. Empirical results demonstrate that for higher-frequency PDE solutions, larger bare-minimum subnetworks are required, reflecting a spectral bias.
Variable Spiking Neurons (VSN): Replaces standard continuous activations with spiking neuron models possessing internal memory, threshold-based event-driven spike emission, and graded outputs (Garg et al., 8 Nov 2025). Pinning discontinuous, event-driven neural computation to PDE solving in a PINN is achieved using a stochastic projection filter for derivatives and surrogate backpropagation for training, enabling direct deployment on neuromorphic or edge hardware with substantial reductions in communication and energy cost.

4. Algorithmic and Training Methodology

NeuroPINN training proceeds as follows:

Define the composite loss as the sum of data, physics-informed (ODE/PDE residual), and optional regularization terms.
Use automatic differentiation (or, for spiking layers, nonlocal stochastic gradient estimators) to compute the required state derivatives at sampled collocation points.
Train all neural-network weights and biophysical parameters jointly via Adam or L-BFGS optimizers, tuning the relative weight $\lambda_{\mathrm{phys}}$ so physics and data losses are comparable.
For fractional-order neuron models, operator splitting and proven L1 finite-difference discretizations support Caputo derivatives; gating and voltage subproblems are solved via separate neural networks (Shekarpaz et al., 2023).
Practical guidelines emphasize careful capacity tuning (depth 3–10, width 20–100 per subnetwork), balancing of loss terms, and robust regularization for parameter constraints and fractional order memory.

5. Empirical Results, Applications, and Quantitative Performance

Single-Neuron and Network System Identification

For synthetic FHN data, NeuroPINNs reconstruct parameters with errors as low as 0.4%–6%, and root-mean-square voltage error below $10^{-3}$ .
For real HH data (e.g., squid axon with 7 spikes), extracted conductances and reversal potentials show coefficients of variation on the order of $10^{-4}–10^{-5}$ , with voltage RMSE ~0.02 (normalized units), and gating variables matching expected kinetic profiles (Ferrante et al., 2022).

Biophysically-Informed EEG Analysis

Embedding the FHN model into deep networks for EEG decoding (NeuroPhysNet) improves classification accuracy, robustness, and generalization in motor imagery tasks. Performance surpasses data-driven baselines under within-session CV-T (76.23%), cross-validation CV-E (78.03%), and session transfer (74.20%) (Xia et al., 16 Jun 2025). The physics-informed variables $\hat{u}$ , $\hat{v}$ alone carry discriminatory power, and regularization via the physics loss mitigates noise, nonstationarity, and inter-subject variability.

Energy-Efficient and Modular PDE Solving

BIMT-derived compact networks achieve nearly full-MLP accuracy (error 0.023 vs. 0.0094) at a fraction ( $\sim$ 10x) of the parameter count for classical PDEs (Markidis, 28 Jan 2024).
Spiking-NeuroPINNs attain relative $L^2$ errors within a factor 2–4 of full-activation PINNs across benchmark PDEs and high-dimensional mechanics, but reduce synaptic energy by factors of 2–4, with measured activity as low as 20% in VSN layers (Garg et al., 8 Nov 2025).

Clinical Neuroimaging and Brain Modeling

Application to infant ASL MRI yields physiologically plausible, smooth spatial maps of cerebral blood flow and arrival time; SUPINN achieves a relative error in CBF of $-0.3 \pm 71.7$ versus 96.0 or even 390.7 for baseline methods. The framework demonstrates robust convergence, statistical rigor, and direct utility in parameter recovery and biomarker estimation for clinical datasets (Galazis et al., 11 Oct 2024).

6. Extensions, Limitations, and Practical Considerations

Extensions: Methods generalize to multi-compartment and spatially extended neuron models, higher-order or fractional neuron dynamics, and multi-scale network or tissue models (e.g., cable equations, cardiac and metabolic systems).
Limitations: Scalability to large-scale networks is challenged by increased parameter counts and stiffness in ODEs; training requires fine-tuning of loss weights; event-driven and modular architectures necessitate specialized optimization and hardware support.
Practical Guidance:
- Operator splitting is essential for handling stiff, multi-timescale neuron models.
- For data-limited or noisy scenarios (clinical, BCI, neuroimaging), NeuroPINNs’ physics loss serves as a strong regularizer and inductive bias.
- Construction of bare-minimum and/or spiking architectures yields computational and energy efficiency, favoring low-power or edge deployments.

7. Theoretical Guarantees, Error Analysis, and Future Directions

Convergence, Stability, and Error Bounds: Generalization error in NeuroPINNs can be bounded by training error and quadrature error via established Sobolev and compactness techniques—see Theorems 2.1–2.4, A.1–A.2 for PDEs in (Murari et al., 9 Apr 2025). For stochastic projection approaches in spiking PINNs, unbiasedness of nonlocal gradient estimators is analytically established (Garg et al., 8 Nov 2025).
Statistical Analysis: Extensive ensemble training and statistical reporting (mean, SD, convergence rates, outlier analysis) are standard for unbiased performance characterization.
Anticipated Developments: Integration of Bayesian neural networks for uncertainty quantification, application to unresolved biophysical and hemodynamic models, and the combination of modular and neuromorphic computing for large-scale brain simulators.

In summary, NeuroPINNs constitute a rigorously principled class of machine learning models at the intersection of computational neuroscience, numerical PDEs/ODEs, and data-driven model discovery, exhibiting demonstrable advantages in robustness, interpretability, parameter fidelity, and, in recent developments, energy and compute efficiency.