cs.NEFeb 27, 2026

Robust Parameter and State Estimation in Multiscale Neuronal Systems Using Physics-Informed Neural Networks

Authors: Changliang Wei, Yangyang Wang, Xueyu Zhu

Organizations: Department of Mathematics, University of Iowa, Iowa City, IA 52246, USA. · Department of Mathematics, Volen National Center for Complex Systems, Brandeis University, Waltham, MA 02453, USA.

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Abstract

Inferring biophysical parameters and hidden state variables from partial and noisy observations is a fundamental challenge in computational neuroscience. This problem is particularly difficult for fast - slow spiking and bursting models, where strong nonlinearities, multiscale dynamics, and limited observational data often lead to severe sensitivity to initial parameter guesses and convergence failure in the methods replying on the traditional numerical forward solvers. In this work, we developed a physics-informed neural network (PINN) framework for the joint reconstruction of unobserved state variables and the estimation of unknown biophysical parameters in neuronal models. We demonstrate the effectiveness of the method on biophysical neuron models, including the Morris-Lecar model across multiple spiking and bursting regimes and a respiratory model neuron. The method requires only partial voltage observations over short observation windows and remains robust even when initialized with non-informative parameter guesses. These results suggest that PINN can deliver robust and accurate parameter inference and state reconstruction, providing a promising alternative for inverse problems in multiscale neuronal dynamics, where traditional techniques often struggle.

Figures & tables

Fig. 1 : Comparison of the signal filtering results under different parameter choices. The left panel corresponds to p=95p=95 , while the right panel corresponds to p=99p=99 . In each subplot, the observed noisy signal is filtered and compared against the ground-truth signal.
Fig. 2 : Illustration of the architecture of the proposed PINN, using model ( 25 )–( 27 ) with state variables (V,n,Ca)(V,n,\mathrm{Ca}) as an illustrative example.
ModelTTData size
Spiking Morris–Lecar0.2 s2001
Bursting Morris–Lecar2 s20001
pBC6 s60001
Table 1: Total number of data points for the observed voltage signal for each model.
ModelRegimeNoise LevelFFT Freq. Count
SML ( p=95p=95 )Hopf1%1\%44
SNIC1%1\%88
HC1%1\%33
Hopf5%5\%44
SNIC5%5\%88
HC5%5\%33
Table 2: Number of dominant Fourier frequencies extracted under different noise levels for the spiking Morris–Lecar (SML), bursting Morris–Lecar (BML), and pBC neuron models.
ModelRegimeNoiseNetwork VVNetwork nnNetwork CaCaNetwork hh
SMLHopf1%1\%35164237——
SNIC1%1\%42305672——
HC1%1\%36184442——
Hopf5%5\%35164237——
SNIC5%5\%43325877——
HC5%5\%37204647——
Table 3: Total number of trainable parameters (network size) for the neural network of the observed voltage VV and of the unobserved variables nn , CaCa , and hh across different models, bifurcation regimes, and noise levels, respectively.
ModelGating NetworksInitial lrTotal Iterations
SMLnn10−410^{-4}800,000800{,}000
BMLn, Can,\,Ca10−410^{-4}1,000,0001{,}000{,}000
pBCn, hn,\,h10−310^{-3}200,000200{,}000
Table 4: Training configurations for the physics-constrained stage of different neuronal models. All gating-variable networks are optimized using the Adam optimizer with staircase exponential decay learning rates (lr), while the biophysical parameters λ\bm{\lambda} are updated using constant learning rates as described in the text.
Fig. 3 : Time trajectories of the solution of the spiking Morris–Lecar model under three distinct bifurcation regimes: Hopf (first column), SNIC (second column) and Homoclinic (third column). Each row corresponds to one state variable: voltage VV , gating variable nn .
 HopfSNICHomoclinic
ϕ\phi0.040.0670.23
gCa444
V3V_{3}21212
V4V_{4}3017.417.4
gK888
gL222
Table 5: The true parameter values of Morris-Lecar model for three different bifurcation regimes, Hope, saddle-node on an invariant circle (SNIC), and Homoclinic.
 Lower boundUpper bound
ϕ\phi0+∞+\infty
gCa0+∞+\infty
V3V_{3}0+∞+\infty
V4V_{4}0+∞+\infty
gK0+∞+\infty
gL0+∞+\infty
Table 6: Bounds for the Spiking Morris-Lecar Model in the training stage of PINNs.
Table 7: Estimated biophysical parameters for the Hopf regime under two observational noise levels ( 1%1\% and 5%5\% ), evaluated using two initial guess strategies (regime-based initial guess and non-informative initial guess where all parameters are set to 11 ), with observed data from bifurcation regime ‘t’ and initial parameter guesses corresponding to bifurcation regime ‘g’.
Fig. 4 : Comparison between the ground-truth trajectories and the PINN-reconstructed trajectories for the Hopf regime under two noise levels based on the non-informative initial guess. PINN predictions are shown as dashed blue lines, while the observation voltage VV and ground-truth gating nn trajectories are shown as solid yellow lines.
Noise LevelTrue RegimeInitial GuesseV^e_{\hat{V}}en^e_{\hat{n}}
1%1\%HopfSNIC0.15%0.15\%0.31%0.31\%
1%1\%HopfHC0.15%0.15\%0.32%0.32\%
1%1\%Hopf110.15%0.15\%0.32%0.32\%
5%5\%HopfSNIC0.62%0.62\%1.21%1.21\%
5%5\%HopfHC0.62%0.62\%1.21%1.21\%
5%5\%Hopf110.62%0.62\%1.25%1.25\%
Table 8: Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the SNIC regime under two noise levels using state reconstruction errors.
Table 9: Estimated biophysical parameters for the SNIC regime under two observational noise levels ( 1%1\% and 5%5\% ), evaluated using two initial guess strategies (regime-based initial guess and non-informative initial guess where all parameters are set to 11 ), with observed data from bifurcation regime ‘t’ and initial parameter guesses corresponding to bifurcation regime ‘g’.
Fig. 5 : Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the SNIC regime under two noise levels based on the non-informative initial guess, where the PINN predictions are shown as dashed blue lines and the observation voltage VV and ground-truth gating nn trajectories are shown as solid yellow lines.
Noise LevelTrue RegimeInitial GuesseV^e_{\hat{V}}en^e_{\hat{n}}
1%1\%SNICHopf0.27%0.27\%0.49%0.49\%
1%1\%SNICHC0.27%0.27\%0.48%0.48\%
1%1\%SNIC110.27%0.27\%0.49%0.49\%
5%5\%SNICHopf1.33%1.33\%1.24%1.24\%
5%5\%SNICHC1.33%1.33\%1.26%1.26\%
5%5\%SNIC111.33%1.33\%1.18%1.18\%
Table 10: Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the SNIC regime under two noise levels using state reconstruction errors.
Table 11: Estimated biophysical parameters for the Homoclinic regime under two observational noise levels ( 1%1\% and 5%5\% ), evaluated using two initial guess strategies (regime-based initial guess and non-informative initial guess where all parameters are set to 11 ), with observed data from bifurcation regime ‘t’ and initial parameter guesses corresponding to bifurcation regime ‘g’.
Fig. 6 : Comparison between the ground-truth trajectories and the PINN-reconstructed trajectories for the Homoclinic regime under two noise levels based on the non-informative initial guess, where the PINN predictions are shown as dashed blue lines and the observation voltage VV and ground-truth gating nn trajectories are shown as solid yellow lines.
Noise LevelTrue RegimeInitial GuesseV^e_{\hat{V}}en^e_{\hat{n}}
1%1\%HCHopf0.21%0.21\%0.37%0.37\%
1%1\%HCSNIC0.21%0.21\%0.38%0.38\%
1%1\%HC110.21%0.21\%0.35%0.35\%
5%5\%HCHopf0.97%0.97\%2.32%2.32\%
5%5\%HCSNIC0.97%0.97\%6.47%6.47\%
5%5\%HC110.97%0.97\%4.45%4.45\%
Table 12: Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the Homoclinic regime under two noise levels using state reconstruction errors.
Fig. 7 : Bifurcation diagrams for the Morris-Lecar Model in three different bifurcation regimes: Hopf (left), SNIC (middle) and Homoclinic (right). Shown are the equilibrium curves and the minimum and maximum VV along families of periodic orbit (PO) solutions as functions of the bifurcation parameter IappI_{\rm app} . Key bifurcation points on the ground-truth bifurcation diagrams are indicated. Gray curves represent the true diagrams, while blue dotted curves correspond to the diagrams produced from PINN-estimated parameters under 5% relative noise levels (Tables 7 , 9 and 11 ). The displayed results correspond to representative worst-case estimates. In the legend, the notation “t:” denotes the true bifurcation regime and “g:” indicates initial parameter guesses.
Parametersquare-waveelliptic
ϕ\phi0.230.230.040.04
gCag_{\mathrm{Ca}}4.04.04.44.4
V3V_{3}12.012.02.02.0
V4V_{4}17.417.430.030.0
gKg_{K}8.08.08.08.0
gLg_{L}2.02.02.02.0
Table 13: Parameters for the bursting Morris–Lecar model ( 50 )–( 52 ).
Fig. 8 : Ground-truth dynamics of the Morris–Lecar model under the two distinct bursting regimes: square-wave (left panel) and elliptic (right panel). Each row corresponds to one state variable: membrane potential VV , gating variable nn , and intracellular calcium concentration CaCa .
Table 14: Estimated biophysical parameters for the SML model in the square-wave bursting regime under two observational noise levels ( 1%1\% and 5%5\% ), evaluated using two initial guess strategies (regime-based initial guess and non-informative initial guess where all parameters are set to 11 ).
Fig. 9 : Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the BML model in the square-wave bursting regime under two noise levels based on the non-informative initial guess. The PINN predictions are shown as dashed blue line, while the true trajectories are shown as solid orange line.
Noise LevelTrue RegimeInitial GuesseV^e_{\hat{V}}en^e_{\hat{n}}eCa^e_{\widehat{Ca}}
1%1\%square-waveElliptic0.09%0.09\%0.79%0.79\%0.85%0.85\%
1%1\%square-wave110.09%0.09\%0.36%0.36\%0.30%0.30\%
5%5\%square-waveElliptic0.25%0.25\%16.79%16.79\%8.93%8.93\%
5%5\%square-wave110.25%0.25\%11.32%11.32\%2.85%2.85\%
Table 15: Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the BML model in the square-wave bursting regime under two noise levels using state reconstruction errors.
Table 16: Estimated biophysical parameters for the Elliptic bursting regime under two observational noise levels ( 1%1\% and 5%5\% ), evaluated using two initial guess strategies (regime-based initial guess and non-informative initial guess where all parameters are set to 11 ).
Fig. 10 : Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the elliptic bursting regime under two noise levels based on the non-informative initial guess, where the PINN predictions shown as dashed blue line and true trajectories shown as solid orange line.
Noise LevelTrue RegimeInitial GuesseV^e_{\hat{V}}en^e_{\hat{n}}eCa^e_{\widehat{Ca}}
1%1\%ellipticsquare-wave0.09%0.09\%0.15%0.15\%0.09%0.09\%
1%1\%elliptic110.09%0.09\%0.14%0.14\%0.10%0.10\%
5%5\%ellipticsquare-wave0.33%0.33\%0.44%0.44\%0.59%0.59\%
5%5\%elliptic110.33%0.33\%0.49%0.49\%0.54%0.54\%
Table 17: Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the elliptic bursting regime under two noise levels using state reconstruction errors.
Fig. 11 : Bifurcation diagrams for the Morris-Lecar Model at two different bursting regimes: Square-Wave (left), and Elliptic (right). The gray lines represent the true diagrams, and the blue dotted lines correspond to the diagrams produced from the PINN-estimated parameters under 5% relative noise levels (Tables 14 and 16 ). The displayed results correspond to representative worst-case estimates. In the legend, the notation “t:” denotes the true bifurcation regime and “g:” indicates initial parameter guesses.
Estimated ParametersFixed Parameters
{\color[rgb]{0,0,0}g_{\rm Na}}28 nS28~{\rm nS}CmC_{m}21 pF21~{\rm pF}σh\sigma_{h}5 mV5~{\rm mV}
{\color[rgb]{0,0,0}g_{\rm K}}11.2 nS11.2~{\rm nS}τˉh\bar{\tau}_{h}10,000 ms10{,}000~{\rm ms}σm\sigma_{m}−5 mV-5~{\rm mV}
{\color[rgb]{0,0,0}g_{\rm L}}2.3 nS2.3~{\rm nS}σmp\sigma_{mp}−6 mV-6~{\rm mV}θm\theta_{m}−34 mV-34~{\rm mV}
{\color[rgb]{0,0,0}g_{\rm NaP}}2.0 nS2.0~{\rm nS}θmp\theta_{mp}−40 mV-40~{\rm mV}θh\theta_{h}−48 mV-48~{\rm mV}
{\color[rgb]{0,0,0}V_{\rm Na}}50 mV50~{\rm mV}τˉn\bar{\tau}_{n}10 ms10~{\rm ms}θn\theta_{n}−29 mV-29~{\rm mV}
{\color[rgb]{0,0,0}V_{\rm K}}−85 mV-85~{\rm mV}σn\sigma_{n}−4 mV-4~{\rm mV}  
Table 18: Ground-truth parameter values used in the pre-Bötzinger complex (pBC) model ( 58 ). Estimated parameters are listed in the first column, while the remaining parameters are assumed to be known.
Fig. 12 : Ground-truth dynamics of the pBC model ( 58 ) over two bursting cycles. Each column corresponds to one state variable: voltage VV , gating variable nn , and gating variable hh .
Table 19: Estimated biophysical parameters for the pBC model under two observational noise levels ( 1%1\% and 10%10\% ), evaluated using non-informative initial guess where all parameters are set to 11 .
Fig. 13 : Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the pBC model under two noise levels based on the non-informative initial guess, where the PINN predictions shown as dashed blue line and true trajectories shown as solid orange line.
Noise LeveleV^e_{\hat{V}}en^e_{\hat{n}}eh^e_{\hat{h}}
1%1\%0.05%0.05\%1.14%1.14\%1.09%1.09\%
10%10\%0.14%0.14\%3.65%3.65\%2.18%2.18\%
Table 20: Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for pBC model under two noise levels using state reconstruction errors.
ParameterTrue ValueInitial Guess: 1Relative Error
gNaPg_{\rm NaP}2.02.01.8071.8079.6%9.6\%
gLg_{\rm L}2.32.32.3192.3190.8%0.8\%
gKg_{\rm K}11.211.211.71011.7104.6%4.6\%
gNag_{\rm Na}28.028.025.15825.15810.2%10.2\%
VLV_{\rm L}−58.0-58.0−57.808-57.8080.3%0.3\%
VKV_{\rm K}−85.0-85.0−81.790-81.7903.8%3.8\%
Table 21: Estimated biophysical parameters for the pBC model under the absolute observational noise level, evaluated using non-informative initial guess where all parameters are set to 11 .
Fig. 14 : Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for the pBC model under the absolute observational noise level, where the PINN predictions shown as dashed blue line and true trajectories shown as solid orange line.
eV^e_{\hat{V}}en^e_{\hat{n}}eh^e_{\hat{h}}
0.58%0.58\%1.88%1.88\%1.32%1.32\%
Table 22: Comparison between the ground-truth trajectories and the PINN-reconstructed solutions for pBC model under the absolute observational noise level using state reconstruction errors.
Fig. 15 : Bifurcation diagrams for the pBC Model. The gray lines represent the true diagram, while the blue dotted lines correspond to the diagrams generated using the PINN-estimated parameters obtained from a noninformative initial guess (g:1) under (left) 1% relative noise, (middle) 10% relative noise, and (right) 3% absolute observational noise levels.
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