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A data-driven approach to modeling brain activity using differential equations

Kuratov Andrey

TL;DR

This work addresses the challenge of recovering governing differential equations for brain activity from incomplete observations by introducing a local, data-driven algorithm that leverages partial knowledge of the system. It uses explicit Runge-Kutta integration to generate candidate trajectories for the known equations, then sequentially infers the unknown equation and possible input terms, selecting the best candidate via convergence criteria and parameter bounds. Key findings show that recovered frequency components $oldsymbol{ ext{omega}}_{2,mk}$ span the true range and can match the target $oldsymbol{ ext{omega}}_2$ when appropriate initial conditions are present, and that external oscillatory inputs can be identified through corresponding coefficients $C_{fi}$ and frequencies $oldsymbol{ ext{omega}}_{fi}$ in both synthetic and real data. The approach demonstrates practical applicability to synthetic pyramidal-cell models and real primate resting-state data, offering a path toward modeling electrophysiological brain signals from partial measurements.

Abstract

This research focuses on an innovative task of extracting equations from incomplete data, moving away from traditional methods used for complete solutions. The study addresses the challenge of extracting equations from data, particularly in the study of brain activity using electrophysiological data, which is often limited by insufficient information. The study provides a brief review of existing open-source equation derivation approaches in the context of modeling brain activity. The section below introduces a novel algorithm that employs incomplete data and prior domain knowledge to recover differential equations. The algorithm's practicality in real-world scenarios is demonstrated through its application on both synthetic and real datasets.

A data-driven approach to modeling brain activity using differential equations

TL;DR

This work addresses the challenge of recovering governing differential equations for brain activity from incomplete observations by introducing a local, data-driven algorithm that leverages partial knowledge of the system. It uses explicit Runge-Kutta integration to generate candidate trajectories for the known equations, then sequentially infers the unknown equation and possible input terms, selecting the best candidate via convergence criteria and parameter bounds. Key findings show that recovered frequency components span the true range and can match the target when appropriate initial conditions are present, and that external oscillatory inputs can be identified through corresponding coefficients and frequencies in both synthetic and real data. The approach demonstrates practical applicability to synthetic pyramidal-cell models and real primate resting-state data, offering a path toward modeling electrophysiological brain signals from partial measurements.

Abstract

This research focuses on an innovative task of extracting equations from incomplete data, moving away from traditional methods used for complete solutions. The study addresses the challenge of extracting equations from data, particularly in the study of brain activity using electrophysiological data, which is often limited by insufficient information. The study provides a brief review of existing open-source equation derivation approaches in the context of modeling brain activity. The section below introduces a novel algorithm that employs incomplete data and prior domain knowledge to recover differential equations. The algorithm's practicality in real-world scenarios is demonstrated through its application on both synthetic and real datasets.
Paper Structure (13 sections, 8 equations, 18 figures, 2 tables)

This paper contains 13 sections, 8 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Methods based on sparse regression
  4. Methods based on neural networks
  5. Other methods
  6. Problem definition
  7. Our approch
  8. Experiments
  9. Data description
  10. System of equations without impact
  11. System of equations with impact
  12. Equation extraction from real data
  13. Conclusion

Figures (18)

  • Figure 1: Schematic representation of Jansen's Jansen1995 model. Currents in pyramidal neurons (blue pyramid), due to gain and attenuation circuits (populations of insertion neurons) generate the signal read on electroencephalograms.
  • Figure 2: Example of synthetic data. Left column of graphs -- hidden variables, upper right graph of the variable visible by the microelectrode, remaining graphs on the right -- graphs of functions $g_i(t)$ and $f_i(t)$
  • Figure 3: Location of microelectrode arrays in visual cortex in areas V1 and V4. (a) -- general position of arrays, (b) -- exact position of microelectrode arrays, (c) -- signal post-processing (filtering, downsampling)) dataset.
  • Figure 4: The model described by the system (\ref{['eq:no_force']}), where $\omega_0 = 7$, $\omega_2 = 20$, $C_0 = 0.13$, $C_1 = 9.0$, $C_2 = 9.0$, $\alpha_0 = 6$, $\alpha_1 = 3$, $\alpha_2 = 0.5$, $v_0 = 0.1$, $v_1 = 2.5$, $v_2 = 1.0$. Number of time steps is 2000, time interval [0, 6], vector of initial conditions $\textbf{y} = [1.5, 1.5, -1.5, 1.0, -1.0, 0.5]$.
  • Figure 5: Solutions $y_{0\;m}$ of the first equation of the model shown in Figure \ref{['fig:ord_1_1']}. The blue dashed line shows the true dependence of $y_0$.
  • ...and 13 more figures