- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Estimating free parameters in stochastic oscillatory models using a weighted cost function

Joseph M. Marcinik1, Dzmitry Vaido1, and Dolores Bozovic1,2,*

  • *Contact author: bozovic@physics.ucla.edu.
PDF

Phys. Rev. Research 8, 013342 – Published 31 March, 2026

DOI: https://doi.org/10.1103/jnk3-cz87

Abstract

In this study, we estimate parameter values in stochastic oscillatory systems by developing a cost function. Our cost function incorporates power spectral density, analytic signal, and position crossings, each weighted to capture distinct characteristics of the oscillatory system such as amplitude, frequency, and shape. By minimizing this cost function, we estimate parameter values in a stochastic biophysical model for auditory mechanics. Furthermore, we develop a procedure to align the phases of two time series, allowing us to compare stochastic phase evolution of two time series. As a broader application of our procedure, we establish a framework for fitting stochastic oscillatory systems and comparing stochastic time series in other systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (120)

  1. R. D. Mortlock, S. K. Georgia, and S. D. Finley, Dynamic regulation of jak-stat signaling through the prolactin receptor predicted by computational modeling, Cell. Mol. Bioeng. 14, 15 (2021).
  2. N. J. Linden, B. Kramer, and P. Rangamani, Bayesian parameter estimation for dynamical models in systems biology, PLoS Comput. Biol. 18, e1010651 (2022).
  3. C. G. Knight, S. H. E. Knight, N. Massey, T. Aina, C. Christensen, D. J. Frame, J. A. Kettleborough, A. Martin, S. Pascoe, B. Sanderson, D. A. Stainforth, and M. R. Allen, Association of parameter, software, and hardware variation with large-scale behavior across 57,000 climate models, Proc. Natl. Acad. Sci. USA 104, 12259 (2007).
  4. G. L. Baker and J. P. Gollub, Chaotic Dynamics: An Introduction (Cambridge University Press, Cambridge, 2002), 2nd ed.
  5. J. C. Sprott, Chaos and Time-Series Analysis (Oxford University Press, Oxford, 2006), 2nd ed.
  6. J. Faber and D. Bozovic, Noise-induced chaos and signal detection by the nonisochronous Hopf oscillator, Chaos 29, 043132 (2019).
  7. J. Faber and D. Bozovic, Criticality and chaos in auditory and vestibular sensing, Sci. Rep. 14, 13073 (2024).
  8. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Springer Series in Synergetics (Springer, Berlin, 2004), 3rd ed.
  9. S. Särkkä, V. Tolvanen, J. Kannala, and E. Rahtu, Adaptive Kalman filtering and smoothing for gravitation tracking in mobile systems, in 2015 International Conference on Indoor Positioning and Indoor Navigation (IPIN) (IEEE, Banff, AB, Canada, 2015), pp. 1–7.
  10. S. Särkkä, A. Solin, A. Nummenmaa, A. Vehtari, T. Auranen, S. Vanni, and F.-H. Lin, Dynamic retrospective filtering of physiological noise in BOLD fMRI: DRIFTER, NeuroImage 60, 1517 (2012).
  11. D. L. Cook, L. S. Satin, and W. F. Hopkins, Pancreatic B cells are bursting, but how? Trends Neurosci. 14, 411 (1991).
  12. N. N. Taleb, Dynamic Hedging: Managing Vanilla and Exotic Options, Series in Financial Engineering (Wiley, New York, 1997).
  13. T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, Measurement of the instantaneous velocity of a brownian particle, Science 328, 1673 (2010).
  14. C. Zimmer and S. Sahle, Deterministic inference for stochastic systems using multiple shooting and a linear noise approximation for the transition probabilities, IET Syst. Biol. 9, 181 (2015).
  15. C. Zimmer, Reconstructing the hidden states in time course data of stochastic models, Math. Biosci. 269, 117 (2015).
  16. G. I. Webb, Naïve bayes, in Encyclopedia of Machine Learning and Data Mining, edited by C. Sammut and G. I. Webb (Springer US, Boston, MA, 2017), pp. 895–896.
  17. K. Cranmer, J. Brehmer, and G. Louppe, The frontier of simulation-based inference, Proc. Natl. Acad. Sci. USA 117, 30055 (2020).
  18. A. J. Hudspeth, Making an effort to listen: Mechanical amplification in the ear, Neuron 59, 530 (2008).
  19. P. Martin, D. Bozovic, Y. Choe, and A. J. Hudspeth, Spontaneous oscillation by hair bundles of the bullfrog's sacculus, J. Neurosci. 23, 4533 (2003).
  20. B. Nadrowski, P. Martin, and F. Jülicher, Active hair-bundle motility harnesses noise to operate near an optimum of mechanosensitivity, Proc. Natl. Acad. Sci. USA 101, 12195 (2004).
  21. D. Ó Maoiléidigh, E. M. Nicola, and A. J. Hudspeth, The diverse effects of mechanical loading on active hair bundles, Proc. Natl. Acad. Sci. USA 109, 1943 (2012).
  22. J. M. Marcinik, M. A. Toderi, and D. Bozovic, Comparing parameter-reduction methods on a biophysical model of an auditory hair cell, Phys. Rev. Res. 6, 033121 (2024).
  23. A. J. Hudspeth, F. Jülicher, and P. Martin, A Critique of the Critical Cochlea: Hopf—a Bifurcation—Is Better Than None, J. Neurophysiol. 104, 1219 (2010).
  24. J. Faber and D. Bozovic, Chaotic dynamics enhance the sensitivity of inner ear hair cells, Sci. Rep. 9, 18394 (2019).
  25. A. J. Hudspeth and P. Martin, The critical thing about the ear's sensory hair cells, J. Neurosci. 44, e1583242024 (2024).
  26. C. Metzler-Winslow, M. A. Toderi, and D. Bozovic, Neural control and innate self-tuning of the hair cell's active process, Biophys. J. 123, 3550 (2024).
  27. W. Marcotti and S. Masetto, Hair cells, in eLS (John Wiley & Sons, Ltd, Chichester, UK, 2010).
  28. M. S. Smotherman and P. M. Narins, Hair cells, hearing, and hopping: A field guide to hair cell physiology in the frog, J. Exp. Biol. 203, 2237 (2000).
  29. H. Ohmori, Mechano-electrical transduction currents in isolated vestibular hair cells of the chick, J. Physiol. 359, 189 (1985).
  30. A. C. Crawford and R. Fettiplace, The mechanical properties of ciliary bundles of turtle cochlear hair cells., J. Physiol. 364, 359 (1985).
  31. R. Belousov, F. Berger, and A. J. Hudspeth, Volterra-series approach to stochastic nonlinear dynamics: The Duffing oscillator driven by white noise, Phys. Rev. E 99, 042204 (2019).
  32. R. Belousov, F. Berger, and A. J. Hudspeth, Volterra-series approach to stochastic nonlinear dynamics: Linear response of the Van der Pol oscillator driven by white noise, Phys. Rev. E 102, 032209 (2020).
  33. Y. Thipmaungprom, L. Saliekh, R. Alonso, É. Roldán, F. Berger, and R. Belousov, Active energy harvesting and work transduction by hair-cell bundles in bullfrog's inner ear, arXiv:2502.14485.
  34. G. Tucci, É. Roldán, A. Gambassi, R. Belousov, F. Berger, R. G. Alonso, and A. J. Hudspeth, Modeling active non-markovian oscillations, Phys. Rev. Lett. 129, 030603 (2022).
  35. J. Howard and A. J. Hudspeth, Mechanical relaxation of the hair bundle mediates adaptation in mechanoelectrical transduction by the bullfrog's saccular hair cell, Proc. Natl. Acad. Sci. USA 84, 3064 (1987).
  36. M. E. Benser, R. E. Marquis, and A. J. Hudspeth, Rapid, active hair bundle movements in hair cells from the bullfrog's sacculus, J. Neurosci. 16, 5629 (1996).
  37. D. Vaido, M. A. Toderi, and D. Bozovic, On spontaneous oscillations of hair bundles in the amphibian papilla, Hear. Res. 460, 109238 (2025).
  38. A. J. Hudspeth, Integrating the active process of hair cells with cochlear function, Nat. Rev. Neurosci. 15, 600 (2014).
  39. R. A. Eatock, D. P. Corey, and A. J. Hudspeth, Adaptation of mechanoelectrical transduction in hair cells of the bullfrog's sacculus., J. Neurosci. 7, 2821 (1987).
  40. J. Howard, W. M. Roberts, and A. J. Hudspeth, Mechanoelectrical transduction by hair cells, Annu. Rev. Biophys. Biophys. Chem. 17, 99 (1988).
  41. R. G. Walker, A. J. Hudspeth, and P. G. Gillespie, Calmodulin and calmodulin-binding proteins in hair bundles, Proc. Natl. Acad. Sci. USA 90, 2807 (1993).
  42. R. G. Walker and A. J. Hudspeth, Calmodulin controls adaptation of mechanoelectrical transduction by hair cells of the bullfrog's sacculus, Proc. Natl. Acad. Sci. USA 93, 2203 (1996).
  43. J. L. Cyr, R. A. Dumont, and P. G. Gillespie, Myosin-1c Interacts with Hair-Cell Receptors through Its Calmodulin-Binding IQ domains, J. Neurosci. 22, 2487 (2002).
  44. J. R. Holt, S. K. H. Gillespie, D. William, K. Shah, K. M. Shokat, D. P. Corey, J. A. Mercer, and P. G. Gillespie, A chemical-genetic strategy implicates myosin-1c in adaptation by hair cells, Cell 108, 371 (2002).
  45. P. G. Gillespie and J. L. Cyr, Myosin-1c, the hair cell's adaptation motor, Annu. Rev. Physiol. 66, 521 (2004).
  46. N. Hacohen, J. A. Assad, W. J. Smith, and D. P. Corey, Regulation of tension on hair-cell transduction channels: Displacement and calcium dependence, J. Neurosci. 9, 3988 (1989).
  47. J. A. Assad, G. M. Shepherd, and D. P. Corey, Tip-link integrity and mechanical transduction in vertebrate hair cells, Neuron 7, 985 (1991).
  48. P. Martin, A. D. Mehta, and A. J. Hudspeth, Negative hair-bundle stiffness betrays a mechanism for mechanical amplification by the hair cell, Proc. Natl. Acad. Sci. USA 97, 12026 (2000).
  49. T. Gowers, The Princeton companion to mathematics, in Mathematical Concepts, edited by J. Barrow-Green and I. Leader (Princeton University Press, Princeton, 2010), pp. 157–314.
  50. Y. Polyanskiy and Y. Wu, Information Theory: From Coding to Learning (Cambridge University Press, Cambridge, 2022), 1st ed.
  51. A. B. Tsybakov and V. Zaiats, Introduction to Nonparametric Estimation, Springer Series in Statistics (Springer, New York, 2009), English ed.
  52. D. E. Vakman and L. A. Vaйnshteйn, Amplitude, phase, frequency—Fundamental concepts ofoscillation theory, Sov. Phys. Usp. 20, 1002 (1977).
  53. J. Justice, Analytic signal processing in music computation, IEEE Trans. Acoust., Speech, Signal Process. 27, 670 (1979).
  54. D. Vakman, On the analytic signal, the Teager-Kaiser energy algorithm, and other methods for defining amplitude and frequency, IEEE Trans. Signal Process. 44, 791 (1996).
  55. M. Feldman, Hilbert transform in vibration analysis, Mech. Syst. Sig. Process. 25, 735 (2011).
  56. J. D. Salvi, D. Ó Maoiléidigh, and A. J. Hudspeth, Identification of Bifurcations from Observations of Noisy Biological Oscillators, Biophys. J. 111, 798 (2016).
  57. S. Kay and R. Sudhaker, A zero crossing-based spectrum analyzer, IEEE Trans. Acoust., Speech, Signal Process. 34, 96 (1986).
  58. S.-Z. Hsue and S. S. Soliman, Automatic modulation classification using zero crossing, IEE Proc. F Radar Signal Process. UK 137, 459 (1990).
  59. S. C. Sekhar and T. V. Sreenivas, Adaptive Window Zero-Crossing-Based Instantaneous Frequency Estimation, EURASIP J. Adv. Signal Process. 2004, 249858 (2004).
  60. D. Ramunno-Johnson, C. E. Strimbu, L. Fredrickson, K. Arisaka, and D. Bozovic, Distribution of frequencies of spontaneous oscillations in hair cells of the bullfrog sacculus, Biophys. J. 96, 1159 (2009).
  61. D. Parr and J. M. Thompson, Solving the multi-objective nurse scheduling problem with a weighted cost function, Ann. Oper. Res. 155, 279 (2007).
  62. K. Veerkamp, N. F. J. Waterval, T. Geijtenbeek, C. P. Carty, D. G. Lloyd, J. Harlaar, and M. M. van der Krogt, Evaluating cost function criteria in predicting healthy gait, J. Biomech. 123, 110530 (2021).
  63. M. D. Adler, Benefit–Cost Analysis and Distributional Weights: An Overview, Rev. Environ. Econ. Policy 10, 264 (2016).
  64. J. Miettinen, M. Ollikainen, M. Nieminen, and L. Valsta, Cost function approach to water protection in forestry, Water Resour. Econ. 31, 100150 (2020).
  65. U.-E.-H. Alvi, W. Ahmed, K.-S. Hong, M. Rehan, and S. Ahmed, Optimality condition and distributed optimization for economic dispatch using a novel weighted incremental cost consensus approach, Front. Energy Res. 10, 722256 (2022).
  66. Z. Wu and Y. Tian, Weighted-loss-function control charts, Int. J. Adv. Manuf. Technol. 31, 107 (2006).
  67. Y. Ho and S. Wookey, The real-world-weight cross-entropy loss function: Modeling the costs of mislabeling, IEEE Access 8, 4806 (2020).
  68. K. Bándy and P. Stumpf, Dynamic weighting factor assignment method for the predictive control of electrical drives, IET Power Electron. 17, 2083 (2024).
  69. R. Storn and K. Price, Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim. 11, 341 (1997).
  70. K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution, Natural Computing Series (Springer, Berlin, 2005).
  71. C. Oguz, T. Laomettachit, K. C. Chen, L. T. Watson, W. T. Baumann, and J. J. Tyson, Optimization and model reduction in the high dimensional parameter space of a budding yeast cell cycle model, BMC Syst. Biol. 7, 53 (2013).
  72. I. De Falco, A. Della Cioppa, U. Scafuri, and E. Tarantino, Differential evolution to estimate the parameters of a SEIAR model with dynamic social distancing: The case of COVID-19 in Italy, in Data Science for COVID-19, edited by U. Kose, D. Gupta, V. H. C. de Albuquerque, and A. Khanna (Academic Press, London, UK, 2021), Chap. 5, pp. 75–90.
  73. R. Heller, N. Klingner, N. Claessens, C. Merckling, and J. Meersschaut, Differential evolution optimization of Rutherford backscattering spectra, J. Appl. Phys. 132, 165302 (2022).
  74. S. Yuan, Y. Ji, Y. Chen, X. Liu, and W. Zhang, An Improved Differential Evolution for Parameter Identification of Photovoltaic Models, Sustainability 15, 13916 (2023).
  75. M. Aljaidi, S. P. Agrawal, P. Jangir, S. B. Pandya, A. Parmar, Arpita, A. F. Alkoradees, B. I. Trivedi, and M. Khishe, A two stage differential evolution algorithm for parameter estimation of proton exchange membrane fuel cell, Sci. Rep. 15, 5354 (2025).
  76. Y. Roongthumskul, L. Fredrickson-Hemsing, A. Kao, and D. Bozovic, Multiple-timescale dynamics underlying spontaneous oscillations of saccular hair bundles, Biophys. J. 101, 603 (2011).
  77. R. Shlomovitz, L. Fredrickson-Hemsing, A. Kao, S. W. Meenderink, R. Bruinsma, and D. Bozovic, Low frequency entrainment of oscillatory bursts in hair cells, Biophys. J. 104, 1661 (2013).
  78. S. W. Meenderink, P. M. Quiñones, and D. Bozovic, Voltage-mediated control of spontaneous bundle oscillations in saccular hair cells, J. Neurosci. 35, 14457 (2015).
  79. M. A. Rutherford and W. M. Roberts, Spikes and membrane potential oscillations in hair cells generate periodic afferent activity in the frog sacculus, J. Neurosci. 29, 10025 (2009).
  80. R. Shlomovitz, Y. Roongthumskul, S. Ji, D. Bozovic, and R. Bruinsma, Phase-locked spiking of inner ear hair cells and the driven noisy Adler equation, Interface Focus. 4, 20140022 (2014).
  81. S. Tymochko, E. Munch, J. Dunion, K. Corbosiero, and R. Torn, Using persistent homology to quantify a diurnal cycle in hurricanes, Pattern Recognit. Lett. 133, 137 (2020).
  82. C. M. Topaz, L. Ziegelmeier, and T. Halverson, Topological data analysis of biological aggregation models, PLoS One 10, e0126383 (2015).
  83. M. Ulmer, L. Ziegelmeier, and C. M. Topaz, A topological approach to selecting models of biological experiments, PLoS One 14, e0213679 (2019).
  84. İ. Güzel, E. Munch, and F. A. Khasawneh, Detecting bifurcations in dynamical systems with CROCKER plots, Chaos 32, 093111 (2022).
  85. F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence, Warwick 1980, edited by D. Rand and L.-S. Young (Springer, Berlin, 1981), pp. 366–381.
  86. J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57, 617 (1985).
  87. A. D. Myers, M. M. Chumley, and F. A. Khasawneh, Delay parameter selection in permutation entropy using topological data analysis, La Matematica 3, 1103 (2024).
  88. https://github.com/jmarcinik3/Stochastic-Oscillation-Fitter.
  89. T. Holton and A. J. Hudspeth, The transduction channel of hair cells from the bull-frog characterized by noise analysis, J. Physiol. 375, 195 (1986).
  90. S. Ji, D. Bozovic, and R. Bruinsma, Amphibian sacculus and the forced Kuramoto model with intrinsic noise and frequency dispersion, Phys. Rev. E 97, 042411 (2018).
  91. W. Denk and W. W. Webb, Forward and reverse transduction at the limit of sensitivity studied by correlating electrical and mechanical fluctuations in frog saccular hair cells, Hear. Res. 60, 89 (1992).
  92. J. Barral, F. Jülicher, and P. Martin, Friction from transduction channels’ gating affects spontaneous hair-bundle oscillations, Biophys. J. 114, 425 (2018).
  93. V. Balakrishnan, Elements of Nonequilibrium Statistical Mechanics (Springer International Publishing, Cham, 2021).
  94. S. M. van Netten, T. Dinklo, W. Marcotti, and C. J. Kros, Channel gating forces govern accuracy of mechano-electrical transduction in hair cells, Proc. Natl. Acad. Sci. USA 100, 15510 (2003).
  95. L. Han and A. B. Neiman, Spontaneous oscillations, signal amplification, and synchronization in a model of active hair bundle mechanics, Phys. Rev. E 81, 041913 (2010).
  96. G. Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo 4, 48 (1955).
  97. S. Särkkä and A. Solin, Applied Stochastic Differential Equations (Cambridge University Press, Cambridge, 2019), 1st ed.
  98. S.-H. Cha, Comprehensive Survey on Distance/Similarity Measures between Probability Density Functions International J. Mathe. Models Methods Appl. Sci. 1, 300 (2007).
  99. I. Sason and S. Verdú, F -divergence inequalities, IEEE Trans. Inf. Theory 62, 5973 (2016).
  100. C. Ericson, Axis-aligned Bounding Boxes (AABBs), in Real-Time Collision Detection (CRC Press, Boca Raton, FL, 2004), pp. 75–124.
  101. Y. Fu, M. Li, X. Zhang, S. Zhang, C. Wei, W. Guo, H. Cai, L. Sun, P. Wang, and F. Zha, Spatial topological relation analysis for cluttered scenes, Sensors (Basel) 20, 7181 (2020).
  102. Y. Qiao and N. Minematsu, F-divergence is a generalized invariant measure between distributions, in Interspeech 2008 (ISCA, Brisbane, Australia, 2008), pp. 1349–1352.
  103. F. Nielsen, On the jensen–shannon symmetrization of distances relying on abstract means, Entropy 21, 485 (2019).
  104. E. B. Page, Ordered hypotheses for multiple treatments: A significance test for linear ranks, J. Am. Stat. Assoc. 58, 216 (1963).
  105. A. P. Piotrowski, Review of differential evolution population size, Swarm Evol. Comput. 32, 1 (2017).
  106. M. Renardy, L. R. Joslyn, J. A. Millar, and D. E. Kirschner, To Sobol or not to Sobol? The effects of sampling schemes in systems biology applications, Math. Biosci. 337, 108593 (2021).
  107. J. Qiang and C. Mitchell, A unified differential evolution algorithm for global optimization, IEEE Trans. Evol. Comput. (2014), https://www.osti.gov/biblio/1163659
  108. M. S. Bartlett, Periodogram analysis and continuous spectra, Biometrika 37, 1 (1950).
  109. P. Welch, The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms, IEEE Trans. Audio Electroacoust. 15, 70 (1967).
  110. S. L. Hahn and K. M. Snopek, Complex and Hypercomplex Analytic Signals: Theory and Applications, Artech House Signal Processing Library (Artech House, Boston, MA; London, UK, 2017).
  111. S. Chang, B. Yu, and M. Vetterli, Adaptive wavelet thresholding for image denoising and compression, IEEE Trans. Image Process. 9, 1532 (2000).
  112. D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika 81, 425 (1994).
  113. S. Arfaoui, A. B. Mabrouk, and C. Cattani, Wavelet Analysis: Basic Concepts and Applications (CRC Press, Boca Raton, FL, 2021), 1st ed.
  114. S. Tanweer, F. A. Khasawneh, and E. Munch, Robust crossings detection in noisy signals using topological signal processing, FoDS 6, 154 (2024).
  115. C. R. Harris, et al., Array programming with NumPy, Nature (London) 585, 357 (2020).
  116. P. Virtanen, et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  117. M. J. Aburn, Critical fluctuations and coupling of stochastic neural mass models, Ph.D. thesis, The University of Queensland, 2017.
  118. S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, Scikit-image: Image processing in Python, PeerJ 2, e453 (2014).
  119. J. D. Hunter, Matplotlib: A 2D Graphics Environment, Comput. Sci. Eng. 9, 90 (2007).
  120. W. McKinney, Data structures for statistical computing in python, in Proceedings of the 9th Python in Science Conference, edited by S. van der Walt and J. Millman (SciPy, Austin, 2010), Vol. 445, pp. 56–61.

Outline

Information

Phys. Rev. Research 8, 013342– Published 31 March, 2026
  • Received 9 July 2025
  • Accepted 20 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation