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Modeling Neuronal Firing in Epilepsy: Fitting Hawkes Processes to Single-Unit Activity

  • György Perczel
  • Loránd Erőss
  • Dániel Fabó
  • László GerencsérEmail author
  • Zsuzsanna Vágó
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 30)

Abstract

Forecasting seizures based on information extracted from neuronal firing has a great potential in controlling closed-loop neurostimulators. For the description of neuronal firing patterns we use self-exiting point processes or Hawkes processes. In fitting them to simulated data, using a large variety of models, we consider both computability and reliability issues related to the maximum likelihood estimation (MLE) method. The models are classified via a single parameter related to stability regimes. The dependence of the accuracy of the individual parameter estimates on different regimes will be explored. We demonstrate the applicability of the MLE method to discriminate between different models with high confidence.

Notes

Acknowledgements

This research has been partially supported by the European Union, co-financed by the European Social Fund (EFOP-3.6.3-VEKOP-16-2017-00002).

References

  1. 1.
    Brémaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, New York (1981)CrossRefGoogle Scholar
  2. 2.
    Chornoboy, E.S., Schramm, L.P., Karr, A.F.: Maximum likelihood identification of neural point process systems. Biol. Cybern. 59, 265–75 (1988)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer Science & Business Media, New York (2013)zbMATHGoogle Scholar
  4. 4.
    Gadhoumi, K., Lina, J.-M., Mormann, F., Gotman, J.: Seizure prediction for therapeutic devices: a review. J. Neurosci. Methods 29, 1–13 (2015)Google Scholar
  5. 5.
    Gerencsér, L., Matias, C., Vágó, Z., Torma, B., Weiss, B.: Self-exciting point processes with applications in finance and medicine. In: Proccedings of the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS2008), Virginia Tech, Blacksburg, Virginia, USA (2008)Google Scholar
  6. 6.
    Gerhard, F., Deger, M., Truccolo, W.: On the stability and dynamics of stochastic spiking neuron models: nonlinear Hawkes process and point process GLMs. PLoS Comp. Biol. 13, e1005390 (2017).  https://doi.org/10.1371/journal.pcbi.1005390 CrossRefGoogle Scholar
  7. 7.
    Hansen, N.R., Reynaud-Bouret, P., Rivoirard, V.: Lasso and probabilistic inequalities for multivariate point processes. Bernoulli 21, 83–143 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hawkes, A.G.: Spectra of some self-exiciting and mutually exciting point processes. Biometrika 58, 83–90 (1971)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kuhlmann, L., Grayden, D.B., Wendling, F., Schiff, S.J.: Role of multiple-scale modeling of epilepsy in seizure forecasting. J. Clin. Neurophysiol. 32, 220–226 (2015)CrossRefGoogle Scholar
  10. 10.
    Merricks, E.M., Smith, E.H., McKhann, G.M., Goodman, R.R., Bateman, L.M., Emerson, R.G., Schevon, C.A., Trevelyan, A.J.: Single unit action potentials in humans and the effect of seizure activity. Brain 138, 2891–2906 (2015)CrossRefGoogle Scholar
  11. 11.
    Møller, J., Rasmussen, J.G..: Perfect simulation of Hawkes processes. Adv. Appl. Probab. 37, 629–646 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ozaki, T.: Maximum likelihood estimation of Hawkes’ self-exciting point processes. Ann. Inst. Stat. Math. 31, 145–155 (1979)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Quiroga, R.Q., Nadasdy, Z., Ben-Shaul, Y.: Unsupervised spike detection and sorting with wavelets and superparamagnetic clustering. Neural. Comput. 16, 1661–1687 (2004)CrossRefGoogle Scholar
  14. 14.
    Schulze-Bonhage, A., Kühn, A.: Unpredictability of seizures and the burden of epilepsy. In: Seizure Prediction in Epilepsy: From Basic Mechanisms to Clinical Applications, p. 1–10 (2008)Google Scholar
  15. 15.
    Truccolo, W., Donoghue, J.A., Hochberg, L.R., Eskandar, E.N., Madsen, J.R., Anderson, W.S., Brown, E.N., Halgren, E., Cash, S.S.: Single-neuron dynamics in human focal epilepsy. Nat. Neurosci. 14, 635–641 (2011)CrossRefGoogle Scholar
  16. 16.
    Ulbert, I., Maglóczky, Z., Erőss, L., Czirják, S., Vajda, J., Bognár, L., Tóth, S., Szabó, Z., Halász, P., Fabó, D., Halgren, E., Freund, T.F., Karmos, G.: In vivo laminar electrophysiology co-registered with histology in the hippocampus of patients with temporal lobe epilepsy. Exp. Neurol. 187, 310–318 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • György Perczel
    • 1
    • 2
  • Loránd Erőss
    • 1
    • 2
  • Dániel Fabó
    • 1
    • 2
  • László Gerencsér
    • 3
    Email author
  • Zsuzsanna Vágó
    • 4
  1. 1.National Institute of Clinical NeurosciencesBudapestHungary
  2. 2.Faculty of Information Technology and BionicsPázmány Péter Catholic UniversityBudapestHungary
  3. 3.Systems and Control LabInstitute for Computer Science and ControlBudapestHungary
  4. 4.Faculty of Information Technology and BionicsPázmány Péter Catholic UniversityBudapestHungary

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