Bibliography

[1]
Atherton, L. A., Prince, L. Y. and Tsaneva-Atanasova, K. (2016). Bifurcation analysis of a two-compartment hippocampal pyramidal cell model. Journal of Computational Neuroscience 41 91–106.
[2]
Bellen, A. and Zennaro, M. (2003). Numerical methods for delay differential equations. Oxford University Press.
[3]
Bezanson, J., Edelman, A., Karpinski, S. and Shah, V. B. (2017). Julia: A fresh approach to numerical computing. SIAM Review 59 65–98.
[4]
Veltz, R. (2020). BifurcationKit.jl.
[5]
Brázdil, M., Pail, M., Halámek, J., Plešinger, F., Cimbálník, J., Roman, R., Klimeš, P., Daniel, P., Chrastina, J., Brichtová, E., Rektor, I., Worrell, G. A. and Jurák, P. (2017). Very high‐frequency oscillations: Novel biomarkers of the epileptogenic zone. Annals of Neurology 82 299–310.
[6]
Brázdil, M., Worrell, G. A., Trávníček, V., Pail, M., Roman, R., Plešinger, F., Klimeš, P., Cimbálník, J., Stacey, W. and Jurák, P. (2023). Ultra fast oscillations in the human brain and their functional significance.
[7]
Breda, D., Maset, S. and Vermiglio, R. (2006). NUMERICAL COMPUTATION OF CHARACTERISTIC MULTIPLIERS FOR LINEAR TIME PERIODIC COEFFICIENTS DELAY DIFFERENTIAL EQUATIONS. IFAC Proceedings Volumes 39 163–8.
[8]
Breda, D. (2006). Solution operator approximations for characteristic roots of delay differential equations. Applied Numerical Mathematics 56 305–17.
[9]
Brin, L. Q. (2020). Tea time numerical analysis.
[10]
Butcher, J. C. (2016). Numerical methods for ordinary differential equations. Wiley.
[11]
Chicone, C. (2006). Ordinary differential equations with applications. Springer, New York, NY.
[12]
Cimbalnik, J., Brinkmann, B., Kremen, V., Jurak, P., Berry, B., Gompel, J. V., Stead, M. and Worrell, G. (2018). Physiological and pathological high frequency oscillations in focal epilepsy. Annals of Clinical and Translational Neurology 5 1062–76.
[13]
Cimbalnik, J., Pail, M., Klimes, P., Travnicek, V., Roman, R., Vajcner, A. and Brazdil, M. (2020). Cognitive processing impacts high frequency intracranial EEG activity of human hippocampus in patients with pharmacoresistant focal epilepsy. Frontiers in Neurology 11.
[14]
Curtain, R. F. and Zwart, H. (1995). An introduction to infinite-dimensional linear systems theory. Springer New York.
[15]
Dhooge, A., Govaerts, W. and A. Kuznetsov, Yu. (2003). MatCont: A MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software 29 141–64.
[16]
Dhooge, A., Govaerts, W., A. Kuznetsov, Yu., Meijer, H. G. E. and Sautois, B. (2008). New features of the software MatCont for bifurcation analysis of dynamical systems. Mathematical and Computer Modelling of Dynamical Systems 14 147–75.
[17]
Diekmann, O., Verduyn Lunel, S. M., Gils, S. A. van and Walther, H.-O. (1995). Delay equations. Springer New York.
[18]
Doedel, E., Keller, H. B. and Kernevez, J. P. (1991). NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (II): BIFURCATION IN INFINITE DIMENSIONS. International Journal of Bifurcation and Chaos 01 745–72.
[19]
Elaydi, S. (2005). An introduction to difference equations. Springer Science+Business Media.
[20]
Engelborghs, K., Luzyanina, T., Hout, K. J. I. ’t. and Roose, D. (2001). Collocation methods for the computation of periodic solutions of delay differential equations. SIAM Journal on Scientific Computing 22 1593–609.
[21]
Engelborghs, K. and Roose, D. (2002). On stability of LMS methods and characteristic roots of delay differential equations. SIAM Journal on Numerical Analysis 40 629–50.
[22]
Ermentrout, G. B. and Terman, D. H. (2010). Mathematical foundations of neuroscience. Springer New York.
[23]
Fairgrieve, T. F. and Jepson, A. D. (1991). O. K. Floquet multipliers. SIAM Journal on Numerical Analysis 28 1446–62.
[24]
Govaerts, W., Kuznetsov, Y. A. and Dhooge, A. (2005). Numerical continuation of bifurcations of limit cycles in MATLAB. SIAM Journal on Scientific Computing 27 231–52.
[25]
Guo, S. and Wu, J. (2013). Bifurcation theory of functional differential equations. Springer New York.
[26]
Hairer, E., Nørsett, S. P. and Wanner, G. (2008). Solving ordinary differential equations I. Springer, Berlin, Germany.
[27]
Halaštová, B. (2025). Analýza signálů v neurovědách. Master's thesis, Masarykova univerzita, Přírodovědecká fakulta, Brno.
[28]
Hale, J. K. (1977). Theory of functional differential equations. Springer New York.
[29]
Hale, J. K. and Lunel, S. M. V. (1993). Introduction to functional differential equations. Springer New York.
[30]
Hartman, P. (2002). Ordinary differential equations. Society for Industrial; Applied Mathematics.
[31]
Hellevik, L. R. (2020). Numerical methods for engineers.Available at https://leifh.folk.ntnu.no/teaching/tkt4140/.
[32]
Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology 117 500–44.
[33]
Hutchinson, G. E. (1948). Circular causal systems in ecology. Annals of the New York Academy of Sciences 50 221–46.
[34]
Izhikevich, E. M. (2006). Dynamical systems in neuroscience: The geometry of excitability and bursting. The MIT Press.
[35]
Jacobs, J., LeVan, P., Chander, R., Hall, J., Dubeau, F. and Gotman, J. (2008). Interictal high‐frequency oscillations (80–500 hz) are an indicator of seizure onset areas independent of spikes in the human epileptic brain. Epilepsia 49 1893–907.
[36]
Jacobs, J., Staba, R., Asano, E., Otsubo, H., Wu, J. Y., Zijlmans, M., Mohamed, I., Kahane, P., Dubeau, F., Navarro, V. and Gotman, J. (2012). High-frequency oscillations (HFOs) in clinical epilepsy. Progress in Neurobiology 98 302–15.
[37]
Jekl, J. (2018). Matematické modely neuronu. Master's thesis, Masarykova univerzita, Přírodovědecká fakulta, Brno.
[38]
Jiruska, P., Curtis, M. de, Jefferys, J. G. R., Schevon, C. A., Schiff, S. J. and Schindler, K. (2013). Synchronization and desynchronization in epilepsy: Controversies and hypotheses. The Journal of Physiology 591 787–97.
[39]
Jiruska, P., Alvarado‐Rojas, C., Schevon, C. A., Staba, R., Stacey, W., Wendling, F. and Avoli, M. (2017). Update on the mechanisms and roles of high‐frequency oscillations in seizures and epileptic disorders. Epilepsia 58 1330–9.
[40]
Keener, J. and Sneyd, J. (2009). Mathematical physiology. Springer New York.
[41]
Kloeden, P. E. and Platen, E. (1992). Numerical solution of stochastic differential equations. Springer Berlin Heidelberg.
[42]
Kochenderfer, M. J. and Wheeler, T. A. (2019). Algorithms for optimization. MIT Press, London, England.
[43]
Kolmanovskii, V. and Myshkis, A. (1992). Applied theory of functional differential equations. Springer Netherlands.
[44]
Kuznetsov, Y. Introduction to numerical bifurcation analysis: Location and continuation of limit cycles.
[45]
Kuznetsov, Y. A. (2023). Elements of applied bifurcation theory. Springer International Publishing.
[46]
Lacerda de Orio, R. (2010). Electromigration modeling and simulation. PhD thesis, echnische Universität Wien.
[47]
Lecar, H. (2007). Morris-Lecar model. Scholarpedia 2 1333.
[48]
Linge, S. and Langtangen, H. P. (2017). Nonlinear problems. In Finite difference computing with PDEs pp 353–407. Springer International Publishing.
[49]
Lust, K. and Roose, D. (1998). An adaptive newton–picard algorithm with subspace iteration for computing periodic solutions. SIAM Journal on Scientific Computing 19 1188–209.
[50]
Danisch, S. and Krumbiegel, J. (2021). Makie.jl: Flexible high-performance data visualization for Julia. Journal of Open Source Software 6 3349.
[51]
Inc., T. M. (2023). MATLAB version: 23.2.0.2365128 (R2023b).
[52]
Mohammed, S. E. A. (1986). Nonlinear flows of stochastic linear delay equations. Stochastics 17 207–13.
[53]
Morris, C. and Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal 35 193–213.
[54]
Murray, J. D. (2002). Mathematical biology: I. An introduction. Springer New York.
[55]
Perko, L. (2001). Differential equations and dynamical systems. Springer New York.
[56]
Pikovsky, A., Rosenblum, M. and Kurths, J. (2001). Synchronization: A universal concept in nonlinear sciences. Cambridge University Press.
[57]
Pinsky, P. F. and Rinzel, J. (1994). Intrinsic and network rhythmogenesis in a reduced traub model for CA3 neurons. Journal of Computational Neuroscience 1 39–60.
[58]
Přibylová, L. (2021). Teorie bifurkací, chaos a fraktály. Masarykova univerzita.
[59]
Přibylová, L. (2021). Nelineární dynamika. Masarykova univerzita.
[60]
Přibylová, L., Ševčík, J., Eclerová, V., Klimeš, P., Brázdil, M. and Meijer, H. G. E. (2024). Weak coupling of neurons enables very high-frequency and ultra-fast oscillations through the interplay of synchronized phase shifts. Network Neuroscience 8 293–318.
[61]
Přibylová, L., Ševčík, J., Halmazňa, T., Husa, Š., Malárik, P., Kajanová, L., Polách, M., Zapadlo, Š. and Eclerová, V. (2025). Chaos links dendritic calcium to bursting in hippocampal pyramidal cells. Chaos, Solitons and Fractals.
[62]
Allaire, J. J., Teague, C., Scheidegger, C., Xie, Y., Dervieux, C. and Woodhull, G. (2025). Quarto.
[63]
Rackauckas, C. and Nie, Q. (2017). DifferentialEquations.jl–a performant and feature-rich ecosystem for solving differential equations in Julia. Journal of Open Research Software 5.
[64]
Roose, D. and Szalai, R. (2007). Continuation and bifurcation analysis of delay differential equations. In Numerical continuation methods for dynamical systems pp 359–99. Springer Netherlands.
[65]
Saperstone, S. H. (1981). Semidynamical systems in infinite dimensional spaces. Springer New York.
[66]
Scheutzow, M. (2018). Stochastic differential equations.
[67]
Ševčík, J. (2021). Synchronizace. Master's thesis, Masarykova univerzita, Přírodovědecká fakulta, Brno.
[68]
Ševčík, J. and Přibylová, L. (2025). Cycle multistability and synchronization mechanisms in coupled interneurons: In-phase and anti-phase dynamics under current stimuli. accepted for publication in Applied Mathematics and Computation.
[69]
Shampine, L. F. and Thompson, S. (2001). Solving DDEs in matlab. Applied Numerical Mathematics 37 441–58.
[70]
Sieber, J., Engelborghs, K., Luzyanina, T., Samaey, G. and Roose, D. (2014). DDE-BIFTOOL manual - bifurcation analysis of delay differential equations.
[71]
Skinner, F. K. (2006). Conductance-based models. Scholarpedia 1 1408.
[72]
Smith, H. (2010). An introduction to delay differential equations with applications to the life sciences. Springer New York.
[73]
Song, H., Mylvaganam, S. M., Wang, J., Mylvaganam, S. M. K., Wu, C., Carlen, P. L., Eubanks, J. H., Feng, J. and Zhang, L. (2018). Contributions of the hippocampal CA3 circuitry to acute seizures and hyperexcitability responses in mouse models of brain ischemia. Frontiers in Cellular Neuroscience 12.
[74]
Staba, R. J. and Bragin, A. (2011). High-frequency oscillations and other electrophysiological biomarkers of epilepsy: Underlying mechanisms. Biomarkers in Medicine 5 545–56.
[75]
Teschl, G. (2012). Ordinary differential equations and dynamical systems. American Mathematical Society, Providence, RI.
[76]
Trefethen, L. N. (1994). Finite difference and spectral methods for ordinary and partial differential equations.
[77]
Verheyden, K. and Lust, K. (2005). A newton-picard collocation method for periodic solutions of delay differential equations. BIT Numerical Mathematics 45 605–25.
[78]
White, J. A., Chow, C. C., Rit, J., Soto-Treviño, C. and Kopell, N. (1998). Journal of Computational Neuroscience 5 5–16.
[79]
Widmann, D. and Rackauckas, C. (2022). DelayDiffEq: Generating delay differential equation solvers via recursive embedding of ordinary differential equation solvers. arXiv preprint arXiv:2208.12879.
[80]
Wiggins, S. (2003). Introduction to applied nonlinear dynamical systems and chaos. Springer, New York, NY.
[81]
Wikipedia contributors. (2024). Grothendieck group — Wikipedia, the free encyclopedia.
[82]
Wikipedia contributors. (2025). Jordan curve theorem — Wikipedia, the free encyclopedia.
[83]
Willms, A. R., Kitanov, P. M. and Langford, W. F. (2017). Huygens’ clocks revisited. Royal Society Open Science 4 170777.
[84]
Worrell, G. and Gotman, J. (2011). High-frequency oscillations and other electrophysiological biomarkers of epilepsy: Clinical studies. Biomarkers in Medicine 5 557–66.
[85]
Záthurecký, J., Eclerová, V., Ševčík, J., Zapadlo, Š. and Přibylová, L. (2025). Phase shifts inside arnold tongues of weakly coupled oscillators. Communications in Nonlinear Science and Numerical Simulation 108729.
[86]
Zemánek, P. (2021). Optimalizace aneb když méně je více.