Open Access
Issue
Math. Model. Nat. Phenom.
Volume 19, 2024
Article Number 3
Number of page(s) 26
Section Mathematical physiology and medicine
DOI https://doi.org/10.1051/mmnp/2024001
Published online 15 February 2024
  1. B. Ambrosio and M.A. Aziz-Alaoui, Synchronization and control of coupled reaction-diffusion systems of the FitzHugh–Nagumo type, Comput. Math. Appl. 64 (2012) 934–943. [CrossRef] [Google Scholar]
  2. B. Ambrosio and M.A. Aziz-Alaoui, Synchronization and Control of a network of coupled reaction-diffusion systems of generalized FitzHugh–Nagumo type, ESAIM Proc. 39 (2013) 15–24. [CrossRef] [EDP Sciences] [Google Scholar]
  3. B. Ambrosio and M.A. Aziz-Alaoui, Basin of attraction of solutions with pattern formation in slow–fast reaction–diffusion systems, Acta Biotheoretica 64 (2013) 311–325. [Google Scholar]
  4. B. Ambrosio, M.A. Aziz-Alaoui and A. Balti, Propagation of bursting oscillations in coupled non-homogeneous Hodgkin–Huxley reaction-diffusion systems. Differ. Equ. Dyn. Syst. 39 (2013) 15–24. [Google Scholar]
  5. B. Ambrosio, M.A. Aziz-Alaoui and V.L.E. Phan, Large time behaviour and synchronization and control of a network of coupled reaction-diffusion systems of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type. IMA J. Appl. Math. 84 (2019) 416–443. [CrossRef] [MathSciNet] [Google Scholar]
  6. B. Ambrosio and J.-P. Francoise, Propagation of bursting oscillations. Philos. Trans. Roy. Soc. A: Math. Phys. Eng. Sci. 367 (2009) 4863–4875. [CrossRef] [PubMed] [Google Scholar]
  7. B. Ambrosio and L-S. Young, The use of reduced models to generate irregular, broad-band signals that resemble brain rhythms Front. Computat. Neurosci. 16 (2022). [Google Scholar]
  8. M.A. Aziz-Alaoui, Synchronization of chaos, in Encyclopedia of Mathematical Physics. (2006) 213–226. [CrossRef] [Google Scholar]
  9. B.V. Atallah and M. Scanziani, Instantaneous modulation of gamma oscillation frequency by balancing excitation with inhibition. Neuron 62 (2009) 566–577. [CrossRef] [PubMed] [Google Scholar]
  10. A. Balti, V. Lanza and M.A. Aziz-Alaoui, A multi-base harmonic balance method applied to Hodgkin–Huxley model. Math. Biosci. Eng. 15 (2018) 807–825. [CrossRef] [MathSciNet] [Google Scholar]
  11. A.-L. Barabasi, Network Science. Cambridge University Press (2016). [Google Scholar]
  12. A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks. Cambridge University Press (2008). [CrossRef] [Google Scholar]
  13. V.N. Belykh, I.V. Belykh, M. Hasler and K.V. Nevidin, Cluster synchronization in three-dimensional lattices of diffusively coupled oscillators. Int. J. Bifurc. Chaos 13 (2003) 755–779. [CrossRef] [Google Scholar]
  14. V.N. Belykh, I.V. Belykh, M. Hasler and K.V. Nevidin, Background gamma rhythmicity and attention in cortical local circuits: a computational study. PNAS 102 (2005) 7002–7007. [CrossRef] [PubMed] [Google Scholar]
  15. C. Borgers, An Introduction to Modeling Neuronal Dynamics. Springer (2017). [CrossRef] [Google Scholar]
  16. N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Computat. Neurosci. 8 (2000) 183–208. [CrossRef] [Google Scholar]
  17. L. Chariker and L.-S. Young, Emergent spike patterns in neuronal populations. J. Computat. Neurosci. 38 (2014) 203–220. [Google Scholar]
  18. L. Chariker, R. Shapley and L.-S. Young, Orientation selectivity from very sparse LGN inputs in a comprehensive model of macaque V1 cortex. J. Neurosci. 36 (2018) 12368–12384. [Google Scholar]
  19. L. Chariker, R. Shapley and L.-S. Young, Rhythm and synchrony in a cortical network model. J. Neurosci. 38 (2018) 8621–8634. [CrossRef] [PubMed] [Google Scholar]
  20. M. Chavez, M. Valencia, V. Latora and J. Martinerie, Complex networks: new trends for the analysis of brain connectivity. Int. J. Bifurc. Chaos 20 (2010) 1–10. [Google Scholar]
  21. J. Cronin, Mathematical Aspects of Hodgkin–Huxley Neural Theory. Cambridge University Press (1987). [CrossRef] [Google Scholar]
  22. P. Dayan and L.F. Abott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. The MIT Press (2001). [Google Scholar]
  23. M. Chavez, M. Valencia, V. Latora and J. Martinerie, Canards, clusters and synchronization in a weakly coupled interneuron model. SIADS 8 (2019) 253–278. [Google Scholar]
  24. G.B. Ermentrout and N. Kopell, Fine structure of neural spiking and synchronization in the presence of conduction delays. PNAS 95 (1998) 1259–1264. [CrossRef] [PubMed] [Google Scholar]
  25. G.B. Ermentrout and D.H. Terman, Mathematical Foundations of Neuroscience. Springer, New York (2010). [CrossRef] [Google Scholar]
  26. E.M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press (2006). [CrossRef] [Google Scholar]
  27. R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1 (1961) 445–466. [CrossRef] [Google Scholar]
  28. W. Gerstner, W.M. Kistler, R. Naud, M. Richard and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition. Cambridge University Press (2014). [CrossRef] [Google Scholar]
  29. J.L. Hindmarsh and R.M. Rose, A model of neuronal bursting using three coupled first order differential equations. Proc. Roy. Soc. Lond. Ser. B 221 (1984) 87–102. [CrossRef] [Google Scholar]
  30. A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (1952) 500–544. [CrossRef] [PubMed] [Google Scholar]
  31. E.M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience). The MIT Press (2006). [CrossRef] [Google Scholar]
  32. E.R. Kandel, J.H. Schwartz, T.M. Jessell, S.A. Siegelbaum, A.J. Hudspeth and S. Mack, Principles of Neural Science. McGraw-Hill (2013). [Google Scholar]
  33. I.C. Lin, D. Xing and R. Shapley, Integrate-and-fire vs Poisson models of LGN input to V1 cortex: noisier inputs reduce orientation selectivity. J. Comput. Neurosci. 33 (2012) 559–572. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  34. B. Hassard, Bifurcation of periodic solutions of the Hodgkin–Huxley model for the squid giant axon. J. Theor. Biol. 71 (1978) 401–420. [CrossRef] [Google Scholar]
  35. J. Guckenheimer and J.S. Labouriau, Bifurcation of the Hodgkin and Huxley equations: a new twist. Bull. Math. Biol. 55 (1993) 937–952. [CrossRef] [Google Scholar]
  36. J. Guckenheimer and R.A. Oliva, Chaos in the Hodgkin–Huxley Model. SIAM J. Appl. Dyn. Syst. 1 (2002) 105–114. [CrossRef] [MathSciNet] [Google Scholar]
  37. C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35 (1981) 193–213. [CrossRef] [Google Scholar]
  38. J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Biophys. J. 50 (1962) 2061–2070. [Google Scholar]
  39. M.E.J. Newman, Networks: An Introduction, Oxford University Press (2010). [Google Scholar]
  40. K.P. O’Keeffe, H. Hong and S.H. Strogatz, Oscillators that sync and swarm., Nat. Commun. 8 (2017). [Google Scholar]
  41. M. Okun and I. Lampl, Instantaneous correlation of excitation and inhibition during ongoing and sensory-evoked activities. Nat. Neurosci. 11 (2008) 535–537. [CrossRef] [PubMed] [Google Scholar]
  42. J. Pantaleone, Oscillators that sync and swarm. Am. J. Phys. 70 (2002) 992–1000. [CrossRef] [Google Scholar]
  43. L.M. Pecora, F. Sorrentino, A.M. Hagerstrom, T.E. Murphy and R. Roy, Cluster synchronization and isolated desynchronization in complex networks with symmetries. Nat. Commun. 5 (2014) 4079. [CrossRef] [Google Scholar]
  44. A.V. Rangan and L.-S. Young, Emergent dynamics in a model of visual cortex. J. Computat. Neurosci. 35 (2013) 155–167. [CrossRef] [PubMed] [Google Scholar]
  45. J. Rinzel and R. N. Miller, Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations. Math. Biosci. 49 (1980) 27–59. [CrossRef] [MathSciNet] [Google Scholar]
  46. Y. Shu, A. Hasenstaub and D. McCormick, Turning on and off recurrent balanced cortical activity. Nature 423 (2003) 288–293. [CrossRef] [PubMed] [Google Scholar]
  47. S. Stogatz and I. Stewart, Coupled Oscillators and biological synchronization. Am. Sci. (1993) 68–75. [Google Scholar]
  48. Y. Sun, D. Zhou, A.V. Rangan and D. Cai, Library-based numerical reduction of the Hodgkin–Huxley neuron for network simulation. J. Computat. Neurosci. 27 (2009) 369–390. [CrossRef] [PubMed] [Google Scholar]
  49. A.Y.Y. Tan, L.I. Zhang, M.M. Merzenich and C.E. Schreiner, Evoked excitatory and inhibitory synaptic conductances of primary auditory cortex neurons. J. Neurophysiol. 92 (2004) 630–643. [CrossRef] [PubMed] [Google Scholar]
  50. R.D. Traub, J.G. Jefferys, R. Miles, M.A. Whittington and K. Tóth, A branching dendritic model of a rodent CA3 pyramidal neuron. J. Physiol. 481 (1994) 79–95. [CrossRef] [PubMed] [Google Scholar]
  51. R.D. Traub, M.A. Whittington, E.H. Buhl, J.G.R. Jefferys and H.J. Faulkner, On the mechanism of the γ → β frequency shift in neuronal oscillations induced in rat hippocampal slices by tetanic stimulation. J. Neurosci. 19 (1999) 1088–1105. [CrossRef] [PubMed] [Google Scholar]
  52. M.A Whittington, R.D. Traub, N. Kopell, B. Ermentrout and E.H. Buhl, Inhibition-based rhythms: experimental and mathematical observations on network dynamics. Int. J. Psychophysiol. 38 (2000) 315–336. [CrossRef] [Google Scholar]
  53. H.R. Wilson and J.D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12 (1972) 1–24. [CrossRef] [Google Scholar]
  54. D. Zhou and A.V. Rangan, D.W. McLaughlin and D. Cai, Spatiotemporal dynamics of neuronal population response in the primary visual cortex. PNAS 110 (2013) 9517–9522. [CrossRef] [PubMed] [Google Scholar]

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