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Propagator, Axonal

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Encyclopedia of Computational Neuroscience

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Definition

The axonal propagator is an operator that is applied to the initial state of a neural field variable and predicts the spatiotemporal spread of neural activity as a function of axonal distribution. It is most commonly represented as an integral operator over the axonal distribution function and the associated time delay via signal transmission. The axonal propagator is used to mathematically capture the connectivity across neuronal ensembles and derive the evolution equations of network activity.

Historical Remarks

Beurle (1956) and Griffith (1963, 1965) were among the first to formulate spatially continuous evolution equations for neural fields. Wilson and Cowan (1972, 1973) and Nunez (1974) extended the neural field populations to include excitatory and inhibitory populations, refractoriness, and signal transmission delays. For specific axonal distributions, Amari (1977) obtained detailed results on pattern formation phenomena when neglecting the time delays of the axonal...

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References

  • Amari S (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybern 27:77–87

    Article  CAS  PubMed  Google Scholar 

  • Beurle RL (1956) Properties of a mass of cells capable of regenerating pulses. Philos Trans R Soc Lond B Biol Sci 240:55–94. doi:10.1098/rstb.1956.0012

    Article  Google Scholar 

  • Bojak I, Liley DTJ (2010) Axonal velocity distributions in neural field equations. PLoS Comput Biol 6:e1000653

    Article  PubMed Central  PubMed  Google Scholar 

  • Bojak I, Oostendorp T, Reid A, Kötter R (2010) Connecting mean field models of neural activity to EEG and fMRI data. Brain Topogr 23(2):139–149

    Article  PubMed  Google Scholar 

  • Bojak I, Liley D (2005) Modeling the effects of anesthesia on the electroencephalogram. Phys Rev E 71:041,902

    Article  CAS  Google Scholar 

  • Bojak I, Oostendorp T, Reid A, Ktter R (2011) Towards a model-based integration of co-registered electroencephalography/functional magnetic resonance imaging data with realistic neural population meshes. Philos Trans R Soc Lond A 369(1952):3785–3801

    Article  CAS  Google Scholar 

  • Breakspear M, Roberts JA, Terry JR, Rodrigues S, Mahant N, Robinson PAA (2006) Unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cereb Cortex 16:1296–1313

    Article  CAS  PubMed  Google Scholar 

  • Breakspear M, Terry J, Friston K (2003) Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a biophysical model of neuronal dynamics. Netw Comput Neural Syst 14:703–732

    Article  Google Scholar 

  • Deco G, Jirsa V, McIntosh A (2011) Emerging concepts for the dynamical organization of resting-state activity in the brain. Nat Rev Neurosci 12(1):43–56

    Article  CAS  PubMed  Google Scholar 

  • Deco G, Ponce-Alvarez A, Mantini D, Romani GL, Hagmann P, Corbetta M (2013) Resting-state functional connectivity emerges from structurally and dynamically shaped slow linear fluctuations. J Neurosci 33(27):11,239–11,252

    Article  CAS  Google Scholar 

  • Freyer F, Roberts JA, Becker R, Robinson PA, Ritter P, Breakspear M (2011) Biophysical mechanisms of multistability in resting-state cortical rhythms. J Neurosci 31(17):6353–6361

    Article  CAS  PubMed  Google Scholar 

  • Griffith JS (1963) A field theory of neural nets: I: Derivation of field equations. Bull Math Biol 25:111–120

    CAS  Google Scholar 

  • Griffith JS (1965) A field theory of neural nets: II: Properties of the field equations. Bull Math Biol 27:187–195

    CAS  Google Scholar 

  • Hutt A (2013) The anaesthetic propofol shifts the frequency of maximum spectral power in eeg during general anaesthesia: analytical insights from a linear model. Front Comput Neurosci 7(2):1–15

    Google Scholar 

  • Jirsa VK, Haken H (1996) Field theory of electromagnetic brain activity. Phys Rev Lett 77:960–963

    Article  CAS  PubMed  Google Scholar 

  • Jirsa VK, Haken H (1997) A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics. Phys D 99:503–526

    Article  Google Scholar 

  • Jirsa VK, Kelso JAS (2000) Spatiotemporal pattern formation in continuous systems with heterogeneous connection topologies. Phys Rev E 62:8462–8465

    Article  CAS  Google Scholar 

  • Jirsa VK, Jantzen KJ, Fuchs A, Kelso JAS (2002) Spatiotemporal forward solution of the EEG and MEG using network modelling. IEEE Trans Med Imaging 21:493–504

    Article  PubMed  Google Scholar 

  • Jirsa VK (2009) Neural field dynamics with local and global connectivity and time delay. Philos Trans R Soc A 367:1131–1143

    Article  Google Scholar 

  • Laing CR, Spiral Waves in Nonlocal Equations (2005) SIAM J Appl Dyn Syst 4(3):588–606

    Google Scholar 

  • Liley DT, Alexander DM, Wright JJ, Aldous MD (1999) Alpha rhythm emerges from large-scale networks of realistically coupled multicompartmental model cortical neurons. Network 10(1):79–92

    Article  CAS  PubMed  Google Scholar 

  • Nunez P (1974) The brainwave equation: a model for the EEG. Math Biosci 21:279–297. doi:10.1016/0025-5564(74)90020-0

    Article  Google Scholar 

  • Robinson PA (2006) Patchy propagators, brain dynamics, and the generation of spatially structured gamma oscillations. Phys Rev E 73:041904

    Article  CAS  Google Scholar 

  • Robinson PA, Loxley PN, O’Connor SC, Rennie CJ (2001a) Modal analysis of corticothalamic dynamics, electroencephalographic spectra, and evoked potentials. Phys Rev E Stat Nonlinear Soft Matter Phys 63(4 Pt 1):041,909

    Article  CAS  Google Scholar 

  • Robinson PA, Rennie CJ, Wright JJ, Bahramali H, Gordon E, Rowe DL (2001b) Prediction of electroencephalographic spectra from neurophysiology. Phys Rev E Stat Nonlinear Soft Matter Phys 63(2 Pt 1):021,903

    Article  CAS  Google Scholar 

  • Robinson PA, Rennie CA, Wright JJ (1997) Propagation and stability of waves of electrical activity in the cerebral cortex. Phys Rev E 56:826–840

    Article  CAS  Google Scholar 

  • Sanz Leon P, Knock S, Woodman M, Domide L, Mersmann J, McIntosh A, Jirsa V (2013) The virtual brain: a simulator of primate brain network dynamics. Front Neuroinf 7:10

    Article  Google Scholar 

  • Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Wilson HR, Cowan JD (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetic 13:55–80

    Article  CAS  Google Scholar 

  • Wright JJ, Liley DTJ (1995) Simulation of electrocortical waves. Biol Cybern 72:347–356

    Article  CAS  PubMed  Google Scholar 

  • Wright JJ, Liley D (1996) Dynamics of the brain at global and microscopic scales: neural networks and the eeg. Behav Brain Sci 19:285–320

    Article  Google Scholar 

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Authors and Affiliations

  1. Institut de Neurosciences des Systèmes, Faculté de Médecine, UMR INSERM 1106, Aix-Marseille Université, 27, Boulevard Jean Moulin, 13005, Marseille, France

    Prof. Viktor Jirsa

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  1. Prof. Viktor Jirsa
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Correspondence to Viktor Jirsa .

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Editors and Affiliations

  1. Atlanta, Georgia, USA

    Dieter Jaeger

  2. Department of Biomedical Engineering, Florida International University, Miami, Florida, USA

    Ranu Jung

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Jirsa, V. (2014). Propagator, Axonal. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7320-6_75-1

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  • DOI: https://doi.org/10.1007/978-1-4614-7320-6_75-1

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  • Online ISBN: 978-1-4614-7320-6

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