References
Amari S (1975) Homogeneous nets of neuron-like elements. Biol Cybern 17:211–220
Amari S (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybern 27:77–87
beim Graben P, Potthast R (2012) A dynamic field account to language-related brain potentials. In: Rabinovich M, Friston K, Varona P (eds) Principles of brain dynamics: global state interactions. MIT Press, Cambridge, MA
beim Graben P, Potthast R (2009) Inverse problems in dynamic cognitive modeling. Chaos 19(1):015103
beim Graben P, Pinotsis D, Saddy D, Potthast R (2008) Language processing with dynamic fields. Cognit Neurodyn 2(2):79–88
Ben-Yishai R, Bar-Or L, Sompolinsky H (1995) Theory of orientation tuning in visual cortex. Proc Natl Acad Sci U S A 92:3844–3848
Berger H (1929) Über das Elektroenkephalogramm des Menschen. Archiv Psychiatr 87:527–570
Beurle RL (1956) Properties of a mass of cells capable of regenerating pulses. Philos Trans R Soc Lond B 240:55–94
Brackley CA, Turner MS (2007) Random fluctuations of the firing rate function in a continuum neural field model. Phys Rev E 75:041,913
Bressloff PC (2001) Traveling fronts and wave propagation failure in an in-homogeneous neural network. Phys D 155:83–100
Bressloff PC (2012) Spatiotemporal dynamics of continuum neural fields. J Phys A 45:033,001
Bressloff PC, Coombes S (1997) Physics of the extended neuron. Int J Mod Phys B 11:2343–2392
Bressloff PC, Cowan JD, Golubitsky M, Thomas PJ, Wiener M (2001) Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. Philos Trans R Soc Lond B 40:299–330
Coombes S, Owen MR (2004) Evans functions for integral neural field equations with Heaviside firing rate function. SIAM J Appl Dyn Syst 34:574–600
Coombes S, Schmidt H (2010) Neural fields with sigmoidal firing rates: approximate solutions. Discret Contin Dyn Syst Ser A 28:1369–1379
Coombes S, Lord GJ, Owen MR (2003) Waves and bumps in neuronal networks with axo-dendritic synaptic interactions. Phys D 178:219–241
Coombes S, Schmidt H, Bojak I (2012) Interface dynamics in planar neural field models. J Math Neurosci 2:9
Coombes S, beim Graben P, Potthast R (2013) Tutorial on neural field theory. In: Coombes S, beim Graben P, Wright J, Potthast R (eds) Neural fields. Theory and applications. Springer, Berlin
Daunizeau J, Kiebel SJ, Friston KJ (2009) Dynamic causal modelling of distributed electromagnetic responses. Neuroimage 47:590–601
Erlhagen W, Bicho E (2006) The dynamic neural field approach to cognitive robotics. J Neural Eng 3:R36–R54
Ermentrout GB, Cowan JD (1979) A mathematical theory of visual hallucination patterns. Biol Cybern 34:137–150
Ermentrout GB, McLeod JB (1993) Existence and uniqueness of travelling waves for a neural network. Proc R Soc Edinb 123A:461–478
Faugeras O, Grimbert F, Slotine JJ (2008) Absolute stability and complete synchronization in a class of neural fields models. SIAM J Appl Math 69:205–250
Folias SE, Bressloff PC (2004) Breathing pulses in an excitatory neural network. SIAM J Appl Dyn Syst 3:378–407
Geise MA (1999) Neural field theory for motion perception. Kluwer Academic, Boston
Griffith JS (1963) A field theory of neural nets: I: derivation of field equations. Bull Math Biophys 25:111–120
Griffith JS (1965) A field theory of neural nets: II: properties of field equations. Bull Math Biophys 27:187–195
Grindrod P, Pinotsis D (2010) On the spectra of certain integro-differential-delay problems with applications in neurodynamics. Phys D Nonlinear Phenom 240(1):13–20. ISSN 0167–2789 doi:10.1016/j.physd.2010.08.002
Hutt A (2004) Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay. Phys Rev E 60(1–4):052,902
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
Jirsa VKV, Jantzen KJ, Fuchs A, Kelso JAS (2001) Information processing in medical imaging, chap. Neural field dynamics on the folded three-dimensional cortical sheet and its forward EEG and MEG. Springer, Berlin, pp 286–299
Jirsa VK, Jantzen KJ, Fuchs A, Kelso JAS (2002) Spatiotemporal forward solution of the EEG and MEG using network modeling. IEEE Trans Med Imaging 21(5):493–504
Kilpatrick ZP, Bressloff PC (2010a) Binocular rivalry in a competitive neural network with synaptic depression. SIAM J Appl Dyn Syst 9:1303–1347
Kilpatrick ZP, Bressloff PC (2010b) Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network. Phys D 239:547–560
Kilpatrick ZP, Bressloff PC (2010c) Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression. J Comput Neurosci 28:193–209
Kishimoto K, Amari S (1979) Existence and stability of local excitations in homogeneous neural fields. J Math Biol 7:303–318
Laing CR (2005) Spiral waves in nonlocal equations. SIAM J Appl Dyn Syst 4:588–606
Laing CR, Troy WC (2003a) PDE methods for nonlocal models. SIAM J Appl Dyn Syst 2:487–516
Laing CR, Troy WC (2003b) Two bump solutions of Amari-type models of working memory. Phys D 178:190–218
Laing CR, Troy WC, Gutkin B, Ermentrout GB (2002) Multiple bumps in a neuronal model of working memory. SIAM J Appl Math 63:62–97
Liley DTJ, Cadusch PJ, Dafilis MP (2002) A spatially continuous mean field theory of electrocortical activity. Netw Comput Neural Syst 13:67–113
Liley DTJ, Foster BL, Bojak I (2011) Sleep and anesthesia, chap. A mesoscopic modelling approach to anaesthetic action on brain electrical activity. Springer, New York, pp 139–166
Nunez PL (1974) The brain wave equation: a model for the EEG. Math Biosci 21:279–297
Nunez PL (1995) Neocortical dynamics and human EEG rhythms. Oxford University Press, New York
Oleynik A, Posnov A, Wyller J (2012) Iterative schemes for bump solutions in a neural field model
Owen MR, Laing CR, Coombes S (2007) Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities. New J Phys 9:378
Pinto DJ, Ermentrout GB (2001a) Spatially structured activity in synaptically coupled neuronal networks: I. Travelling fronts and pulses. SIAM J Appl Math 62:206–225
Pinto DJ, Ermentrout GB (2001b) Spatially structured activity in synaptically coupled neuronal networks: II. lateral inhibition and standing pulses. SIAM J Appl Math 62:226–243
Potthast R, beim Graben P (2009) Inverse problems in neural field theory. SIAM J Appl Dyn Syst 8(4):1405–1433
Potthast R, beim Graben P (2010) Existence and properties of solutions for neural field equations. Math Methods Appl Sci 33(8):935–949
Rabinovich M, Friston K, Varona P (eds) (2012) Principles of brain dynamics: global state interactions. MIT Press, Cambridge, MA
Schmidt H, Hutt A, Schimansky-Geier L (2009) Wave fronts in inhomogeneous neural field models. Phys D 238:1101–1112
Schöner G, Dineva E (2007) Dynamic instabilities as mechanisms for emergence. Dev Sci 10:69–74
Tass P (1995) Cortical pattern formation during visual hallucinations. J Biol Phys 21:177–210
Taylor JG (1999) Neural ‘bubble’ dynamics in two dimensions: foundations. Biol Cybern 80:393–409 Structure
Venkov NA, Coombes S, Matthews PC (2007) Dynamic instabilities in scalar neural field equations with space-dependent delays. Phys D 232:1–15
Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24
Wilson HR, Cowan JD (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13:55–80
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this entry
Cite this entry
Potthast, R. (2013). The Amari Model in Neural Field Theory. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7320-6_51-1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7320-6_51-1
Received:
Accepted:
Published:
Publisher Name: Springer, New York, NY
Online ISBN: 978-1-4614-7320-6
eBook Packages: Springer Reference Biomedicine and Life SciencesReference Module Biomedical and Life Sciences
Publish with us
Chapter history
-
Latest
Amari Model- Published:
- 07 August 2014
DOI: https://doi.org/10.1007/978-1-4614-7320-6_51-2
-
Original
The Amari Model in Neural Field Theory- Published:
- 07 February 2014
DOI: https://doi.org/10.1007/978-1-4614-7320-6_51-1