Encyclopedia of Computational Neuroscience

2015 Edition
| Editors: Dieter Jaeger, Ranu Jung

Inverse Problems in Neural Population Models

  • Roland Potthast
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-6675-8_64


Inverse problems are usually based on some direct or forward model, where the inversion aims to determine either parameter distributions of the model or states of the dynamics. The application and development of inverse techniques to neural population models will be discussed here with a focus on neural field theory. The goal is the construction and reconstruction of neural connectivity and parameters of neural activity such as the local activation of pulses or spike trains.

Detailed Description

Neural Field Models such as the Cowan-Wilson Model (Wilson and Cowan 1972, 1973; Nunez 1974) or the Amari Neural Field Model (cf. (Amari 1975, 1977)) establish a field-theoretic approach to the dynamics of neural activity in the brain. The models use excitations and inhibitions over some distance as an effective model of mixed inhibitory and excitatory neurons with typical cortical connectivities, for example, by a simple dynamical equation for the voltage or activity u( x, t) of the...
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  1. Amari S (1975) Homogeneous nets of neuron-like elements. Biol Cybern 17:211–220PubMedGoogle Scholar
  2. Amari S (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybern 27:77–87PubMedGoogle Scholar
  3. beim Graben P, Potthast R (2009) Inverse problems in dynamic cognitive modeling. Chaos 19(1):015103PubMedGoogle Scholar
  4. beim Graben P, Potthast R (2012a) 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, MAGoogle Scholar
  5. beim Graben P, Potthast R (2012b) Implementing turing machines in dynamic field architectures AISB/IACAP world congress 2012, BirminghamGoogle Scholar
  6. beim Graben P, Potthast R (2013) Universal neural field computation. In: beim Graben P, Coombes S, Potthast R, Wright JJ (eds) Neural field theory, SpringerGoogle Scholar
  7. beim Graben P, Pinotsis D, Saddy D, Potthast R (2008) Language processing with dynamic fields. Cognit Neurodyn 2(2):79–88Google Scholar
  8. Berger JO (1985) Statistical decision theory and Bayesian analysis, 2nd edn. Springer, New YorkGoogle Scholar
  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, BerlinGoogle Scholar
  10. Freitag M, Potthast R. Synergy of inverse problems and data assimilation techniques in large scale inverse problems – computational methods and applications in the earth sciences, radon series on computational and applied mathematics 13, Hrsg. v. Cullen, Mike/Freitag, Melina A/Kindermann, Stefan/Scheichl, RobertGoogle Scholar
  11. Friston K (2009) Causal modelling and brain connectivity in functional magnetic resonance imaging. PLoS Biol 7:e33PubMedGoogle Scholar
  12. Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. Neuroimage 19:1273–1302PubMedGoogle Scholar
  13. Friston K, Ashburner J, Kiebel S, Nichols T, Penny W (2006) Statistical parametric mapping: the analysis of functional brain images. Elsevier, LondonGoogle Scholar
  14. Geise MA (1999) Neural field theory for motion perception. Kluwer Academic, BostonGoogle Scholar
  15. Kaipio J, Somersalo E. Statistical and computational inverse problems. Springer, 1010Google Scholar
  16. Kiebel SJ, Garrido MI, Moran RJ, Friston KJ (2008) Dynamic causal modelling for EEG and MEG. Cognit Neurodyn 2:121–136Google Scholar
  17. Kirsch A (1996) An introduction to the mathematical theory of inverse problems, vol 120, Applied mathematical sciences. Springer, New YorkGoogle Scholar
  18. Kress R (1989) Linear integral equations. Springer, BerlinGoogle Scholar
  19. Nunez PL (1974) The brain wave equation: a model for the EEG. Math Biosci 21:279–297Google Scholar
  20. Penny WD, Stephan KE, Mechelli A, Friston KJ (2004) Modelling functional integration: a comparison of structural equation and dynamic causal models. Neuroimage 23:S264–S274PubMedGoogle Scholar
  21. Potthast R, beim Graben P (2009) Inverse problems in neural field theory. SIAM J Appl Dyn Syst 8(4):1405–1433Google Scholar
  22. Potthast R, beim Graben P (2010) Existence and properties of solutions for neural field equations. Math Methods Appl Sci 33(8):935–949Google Scholar
  23. Potthast R, beim Graben P (2009) Dimensional reduction for the inverse problem of neural field theory. Front Neurosci 3. doi:10.3389/neuro.10/017.2009Google Scholar
  24. Rabinovich M, Friston K, Varona P (eds) (2012) Principles of brain dynamics: global state interactions. MIT Press, Cambridge, MAGoogle Scholar
  25. Stephan KE, Harrison LM, Kiebel SJ, David O, Penny WD, Friston KJ (2007) Dynamic causal models of neural system dynamics: current state and future extensions. J Biosci 32:129–144PubMedCentralPubMedGoogle Scholar
  26. Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24PubMedCentralPubMedGoogle Scholar
  27. Wilson HR, Cowan JD (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13:55–80PubMedGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ReadingReadingUK