Skip to main content
SpringerLink
Log in

Inverse Problems in Neural Population Models

  • Living reference work entry
  • First Online:
Encyclopedia of Computational Neuroscience

  • 299 Accesses

Definition

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.

Introduction

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 form

$$...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • Amari S (1975) Homogeneous nets of neuron-like elements. Biol Cybern 17:211–220

    Article  CAS  PubMed  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  • 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

    Google Scholar 

  • beim Graben P, Potthast R (2012) Implementing turing machines in dynamic field architectures AISB/IACAP world congress 2012, Birmingham

    Google Scholar 

  • beim Graben P, Potthast R (2013) Universal neural field computation. In: beim Graben P, Coombes S, Potthast R, Wright JJ (eds) Neural field theory, Springer

    Google Scholar 

  • beim Graben P, Potthast R (2009) Inverse problems in dynamic cognitive modeling. Chaos 19(1):015103

    Article  PubMed  Google Scholar 

  • beim Graben P, Pinotsis D, Saddy D, Potthast R (2008) Language processing with dynamic fields. Cognit Neurodyn 2(2):79–88

    Article  Google Scholar 

  • Berger JO (1985) Statistical decision theory and Bayesian analysis, 2nd edn. Springer, New York

    Book  Google Scholar 

  • 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

    Google Scholar 

  • 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, Robert

    Google Scholar 

  • Friston K (2009) Causal modelling and brain connectivity in functional magnetic resonance imaging. PLoS Biol 7:e33

    Article  PubMed  Google Scholar 

  • Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. Neuroimage 19:1273–1302

    Article  CAS  PubMed  Google Scholar 

  • Friston K, Ashburner J, Kiebel S, Nichols T, Penny W (2006) Statistical parametric mapping: the analysis of functional brain images. Elsevier, London

    Google Scholar 

  • Geise MA (1999) Neural field theory for motion perception. Kluwer Academic, Boston

    Book  Google Scholar 

  • Kaipio J, Somersalo E. Statistical and computational inverse problems. Springer, 1010

    Google Scholar 

  • Kiebel SJ, Garrido MI, Moran RJ, Friston KJ (2008) Dynamic causal modelling for EEG and MEG. Cognit Neurodyn 2:121–136

    Article  Google Scholar 

  • Kirsch A (1996) An introduction to the mathematical theory of inverse problems, vol 120, Applied mathematical sciences. Springer, New York

    Book  Google Scholar 

  • Kress R (1989) Linear integral equations. Springer, Berlin

    Book  Google Scholar 

  • Nunez PL (1974) The brain wave equation: a model for the EEG. Math Biosci 21:279–297

    Article  Google Scholar 

  • Penny WD, Stephan KE, Mechelli A, Friston KJ (2004) Modelling functional integration: a comparison of structural equation and dynamic causal models. Neuroimage 23:S264–S274

    Article  PubMed  Google Scholar 

  • Potthast R, beim Graben P (2009) Inverse problems in neural field theory. SIAM J Appl Dyn Syst 8(4):1405–1433

    Article  Google Scholar 

  • Potthast R, beim Graben P (2010) Existence and properties of solutions for neural field equations. Math Methods Appl Sci 33(8):935–949

    Google Scholar 

  • Potthast R, beim Graben P (2009) Dimensional reduction for the inverse problem of neural field theory. Front Neurosci 3. doi:10.3389/neuro.10/017.2009

    Google Scholar 

  • Rabinovich M, Friston K, Varona P (eds) (2012) Principles of brain dynamics: global state interactions. MIT Press, Cambridge, MA

    Google Scholar 

  • 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 32:129–144

    Article  PubMed Central  PubMed  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. Kybernetik 13:55–80

    Article  CAS  PubMed  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Reading, Whiteknights, 220, Berkshire, UK

    Prof. Roland Potthast

Authors
  1. Prof. Roland Potthast
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roland Potthast .

Editor information

Editors and Affiliations

  1. Atlanta, Georgia, USA

    Dieter Jaeger

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

    Ranu Jung

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this entry

Cite this entry

Potthast, R. (2013). Inverse Problems in Neural Population Models. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7320-6_64-1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-7320-6_64-1

  • 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

Policies and ethics

Search

Navigation