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Firing Rate Models

  • G. Bard Ermentrout
  • David H. Terman
Chapter
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 35)

Abstract

One of the most common ways to model large networks of neurons is to use a simplification called a firing rate model. Rather than track the spiking of every neuron, instead one tracks the averaged behavior of the spike rates of groups of neurons within the circuit. These models are also called population models since they can represent whole populations of neurons rather than single cells. In this book, we will call them rate models although their physical meaning may not be the actual firing rate of a neuron. In general, there will be some invertible relationship between the firing rate of the neuron and the variable at hand. We derive the individual model equation in several different ways, some of the derivations are rigorous and are directly related to some biophysical model and other derivations are ad hoc. After deriving the rate models, we apply them to a number of interesting phenomena, including working memory, hallucinations, binocular rivalry, optical illusions, and traveling waves. We also describe a number of theorems about asymptotic states as well as some of the now classical work on attractor networks.

Keywords

Firing Rate Hopf Bifurcation Bifurcation Diagram Spike Train Stable Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 5.
    J. Anderson and E. Rosenfeld. Talking Nets: An Oral History of Neural Networks. MIT, Cambridge, MA, 1998.Google Scholar
  2. 11.
    R. Beer. On the dynamics of small continuous-time recurrent neural networks. Adapt. Behav., 3:471–511, 1995.CrossRefGoogle Scholar
  3. 13.
    R. Bellman and K. L. Cooke. Differential-Difference Equations. Academic, New York, 1963.MATHGoogle Scholar
  4. 18.
    R. M. Borisyuk and A. B. Kirillov. Bifurcation analysis of a neural network model. Biol. Cybern., 66:319–325, 1992.MATHCrossRefGoogle Scholar
  5. 21.
    P. Bressloff. Stochastic neural field theory and the system-size expansion. preprint, 2010.Google Scholar
  6. 29.
    M. A. Buice and J. D. Cowan. Statistical mechanics of the neocortex. Prog. Biophys. Mol. Biol., 99:53–86, 2009.CrossRefGoogle Scholar
  7. 32.
    D. Cai, L. Tao, A. V. Rangan, and D. W. McLaughlin. Kinetic theory for neuronal network dynamics. Commun. Math. Sci., 4(1):97–127, 2006.MathSciNetMATHGoogle Scholar
  8. 49.
    J. Cowan and G. Ermentrout. Some aspects of the eigenbehavior of neural nets. In S. Levin, editor, Studies Mathematical Biology, volume 15, pages 67–117. Mathematical Association of America, Providence, RI, 1978.Google Scholar
  9. 50.
    J. Cowan and D. Sharp. Neural networks and artificial intelligence. Daedalus, 117:85–121, 1988.Google Scholar
  10. 52.
    R. Curtu and B. Ermentrout. Oscillations in a refractory neural net. J. Math. Biol., 43:81–100, 2001.MathSciNetMATHCrossRefGoogle Scholar
  11. 69.
    G. B. Ermentrout and J. D. Cowan. Large scale spatially organized activity in neural nets. SIAM J. Appl. Math., 38(1):1–21, 1980.MathSciNetMATHCrossRefGoogle Scholar
  12. 90.
    N. Fourcaud and N. Brunel. Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Comput., 14:2057–2110, 2002.MATHCrossRefGoogle Scholar
  13. 98.
    S. Grossberg, G. A. Carpenter. A massively parallel architecture for a self-organizing neural pattern recognition machine. In Computer Vision, Graphics, and Image Processing, pages 54–115, Academic, New York, 1987.Google Scholar
  14. 99.
    W. Gerstner and W. M. Kistler. Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge, 2002.MATHCrossRefGoogle Scholar
  15. 125.
    D. Holcman and M. Tsodyks. The emergence of up and down states in cortical networks. PLoS Comput. Biol., 2:e23, 2006.CrossRefGoogle Scholar
  16. 128.
    F. C. Hoppensteadt and E. M. Izhikevich. Weakly Connected Neural Networks, volume 126 of Applied Mathematical Sciences. Springer, New York, 1997.CrossRefGoogle Scholar
  17. 168.
    H. T. Kyriazi and D. J. Simons. Thalamocortical response transformations in simulated whisker barrels. J. Neurosci., 13:1601–1615, 1993.Google Scholar
  18. 178.
    S. R. Lehky. An astable multivibrator model of binocular rivalry. Perception, 17:215–228, 1988.CrossRefGoogle Scholar
  19. 196.
    J. McClelland and D. Rumelhart. Parallel Distributed Processes. MIT, Cambridge, MA, 1987.Google Scholar
  20. 217.
    D. J. Pinto, J. C. Brumberg, D. J. Simons, and G. B. Ermentrout. A quantitative population model of whisker barrels: re-examining the Wilson-Cowan equations. J. Comput. Neurosci., 3:247–264, 1996.CrossRefGoogle Scholar
  21. 218.
    D. J. Pinto, J. A. Hartings, J. C. Brumberg, and D. J. Simons. Cortical damping: analysis of thalamocortical response transformations in rodent barrel cortex. Cereb. Cortex, 13:33–44, Jan 2003.CrossRefGoogle Scholar
  22. 219.
    D. J. Pinto, S. L. Patrick, W. C. Huang, and B. W. Connors. Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms. J. Neurosci., 25:8131–8140, 2005.CrossRefGoogle Scholar
  23. 224.
    A. D. Reyes. Synchrony-dependent propagation of firing rate in iteratively constructed networks in vitro. Nat. Neurosci., 6:593–599, 2003.CrossRefGoogle Scholar
  24. 246.
    A. Shpiro, R. Curtu, J. Rinzel, and N. Rubin. Dynamical characteristics common to neuronal competition models. J. Neurophysiol., 97:462–473, 2007.CrossRefGoogle Scholar
  25. 247.
    Y. Shu, A. Hasenstaub, and D. A. McCormick. Turning on and off recurrent balanced cortical activity. Nature, 423:288–293, 2003.CrossRefGoogle Scholar
  26. 257.
    J. Tabak, W. Senn, M. J. O’Donovan, and J. Rinzel. Modeling of spontaneous activity in developing spinal cord using activity-dependent depression in an excitatory network. J. Neurosci., 20:3041–3056, 2000.Google Scholar
  27. 273.
    M. Tsodyks, A. Uziel, and H. Markram. Synchrony generation in recurrent networks with frequency-dependent synapses. J. Neurosci., 20:RC50, 2000.Google Scholar
  28. 280.
    M. Volgushev, S. Chauvette, M. Mukovski, and I. Timofeev. Precise long-range synchronization of activity and silence in neocortical neurons during slow-wave oscillations [corrected]. J. Neurosci., 26:5665–5672, 2006.CrossRefGoogle Scholar
  29. 287.
    H. R. Wilson and J. D. Cowan. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J., 12:1–24, 1972.CrossRefGoogle Scholar
  30. 289.
    H. R. Wilson, R. Blake, and S. H. Lee. Dynamics of travelling waves in visual perception. Nature, 412:907–910, 2001.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Dept. MathematicsOhio State UniversityColumbusUSA

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