Skip to main content
Top
Published in: Journal of Computational Neuroscience 1/2013

01-08-2013

Bifurcations of large networks of two-dimensional integrate and fire neurons

Authors: Wilten Nicola, Sue Ann Campbell

Published in: Journal of Computational Neuroscience | Issue 1/2013

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Recently, a class of two-dimensional integrate and fire models has been used to faithfully model spiking neurons. This class includes the Izhikevich model, the adaptive exponential integrate and fire model, and the quartic integrate and fire model. The bifurcation types for the individual neurons have been thoroughly analyzed by Touboul (SIAM J Appl Math 68(4):1045–1079, 2008). However, when the models are coupled together to form networks, the networks can display bifurcations that an uncoupled oscillator cannot. For example, the networks can transition from firing with a constant rate to burst firing. This paper introduces a technique to reduce a full network of this class of neurons to a mean field model, in the form of a system of switching ordinary differential equations. The reduction uses population density methods and a quasi-steady state approximation to arrive at the mean field system. Reduced models are derived for networks with different topologies and different model neurons with biologically derived parameters. The mean field equations are able to qualitatively and quantitatively describe the bifurcations that the full networks display. Extensions and higher order approximations are discussed.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Appendix
Available only for authorised users
Literature
go back to reference Abbott, L.F., & van Vreeswijk, C. (1993). Asynchronous states in networks of pulse-coupled oscillators. Learning and Memory, 48(2), 1483–1490. Abbott, L.F., & van Vreeswijk, C. (1993). Asynchronous states in networks of pulse-coupled oscillators. Learning and Memory, 48(2), 1483–1490.
go back to reference Apfaltrer, F., Ly, C., Tranchina, D. (2006). Population density methods for stochastic neurons with realistic synaptic kinetics: firing rate dynamics and fast computational methods. Network: Computation in Neural Systems, 17(4), 373–418.CrossRef Apfaltrer, F., Ly, C., Tranchina, D. (2006). Population density methods for stochastic neurons with realistic synaptic kinetics: firing rate dynamics and fast computational methods. Network: Computation in Neural Systems, 17(4), 373–418.CrossRef
go back to reference di Bernardo, M., Budd, C., Champneys, A., Kowalczyk, P., Nordmark, A., Tost, G., Piiroinen, P. (2008). Bifurcations in non-smooth dynamical systems. SIAM Review, 50(4), 629–701.CrossRef di Bernardo, M., Budd, C., Champneys, A., Kowalczyk, P., Nordmark, A., Tost, G., Piiroinen, P. (2008). Bifurcations in non-smooth dynamical systems. SIAM Review, 50(4), 629–701.CrossRef
go back to reference Brette, R., & Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Neurophysiology, 94(5), 3637–3642.PubMedCrossRef Brette, R., & Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Neurophysiology, 94(5), 3637–3642.PubMedCrossRef
go back to reference Casti, A., Omurtag, A., Sornborger, A., Kaplan, E., Knight, B.W., Victor, J., Sirovich, L. (2002). A population study of integrate-and-fire-or-burst neurons. Neural Computation, 14(5), 957–986.PubMedCrossRef Casti, A., Omurtag, A., Sornborger, A., Kaplan, E., Knight, B.W., Victor, J., Sirovich, L. (2002). A population study of integrate-and-fire-or-burst neurons. Neural Computation, 14(5), 957–986.PubMedCrossRef
go back to reference Destexhe, A., Mainen, Z., Sejnowski, T. (1998). Kinetic models of synaptic transmission. In C. Koch & I. Segev (Eds.), Methods in neuronal modeling: From synapses to networks (chap. 1). Cambridge, MA: MIT Press. Destexhe, A., Mainen, Z., Sejnowski, T. (1998). Kinetic models of synaptic transmission. In C. Koch & I. Segev (Eds.), Methods in neuronal modeling: From synapses to networks (chap. 1). Cambridge, MA: MIT Press.
go back to reference Dhooge, A., Govaerts, W., Kuznetsov, Y.A. (2003). MatCont: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software, 29, 141–164.CrossRef Dhooge, A., Govaerts, W., Kuznetsov, Y.A. (2003). MatCont: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software, 29, 141–164.CrossRef
go back to reference Dur-e-Ahmad, M., Nicola, W., Campbell, S.A., Skinner, F. (2012). Network bursting using experimentally constrained single compartment CA3 hippocampal neuron models with adaptation. Journal of Computational Neuroscience, 33(1), 21–40.PubMedCrossRef Dur-e-Ahmad, M., Nicola, W., Campbell, S.A., Skinner, F. (2012). Network bursting using experimentally constrained single compartment CA3 hippocampal neuron models with adaptation. Journal of Computational Neuroscience, 33(1), 21–40.PubMedCrossRef
go back to reference Ermentrout, G.B., & Terman, D.H. (2010). Mathematical Foundations of Neuroscience. New York, NY: Springer.CrossRef Ermentrout, G.B., & Terman, D.H. (2010). Mathematical Foundations of Neuroscience. New York, NY: Springer.CrossRef
go back to reference Fitzhugh, R. (1952). Impulses and phsyiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445–466.CrossRef Fitzhugh, R. (1952). Impulses and phsyiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445–466.CrossRef
go back to reference Gerstner, W., & Kistler, W. (2002). Spiking Neuron Models. Cambridge, UK: Cambridge University Press.CrossRef Gerstner, W., & Kistler, W. (2002). Spiking Neuron Models. Cambridge, UK: Cambridge University Press.CrossRef
go back to reference Hines, M.L., Morse, T., Migliore, M., Carnevale, N.T., Shepherd, G.M. (2004). ModelDB: a database to support computational neuroscience. Journal of Computational Neuroscience, 17(1), 7–11.PubMedCrossRef Hines, M.L., Morse, T., Migliore, M., Carnevale, N.T., Shepherd, G.M. (2004). ModelDB: a database to support computational neuroscience. Journal of Computational Neuroscience, 17(1), 7–11.PubMedCrossRef
go back to reference Hemond, P., Epstein, D., Boley, A., Migliore, M., Ascoli, G., Jaffe, D. (2008). Distinct classes of pyramidal cells exhibit mutually exclusive firing patterns in hippocampal area CA3b. Hippocampus, 18(4), 411–424.PubMedCrossRef Hemond, P., Epstein, D., Boley, A., Migliore, M., Ascoli, G., Jaffe, D. (2008). Distinct classes of pyramidal cells exhibit mutually exclusive firing patterns in hippocampal area CA3b. Hippocampus, 18(4), 411–424.PubMedCrossRef
go back to reference Izhikevich, E. (2003). Simple model of spiking neurons. Neural Networks, IEEE Transactions, 14(6), 1569–1572.CrossRef Izhikevich, E. (2003). Simple model of spiking neurons. Neural Networks, IEEE Transactions, 14(6), 1569–1572.CrossRef
go back to reference Knight, B.W. (2000). Dynamics of encoding in neuron populations: some general mathematical features. Neural Computation, 12, 473–518.PubMedCrossRef Knight, B.W. (2000). Dynamics of encoding in neuron populations: some general mathematical features. Neural Computation, 12, 473–518.PubMedCrossRef
go back to reference La, Camera, G., Rauch, A., Luscher, H.R., Senn, W., Fusi, S. (2004). Minimal models of adapted neuronal response to in-vivo like input currents. Neural Computation, 16, 2101–2124.PubMedCrossRef La, Camera, G., Rauch, A., Luscher, H.R., Senn, W., Fusi, S. (2004). Minimal models of adapted neuronal response to in-vivo like input currents. Neural Computation, 16, 2101–2124.PubMedCrossRef
go back to reference La Camera, G., Giugliano, M., Senn, W., Fusi, S. (2008). The response of cortical neurons to in vivo-like input current: theory and experiment. Biological Cybernetics, 99, 279–301.PubMedCrossRef La Camera, G., Giugliano, M., Senn, W., Fusi, S. (2008). The response of cortical neurons to in vivo-like input current: theory and experiment. Biological Cybernetics, 99, 279–301.PubMedCrossRef
go back to reference Ly, C., & Tranchina, D. (2007). A critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling. Neural Computation, 19(8), 2032–2092.PubMedCrossRef Ly, C., & Tranchina, D. (2007). A critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling. Neural Computation, 19(8), 2032–2092.PubMedCrossRef
go back to reference Markram, H., Toledo-Rodriguez, M., Wang, Y., Gupta, A., Silberberg, G., Wu, C. (2004). Interneurons of the neocortical inhibitory system. Nature Reviews: Neuroscience, 5(10), 793–807.PubMedCrossRef Markram, H., Toledo-Rodriguez, M., Wang, Y., Gupta, A., Silberberg, G., Wu, C. (2004). Interneurons of the neocortical inhibitory system. Nature Reviews: Neuroscience, 5(10), 793–807.PubMedCrossRef
go back to reference MATLAB (2012). Version 7.10.0 (R2012a). The MathWorks Inc. Massachusetts: Natick. MATLAB (2012). Version 7.10.0 (R2012a). The MathWorks Inc. Massachusetts: Natick.
go back to reference Naud, R., Marcille, N., Clopath, C., Gerstner, W. (2008). Firing patterns in the adaptive exponential integrate-and-fire model. Biological Cybernetics, 99, 335–347.PubMedCrossRef Naud, R., Marcille, N., Clopath, C., Gerstner, W. (2008). Firing patterns in the adaptive exponential integrate-and-fire model. Biological Cybernetics, 99, 335–347.PubMedCrossRef
go back to reference Nykamp, D., & Tranchina, D. (2000). A population density approach that facilitates large-scale modeling of neural networks: analysis and an application to orientation tuning. Journal of Computational Neuroscience, 8, 19–50.PubMedCrossRef Nykamp, D., & Tranchina, D. (2000). A population density approach that facilitates large-scale modeling of neural networks: analysis and an application to orientation tuning. Journal of Computational Neuroscience, 8, 19–50.PubMedCrossRef
go back to reference Omurtag, A., Knight, B.W., Sirovich, L. (2000). On the simulation of large populations of neurons. Journal of Computational Neuroscience, 8, 51–63.PubMedCrossRef Omurtag, A., Knight, B.W., Sirovich, L. (2000). On the simulation of large populations of neurons. Journal of Computational Neuroscience, 8, 51–63.PubMedCrossRef
go back to reference Sirovich, L., Omurtag, A., Knight, B.W. (2000). Dynamics of neuronal populations: the equilibrium solution. SIAM Journal on Applied Mathematics, 60(6), 2009–2028.CrossRef Sirovich, L., Omurtag, A., Knight, B.W. (2000). Dynamics of neuronal populations: the equilibrium solution. SIAM Journal on Applied Mathematics, 60(6), 2009–2028.CrossRef
go back to reference Sirovich, L., Omurtag, A., Lubliner, K. (2006). Dynamics of neural populations: stability and synchrony. Network: Computation in Neural Systems, 17, 3–29.CrossRef Sirovich, L., Omurtag, A., Lubliner, K. (2006). Dynamics of neural populations: stability and synchrony. Network: Computation in Neural Systems, 17, 3–29.CrossRef
go back to reference Strogatz, S., & Mirollo, R.E. (1991). Stability of incoherence in a population of coupled oscillators. Journal of Statistical Physics, 63, 613–635.CrossRef Strogatz, S., & Mirollo, R.E. (1991). Stability of incoherence in a population of coupled oscillators. Journal of Statistical Physics, 63, 613–635.CrossRef
go back to reference Tikhonov, A. (1952). Systems of differential equations containing small parameters in the derivatives (in Russian). Matematicheskii Sbornik (NS), 31(73), 575–586. Tikhonov, A. (1952). Systems of differential equations containing small parameters in the derivatives (in Russian). Matematicheskii Sbornik (NS), 31(73), 575–586.
go back to reference Touboul, J. (2008). Bifurcation analysis of a general class ofnonlinear integrate-and-fire neurons. SIAM Journal on Applied Mathematics, 68(4), 1045–1079.CrossRef Touboul, J. (2008). Bifurcation analysis of a general class ofnonlinear integrate-and-fire neurons. SIAM Journal on Applied Mathematics, 68(4), 1045–1079.CrossRef
go back to reference Treves, A. (1993). Mean-field analysis of neuronal spike dynamics. Network: Computation in Neural Systems, 4(3), 259–284.CrossRef Treves, A. (1993). Mean-field analysis of neuronal spike dynamics. Network: Computation in Neural Systems, 4(3), 259–284.CrossRef
go back to reference van Vreeswijk, C., & Hansel, D. (2001). Patterns of synchrony in neural networks with spike adaptation. Neural Computation, 13(5), 959–992.PubMedCrossRef van Vreeswijk, C., & Hansel, D. (2001). Patterns of synchrony in neural networks with spike adaptation. Neural Computation, 13(5), 959–992.PubMedCrossRef
go back to reference van Vreeswijk, C., Abbott, L.F., Ermentrout, G.B. (1994). When inhibition not excitation synchronizes neural firing. Journal of Computational Neuroscience, 1, 313–321.PubMedCrossRef van Vreeswijk, C., Abbott, L.F., Ermentrout, G.B. (1994). When inhibition not excitation synchronizes neural firing. Journal of Computational Neuroscience, 1, 313–321.PubMedCrossRef
go back to reference Vladimirski, B.B., Tabak, J., O’Donovan, M.J., Rinzel, J. (2008). Episodic activity in a heterogeneous excitatory network, from spiking neurons to mean field. Journal of Computational Neuroscience, 25, 39–63.PubMedCrossRef Vladimirski, B.B., Tabak, J., O’Donovan, M.J., Rinzel, J. (2008). Episodic activity in a heterogeneous excitatory network, from spiking neurons to mean field. Journal of Computational Neuroscience, 25, 39–63.PubMedCrossRef
go back to reference Wu, Y., Lu, W., Lin, W., Leng, G., Feng, J. (2012). Bifurcations of emergent bursting in a neuronal network. PLoS ONE, 7(6), e38402.PubMedCrossRef Wu, Y., Lu, W., Lin, W., Leng, G., Feng, J. (2012). Bifurcations of emergent bursting in a neuronal network. PLoS ONE, 7(6), e38402.PubMedCrossRef
Metadata
Title
Bifurcations of large networks of two-dimensional integrate and fire neurons
Authors
Wilten Nicola
Sue Ann Campbell
Publication date
01-08-2013
Publisher
Springer US
Published in
Journal of Computational Neuroscience / Issue 1/2013
Print ISSN: 0929-5313
Electronic ISSN: 1573-6873
DOI
https://doi.org/10.1007/s10827-013-0442-z

Other articles of this Issue 1/2013

Journal of Computational Neuroscience 1/2013 Go to the issue

Premium Partner