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

Journal of Computational Neuroscience

Volumn 35, Issue 1, 2013, Pages 87-108

  Nicola, Wilten ^a   Campbell, Sue Ann ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Bifurcation theory; Bursting; Integrate and fire; Large networks; Mean field; Population density methods

Indexed keywords

BIFURCATION (MATHEMATICS); FIRES; MEAN FIELD THEORY; ORDINARY DIFFERENTIAL EQUATIONS; POPULATION DISTRIBUTION; POPULATION DYNAMICS; POPULATION STATISTICS; TOPOLOGY;

BIFURCATIONS THEORY; BURSTING; DENSITY METHODS; INTEGRATE AND FIRES; INTEGRATE-AND-FIRE MODEL; LARGER NETWORKS; MEAN-FIELD; POPULATION DENSITIES; POPULATION DENSITY METHOD; TWO-DIMENSIONAL;

NEURONS;

ACCURACY; ALGORITHM; ARTICLE; HIPPOCAMPAL CA3 REGION; LINEAR SYSTEM; MATHEMATICAL ANALYSIS; MATHEMATICAL MODEL; NERVE CELL EXCITABILITY; NERVE CELL MEMBRANE POTENTIAL; NERVE CELL NETWORK; PRIORITY JOURNAL; PYRAMIDAL NERVE CELL; QUALITATIVE ANALYSIS; QUANTITATIVE ANALYSIS; SENSORY GATING; STEADY STATE; SYNAPTIC TRANSMISSION;

EID: 84880193048     PISSN: 09295313     EISSN: 15736873     Source Type: Journal    
DOI: 10.1007/s10827-013-0442-z     Document Type: Article

References (36)

1. Abbott, L.F.; van Vreeswijk, C. (1993). Asynchronous states in networks of pulse-coupled oscillators. Learning and Memory, 48(2), 1483-1490.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. Ermentrout, G.B.; Terman, D.H. (2010). Mathematical Foundations of Neuroscience. New York, NY: Springer.
10. Fitzhugh, R. (1952). Impulses and phsyiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445-466.
11. Gerstner, W.; Kistler, W. (2002). Spiking Neuron Models. Cambridge, UK: Cambridge University Press.
12. 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.
13. 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.
14. Ho, E.C.; Zhang, L.; Skinner, F.K. (2009). Hippocampus, 19(2), 152-165.
15. Izhikevich, E. (2003). Simple model of spiking neurons. Neural Networks, IEEE Transactions, 14(6), 1569-1572.
16. Knight, B.W. (2000). Dynamics of encoding in neuron populations: some general mathematical features. Neural Computation, 12, 473-518.
17. 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.
18. 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.
19. 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.
20. 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.
21. MATLAB (2012). Version 7.10.0 (R2012a). The MathWorks Inc. Massachusetts: Natick.
22. Naud, R.; Marcille, N.; Clopath, C.; Gerstner, W. (2008). Firing patterns in the adaptive exponential integrate-and-fire model. Biological Cybernetics, 99, 335-347.
23. Nesse, W.; Borisyuk, A.; Bressloff, P. (2008). Fluctuation-driven rhythmogenesis in an excitatory neuronal network with slow adaptation. Journal of Computational Neuroscience, 25, 317-333. doi: 10.1007/s10827-008-0081-y.
24. 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.
25. Omurtag, A.; Knight, B.W.; Sirovich, L. (2000). On the simulation of large populations of neurons. Journal of Computational Neuroscience, 8, 51-63.
26. Sirovich, L.; Omurtag, A.; Knight, B.W. (2000). Dynamics of neuronal populations: the equilibrium solution. SIAM Journal on Applied Mathematics, 60(6), 2009-2028.
27. Sirovich, L.; Omurtag, A.; Lubliner, K. (2006). Dynamics of neural populations: stability and synchrony. Network: Computation in Neural Systems, 17, 3-29.
28. Strogatz, S.; Mirollo, R.E. (1991). Stability of incoherence in a population of coupled oscillators. Journal of Statistical Physics, 63, 613-635.
29. Tikhonov, A. (1952). Systems of differential equations containing small parameters in the derivatives (in Russian). Matematicheskii Sbornik (NS), 31(73), 575-586.
30. Touboul, J. (2008). Bifurcation analysis of a general class ofnonlinear integrate-and-fire neurons. SIAM Journal on Applied Mathematics, 68(4), 1045-1079.
31. Treves, A. (1993). Mean-field analysis of neuronal spike dynamics. Network: Computation in Neural Systems, 4(3), 259-284.
32. van Vreeswijk, C. (1996). Partial synchronization in populations of pulse-coupled oscillators. Physical Review E, 54, 5522-5537. doi: 10.1103/PhysRevE.54.5522.
33. van Vreeswijk, C.; Hansel, D. (2001). Patterns of synchrony in neural networks with spike adaptation. Neural Computation, 13(5), 959-992.
34. van Vreeswijk, C.; Abbott, L.F.; Ermentrout, G.B. (1994). When inhibition not excitation synchronizes neural firing. Journal of Computational Neuroscience, 1, 313-321.
35. 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.
36. Wu, Y.; Lu, W.; Lin, W.; Leng, G.; Feng, J. (2012). Bifurcations of emergent bursting in a neuronal network. PLoS ONE, 7(6), e38402.