Mean-field models for heterogeneous networks of two-dimensional integrate and fire neurons

Frontiers in Computational Neuroscience

Volumn , Issue DEC, 2013, Pages

  Nicola, Wilten ^a   Campbell, Sue Ann ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Bifurcation analysis; Burstin; Hippocampus; Integrate and fire neuron; Mean field model

Indexed keywords

BIFURCATION ANALYSIS; BURSTIN; HIPPOCAMPUS; INTEGRATE-AND-FIRE NEURONS; MEAN FIELD MODELING;

BRAIN; FIRES; HETEROGENEOUS NETWORKS; HOPF BIFURCATION; NEURONS; POPULATION STATISTICS; TWO DIMENSIONAL;

NEURAL NETWORKS;

ANALYTICAL ERROR; ARTICLE; BRAIN REGION; CORTICAL SYNCHRONIZATION; HIPPOCAMPAL CA3 REGION; INTERMETHOD COMPARISON; MATHEMATICAL MODEL; NERVE CELL MEMBRANE CONDUCTANCE; NERVE CELL NETWORK; NONLINEAR SYSTEM; POPULATION DENSITY; PREDICTION; PRESYNAPTIC NERVE; QUALITATIVE ANALYSIS; QUANTITATIVE ANALYSIS; STEADY STATE;

EID: 84891504526     PISSN: 16625188     EISSN: None     Source Type: Journal    
DOI: 10.3389/fncom.2013.00184     Document Type: Article

References (39)

1. Abbott, L. F., and van Vreeswijk, C. (1993). Asynchronous states in networks of pulse-coupled oscillators. Learn Mem. 48, 1483-1490.
2. Andersen, P., Morris, R., Amaral, D., Bliss, T., and O'Keefe, J. (eds.). (2006). The Hippocampus Book. Oxford: Oxford University Press.
3. Bressloff, P. C. (2012). Spatiotemporal dynamics ofcontinuum neural fields. J. Phys. Math. Theor. 45:033001. doi: 10.1088/1751-8113/45/3/033001
4. Brette, R., and Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J. Neurophys. 94, 3637-3642. doi: 10.1152/jn.00686.2005
5. Buzsaki, G. (2004). Large-scale recordings of neuronal ensembles. Nat. Neurosci. 7, 446-451. doi: 10.1038/nn1233
6. Dayan, P., and Abbott, L. F. (2001). Theoretical Neuroscience. Cambridge MA: MIT Press. doi: 10.1016/S0306-4522(00)00552-2
7. Dhooge, A., Govaerts, W., and Kuznetsov, Yu. A. (2003). MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141-164. doi: 10.1145/779359.779362
8. di Bernardo, M., Budd, C. J., Champneys, A. R., Kowalczyk, P., Nordmark, A. B., Tost, G. O., et al. (2008). Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50, 629-701. doi: 10.1137/050625060
9. Dur-e-Ahmad, M., Nicola, W., Campbell, S. A., and Skinner, F. (2012). Network bursting using experimentally constrained single compartment CA3 hippocampal neuron models with adaptation. J. Comput. Neurosci. 33, 21-40. doi: 10.1007/s10827-011-0372-6
10. Ermentrout, G. B. (1998). Neural networks as spatio-temporal pattern-forming systems. Rep. Prog. Phys. 61, 353-430. doi: 10.1088/0034-4885/61/4/002
11. Ermentrout, G. B., and Terman, D. H. (2010). Mathematical Foundations of Neuroscience. New York, NY: Springer.
12. Grewe, B. F., Langer, D., Kasper, H., Kampa, B. M., and Helmchen, F. (2010). High-speed in-vivo calcium imaging reveals neuronal network activity with near-millisecond precision Nat. Meth. 7, 399-405. doi: 10.1038/nmeth06 10-479d
13. Hansel, D., and Mato, G. (2001). Existence and stability of persistent states in large neuronal networks. Phys. Rev. Lett. 86, 4175-4718. doi: 10.1103/PhysRevLett.86.4175
14. Hansel, D., and Mato, G. (2003). Asynchronous states and the emergence of synchrony in large networks of excitatory and inhibitory neurons. Neural Comput. 15, 1-56. doi: 10.1162/089976603321043685
15. Hemond, P., Epstein, D., Boley, A., Migliore, M., Ascoli, G. A., and Jaffe, D. (2008). Distinct classes of pyramidal cells exhibit mutually exclusive firing patterns in hippocampal area CA3b. Hippocampus 18, 411-424. doi: 10.1002/hipo.20404
16. Hermann, G., and Touboul, J. (2012). Heterogeneous connect 2012 ions induced oscillations in large-scale networks. Phys. Rev. Lett. 109:018702. doi: 10.1103/PhysRevLett.109.018702
17. Izhikevich, E. M. (2003). Simple model of spiking neurons. IEEE Trans. Neural Netw. 14, 1569-1572. doi: 10.1109/TNN.2003.820440
18. Izhikevich, E. M., and Edelman, G. M. (2008). Large-scale model of mammalian thalamocortical systems. PNAS 105, 3593-3598. doi: 10.1073/pnas.0712231105
19. Knight, B. W. (2000). Dynamics of encoding in neuron populations: some general mathematical features. Neural Comput. 12, 473-518. doi: 10.1162/089976600300015673
20. Ko, T., and Ermentrout, G. B. (2007). Effects of axonal time delay on synchronization and wave formation in sparsely coupled neuronal oscillators. Phys. Rev. E 76, 0562061-0562067. doi: 10.1103/PhysRevE.76.056206
21. Kuznetsov, Y. A., Rinaldi, S., and Gragnani, A. (2003). One-parameter bifurcations in planar fillipov systems. Int. J. Bifurcat. Chaos 13, 2157-2188. doi: 10.1142/S0218127403007874
22. Lisman, J. E. (1997). Bursts as a unit of neural information: making unreliable synapses reliable Trends Neurosci. 20, 38-43. doi: 10.1016/S0166- 2236(96)10070-9
23. Ly, C., and Tranchina, D. (2007). Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling. Neural Comput. 19, 2032-2092. doi: 10.1162/neco.2007.19. 8.2032
24. MATLAB. (2012). Version 7.10.0 (R2012a). Natick, MA: The MathWorks Inc.
25. Naud, R., Marcille, N., Clopath, C., and Gerstner, W. (2008). Firing patterns in the adaptive exponential integrate-and-fire model. Biol. Cybern. 99, 335-347. doi: 10.1007/s00422-008-0264-7
26. Nicola, W., and Campbell, S. A. (2013). Bifurcations of large networks of twodimensional integrate and fire neurons J. Comput. Neurosci. 35, 87-108. doi: 10.1007/s10827-013-0442-z
27. Nykamp, D. Q., and Tranchina, D. (2000). A population density approach that facilitates large-scale modeling of neural networks: analysis and an application to orientation tuning. J. Comput. Neurosci. 8, 19-50. doi: 10.1023/A:1008912914816
28. Omurtag, A., Knight, B. W., and Sirovich, L. (2000). On the simulation of large populations of neurons. J. Comput. Neurosci. 8, 51-63. doi: 10.1023/A:1008964915724
29. Renyi, L. (1970). Probability Theory. Amsterdam: North-Holland Publishing Company.
30. Sirovich, L., Omurtag, A., and Knight, B. W. (2000). Dynamics of neuronal populations: the equilibrium solution. SIAM J. Appl. Math. 60, 2009-2028. doi: 10.1137/S0036139998344921
31. Sirovich, L., Omurtag, A., and Lubliner, K. (2006). Dynamics of neural populations: stability and synchrony. Netw. Comput. Neural Syst. 17, 3-29. doi: 10.1080/09548980500421154
32. Skinner, F. K., Bazzazi, H., and Campbell, S. A. (2005). Two-cell to N-cell heterogeneous, inhibitory networks: precise linking of multistable and coherent properties. J. Comput. Neurosci. 18,343-352.doi: 10.1007/s10827-005-0331-1
33. Strogatz, S. H., and Mirollo, R. E. (1991). Stability of incoherence in a population ofcoupled oscillators. J. Stat. Phys. 63, 613-635. doi: 10.1007/BF01029202
34. Touboul, J. (2008). Bifurcation analysis of a general class of nonlinear integrate- and-fire neurons. SIAM J. Appl. Math. 68, 1045-1079. doi: 10.1137/070687268
35. van Vreeswijk, C. (1996). Partial synchronization in populations of pulse-coupled oscillators. Phys. Rev. E 54, 5522-5537. doi: 10.1103/PhysRevE.54.5522
36. van Vreeswijk, C., Abbott, L. F., and Ermentrout, B. G. (1994). When inhibition not excitation synchronizes neural firing. J. Comput. Neurosci. 1, 313-321. doi: 10.1007/BF00961879
37. van Vreeswijk, C., and Hansel, D. (2001). Patterns of synchrony in neural networks with spike adaptation. Neural Comput. 13, 959-992. doi: 10.1162/08997660151134280
38. Vladimirski, B. B., Tabak, J., O'Donovan, M. J., andRinzel, J. (2008). Episodic activity in a heterogeneous excitatory network, from spiking neurons to mean field. J. Comput. Neurosci. 25, 39-63. doi: 10.1007/s10827-007-0064-4
39. Zilli, E. A., and Hasselmo, M. E. (2010). Coupled noisy spiking neurons as velocity controlled oscillators in a model of grid cell spatial firing. J. Neurosci. 30, 13850-14860. doi: 10.1523/JNEUROSCI.0547-10.2010