Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity

Journal of Theoretical Biology

Volumn 350, Issue , 2014, Pages 81-89

  Cáceres, María J ^a   Perthame, Benoît ^b,c  

^a UNIVERSITY OF GRANADA   (Spain)

^b SORBONNE UNIVERSITÉ   (France)

^c INRIA ROCQUENCOURT   (France)

Author keywords

Blow up; Integrate and fire models; Neural networks; Noise; Spontaneous activity

Indexed keywords

ARTIFICIAL NEURAL NETWORK; EQUATION; NOISE; NUMERICAL MODEL;

ARTICLE; CONTROLLED STUDY; FINITE ELEMENT ANALYSIS; MATHEMATICAL COMPUTING; MATHEMATICAL MODEL; NERVE CELL MEMBRANE POTENTIAL; NERVE CELL NETWORK; NERVE CELL STIMULATION; NERVOUS SYSTEM PARAMETERS; NETWORK NOISY LEAKY INTEGRATE AND FIRE MODEL; PRIORITY JOURNAL; REFRACTORY PERIOD; SPONTANEOUS ACTIVITY; STIMULUS RESPONSE; ACTION POTENTIAL; BIOLOGICAL MODEL; PHYSIOLOGY;

ACTION POTENTIALS; MODELS, NEUROLOGICAL; NERVE NET; NUMERICAL ANALYSIS, COMPUTER-ASSISTED; REFRACTORY PERIOD, ELECTROPHYSIOLOGICAL;

EID: 84896763570     PISSN: 00225193     EISSN: 10958541     Source Type: Journal    
DOI: 10.1016/j.jtbi.2014.02.005     Document Type: Article

References (25)

1. Abbott L.F., Vreeswijk C.v. Asynchronous states in networks of pulse-coupled oscillators. Phys. Rev. E 1993, 48:1483-1490.
2. Brette R., Gerstner W. Adaptive exponential integrate-and-fire model as an effective description of neural activity. J. Neurophysiol. 2005, 94:3637-3642.
3. Brunel N. Dynamics of sparsely connected networks of excitatory and inhibitory spiking networks. J. Comput. Neurosci. 2000, 8:183-208.
4. Brunel N., Hakim V. Fast global oscillations in networks of integrate-and-fire neurons with long firing rates. Neural Comput. 1999, 11:1621-1671.
5. Cáceres M.J., Carrillo J.A., Perthame B. Analysis of nonlinear noisy integrate & fire neuron models. blow-up and steady states. J. Math. Neurosci. 2011, 1-7.
6. Cáceres M.J., Carrillo J.A., Tao L. A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics. J. Comput. Phys. 2011, 230:1084-1099.
7. Carrillo J.A., González M.d.M., Gualdani M.P., Schonbek M.E. Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience. Comm. Partial Differ. Equat. 2013, 38:385-409.
8. Delarue, F., Inglis, J., Rubenthaler, S., Tanré, E., 2012. Global solvability of a networked integrate-and-fire model of McKean-Vlasov type. 42p., arxiv:1211.0299.
9. Dumont G., Henry J. Population density models of integrate-and-fire neurons with jumps: well-posedness. J. Math. Biol. 2013, 67:453-481.
10. Dumont G., Henry J. Synchronization of an excitatory integrate-and-fire neural network. Bull. Math. Biol. 2013, 75:629-648.
11. Gerstner W., Kistler W. Spiking Neuron Models 2002, Cambridge University Press, Cambridge.
12. Guillamon T. An introduction to the mathematics of neural activity. Butl. Soc. Catalana Mat. 2004, 19:25-45.
13. Ledoux, M., 2001. The concentration of measure phenomenon. In: AMS Math. Surveys and Monographs, vol. 89, AMS.
14. Michel P., Mischler S., Perthame B. General relative entropy inequality. an illustration on growth models. J. Math. Pures Appl. 2005, 84:1235-1260.
15. Newhall K., Kovačič G., Kramer P., Rangan A.V., Cai D. Cascade-induced synchrony in stochastically driven neuronal networks. Phys. Rev. E 2010, 82:041903.
16. Newhall K., Kovačič G., Kramer P., Zhou D., Rangan A.V., Cai D. Dynamics of current-based, poisson driven, integrate-and-fire neuronal networks. Comm. Math. Sci. 2010, 8:541-600.
17. Pakdaman K., Perthame B., Salort D. Relaxation and self-sustained oscillations in the time elapsed neuron network model. SIAM. J. Appl. Math. 2013, 73(3):1260-1279.
18. Pakdaman K., Thieullen M., Wainrib G. Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv. Appl. Probab. 2010, 42:761-794.
19. Perthame B. Transport Equations in Biology, Frontiers in Mathematics 2007, Birkhäuser, Verlag, Basel.
20. Rangan A.V., Kovaîiî G., Cai D. Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train. Phys. Rev. E 2008, 77:1-13.
21. Renart A., Brunel N., Wang X.-J. Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks. Computational Neuroscience: A Comprehensive Approach, Mathematical Biology and Medicine Series 2004, Chapman & Hall/CRC. J. Feng (Ed.).
22. Rossant C., Goodman D.F.M., Fontaine B., Platkiewicz J., Magnusson A.K., Brette R. Fitting neuron models to spike trains. Front. Neurosci. 2011, 5:1-8.
23. Touboul J. Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons. SIAM J. Appl. Math. 2008, 68:1045-1079.
24. Touboul J. Importance of the cutoff value in the quadratic adaptive integrate-and-fire model. Neural Comput. 2009, 21:2114-2122.
25. Tuckwell H. Introduction to Theoretical Neurobiology 1988, Cambridge University Press, Cambridge.