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Microscopic approach of a time elapsed neural model

Laboratoire J. A. Dieudonné, UMR CNRS 6621, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 2, France

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María José Cáceres

Departamento de Matemática Aplicada, Universidad de Granada, Campus de Fuentenueva, E-18071 Granada, Spain

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Marie Doumic

Inria Paris-Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay, France

UPMC University of Paris 6, JL Lions Lab., 4 Place Jussieu, 75005 Paris, France

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, and

Patricia Reynaud-Bouret

Laboratoire J. A. Dieudonné, UMR CNRS 6621, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 2, France

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https://doi.org/10.1142/S021820251550058XCited by:45 (Source: Crossref)

Abstract

The spike trains are the main components of the information processing in the brain. To model spike trains several point processes have been investigated in the literature. And more macroscopic approaches have also been studied, using partial differential equation models. The main aim of the present paper is to build a bridge between several point processes models (Poisson, Wold, Hawkes) that have been proved to statistically fit real spike trains data and age-structured partial differential equations as introduced by Pakdaman, Perthame and Salort.

Keywords:

AMSC: 35F15, 35B10, 92B20, 60G57, 60K15

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References

M. Bossyet al., Encyclopedia of Quantitative Finance (Wiley, 2010) pp. 1142–1159. Google Scholar
O. Boxma et al. , Adv. Appl. Probab. 38 , 221 ( 2006 ) . Crossref, Web of Science, Google Scholar
P. Brémaud , Point Processes and Queues: Martingale Dynamics , Springer Series in Statistics ( Springer , 1981 ) . Crossref, Google Scholar
P. Brémaud and L. Massoulié , Ann. Probab. 24 , 1563 ( 1996 ) . Crossref, Web of Science, Google Scholar
D. R. Brillinger , H. L. Bryant and J. P. Segundo , Biol. Cybern. 22 , 213 ( 1976 ) . Crossref, Web of Science, Google Scholar
E. N. Brown et al. , Neural Comput. 14 , 325 ( 2002 ) . Crossref, Web of Science, Google Scholar
N. Brunel , J. Comput. Neurosci. 8 , 183 ( 2000 ) . Crossref, Web of Science, Google Scholar
N. Brunel and V. Hakim , Neural Comput. 11 , 1621 ( 1999 ) . Crossref, Web of Science, Google Scholar
M. J. Cáceres , J. A. Carrillo and B. Perthame , J. Math. Neurosci. 1 , 1 ( 2011 ) . Crossref, Google Scholar
M. J. Cáceres , J. A. Carrillo and L. Tao , J. Comput. Phys. 230 , 1084 ( 2011 ) . Crossref, Web of Science, Google Scholar
M. J. Cáceres and B. Perthame , J. Theor. Biol. 350 , 81 ( 2014 ) . Crossref, Web of Science, Google Scholar
J. A. Cañizo , J. A. Carrillo and S. Cuadrado , Acta Appl. Math. 123 , 141 ( 2013 ) . Crossref, Web of Science, Google Scholar
E. S. Chornoboy , L. P. Schramm and A. F. Karr , Biol. Cybern. 59 , 265 ( 1988 ) . Crossref, Web of Science, Google Scholar
B. Cloez, Limit theorems for some branching measure-valued processes, preprint (2012) , arXiv: 1106.0660v2 . Google Scholar
A. Compte et al. , Cereb. Cortex 10 , 910 ( 2000 ) . Crossref, Web of Science, Google Scholar
D. J. Daley and D. Vere-Jones , An Introduction to the Theory of Point Processes 2 ( Springer , 1988 ) . Google Scholar
F. Delarue et al. , Ann. Appl. Probab. 25 , 2096 ( 2015 ) . Crossref, Web of Science, Google Scholar
S. Ditlevsen and A. Samson , Ann. Appl. Statist. 8 , 674 ( 2014 ) . Crossref, Web of Science, Google Scholar
M. Doumic et al. , Bernoulli 21 , 1760 ( 2015 ) . Crossref, Web of Science, Google Scholar
A. G. Hawkes and D. Oakes , J. Appl. Probab. 11 , 493 ( 1974 ) . Crossref, Web of Science, Google Scholar
P. Jahn et al. , J. Comput. Neurosci. 31 , 563 ( 2011 ) . Crossref, Web of Science, Google Scholar
J. F. C. Kingman , Poisson Processes ( Oxford Science Publications , 1993 ) . Google Scholar
C. D. Lai , J. Appl. Probab. 15 , 748 ( 1978 ) . Crossref, Web of Science, Google Scholar
P. A. W. Lewis and G. S. Shedler , Naval Res. Logistics Quart. 26 , 403 ( 1979 ) . Crossref, Google Scholar
M. Mattia and P. Del Giudice , Phys. Rev. E 66 , 051917 ( 2002 ) . Crossref, Web of Science, Google Scholar
P. Mattila , Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability , Cambridge Studies in Advanced Mathematics ( Cambridge Univ. Press , 1999 ) . Google Scholar
J. Møller and J. G. Rasmussen , Adv. Appl. Probab. 37 , 629 ( 2005 ) . Crossref, Web of Science, Google Scholar
Y. Ogata , IEEE Trans. Inform. Theory 27 , 23 ( 1981 ) . Crossref, Web of Science, Google Scholar
A. Omurtag , B. W. Knight and L. Sirovich , J. Comp. Neurosci. 8 , 51 ( 2000 ) . Crossref, Web of Science, Google Scholar
K. Pakdaman , B. Perthame and D. Salort , Nonlinearity 23 , 55 ( 2010 ) . Crossref, Web of Science, Google Scholar
K. Pakdaman , B. Perthame and D. Salort , SIAM J. Appl. Math. 73 , 1260 ( 2013 ) . Crossref, Web of Science, Google Scholar
K. Pakdaman , B. Perthame and D. Salort , J. Math. Neurosci. 4 , 1 ( 2014 ) . Crossref, Google Scholar
B. Perthame , Transport Equations in Biology , Frontiers in Mathematics ( Birkhäuser , 2007 ) . Crossref, Google Scholar
J. Pham et al. , Neural Netw. 11 , 415 ( 1998 ) . Crossref, Web of Science, Google Scholar
J. W. Pillow et al. , Nature 454 , 995 ( 2008 ) . Crossref, Web of Science, Google Scholar
G. Pipa , S. Grün and C. van Vreeswijk , Neural Comput. 25 , 1123 ( 2013 ) . Crossref, Web of Science, Google Scholar
C. Pouzat and A. Chaffiol , J. Neurosci. Methods 181 , 119 ( 2009 ) . Crossref, Web of Science, Google Scholar
A. Renart , N. Brunel and X.-J. Wang , Computational Neuroscience: A Comprehensive Approach , ed. J. Feng ( Chapman & Hall/CRC , 2004 ) . Google Scholar
P. Reynaud-Bouret et al. , J. Math. Neurosci. 4 , 1 ( 2014 ) . Crossref, Google Scholar
P. Reynaud-Bouret , V. Rivoirard and C. Tuleau-Malot , Inference of functional connectivity in neurosciences via Hawkes processes , 1st IEEE Global Conf. Signal and Information Processing ( 2013 ) . Google Scholar
L. Sirovich , A. Omurtag and K. Lubliner , Netw. Comput. Neural Syst. 17 , 3 ( 2006 ) . Crossref, Web of Science, Google Scholar
V. Ventura et al. , Biostatistics 3 , 1 ( 2002 ) . Crossref, Web of Science, Google Scholar