Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons
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Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli
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2.
Istituto per le Appplicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, Via Pietro Castellino, Napoli
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Received:
01 October 2012
Accepted:
29 June 2018
Published:
01 October 2013
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MSC :
Primary: 60J60, 60J70; Secondary: 92-08.
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With the aim to describe the interaction between a couple of neurons a stochastic model is proposed and formalized. In such a model, maintaining statements of the Leaky Integrate-and-Fire framework, we include a random component in the synaptic current, whose role is to modify the equilibrium point of the membrane potential of one of the two neurons and when a spike of the other one occurs it is turned on. The initial and after spike reset positions do not allow to identify the inter-spike intervals with the corresponding first passage times. However, we are able to apply some well-known results for the first passage time problem for the Ornstein-Uhlenbeck process in order to obtain (i) an approximation of the probability density function of the inter-spike intervals in one-way-type interaction and (ii) an approximation of the tail of the probability density function of the inter-spike intervals in the mutual interaction. Such an approximation is admissible for small instantaneous firing rates of both neurons.
Citation: Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons[J]. Mathematical Biosciences and Engineering, 2014, 11(2): 189-201. doi: 10.3934/mbe.2014.11.189
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Abstract
With the aim to describe the interaction between a couple of neurons a stochastic model is proposed and formalized. In such a model, maintaining statements of the Leaky Integrate-and-Fire framework, we include a random component in the synaptic current, whose role is to modify the equilibrium point of the membrane potential of one of the two neurons and when a spike of the other one occurs it is turned on. The initial and after spike reset positions do not allow to identify the inter-spike intervals with the corresponding first passage times. However, we are able to apply some well-known results for the first passage time problem for the Ornstein-Uhlenbeck process in order to obtain (i) an approximation of the probability density function of the inter-spike intervals in one-way-type interaction and (ii) an approximation of the tail of the probability density function of the inter-spike intervals in the mutual interaction. Such an approximation is admissible for small instantaneous firing rates of both neurons.
References
[1]
|
Neural Comp., 13 (2001), 2763-2797.
|
[2]
|
Advances in Applied Probability, 19 (1987), 784-990.
|
[3]
|
Neural Comput., 22 (2010), 2558-2585.
|
[4]
|
Methodol. Comput. Appl. Probab., 13 (2011), 29-57.
|
[5]
|
Biol. Cybern., 95 (2006), 1-19.
|
[6]
|
in Computer Aided Systems Theory – EUROCAST 2005, Lecture Notes in Computer Science, 3643, Springer, Berlin-Heidelberg, 2005, 436-446.
|
[7]
|
BioSystems, 79 (2005), 109-116.
|
[8]
|
Adv. Appl. Prob., 33 (2001), 453-482.
|
[9]
|
Neural Computation, 23 (2011), 421-434.
|
[10]
|
Adv. Appl. Prob., 22 (1990), 883-914.
|
[11]
|
Phys. Rev. Lett., 68 (2001), 4179-4182.
|
[12]
|
in Computer Aided Systems Theory – EUROCAST 2007, Lecture Notes in Computer Science, 4739, Springer, Berlin-Heidelberg, 2007, 146-153.
|
[13]
|
Scientiae Mathematicae Japonicae, 67 (2008), 241-266. Available from: http://www.jams.or.jp/scm/contents/e-2008-2/2008-12.pdf.
|
[14]
|
in Network Science, Springer, London, 2010, 217-242.
|
[15]
|
Brain Research, 1434 (2012), 243-256.
|
[16]
|
Phys. Rev. E., 70 (2004), 1-4.
|
[17]
|
Progress of Theoretical Physics, 114 (2005), 1-18.
|
[18]
|
in Computational and Ambient Intelligence, Lecture Notes in Computer Science, 4507, Springer, Berlin-Heidelberg, 2007, 23-30.
|
[19]
|
Neural Comput., 19 (2007), 3262-3292.
|
-
-
-
-