Moment-Generating Function of Output Stream of Leaky Integrate-and-Fire Neuron


  • A.K. Vidybida Bogolyubov Institute for Theoretical Physics of the Nat. Acad. of Scie of Ukraine
  • O.V. Shchur Bogolyubov Institute for Theoretical Physics of the Nat. Acad. of Scie of Ukraine



leaky integrate-and-fire neuron, stochastic Poisson process, interspike interval, moments of probability distribution, moment-generating function


The statistics of the output activity of a neuron during its stimulation by the stream of input impulses that forms the stochastic Poisson process is studied. The leaky integrate-and-fire neuron is considered as a neuron model. A new representation of the probability distribution function of the output interspike interval durations is found. Based on it, the moment-generating function of the probability distribution is calculated explicitly. The latter, according to the Curtiss theorem, completely determines the distribution itself. In particular, explicit expressions are derived from the moment-generating function for the moments of all orders. The first moment coincides with the one found earlier. Formulas for the second and third moments have been checked numerically by direct modeling of the stochastic dynamics of a neuron with specific physical parameters.


J. Werner. Electrical activities in the body. In Comprehensive Biomedical Physics. Edited by A. Brahme (Elsevier, 2014), Vol. 5.

J.C. Brookes. Olfaction: The physics of how smell works? Contemp. Phys. 52, 385 (2011).

M. Denny. The physics of bat echolocation: Signal processing techniques. Am. J. Phys. 72, 1465 (2004).

C.W. Gardiner. Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences. (Springer, 1985).

L. Sacerdote, M.T. Giraudo. Stochastic integrate and fire models: A review on mathematical methods and their applications. In: Stochastic Biomathematical Models. Edited by M. Bachar, J. J. Batzel, S. Ditlevsen (Springer, 2013).

O.K. Vidybida. Output stream of leaky integrate and fire neuron. Dopov. Nac. akad. nauk Ukr. 2014 (12), 18 (2014).

A.K. Vidybida. Output stream of leaky integrate-and-fire neuron without diffusion approximation. J. Stat. Phys. 166, 267 (2017).

J.H. Curtiss. A note on the theory of moment generating functions. Ann. Math. Stat. 13, 430 (1942).

A.N. Burkitt. A review of the integrate-and-fire neuron model: I. homogeneous synaptic input. Biol. Cybern. 95, 1 (2006).

K. Kravchuk. Leaky integrate-and-fire neuron under Poisson stimulation. In: Proceedings of 2016 II International Young Scientists Forum on Applied Physics and Engineering (YSF) (IEEE, 2016), p. 203.



How to Cite

Vidybida, A., & Shchur, O. (2021). Moment-Generating Function of Output Stream of Leaky Integrate-and-Fire Neuron. Ukrainian Journal of Physics, 66(3), 254.



Physics of liquids and liquid systems, biophysics and medical physics