Skip to main content
SpringerLink
Log in

Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Providing an analytical treatment to the stochastic feature of neurons’ dynamics is one of the current biggest challenges in mathematical biology. The noisy leaky integrate-and-fire model and its associated Fokker–Planck equation are probably the most popular way to deal with neural variability. Another well-known formalism is the escape-rate model: a model giving the probability that a neuron fires at a certain time knowing the time elapsed since its last action potential. This model leads to a so-called age-structured system, a partial differential equation with non-local boundary condition famous in the field of population dynamics, where the age of a neuron is the amount of time passed by since its previous spike. In this theoretical paper, we investigate the mathematical connection between the two formalisms. We shall derive an integral transform of the solution to the age-structured model into the solution of the Fokker–Planck equation. This integral transform highlights the link between the two stochastic processes. As far as we know, an explicit mathematical correspondence between the two solutions has not been introduced until now.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Abbott L (1999) Lapique’s introduction of the integrate-and-fire model neuron (1907). Brain Res Bull 50(5):303–304

    Article  Google Scholar 

  • Abbott LF, van Vreeswijk C (1993) Asynchronous states in networks of pulse-coupled oscillators. Phys Rev E 48:1483–1490

    Article  Google Scholar 

  • Bressloff Newby JM (2013) Stochastic models of intracellular transport. Rev Mod Phys 85:135

    Article  Google Scholar 

  • Brunel N (2000) Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J Comput Neurosci 8:183–208

    Article  MATH  Google Scholar 

  • Brunel N, Hakim V (1999) Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput 11:1621–1671

    Article  Google Scholar 

  • Brunel N, van Rossum M (2007) Lapicque’s 1907 paper: from frogs to integrate-and-fire. Biol Cybern 97:341–349

    Article  MATH  Google Scholar 

  • Burkitt AN (2006) A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input. Biol Cybern 95:1–19

    Article  MathSciNet  MATH  Google Scholar 

  • Cáceres MJ, Carrillo JA, Perthame B (2011) Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states. J Math Neurosci 1(1):7. 10.1186/2190-8567-1-7

  • Carrillo JA, d González M, Gualdani MP, Schonbek ME (2013) Classical solutions for a nonlinear Fokker–Planck equation arising in computational neuroscience. Commun PDEs 38:385–409

    Article  MathSciNet  MATH  Google Scholar 

  • Cox DR (1962) Renewal theory. Mathuen, London

    MATH  Google Scholar 

  • Dumont G, Henry J (2013a) Population density models of integrate-and-fire neurons with jumps, well-posedness. J Math Biol 67(3):453–481

    Article  MathSciNet  MATH  Google Scholar 

  • Dumont G, Henry J (2013b) Synchronization of an excitatory integrate-and-fire neural network. Bull Math Biol 75(4):629–648

    Article  MathSciNet  MATH  Google Scholar 

  • Dumont G, Henry J, Tarniceriu CO (2014) Well-posedness of a density model for a population of theta neurons. J Math Neurosci 4(1):2

    Article  MathSciNet  MATH  Google Scholar 

  • Dumont G, Henry J, Tarniceriu CO (2016) Theoretical connections between mathematical neuronal models corresponding to different expressions of noise. arXiv:1602.03764v1 [q-bio.NC]

  • Ermentrout GB, Terman D (2010) Mathematical foundations of neuroscience. Springer, New York

    Book  MATH  Google Scholar 

  • Faisal A, Selen L, Wolpert D (2008) Noise in the nervous system. Nat Rev Neurosci 9(4):292–303

    Article  Google Scholar 

  • Gardiner CW (1996) Handbook of stochastic method for physics, chemistry and natural sciences. Springer, New York

    MATH  Google Scholar 

  • Gerstner W (1995) Time structure of the activity in neural network models. Phys Rev E 51:738–758

    Article  Google Scholar 

  • Gerstner W (2000) Population dynamics of spiking neurons: fast transients, asynchronous states, and locking. Neural Comput 12:43–89

    Article  Google Scholar 

  • Gerstner W, Kistler W (2002) Spiking neuron models. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Gerstner W, Naud R (2009) How good are neuron models? Science 326(5951):379–380

    Article  Google Scholar 

  • Gillespie DT, Seitaridou E (2012) Simple Brownian diffusion: an introduction to the standard theoretical models. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

  • Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500–544

    Article  Google Scholar 

  • Holcman D, Schuss Z (2014) The narrow escape problem. SIAM Rev 56(2):213–257

    Article  MathSciNet  MATH  Google Scholar 

  • Izhikevich EM (2007) Dynamical systems in neuroscience. The MIT Press, Cambridge

    Google Scholar 

  • Knight B (2000) Dynamics of encoding in neuron populations: some general mathematical features. Neural Comput 12(3):473–518

    Article  Google Scholar 

  • Knight B, Manin D, Sirovich L (1996) Dynamical models of interacting neuron populations in visual cortex. Robot Cybern 54:4–8

    Google Scholar 

  • Longtin A (2010) Stochastic dynamical systems. Scholarpedia 5(4):1619

    Article  Google Scholar 

  • Longtin A (2013) Neuronal noise. Scholarpedia 8(9):1618

    Article  MathSciNet  Google Scholar 

  • Millman D, Mihalas S, Kirkwood A, Niebur E (2010) Self-organized criticality occurs in non-conservative neuronal networks during ‘up’ states. Nat Phys 6:801–805

    Article  Google Scholar 

  • Newhall KA, Kovačič G, Kramer PR, Cai D (2010a) Cascade-induced synchrony in stochastically driven neuronal networks. Phys Rev E Stat Nonlin Soft Matter Phys 82(4 Pt 1):041903

    Article  MathSciNet  Google Scholar 

  • Newhall KA, Kovacic G, Kramer PR, Zhou D, Rangan AV, Cai D (2010b) Dynamics of current-based, poisson driven, integrate-and-fire neuronal networks. Commun Math Sci 8:541–600

    Article  MathSciNet  MATH  Google Scholar 

  • Nykamp DQ, Tranchina D (2000) A population density appraoch that facilitates large-scale modeling of neural networks : analysis and an application to orientation tuning. J Comput Neurosci 8:19–50

    Article  MATH  Google Scholar 

  • Omurtag A, Knight B, Sirovich L (2000) On the simulation of large population of neurons. J Comput 8:51–63

    MATH  Google Scholar 

  • Ostojic S (2011) Interval interspike distributions of spiking neurons driven by fluctuating inputs. J Neurophysiol 106:361–373

    Article  Google Scholar 

  • Ostojic S, Brunel N, Hakim V (2009) Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities. J Comput Neurosci 26:369–392

    Article  MathSciNet  Google Scholar 

  • Pakdaman K, Perthame B, Salort D (2009) Dynamics of a structured neuron population. Nonlinearity 23:23–55

    MathSciNet  MATH  Google Scholar 

  • Pakdaman K, Perthame B, Salort D (2013) Relaxation and self-sustained oscillations in the time elapsed neuron network model. SIAM J Appl Math 73(3):1260–1279

    Article  MathSciNet  MATH  Google Scholar 

  • Plesser HE, Gerstner W (2000) Noise in integrate-and-fire neurons: from stochastic input to escape rates. Neural Comput 12(2):367–384

    Article  Google Scholar 

  • Schuss Z, Singer A, Holcman D (2007) The narrow escape problem for diffusion in cellular domains. Proc Natl Acad Sci 104(41):16098–16103

    Article  Google Scholar 

  • Wilbur WJ, Rinzel J (1983) A theoretical basis for large coefficient of variation and bimodality in interspike interval distributions. J Theor Biol 105:345–368

    Article  Google Scholar 

Download references

Acknowledgments

We would like to thank both anonymous reviewers whose valuable comments helped us improve our paper. In particular, some important observations over the nature of a possible inverse transform have been made that were included in the last section of the paper.

Author information

Authors and Affiliations

  1. Group for Neural Theory, Ecole Normale Sperieure, 29 rue d’Ulm, 75005, Paris, France

    G. Dumont

  2. INRIA Bordeaux Sud Ouest, 200 avenue de la vieille tour, 33405, Talence Cedex, France

    J. Henry

  3. Interdisciplinary Research Department-Field Science, Alexandru Ioan Cuza University of Iaşi, Lascăr Catargi 54, 700107, Iasi, Romania

    C. O. Tarniceriu

Authors
  1. G. Dumont
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Henry
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. O. Tarniceriu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Dumont.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dumont, G., Henry, J. & Tarniceriu, C.O. Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model. J. Math. Biol. 73, 1413–1436 (2016). https://doi.org/10.1007/s00285-016-1002-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-016-1002-8

Mathematics Subject Classification

Search

Navigation