Skip to main content
Log in

The Narrow Escape Problem—A Short Review of Recent Results

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The narrow escape problem in diffusion theory, which goes back to Lord Rayleigh, is to calculate the mean first passage time, also called the narrow escape time (NET), of a Brownian particle to a small absorbing window on the otherwise reflecting boundary of a bounded domain. The renewed interest in the NET problem is due to its relevance in molecular biology and biophysics. The small window often represents a small target on a cellular membrane, such as a protein channel, which is a target for ions, a receptor for neurotransmitter molecules in a neuronal synapse, a narrow neck in the neuronal spine, which is a target for calcium ions, and so on. The leading order singularity of the Neumann function for a regular domain strongly depends on the geometric properties of the boundary. It can give a smaller contribution than the regular part to the absorption flux through the small window when it is located near a boundary cusp. We review here recent results on the dependence of the absorption flux on the geometric properties of the domain and thus reveal geometrical features that can modulate the flux. This indicates a possible way to code information physiologically.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Notes

  1. Figure 3 is based on

References

  1. Baron Rayleigh, J.W.S.: The Theory of Sound vol. 2, 2nd edn. Dover, New York (1945)

    Google Scholar 

  2. Gardiner, C.W.: Handbook of Stochastic Methods, 2nd edn. Springer, New York (1985)

    Google Scholar 

  3. Schuss, Z.: Diffusion and Stochastic Processes: an Analytical Approach. Springer, New York (2010)

    Google Scholar 

  4. Fabrikant, V.I.: Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering (Mathematics and Its Applications). Kluwer Academic, Dordrecht (1991)

    Google Scholar 

  5. Ward, M.J., Keller, J.B.: Strong localized perturbations of eigenvalue problems. SIAM J. Appl. Math. 53, 770–798 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ward, M.J., Henshaw, W.D., Keller, J.B.: Summing logarithmic expansions for singularly perturbed eigenvalue problems. SIAM J. Appl. Math. 53, 799–828 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ward, M.J., Van de Velde, E.: The onset of thermal runaway in partially insulated or cooled reactors. IMA J. Appl. Math. 48, 53–85 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kolokolnikov, T., Titcombe, M., Ward, M.J.: Optimizing the fundamental neumann eigenvalue for the Laplacian in a domain with small traps. Eur. J. Appl. Math. 16, 161–200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheviakov, A.F., Ward, M.J., Straube, R.: An asymptotic analysis of the mean first passage time for narrow escape problems. Part II: The sphere. Multiscale Model. Simul. 8(3), 803–835 (2010)

    Article  MathSciNet  Google Scholar 

  10. Pillay, S., Ward, M.J., Peirce, A., Kolokolnikov, T.: An asymptotic analysis of the mean first passage time for narrow escape problems. Part I: Two-dimensional domains. Multiscale Model. Simul. 8(3), 836–870 (2010)

    Article  MathSciNet  Google Scholar 

  11. Cheviakov, A.F., Reimer, A.S., Ward, M.J.: Mathematical modeling and numerical computation of narrow escape problems. Phys. Rev. E 85, 021131 (2012)

    Article  Google Scholar 

  12. Holcman, D., Schuss, Z.: Escape through a small opening: receptor trafficking in a synaptic membrane. J. Stat. Phys. 117(5–6), 975–1014 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Singer, Z., Schuss, A., Holcman, D., Eisenberg, B.: Narrow escape I. J. Stat. Phys. 122(3), 437–463 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Singer, Z., Schuss, A., Holcman, D.: Narrow escape II. J. Stat. Phys. 122(3), 465–489 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Singer, Z., Schuss, A., Holcman, D.: Narrow escape III. J. Stat. Phys. 122(3), 491–509 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grigoriev, I.V., Makhnovskii, Y.A., Berezhkovskii, A.M., Zitserman, V.Y.: Kinetics of escape through a small hole. J. Chem. Phys. 116(22), 9574–9577 (2002)

    Article  Google Scholar 

  17. Chevalier, C., Bénichou, O., Meyer, B., Voituriez, R.: First-passage quantities of brownian motion in a bounded domain with multiple targets: a unified approach. J. Phys. A, Math. Theor. 44, 025002 (2011)

    Article  Google Scholar 

  18. Harris, K.M., Stevens, J.K.: Dendritic spines of rat cerebellar purkinje cells: serial electron microscopy with reference to their biophysical characteristics. J. Neurosci. 12, 4455–4469 (1988)

    Google Scholar 

  19. Bourne, J.N., Harris, K.M.: Balancing structure and function at hippocampal dendritic spines. Annu. Rev. Neurosci. 31, 47–67 (2008)

    Article  Google Scholar 

  20. Korkotian, E., Holcman, D., Segal, M.: Dynamic regulation of spine-dendrite coupling in cultured hippocampal neurons. Eur. J. Neurosci. 20(10), 2649–2663 (2004)

    Article  Google Scholar 

  21. Hotulainen, P., Hoogenraad, C.C.: Actin in dendritic spines: connecting dynamics to function. J. Cell Biol. 189(4), 619–629 (2010)

    Article  Google Scholar 

  22. Newpher, T.M., Ehlers, M.D.: Spine microdomains for postsynaptic signaling and plasticity. Trends Cell Biol. 5, 218–227 (2009)

    Article  Google Scholar 

  23. von Helmholtz, H.L.F.: Crelle, Bd. 7 (1860)

  24. Hille, B.: Ionic Channels of Excitable Membranes, 2nd edn. Sinauer, Sunderland (1992)

    Google Scholar 

  25. Singer, A., Schuss, Z., Holcman, D.: Narrow escape and leakage of brownian particles. Phys. Rev. E 78, 051111 (2008)

    Article  MathSciNet  Google Scholar 

  26. Yuste, R., Majewska, A., Holthoff, K.: From form to function: calcium compartmentalization in dendritic spines. Nat. Neurosci. 7, 653–659 (2000)

    Article  Google Scholar 

  27. Svoboda, K., Tank, D.W., Denk, W.: Direct measurement of coupling between dendritic spines and shafts. Science 272(5262), 716–719 (1996)

    Article  Google Scholar 

  28. Biess, A., Korkotian, E., Holcman, D.: Diffusion in a dendritic spine: the role of geometry. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 021922 (2007)

  29. Holcman, D., Kupka, I.: Some questions in computational cellular biology. J. Fixed Point Theory Appl. 7(1), 67–83 (2010). doi:10.1007/s11784-010-0012-1

    Article  MathSciNet  MATH  Google Scholar 

  30. Borgdorff, A.J., Choquet, D.: Regulation of AMPA receptor lateral movements. Nature 417(6889), 649–653 (2002)

    Article  Google Scholar 

  31. Choquet, D., Triller, A.: The role of receptor diffusion in the organization of the postsynaptic membrane. Nat. Rev., Neurosci. 4, 251–265 (2003)

    Article  Google Scholar 

  32. Holcman, D., Triller, A.: Modeling synaptic dynamics and receptor trafficking. Biophys. J. 91(7), 2405–2415 (2006)

    Article  Google Scholar 

  33. Holcman, D., Korkotian, E., Segal, M.: Calcium dynamics in dendritic spines, modeling and experiments. Cell Calcium 37(5), 467–475 (2005)

    Article  Google Scholar 

  34. Holcman, D., Hoze, N., Schuss, Z.: Narrow escape through a funnel and effective diffusion on a crowded membrane. Phys. Rev. E 84, 021906 (2011)

    Article  Google Scholar 

  35. Holcman, D., Schuss, Z.: Brownian needle in dire straits: stochastic motion of a rod in very confined narrow domains. Phys. Rev. E 85, 010103(R) (2012)

    Google Scholar 

  36. Berezhkovskii, A.M., Barzykin, A.V., Zitserman, V.Y.: Escape from cavity through narrow tunnel. J. Chem. Phys. 130(24), 245104 (2009)

    Article  Google Scholar 

  37. Popov, I.Yu.: Extension theory and localization of resonances for domains of trap type. Math. USSR Sb. 71(1), 209–234 (1992)

    Article  MathSciNet  Google Scholar 

  38. Schuss, Z.: Singular perturbation methods for stochastic differential equations of mathematical physics. SIAM Rev. 22, 116–155 (1980)

    Article  MathSciNet  Google Scholar 

  39. Dagdug, L., Berezhkovskii, A.M., Shvartsman, S.Y., Weiss, G.H.: Equilibration in two chambers connected by a capillary. J. Chem. Phys. 119(23), 12473 (2003)

    Article  Google Scholar 

  40. Matkowsky, B.J., Schuss, Z., Tier, C.: Uniform expansions of the transition rate in Kramers’ problem. J. Stat. Phys. 35(3–4), 443–456 (1984)

    Article  MathSciNet  Google Scholar 

  41. Holcman, D., Schuss, Z.: Diffusion laws in dendritic spines. J. Math. Neurosci. 1, 10 (2011). doi:10.1186/2190-8567-1-10

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Z. Schuss.

Additional information

Dedicated to S. Abarbanel on his 80th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schuss, Z. The Narrow Escape Problem—A Short Review of Recent Results. J Sci Comput 53, 194–210 (2012). https://doi.org/10.1007/s10915-012-9590-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9590-y

Keywords

Navigation