Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Calcium Dynamics in Neuronal Microdomains: Modeling, Stochastic Simulations, and Data Analysis

Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_179-1


Calcium is a key but a ubiquitous messenger in cell physiology. Yet direct electrophysiological or light imaging measurements are limited by the intrinsic small nano- to micrometer space where chemical reactions occur and also by the small number of molecules. Thus any fluorescence dye molecule added to measure the number of calcium ions can severely perturb the endogenous chemical reactions. Over the years, an alternative approach based on modeling, mathematical analysis, and numerical simulations has demonstrated that it can be used to obtain precise quantitative results about the order of magnitude, rate constants, the role of the cell geometry, and flux regulation across scales from channels to the cell level.

The aim of this ECN is to present physical models of calcium ions from the molecular description to the concentration level and to present the mathematical tools used to analyze the model equations. From such analysis, asymptotic formulas can be obtained, which are...


Dendritic Spine Brownian Particle Robin Boundary Condition Spine Head Spine Neck 
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  1. Andrews S, Bray D (2004) Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Phys Biol 1(3-4):137–151PubMedCrossRefGoogle Scholar
  2. Batsilas L, Berezhkovskii AM, Shvartsman SY (2003) Stochastic model of autocrine and paracrine signals in cell culture assays. Biophys J 85:3659–3665PubMedCrossRefPubMedCentralGoogle Scholar
  3. Benazilla F (2000) The voltage sensor in voltage-dependent ion channels. Physiol Rev 80(2):555–592Google Scholar
  4. Berezhkovskii AM, Makhnovskii YA, Monine MI, Zitserman VY, Shvartsman SY (2004) Boundary homogenization for trapping by patchy surfaces. J Chem Phys 121(22):11390–11394PubMedCrossRefGoogle Scholar
  5. Berg HC, Purcell EM (1977) Physics of chemoreception. Biophys J 20(2):193–219PubMedCrossRefPubMedCentralGoogle Scholar
  6. Biess A, Korkotian E, Holcman D (2007) Diffusion in a dendritic spine: the role of geometry. Phys Rev E 76(2 Pt 1):021922CrossRefGoogle Scholar
  7. Biess A, Korkotian E, Holcman D (2011) Barriers to diffusion on dendrites and estimation of calcium spread following synaptic inputs. PLoS Comput Biol 7(10):e1002182PubMedCrossRefPubMedCentralGoogle Scholar
  8. Bliss TVP, Collingridge GL (1993) A synaptic model of memory: long-term potentiation in the hippocampus. Nature 361(6407):31–39PubMedCrossRefGoogle Scholar
  9. Blomberg F, Cohen RS, Siekevitz P (1977) The structure of postsynaptic densities isolated from dog cerebral cortex. II. Characterization and arrangement of some of the major proteins within the structure. J Cell Biol 74(1):204–225PubMedCrossRefPubMedCentralGoogle Scholar
  10. Bressloff PC, Earnshaw BA (2009) A dynamical corral model of protein trafficking in spines. Biophys J 96:1786–1802PubMedCrossRefPubMedCentralGoogle Scholar
  11. Carslaw HS, Jaeger JC (1959) Conduction of heat in solids. Oxford University Press, New YorkGoogle Scholar
  12. Cheviakov AF, Ward MJ, Straube R (2010) An asymptotic analysis of the mean first passage time for narrow escape problems: Part II: the sphere. Multiscale Model Simulat 8(3):836–870CrossRefGoogle Scholar
  13. Collins FC, Kimball GE (1949) Diffusion-controlled reaction rates. J Colloid Sci 4(7–8):425–437CrossRefGoogle Scholar
  14. Crank J (1975) The mathematics of diffusion, 2nd edn. Oxford University Press, LondonGoogle Scholar
  15. Crick F (1982) Do dendritic spines twitch? Trends Neurosci 5:44–46CrossRefGoogle Scholar
  16. Dao Duc K, Holcman D (2010) Threshold activation for stochastic chemical reactions in microdomains. Phys Rev E 81(4 Pt 1):041107CrossRefGoogle Scholar
  17. Dao Duc K, Holcman D (2012) Using default constraints of the spindle assembly checkpoints to estimate the associate chemical rates. BMC Biophys 5:1PubMedCrossRefPubMedCentralGoogle Scholar
  18. Eisenberg RS, Klosek MM, Schuss Z (1995) Diffusion as a chemical reaction: stochastic trajectories between fixed concentrations. J Chem Phys 102:1767–1780CrossRefGoogle Scholar
  19. Erban R, Chapman J (2007) Reactive boundary conditions for stochastic simulations of reaction diffusion processes. Phys Biol 4:16–28PubMedCrossRefGoogle Scholar
  20. Fischer M, Kaech S, Knutti D, Matus A (1998) Rapid actin-based plasticity in dendritic spines. Neuron 20(5):847–854PubMedCrossRefGoogle Scholar
  21. Flegg M, Chapman J, Erban R (2012) Two regime method for optimizing stochastic reaction-diffusion simulations. J R Soc Interface 9(70):859–868PubMedCrossRefPubMedCentralGoogle Scholar
  22. Franz B, Flegg M, Chapman J, Erban R (2013) Multiscale reaction-diffusion algorithms: PDE-assisted brownian dynamics. SIAM J Appl Math 73(3):1224–1247CrossRefGoogle Scholar
  23. Ghosh PK, Hanggi P, Marchesoni F, Nori F, Schmid G (2012) Brownian transport in corrugated channels with inertia. Phys Rev E 86(2):021112CrossRefGoogle Scholar
  24. Goldberg JH, Tamas G, Aronov D, Yuste R (2003) Calcium microdomains in aspiny dendrites. Neuron 40(4):807–821PubMedCrossRefGoogle Scholar
  25. Guerrier C, Holcman D (2014a) Activation of CaMKII in dendritic spines: a stochastic model (in preparation)Google Scholar
  26. Guerrier C, Holcman D (2014b) The dire strait time for hidden targets (in preparation)Google Scholar
  27. Hille B (2001) Ion channels of excitable membranes, 3rd edn. Sinauer Associates, MassachusettsGoogle Scholar
  28. Holcman D, Schuss Z (2004) Modeling calcium dynamics in dendritic spines. SIAM J Appl Math 65(3):1006–1026CrossRefGoogle Scholar
  29. Holcman D, Schuss Z (2005) Stochastic chemical reactions in microdomains. J Chem Phys 122:114710PubMedCrossRefGoogle Scholar
  30. Holcman D, Schuss Z (2011) Diffusion laws in dendritic spines. J Math Neuroscience 1:10CrossRefGoogle Scholar
  31. Holcman D, Schuss Z (2012) Brownian motion in dire straits. SIAM J Multiscale Model Simulat 1:10Google Scholar
  32. Holcman D, Schuss Z (2013) Control of flux by narrow passages and hidden targets in cellular biology. Phys Progr Report 76(7) 074601Google Scholar
  33. Holcman D, Schuss Z (2014) The narrow escape problem. SIAM Rev 56(2): 213–257Google Scholar
  34. Holcman D, Triller A (2006) Modeling synaptic dynamics driven by receptor lateral diffusion. Biophys J 91(7):2405–2415Google Scholar
  35. Holcman D, Schuss Z, Korkotian E (2004) Calcium dynamics in dendritic spines and spine motility. Biophys J 87:81–91PubMedCrossRefPubMedCentralGoogle Scholar
  36. Holcman D, Marchewka A, Schuss Z (2005) The survival probability of diffusion with trapping in cellular biology. Phys Rev E, Stat Nonlin Soft Matter Phys 72(3):031910CrossRefGoogle Scholar
  37. Holcman D, Hoze N, Schuss Z (2011) Narrow escape through a funnel and effective diffusion on a crowded membrane. Phys Rev E 84:021906CrossRefGoogle Scholar
  38. Holcman D, Daoduc K, Burrage K (2014) Successful delivery of pten in the cytoplasm escaping from micrornas degradation (pre-print)Google Scholar
  39. Holderith N, Lorincz A, Katona G, Rózsa B, Kulik A, Masahoki W, Nusser Z (2013) Release probability of hippocampal glutamatergic terminals scales with the size of the active zone. Nat Neurosci 15(7):988–997CrossRefGoogle Scholar
  40. Koch C (1999) Biophysics of computation, information processing in single neurons. Oxford University Press, New YorkGoogle Scholar
  41. Kochubey O, Lou X, Schneggenburger R (2011) Regulation of transmitter release by ca2+ and synaptotagmin: insight from large synapse. Trends Neuro 34(5):237–46CrossRefGoogle Scholar
  42. Korkotian E, Segal M (2006) Spatially confined diffusion of calcium in dendrites of hippocampal neurons revealed by flash photolysis of caged calcium. Cell Calcium 40(5–6):441–449PubMedCrossRefGoogle Scholar
  43. Korkotian E, Holcman D, Segal M (2004) Dynamic regulation of spine-dendrite coupling in cultured hippocampal neurons. Eur J Neuroscience 20(10):2649–2663CrossRefGoogle Scholar
  44. Lamm G, Schulten K (1983) Extended brownian dynamics. II. Reactive, nonlinear diffusion. J Chem Phys 78(5):2713–2734CrossRefGoogle Scholar
  45. Landau LD, Lifshitz EM (1975) Fluid mechanics. Pergamon Press, ElmsfordGoogle Scholar
  46. Lee SR, Escobedo-Lozoya J, Szatmari EM, Yasuda R (2009) Activation of CaMKII in single dendritic spines during long-term potentiation. Nature 458(7236):299PubMedCrossRefPubMedCentralGoogle Scholar
  47. Lisman J, Yasuda R, Raghavachari S (2012) Mechanisms of CaMKII action in long-term potentiation. Nat Rev Neurosci 13(3):169–182PubMedPubMedCentralGoogle Scholar
  48. Majewska A, Tashiro A, Yuste R (2000) Regulation of spine calcium dynamics by rapid spine motility. J Neurosci 20(22):8262–8268PubMedGoogle Scholar
  49. Malenka RC, Nicoll RA (1999) Long-term potentiation–a decade of progress? Science 285(5435):1870–1874PubMedCrossRefGoogle Scholar
  50. Matkowsky BJ, Schuss Z, Ben-Jacob E (1982) A singular perturbation approach to Kramers’ diffusion problem. SIAM J Appl Math 42(4):835–849CrossRefGoogle Scholar
  51. Matveev V, Zucker RS, Sherman A (2004) Facilitation through buffer saturation: constraints on endogenous buffering properties. Biophys J 86(5):2691–2709PubMedCrossRefPubMedCentralGoogle Scholar
  52. Monine MI, Haugh JM (2005) Reactions on cell membranes: comparison of continuum theory and Brownian dynamics simulations. J Chem Phys 123(7):074908PubMedCrossRefPubMedCentralGoogle Scholar
  53. Neher E (2010) Complexin: does it deserve its name ? Neuron Prev 68(5):803–806CrossRefGoogle Scholar
  54. Pontryagin LS, Andronovn AA, Vitt AA (1933) On the statistical treatment of dynamical systems. J Theor Exper Phys (Russian) 3:165–180Google Scholar
  55. Pontryagin LS, Andronov AA, Vitt AA (1989) On the statistical treatment of dynamical systems. Noise Nonlinear Dynamics 1:329–340Google Scholar
  56. Roux B, Prod’hom B, Karplus M (1995) Ion transport in the gramicidin channel: molecular dynamics study of single and double occupancy. Biophys J 68(3):876–892PubMedCrossRefPubMedCentralGoogle Scholar
  57. Sabatini BL, Maravall M, Svoboda K (2001) Ca2+ signalling in dendritic spines. Curr Opin Neurobiol 11(3):349–356PubMedCrossRefGoogle Scholar
  58. Schneggenburger R, Han Y, Kochubey O (2012) Ca2+ channels and transmitter release at active zone. Cell Calcium 52(3-4):199–207PubMedCrossRefGoogle Scholar
  59. Schuss Z (1980) Theory and applications of stochastic differential equations, Wiley series in probability and statistics. Wiley, New YorkGoogle Scholar
  60. Schuss Z (2010a) Diffusion and stochastic processes: an analytical approach. Springer, New YorkGoogle Scholar
  61. Schuss Z (2010b) Theory and applications of stochastic processes, an analytical approach, vol 170, Springer series on applied mathematical sciences. Springer, New YorkGoogle Scholar
  62. Schuss Z, Holcman D (2013) The narrow escape problem and its applications in cellular and molecular biology. SIAM Rev, SIREVGoogle Scholar
  63. Schuss Z, Holcman D (2014) Time scales of Diffusion for Molecular and Cellular processes, J.Phys A: Mathematical and Theoretical 47(17), 173001Google Scholar
  64. Schuss Z, Singer A, Holcman D (2007) The narrow escape problem for diffusion in cellular microdomains. Proc Natl Acad Sci USA 104(41):16098–16103PubMedCrossRefPubMedCentralGoogle Scholar
  65. Singer A, Schuss Z, Osipov A, Holcman D (2008) Partially reflected diffusion. SIAM J Appl Math 68:98–108CrossRefGoogle Scholar
  66. Svoboda K, Tank DW, Denk W (1996) Direct measurement of coupling between dendritic spines and shafts. Science 272(5262):716–719PubMedCrossRefGoogle Scholar
  67. Taflia A, Holcman D (2011) Estimating the synaptic current in a multiconductance ampa receptor model. Biophys J 101:781–792PubMedCrossRefPubMedCentralGoogle Scholar
  68. Tai K, Bond SD, MacMillan HR, Baker NA, Holst MJ, McCammon JA (2003) Finite element simulations of acetylcholine diffusion in neuromuscular junctions. Biophys J 84(4):2234–2241PubMedCrossRefPubMedCentralGoogle Scholar
  69. Zucker RS (1993) Calcium and transmitter release. J Physiol Paris 87(1):25–36PubMedCrossRefGoogle Scholar
  70. Zucker RS, Regehr WG (2002) Short-term synaptic plasticity. Annu Rev Physiol 64:355–405PubMedCrossRefGoogle Scholar
  71. Zwanzig R (1990) Diffusion-controlled ligand binding to spheres partially covered by receptors: an effective medium treatment. Proc Natl Acad Sci U S A 87:5856–5857PubMedCrossRefPubMedCentralGoogle Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Ecole Normale SupérieureInstitute for Biology, IBENS, INSERM 1024 and CNRS Group of Computational Biology and Applied MathematicsParisFrance
  2. 2.University Paris 6, Laboratoire Jacques-Louis LionsParisFrance
  3. 3.Department of NeurobiologyWeizmann Institute of scienceRehovotIsrael