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Electrodiffusion with Calcium-Activated Potassium Channels in Dendritic Spine

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Abstract

We investigate calcium signaling feedback through calcium-activated potassium channels of a dendritic spine by applying the immersed boundary method with electrodiffusion. We simulate the stochastic gating of such ion channels and the resulting spatial distribution of concentration, current, and membrane voltage within the dendritic spine. In this simulation, the permeability to ionic flow across the membrane is regulated by the amplitude of chemical potential barriers. With spatially localized ion channels, chemical potential barriers are locally and stochastically regulated. This regulation represents the ion channel gating with multiple subunits, the open and closed states governed by a continuous-time Markov process. The model simulation recapitulates an inhibitory action on voltage-sensitive calcium channels by the calcium-activated potassium channels in a stochastic manner as a non-local feedback loop. The model predicts amplified calcium influx with more closely placed channel complexes, proposing a potential mechanism of differential calcium handling by channel distributions. This work provides a foundation for future computer simulation studies of dendritic spine motility and structural plasticity.

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References

  • Balay S, Eijkhout V, Gropp WD, McInnes LC, Smith BF (1997) Efficient management of parallelism in object-oriented numerical software libraries. In: Arge E, Bruaset AM, Langtangen HP (eds) Modern software tools in scientific computing. Birkhauser Press, Basel, pp 163–202

    Chapter  Google Scholar 

  • Balay S, Abhyankar S, Adams M-F, Brown J, Brune P, Buschelman K, Dalcin L, Dener A, Eijkhout V, Gropp WD, Karpeyev D, Kaushik D, Knepley MG, May D-A, McInnes LC, Smith BF, Zampini S, Zhang H (2019) PETSc users manual, technical report, ANL-95/11 Revision 3.12, Argonne National Laboratory

  • Balay S, Abhyankar S, Adams M-F, Brown J, Bruen P, Buschelman K, Dalcin L, Dener A, Eijkhout V, Gropp WD, Karpeyev D, Kaushik D, Knepley MG, McInnes LC, Mills RT, Munson T, Rupp K, Sanan P, Smith BF, Zampini S, Zhang H, PETSc, http://www.mcs.anl.gov/petsc (2019)

  • Berkefeld H, Sailer CA, Bildl W, Rohde V, Thumfart J-O, Eble S, Klugbauer N, Reisinger E, Bischofberger J, Oliver D, Knaus H-G, Schulte U, Fakler B (2006) BK\({_{{\rm Ca}}}\)-CaV channel complexes mediate rapid and localized \(\text{ Ca}^{2+}\)-activated \(\text{ K}^+\) signaling. Science 314:615–620

    Article  Google Scholar 

  • Bloodgood BL, Sabatini BL (2007a) Nonlinear regulation of unitary synaptic signals by CaV2.3 voltage-sensitive calcium channels located in dendritic spines. Neuron 53:249–260

    Article  Google Scholar 

  • Bloodgood BL, Sabatini BL (2007b) \(\text{ Ca}^{2+}\) signalling in dendritic spines. Curr Opin Neurobiol 17:345–351

    Article  Google Scholar 

  • Bonhoeffer T, Yuste R (2002) Spine motility: phenomenology, mechanisms, and function. Neuron 35:1019–1027

    Article  Google Scholar 

  • Bosch M, Hayashi Y (2011) Structural plasticity of dendritic spines. Curr Opin Neurobiol 22:1–6

    Google Scholar 

  • Brunig I, Kaech S, Brinkhaus H, Oertner TG, Matus A (2004) Influx of extracellular calcium regulates actin-dependent morphological plasticity in dendritic spines. Neuropharmacology 47:669–676

    Article  Google Scholar 

  • Cartailler J, Holcman D (2019) Steady-state voltage distribution in three-dimensional cups-shaped funnels modeled by PNP. J Math Biol 79:155–185

    Article  MathSciNet  Google Scholar 

  • Cox DH (2014) Modeling a \(\text{ Ca}^{2+}\) channel/\(\text{ BK}_{{\rm Ca}}\) channel complex at the single-complex level. Biophys J 107:2797–2814

    Article  Google Scholar 

  • Falgout RD, Yang UM (2002) hypre: a library of high performance preconditioners. In: Sloot PMA, Tan CJK, Dongarra JJ, Hoekstra AG (eds) Computational science—ICCS 2002 Part III, Vol. 2331 of Lecture Notes in Computer Science, Springer-Verlag, pp 632–641, also available as LLNL Technical Report UCRL-JC-146175

  • Gardner CL, Jones JR, Baer SM, Crook SM (2015) Drift-diffusion simulation of the ephaptic effect in the triad synapse of the retina. J Comput Neuro 38:129–142

    Article  MathSciNet  Google Scholar 

  • Griffith T, Tsaneva-Atanasova K, Mellor JR (2016) Control of \(\text{ Ca}^{2+}\) influx and calmodulin activation by SK-channels in dendritic spines. PLoS Comput Biol 12:e1004949

    Article  Google Scholar 

  • Gutzmann JJ, Lin L, Hoffman DA (2019) Functional coupling of Cav2.3 and BK potassium channels regulates action potential repolarization and short-term plasticity in the mouse hippocampus. Front Cell Neurosci 34:5261–5272

    Google Scholar 

  • Hage TA, Khaliq ZM (2015) Tonic firing rate controls dendritic \(\text{ Ca}^{2+}\) signaling ad synaptic gain in substantia nigra dopamine neurons. J Neurosci 35:5823–5836

    Article  Google Scholar 

  • Harris KM, Kater SB (1993) Dendritic spines: cellular specializations imparting both stability and flexibility to synaptic function. Annu Rev Neurosci 17:341–371

    Article  Google Scholar 

  • He S, Wang Y-X, Petralia RS, Brenowitz SD (2014) Cholinergic modulation of large-conductance calcium-activated potassium channels regulates synaptic strength and spine calcium in cartwheel cells of the dorsal cochlear nucleus. J Neurosci 34:5261–5272

    Article  Google Scholar 

  • Holcman D, Yuste R (2015) The new nanophysiology: regulation of ionic flow in neuronal subcompartments. Nat Rev Neurosci 16:685–692

    Article  Google Scholar 

  • HYPRE: high performance preconditioners, http://www.llnl.gov/CASC/hypre

  • Jones SL, To M-S, Stuart GJ (2017) Dendritic small conductance calcium-activated potassium channels activated by action potentials supress EPSPs and gate spike-timing dependent synaptic plasticity. eLife 6:e30333

    Article  Google Scholar 

  • Kandel ER, Schwartz JH, Jessel TM (2000) Principles of neural science, 4th edn. McGraw-Hill Medical, New York

    Google Scholar 

  • Kim IH, Rossi MA, Aryal DK, Racz B, Kim N, Uezu A, Wang F, Wetsel WC, Weinberg R, Yin H, Soderling SH (2015) Spine pruning drives antipsychotic-sensitive locomotion via circuit control of striatal dopamine. Nat Neurosci 18:883–891

    Article  Google Scholar 

  • Koleske AJ (2013) Molecular mechanisms of dendrite stability. Nat Rev Neurosci 14:536–550

    Article  Google Scholar 

  • Kuznetsov AS, Kopell NJ, Wilson CJ (2006) Transient high-frequency firing in a coupled-oscillator model of the mesencephalic dopaminergic neuron. J Neurophysiol 95:932–947

    Article  Google Scholar 

  • Lagache T, Hanson A, Fairhall A, Yuste R (2020) Robust single neuron tracking of calcium imaging in behaving Hydra, bioRxiv, https://doi.org/10.1101/2020.06.22.165696

  • Lamprecht R, Farb CR, Rodrigues SM, LeDoux JE (2006) Fear conditioning drives prolifin into amygdala dendritic spines. Nat Neurosci 9:481–483

    Article  Google Scholar 

  • Lee P (2007) The immersed boundary method with advection-electrodiffusion, Ph.D. thesis. New York University, Courant Institute of Mathematical Sciences

  • Lee P, Griffith BE, Peskin CS (2010) The immersed boundary method for advection-electrodiffusion with implicit timestepping and local mesh refinement. J Comp Phys 229:5208–5227

    Article  MathSciNet  Google Scholar 

  • Lee P, Sobie EA, Peskin CS (2013) Computer simulation of voltage sensitive calcium ion channels in a dendritic spine. J Theor Biol 338:87–93

    Article  MathSciNet  Google Scholar 

  • Lujan R, Aguado C, Ciruela F, Arus XM, Martin-Belmonte A, Alfaro-Ruiz R, Martinez-Gomez J, de la Ossa L, Watanabe M, Adelman JP, Shigemoto R, Fukazawa Y (2018) SK2 channels associate with \(\text{ mGlu}_{1\alpha }\) receptors and \(\text{ Ca}_{{V}}\)2.1 channnels in purkinje cells. Front Cell Neurosci 12:1–16

    Article  Google Scholar 

  • Mori Y, Jerome JW, Peskin CS (2007) A three-dimensional model of cellular electrical activity. Bull Inst Math Acad 2:367–390

    MathSciNet  MATH  Google Scholar 

  • Peskin CS (2002) The immersed boundary method. Acta Numer 11:479–517

    Article  MathSciNet  Google Scholar 

  • Peskin CS (2000) Mathematical aspects of neurophysiology, lecture note, Courant Institute of Mathematical Sciences, New York University

  • Piochon C, Kano M, Hansel C (2016) LTD-like molecular pathways in developmental synaptic pruning. Nat Neurosci 19:1299–1310

    Article  Google Scholar 

  • Rice RA, Spangenberg EE, Yamate-Morgan H, Lee RJ, Arora RPS, Hernandez MX, Tenner AJ, West BL, Green KN (2015) Elimination of microglia improves functional outcomes following extensive neuronal loss in the hippocampus. J Neurosci 35:9977–9989

    Article  Google Scholar 

  • Sabatini BL, Svoboda K (2000) Analysis of calcium channels in single spines using optical fluctuation analysis. Nature 408:589–593

    Article  Google Scholar 

  • Sabatini BL, Oertner TG, Svoboda K (2002) The life cycle of \(\text{ Ca}^{2+}\) ions in dendritic spines. Neuron 33:439–452

    Article  Google Scholar 

  • Sah P, Faber ESL (2002) Channels underlying neuronal calcium-activated potassium currents. Prog Neurobiol 66:345–353

    Article  Google Scholar 

  • Sheng M, Kim M (2002) Postsynaptic signaling and plasticity mechanisms. Science 298:776–780

    Article  Google Scholar 

  • Shepherd GM (1996) The dendritic spine: a multifunctional integrative unit. J Neurophysiol 75:2197–2210

    Article  Google Scholar 

  • Yasuda R, Sabatini BL, Svoboda K (2003) Plasticity of calcium channels in dendritic spines. Nat Neurosci 6:948–955

    Article  Google Scholar 

  • Tada T, Seung M (2006) Molecular mechanisms of dendritic spine morphogenesis. Curr Opin Neurobiol 16:95–101

    Article  Google Scholar 

Download references

Acknowledgements

The author appreciates careful comments from Charles Peskin.

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Correspondence to Pilhwa Lee.

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Lee, P. Electrodiffusion with Calcium-Activated Potassium Channels in Dendritic Spine. Bull Math Biol 83, 30 (2021). https://doi.org/10.1007/s11538-020-00854-4

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