Bulletin of Mathematical Biology

, Volume 80, Issue 1, pp 130–150 | Cite as

Slow Passage Through a Hopf Bifurcation in Excitable Nerve Cables: Spatial Delays and Spatial Memory Effects

Original Article
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Abstract

It is well established that in problems featuring slow passage through a Hopf bifurcation (dynamic Hopf bifurcation) the transition to large-amplitude oscillations may not occur until the slowly changing parameter considerably exceeds the value predicted from the static Hopf bifurcation analysis (temporal delay effect), with the length of the delay depending upon the initial value of the slowly changing parameter (temporal memory effect). In this paper we introduce new delay and memory effect phenomena using both analytic (WKB method) and numerical methods. We present a reaction–diffusion system for which slowly ramping a stimulus parameter (injected current) through a Hopf bifurcation elicits large-amplitude oscillations confined to a location a significant distance from the injection site (spatial delay effect). Furthermore, if the initial current value changes, this location may change (spatial memory effect). Our reaction–diffusion system is Baer and Rinzel’s continuum model of a spiny dendritic cable; this system consists of a passive dendritic cable weakly coupled to excitable dendritic spines. We compare results for this system with those for nerve cable models in which there is stronger coupling between the reactive and diffusive portions of the system. Finally, we show mathematically that Hodgkin and Huxley were correct in their assertion that for a sufficiently slow current ramp and a sufficiently large cable length, no value of injected current would cause their model of an excitable cable to fire; we call this phenomenon “complete accommodation.”

Keywords

Hopf bifurcation Reaction–diffusion Delayed bifurcation WKB FitzHugh–Nagumo Hodgkin–Huxley 

Mathematics Subject Classification

74H60 34E20 35K57 

References

  1. Baer SM, Gaekel EM (2008) Slow acceleration and deacceleration through a Hopf bifurcation: power ramps, target nucleation, and elliptic bursting. Phys Rev E 78:036205MathSciNetCrossRefGoogle Scholar
  2. Baer SM, Rinzel J (1991) Propagation of dendritic spikes mediated by excitable spines: a continuum theory. J Neurophysiol 65:874–890CrossRefGoogle Scholar
  3. Baer SM, Erneux T, Rinzel J (1989) The slow passage through a Hopf bifurcation: delay, memory effects, and resonance. SIAM J Appl Math 49:55–71MathSciNetCrossRefMATHGoogle Scholar
  4. Bilinsky LM (2012) Dynamic Hopf bifurcation in spatially extended excitable systems from neuroscience. Ph.D. thesis, Arizona State University, Tempe, AZGoogle Scholar
  5. Chen YX, Kolokolnikov T, Tzou J, Gai CY (2015) Patterned vegetation, tipping points, and the rate of climate change. Eur J Appl Math 26:945–958MathSciNetCrossRefMATHGoogle Scholar
  6. Cooley JW, Dodge FA (1966) Digital computer solutions for excitation and propagation of nerve impulse. Biophys J 6:583–599CrossRefGoogle Scholar
  7. Doedel EJ (1981) AUTO, a program for the automatic bifurcation analysis of autonomous systems. Cong Numer 30:265–384MathSciNetMATHGoogle Scholar
  8. du Bois-Reymond E (1849) Untersuchungen ueber thierische Elektricitaet. G. ReimerGoogle Scholar
  9. Hayes MG, Kaper TJ, Szmolyan P, Wechselberger M (2015) Geometric desingularization of degenerate singularities in the presence of fast rotation: a new proof of known results for slow passage through Hopf bifurcations. Indagationes Math.  https://doi.org/10.1016/j.indag.2015.11.005 MATHGoogle Scholar
  10. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol Lond 117:500–544CrossRefGoogle Scholar
  11. Jakobsson E, Guttman R (1980) The standard Hodgkin–Huxley model and squid axons in reduced external \(C^{++}\) fail to accommodate to slowly rising currents. Biophys J 31:293–298CrossRefGoogle Scholar
  12. Keener J, Sneyd J (1998) Mathematical physiology. Springer, BerlinMATHGoogle Scholar
  13. Kovalsky Y, Amir R, Devor M (2008) Subthreshold oscillations facilitate neuropathic spike discharge by overcoming membrane accommodation. Exp Neurol 210:194–206CrossRefGoogle Scholar
  14. Kuske R (1999) Probability densities for noisy delay bifurcations. J Stat Phys 96:797–816MathSciNetCrossRefMATHGoogle Scholar
  15. Kuske R, Baer SM (2002) Asymptotic analysis of noise sensitivity in a neuronal burster. Bull Math Biol 64:447–481CrossRefMATHGoogle Scholar
  16. Lester RAJ, Clements JD, Westbrook GL, Jahr CE (1990) Channel kinetics determine the time course of NMDA receptor-mediated synaptic currents. Nature 346:565–567CrossRefGoogle Scholar
  17. Neishtadt AI (1987) Persistence of stability loss for dynamical bifurcations I. Differ Equ 23:1385–1391MATHGoogle Scholar
  18. Rinzel J, Baer SM (1988) Threshold for repetitive activity for a slow stimulus ramp. Biophys J 54:551–555CrossRefGoogle Scholar
  19. Rinzel J, Keener JP (1983) Hopf bifurcation to repetitive activity in nerve. SIAM J Appl Math 43:907–921MathSciNetCrossRefMATHGoogle Scholar
  20. Segev I, Rall W (1988) Computational study of an excitable dendritic spine. J Neurophysiol 60:499–523CrossRefGoogle Scholar
  21. Sobel SG, Hastings HM, Field RJ (2006) Oxidation state of BZ reaction mixtures. J Phys Chem A 110:5–7CrossRefGoogle Scholar
  22. Strizhak P, Menzinger M (1996) Slow passage through a supercritical hopf bifurcation: time-delayed response in the belousov-zhabotinsky reaction in a batch reactor. J Chem Phys 105:10905–10910CrossRefGoogle Scholar
  23. Su J (1994) On delayed oscillation in nonspatially uniform Fitzhugh–Nagumo equation. J Differ Equ 110:38–52CrossRefMATHGoogle Scholar
  24. Su J (2003) Effects of noise on elliptic bursters. Nonlinearity 17:133–157MathSciNetCrossRefMATHGoogle Scholar
  25. Tzou JC, Ward MJ, Kolokolnikov T (2015) Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction-diffusion systems. Phys D 290:24–43MathSciNetCrossRefMATHGoogle Scholar
  26. Vallbo AB (1964) Accommodation related to inactivation of sodium permeability in single myelinated nerve fibres from Xenopus laevis. Acta Physiol Scand 61:429–444Google Scholar
  27. Verzi DW, Rheuben MB, Baer SM (2005) Impact of time-dependent changes in spine density and spine shape on the input–output properties of a dendritic branch: a computational study. J Neurophysiol 93:2073–2089CrossRefGoogle Scholar
  28. Vo T, Tabak J, Bertram R, Wechselberger M (2014) A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells. J Comput Neurosci 36:259–278MathSciNetCrossRefGoogle Scholar
  29. Younghae D, Lopez JM (2013) Slow passage through multiple bifurcation points. Discrete Contin Dyn Syst Ser B 18:95–107MathSciNetMATHGoogle Scholar
  30. Zhou Y (1998) Unique wave front for dendritic spines with Nagumo dynamics. Math Biosci 148:205–225MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

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