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

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 (te...

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Bibliographic Details
Published in:Bulletin of mathematical biology 2018, Vol.80 (1), p.130-150
Main Authors: Bilinsky, L. M., Baer, S. M.
Format: Article
Language:English
Subjects:
Cables
Cell Biology
Continuum modeling
Delay
Dendritic spines
Diffusion effects
Diffusion rate
Hopf bifurcation
Life Sciences
Mathematical and Computational Biology
Mathematical models
Mathematics
Mathematics and Statistics
Memory tasks
Numerical methods
Original Article
Oscillations
Spatial analysis
Spatial memory
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Summary: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.”
ISSN:0092-8240
1522-9602
DOI:10.1007/s11538-017-0366-2