Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

Solitary Waves in a Model of Dendritic Cable with Active Spines

S. Coombes and P. C. Bressloff
SIAM Journal on Applied Mathematics
Vol. 61, No. 2 (Aug., 2000), pp. 432-453
Stable URL: http://www.jstor.org/stable/3061734
Page Count: 22
  • More info
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
Solitary Waves in a Model of Dendritic Cable with Active Spines
Preview not available

Abstract

We consider a continuum model of dendritic spines with active membrane dynamics uniformly distributed along a passive dendritic cable. By considering a systematic reduction of the Hodgkin-Huxley dynamics that is valid on all but very short time-scales we derive two-dimensional and one-dimensional systems for excitable tissue, both of which may be used to model the active processes in spine-heads. In the first case the coupling of the spine-head dynamics to a passive dendritic cable via a direct electrical connection yields a model that may be regarded as a simplification of the Baer and Rinzel cable theory of excitable spiny nerve tissue [J. Neurophysiology, 65 (1991), pp. 874-890]. This model is computationally simple with few free parameters. Importantly, as in the full model, numerical simulation illustrates the possibility of a traveling wave. We present a systematic numerical investigation of the speed and stability of the wave as a function of physiologically important parameters. A further reduction of this model suggests that active spine-head dynamics may be modeled by an all-or-none type process which we take to be of the integrate-and-fire (IF) type. The model is analytically tractable allowing the explicit construction of the shape of traveling waves as well as the calculation of wave speed as a function of system parameters. In general a slow and fast wave are found to coexist. The behavior of the fast wave is found to closely reproduce the behavior of the wave seen in simulations of the more detailed model. Importantly a linear stability theory is presented showing that it is the faster of the two solutions that are stable. Beyond a critical value the speed of the stable wave is found to decrease as a function of spine density. Moreover, the speed of this wave is found to decrease as a function of the strength of the electrical resistor coupling the spine-head and the cable, such that beyond some critical value there is propagation failure. Finally, we discuss the importance of a model of passive electrical cable coupled to a system of IF units for physiological studies of branching dendritic tissue with active spines.

Page Thumbnails

  • Thumbnail: Page 
432
    432
  • Thumbnail: Page 
433
    433
  • Thumbnail: Page 
434
    434
  • Thumbnail: Page 
435
    435
  • Thumbnail: Page 
436
    436
  • Thumbnail: Page 
437
    437
  • Thumbnail: Page 
438
    438
  • Thumbnail: Page 
439
    439
  • Thumbnail: Page 
440
    440
  • Thumbnail: Page 
441
    441
  • Thumbnail: Page 
442
    442
  • Thumbnail: Page 
443
    443
  • Thumbnail: Page 
444
    444
  • Thumbnail: Page 
445
    445
  • Thumbnail: Page 
446
    446
  • Thumbnail: Page 
447
    447
  • Thumbnail: Page 
448
    448
  • Thumbnail: Page 
449
    449
  • Thumbnail: Page 
450
    450
  • Thumbnail: Page 
451
    451
  • Thumbnail: Page 
452
    452
  • Thumbnail: Page 
453
    453