Abstract
From a theoretical point of view we investigate cortical activity patterns causing dynamic visual hallucinations. For this reason we analyze an oscillatory instability of the dynamics ofthe activator-inhibitor model of Ermentrout and Cowan.Such an oscillatory instability occurs as a result of several disease mechanisms.We explicitly derive the order parameter equation. By means of the averaging theorem, we obtain the averaged order parameter equation.The latter enables us to determine stable and unstable bifurcating cortical activity patterns analytically in lowest order.Depending on model parameters as well as on initial conditions two types of cortical activity patterns occur: travelling waves and ’blinking rolls‘, i.e.standing waves oscillating with the same frequency and with a phase shift of π/2. In contrast to cortical activitypatterns caused by non-oscillatory instabilitiesanalyzed by Ermentrout and Cowan and by the author the travelling waves and the blinking rolls lead to a variety of dynamic visual hallucinations which are discussed indetail.
Keywords: Activator-inhibitor network, Pattern formation, Bifurcation, Center manifold, Order parameter equation, Blinking rolls, Visual hallucinations.
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References
- 1.Arnold V.I. Geometrical methods in the theory of ordinary differential equations. Berlin, Heidelberg, New York: Springer; 1983. [Google Scholar]
- 2.Bestehorn M., Friedrich R., Haken H. Modulated traveling waves in nonequilibrium systems: the ‘blinking state’. Z. Phys. B. 1989;77:151–155. [Google Scholar]
- 3.Coddington E.A., Levinson N. Ordinary differential equations. New York: McGraw-Hill; 1955. [Google Scholar]
- 4.Cowan J.D. Some remarks on channel bandwidths for visual contrast detection. Neurosci. Res. Bull. 1977;15:492–515. [Google Scholar]
- 5.Cowan J.D. Brain mechanisms underlying visual hallucinations. In: Paines D., editor. Emerging syntheses in science. New York: Addison-Wesley; 1987. pp. 123–131. [Google Scholar]
- 6.Ermentrout G.B., Cowan J. A mathematical theory of visual hallucination patterns. Biol. Cybern. 1979;34:137–150. doi: 10.1007/BF00336965. [DOI] [PubMed] [Google Scholar]
- 7.Fineberg J., Moses E., Steinberg V. Spatially and temporally modulated traveling-wave pattern in convecting binary mixtures. Phys. Rev. Lett. 1988;61:838–841. doi: 10.1103/PhysRevLett.61.838. [DOI] [PubMed] [Google Scholar]
- 8.Guckenheimer J., Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Berlin, Heidelberg, New York: Springer; 1983. [Google Scholar]
- 9.Haken H. Synergetics: An introduction. Berlin, Heidelberg, New York: Springer; 1977. [Google Scholar]
- 10.Haken H. Advanced synergetics. Berlin, Heidelberg, New York: Springer; 1983. [Google Scholar]
- 11.Haken H. Synergetic computers and cognition. Berlin, Heidelberg, New York: Springer; 1991. [Google Scholar]
- 12.Haken H. Principles of brain functioning: A synergetic approach to brain activity, behaviour and cognition. Berlin, Heidelberg, New York: Springer; 1995. [Google Scholar]
- 13.Hale J.K. Ordinary differential equations. New York: Wiley; 1969. [Google Scholar]
- 14.Horowitz M.J., Adams J.E., Rutkin B.B. Evoked hallucinations in epilepsy. Psychiatr. Speculator. 1967;11:4. [Google Scholar]
- 15.Kelley A. The stable, center-stable, center, center-unstable and unstable manifolds. J. Diff. Equ. 1967;3:546–570. [Google Scholar]
- 16.Klüver, H.: Mescal and mechanisms of hallucination, University of Chicago Press, 1967.
- 17.Markus M., Schepers H. Pattern formation in neural activator–inhibitor networks. In: Müller S.C., Plesser T., editors. Contributions to the Dortmunder Dynamische Woche June 1992. Dortmund: Projekt-Verlag; 1992. [Google Scholar]
- 18.Murray J.D. Mathematical biology. Berlin, Heidelberg, New York: Springer; 1989. [Google Scholar]
- 19.Pego R.L., Weinstein M.I. Asymptotic stability of solitary waves. Comm. Math. Phys. 1994;22:1–50. [Google Scholar]
- 20.Pliss V. Principal reduction in the theory of stability of motion. Izv. Akad. Nauk. SSSR Math. Ser. 1964;28:1297–1324. [Google Scholar]
- 21.Sattinger D.H. On the stability of waves of nonlinear parabolic systems. Advances in Mathematics. 1976;22:312–355. [Google Scholar]
- 22.Schwartz E. Spatial mapping in the primate sensory projection: Analytic structure and relevance to perception. Biol. Cybernetics. 1977;25:181–194. doi: 10.1007/BF01885636. [DOI] [PubMed] [Google Scholar]
- 23.Siegel R.K., West L.J., editors. Hallucinations. New York: Wiley; 1975. [Google Scholar]
- 24.Tass P. Cortical pattern formation during visual hallucinations. J. Biol. Phys. 1995;21:177–210. doi: 10.1023/A:1004990707739. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 25.Winfree A.T. The geometry of biological time. Berlin, Heidelberg, New York: Springer; 1980. [Google Scholar]