Abstract
Quantification of a chaotic system can be made by calculating the correlation dimension (D2) of the data that the system generates (Packard et al., 1980). The D2 algorithm, however, requires stationarity of the generator, a feature that biological data rarely reflect (Mayer-Kress et al., 1988). So we developed the “point correlation dimension” (PD2), an algorithm that accurately tracks D2 in linked data of different dimensions (Carpeggiani et al., 1991). We now present a mathematical argument that, for stationary data, individual PD2s converge to D2 and we demonstrate that the algorithm rejects contributions made by bursts of noise. Data were obtained from the surface of the olfactory bulb of the conscious rabbit (64 electrodes, 640 Hz each, 1.3 sec epochs) before and after presentation of a novel or habituated odor. D2 could be calculated in only 1 of 10 novel-odor trials, whereas PD2 could be calculated in all. Both algorithms indicated that a novel odor evokes a spatially uniform dimensional increase. The PD2 uniquely exhibited the dimensional decreases that occur during inspiration and the gradients of mean dimension present during the nonstimulated control state. These control gradients remained unchanged without odor experience, but showed spatially specific PD2 increases following odor habituation. It is interpreted that, 1) the PD2 issensitive, accurate, and appropriate for dimensional assessment of biological data, 2) that during analysis of unfamiliar information a singleglobal process is transiently evoked in the neuropil, and 3) after experience multiplespatially specific processes tonically map the sites of learning.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adrian, E.D. (1950). Sensory discrimination. British medical bulletin, 6: 330–333.
Babloyantz, A. (1985). Strange attractors in the dynamics of brain activity. In H. Haken (Ed.), Complex Systems—operational approaches in neurobiology, physics, and computers (pp. 116–122). Berlin: Springer.
Babloyantz, A. (1988). Evidence of chaotic dynamics of brain activity during the sleep cycle. In G. Mayer-Kress (Ed.), dimensions and entropies in chaotic systems (pp. 241–245). Berlin: Springer.
Baloyantz, A. & Destexhe, A. (1986). Low-dimensional chaos in an instance of epilepsy. Proceedings of the National Academy of Sciences, USA, 83: 3513–3517.
Babloyantz, A. (1990). Chaotic dynamics in brain activity. In E. Basar, (Ed.), Chaos in brain function (pp. 42–48.). Berlin: Springer.
Basar, E. (1990).Chaos in Brain Function. Springer-Verlag, New York.
Elbert, T. & Rockstroh, B. (1987). Threshold regulation—a key to the understanding of the combined dynamics of EEG and event-related potentials.Journal of Psychophysiology, 4: 317–333.
Farmer, J.D., Ott, E. & Yorke, J.A. (1983). Dimension of chaotic attractors. Physica 7D, 153–180.
Freeman, W.J., & Schneider, W. (1982). Changes in spatial pattern of rabbit olfactory EEG with conditioning to odors.Psychophysiology, 19: 44–56.
Graf, K.E. & Elbert, T. (1990). Dimensional analysis of the waking EEG. In E. Basar (Ed.), Chaos in brain function. (pp. 135–152). Berlin: Springer.
Grajski, K.A. & Freeman, W.J. (1989). Spatial EEG correlates of nonassociative and associative olfactory learning in rabbits.Behavioral Neuroscience, 103: 790–804.
Grassberger, P., & Procaccia, I. (1983). Measuring the strangeness of strange attractors.Physica. 9D: 183–208.
Gray, C.M., Freeman, W.J., & Skinner, J.E. (1986). Chemical dependencies of learning in the rabbit olfactory bulb: Acquisition of the transient spatial-pattern change depends on norepinephrine.Behavioral Neuroscience, 100: 585–596.
Gray, C.M. Konig, P., Engel, A.K. & Singer, W. (1989). Oscillatory response in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties.Nature, 338: 334–337.
Gray, C.M. & Singer, W. (1989). Stimulus specific neuronal oscillations in orientation coloumns of cat visual cortex. Proceedings of the National Academy of Sciences, USA, 86: 1968–1702.
Haken, H. (1983).Advanced synergetics. Berlin: Springer.
Hopkins, W.F., & Johnston, D. (1988). Noradrenergic enhancement of long-term potentiation at mossy fiber synapses in the hippocampus.Journal of Neurophysiology. 59: 667–687.
Kasamatsu, T., Watanabe, K., & Heggelund, P. (1985). Plasticity in cat visual cortex respotred by electrical stimulation of the locus coeruleus.Neuroscience Research. 2: 365–386.
Loughlin, S.E., Foote S.L., & Bloom F.E. (1986). Efferent projections of nucleus locus coeruleus: topographic organization of cells of origin demonstrated by three-dimensional reconstruction. Neuroscience, 18: 291–306.
Mandelbrot, B.B. (1983).The fractal geometry of nature. New York: Freeman and Co.
Mayer-Kress, G., Yates, F.E., Benton, L., Keidel, M., Tirsch, W., Poppl, S.J. & Geist, K. (1988). Dimensional analysis of non-linear oscillations in brain, heart and muscle.Mathematical Biosciences. 90: 155–182.
Molnar M. & Skinner J. (1992). Low-dimensional chaos in event-related potentials.International Journal of Neuroscience. (in press).
Packard, N.H., Crutchfield, J.P., Farmer, J.D. & Shaw, R.S. (1980) Geometry from a time series.Physical Review Letters. 45: 712–716.
Rapp, P.E., Bashore, T.R., Martineire, J.M., Albano, A.M., Zimmerman, I.D. & Mees, A.I. (1989). Dynamics of brain electrical activity.Brain Topography, 2: 99–118.
Rapp, P.E., Bashore, T.R., Zimmerman, I.D., Martinerie, J.M., Albano, A.M. & Mees, A.I. (1990). Dynamical characterization of brain electrical activity. In S. Krasner (Ed.), The ubiquity of chaos (pp. 10–22). American Association for the Advancement of Sciences. Washington, DC.
Roschke, J. & Basar, E. (1990). The EEG is not a simple noise: strange attractors in intracranial structures. In E. Basar (Ed.), Chaos in brain function (pp. 49–62). Berlin: Springer.
Skarda, C.A. & Freeman, W.J. (1987). How brains make chaos in order to make sense of the world.Behavioral and Brain Sciences, 10: 161–195.
Skinner, J.E., Carpeggiani, C., Landisman, C.E. & Fulton, K.W. (1991a). The correlation-dimension of the heartbeat is reduced by myocardial ischemia in conscious pigs.Circulation Research, 68: 966–976.
Skinner, J.E.; Goldberger, A.L., Mayer-Kress, G. & Ideker, R.E. (1990a). Chaos in the heart: implications for clinical cardiology.Biotechnology, 8: 1018–1024.
Skinner, J.E., Martin, J.L., Landisman, C.E., Mommer, M.M., Fulton, K., Mitra, M., Burton, W.D., & Saltzberg, B. (1990b). Chaotic attractors in a model of neocortex: dimensionalities of olfactory bulb surface potentials are spatially uniform and event-related. In Brain Dynamics Progress and Perspectives. Edited by E. Basar and T.H. Bullock. Springer-Verlag, Berlin, pp. 119–134.
Skinner, J.E., Mitra, M. & Fulton, K. (1991b). Low-dimensional chaos in a simple biological model of neocortex: implications for cardiovascular and cognitive disorders. In J.G. Carlson and A.R. Seifert (Eds.), An international perspective on self-regulation and health (pp. 95–117). New York: Plenum.
Skinner, J.E., Pratt, C.M., & Vybiral, T. (1992). Low-dimensional chaos in heartbeat intervals may predict sudden cardiac death. (in press).
Skinner J.E., Reed, J.C., Welch, K.M.A., & Nell, J.H. (1978). Cutaneous shock produces correlated shifts in slow potential amplitude and cyclic 3′, 5′-adenosine monophosphate level in the parietal coertex of the conscious rat.J. Neurochem, 30: 699–704.
Schuster, H.G. (1988).Deterministic chaos. VCH. Weinheim.
Takens, F. (1981) Detecting strange attractors in turbulence. Lecture Notes in Mathematics, 898: 366–381.
Takens, F. (1985). On the numerical determination of the dimension of an attractor. Lecture Notes in Mathematics, 1125: 99–106.
Theiler, J. (1988). Quantifying chaos: practical estimation of the correlation dimension. Thesis. California Institute of Technology, Pasadena, California.
Wilson, D.A., & Leon, M. (1988). Spatial patterns of olfactory bulb single-unit responses to learned olfactory cues in young rats.Journal of Neurophysiology, 59: 1770–1782.
Author information
Authors and Affiliations
Additional information
Grant Support: National Institutes of Health, HL 31164 and NS27745
The authors of the following article, Mima Mitra and James E. Skinner, have taken advantage of a recent technical revelation showing that cognative activity (in this instance evaluating odors) is associated with small event-related potentials in the electroencephalogram. These aperiodic brain waves were previously thought to represent “noise” but now have been identified as low dimensional “chaos” reflecting precise patterns of low intensity impulses involved in the central processing and storage of sensory input. This work contributes importantly to the effort to document how the central nervous system assesses and stores sensory information and how the pattern of the sensory processing reflects on the significance of the stimulus (novelty in this instance).
The Mitra-Skinner article and the article by Bjom Nordenstrom that precedes it serve to remind us that a strong base in physics and mathematics is essential to the understanding of human biology and hence to education in medicine. Each wave of new knowledge about enzymes, antibodies, neurotransmitters, growth factors, and about secretion, absorption, traffic across cell membranes, antigen-antibody, interactions, genetics, sensation, memory, language, and indeed all the functions that characterize life and conciousness, the more we need to know about subtle physical forces and small electrical charges.
Rights and permissions
About this article
Cite this article
Mitra, M., Skinner, J.E. Low-dimensional chaos maps learning in a model neuropil (olfactory bulb). Integrative Physiological and Behavioral Science 27, 304–322 (1992). https://doi.org/10.1007/BF02691166
Issue Date:
DOI: https://doi.org/10.1007/BF02691166