Noise-Induced Spiral Waves in Astrocyte Syncytia Show Evidence
of Self-Organized Criticality
Peter Jung1,
Ann Cornell-Bell2,
Kathleen Shaver Madden3, and
Frank Moss4
1 School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332; 2 Viatech Imaging, Ivoryton, Connecticut 06442; 3 Foundation for International Nonlinear Dynamics, Bethesda, Maryland 20816; and 4 Center for Neurodynamics, University of Missouri, St. Louis, Missouri 63121
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ABSTRACT |
Jung, Peter, Ann Cornell-Bell, Kathleen Shaver Madden, and Frank Moss. Noise-induced spiral waves in astrocyte syncytia show evidence of self-organized criticality. J. Neurophysiol. 79: 1098-1101, 1998. Long range (a few centimeters), long lived (many seconds), spiral chemical waves of calcium ions (Ca2+) are observed in cultured networks of glial cells for normal concentrations of the neurotransmitter kainate. A new method for quantitatively measuring the spatiotemporal size of the waves is described. This measure results in a power law distribution of wave sizes, meaning that the process that creates the waves has no preferred spatial or temporal (size or lifetime) scale. This power law is one signature of self-organized critical phenomena, a class of behaviors found in many areas of science. The physiological results for glial networks are fully supported by numerical simulations of a simple network of noisy, communicating threshold elements. By contrast, waves observed in astrocytes cultured from human epileptic foci exhibited radically different behavior. The background random activity, or "noise", of the network is controlled by the kainate concentration. The mean rate of wave nucleation is mediated by the network noise. However, the power law distribution is invariant, within our experimental precision, over the range of noise intensities tested. These observations indicate that spatially and temporally coherent Ca2+ waves, mediated by network noise may play and important role in generating correlated neural activity (waves) over long distances and times in the healthy vertebrate central nervous system.
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INTRODUCTION |
Self-organized criticality (SOC) is an established dynamical behavior (Bak et al. 1987
) of numerous physical systems, including avalanches and earthquakes (Bak 1996), magnetic noise (Spasojevic et al. 1996
), and an economic index (Mantegna and Stanley 1995
) to name only a few. A common feature of these processes are power law distributions of event sizes and/or lifetimes. By contrast, processes with characteristic times, for example, the discharge of a capacitor through a resistor or the decay of a population of radioactive nuclides (and innumerable similar phenomena, for example those described by linear differential equations) proceed according to exponential laws. Systems showing SOC tend naturally toward a critical state where power law scaling is the rule. Moreover SOC arises in some complex systems that are far from equilibrium. Complexity means that an enormously large number of events are possible, the size or lifetime of any particular event being unpredictable. Thus a fundamental randomness, or noise, is characteristic of SOC. Far from equilibrium means that the systems are continuously driven from the external environment, for example, the relentless build up of strain along a fault line, ultimately leading to an earthquake (of unpredictable size to occur at an unpredictable time).
The human brain is a complex object that operates far from equilibrium, owing to an incessant stream of stimuli to "think about." Thus one might speculate that brain function may also be an example of SOC, and indeed, interesting and suggestive numerical simulations have been reported (Bak 1996; Stassinopoulos and Bak 1995
).
We report here the first experimental evidence supporting a role for SOC in a vertebrate CNS preparation. In a culture of glial cells, time-lapse images of Fluo-3-AM fluorescence, obtained during dose-response studies with kainate, revealed well-defined spiral and other waves. The waves were born in the background network noise, grew and propagated some distance before dying again in the noise. The spatiotemporal size, s, of a single wave is comprised not only of its physical size at any given time, but also of its growth during its lifetime and its propagation distance. Motivated by our physiological spiral wave observations, one of us developed a novel statistical analysis suitable for quantitatively characterizing such waves (Jung 1997
). The analysis yields size distributions, p(s)
s
a, which, for both numerical and biological data, are accurately described by power-laws with quite similar exponents, a. In the healthy CNS, long-range signaling by calcium ions probably occurs via such coherent waves.
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METHODS |
Experimental methods
An expanded view of astrocyte physiology implicates astrocyte syncytia in long-distance signaling systems based on propagating, intercellular waves of calcium ions (Ca2+). Waves are revealed with the fluoroprobe, Fluo-3-AM, and time-lapse imaging (Boitano et al. 1992
; Cornell-Bell et al. 1990
; Cornell-Bell and Finkbeiner 1991
; Jensen and Chiu 1991
; McCarthy and Salam 1991) and may carry information. Previous models of wave propagation require a Ca2+-dependent autocatalytic step and diffusion of inositol trisphosphate (IP3) and/or Ca2+ through gap junctions into neighboring cells (Berridge 1993
; Finkbeiner 1992
). In contrast, the propagating waves elicited by the excitatory amino acid kainate (1-100 mM) involve the Na+/ Ca2+ exchanger and a different mechanism. Kainate binds to the ionotropic receptor and extracellular Na+ ions enter an astrocyte upsetting normal Na+/ Ca2+ exchange (Goldman et al. 1994
; Golovina and Blaustein 1997). The Na+ concentration increases with time (>100 s) eventually reversing the Na+/Ca2+ exchanger. This favors the exchange of extracellular Ca2+ for intracellular Na+, which generates a propagating wave that conserves the Na+/ Ca2+ imbalance. The waves are sensitive to extracellular neurotransmitter levels, suggesting that they carry information (Smith 1992
) about the excitability of the astrocyte syncitia over long distances (A. H. Cornell-Bell, R. Villalba, W. T. Kim, M. G. Rioult, and V. Trinkaus-Randall, unpublished manuscript).

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| FIG. 1.
Two sequences of difference-pictures between consecutive frames. Time advances in both series from top left to bottom right (8 s, total). A darker color encodes higher concentration of calcium ions. There are many dark spots of many sizes visible on panels. They represent waves of various spatial sizes and lifetimes. Most significant is large-scale wave in top right corner in bottom series. Sequence of snapshots indicates a not-closed wave front with a rotation the fingerprint of a rotating spiral wave. In seventh frame spiral wave dies in noise and consequently it is geometrically not fully developed. Kainate dose was 10 mM in 1st series and 50 mM in 2nd series. At smaller kainate dose, general calcium activity (background noise) is lower than at higher kainate dose, indicating that kainate reduces threshold of chemical imbalance, which results in greater network noise. Within observation time, we have not observed large scale waves at 1 mM kainate, in qualitative agreement with power-law for occurrence frequency discussed in METHODS.
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| FIG. 2.
Top: snapshots of evolving patterns produced by numerical simulation. System reached its statistical steady state during a transient period of about 1,000 time steps. These snap-shots were taken equidistantly with 8 time-steps between each consecutive pair commencing after transient period. Array consisted of 100 × 100 elements. Time increases from a to d. Bottom: snapshots of a calcuim wave in experiment. Time step is 8 s between frames. Kainate dose was 50 mM. Large background noise (evident in experiment) is suppressed in numerical simulation.
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Time-lapse images were analyzed to identify waves. Although most events are single fluorescent flashes, some initiate propagating waves (Fig. 1). Similar waves are found in certain chemical systems such as the Belousov-Zhabotinsky (B-Z) reaction (Belousov 1985
). In contrast to the typically crisp, smooth B-Z wave fronts, the advancing edges of kainate waves are exceptionally noisy, indicating local fluctuations in kainate concentration.
Stochastic model
On the basis of these observations, we constructed a simple stochastic model consisting of a two-dimensional (2-D) square array of elements (Jung 1997
). Each element presents a threshold for the fluctuating local concentration of neurotransmitter. A threshold crossing causes the element to enter an unbalanced state returning to the balanced state after a short refractory time. If an element is excited because of local kainate concentration unbalance, it reduces the thresholds of nearby elements by an amount that decays exponentially with the square of the distance between itself and every other element. This reduction is small enough that the state of imbalance cannot propagate to its neighbors in the absence of noise; the propagation thus becomes probabilistic (see Jung 1997
for details). The different sizes of dark spots in Figs. 1 and 2 indicate a similar behavior of the calcium waves. Thus network noise is necessary for the generation and propagation of the waves. An identical phenomenon has recently been observed with chemical waves in the Belousov-Zhabotinsky chemical reaction (Kádár et al. 1997
). Local fluctuations are modeled as Gaussian noise sources (zero mean, uniform variance) causing each threshold to fluctuate incoherently with respect to its neighbors (and to all other elements in the array). The waves produced by the model closely resemble the physiologically observed calcium waves (see Fig. 2). Waves produced in 2-D media typically have the form of rotating spirals. This is simply the geometric consequence of a nonrotation symmetric nucleus and a refractory layer behind the wave front. Spiral waves have been also observed in noise-free simulations of coupled neuron-models (Milton et al. 1993
).
Data analysis method
To compare model and biological data, we measured the distribution of wave sizes (Jung 1997
). We evaluated time lapsed sequences of images (10,000 pixels per image, 20-50 time lapse images) from three separate experiments. Each image was stacked along a time-axis (50 s total) forming a spatiotemporal cube. Active sites (gray-scale 400% of background) within the cube were joined, forming a 3-D structure (s = volume), which tracked the wave from birth to death. Analysis of many structures, for fixed conditions (kainate concentration in the experiment, noise level in the model) determined the statistical size distribution, p(s)(Fig. 3).

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| FIG. 3.
Cluster-size distribution, p(s), versus size, s, of biological data ( ) fits a power law with an exponent, a = 2.2 ± 0.2 ( ). Statistical significance of data points at both small (s 5) and large (s 50) sizes is not great, because of poor precision of s for small-s events and few number of large-s events. Cluster-size distribution of simulated data ( ) can be fit with an exponent, a = 2.0.
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RESULTS AND DISCUSSION |
The calcium imaging data are describable by a power law with exponent, a = 2.2 ± 0.2 (uncertainty determined by multiple straight line fits) in agreement with a = 2.0 for the numerically simulated model (Fig. 3). Thus no preferred temporal or spatial size scale can characterize the Ca2+ waves. From the neurophysiological point of view, it is difficult to conceive how this result could have been obtained with single, or even multiple, unit recordings. Yet it is well known from such recordings that kainate concentrations do influence and correlate the activities of nearby cells as they communicate through the synapses. What we have shown here is that such correlations, which in fact result in the waves, can be long range both in space and time. In our experiment, no fine-tuning of the kainate concentration was necessary to observe the power law distribution, thus the system moves spontaneously toward the critical state. Spiral wave phenomena in astrocyte syncytia thus satisfy the conditions required for SOC, which must therefore be a normal occurrence. In contrast, Ca2+ waves in human epileptic tissue cultures show dramatic differences in propagation and lifetime characteristics. These results suggest that the power law scaling, a signature of SOC phenomena, is characteristic of healthy cerebral activity, and that breakdowns of this scaling signal pathologic conditions. Ongoing analysis is revealing both qualitative and quantitative differences in the structures observed for pathologic tissues.
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ACKNOWLEDGEMENTS |
This work was supported in part by a Heisenberg Fellowship to P. Jung from Deutsche Forschungs Gemeinschaft and by the U.S. Office of Naval Research, Physics Division to F. Moss.
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FOOTNOTES |
Present address of P. Jung: Dept. of Physics, Ohio University, Athens, OH 45701.
Address reprint requests to F. Moss.
Received 2 July 1997; accepted in final form 3 November 1997.
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