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Munch, SB, Giron-Nava A, Sugihara G.  2018.  Nonlinear dynamics and noise in fisheries recruitment: A global meta-analysis. Fish and Fisheries. 19:964-973.   10.1111/faf.12304   AbstractWebsite

The relative importance of environmental and intrinsic controls on recruitment in fishes has been studied for over a century. Despite this, we are not much closer to predicting recruitment. Rather, recent analyses suggest that recruitment is virtually independent of stock size and, instead, seems to occur in distinct environmental regimes. This issue of whether or not recruitment and subsequent production are coupled to stock size is highly relevant to management. Here, we apply empirical dynamical modelling (EDM) to a global database of 185 fish populations to address the questions of whether or not variation in recruitment is (a) predictable and (b) coupled to stock size. We find that a substantial fraction of recruitment variation is predictable using only the observed history of fluctuations (similar to 40% on average). In addition, although recruitment is often coupled to stock size (107 of 185 stocks), stock size alone explains very little of the variation in recruitment; In similar to 90% of the stocks analysed, EDM forecasts have substantially lower prediction error than models based solely on stock size. We find that predictability varies across taxa and improves with the number of generations that have been sampled. In the light of these results, we suggest that EDM will be of greatest use in managing relatively short-lived species.

Deyle, E, Schueller AM, Ye H, Pao GM, Sugihara G.  2018.  Ecosystem-based forecasts of recruitment in two menhaden species. Fish and Fisheries. 19:769-781.   10.1111/faf.12287   AbstractWebsite

Gulf (Brevoortia patronus, Clupeidae) and Atlantic menhaden (Brevoortia tyrannus, Clupeidae) support large fisheries that have shown substantial variability over several decades, in part, due to dependence on annual recruitment. Nevertheless, traditional stock-recruitment relationships lack predictive power for these stocks. Current management of Atlantic menhaden explicitly treats recruitment as a random process. However, traditional methods for understanding recruitment variability carry the very specific hypothesis that the effect of adult biomass on subsequent recruitment occurs independently of other ecosystem factors such as food availability and predation. Here, we evaluate the predictability of menhaden recruitment using a model-free approach that is not restricted by these strong assumptions. We find that menhaden recruitment is predictable, but only when allowing for interdependence of stock with other ecological factors. Moreover, while the analysis confirms the presence of environmental effects, the environment alone does not readily account for the complexity of menhaden recruitment dynamics. The findings set the stage for revisiting recruitment prediction in management and serve as an instructive example in the ongoing debate about how to best treat and understand recruitment variability across species and fisheries.

Dakos, V, Glaser SM, Hsieh CH, Sugihara G.  2017.  Elevated nonlinearity as an indicator of shifts in the dynamics of populations under stress. Journal of the Royal Society Interface. 14   10.1098/risf.2016.0845   AbstractWebsite

Populations occasionally experience abrupt changes, such as local extinctions, strong declines in abundance or transitions from stable dynamics to strongly irregular fluctuations. Although most of these changes have important ecological and at times economic implications, they remain notoriously difficult to detect in advance. Here, we study changes in the stability of populations under stress across a variety of transitions. Using a Ricker- type model, we simulate shifts from stable point equilibrium dynamics to cyclic and irregular boom- bust oscillations as well as abrupt shifts between alternative attractors. Our aim is to infer the loss of population stability before such shifts based on changes in nonlinearity of population dynamics. We measure nonlinearity by comparing forecast performance between linear and nonlinear models fitted on reconstructed attractors directly from observed time series. We compare nonlinearity to other suggested leading indicators of instability (variance and autocorrelation). We find that nonlinearity and variance increase in a similar way prior to the shifts. By contrast, autocorrelation is strongly affected by oscillations. Finally, we test these theoretical patterns in datasets of fisheries populations. Our results suggest that elevated nonlinearity could be used as an additional indicator to infer changes in the dynamics of populations under stress.

Deyle, ER, Maher MC, Hernandez RD, Basu S, Sugihara G.  2016.  Global environmental drivers of influenza. Proceedings of the National Academy of Sciences of the United States of America. 113:13081-13086.   10.1073/pnas.1607747113   AbstractWebsite

In temperate countries, influenza outbreaks are well correlated to seasonal changes in temperature and absolute humidity. However, tropical countries have much weaker annual climate cycles, and outbreaks show less seasonality and are more difficult to explain with environmental correlations. Here, we use convergent cross mapping, a robust test for causality that does not require correlation, to test alternative hypotheses about the global environmental drivers of influenza outbreaks from country-level epidemic time series. By moving beyond correlation, we show that despite the apparent differences in outbreak patterns between temperate and tropical countries, absolute humidity and, to a lesser extent, temperature drive influenza outbreaks globally. We also find a hypothesized U-shaped relationship between absolute humidity and influenza that is predicted by theory and experiment, but hitherto has not been documented at the population level. The balance between positive and negative effects of absolute humidity appears to be mediated by temperature, and the analysis reveals a key threshold around 75 degrees F. The results indicate a unified explanation for environmental drivers of influenza that applies globally.

Ye, H, Sugihara G.  2016.  Information leverage in interconnected ecosystems: Overcoming the curse of dimensionality. Science. 353:922-925.   10.1126/science.aag0863   AbstractWebsite

In ecological analysis, complexity has been regarded as an obstacle to overcome. Here we present a straightforward approach for addressing complexity in dynamic interconnected systems. We show that complexity, in the form of multiple interacting components, can actually be an asset for studying natural systems from temporal data. The central idea is that multidimensional time series enable system dynamics to be reconstructed from multiple viewpoints, and these viewpoints can be combined into a single model. We show how our approach, multiview embedding (MVE), can improve forecasts for simulated ecosystems and a mesocosm experiment. By leveraging complexity, MVE is particularly effective for overcoming the limitations of short and noisy time series and should be highly relevant for many areas of science.

Tsonis, AA, Deyle ER, May RM, Sugihara G, Swanson K, Verbeten JD, Wang GL.  2015.  Dynamical evidence for causality between galactic cosmic rays and interannual variation in global temperature. Proceedings of the National Academy of Sciences of the United States of America. 112:3253-3256.   10.1073/pnas.1420291112   AbstractWebsite

As early as 1959, it was hypothesized that an indirect link between solar activity and climate could be mediated by mechanisms controlling the flux of galactic cosmic rays (CR) [Ney ER (1959) Nature 183:451-452]. Although the connection between CR and climate remains controversial, a significant body of laboratory evidence has emerged at the European Organization for Nuclear Research [Duplissy J, et al. (2010) Atmos Chem Phys 10:1635-1647; Kirkby J, et al. (2011) Nature 476(7361):429-433] and elsewhere [Svensmark H, Pedersen JOP, Marsh ND, Enghoff MB, Uggerhoj Ul (2007) Proc R Soc A 463:385-396; Enghoff MB, Pedersen JOP, Uggerhoj Ul, Paling SM, Svensmark H (2011) Geophys Res Lett 38:L09805], demonstrating the theoretical mechanism of this link. In this article, we present an analysis based on convergent cross mapping, which uses observational time series data to directly examine the causal link between CR and year-to-year changes in global temperature. Despite a gross correlation, we find no measurable evidence of a causal effect linking CR to the overall 20th-century warming trend. However, on short interannual timescales, we find a significant, although modest, causal effect between CR and short-term, year-to-year variability in global temperature that is consistent with the presence of nonlinearities internal to the system. Thus, although CR do not contribute measurably to the 20th-century global warming trend, they do appear as a nontraditional forcing in the climate system on short interannual timescales.

Ye, H, Beamish RJ, Glaser SM, Grant SCH, Hsieh CH, Richards LJ, Schnute JT, Sugihara G.  2015.  Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling. Proceedings of the National Academy of Sciences of the United States of America. 112:E1569-E1576.   10.1073/pnas.1417063112   AbstractWebsite

It is well known that current equilibrium-based models fall short as predictive descriptions of natural ecosystems, and particularly of fisheries systems that exhibit nonlinear dynamics. For example, model parameters assumed to be fixed constants may actually vary in time, models may fit well to existing data but lack out-of-sample predictive skill, and key driving variables may be misidentified due to transient (mirage) correlations that are common in nonlinear systems. With these frailties, it is somewhat surprising that static equilibrium models continue to be widely used. Here, we examine empirical dynamic modeling (EDM) as an alternative to imposed model equations and that accommodates both nonequilibrium dynamics and nonlinearity. Using time series from nine stocks of sockeye salmon (Oncorhynchus nerka) from the Fraser River system in British Columbia, Canada, we perform, for the the first time to our knowledge, real-data comparison of contemporary fisheries models with equivalent EDM formulations that explicitly use spawning stock and environmental variables to forecast recruitment. We find that EDM models produce more accurate and precise forecasts, and unlike extensions of the classic Ricker spawner-recruit equation, they show significant improvements when environmental factors are included. Our analysis demonstrates the strategic utility of EDM for incorporating environmental influences into fisheries forecasts and, more generally, for providing insight into how environmental factors can operate in forecast models, thus paving the way for equation-free mechanistic forecasting to be applied in management contexts.

Glaser, SM, Ye H, Sugihara G.  2014.  A nonlinear, low data requirement model for producing spatially explicit fishery forecasts. Fisheries Oceanography. 23:45-53.   10.1111/fog.12042   AbstractWebsite

Spatial variability can confound accurate estimates of catch per unit effort (CPUE), especially in highly migratory species. The incorporation of spatial structure into fishery stock assessment models should ultimately improve forecasts of stock biomass. Here, we describe a nonlinear time series model for producing spatially explicit forecasts of CPUE that does not require ancillary environmental or demographic data, or specification of a model functional form. We demonstrate this method using spatially resolved (1 degrees x1 degrees cells) CPUE time series of North Pacific albacore in the California Current System. The spatial model is highly significant (P<0.00001) and outperforms two spatial null models. We then create a spatial forecast map for years beyond the range of data. Such approaches can guide spatial management of resources and provide a complement to more data-intensive, highly parameterized population dynamics and ecosystem models currently in use.

Deyle, ER, Fogarty M, Hsieh CH, Kaufman L, MacCall AD, Munch SB, Perretti CT, Ye H, Sugihara G.  2013.  Predicting climate effects on Pacific sardine. Proceedings of the National Academy of Sciences of the United States of America. 110:6430-6435.   10.1073/pnas.1215506110   AbstractWebsite

For many marine species and habitats, climate change and overfishing present a double threat. To manage marine resources effectively, it is necessary to adapt management to changes in the physical environment. Simple relationships between environmental conditions and fish abundance have long been used in both fisheries and fishery management. In many cases, however, physical, biological, and human variables feed back on each other. For these systems, associations between variables can change as the system evolves in time. This can obscure relationships between population dynamics and environmental variability, undermining our ability to forecast changes in populations tied to physical processes. Here we present a methodology for identifying physical forcing variables based on nonlinear forecasting and show how the method provides a predictive understanding of the influence of physical forcing on Pacific sardine.

Perretti, CT, Munch SB, Sugihara G.  2013.  Model-free forecasting outperforms the correct mechanistic model for simulated and experimental data. Proceedings of the National Academy of Sciences of the United States of America. 110:5253-5257.   10.1073/pnas.1216076110   AbstractWebsite

Accurate predictions of species abundance remain one of the most vexing challenges in ecology. This observation is perhaps unsurprising, because population dynamics are often strongly forced and highly nonlinear. Recently, however, numerous statistical techniques have been proposed for fitting highly parameterized mechanistic models to complex time series, potentially providing the machinery necessary for generating useful predictions. Alternatively, there is a wide variety of comparatively simple model-free forecasting methods that could be used to predict abundance. Here wepose a rather conservative challenge and ask whether a correctly specified mechanistic model, fit with commonly used statistical techniques, can provide better forecasts than simple model-free methods for ecological systems with noisy nonlinear dynamics. Using four different control models and seven experimental time series of flour beetles, we found that Markov chain Monte Carlo procedures for fitting mechanistic models often converged on best-fit parameterizations far different from the known parameters. As a result, the correctly specified models provided inaccurate forecasts and incorrect inferences. In contrast, a model-free method based on state-space reconstruction gave the most accurate short-term forecasts, even while using only a single time series from the multivariate system. Considering the recent push for ecosystem-based management and the increasing call for ecological predictions, our results suggest that a flexible model-free approach may be the most promising way forward.

Sugihara, G, May R, Ye H, Hsieh CH, Deyle E, Fogarty M, Munch S.  2012.  Detecting causality in complex ecosystems. Science. 338:496-500.   10.1126/science.1227079   Abstract

Identifying causal networks is important for effective policy and management recommendations on climate, epidemiology, financial regulation, and much else. We introduce a method, based on nonlinear state space reconstruction, that can distinguish causality from correlation. It extends to nonseparable weakly connected dynamic systems (cases not covered by the current Granger causality paradigm). The approach is illustrated both by simple models (where, in contrast to the real world, we know the underlying equations/relations and so can check the validity of our method) and by application to real ecological systems, including the controversial sardine-anchovy-temperature problem.

Hsieh, CH, Kim HJ, Watson W, Di Lorenzo E, Sugihara G.  2009.  Climate-driven changes in abundance and distribution of larvae of oceanic fishes in the southern California region. Global Change Biology. 15:2137-2152.   10.1111/j.1365-2486.2009.01875.x   AbstractWebsite

We examined climatic effects on the geographic distribution and abundance of 34 dominant oceanic fishes in the southern California region using larval fish data collected from the 50-year long California Cooperative Oceanic Fisheries Investigations (CalCOFI) surveys. The oceanic species responses to environmental changes in their geographic distributions were not very pronounced, perhaps because they lived in the deep layer where temperature change was relatively small or because the environmental variation of the CalCOFI region is not strong enough (with an average temperature gradient of the upper 100 m around 91 km degrees C(-1)). Among the 34 taxa, 16 showed a significant distributional shift (median latitude or boundaries) in relation to environmental variables, and eight species significantly shifted their geographic distribution from the 1951-1976 cold period to the 1977-1998 warm period. Interestingly, the vertically migrating taxa more often showed a significant response to environmental variables than the nonmigrating mesopelagic taxa, reflecting the more significant increase in heat content of the upper ocean (< 200 m), compared with the deeper zone (300-500 m) where the mesopelagic fishes typically remain. Climate change has significant effects on the abundances of oceanic fishes. Twenty-four taxa exhibited a significant change in abundance in relation to environmental variables, and 25 taxa, including both warm and cold-water taxa, showed a significant increase in abundance from the cold to warm period. Analysis of physical data indicated that the surface-layer (20-200 m) warmed significantly and the isotherms approached shoreward from the cold to the warm period. We further show that the spatial distribution of coastal-neritic fish retreated shoreward and oceanic fish extended shoreward from the cold to warm period. Our results suggest intensified stratification of the southern California region during the warm period may create a suitable habitat for the oceanic species. Moreover, such an unfavorable condition (e.g. changes in food habitat) for coastal-neritic species might result in competitive release for the oceanic fishes to flourish.

Scheffer, M, Bascompte J, Brock WA, Brovkin V, Carpenter SR, Dakos V, Held H, van Nes EH, Rietkerk M, Sugihara G.  2009.  Early-warning signals for critical transitions. Nature. 461:53-59.   10.1038/nature08227   AbstractWebsite

Complex dynamical systems, ranging from ecosystems to financial markets and the climate, can have tipping points at which a sudden shift to a contrasting dynamical regime may occur. Although predicting such critical points before they are reached is extremely difficult, work in different scientific fields is now suggesting the existence of generic early-warning signals that may indicate for a wide class of systems if a critical threshold is approaching.

Kilcik, A, Anderson CNK, Rozelot JP, Ye H, Sugihara G, Ozguc A.  2009.  Nonlinear prediction of solar cycle 24. Astrophysical Journal. 693:1173-1177.   10.1088/0004-637x/693/2/1173   AbstractWebsite

Sunspot activity is highly variable and challenging to forecast. Yet forecasts are important, since peak activity has profound effects on major geophysical phenomena including space weather (satellite drag, telecommunications outages) and has even been correlated speculatively with changes in global weather patterns. This paper investigates trends in sunspot activity, using new techniques for decadal-scale prediction of the present solar cycle (cycle 24). First, Hurst exponent H analysis is used to investigate the autocorrelation structure of the putative dynamics; then the Sugihara-May algorithm is used to predict the ascension time and the maximum intensity of the current sunspot cycle. Here we report H = 0.86 for the complete sunspot number data set (1700-2007) and H = 0.88 for the reliable sunspot data set (1848-2007). Using the Sugihara-May algorithm analysis, we forecast that cycle 24 will reach its maximum in 2012 December at approximately 87 sunspot units.

Maye, A, Hsieh CH, Sugihara G, Brembs B.  2007.  Order in spontaneous behavior. Plos One. 2   10.1371/journal.pone.0000443   AbstractWebsite

Brains are usually described as input/output systems: they transform sensory input into motor output. However, the motor output of brains (behavior) is notoriously variable, even under identical sensory conditions. The question of whether this behavioral variability merely reflects residual deviations due to extrinsic random noise in such otherwise deterministic systems or an intrinsic, adaptive indeterminacy trait is central for the basic understanding of brain function. Instead of random noise, we find a fractal order (resembling Levy flights) in the temporal structure of spontaneous flight maneuvers in tethered Drosophila fruit flies. Levy-like probabilistic behavior patterns are evolutionarily conserved, suggesting a general neural mechanism underlying spontaneous behavior. Drosophila can produce these patterns endogenously, without any external cues. The fly's behavior is controlled by brain circuits which operate as a nonlinear system with unstable dynamics far from equilibrium. These findings suggest that both general models of brain function and autonomous agents ought to include biologically relevant nonlinear, endogenous behavior-initiating mechanisms if they strive to realistically simulate biological brains or out-compete other agents.

Hsieh, CH, Glaser SM, Lucas AJ, Sugihara G.  2005.  Distinguishing random environmental fluctuations from ecological catastrophes for the North Pacific Ocean. Nature. 435:336-340.   10.1038/nature03553   AbstractWebsite

The prospect of rapid dynamic changes in the environment is a pressing concern that has profound management and public policy implications(1,2). Worries over sudden climate change and irreversible changes in ecosystems are rooted in the potential that nonlinear systems have for complex and 'pathological' behaviours(1,2). Nonlinear behaviours have been shown in model systems(3) and in some natural systems(1,4-8), but their occurrence in large-scale marine environments remains controversial(9,10). Here we show that time series observations of key physical variables(11-14) for the North Pacific Ocean that seem to show these behaviours are not deterministically nonlinear, and are best described as linear stochastic. In contrast, we find that time series for biological variables(5,15-17) having similar properties exhibit a low-dimensional nonlinear signature. To our knowledge, this is the first direct test for nonlinearity in large-scale physical and biological data for the marine environment. These results address a continuing debate over the origin of rapid shifts in certain key marine observations as coming from essentially stochastic processes or from dominant nonlinear mechanisms(1,9,10,18-20). Our measurements suggest that large-scale marine ecosystems are dynamically nonlinear, and as such have the capacity for dramatic change in response to stochastic fluctuations in basin-scale physical states.

Hsieh, CH, Reiss C, Watson W, Allen MJ, Hunter JR, Lea RN, Rosenblatt RH, Smith PE, Sugihara G.  2005.  A comparison of long-term trends and variability in populations of larvae of exploited and unexploited fishes in the Southern California region: A community approach. Progress in Oceanography. 67:160-185.   10.1016/j.pocean.2005.05.002   AbstractWebsite

We have constructed an "expert-knowledge classification system" to categorize 309 fish taxa in the California Cooperative Oceanic Fisheries Investigations ichthyoplankton database into primary (coastal, coastal-oceanic, and oceanic) assemblages based on their principal ecological domains and subsequently, secondary assemblages according to the habitat affinities of adults. We examined effects of fishing, climate, adult habitat, and age-at-maturation on long-term variation of fish populations. We tested the hypothesis that populations of unexploited taxa, track climatic trends more closely than those of exploited taxa insofar as climatic signals may be confounded by fishing effects. Most oceanic taxa (23/34) showed a significant relationship with environmental variables and followed the trend of the Pacific Decadal Oscillation. Very few coastal (3/10) and coastal-oceanic (3/23) taxa exhibited a significant relationship with environmental signals; however, several fluctuated coherently, and age-at-maturation is an important factor. The lack of close correlation between fish populations and environmental signals in the coastal and coastal-oceanic assemblages indicates that these species might show nonlinear biological responses to external forcing rather than a simple linear tracking of environmental variables. We did not find a systematic pattern indicating that fishing influenced population fluctuation of exploited species. Constrained comparisons of exploited to unexploited species living in the same habitat and reaching maturity at the same age revealed evidence of overexploitation for some species but not for all. Our results suggest that considering life history and ecological characteristics of fish species and applying a community approach are important in understanding fishing effects on fish populations in the context of a changing environment. (c) 2005 Elsevier Ltd. All rights reserved.

Dixon, PA, Milicich MJ, Sugihara G.  1999.  Episodic fluctuations in larval supply. Science. 283:1528-1530.   10.1126/science.283.5407.1528   AbstractWebsite

The lack of a clear relationship between spawning output and recruitment success continues to confound attempts to understand and manage temporally variable fish populations. This relationship for a common reef fish is shown to be obscured by nonlinear processes in operation during the larval phase. Non-linear responses of larval fish to their noisy physical environment may offer a general explanation for the erratic, often episodic, replenishment of open marine populations.

Segundo, JP, Sugihara G, Dixon P, Stiber M, Bersier LF.  1998.  The spike trains of inhibited pacemaker neurons seen through the magnifying glass of nonlinear analyses. Neuroscience. 87:741-766. AbstractWebsite

This communication describes the new information that may be obtained by applying nonlinear analytical techniques to neurobiological time-series. Specifically, we consider the sequence of interspike intervals T-i (the "timing") of trains recorded from synaptically inhibited crayfish pacemaker neurons. As reported earlier, different postsynaptic spike train forms (sets of timings with shared properties) are generated by varying the average rate and/or pattern (implying interval dispersions and sequences) of presynaptic spike trains. When the presynaptic train is Poisson (independent exponentially distributed intervals), the form is "Poisson-driven" (unperturbed and lengthened intervals succeed each other irregularly). When presynaptic trains are pacemaker (intervals practically equal), forms are either "p:q locked" (intervals repeat periodically), "intermittent" (mostly almost locked but disrupted irregularly), "phase walk throughs" (intermittencies with briefer regular portions), or "messy" (difficult to predict or describe succinctly). Messy trains are either "erratic" (some intervals natural and others lengthened irregularly) or "stammerings" (intervals are integral multiples of presynaptic intervals). The individual spike train forms were analysed using attractor reconstruction methods based on the lagged coordinates provided by successive intervals from the time-series T-i. Numerous models were evaluated in terms of their predictive performance by a trial-and-error procedure: the most successful model was taken as best reflecting the true nature of the system's attractor. Each form was characterized in terms of its dimensionality, nonlinearity and predictability. (1) The dimensionality of the underlying dynamical attractor was estimated by the minimum number of variables (coordinates T-i) required to model acceptably the system's dynamics, i.e. by the system's degrees of freedom. Each model tested was based on a different number of T-i; the smallest number whose predictions were judged successful provided the best integer approximation of the attractor's true dimension (not necessarily an integer). Dimensionalities from three to five provided acceptable fits. (2) The degree of nonlinearity was estimated by: (i) comparing the correlations between experimental results and data from linear and nonlinear models, and (ii) tuning model nonlinearity via a distance-weighting function and identifying the either local or global neighborhood size. Lockings were compatible with linear models and stammerings were marginal; nonlinear models were best for Poisson-driven, intermittent and erratic forms. (3) Finally, prediction accuracy was plotted against increasingly long sequences of intervals forecast: the accuracies for Poisson-driven, locked and stammering forms were invariant, revealing irregularities due to uncorrelated noise, but those of intermittent and messy erratic forms decayed rapidly, indicating an underlying deterministic process. The excellent reconstructions possible for messy erratic and for some intermittent forms are especially significant because of their relatively low dimensionality (around 4), high degree of nonlinearity and prediction decay with time. This is characteristic of chaotic systems, and provides evidence that nonlinear couplings between relatively few variables are the major source of the apparent complexity seen in these cases. This demonstration of different dimensions. degrees of nonlinearity and predictabilities provides rigorous support for the categorization of different synaptically driven discharge forms proposed earlier on the basis of more heuristic criteria. This has significant implications. (1) It demonstrates that heterogeneous postsynaptic forms can indeed be induced by manipulating a few presynaptic variables. (2) Each presynaptic timing induces a form with characteristic dimensionality, thus breaking up the preparation into subsystems such that the physical variables in each operate as one formal parameter or degree of freedom. A system's partitions differ because of component subsystems and/or dynamics: the set of all partitions is probably large and continuous. Driver-induced partitions have general theoretical interest, and provide guidelines for identifying the responsible physical variables. (3) Because forms tolerate changing conditions and are encountered widely (e.g., along transients), it is hypothesized that they are elementary building blocks for many synaptic codings. Codings are linear if postsynaptic forms have the same spectral components as the presynaptic pacemaker, or nonlinear if novel components arise as with, respectively, 1:1 locked or erratic trains. This is relevant to network operations where regularity and irregularity are often vital. (4) Rigorously identifying spike train forms in experimental data from living preparations allowed matchings with available theoretical computations and considerations. Relevant models are based either on iterations of maps derived from rhythm resettings by isolated arrivals or on Bonhoeffer-van der Pol formulations: such models generate, respectively, only periodic locking and phase walk throughs, or all forms. This precise and broad conceptual context explains and predicts outcomes, recognizes data/theory discrepancies, and identifies their reasons (e.g., after-effects, noise). (5) Accordingly, forms pertain to universal behavior categories called "noisy", "periodic", "intermittent", "quasiperiodic" or "chaotic" whose available theories provide valuable contexts For genuinely physiological issues. Thus, experimental design and thinking benefit from significant insights about the dynamics of pacemaker-driven pacemakers, the simplest of all synaptic codings. (C) 1998 IBRO. Published by Elsevier Science Ltd.

Sugihara, G, Allan W, Sobel D, Allan KD.  1996.  Nonlinear control of heart rate variability in human infants. Proceedings of the National Academy of Sciences of the United States of America. 93:2608-2613.   10.1073/pnas.93.6.2608   AbstractWebsite

Nonlinear analyses of infant heart rhythms reveal a marked rise in the complexity of the electrocardiogram with maturation, We find that normal mature infants (gestation greater than or equal to 35 weeks) have complex and distinctly nonlinear heart rhythms (consistent with recent reports for healthy adults) but that such nonlinearity is lacking in preterm infants (gestation less than or equal to 27 weeks) where parasympathetic-sympathetic interaction and function are presumed to be less well developed. Our study further shows that infants with clinical brain death and those treated with atropine exhibit a similar lack of nonlinear feedback control, These three lines of evidence support the hypothesis championed by Goldberger et nl, [Goldberger, A. L., Rigney, D. R. & West, B. J. (1990) Sci, Am, 262, 43-49] that autonomic nervous system control underlies the nonlinearity and possible chaos of normal heart rhythms. This report demonstrates the acquisition of nonlinear heart rate dynamics and possible chaos in developing human infants and its loss in brain death and with the administration of atropine. It parallels earlier work documenting changes in the variability of heart rhythms in each of these cases and suggests that nonlinearity may provide additional power in characterizing physiological states.

Yamazaki, H, Sugihara G, Kirkpatrick GJ, Kamykowski D.  1993.  Is the photosynthetic process nonlinear? Journal of Plankton Research. 15:1297-1308.   10.1093/plankt/15.11.1297   AbstractWebsite

We applied two non-linear time series analysis methods to photosynthetic data obtained from a single-species population of Thalassiosira pseudonana incubated in situ in order to identify whether the time series were generated predominantly by linear or non-linear processes. The tests used to make these distinctions involved a comparison of the predictability of the observed data under a non-linear hypothesis versus a linear hypothesis. Two records were analyzed. For the first data segment taken in the morning, the linear method performed as well or better than the nonlinear methods. Although a weak non-linearity was detected in the second data set observed in the afternoon, the time series is dominantly linear. The best embedding dimension, whose value suggests the number of participating independent parameters in the system, is 2 for the morning data and 7 for the afternoon data. These results are true for aggregate productivity measures (1.8 x 10(8) cells) on a time scale of 1-5 min.