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Sort order. Oct 29, Adam Lantos rated it it was amazing. This book always offers an intuitive approach to waves. Although it delves into some subtle themes sometimes and give a nice insight to some application of the phenomena, at some other occasions I would want a bit more discussion on some small details that are a bit more subtle. It has a nice balance of mathematical proofs and discussion.

There are some errors in it, but nothing important. I would recommend Crawford's book over this, but this is very good too. Based on the things I already mentione This book always offers an intuitive approach to waves. Based on the things I already mentioned, I would give it a 4-star rating. But, as this gives a nice introduction to Quantum Mechanics, it gets the extra star.

Oct 30, Mike rated it liked it. Really 3. A decent alternative to A. French's text, which while gets things "rigorously right" is not the most readable text for someone's introduction to the subject. This text is readable, although like most waves texts out there the subjects jump around quite a bit. The book also now suffers from most of the online content no longer being available online due to moved and dead links, and the continuing saga of Java updates.

Mar 15, geranimo rated it liked it. I didn't enjoy the read as Walter likes to joke and I hate jokes in science books. Jovany Agathe rated it liked it Feb 08, Yueru Li rated it it was amazing Mar 23, Angie rated it it was amazing May 27, Sachin rated it liked it Oct 03, Vincent Gammill rated it liked it Jan 11, Orionic rated it really liked it Nov 14, Ali marked it as to-read Feb 24, Conor marked it as to-read Dec 03, Chris marked it as to-read Apr 28, T Dodson is currently reading it Jan 07, The use of multiple nonlinear time series techniques, however, suggests an alternative interpretation, namely deterministic chaos.

We provide an extensive, nonlinear time series analysis on the nature of this calcium oscillation response. We build up evidence through a series of techniques that test for determinism, quantify linear and nonlinear components, and measure the local divergence of the system. Chaos is common in nature and it seems plausible that properties of chaotic dynamics might be exploited by biological systems to control processes within the cell.

Systems possessing chaotic control mechanisms are more robust in the sense that the enhanced flexibility allows more rapid response to environmental changes with less energetic costs. The desired behaviour could be most efficiently targeted in this manner, supporting some intriguing speculations about nonlinear mechanisms in biological signaling. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: S. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist. During this interaction, bacteria called rhizobia invade the plant roots and are accommodated in membrane bound compartments within plant cells of a specialized organ on the root: the nodule.

Within the nodule the bacteria convert atmospheric dinitrogen into ammonia, a form of nitrogen readily available to the plant. Perception of Nod factor by legumes activates most of the developmental processes associated with the formation of a nodule. The Nod factor signal transduction pathway of legumes has been well characterized and involves calcium oscillations, termed calcium spiking.

An example of calcium spiking is given in Figure 1. Receptor-like kinases are involved in the perception of Nod factor and this leads to induction of calcium spiking via cation channels, that appear to regulate potassium movement and components of the nuclear-pore complex [1]. This signal transduction pathway has also been shown to function in the establishment of a second symbiotic interaction: the mycorrhizal symbiosis.

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This interaction involves the colonization of the plant root by mycorrhizal fungi that aid the plant in the uptake of nutrients from the soil. Mycorrhizal fungi have been shown to activate calcium oscillations, but with a different structure to Nod factor induced calcium spiking [2]. This suggests that the symbiosis signaling pathway can be differentially activated by both rhizobia and mycorrhizal fungi. The X axis is time in seconds. The nature of biological systems and the challenges inherent in experimentation often result in seemingly erratic time-series behaviour with little apparent structure.

Despite advances in signal processing methodology, the extraction of information from such data remains a challenge. Erratic behaviour is often thought to be the consequence of noise or stochastic effects, but apparent randomness can also be generated by a deterministic system operating in the chaotic regime. A universally accepted definition of chaos is still outstanding, however, a number of key features are held in common: A chaotic system is deterministic, nonlinear, and highly sensitive to the initial conditions.

The exponential divergence of nearby trajectories implies that the predictability is limited to short time scales. Long-term forecasts become impossible despite the underlying deterministic nature. Unpredictable systems are frequently handled with the methods of probability theory and termed stochastic. Sophisticated techniques exist for distinguishing between linear, nonlinear, deterministic, stochastic and chaotic systems.

However, disentangling experimental noise, stochastic effects, and underlying deterministic laws is non-trivial and the initial data derived from biological processes are not often of sufficient quality to allow such analyses. Initial chaotic models were inspired by the bursting behaviour observed in experiments on hepatocytes [8] , [9] , [10]. Further analysis revealed that the statistics of the interspike intervals are in agreement with a stochastic model. In plants, moreover, little is known about the secondary messengers or calcium channels that may direct Nod factor induced calcium spiking [14] , also it is apparent that there are major differences in the proteins that activate or perceive well-characterized animal secondary messengers such as inositol trisphosphate and cADPR [15].

Given these unknowns and differences, we are reluctant to bias our analysis towards the models and conclusions drawn from animal systems. Instead, a more appropriate approach to understand Nod factor signaling is to analyse the experimentally obtained calcium oscillations using methods from nonlinear time series analysis. This observation suggested an alternative explanation to a stochastic interpretation and prompted us to validate our methodology using negative and positive controls. These models were tested alongside our experimental data.

In the following we describe the results of a number of nonlinear time series analyses. In order to check whether a stream of data has arisen from a chaotic system, a number of tests must be carried out. Definitive answers are rare unless the system of underlying equations or map is known. Plotting system observables as a function of themselves at an earlier time gives rise to the return map, which often appears as a simple curve for deterministic systems. The shape of such a curve strongly indicates the classification of the dynamics. This technique is in fact a form of state space reconstruction, in which typical deterministic trajectories should establish themselves upon a low-dimensional attractor.

A further test is for exponential divergence and the calculation of Lyapunov exponents, which if positive indicates chaos. These tests are sensitive to noise, which is always present especially in biological data, and hence rarely provide definitive answers. One of the key steps for such analyses is proper embedding and the determination of attractor dimensionality.

Thus, we are limited in the application of such methods and as a result could not determine the dimensionality reliably, and the return map computations did not produce convincing results.

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However, as can be seen in Figure 2 , the attractor does appear to unfold well in three dimensions. Additionally, a number of tests did provide useful results with a good confidence level. The following sections describe the application of a number of different tests, which taken together certainly do not prove but provide evidence for chaos. A: The experimental series is clearly noisy, prohibiting accurate dimensionality determination, but it unfolds well to the eye in three dimensions.

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B: The data from the stochastic spiking model, however, appears to cross itself in many places and coalesces, violating the uniqueness property of ODE. The results are described in two sections. The full time series are used and not just interspike times. However, because the low frequency components of the signal may not be significant, as an alternative to EMD we also detrended the data using a moving average.

A summary of results on the left of the figure are after processing with a moving average. If nonlinearity was detected, a noise titration was used to test for chaos and the Lyapunov exponent was calculated using a direct method. The direct method calculates the maximal Lyapunov exponent and inspection of the resulting divergence data can help one to discern if the divergence of the system is due to chaotic or stochastic effects.

Nonlinear oscillations of a fluttering plate.

An indirect method was also used where multiple nonlinear models were fitted to the experimental data and a maximal exponent calculated for each model. The indirect method gave a selection of Lyapunov exponents and if a clear majority of well fitting models had positive exponents then we take this as evidence that the divergence is more likely due to deterministic chaos rather than stochasticity Figure S Evidence of chaos was suggested in the majority of traces 16 out of 21 using a noise titration with the surrogates nonlinear test Table 1.

Applied to linear autoregressive AR models fitted to the experimental data, this test correctly identified forty true negatives and only two false positives. In some cases, the results of the experimental time series vary depending on the method of detrending, with some 4 out of 10 EMD detrended time series failing the test for nonlinearity. The nonlinear surrogates test exhibits a length dependence and so the shorter time series failed Table 1. The nonlinear predictability was computed for a long time series that was steadily truncated to provide a comparison of p -value against series length Figure 4.

The p -values do not consistently indicate nonlinearity for times shorter than samples. Each p -value was calculated using surrogates. Signs of nonlinearity are not detected until the length of series being tested is greater than samples. P-values do not drop to show significant nonlinearity, which is marked with a red line, until the time series is approximately samples long. All negative controls correctly gave negative maximal Lyapunov exponents.

Since the majority of the traces passed a test for nonlinearity the system can be considered nonlinear, justifying the application of nonlinear noise reduction techniques. The logarithm of the divergence of neighbouring points in phase space against time revealed a clear linear trend in the majority of the time series, indicating exponential divergence. This is shown in Figure 5. Taking an average gradient gave a Lyapunov exponent of 0. The short term fluctuations are due to the periodicity of the signal, but the average distance clearly grows exponentially.

The data points pictured are for an embedding dimension of seven, and consistent values for the maximal exponent are achieved once the dimension is greater than or equal to six. The exponent is 0. These values are given by the slope of the black dashed lines. For each trace, the exponent was computed as the average of three slopes: 1 the slope through local maxima; 2 the slope through local minima; and 3 the slope through the average of local minima and maxima. The final value of the exponent was computed by averaging over all traces. Computations were not done for traces Nod6 and Nod9 for either detrending because of the short series length.

Properties of the autocorrelation of interspike intervals have been used to support the idea of stochastic spike activation in four cell types from mice and humans [13]. For a purely random time series white noise the autocorrelation is close to zero. Both the mathematical models and the experimental data show a rapid drop in autocorrelation indicating that successive intervals are not correlated. However, the mathematical models act as positive controls revealing that the drop in autocorrelation is not necessarily down to stochastic effects.

The X axis is the lag measured in number of samples sample time is 5 seconds. It must be pointed out that nonlinear time series analyses cannot provide a definite answer regarding the nature of spike activation and interspike times in the system. We considered a nonlinear deterministic model for the spike waveforms, with randomly chosen interspike intervals.

As expected, this signal clearly appears nonlinear; however, it also appears chaotic using a noise titration. This demonstrates that some conventional tests used to detect chaos are unable to discern between purely chaotic systems and a carefully designed deterministic spiking system with stochastic activation. A direct Lyapunov calculation for the time series with stochastic interspike times does not exhibit a clear exponential divergence. The indirect method also indicates that the majority of models fitted to the time series with stochastic interspike intervals have a negative Lyapunov exponent.

The results of determinism tests are somewhat subjective and were therefore not used to support our conclusions. All traces obtained from our experiments pass the three statistical tests for determinism proposed by Aparicio [16] without the use of noise reduction. A negative control using random numbers fails the three determinism tests. This method is based on a vector reconstruction of the attractor over a grid of 5 6 boxes. More than highlighting deficiencies of the Kaplan Glass test, these results show the limitations in using only one metric to characterize noisy data sets.

Chaos is common in nature.

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