The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical observables are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest's theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the . The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie-Poisson structure associated with another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models that have previously appeared in the chemical physics literature.

Attractor reconstruction analysis has previously been applied to analyse arterial blood pressure and photoplethysmogram signals. This study extends this novel technique to ECG signals. We show that the method gives high accuracy in identifying gender from ECG signals, performing significantly better than the same classification by interval measures. IntroductionTraditional analysis of electrocardiogram (ECG) signals consists of identifying particular points and deriving lengths of various intervals from these points [1]. For instance, Heart Rate Variability (HRV) methods are used to analyse beat-to-beat intervals [2]. However, such approaches exclude the wealth of diagnostic information contained in the remainder of the waveform shape.Our novel attractor reconstruction (AR) method uses all of the available data and so retains the underlying waveform information. It therefore goes "beyond HRV" and can characterise changes in signal morphology that are not detected by HRV [3]. AR analysis has previously been applied to arterial blood pressure (ABP) [3,4] and photoplethysmogram (PPG) [5] signals, where it has been shown to supplement standard cardiovascular assessment.The aim of this study is to demonstrate how the AR method can be applied to ECG signals. We illustrate this by comparing the performance of our AR method with standard interval analysis for identifying gender from ECG signals, both in a normal state and when dosed with drugs that impact ventricular repolarisation. It is well known that there are significant differences in various ECG parameters between females and males, including the PR interval, QRS duration and QT interval [6,7]. Normal interval ranges may be gender-specific and certain cardiovascular diseases have a higher prevalence in one sex. Differences in ventricular repolarisation are of key interest, especially in the development and application of drugs that have a QT-prolonging effect [8]. Attractor Reconstruction AnalysisThe attractor reconstruction method has been described previously [3][4][5], so only an overview is given here. The AR approach provides a means of analysing approximately periodic signals that may be irregular, strongly nonstationary and noisy, supporting its application to ECG signals. Moreover, it uses all of the data and so can detect changes in the morphology of the signal.The first step of the AR method consists of embedding the one-dimensional ECG time series, which we denote by x(t), in a three-dimensional phase space by using Takens' delay coordinate method [9]. From the original time series, we generate two further time series y(t) = x(t − τ ) and z(t) = x(t − 2τ ), where τ > 0 is a fixed time delay, taken as one third of the average cycle length of the data, (i.e. one third of the cardiac cycle duration). A plot of (x, y, z) gives the reconstructed attractor in the bounded three-dimensional phase space (see Fig. 1(ii)).Baseline variation, e.g. due to respiration or motion, in x(t) is removed by projecting the attractor onto a plane perpendicular to the...

The aim of this work is to distinguish between wild-type mice and Scn5a +/− mutant mice using short ECG signals. This mutation results in impaired cardiac sodium channel function and is associated with increased ventricular arrhythmogenic risk which can result in sudden cardiac death. Lead I and Lead II ECG signals from wild-type and Scn5a +/− mice are used and the mice are also grouped as female/male and young/old.We use our novel Symmetric Projection Attractor Reconstruction (SPAR) method to generate an attractor from the ECG signal using all of the available waveform data. We have previously manually extracted a variety of quantitative measures from the attractor and used machine learning to classify each animal as either wild-type or mutant. In this work, we take the attractor images and use these as input to a deep learning algorithm in order to perform the same classification. As there is only data available from 42 mice, we use a transfer learning approach in which a network that has been pretrained on millions of images is used as a starting point and the last few layers are changed in order to fine tune the network for the attractor images.The results for the transfer learning approach are not as good as for the manual features, which is not too surprising as the networks have not been trained on attractor images. However, this approach shows the potential for using deep learning for classification of attractor images.

BACKGROUND Life-threatening arrhythmias resulting from genetic mutations are often missed in current electrocardiogram (ECG) analysis. We combined a new method for ECG analysis that uses all the waveform data with machine learning to improve detection of such mutations from short ECG signals in a mouse model. OBJECTIVE We sought to detect consequences of Na 1 channel deficiencies known to compromise action potential conduction in comparisons of Scn5a 1/mutant and wild-type mice using short ECG signals, examining novel and standard features derived from lead I and II ECG recordings by machine learning algorithms. METHODS Lead I and II ECG signals from anesthetized wild-type and Scn5a 1/mutant mice of length 130 seconds were analyzed by extracting various groups of features, which were used by machine learning to classify the mice as wild-type or mutant. The features used were standard ECG intervals and amplitudes, as well as features derived from attractors generated using the novel Symmetric Projection Attractor Reconstruction method, which reformulates the whole signal as a bounded, symmetric 2-dimensional attractor. All the features were also combined as a single feature group. RESULTS Classification of genotype using the attractor features gave higher accuracy than using either the ECG intervals or the intervals and amplitudes. However, the highest accuracy (96%) was obtained using all the features. Accuracies for different subgroups of the data were obtained and compared. CONCLUSION Detection of the Scn5a 1/mutation from short mouse ECG signals with high accuracy is possible using our Symmetric Projection Attractor Reconstruction method.