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Concept: Dynamical system


There is a popular belief in neuroscience that we are primarily data limited, and that producing large, multimodal, and complex datasets will, with the help of advanced data analysis algorithms, lead to fundamental insights into the way the brain processes information. These datasets do not yet exist, and if they did we would have no way of evaluating whether or not the algorithmically-generated insights were sufficient or even correct. To address this, here we take a classical microprocessor as a model organism, and use our ability to perform arbitrary experiments on it to see if popular data analysis methods from neuroscience can elucidate the way it processes information. Microprocessors are among those artificial information processing systems that are both complex and that we understand at all levels, from the overall logical flow, via logical gates, to the dynamics of transistors. We show that the approaches reveal interesting structure in the data but do not meaningfully describe the hierarchy of information processing in the microprocessor. This suggests current analytic approaches in neuroscience may fall short of producing meaningful understanding of neural systems, regardless of the amount of data. Additionally, we argue for scientists using complex non-linear dynamical systems with known ground truth, such as the microprocessor as a validation platform for time-series and structure discovery methods.

Concepts: Brain, Mathematics, Systems, Neuroscience, Information, Knowledge, Logic, Dynamical system


A new tool for visualization and analysis of system dynamics is introduced: the phasegram. Its application is illustrated with both classical nonlinear systems (logistic map and Lorenz system) and with biological voice signals. Phasegrams combine the advantages of sliding-window analysis (such as the spectrogram) with well-established visualization techniques from the domain of nonlinear dynamics. In a phasegram, time is mapped onto the x-axis, and various vibratory regimes, such as periodic oscillation, subharmonics or chaos, are identified within the generated graph by the number and stability of horizontal lines. A phasegram can be interpreted as a bifurcation diagram in time. In contrast to other analysis techniques, it can be automatically constructed from time-series data alone: no additional system parameter needs to be known. Phasegrams show great potential for signal classification and can act as the quantitative basis for further analysis of oscillating systems in many scientific fields, such as physics (particularly acoustics), biology or medicine.

Concepts: Fundamental physics concepts, Physics, Logistic map, Chaos theory, Dynamical system, Nonlinear system, Bifurcation diagram, Seasonality


Most natural odors have sparse molecular composition. This makes the principles of compressed sensing potentially relevant to the structure of the olfactory code. Yet, the largely feedforward organization of the olfactory system precludes reconstruction using standard compressed sensing algorithms. To resolve this problem, recent theoretical work has shown that signal reconstruction could take place as a result of a low dimensional dynamical system converging to one of its attractor states. However, the dynamical aspects of optimization slowed down odor recognition and were also found to be susceptible to noise. Here we describe a feedforward model of the olfactory system that achieves both strong compression and fast reconstruction that is also robust to noise. A key feature of the proposed model is a specific relationship between how odors are represented at the glomeruli stage, which corresponds to a compression, and the connections from glomeruli to third-order neurons (neurons in the olfactory cortex of vertebrates or Kenyon cells in the mushroom body of insects), which in the model corresponds to reconstruction. We show that should this specific relationship hold true, the reconstruction will be both fast and robust to noise, and in particular to the false activation of glomeruli. The predicted connectivity rate from glomeruli to third-order neurons can be tested experimentally.

Concepts: Mathematics, Olfactory bulb, Olfaction, Olfactory system, Odor, Attractor, Dynamical system, Lorenz attractor


The characterization of heart dynamics with a view to distinguish abnormal from normal behavior is an interesting topic in clinical sciences. Here we present an analysis of the Electro-cardiogram (ECG) signals from several healthy and unhealthy subjects using the framework of dynamical systems approach to multifractal analysis. Our analysis differs from the conventional nonlinear analysis in that the information contained in the amplitude variations of the signal is being extracted and quantified. The results thus obtained reveal that the attractor underlying the dynamics of the heart has multifractal structure and the variations in the resultant multifractal spectra can clearly separate healthy subjects from unhealthy ones. We use supervised machine learning approach to build a model that predicts the group label of a new subject with very high accuracy on the basis of the multifractal parameters. By comparing the computed indices in the multifractal spectra with that of beat replicated data from the same ECG, we show how each ECG can be checked for variations within itself. The increased variability observed in the measures for the unhealthy cases can be a clinically meaningful index for detecting the abnormal dynamics of the heart.

Concepts: Scientific method, Psychology, Behavior, Normality, Attractor, Dynamical system, Supervised learning, Lorenz attractor


This paper presents a heuristic proof (and simulations of a primordial soup) suggesting that life-or biological self-organization-is an inevitable and emergent property of any (ergodic) random dynamical system that possesses a Markov blanket. This conclusion is based on the following arguments: if the coupling among an ensemble of dynamical systems is mediated by short-range forces, then the states of remote systems must be conditionally independent. These independencies induce a Markov blanket that separates internal and external states in a statistical sense. The existence of a Markov blanket means that internal states will appear to minimize a free energy functional of the states of their Markov blanket. Crucially, this is the same quantity that is optimized in Bayesian inference. Therefore, the internal states (and their blanket) will appear to engage in active Bayesian inference. In other words, they will appear to model-and act on-their world to preserve their functional and structural integrity, leading to homoeostasis and a simple form of autopoiesis.

Concepts: Mathematics, Systems, Systems theory, Dynamical system, Dynamical systems theory, Ergodic theory, Poincaré recurrence theorem, Ergodic hypothesis


Extreme events are ubiquitous in a wide range of dynamical systems, including turbulent fluid flows, nonlinear waves, large-scale networks, and biological systems. We propose a variational framework for probing conditions that trigger intermittent extreme events in high-dimensional nonlinear dynamical systems. We seek the triggers as the probabilistically feasible solutions of an appropriately constrained optimization problem, where the function to be maximized is a system observable exhibiting intermittent extreme bursts. The constraints are imposed to ensure the physical admissibility of the optimal solutions, that is, significant probability for their occurrence under the natural flow of the dynamical system. We apply the method to a body-forced incompressible Navier-Stokes equation, known as the Kolmogorov flow. We find that the intermittent bursts of the energy dissipation are independent of the external forcing and are instead caused by the spontaneous transfer of energy from large scales to the mean flow via nonlinear triad interactions. The global maximizer of the corresponding variational problem identifies the responsible triad, hence providing a precursor for the occurrence of extreme dissipation events. Specifically, monitoring the energy transfers within this triad allows us to develop a data-driven short-term predictor for the intermittent bursts of energy dissipation. We assess the performance of this predictor through direct numerical simulations.

Concepts: Mathematics, Fundamental physics concepts, Fluid dynamics, Logistic map, Systems theory, Dynamical system, Dynamical systems, Dynamical systems theory


Physiological processes are regulated by nonlinear dynamical systems. Various nonlinear measures have frequently been used for characterizing the complexity of fractal time signals to detect system features that cannot be derived from linear analyses. We analysed human balance dynamics ranging from simple standing to balancing on one foot with closed eyes to study the inherent methodological problems when applying fractal dimension analysis to real-world signals. Higuchi dimension was used as an example. Choice of measurement and analysis parameters has a distinct influence on the computed dimension. Noise increases the fractional dimension which may be misinterpreted as a higher complexity of the signal. Publications without specifying the parameter setting, or without analysing the noise-sensitivity are not comparable to findings of others and therefore of limited scientific value.

Concepts: Scientific method, Mathematics, Chaos theory, Measurement, Systems theory, Attractor, Dynamical system, Dynamical systems


Quantitative biomechanical models can identify control parameters that are used during movements, and movement parameters that are encoded by premotor neurons. We fit a mathematical dynamical systems model including subsyringeal pressure, syringeal biomechanics and upper-vocal-tract filtering to the songs of zebra finches. This reduces the dimensionality of singing dynamics, described as trajectories (motor ‘gestures’) in a space of syringeal pressure and tension. Here we assess model performance by characterizing the auditory response ‘replay’ of song premotor HVC neurons to the presentation of song variants in sleeping birds, and by examining HVC activity in singing birds. HVC projection neurons were excited and interneurons were suppressed within a few milliseconds of the extreme time points of the gesture trajectories. Thus, the HVC precisely encodes vocal motor output through activity at the times of extreme points of movement trajectories. We propose that the sequential activity of HVC neurons is used as a ‘forward’ model, representing the sequence of gestures in song to make predictions on expected behaviour and evaluate feedback.

Concepts: Time, Mathematics, Cerebral cortex, Force, Biomechanics, Song, Dynamical system, Vocal music


We consider the problem of categorizing video sequences of dynamic textures, i.e., nonrigid dynamical objects such as fire, water, steam, flags, etc. This problem is extremely challenging because the shape and appearance of a dynamic texture continuously change as a function of time. State-of-the-art dynamic texture categorization methods have been successful at classifying videos taken from the same viewpoint and scale by using a Linear Dynamical System (LDS) to model each video, and using distances or kernels in the space of LDSs to classify the videos. However, these methods perform poorly when the video sequences are taken under a different viewpoint or scale. In this paper, we propose a novel dynamic texture categorization framework that can handle such changes. We model each video sequence with a collection of LDSs, each one describing a small spatiotemporal patch extracted from the video. This Bag-of-Systems (BoS) representation is analogous to the Bag-of-Features (BoF) representation for object recognition, except that we use LDSs as feature descriptors. This choice poses several technical challenges in adopting the traditional BoF approach. Most notably, the space of LDSs is not euclidean; hence, novel methods for clustering LDSs and computing codewords of LDSs need to be developed. We propose a framework that makes use of nonlinear dimensionality reduction and clustering techniques combined with the Martin distance for LDSs to tackle these issues. Our experiments compare the proposed BoS approach to existing dynamic texture categorization methods and show that it can be used for recognizing dynamic textures in challenging scenarios which could not be handled by existing methods.

Concepts: Mathematics, Logistic map, Dimension, Distance, Dynamical system, Dynamical systems, Dynamical systems theory


Many genetic oscillators (circadian clocks, synthetic oscillators) continue to oscillate accross the cell division cycle. Since cell divisions create discontinuities in the dynamics of genetic oscillators the question about the resilience of oscillations and the factors that contribute to the robustness of the oscillations may be raised. We study here, through stochastic simulations, the effect of the cell division cycle on genetic oscillations using the Repressilator - a genetic oscillator developed in the context of synthetic biology. We consider intrinsic noise (molecular noise due to the limited number of molecules) and extrinsic noise (variability in the cell division time and in the partition of the molecules into daughter cells, cell-cell variability in kinetic parameters, etc). Our numerical simulations show that, although noisy, oscillations are quite resilient to cell division and that cell-cell heterogeneity may be the main source of variability observed experimentally. Finally, similar simulations performed with another model, the Goodwin model, show that oscillations may be entrained and synchronized by cell division. This highlights the influence of the clock architecture on the robustness of genetic oscillations. Our approach provides a general framework to study the effect of cell division on dynamical systems and several possible extensions are described.

Concepts: Cell nucleus, Mathematics, Cell division, Cell cycle, Oscillation, Mitosis, Dynamical system, Structural stability