Henrik Jeldtoft Jensen

Universal fluctuations in correlated systems

The probability density function (PDF) of a global measure in a large class of highly correlated systems has been suggested to be of the same functional form. Here, we identify the analytical form of the PDF of one such measure, the order parameter in the low temperature phase of the 2D XY model. We demonstrate that this function describes the fluctuations of global quantities in other correlated equilibrium and nonequilibrium systems. These include a coupled rotor model, Ising and percolation models, models of forest fires, sandpiles, avalanches, and granular media in a self-organized critical state. We discuss the relationship with both Gaussian and extremal statistics.

Self-similar correlation function in brain resting-state functional magnetic resonance imaging.

Adaptive behaviour, cognition and emotion are the result of a bewildering variety of brain spatio-temporal activity patterns. An important problem in neuroscience is to understand the mechanism by which the human brains 100 billion neurons and 100 trillion synapses manage to produce this large repertoire of cortical configurations in a flexible manner. In addition, it is recognized that temporal correlations across such configurations cannot be arbitrary, but they need to meet two conflicting demands: while diverse cortical areas should remain functionally segregated from each other, they must still perform as a collective, i.e. they are functionally integrated. Here, we investigate these large-scale dynamical properties by inspecting the character of the spatio-temporal correlations of brain resting-state activity. In physical systems, these correlations in space and time are captured by measuring the correlation coefficient between a signal recorded at two different points in space at two different times. We show that this two-point correlation function extracted from resting-state functional magnetic resonance imaging data exhibits self-similarity in space and time. In space, self-similarity is revealed by considering three successive spatial coarse-graining steps while in time it is revealed by the 1/f frequency behaviour of the power spectrum. The uncovered dynamical self-similarity implies that the brain is spontaneously at a continuously changing (in space and time) intermediate state between two extremes, one of excessive cortical integration and the other of complete segregation. This dynamical property may be seen as an important marker of brain well-being in both health and disease.

Open questions in the magnetic behaviour of high-temperature superconductors

A principally experimental review of vortex behaviour in high-temperature superconductors is presented. The reader is first introduced to the basic concepts needed to understand the magnetic properties of type II superconductors. The concepts of vortex melting, the vortex glass, vortex creep, etc are also discussed briefly. The bulk part of the review relates the theoretical predictions proposed for the vortex system in high temperature superconductors to experimental findings. The review ends with an attempt to direct the reader to those areas which still require further clarification.

On the critical behaviour of simple epidemics

We show how ideas and models which were originally introduced to gain an understanding of critical phenomena can be used to interpret the dynamics of epidemics of communicable disease in real populations. Specifically, we present an analysis of the dynamics of disease outbreaks for three common communicable infections from a small isolated island population. The strongly fluctuating nature of the temporal incidence of disease is captured by the model, and comparisons between exponents calculated from the data and from simulations are made. A forest–fire model with sparks is used to classify the observed scaling dynamics of the epidemics and provides a unified picture of the epidemiology which conventional epidemiological analysis is unable to reproduce. This study suggests that power–law scaling can emerge in natural systems when they are driven on widely separated time–scales, in accordance with recent analytic renormalization group calculations.

Evolution in complex systems

What features characterize complex system dynamics? Power laws and scale invariance of fluctuations are often taken as the hallmarks of complexity, drawing on analogies with equilibrium critical phenomena. Here we argue that slow, directed dynamics, during which the system’s properties change significantly, is fundamental. The underlying dynamics is related to a slow, decelerating but spasmodic release of an intrinsic strain or tension. Time series of a number of appropriate observables can be analyzed to confirm this effect. The strain arises from local frustration. As the strain is released through “quakes,” some system variable undergoes record statistics with accompanying log-Poisson statistics for the quake event times. We demonstrate these phenomena via two very different systems: a model of magnetic relaxation in type II superconductors and the Tangled Nature model of evolutionary ecology and show how quantitative indications of aging can be found.

Dynamical and spatial aspects of sandpile cellular automata

The Bak, Tang, and Wiesenfeld cellular automaton is simulated in 1, 2, 3, 4, and 5 dimensions. We define a (new) set of scaling exponents by introducing the concept of conditional expectation values. Scaling relations are derived and checked numerically and the critical dimension is discussed. We address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2. The scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation. Primarily, the results of our work show that the flow of sand down the slope does not have a 1/f power spectrum in any dimension, although the model does show clear critical behavior with scaling exponents depending on the dimension. The power spectrum behaves as 1/f2 in all the dimensions considered.

Network properties, species abundance and evolution in a model of evolutionary ecology

We study the evolution of the network properties of a populated network embedded in a genotype space characterized by either a low or a high number of potential links, with particular emphasis on the connectivity and clustering. Evolution produces two distinct types of network. When a specific genotype is only able to influence a few other genotypes, the ecosystem consists of separate non-interacting clusters (i.e. isolated compartments) in genotype space. When different types may influence a large number of other sites, the network becomes one large interconnected cluster. The distribution of interaction strengths--but not the number of connections--changes significantly with time. We find that the species abundance is only realistic for a high level of species connectivity. This suggests that real ecosystems form one interconnected whole in which selection leads to stronger interactions between the different types. Analogies with niche and neutral theory and assembly models are also considered.

Universal fluctuations and extreme-value statistics

We study the effect of long-range algebraic correlations on extreme-value statistics and demonstrate that correlations can produce a limit distribution which is indistinguishable from the ubiquitous Bramwell–Holdsworth–Pinton distribution. We also consider the square-width fluctuations of the avalanche signal. We find, as recently predicted by Antal et al for logarithmic correlated 1/f signals, that these fluctuations follow the Fisher–Tippett–Gumbel distribution from uncorrelated extreme-value statistics.

Vortex Fluctuations, Negative Hall Effect, and Thermally Activated Resistivity in Layered and Thin-Film Superconductors in an External Magnetic Field

The thermally activated resistivity, rxx, and the negative Hall resistivity, rxy are explained as two consequences of the same effect, namely the unbinding of vortex pairs in the vicinity of Tc. Both rxx and rxy exhibit a thermally activated behaviour. The activation energy depends logarithmically on the magnetic field. Our explanation suggests rxy ~ rxxa with a = 1 in accordance with recent measurements.

Criticality as a signature of healthy neural systems

This Research Topic in “Frontiers in Systems Neuroscience” contains a collection of original contributions and review articles on the hypothesis that the normal, healthy brain resides in a critical state. The hypothesis that brain activity, or specifically, neuronal activity in the cortex, might be critical arose from the premise that a critical brain can show the fastest and most flexible adaptation to a rather unpredictable environment (for review see Chialvo, 2010). Over the last decade, numerous signatures of criticality have been identified in brain activity. Some of the most striking examples are the probability distributions of size and duration for intermittent spontaneous activity bursts during ongoing activity in the cortex (Beggs and Plenz, 2003). These distributions have been found to follow power laws, which are conserved across species [rat: (Gireesh and Plenz, 2008); non-human primate: (Petermann et al., 2009; Yu et al., 2011); MEG: (Poil et al., 2012; Palva et al., 2013; Shriki et al., 2013); EEG: (Meisel et al., 2013); fMRI: (Tagliazucchi et al., 2012; Haimovici et al., 2013)] and experimental preparations, spanning from reduced in vitro models [i.e., acute and organotypic slices (Beggs and Plenz, 2003) and dissociated cultures (Pasquale et al., 2008)] to in vivo animal models (Gireesh and Plenz, 2008; Petermann et al., 2009; Ribeiro et al., 2010). These scale-free activation patterns, called neuronal avalanches, provide evidence for criticality in the brain. The bold claim that the brain is critical has elicited a healthy dose of skepticism and critique, often on technical grounds. For example, power laws are ubiquitous in nature, can potentially emerge from noise, and might not be particular to brain function (Touboul and Destexhe, 2010). This critique has refocused the debate on the specific, shallow exponents found for avalanche power laws, which demonstrate that unique, long-range spatial correlations are introduced by these dynamics, which require precisely balanced, weak interactions that differ from noise (Klaus et al., 2011). Similarly, discussion about the proper power law model and functional fit (Clauset et al., 2009; Dehghani et al., 2012) has highlighted the importance of careful identification of power law cut-offs in avalanche distributions, and their correct incorporation into appropriate statistical models (e.g., Langlois et al., 2014; Yu et al., 2014). Importantly, alternative approaches to avalanche dynamics using temporal scaling (Hardstone et al., 2012) and spatial scaling of fluctuations in ongoing human brain activity (Haimovici et al., 2013) have brought further support to the hypothesis of criticality in the brain. As evidenced in the contributions to this Special Topic issue, the exploration and examination of brain activity in the framework of criticality represents a highly active, ongoing field of research. It has been shown that the distribution of silent times between consecutive avalanches displays a non-monotonic behavior (due to the slow alternation between up- and down-states, Scarpetta and De Candia, 2014), with a power law decay at short time scales (Lombardi et al., 2014). Further analyses (Lombardi et al., 2012) demonstrate that avalanche size and inter-avalanche silent times are correlated, and highlight that avalanche occurrence exhibits the characteristic periodicity of θ and β/γ oscillations. This observation is in line with pharmacological results that connect nested oscillations and neuronal avalanches in cortex (Gireesh and Plenz, 2008). Experimental observations of long-term temporal correlations (Botcharova et al., 2014) in fluctuations of phase synchronization in EEG and MEG signals suggest that the driving mechanisms behind avalanche activity are non-local, with all scales contributing to system behavior. Indeed, an important “keyword” that characterizes such scale-free systems is the presence of a critical point, indicating the existence of a critical branching process as underlying structure that sustains this kind of dynamics. As it emerges in Yu et al. (2013), the ongoing resting activity in cortical networks organizes close to an effective thermodynamic critical point, suggesting the possibility that a critical state may in effect be described by methodology from thermodynamic equilibrium. As reviewed in Hesse and Gross (2014), a critical system displays optimal computational properties, indicating that criticality has been evolutionarily selected as a useful feature for the nervous system. The progress in the field of criticality and brain dynamics is further demonstrated by the fact that current discussions, rather than rejecting criticality altogether, are often focused on the proximity of brain dynamics to the critical point under different conditions. Based on in vivo recordings of extracellular spiking activity and modeling work, it has been concluded that the brain does not reflect a critical state, but its emergent dynamics might self-organize to a slightly sub-critical regime (Priesemann et al., 2014). To reside in such a regime can be considered an advantage, since it might prevent brain activity from becoming epileptic, which has been associated with supercritical dynamics (Meisel et al., 2012). Based on modeling (Tomen et al., 2014), it has been suggested that cortical networks, by operating at the sub-critical to critical transition region, could dramatically enhance stimulus representation. Thus, if the brain works close to or at a critical point, it is interesting to investigate the role of criticality on cognition and long-term temporal correlations observed in behavioral studies (Papo, 2014). Moreover, little is known about the causes and/or consequences of a loss of criticality, and its relation with brain diseases (e.g., epilepsy). The study of how pathogenic mechanisms are related to the critical/non-critical behavior of neuronal networks would likely provide new insights into the cellular and synaptic determinants supporting the emergence of critical-like dynamics and structures in neural systems. At the same time, the relationship between disrupted criticality and impaired behavior would help clarify the role of critical dynamics in normal brain functioning. In this Research Topic, (Tinker and Perez Velazquez, 2014) studied whether power law scaling can be achieved in the distribution of phase synchronization derived from MEG recordings, acquired from children with or without autism performing executive function tasks. Interestingly, (Roberts et al., 2014) point out an issue not well explored in previous works: i.e., that existing models lack precise physiological descriptions for how the brain maintains its tuning near a critical point. The authors claim that a missing fundamental ingredient is a formulation of the reciprocal coupling between neural activity and metabolic resources. Recent findings are aligned with the authors idea, which emerged from the analysis of disorders involving severe metabolic disturbances and altered scale-free properties of brain dynamics. The hypothesis that cortical dynamics resides at a critical point, at which information processing is optimized, has refocused attempts to explain the tremendous variability in neuronal activity patterns observed in the brain at all scales. Over the last several years, this hypothesis has given rise to numerous conferences and workshops on the brain and criticality (Plenz and Niebur, 2014). The current Research Topic continues the endeavor to explore one of the most exciting current concepts on brain function.