Tom Lorimer

Two universal physical principles shape the power-law statistics of real-world networks

The study of complex networks has pursued an understanding of macroscopic behaviour by focusing on power-laws in microscopic observables. Here, we uncover two universal fundamental physical principles that are at the basis of complex network generation. These principles together predict the generic emergence of deviations from ideal power laws, which were previously discussed away by reference to the thermodynamic limit. Our approach proposes a paradigm shift in the physics of complex networks, toward the use of power-law deviations to infer meso-scale structure from macroscopic observations.

Macroscopic bursting in physiological networks: node or network property?

Activity pattern modalities of neuronal ensembles are determined by node properties as well as network structure. For many purposes, it is of interest to be able to relate activity patterns to either node properties or to network properties (or to a combination of both). When in physiological neural networks we observe bursting on a coarse-grained time and space scale, a proper decision on whether bursts are the consequence of individual neurons with an inherent bursting property or whether we are dealing with a genuine network effect has generally not been possible because of the noise in these systems. Here, by linking different orders of time and space scales, we provide a simple coarse-grained criterion for deciding this question.

Mammalian cochlea as a physics guided evolution-optimized hearing sensor

Nonlinear physics plays an essential role in hearing. We demonstrate on a mesoscopic description level that during the evolutionary perfection of the hearing sensor, nonlinear physics led to the unique design of the cochlea observed in mammals, and that this design requests as a consequence the perception of pitch. Our insight challenges the view that mostly genetics is responsible for the uniformity of the construction of the mammalian hearing sensor. Our analysis also suggests that scaleable and non-scaleable arrangements of nonlinear sound detectors may be at the origin of the differences between hearing sensors in amniotic lineages.

Avalanche and edge-of-chaos criticality do not necessarily co-occur in neural networks

There are indications that for optimizing neural computation, neural networks may operate at criticality. Previous approaches have used distinct fingerprints of criticality, leaving open the question whether the different notions would necessarily reflect different aspects of one and the same instance of criticality, or whether they could potentially refer to distinct instances of criticality. In this work, we choose avalanche criticality and edge-of-chaos criticality and demonstrate for a recurrent spiking neural network that avalanche criticality does not necessarily entrain dynamical edge-of-chaos criticality. This suggests that the different fingerprints may pertain to distinct phenomena.

Frequency sensitivity in mammalian hearing from a fundamental nonlinear physics model of the inner ear

A dominant view holds that the outer and middle ear are the determining factors for the frequency dependence of mammalian hearing sensitivity, but this view has been challenged. In the ensuing debate, there has been a missing element regarding in what sense and to what degree the biophysics of the inner ear might contribute to this frequency dependence. Here, we show that a simple model of the inner ear based on fundamental physical principles, reproduces, alone, the experimentally observed frequency dependence of the hearing threshold. This provides direct cochlea modeling support of the possibility that the inner ear could have a substantial role in determining the frequency dependence of mammalian hearing.

Clustering: how much bias do we need?

Scientific investigations in medicine and beyond increasingly require observations to be described by more features than can be simultaneously visualized. Simply reducing the dimensionality by projections destroys essential relationships in the data. Similarly, traditional clustering algorithms introduce data bias that prevents detection of natural structures expected from generic nonlinear processes. We examine how these problems can best be addressed, where in particular we focus on two recent clustering approaches, Phenograph and Hebbian learning clustering, applied to synthetic and natural data examples. Our results reveal that already for very basic questions, minimizing clustering bias is essential, but that results can benefit further from biased post-processing. This article is part of the themed issue ‘Mathematical methods in medicine: neuroscience, cardiology and pathology’.

Power Laws in Neuronal Culture Activity from Limited Availability of a Shared Resource

We record spontaneous activity from a developing culture of dissociated rat hippocampal neurons in vitro using a multi electrode array. To statistically characterize activity, we look at the time intervals between recorded spikes, which, unlike neuronal avalanche sizes, do not require the selection of a time bin. The distribution of inter event intervals in our data approximate power laws at all recorded stages of development, with exponents that can be used to characterize the development of the culture. Synchronized bursting emerges as the culture matures, and these bursts show activity that decays approximately exponentially. From this, we propose a model for neuronal activity within bursts based on the consumption of a shared resource. Our model produces power law distributed avalanches in simulations, and is analytically demonstrated to produce power law distributed inter event intervals with an exponent close to that observed in our data. This indicates that power law distributions in neuronal avalanche size and other observables, can be also an artefact of exponentially decaying activity within synchronized bursts.

Complex Networks of Harmonic Structure in Classical Music

Music is a ubiquitous, complex and defining phenomenon of human culture. We create and analyze complex networks representing harmonic transitions in eight selected compositions of Johann Sebastian Bach’s Well-Tempered Clavier. While all resulting networks exhibit the typical ‘small-world’-characteristics, they clearly differ in their degree distributions. Some of the degree distributions are well fit by a power-law, others by an exponential, and some by neither. This seems to preclude the necessity of a scale-free degree distribution for music to be appealing. To obtain a quality measure for the network representation, we design a simple algorithm that generates artificial polyphonic music, which also exhibits the different styles of composition underlying the various pieces.

Hebbian Learning Clustering with Rulkov Neurons

The recent explosion of high dimensional, high resolution ‘big-data’ from automated bioinformatics measurement techniques demands new methods for unsupervised data processing. An essential analysis step is the identification of groups of similar data, or ‘clusters’, in noisy high-dimensional data spaces, as this permits to perform some analysis steps at the group level. Popular clustering algorithms introduce an undesired cluster shape bias, require prior knowledge of the number of clusters, and are unable to properly deal with noise. Manual data gating, often used to assist these methods, is based on low-dimensional projection techniques, which is prone to obscure the underlying data structure. While Hebbian Learning Clustering successfully overcomes all of these limitations (by using only local similarities to infer global structure), previous implementations were unsuited to deal with big data sets. Here, we present a novel implementation based on realistic neuronal dynamics that removes also this obstacle. By a performance that scales favourably compared to all standard clustering algorithms, unbiased large data analysis becomes feasible on standard desktop hardware.

Deviation from Criticality in Functional Biological Networks

Claims based on power laws that cognition occurs in a critical state often rely on the assumption that the network observables studied are observables of cognition, however this relationship to function is not clear. Our novel approach to investigate this problem is instead to consider functional output during (goal-directed) pre-copulatory courtship of Drosophila melanogaster, which we study as a complex network. This courtship body language, expressed through a symbolic dynamics, has previously been shown to be situation specific and grammatically complex; here, we show that the networks underlying it deviate from a scale-free structure when recursive grammars are included. This structural deviation is modelled by a simple network growth algorithm which adds internal edge saturation to the preferential attachment paradigm. From this, we suggest that a critical state may not be compatible with higher level cognition.