arXiv subjects
Publications and source records attributed to Anna Levina.
We propose a nonparametric parameter estimation of confidence intervals when the underlying has large or infinite variance. We explain the method by a simple numerical example and provide an application to estimate the coupling strength in neuronal networks.
Weights and directionality of the edges carry a large part of the information we can extract from a complex network. However, many network measures were formulated initially for undirected binary networks. The necessity to incorporate information about the weights led to the conception of the multiple extensions, particularly for definitions of the local clustering coefficient discussed here. We uncover that not all of these extensions are fully-weighted; some depend on the degree and thus change a lot when an infinitely small weight edge is exchanged for the absence of an edge, a feature that is not always desirable. We call these methods ``hybrid'' and argue that, in many situations, one should prefer fully-weighted definitions. After listing the necessary requirements for a method to analyze many various weighted networks properly, we propose a fully-weighted continuous clustering coefficient that satisfies all the previously proposed criteria while also being continuous with respect to vanishing weights. We demonstrate that the behavior and meaning of the Zhang--Horvath clustering and our new continuous definition provide complementary results and significantly outperform other definitions in multiple relevant conditions. Using synthetic and real-world examples, we show that when the network is inferred, noisy, or very heterogeneous, it is essential to use the fully-weighted clustering definitions.
Can we generate abstract aesthetic images without bias from natural or human selected image corpi? Are aesthetic images singled out in their correlation functions? In this paper we give answers to these and more questions. We generate images using compositional pattern-producing networks with random weights and varying architecture. We demonstrate that even with the randomly selected weights the correlation functions remain largely determined by the network architecture. In a controlled experiment, human subjects picked aesthetic images out of a large dataset of all generated images. Statistical analysis reveals that the correlation function is indeed different for aesthetic images.
It has long been hypothesized that operating close to the critical state is beneficial for natural and artificial systems. We test this hypothesis by evolving foraging agents controlled by neural networks that can change the system's dynamical regime throughout evolution. Surprisingly, we find that all populations, regardless of their initial regime, evolve to be subcritical in simple tasks and even strongly subcritical populations can reach comparable performance. We hypothesize that the moderately subcritical regime combines the benefits of generalizability and adaptability brought by closeness to criticality with the stability of the dynamics characteristic for subcritical systems. By a resilience analysis, we find that initially critical agents maintain their fitness level even under environmental changes and degrade slowly with increasing perturbation strength. On the other hand, subcritical agents originally evolved to the same fitness, were often rendered utterly inadequate and degraded faster. We conclude that although the subcritical regime is preferable for a simple task, the optimal deviation from criticality depends on the task difficulty: for harder tasks, agents evolve closer to criticality. Furthermore, subcritical populations cannot find the path to decrease their distance to criticality. In summary, our study suggests that initializing models near criticality is important to find an optimal and flexible solution.
Self-organized criticality has been proposed to be a universal mechanism for the emergence of scale-free dynamics in many complex systems, and possibly in the brain. While such scale-free patterns were identified experimentally in many different types of neural recordings, the biological principles behind their emergence remained unknown. Utilizing different network models and motivated by experimental observations, synaptic plasticity was proposed as a possible mechanism to self-organize brain dynamics towards a critical point. In this review, we discuss how various biologically plausible plasticity rules operating across multiple timescales are implemented in the models and how they alter the network's dynamical state through modification of number and strength of the connections between the neurons. Some of these rules help to stabilize criticality, some need additional mechanisms to prevent divergence from the critical state. We propose that rules that are capable of bringing the network to criticality can be classified by how long the near-critical dynamics persists after their disabling. Finally, we discuss the role of self-organization and criticality in computation. Overall, the concept of criticality helps to shed light on brain function and self-organization, yet the overall dynamics of living neural networks seem to harnesses not only criticality for computation, but also deviations thereof.
Spreading processes are conventionally monitored on a macroscopic level by counting the number of incidences over time. The spreading process can then be modeled either on the microscopic level, assuming an underlying interaction network, or directly on the macroscopic level, assuming that microscopic contributions are negligible. The macroscopic characteristics of both descriptions are commonly assumed to be identical. In this work, we show that these characteristics of microscopic and macroscopic descriptions can be different due to coalescence, i.e., a node being activated at the same time by multiple sources. In particular, we consider a (microscopic) branching network (probabilistic cellular automaton) with annealed connectivity disorder, record the macroscopic activity, and then approximate this activity by a (macroscopic) branching process. In this framework, we analytically calculate the effect of coalescence on the collective dynamics. We show that coalescence leads to a universal non-linear scaling function for the conditional expectation value of successive network activity. This allows us to quantify the difference between the microscopic model parameter and established macroscopic estimates. To overcome this difference, we propose a non-linear estimator that correctly infers the model branching parameter for all system sizes.
The dynamic range of stimulus processing in living organisms is much larger than a single neural network can explain. For a generic, tunable spiking network we derive that while the dynamic range is maximal at criticality, the interval of discriminable intensities is very similar for any network tuning due to coalescence. Compensating coalescence enables adaptation of discriminable intervals. Thus, we can tailor an ensemble of networks optimized to the distribution of stimulus intensities, e.g., extending the dynamic range arbitrarily. We discuss potential applications in machine learning.
Power laws in nature are considered to be signatures of complexity. The theory of self-organized criticality (SOC) was proposed to explain their origins. A longstanding principle of SOC is the \emph{separation of timescales} axiom. It dictates that external input is delivered to the system at a much slower rate compared to the timescale of internal dynamics. The statistics of neural avalanches in the brain was demonstrated to follow a power law, indicating closeness to a critical state. Moreover, criticality was shown to be a beneficial state for various computations leading to the hypothesis, that the brain is a SOC system. However, for neuronal systems that are constantly bombarded by incoming signals, separation of timescales assumption is unnatural. Recently it was experimentally demonstrated that a proper correction of the avalanche detection algorithm to account for the increased drive during task performance leads to a change of the power-law exponent from $1.5$ to approximately $1.3$, but there is so far no theoretical explanation for this change. Here we investigate the importance of timescales separation, by partly abandoning it in various models. We achieve it by allowing for external input during the avalanche, without compromising the separation of avalanches. We develop an analytic treatment and provide numerical simulations of a simple neuronal model.
We define the Abelian distribution and study its basic properties. Abelian distributions arise in the context of neural modeling and describe the size of neural avalanches in fully-connected integrate-and-fire models of self-organized criticality in neural systems.
In real-world applications, observations are often constrained to a small fraction of a system. Such spatial subsampling can be caused by the inaccessibility or the sheer size of the system, and cannot be overcome by longer sampling. Spatial subsampling can strongly bias inferences about a system's aggregated properties. To overcome the bias, we derive analytically a subsampling scaling framework that is applicable to different observables, including distributions of neuronal avalanches, of number of people infected during an epidemic outbreak, and of node degrees. We demonstrate how to infer the correct distributions of the underlying full system, how to apply it to distinguish critical from subcritical systems, and how to disentangle subsampling and finite size effects. Lastly, we apply subsampling scaling to neuronal avalanche models and to recordings from developing neural networks. We show that only mature, but not young networks follow power-law scaling, indicating self-organization to criticality during development.
How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness upon gradual addition of links, was long believed to be continuous but recent numerical evidence on "explosive percolation" suggests that it might as well be discontinuous if links compete for addition. Here we analyze the microscopic mechanisms underlying discontinuous percolation processes and reveal a strong impact of single link additions. We show that in generic competitive percolation processes, including those displaying explosive percolation, single links do not induce a discontinuous gap in the largest cluster size in the thermodynamic limit. Nevertheless, our results highlight that for large finite systems single links may still induce observable gaps because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, thus announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively.
We show that a network of spiking neurons exhibits robust self-organized criticality if the synaptic efficacies follow realistic dynamics. Deriving analytical expressions for the average coupling strengths and inter-spike intervals, we demonstrate that networks with dynamical synapses exhibit critical avalanche dynamics for a wide range of interaction parameters. We prove that in the thermodynamical limit the network becomes critical for all large enough coupling parameters. We thereby explain experimental observations in which cortical neurons show avalanche activity with the total intensity of firing events being distributed as a power-law.