Subsampling scaling

Anna Levina, Viola Priesemann
Nature Communications, Vol. 8, 15140
Publication year: 2017

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.

Avalanche dynamics

Manfred Denker, Anna Levina
Stochastics and Dynamics, Vol. 16, No. 2, 1660005
Publication year: 2016

The avalanche transformation as a model for avalanches in neural dynamics was introduced in my dissertation in 2008. Here we discuss this transformation in terms of group actions, random dynamics and skew products with a finite invariant measure. Some open problems are mentioned.

Self-organization in balanced state networks by STDP and homeostatic plasticity

Felix Effenberger, Jürgen Jost, Anna Levina
PLoS Comput Biol., Vol. 11, No. 9, e1004420
Publication year: 2015

Structural inhomogeneities in synaptic efficacies have a strong impact on population response dynamics of cortical networks and are believed to play an important role in their functioning. However, little is known about how such inhomogeneities could evolve by means of synaptic plasticity. Here we present an adaptive model of a balanced neuronal network that combines two different types of plasticity, STDP and synaptic scaling. The plasticity rules yield both long-tailed distributions of synaptic weights and firing rates. Simultaneously, a highly connected subnetwork of driver neurons with strong synapses emerges. Coincident spiking activity of several driver cells can evoke population bursts and driver cells have similar dynamical properties as leader neurons found experimentally. Our model allows us to observe the delicate interplay between structural and dynamical properties of the emergent inhomogeneities. It is simple, robust to parameter changes and able to explain a multitude of different experimental findings in one basic network.

Theoretical neuroscience of self‐organized criticality: from formal approaches to realistic models

book (or chapter)
Anna Levina, J.Michael Herrmann, Theo Geisel
Criticality in Neural Systems, 417-436
Publication year: 2014

Book chapter, summarizing my the papers of previous years, but including also some unpublished results

The Abelian distribution

Anna Levina, J.Michael Herrmann
Stochastics and Dynamics, Vol. 14, No 03, 1450001
Publication year: 2014

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.

Critical dynamics in associative memory networks

Maximilian Uhlig, Anna Levina, J.Michael Herrmann, Theo Geisel
Frontiers in computational neuroscience, Vol. 7
Publication year: 2013

Critical behavior in neural networks is characterized by scale-free avalanche size distributions and can be explained by self-regulatory mechanisms. Theoretical and experimental evidence indicates that information storage capacity reaches its maximum in the critical regime. We study the effect of structural connectivity formed by Hebbian learning on the criticality of network dynamics. The network only endowed with Hebbian learning does not allow for simultaneous information storage and criticality. However, the critical regime can be stabilized by short-term synaptic dynamics in the form of synaptic depression and facilitation or, alternatively, by homeostatic adaptation of the synaptic weights. We show that a heterogeneous distribution of maximal synaptic strengths does not preclude criticality if the Hebbian learning is alternated with periods of critical dynamics recovery. We discuss the relevance of these findings for the flexibility of memory in aging and with respect to the recent theory of synaptic plasticity.

Impact of single links in competitive percolation

Jan Nagler, Anna Levina, Marc Timme
Nature Physics, 7 (3), 265-270
Publication year: 2011

How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness on gradual addition of links, was long believed to be continuous, but recent numerical evidence of ‘explosive percolation’ suggests that it might also be discontinuous if links compete for addition. Here we analyse 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 showing 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 substantial gaps, because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively.

Phase transitions towards criticality in a neural system with adaptive interactions

Anna Levina, J.Michael Herrmann, Theo Geisel
Physical Review Letters, 102 (11), 118110
Publication year: 2009

We analytically describe a transition scenario to self-organized criticality (SOC) that is new for physics as well as neuroscience; it combines the criticality of first and second-order phase transitions with a SOC phase. We consider a network of pulse-coupled neurons interacting via dynamical synapses, which exhibit depression and facilitation as found in experiments. We analytically show the coexistence of a SOC phase and a subcritical phase connected by a cusp bifurcation. Switching between the two phases can be triggered by varying the intensity of noisy inputs.

Limit correlation functions at zero for fixed trace random matrix ensembles

Friedrich Götze, Mikhail Gordin, Anna Levina
Journal of Mathematical Sciences, Vol. 147, No. 4,
Publication year: 2007

Dynamical synapses causing self-organized criticality in neural networks

Anna Levina, J.Michael Herrmann, Theo Geisel
Nature physics, Vol. 3, No. 12, 857-860
Publication year: 2007

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.

Criticality of avalanche dynamics in adaptive recurrent networks

Anna Levina, Udo Ernst, J. Michael Herrmann
Neurocomputing, 70, 1877–1881
Publication year: 2007

In many studies of self-organized criticality (SOC), branching processes were used to model the dynamics of the activity of the system during avalanches. This mathematical simplification was also adopted when investigating systems with a complicated connection topology including recurrent and subthreshold interactions. However, none of these studies really analyzed whether this convenient approximation was indeed applicable. In present paper we study the correspondences between avalanches generated by branching processes and by a fully connected neural network. The benefit from the analysis is not only the justification of such correspondence but also a simple learning rule, which allows self-organization of the network towards a critical state as recently observed in slice experiments

Critical branching processes in neural networks

Anna Levina, J.Michael Herrmann, Manfred Denker
PAMM, Vol. 7, No. 1, 1030701-1030702
Publication year: 2007

Self-organized criticality generates complex behavior in systems of simple  elements. It is observed in various biological neural systems and has been analyzed in simplified model systems. Branching processes often considered to be a mean-field approximation to the dynamics of critical systems. Here we study the validity of such an approximation for the case of a neural network.

Dynamical synapses give rise to a power-law distribution of neuronal avalanches

Anna Levina, Theo Geisel, J. Michael Herrmann
NIPS, 18, 771
Publication year: 2006

There is experimental evidence that cortical neurons show avalanche activity with the intensity of firing events being distributed as a power-law. We present a biologically plausible extension of a neural network which exhibits a power-law avalanche distribution for a wide range of connectivity parameters.