The interplay between structural and functional networks is an urgent topic in neuroscientific research. The discrepancy in the respective topologies may help to unravel details how network structure facilitates or constrains function, i.e. information transfer. In a combined neural mass and graph theoretical model, it was found that patterns of functional connectivity are influenced by the corresponding structural level [1]. Functional connectivity is often being defined through the synchronization between activities at different nodes. These activities are considered to stem from meso-scale neural populations that oscillate at certain frequencies with certain amplitudes. The amplitude of a local neural population reflects the degree of synchronization across its members. It will be illustrated how this amplitude can affect the phase dynamics in neural networks by approximating the node dynamics as self-sustaining, weakly non-linear oscillators. The dynamics of these populations can be derived from networks of spiking neurons in mean-field approximation. The resulting neural mass models allow for deducing the corresponding phase dynamics proper [2,3]. Dependent on the type of neural mass, the phase dynamics may be influenced by the amplitudes of the individual oscillators. The corresponding phases obeys the form of a Kuramoto-like network [4]. It will be discussed how the functional connectivity between phases depends on the structural connectivity but also on the oscillators’ amplitudes. In consequence, phase dynamics and, hence, synchrony patterns should always be analyzed in conjunction with the corresponding changes in amplitude [4]. These results will be extended to the case of two or more coupled networks [5] to bring the theoretical findings closer to neuro-imaging data and corresponding numerical studies. References [1] Ponten, Daffertshofer, Hillebrand, Stam (2010) The relationship between structural and functional connectivity: Graph theoretical analysis of an EEG neural mass model. NeuroImage, 52(3):985. [2] Ton, Deco, Daffertshofer (2014). Structure-function discrepancy: inhomogeneity and delays in synchronized neural networks. PLoS Comp. Biol., 10(7): e1003736. [3] Daffertshofer, van Wijk (2011). On the influence of amplitude on the connectivity between phases. Frontiers in Neuroinf., 5, art. 6. [4] Breakspear, Heitmann, Daffertshofer (2010) Generative models of cortical oscillations: From Kuramoto to the nonlinear Fokker–Planck equation, Frontiers Human Neurosci., 4, art. 190. [5] Pietras, Deschle, Daffertshofer (2016) Equivalence of coupled networks and networks with multimodal frequency distributions: Conditions for the bimodal and trimodal case, Phys. Rev. E 94, 052211.

The brain of mammalians, including humans, have the remarkable property of being endogenously active, even in the absence of any task or stimuli. In vivo and in vitro experiments at different resolution scales and employing diverse experimental techniques, consistently reveal the existence of intermittent outbursts of electro-chemical activity that propagate through neural networks in the form of "avalanches". The sizes and durations of neural avalanches are distributed as power laws, obey scaling, exhibit long-range correlations, and other characteristic features of critical points. These observations have elicited enthusiasm and attracted much interest among theoretitians, who took them as possible empirical evidence that some aspects of living systems (or parts or groups of them) could extract important functional benefits from operating at the edge of a continuous phase transition between two radically different phases, order and disorder. Criticality has been claimed to provide such systems with large susceptibility, huge dynamics ranges, large information processing and storing power, optimal computational capabilities, etc. In this talk I will discuss possible alternatives and/or complementary explanations to criticality for the empirically observed scale-free avalanches of neuronal activity. In particular, I will introduce two novel concepts in the field: self-organized bistability (SOB) which naturally extends the idea of self-organized criticality (SOC) to discontinuous phase transitions, and the neutral theory of neural dynamics, that borrows from important development in molecular and population genetics. Both of these concepts have shed light on the nature of neural dynamics and have opened new perspectives and novel research lines in this fascinating field, whose ultimate goal is to undertand how the amazingly complex behavior of the brain can possibly emerge from its underlying network of neurons and plastic synapses.

There is large consensus that the complex, self-organizing structure of the human brain can be well described by the mathematical framework based on network theory. Recently1, it has been proposed to characterize brain networks in terms of their “controllability”, drawing on concepts and methods of control theory. The analysis of controllability has the potential to unveil how specific nodes and/or sets of nodes control the dynamics of the entire network and thus might provide insights on whether manipulating the local activity of specific nodes would fully or partially restore network functions after brain damage. In this talk I will briefly summarise the theoretical framework of brain network controllability, and I will present results that challenge the state of the art knowledge on brain controllability. We conclude that, though theoretically intriguing, our understanding of the relationship between controllability and structural brain network remains elusive and we are far to translational applications of this concept.

Availability of multi-unit recordings from neurons has opened interesting approaches to neuronal coding. While single neuron recordings made it possible to look into neuronal coding at the single unit level, there is evidence for information being transmitted on the population level that cannot be seen from single units alone. In order to tackle this problem we have designed algorithms to identify the subpopulation among the larger group of recorded neurons that discriminates best between the presented stimuli.

It is well known that complex dynamical evolution may originate from simple, low-dimensional dynamical systems when a time-delayed feedback mechanism is considered. These may typically happen in systems where the propagation time of a signal is not negligible with respect to the typical timescale of the local dynamics. Examples include laser physics, where a long delayed feedback may be easily obtained by optical or electronic devices. In these systems, a deep analogy exists between delayed feedback and spatially extended dynamical systems. In particular, it is well known that deterministic systems with long time-delay may be interpreted in terms of a suitable spatiotemporal dynamics. In this work, we extend this interpretation to stochastic, time delayed dynamics. In particular, we consider a simple bistable system with long delay, and perturb it with a multiplicative noise that preserves one of the two minima. Comparing experimental data — obtained from a bistable semiconductor laser with long delay — with numerical simulations of a simple effective model, we argue that these systems exhibit a non-equilibrium phase transition into an absorbing state, possibly belonging to the well known Directed Percolation universality class.

The detection of interdependence of unknown dynamics from their signals is an important problem in many different fields. In particular in neuroscience, assessing the connectivity between neurons is a crucial task in order to understand the architecture of the brain. Initially, continuous signals like EEG and MEG were studied to reconstruct connectivity on a large scale. More recently the focus of attention has shifted to the analysis of microscopic data recorded from individual neurons in the form of discrete spike trains. Here we describe an approach based on the asymmetric state similarity criterion, in the formulation of the interdependence measure L [1]. It is an extension of a method for continuous signals [2], and it can detect both linear and nonlinear coupling. The approach is modular, for example different spike train distances can be used to assess similarity [3]. With tests on the Hindmarsh-Rose model system we show that this method is robust to noise and versatile with respect to different spike train regimes. Furthermore, its modularity leads to sensitivity to different coupling intensities. In closing we discuss the necessity of surrogate techniques for assessing significance in the application to real data. The implementations of the measure L and of the spike train distances are available online [4]. References: [1] R.G. Andrzejak, T. Kreuz. EPL (Europhysics Letters) 96, 50012 (2011) [2] D. Chicharro, R.G. Andrzejak. Physical Review E 80, 026217 (2009) [3] T. Kreuz. Scholarpedia 6, 11934 (2011) [4] http://ntsa.upf.edu/downloads; http://www.fi.isc.cnr.it/users/thomas.kreuz/sourcecode.html

Living systems execute an extraordinary plethora of complex functions, that result from the intertwined interactions among key microscopic actors. Positive and negative feedbacks appear to orchestrate the necessary degree of macroscopic coordination, by propagating information to distant sites while supporting the processing steps that underly categorization and decision making. Excitatory and inhibitory circuits play, in this respect, a role of paramount importance. As an example, networks of excitatory and inhibitory neurons constitute the primary computational units in the brain cortex and can adjust to dierent computational modalities, as triggered by distinct external stimuli. Genetic regulation also relies on sophisticated inhibitory and excitatory loops. Working in this context, we shall here discuss a minimal model for a discrete collections of agents in mutual interaction via excitatory and inhibitory loops, bearing universality traits in light of its inherent simplicity. Endogenous-noise stemming from finite size corrections induces quasi-cyclic dynamics that display unusual long range correlations, persisting over arbitrary large network structures. When the excitatory and inhibitory species are distributed on a directed network, the internal noise seeds giant quasi-cycles, with tunable frequency and amplitude. The system spontaneously behaves as an effective, stochastic driven pacemaker, a non trivial self-organized dynamics that holds general interest, for its fundamental to applied implications. The phenomenon is characterized analytically. The theory prediction are corroborated by direct stochastic simulations.

Voltage-sensitive dye imaging (VSDi) is able to reveal fundamental proprieties of cortical activity (Arieli et al. 1996). According to its spatial resolution it is reasonable to investigate results from such measurements through mean field models of neural activity, describing the collective state of a large number of neural units. In this talk I will describe results from VSDi measures in fixating monkeys and compare them with mean field models of asynchronous irregular activity in the cortex.

Large-scale neuronal dynamics models (i.e, whole-brain models) have attracted great attention of the scientific community to unveil key mechanisms about the relationship between brain function and brain structure. Despite the vast amount of whole-brain models found in literature with different degrees of complexity, some empirical signatures of brain at rest, such as the functional connections (FC) between brain regions and the emergence of resting state networks (RSNs) are only partially understood and still poorly reproduced. The increase in the predictive power of whole-brain models is a fundamental step that can trigger many fundamental applications, such as quantitative discriminations between healthy and lesioned subjects. It has been shown [1] that brain at rest may be poised near a critical state, defined as the special point in the space of parameters where the system displays the maximum susceptibility/complexity, i.e. it optimally and collectively responds to external inputs [2]. Exploiting this notion of criticality and following seminal work of Haimovici et al. [2], we propose a biological meaningful yet simple stochastic model which is able to predict the brain organization into resting state networks through only few independent parameters. In this study we developed biological meaningful parametrization of the structural connectivity matrix (SC) (i.e., the human connectome) that increases the match between simulated and empirical data. In addition, using our modeling approach we are able to distinguish between simulated healthy and lesioned brains. [1] J. M. Beggs and D. Plenz, Neuronal avalanches in neocortical circuits (2003). [2] A. Haimovici, E. Tagliazucchi, P. Balenzuela, D. R. Chialvo, Brain organization into resting state networks emerges at criticality on a model of the human connectome (2013).

A stochastic version of the Wilson Cowan model is considered, which accommodates for a discrete population of excitatory and inhibitory neurons. The model assumes a finite carrying capacity, the two populations being hence constant in size. The master equation that governs the dynamics of the stochastic model is expanded in powers of the inverse population size, yielding a coupled pair of non linear Langevin equations with multiplicative noise. Numerical simulations point to the validity of the obtained fluctuating hydrodynamics approximation, in the region of dynamical bistability. Analytical progress is possible when silencing the retroaction of the activators on the inhibitors, while still assigning the parameters so to fall in the region of deterministic bistability for the excitatory species. The proposed approach forms the basis of a perturbative generalization which applies to the case where a modest degree of coupling is restored.