From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior

The study of neuronal interactions is currently at the center of several neuroscience big collaborative projects (including the Human Connectome, the Blue Brain, the Brainome, etc.) which attempt to obtain a detailed map of the entire brain matrix. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of such synaptic interaction matrix. This work explores the application of free random variables (FRV) to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra of neural networks, we extend them to heavy-tailed distributions of interactions. More importantly, we derive analytically the behavior of eigenvector overlaps, which determine stability of the spectra. We observe that upon imposing the neuronal excitation/inhibition balance, although the eigenvalues remain unchanged, their stability dramatically decreases due to strong non-orthogonality of associated eigenvectors. It leads us to the conclusion that the understanding of the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of FRV analysis in a wide variety of branches disciplines, we hope that the results presented here foster additional application of these ideas in the area of brain sciences.Comment: 24 pages + 4 pages of refs, 8 figure

Evolutionary multiplayer games on graphs with edge diversity

Evolutionary game dynamics in structured populations has been extensively explored in past decades. However, most previous studies assume that payoffs of individuals are fully determined by the strategic behaviors of interacting parties and social ties between them only serve as the indicator of the existence of interactions. This assumption neglects important information carried by inter-personal social ties such as genetic similarity, geographic proximity, and social closeness, which may crucially affect the outcome of interactions. To model these situations, we present a framework of evolutionary multiplayer games on graphs with edge diversity, where different types of edges describe diverse social ties. Strategic behaviors together with social ties determine the resulting payoffs of interactants. Under weak selection, we provide a general formula to predict the success of one behavior over the other. We apply this formula to various examples which cannot be dealt with using previous models, including the division of labor and relationship- or edge-dependent games. We find that labor division facilitates collective cooperation by decomposing a many-player game into several games of smaller sizes. The evolutionary process based on relationship-dependent games can be approximated by interactions under a transformed and unified game. Our work stresses the importance of social ties and provides effective methods to reduce the calculating complexity in analyzing the evolution of realistic systems.Comment: 50 pages, 7 figure

Variability of X-ray binaries from an oscillating hot corona

The spectral and timing properties of an oscillating hot thermal corona are investigated. This oscillation is assumed to be due to a magneto-acoustic wave propagating within the corona and triggered by an external, non specified, excitation. A cylindrical geometry is adopted and, neglecting the rotation, the wave equation is solved in for different boundary conditions. The resulting X-ray luminosity, through thermal comptonization of embedded soft photons, is then computed, first analytically, assuming linear dependence between the local pressure disturbance and the radiative modulation. These calculations are also compared to Monte-Carlo simulations. The main results of this study are: (1) the corona plays the role of a low band-pass medium, its response to a white noise excitation being a at top noise Power Spectral Density (PSD) at low frequencies and a red noise at high frequency, (2) resonant peaks are present in the PSD. Their powers depend on the boundary conditions chosen and, more specifically, on the impedance adaptation with the external medium at the corona inner boundary. (3) The flat top noise level and break as well as the resonant peak frequencies are inversely proportional to the external radius rj. (4) Computed rms and f-spectra exhibit an overall increase of the variability with energy. Comparison with observed variability features, especially in the hard intermediate states of X-ray binaries are discussed.Comment: 12 pages, 7 figures, accepted for publication in MNRA

Active Learning of Multiple Source Multiple Destination Topologies

We consider the problem of inferring the topology of a network with $M$ sources and $N$ receivers (hereafter referred to as an $M$-by-$N$ network), by sending probes between the sources and receivers. Prior work has shown that this problem can be decomposed into two parts: first, infer smaller subnetwork components (i.e., $1$-by-$N$'s or $2$-by-$2$'s) and then merge these components to identify the $M$-by-$N$ topology. In this paper, we focus on the second part, which had previously received less attention in the literature. In particular, we assume that a $1$-by-$N$ topology is given and that all $2$-by-$2$ components can be queried and learned using end-to-end probes. The problem is which $2$-by-$2$'s to query and how to merge them with the given $1$-by-$N$, so as to exactly identify the $2$-by-$N$ topology, and optimize a number of performance metrics, including the number of queries (which directly translates into measurement bandwidth), time complexity, and memory usage. We provide a lower bound, $\lceil \frac{N}{2} \rceil$, on the number of $2$-by-$2$'s required by any active learning algorithm and propose two greedy algorithms. The first algorithm follows the framework of multiple hypothesis testing, in particular Generalized Binary Search (GBS), since our problem is one of active learning, from $2$-by-$2$ queries. The second algorithm is called the Receiver Elimination Algorithm (REA) and follows a bottom-up approach: at every step, it selects two receivers, queries the corresponding $2$-by-$2$, and merges it with the given $1$-by-$N$; it requires exactly $N-1$ steps, which is much less than all $\binom{N}{2}$ possible $2$-by-$2$'s. Simulation results over synthetic and realistic topologies demonstrate that both algorithms correctly identify the $2$-by-$N$ topology and are near-optimal, but REA is more efficient in practice

The mechanics of stochastic slowdown in evolutionary games

We study the stochastic dynamics of evolutionary games, and focus on the so-called `stochastic slowdown' effect, previously observed in (Altrock et. al, 2010) for simple evolutionary dynamics. Slowdown here refers to the fact that a beneficial mutation may take longer to fixate than a neutral one. More precisely, the fixation time conditioned on the mutant taking over can show a maximum at intermediate selection strength. We show that this phenomenon is present in the prisoner's dilemma, and also discuss counterintuitive slowdown and speedup in coexistence games. In order to establish the microscopic origins of these phenomena, we calculate the average sojourn times. This allows us to identify the transient states which contribute most to the slowdown effect, and enables us to provide an understanding of slowdown in the takeover of a small group of cooperators by defectors: Defection spreads quickly initially, but the final steps to takeover can be delayed substantially. The analysis of coexistence games reveals even more intricate behavior. In small populations, the conditional average fixation time can show multiple extrema as a function of the selection strength, e.g., slowdown, speedup, and slowdown again. We classify two-player games with respect to the possibility to observe non-monotonic behavior of the conditional average fixation time as a function of selection strength.Comment: Accepted for publication in the Journal of Theoretical Biology. Includes changes after peer revie

The global minima of the communicative energy of natural communication systems

Until recently, models of communication have explicitly or implicitly assumed that the goal of a communication system is just maximizing the information transfer between signals and `meanings'. Recently, it has been argued that a natural communication system not only has to maximize this quantity but also has to minimize the entropy of signals, which is a measure of the cognitive cost of using a word. The interplay between these two factors, i.e. maximization of the information transfer and minimization of the entropy, has been addressed previously using a Monte Carlo minimization procedure at zero temperature. Here we derive analytically the globally optimal communication systems that result from the interaction between these factors. We discuss the implications of our results for previous studies within this framework. In particular we prove that the emergence of Zipf's law using a Monte Carlo technique at zero temperature in previous studies indicates that the system had not reached the global optimum.Peer ReviewedPostprint (author's final draft

A multi-flow model for microquasars

We present a new picture for the central regions of Black Hole X-ray Binaries. In our view, these central regions have a multi-flow configuration which consists in (1) an outer standard accretion disc down to a transition radius r_J, (2) an inner magnetized accretion disc below r_J driving (3) a non relativistic self-collimated electron-proton jet surrounding, when adequate conditions for pair creation are met, (4) a ultra relativistic electron-positron beam. This accretion-ejection paradigm provides a simple explanation to the canonical spectral states, from radio to X/gamma-rays, by varying the transition radius r_J and disc accretion rate independently. Large values of r_J and low accretion rate correspond to Quiescent and Hard states. These states are characterized by the presence of a steady electron-proton MHD jet emitted by the disc below r_J. The hard X-ray component is expect to form at the jet basis. When r_J becomes smaller than the marginally stable orbit r_i, the whole disc resembles a standard accretion disc with no jet, characteristic of the Soft state. Intermediate states correspond to situations where r_J ~ r_i. At large accretion rate, an unsteady pair cascade process is triggered within the jet axis, giving birth to flares and ejection of relativistic pair blobs. This would correspond to the luminous intermediate state, with its associated superluminal motions.Comment: 12 pages, 3 figures. Proceedings of ``High Energies in the Highlands'', Fort-William, 27 June-1 July 200