Optimal input reverberation and homeostatic self-organization toward the edge of synchronization.

Transient or partial synchronization can be used to do computations, although a fully synchronized network is sometimes related to the onset of epileptic seizures. Here, we propose a homeostatic mechanism that is capable of maintaining a neuronal network at the edge of a synchronization transition, thereby avoiding the harmful consequences of a fully synchronized network. We model neurons by maps since they are dynamically richer than integrate-and-fire models and more computationally efficient than conductance-based approaches. We first describe the synchronization phase transition of a dense network of neurons with different tonic spiking frequencies coupled by gap junctions. We show that at the transition critical point, inputs optimally reverberate through the network activity through transient synchronization. Then, we introduce a local homeostatic dynamic in the synaptic coupling and show that it produces a robust self-organization toward the edge of this phase transition. We discuss the potential biological consequences of this self-organization process, such as its relation to the Brain Criticality hypothesis, its input processing capacity, and how its malfunction could lead to pathological synchronization and the onset of seizure-like activity.

Less is different: Why sparse networks with inhibition differ from complete graphs.

In neuronal systems, inhibition contributes to stabilizing dynamics and regulating pattern formation. Through developing mean-field theories of neuronal models, using complete graph networks, inhibition is commonly viewed as one "control parameter" of the system, promoting an absorbing phase transition. Here, we show that, for low connectivity sparse networks, inhibition weight is not a control parameter of the absorbing transition. We present analytical and simulation results using generic stochastic integrate-and-fire neurons that, under specific restrictions, become other simpler stochastic neuron models common in literature, which allows us to show that our results are valid for those models as well. We also give a simple explanation about why the inhibition role depends on topology, even when the topology has a dimensionality greater than the critical one. The absorbing transition independence of the inhibitory weight may be an important feature of a sparse network, as it will allow the network to maintain a near-critical regime, self-tuning average excitation, but at the same time have the freedom to adjust inhibitory weights for computation, learning, and memory, exploiting the benefits of criticality.

Prime numbers and random walks in a square grid.

In recent years, computer simulations have played a fundamental role in unveiling some of the most intriguing features of prime numbers. In this paper, we define an algorithm for a deterministic walk through a two-dimensional grid, which we refer to as a prime walk. The walk is constructed from a sequence of steps dictated by and dependent on the sequence of the last digits of the primes. Despite the apparent randomness of this generating sequence, the resulting structure-in both two and three dimensions-created by the algorithm presents remarkable properties and regularities in its pattern, which we proceed to analyze in detail.

Neuronal avalanches in Watts-Strogatz networks of stochastic spiking neurons.

Networks of stochastic leaky integrate-and-fire neurons, both at the mean-field level and in square lattices, present a continuous absorbing phase transition with power-law neuronal avalanches at the critical point. Here we complement these results showing that small-world Watts-Strogatz networks have mean-field critical exponents for any rewiring probability p>0. For the ring (p=0), the exponents are the same from the dimension d=1 of the directed-percolation class. In the model, firings are stochastic and occur in discrete time steps, based on a sigmoidal firing probability function. Each neuron has a membrane potential that integrates the signals received from its neighbors. The membrane potentials are subject to a leakage parameter. We study topologies with a varied number of neuron connections and different values of the leakage parameter. Results indicate that the dynamic range is larger for p=0. We also study a homeostatic synaptic depression mechanism to self-organize the network towards the critical region. These stochastic oscillations are characteristic of the so-called self-organized quasicriticality.

Stochastic oscillations and dragon king avalanches in self-organized quasi-critical systems.

In the last decade, several models with network adaptive mechanisms (link deletion-creation, dynamic synapses, dynamic gains) have been proposed as examples of self-organized criticality (SOC) to explain neuronal avalanches. However, all these systems present stochastic oscillations hovering around the critical region that are incompatible with standard SOC. Here we make a linear stability analysis of the mean field fixed points of two self-organized quasi-critical systems: a fully connected network of discrete time stochastic spiking neurons with firing rate adaptation produced by dynamic neuronal gains and an excitable cellular automata with depressing synapses. We find that the fixed point corresponds to a stable focus that loses stability at criticality. We argue that when this focus is close to become indifferent, demographic noise can elicit stochastic oscillations that frequently fall into the absorbing state. This mechanism interrupts the oscillations, producing both power law avalanches and dragon king events, which appear as bands of synchronized firings in raster plots. Our approach differs from standard SOC models in that it predicts the coexistence of these different types of neuronal activity.

Correlations induced by depressing synapses in critically self-organized networks with quenched dynamics.

In a recent work, mean-field analysis and computer simulations were employed to analyze critical self-organization in networks of excitable cellular automata where randomly chosen synapses in the network were depressed after each spike (the so-called annealed dynamics). Calculations agree with simulations of the annealed version, showing that the nominal branching ratio σ converges to unity in the thermodynamic limit, as expected of a self-organized critical system. However, the question remains whether the same results apply to the biological case where only the synapses of firing neurons are depressed (the so-called quenched dynamics). We show that simulations of the quenched model yield significant deviations from σ=1 due to spatial correlations. However, the model is shown to be critical, as the largest eigenvalue of the synaptic matrix approaches unity in the thermodynamic limit, that is, λ_{c}=1. We also study the finite size effects near the critical state as a function of the parameters of the synaptic dynamics.

Phase transitions and self-organized criticality in networks of stochastic spiking neurons.

Phase transitions and critical behavior are crucial issues both in theoretical and experimental neuroscience. We report analytic and computational results about phase transitions and self-organized criticality (SOC) in networks with general stochastic neurons. The stochastic neuron has a firing probability given by a smooth monotonic function Φ(V) of the membrane potential V, rather than a sharp firing threshold. We find that such networks can operate in several dynamic regimes (phases) depending on the average synaptic weight and the shape of the firing function Φ. In particular, we encounter both continuous and discontinuous phase transitions to absorbing states. At the continuous transition critical boundary, neuronal avalanches occur whose distributions of size and duration are given by power laws, as observed in biological neural networks. We also propose and test a new mechanism to produce SOC: the use of dynamic neuronal gains - a form of short-term plasticity probably located at the axon initial segment (AIS) - instead of depressing synapses at the dendrites (as previously studied in the literature). The new self-organization mechanism produces a slightly supercritical state, that we called SOSC, in accord to some intuitions of Alan Turing.

Nonsynchronous updating in the multiverse of cellular automata.

In this paper we study updating effects on cellular automata rule space. We consider a subset of 6144 order-3 automata from the space of 262144 bidimensional outer-totalistic rules. We compare synchronous to asynchronous and sequential updatings. Focusing on two automata, we discuss how update changes destroy typical structures of these rules. Besides, we show that the first-order phase transition in the multiverse of synchronous cellular automata, revealed with the use of a recently introduced control parameter, seems to be robust not only to changes in update schema but also to different initial densities.

Conway's Game of Life is a near-critical metastable state in the multiverse of cellular automata.

Conway's cellular automaton Game of Life has been conjectured to be a critical (or quasicritical) dynamical system. This criticality is generally seen as a continuous order-disorder transition in cellular automata (CA) rule space. Life's mean-field return map predicts an absorbing vacuum phase (ρ = 0) and an active phase density, with ρ = 0.37, which contrasts with Life's absorbing states in a square lattice, which have a stationary density of ρ(2D) ≈ 0.03. Here, we study and classify mean-field maps for 6144 outer-totalistic CA and compare them with the corresponding behavior found in the square lattice. We show that the single-site mean-field approach gives qualitative (and even quantitative) predictions for most of them. The transition region in rule space seems to correspond to a nonequilibrium discontinuous absorbing phase transition instead of a continuous order-disorder one. We claim that Life is a quasicritical nucleation process where vacuum phase domains invade the alive phase. Therefore, Life is not at the "border of chaos," but thrives on the "border of extinction."

Single-neuron criticality optimizes analog dendritic computation.

Active dendritic branchlets enable the propagation of dendritic spikes, whose computational functions remain an open question. Here we propose a concrete function to the active channels in large dendritic trees. Modelling the input-output response of large active dendritic arbors subjected to complex spatio-temporal inputs and exhibiting non-stereotyped dendritic spikes, we find that the dendritic arbor can undergo a continuous phase transition from a quiescent to an active state, thereby exhibiting spontaneous and self-sustained localized activity as suggested by experiments. Analogously to the critical brain hypothesis, which states that neuronal networks self-organize near criticality to take advantage of its specific properties, here we propose that neurons with large dendritic arbors optimize their capacity to distinguish incoming stimuli at the critical state. We suggest that "computation at the edge of a phase transition" is more compatible with the view that dendritic arbors perform an analog rather than a digital dendritic computation.

Speech graphs provide a quantitative measure of thought disorder in psychosis.

BACKGROUND: Psychosis has various causes, including mania and schizophrenia. Since the differential diagnosis of psychosis is exclusively based on subjective assessments of oral interviews with patients, an objective quantification of the speech disturbances that characterize mania and schizophrenia is in order. In principle, such quantification could be achieved by the analysis of speech graphs. A graph represents a network with nodes connected by edges; in speech graphs, nodes correspond to words and edges correspond to semantic and grammatical relationships. METHODOLOGY/PRINCIPAL FINDINGS: To quantify speech differences related to psychosis, interviews with schizophrenics, manics and normal subjects were recorded and represented as graphs. Manics scored significantly higher than schizophrenics in ten graph measures. Psychopathological symptoms such as logorrhea, poor speech, and flight of thoughts were grasped by the analysis even when verbosity differences were discounted. Binary classifiers based on speech graph measures sorted schizophrenics from manics with up to 93.8% of sensitivity and 93.7% of specificity. In contrast, sorting based on the scores of two standard psychiatric scales (BPRS and PANSS) reached only 62.5% of sensitivity and specificity. CONCLUSIONS/SIGNIFICANCE: The results demonstrate that alterations of the thought process manifested in the speech of psychotic patients can be objectively measured using graph-theoretical tools, developed to capture specific features of the normal and dysfunctional flow of thought, such as divergence and recurrence. The quantitative analysis of speech graphs is not redundant with standard psychometric scales but rather complementary, as it yields a very accurate sorting of schizophrenics and manics. Overall, the results point to automated psychiatric diagnosis based not on what is said, but on how it is said.

Statistical physics approach to dendritic computation: the excitable-wave mean-field approximation.

We analytically study the input-output properties of a neuron whose active dendritic tree, modeled as a Cayley tree of excitable elements, is subjected to Poisson stimulus. Both single-site and two-site mean-field approximations incorrectly predict a nonequilibrium phase transition which is not allowed in the model. We propose an excitable-wave mean-field approximation which shows good agreement with previously published simulation results [Gollo et al., PLoS Comput. Biol. 5, e1000402 (2009)] and accounts for finite-size effects. We also discuss the relevance of our results to experiments in neuroscience, emphasizing the role of active dendrites in the enhancement of dynamic range and in gain control modulation.

A reliable measure of similarity based on dependency for short time series: an application to gene expression networks.

BACKGROUND: Microarray techniques have become an important tool to the investigation of genetic relationships and the assignment of different phenotypes. Since microarrays are still very expensive, most of the experiments are performed with small samples. This paper introduces a method to quantify dependency between data series composed of few sample points. The method is used to construct gene co-expression subnetworks of highly significant edges. RESULTS: The results shown here are for an adapted subset of a Saccharomyces cerevisiae gene expression data set with low temporal resolution and poor statistics. The method reveals common transcription factors with a high confidence level and allows the construction of subnetworks with high biological relevance that reveals characteristic features of the processes driving the organism adaptations to specific environmental conditions. CONCLUSION: Our method allows a reliable and sophisticated analysis of microarray data even under severe constraints. The utilization of systems biology improves the biologists ability to elucidate the mechanisms underlying cellular processes and to formulate new hypotheses.

Active dendrites enhance neuronal dynamic range.

Since the first experimental evidences of active conductances in dendrites, most neurons have been shown to exhibit dendritic excitability through the expression of a variety of voltage-gated ion channels. However, despite experimental and theoretical efforts undertaken in the past decades, the role of this excitability for some kind of dendritic computation has remained elusive. Here we show that, owing to very general properties of excitable media, the average output of a model of an active dendritic tree is a highly non-linear function of its afferent rate, attaining extremely large dynamic ranges (above 50 dB). Moreover, the model yields double-sigmoid response functions as experimentally observed in retinal ganglion cells. We claim that enhancement of dynamic range is the primary functional role of active dendritic conductances. We predict that neurons with larger dendritic trees should have larger dynamic range and that blocking of active conductances should lead to a decrease in dynamic range.

Deterministic walks as an algorithm of pattern recognition.

New tools for automatically finding data clusters that share statistical properties in a heterogeneous data set are imperative in pattern recognition research. Here we introduce a deterministic procedure as a tool for pattern recognition in a hierarchical way. The algorithm finds attractors of mutually close points based on the neighborhood ranking. A memory parameter mu acts as a hierarchy parameter, in which the clusters are identified. The final result of the method is a general tree that represents the nesting structure of the data in an invariant way by scale transformation.

Exploratory behavior, trap models, and glass transitions.

A random walk is performed on a disordered landscape composed of N sites randomly and uniformly distributed inside a d-dimensional hypercube. The walker hops from one site to another with probability proportional to exp[-betaE(D)], where beta=1/T is the inverse of a formal temperature and E(D) is an arbitrary cost function which depends on the hop distance D. Analytic results indicate that, if E(D)=D(d) and N--> infinity, there exists a glass transition at beta(d)=pi(d/2)/[(d/2)Gamma(d/2)]. Below T(d), the average trapping time diverges and the system falls into an out-of-equilibrium regime with aging phenomena. A Lévy flight scenario and applications of exploratory behavior are considered.

Escaping from cycles through a glass transition.

A random walk is performed over a disordered media composed of N sites random and uniformly distributed inside a d-dimensional hypercube. The walker cannot remain in the same site and hops to one of its n neighboring sites with a transition probability that depends on the distance D between sites according to a cost function E(D). The stochasticity level is parametrized by a formal temperature T. In the case T=0, the walk is deterministic and ergodicity is broken: the phase space is divided in a O(N) number of attractor basins of two-cycles that trap the walker. For d=1, analytic results indicate the existence of a glass transition at T(1)=1/2 as N--> infinity. Below T1, the average trapping time in two-cycles diverges and an out-of-equilibrium behavior appears. Similar glass transitions occur in higher dimensions when the right cost function is chosen. We also present some results for the statistics of distances for Poisson spatial point processes.

Physics of psychophysics: Stevens and Weber-Fechner laws are transfer functions of excitable media.

Sensory arrays made of coupled excitable elements can improve both their input sensitivity and dynamic range due to collective nonlinear wave properties. This mechanism is studied in a neural network of electrically coupled (e.g., via gap junctions) elements subject to a Poisson signal process. The network response interpolates between a Weber-Fechner logarithmic law, and a Stevens power law depending on the relative refractory period of the cell. Therefore, these nonlinear transformations of the input level could be performed in the sensory periphery simply due to a basic property: the transfer function of excitable media.