From Synaptic Interactions to Collective Dynamics in Random Neuronal Networks Models: Critical Role of Eigenvectors and Transient Behavior.

The study of neuronal interactions is at the center of several big collaborative neuroscience projects (including the Human Connectome Project, the Blue Brain Project, and the Brainome) that attempt to obtain a detailed map of the entire brain. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of the synaptic interaction matrix. This work explores the application of free random variables to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra in types of models of neural networks proposed by Rajan and Abbott (2006), we extend them to heavy-tailed distributions of interactions. More important, we analytically derive the behavior of eigenvector overlaps, which determine the stability of the spectra. We observe that on imposing the neuronal excitation/inhibition balance, despite the eigenvalues remaining unchanged, their stability dramatically decreases due to the strong nonorthogonality of associated eigenvectors. This leads us to the conclusion that understanding 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 free random variables theory in a wide variety of disciplines, we hope that the results presented here foster the additional application of these ideas in the area of brain sciences.

Evolutionary games on cycles with strong selection.

Evolutionary games on graphs describe how strategic interactions and population structure determine evolutionary success, quantified by the probability that a single mutant takes over a population. Graph structures, compared to the well-mixed case, can act as amplifiers or suppressors of selection by increasing or decreasing the fixation probability of a beneficial mutant. Properties of the associated mean fixation times can be more intricate, especially when selection is strong. The intuition is that fixation of a beneficial mutant happens fast in a dominance game, that fixation takes very long in a coexistence game, and that strong selection eliminates demographic noise. Here we show that these intuitions can be misleading in structured populations. We analyze mean fixation times on the cycle graph under strong frequency-dependent selection for two different microscopic evolutionary update rules (death-birth and birth-death). We establish exact analytical results for fixation times under strong selection and show that there are coexistence games in which fixation occurs in time polynomial in population size. Depending on the underlying game, we observe inherence of demographic noise even under strong selection if the process is driven by random death before selection for birth of an offspring (death-birth update). In contrast, if selection for an offspring occurs before random removal (birth-death update), then strong selection can remove demographic noise almost entirely.

Spatial evolution of tumors with successive driver mutations.

We study the influence of driver mutations on the spatial evolutionary dynamics of solid tumors. We start with a cancer clone that expands uniformly in three dimensions giving rise to a spherical shape. We assume that cell division occurs on the surface of the growing tumor. Each cell division has a chance to give rise to a mutation that activates an additional driver gene. The resulting clone has an enhanced growth rate, which generates a local ensemble of faster growing cells, thereby distorting the spherical shape of the tumor. We derive formulas for the abundance and diversity of additional driver mutations as function of time. Our model is semi-deterministic: the spatial growth of cancer clones is deterministic, while mutants arise stochastically.

Incidence of Circulating Antibodies Against Hemagglutinin of Influenza Viruses in the Epidemic Season 2013/2014 in Poland.

The aim of this study was to determine the level of antibodies against hemagglutinin of influenza viruses in the sera of people in different age groups in the epidemic season 2013/2014 in Poland. The level of anti-hemagglutinin antibodies was determined by hemagglutination inhibition test (HAI). A total number of 1,050 randomly selected sera was tested in seven age groups. The level of antibodies against influenza viruses was very low, which indicates that the people have not been vaccinated against influenza in the epidemic season 2013/2014. The value of protection rate against influenza in the Polish population is very low. These results are worrying, because complications of influenza may be harmful to health and even life-threatening to persons who are not vaccinated. Furthermore, these results confirm the circulation of three antigenically different influenza virus strains, two subtypes of influenza A virus--A/California/7/2009/(H1N1)pdm09 and A/Victoria/361/2011(H3N2)--and B/Massachusetts/2/2012.

Dissecting x-ray-emitting gas around the center of our galaxy.

Most supermassive black holes (SMBHs) are accreting at very low levels and are difficult to distinguish from the galaxy centers where they reside. Our own Galaxy's SMBH provides an instructive exception, and we present a close-up view of its quiescent x-ray emission based on 3 megaseconds of Chandra observations. Although the x-ray emission is elongated and aligns well with a surrounding disk of massive stars, we can rule out a concentration of low-mass coronally active stars as the origin of the emission on the basis of the lack of predicted iron (Fe) Kα emission. The extremely weak hydrogen (H)-like Fe Kα line further suggests the presence of an outflow from the accretion flow onto the SMBH. These results provide important constraints for models of the prevalent radiatively inefficient accretion state.

Spectral relations between products and powers of isotropic random matrices.

We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble is equal to the eigenvalue density of nth power of a single matrix from this ensemble, in the limit when the size of the matrix tends to infinity. Using this observation, one can derive the limiting density of the product of n independent identically distributed non-Hermitian matrices with unitary invariant measures. In this paper we discuss two examples: the product of n Girko-Ginibre matrices and the product of n truncated unitary matrices. We also provide evidence that the result holds also for isotropic orthogonal ensembles.

Multiplication law and S transform for non-Hermitian random matrices.

We derive a multiplication law for free non-Hermitian random matrices allowing for an easy reconstruction of the two-dimensional eigenvalue distribution of the product ensemble from the characteristics of the individual ensembles. We define the corresponding non-Hermitian S transform being a natural generalization of the Voiculescu S transform. In addition, we extend the classical Hermitian S transform approach to deal with the situation when the random matrix ensemble factors have vanishing mean including the case when both of them are centered. We use planar diagrammatic techniques to derive these results.

Eigenvalues and singular values of products of rectangular gaussian random matrices.

We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions. We show that they both have power-law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions.

Wealth condensation in pareto macroeconomies.

We discuss a Pareto macroeconomy (a) in a closed system with fixed total wealth and (b) in an open system with average mean wealth, and compare our results to a similar analysis in a super-open system (c) with unbounded wealth [J.-P. Bouchaud and M. Mézard, Physica A 282, 536 (2000)]. Wealth condensation takes place in the social phase for closed and open economies, while it occurs in the liberal phase for super-open economies. In the first two cases, the condensation is related to a mechanism known from the balls-in-boxes model, while in the last case, to the nonintegrable tails of the Pareto distribution. For a closed macroeconomy in the social phase, we point to the emergence of a "corruption" phenomenon: a sizeable fraction of the total wealth is always amassed by a single individual.

Role of CD8(+) lymphocytes in control of simian immunodeficiency virus infection and resistance to rechallenge after transient early antiretroviral treatment.

Transient antiretroviral treatment with tenofovir, (R)-9-(2-phosphonylmethoxypropyl)adenine, begun shortly after inoculation of rhesus macaques with the highly pathogenic simian immunodeficiency virus (SIV) isolate SIVsmE660, facilitated the development of SIV-specific lymphoproliferative responses and sustained effective control of the infection following drug discontinuation. Animals that controlled plasma viremia following transient postinoculation treatment showed substantial resistance to subsequent intravenous rechallenge with homologous (SIVsmE660) and highly heterologous (SIVmac239) SIV isolates, up to more than 1 year later, despite the absence of measurable neutralizing antibody. In some instances, resistance to rechallenge was observed despite the absence of detectable SIV-specific binding antibody and in the face of SIV lymphoproliferative responses that were low or undetectable at the time of challenge. In vivo monoclonal antibody depletion experiments demonstrated a critical role for CD8(+) lymphocytes in the control of viral replication; plasma viremia rose by as much as five log units after depletion of CD8(+) cells and returned to predepletion levels (as low as <100 copy Eq/ml) as circulating CD8(+) cells were restored. The extent of host control of replication of highly pathogenic SIV strains and the level of resistance to heterologous rechallenge achieved following transient postinoculation treatment compared favorably to the results seen after SIVsmE660 and SIVmac239 challenge with many vaccine strategies. This impressive control of viral replication was observed despite comparatively modest measured immune responses, less than those often achieved with vaccination regimens. The results help establish the underlying feasibility of efforts to develop vaccines for the prevention of AIDS, although the exact nature of the protective host responses involved remains to be elucidated.

Reward and punishment.

Minigames capturing the essence of Public Goods experiments show that even in the absence of rationality assumptions, both punishment and reward will fail to bring about prosocial behavior. This result holds in particular for the well-known Ultimatum Game, which emerges as a special case. But reputation can induce fairness and cooperation in populations adapting through learning or imitation. Indeed, the inclusion of reputation effects in the corresponding dynamical models leads to the evolution of economically productive behavior, with agents contributing to the public good and either punishing those who do not or rewarding those who do. Reward and punishment correspond to two types of bifurcation with intriguing complementarity. The analysis suggests that reputation is essential for fostering social behavior among selfish agents, and that it is considerably more effective with punishment than with reward.

A generalized adaptive dynamics framework can describe the evolutionary Ultimatum Game.

Adaptive dynamics describes the evolution of games where the strategies are continuous functions of some parameters. The standard adaptive dynamics framework assumes that the population is homogeneous at any one time. Differential equations point to the direction of the mutant that has maximum payoff against the resident population. The population then moves towards this mutant. The standard adaptive dynamics formulation cannot deal with games in which the payoff is not differentiable. Here we present a generalized framework which can. We assume that the population is not homogeneous but distributed around an average strategy. This approach can describe the long-term dynamics of the Ultimatum Game and also explain the evolution of fairness in a one-parameter Ultimatum Game.

Cytotoxic T-cell abundance and virus load in human immunodeficiency virus type 1 and human T-cell leukaemia virus type 1.

The correlation between virus load and specific cytotoxic T-lymphocyte (CTL) frequency during the chronic phase in human immunodeficiency virus type 1 (HIV-1) infection has been found to be negative in cross-sectional studies. We report here that, in infection with the related retrovirus human T-cell leukaemia virus type 1 (HTLV-1), the correlation is positive in asymptomatic carriers and zero in patients with the associated inflammatory disease HTLV-1-associated myelopathy/tropical spastic paraparesis (HAM/TSP). We demonstrate that the direction of the correlation may depend on the efficacy of the CTL response using mathematical models. We conclude that the CTL response is effective in asymptomatic carriers of HTLV-1, but ineffective in patients with HAM/TSP. Virus-mediated impairment of specific CTL production in HIV-1 infection can account for the negative correlation observed.

Towards an evolutionary theory of language.

Language is a biological trait that radically changed the performance of one species and the appearance of the planet. Understanding how human language came about is one of the most interesting tasks for evolutionary biology. Here we discuss how natural selection can guide the emergence of some basic features of human language, including arbitrary signs, words, syntactic communication and grammar. We show how natural selection can lead to the duality of patterning of human language: sequences of phonemes form words; sequences of words form sentences. Finally, we present a framework for the population dynamics of grammar acquisition, which allows us to study the cultural evolution of grammar and the biological evolution of universal grammar.

The evolutionary dynamics of the lexical matrix.

The lexical matrix is an integral part of the human language system. It provides the link between word form and word meaning. A simple lexical matrix is also at the center of any animal communication system, where it defines the associations between form and meaning of animal signals. We study the evolution and population dynamics of the lexical matrix. We assume that children learn the lexical matrix of their parents. This learning process is subject to mistakes: (i) children may not acquire all lexical items of their parents (incomplete learning); and (ii) children might acquire associations between word forms and word meanings that differ from their parents' lexical items (incorrect learning). We derive an analytic framework that deals with incomplete learning. We calculate the maximum error rate that is compatible with a population maintaining a coherent lexical matrix of a given size. We calculate the equilibrium distribution of the number of lexical items known to individuals. Our analytic investigations are supplemented by numerical simulations that describe both incomplete and incorrect learning, and other extensions.

Natural selection of the critical period for language acquisition.

The language acquisition period in humans lasts about 13 years. After puberty it becomes increasingly difficult to learn a language. We explain this phenomenon by using an evolutionary framework. We present a dynamical system describing competition between language acquisition devices, which differ in the length of the learning period. There are two selective forces that play a role in determining the critical learning period: (i) having a longer learning period increases the accuracy of language acquisition; (ii) learning is associated with certain costs that affect fitness. As a result, there exists a limited learning period which is evolutionarily stable. This result is obtained analytically by means of a Nash equilibrium analysis of language acquisition devices. Interestingly, the evolutionarily stable learning period does not maximize the average fitness of the population.

The evolutionary dynamics of grammar acquisition.

Grammar is the computational system of language. It is a set of rules that specifies how to construct sentences out of words. Grammar is the basis of the unlimited expressibility of human language. Children acquire the grammar of their native language without formal education simply by hearing a number of sample sentences. Children could not solve this learning task if they did not have some pre-formed expectations. In other words, children have to evaluate the sample sentences and choose one grammar out of a limited set of candidate grammars. The restricted search space and the mechanism which allows to evaluate the sample sentences is called universal grammar. Universal grammar cannot be learned; it must be in place when the learning process starts. In this paper, we design a mathematical theory that places the problem of language acquisition into an evolutionary context. We formulate equations for the population dynamics of communication and grammar learning. We ask how accurate children have to learn the grammar of their parents' language for a population of individuals to evolve and maintain a coherent grammatical system. It turns out that there is a maximum error tolerance for which a predominant grammar is stable. We calculate the maximum size of the search space that is compatible with coherent communication in a population. Thus, we specify the conditions for the evolution of universal grammar.

Evolution of universal grammar.

Universal grammar specifies the mechanism of language acquisition. It determines the range of grammatical hypothesis that children entertain during language learning and the procedure they use for evaluating input sentences. How universal grammar arose is a major challenge for evolutionary biology. We present a mathematical framework for the evolutionary dynamics of grammar learning. The central result is a coherence threshold, which specifies the condition for a universal grammar to induce coherent communication within a population. We study selection of grammars within the same universal grammar and competition between different universal grammars. We calculate the condition under which natural selection favors the emergence of rule-based, generative grammars that underlie complex language.

Evolutionary biology of language.

Language is the most important evolutionary invention of the last few million years. It was an adaptation that helped our species to exchange information, make plans, express new ideas and totally change the appearance of the planet. How human language evolved from animal communication is one of the most challenging questions for evolutionary biology The aim of this paper is to outline the major principles that guided language evolution in terms of mathematical models of evolutionary dynamics and game theory. I will discuss how natural selection can lead to the emergence of arbitrary signs, the formation of words and syntactic communication.