Instabilities and oscillation in the deterministic Boltzmann machine

Simulations indicate that the deterministic Boltzmann machine, unlike the stochastic Boltzmann machine from which it is derived, exhibits unstable behavior during contrastive Hebbian learning of nonlinear problems, including oscillation in the learning algorithm and extreme sensitivity to small weight perturbations. Although careful choice of the initial weight magnitudes, the learning rate, and the annealing schedule will produce convergence in most cases, the stability of the resulting solution depends on the parameters in a complex and generally indiscernible way. We show that this unstable behavior is the result of over parameterization (excessive freedom in the weights), which leads to continuous rather than isolated optimal weight solution sets. This allows the weights to drift without correction by the learning algorithm until the free energy landscape changes in such a way that the settling procedure employed finds a different minimum of the free energy function than it did previously and a gross output error occurs. Because all the weight sets in a continuous optimal solution set produce exactly the same network outputs, we define reliability, a measure of the robustness of the network, as a new performance criterion.     

The phase response of the cortical slow oscillation

Cortical slow oscillations occur in the mammalian brain during deep sleep and have been shown to contribute to memory consolidation, an effect that can be enhanced by electrical stimulation. As the precise underlying working mechanisms are not known it is desired to develop and analyze computational models of slow oscillations and to study the response to electrical stimuli. In this paper we employ the conductance based model of Compte et al. (J Neurophysiol 89:2707–2725, 2003) to study the effect of electrical stimulation. The population response to electrical stimulation depends on the timing of the stimulus with respect to the state of the slow oscillation. First, we reproduce the experimental results of electrical stimulation in ferret brain slices by Shu et al. (Nature 423:288–293, 2003) from the conductance based model. We then numerically obtain the phase response curve for the conductance based network model to quantify the network’s response to weak stimuli. Our results agree with experiments in vivo and in vitro that show that sensitivity to stimulation is weaker in the up than in the down state. However, we also find that within the up state stimulation leads to a shortening of the up state, or phase advance, whereas during the up–down transition a prolongation of up states is possible, resulting in a phase delay. Finally, we compute the phase response curve for the simple mean-field model by Ngo et al. (EPL Europhys Lett 89:68002, 2010) and find that the qualitative shape of the PRC is preserved, despite its different mechanism for the generation of slow oscillations.     

Stochastic and deterministic simulations of heterogeneous cell population dynamics

A Monte Carlo algorithm, which can accurately simulate the dynamics of entire heterogeneous cell populations, was developed. The algorithm takes into account the random nature of cell division as well as unequal partitioning of cellular material at cell division. Moreover, it is general in the sense that it can accommodate a variety of single-cell, deterministic reaction kinetics as well as various stochastic division and partitioning mechanisms. The validity of the algorithm was assessed through comparison of its results with those of the corresponding deterministic cell population balance model in cases where stochastic behavior is expected to be quantitatively negligible. Both algorithms were applied to study: (a) linear intracellular kinetics and (b) the expression dynamics of a genetic network with positive feedback architecture, such as the lac operon. The effects of stochastic division as well as those of different division and partitioning mechanisms were assessed in these systems, while the comparison of the stochastic model with a continuum model elucidated the significance of cell population heterogeneity even in cases where only the prediction of average properties is of primary interest.     

Subtle interplay of stochasticity and deterministic dynamics pervades an evolutionary plausible genetic circuit for Bacillus subtilis competence

Here we study the interplay of stochastic and deterministic dynamics in an evolutionary plausible candidate core genetic circuit for Bacillus subtilis competence. We find that high noise would not necessarily be detrimental to the circuit’s ability to deliver the phenotype, due to an unexpected built-in robustness that we further investigate. Also, we find that seemingly subtle deterministic dynamical features of the regulation, unstable and stable limit cycles, while in the presence of biochemical noise, would result in a distinctive new observable in the phenotype. We conduct mathematical analyses of the system’s stability at the fixed points and derive some general model-independent consequences. We also show how imperfect time-scale separation in the system would result in observables detrimental to the phenotype, that nature could have harnessed for selection.     

The K-complex and slow oscillation in terms of a mean-field cortical model

We use a mean-field macrocolumn model of the cerebral cortex to offer an interpretation of the K-complex of the electroencephalogram to complement those of more detailed neuron-by-neuron models. We interpret the K-complex as a momentary excursion of the cortex from a stable low-firing state to an unstable high-firing state, and hypothesize that the related slow oscillation can be considered as the periodic oscillation between two meta-stable solutions of the mean-field model. By incorporating a Hebbian-style learning rule that links the growth in synapse strength to fluctuations in soma potential, we demonstrate a self-organization behaviour that draws the modelled cortex close to the edge of stability of the low-firing state. Furthermore, a very slow oscillation can occur in the excitability of the cortex that has similarities with the infra-slow oscillation of sleep. Action Editor: Nicolas Brunel     

Spatial heterogeneity in mortality and its impact on the population dynamics of Eurasian woodcocks

Spatial heterogeneity, especially in mortality risk, is a major factor shaping population dynamics. Here we study the impacts of spatial heterogeneity in hunting pressure on the demography of Eurasian woodcock Scolopax rusticola, a relatively long-lived migratory game bird. We develop capture–recapture–recovery models in which both seasonality and spatial variation in hunting pressure are accounted for, and fit them to individual-based data collected across the French wintering range (>44000 banded individuals) as well as recoveries from spring stopovers and breeding grounds in Europe. Our results quantify spatial variation in survival probability in the wintering areas. They highlight the role of source-sink dynamics involving juvenile settlement decisions, as well as the importance of mortality outside the winter quarters. We also discuss the impact of spatial heterogeneity for demographic parameter estimation and data collection at the range scale.     

Adhesive dynamics simulation of neutrophil arrest with deterministic activation

The transition from rolling to firm adhesion is a key element of neutrophil activation and essential to the inflammatory response. Although the molecular mediators of rolling and firm adhesion are known to be selectins and beta2 -integrins, respectively, the precise dynamic mechanism by which these ligands facilitate neutrophil arrest remains unknown. Recently, it has been shown that ligation of E-selectin can stimulate the firm adhesion of neutrophils via a MAP-kinase cascade. To study the possible mechanism by which neutrophil arrest could occur, we created an integrated model by combining two methodologies from computational biology: a mechanics-based modeling of leukocyte adhesion (adhesive dynamics) and signal transduction pathway modeling. Within adhesive dynamics, a computational method our group has shown to accurately recreate rolling dynamics, we include a generic, tunable integrin activation module that links selectin engagement to integrin and activity. This model allows us to relate properties of the activation function to the dynamics of rolling and the time and distance rolled before arrest. This integrated model allows us to understand how intracellular signaling activity can set the timescale of neutrophil activation, adhesion, and diapedesis.     

Blowing-up of deterministic fixed points in stochastic population dynamics

We discuss the stochastic dynamics of biological (and other) populations presenting a limit behaviour for large environments (called deterministic limit) and its relation with the dynamics in the limit. The discussion is circumscribed to linearly stable fixed points of the deterministic dynamics, and it is shown that the cases of extinction and non-extinction equilibriums present different features. Mainly, non-extinction equilibria have associated a region of stochastic instability surrounded by a region of stochastic stability. The instability region does not exist in the case of extinction fixed points, and a linear Lyapunov function can be associated with them. Stochastically sustained oscillations of two subpopulations are also discussed in the case of complex eigenvalues of the stability matrix of the deterministic system.     

Protozoan grazing and its impact upon population dynamics in biofilm communities

AIMS: To determine the impact of protozoan grazing on the population dynamics of a multispecies bacterial biofilm community. METHODS AND RESULTS: Grazing by Acanthamoeba castellanii and the ciliate Colpoda maupasi upon biofilm and planktonic communities, composed of Klebsiella pneumoniae, Pseudomonas fluorescens and Staphylococcus epidermidis was investigated. Biofilms were formed using glass coverslips, held in a carousel device, as substrata for biofilm formation or in glass flow cells. The predatory effects of the amoeba were generally confined to the biofilm, where grazing rates corresponded to losses from the biofilm equivalent to ca 30,000 biofilm cells cm(-2) h(-1), with the amoeba becoming an integral part of the community. C. maupasi reduced the thickness of mature multispecies biofilms at steady-state from 500 to <200 microm. CONCLUSIONS: We report that the presence of the protozoa A. castellanii and C. maupasi markedly influence population dynamics within defined biofilm communities. SIGNIFICANCE AND IMPACT OF THE STUDY: The current study dispels the popular opinion that biofilms are protected against predation by protozoa. A. castellanii clearly has the capacity to graze mixed biofilm communities and to become integrally associated with them, whereas the ciliate C. maupasi reduced biofilm thickness by up to 60%.     

Large-scale cortical dynamics of sleep slow waves

Slow waves constitute the main signature of sleep in the electroencephalogram (EEG). They reflect alternating periods of neuronal hyperpolarization and depolarization in cortical networks. While recent findings have demonstrated their functional role in shaping and strengthening neuronal networks, a large-scale characterization of these two processes remains elusive in the human brain. In this study, by using simultaneous scalp EEG and intracranial recordings in 10 epileptic subjects, we examined the dynamics of hyperpolarization and depolarization waves over a large extent of the human cortex. We report that both hyperpolarization and depolarization processes can occur with two different characteristic time durations which are consistent across all subjects. For both hyperpolarization and depolarization waves, their average speed over the cortex was estimated to be approximately 1 m/s. Finally, we characterized their propagation pathways by studying the preferential trajectories between most involved intracranial contacts. For both waves, although single events could begin in almost all investigated sites across the entire cortex, we found that the majority of the preferential starting locations were located in frontal regions of the brain while they had a tendency to end in posterior and temporal regions.     

A stochastic model of evolutionary dynamics with deterministic large-population asymptotics

An evolutionary birth-death process is proposed as a model of evolutionary dynamics. Agents residing in a continuous spatial environment X, play a game G, with a continuous strategy set S, against other agents in the environment. The agents’ positions and strategies continuously change in response to other agents and to random effects. Agents spawn asexually at rates that depend on their current fitness, and agents die at rates that depend on their local population density. Agents’ individual evolutionary trajectories in X and S are governed by a system of stochastic ODEs. When the number of agents is large and distributed in a smooth density on (X,S), the collective dynamics of the entire population is governed by a certain (deterministic) PDE, which we call a fitness-diffusion equation.