On the spectra of certain integro-differential-delay problems with applications in neurodynamics

We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs.

A mathematical model, implementation and study of a swarm system

The work reported in this paper is motivated towards the development of a mathematical model for swarm systems based on macroscopic primitives. A pattern formation and transformation model is proposed. The pattern transformation model comprises two general methods for pattern transformation, namely a macroscopic transformation method and a mathematical transformation method. The problem of transformation is formally expressed and four special cases of transformation are considered. Simulations to confirm the feasibility of the proposed models and transformation methods are presented. Comparison between the two transformation methods is also reported.

Axonal velocity distributions in neural feld equations

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

Interface dynamics in planar neural field models

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.

Design of associative memories using cellular neural networks

Cellular neural networks (CNNs) have locally connected neurons. This characteristic makes CNNs adequate for hardware implementation and, consequently, for their employment on a variety of applications as real-time image processing and construction of efficient associative memories. Adjustments of CNN parameters is a complex problem involved in the configuration of CNN for associative memories. This paper reviews methods of associative memory design based on CNNs, and provides comparative performance analysis of these approaches.

Time Evolution of the Activation Energy in a Batch Chemical Oscillator

The batch-operated bromate/phosphate/acetone/dual catalyst system was studied at four temperatures between 5 and 35 degrees C. The dynamics was simultaneously followed by potential measurements with platinum and bromide selective electrodes, and spectroscopically at two different wavelengths. By simultaneously recording these four time series it was possible to characterize the dynamics of the sequential oscillations that evolve in time. The existence of three sequential oscillatory patterns at each temperature allowed estimating the activation energies in each case. Along with the activation energy of the induction period, it was possible to trace the time evolution of the overall activation energy at four different stages as the reaction proceeds. The study was carried out for two different sets of initial concentrations and it was observed that the overall activation energy increases as reactants turn into products. This finding was propounded as a result of the decrease in the driving force, or the system`s affinity, of the catalytic oxidative bromination of acetone with acidic bromate, as the closed system evolves toward the thermodynamic equilibrium.

Formations of robotic swarm: an artificial force based approach

Cooperative control of multiple mobile robots is an attractive and challenging problem which has drawn considerable attention in the recent past. This paper introduces a scalable decentralized control algorithm to navigate a group of mobile robots (swarm) into a predefined shape in 2D space. The proposed architecture uses artificial forces to control mobile agents into the shape and spread them inside the shape while avoiding intermember collisions. The theoretical analysis of the swarm behavior describes the motion of the complete swarm and individual members in relevant situations. We use computer simulated case studies to verify the theoretical assertions and to demonstrate the robustness of the swarm under external disturbances such as death of agents, change of shape etc. Also the performance of the proposed distributed swarm control architecture was investigated in the presence of realistic implementation issues such as localization errors, communication range limitations, boundedness of forces etc.

Fenologia e dendrocronologia de duas espécies de Fabaceae em uma área de cerrado no sudeste do Brasil

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Using complex networks to quantify consistency in the use of words

In this paper we have quantified the consistency of word usage in written texts represented by complex networks, where words were taken as nodes, by measuring the degree of preservation of the node neighborhood. Words were considered highly consistent if the authors used them with the same neighborhood. When ranked according to the consistency of use, the words obeyed a log-normal distribution, in contrast to Zipf's law that applies to the frequency of use. Consistency correlated positively with the familiarity and frequency of use, and negatively with ambiguity and age of acquisition. An inspection of some highly consistent words confirmed that they are used in very limited semantic contexts. A comparison of consistency indices for eight authors indicated that these indices may be employed for author recognition. Indeed, as expected, authors of novels could be distinguished from those who wrote scientific texts. Our analysis demonstrated the suitability of the consistency indices, which can now be applied in other tasks, such as emotion recognition.

Network-based high level data classification

Traditional supervised data classification considers only physical features (e. g., distance or similarity) of the input data. Here, this type of learning is called low level classification. On the other hand, the human (animal) brain performs both low and high orders of learning and it has facility in identifying patterns according to the semantic meaning of the input data. Data classification that considers not only physical attributes but also the pattern formation is, here, referred to as high level classification. In this paper, we propose a hybrid classification technique that combines both types of learning. The low level term can be implemented by any classification technique, while the high level term is realized by the extraction of features of the underlying network constructed from the input data. Thus, the former classifies the test instances by their physical features or class topologies, while the latter measures the compliance of the test instances to the pattern formation of the data. Our study shows that the proposed technique not only can realize classification according to the pattern formation, but also is able to improve the performance of traditional classification techniques. Furthermore, as the class configuration's complexity increases, such as the mixture among different classes, a larger portion of the high level term is required to get correct classification. This feature confirms that the high level classification has a special importance in complex situations of classification. Finally, we show how the proposed technique can be employed in a real-world application, where it is capable of identifying variations and distortions of handwritten digit images. As a result, it supplies an improvement in the overall pattern recognition rate.

The tangential variation of a localized flux-type eigenvalue problem

In this work the differentiability of the principal eigenvalue lambda = lambda(1)(Gamma) to the localized Steklov problem -Delta u + qu = 0 in Omega, partial derivative u/partial derivative nu = lambda chi(Gamma)(x)u on partial derivative Omega, where Gamma subset of partial derivative Omega is a smooth subdomain of partial derivative Omega and chi(Gamma) is its characteristic function relative to partial derivative Omega, is shown. As a key point, the flux subdomain Gamma is regarded here as the variable with respect to which such differentiation is performed. An explicit formula for the derivative of lambda(1) (Gamma) with respect to Gamma is obtained. The lack of regularity up to the boundary of the first derivative of the principal eigenfunctions is a further intrinsic feature of the problem. Therefore, the whole analysis must be done in the weak sense of H(1)(Omega). The study is of interest in mathematical models in morphogenesis. (C) 2011 Elsevier Inc. All rights reserved.

Multi-level models and infrastructures for simulating biological system development

The hierarchical organisation of biological systems plays a crucial role in the pattern formation of gene expression resulting from the morphogenetic processes, where autonomous internal dynamics of cells, as well as cell-to-cell interactions through membranes, are responsible for the emergent peculiar structures of the individual phenotype. Being able to reproduce the systems dynamics at different levels of such a hierarchy might be very useful for studying such a complex phenomenon of self-organisation. The idea is to model the phenomenon in terms of a large and dynamic network of compartments, where the interplay between inter-compartment and intra-compartment events determines the emergent behaviour resulting in the formation of spatial patterns. According to these premises the thesis proposes a review of the different approaches already developed in modelling developmental biology problems, as well as the main models and infrastructures available in literature for modelling biological systems, analysing their capabilities in tackling multi-compartment / multi-level models. The thesis then introduces a practical framework, MS-BioNET, for modelling and simulating these scenarios exploiting the potential of multi-level dynamics. This is based on (i) a computational model featuring networks of compartments and an enhanced model of chemical reaction addressing molecule transfer, (ii) a logic-oriented language to flexibly specify complex simulation scenarios, and (iii) a simulation engine based on the many-species/many-channels optimised version of Gillespie’s direct method. The thesis finally proposes the adoption of the agent-based model as an approach capable of capture multi-level dynamics. To overcome the problem of parameter tuning in the model, the simulators are supplied with a module for parameter optimisation. The task is defined as an optimisation problem over the parameter space in which the objective function to be minimised is the distance between the output of the simulator and a target one. The problem is tackled with a metaheuristic algorithm. As an example of application of the MS-BioNET framework and of the agent-based model, a model of the first stages of Drosophila Melanogaster development is realised. The model goal is to generate the early spatial pattern of gap gene expression. The correctness of the models is shown comparing the simulation results with real data of gene expression with spatial and temporal resolution, acquired in free on-line sources.

Modeling and simulation of a synthetic genetic circuit that implements a multicelled behavior in a growing microcolony of E. coli

Synthetic Biology is a relatively new discipline, born at the beginning of the New Millennium, that brings the typical engineering approach (abstraction, modularity and standardization) to biotechnology. These principles aim to tame the extreme complexity of the various components and aid the construction of artificial biological systems with specific functions, usually by means of synthetic genetic circuits implemented in bacteria or simple eukaryotes like yeast. The cell becomes a programmable machine and its low-level programming language is made of strings of DNA. This work was performed in collaboration with researchers of the Department of Electrical Engineering of the University of Washington in Seattle and also with a student of the Corso di Laurea Magistrale in Ingegneria Biomedica at the University of Bologna: Marilisa Cortesi. During the collaboration I contributed to a Synthetic Biology project already started in the Klavins Laboratory. In particular, I modeled and subsequently simulated a synthetic genetic circuit that was ideated for the implementation of a multicelled behavior in a growing bacterial microcolony. In the first chapter the foundations of molecular biology are introduced: structure of the nucleic acids, transcription, translation and methods to regulate gene expression. An introduction to Synthetic Biology completes the section. In the second chapter is described the synthetic genetic circuit that was conceived to make spontaneously emerge, from an isogenic microcolony of bacteria, two different groups of cells, termed leaders and followers. The circuit exploits the intrinsic stochasticity of gene expression and intercellular communication via small molecules to break the symmetry in the phenotype of the microcolony. The four modules of the circuit (coin flipper, sender, receiver and follower) and their interactions are then illustrated. In the third chapter is derived the mathematical representation of the various components of the circuit and the several simplifying assumptions are made explicit. Transcription and translation are modeled as a single step and gene expression is function of the intracellular concentration of the various transcription factors that act on the different promoters of the circuit. A list of the various parameters and a justification for their value closes the chapter. In the fourth chapter are described the main characteristics of the gro simulation environment, developed by the Self Organizing Systems Laboratory of the University of Washington. Then, a sensitivity analysis performed to pinpoint the desirable characteristics of the various genetic components is detailed. The sensitivity analysis makes use of a cost function that is based on the fraction of cells in each one of the different possible states at the end of the simulation and the wanted outcome. Thanks to a particular kind of scatter plot, the parameters are ranked. Starting from an initial condition in which all the parameters assume their nominal value, the ranking suggest which parameter to tune in order to reach the goal. Obtaining a microcolony in which almost all the cells are in the follower state and only a few in the leader state seems to be the most difficult task. A small number of leader cells struggle to produce enough signal to turn the rest of the microcolony in the follower state. It is possible to obtain a microcolony in which the majority of cells are followers by increasing as much as possible the production of signal. Reaching the goal of a microcolony that is split in half between leaders and followers is comparatively easy. The best strategy seems to be increasing slightly the production of the enzyme. To end up with a majority of leaders, instead, it is advisable to increase the basal expression of the coin flipper module. At the end of the chapter, a possible future application of the leader election circuit, the spontaneous formation of spatial patterns in a microcolony, is modeled with the finite state machine formalism. The gro simulations provide insights into the genetic components that are needed to implement the behavior. In particular, since both the examples of pattern formation rely on a local version of Leader Election, a short-range communication system is essential. Moreover, new synthetic components that allow to reliably downregulate the growth rate in specific cells without side effects need to be developed. In the appendix are listed the gro code utilized to simulate the model of the circuit, a script in the Python programming language that was used to split the simulations on a Linux cluster and the Matlab code developed to analyze the data.

The bovine aristaless-like homeobox 4 (ALX4) as a candidate gene for syndactyly

The ALX4 (aristaless-like homeobox 4) gene encodes a paired-type homeodomain transcriptional activator and plays a major role in anterior-posterior pattern formation during limb development. Here, the cloning, genomic structure and expression of the bovine ortholog of the ALX4 gene are reported. The bovine ALX4 gene consists of four exons and is located on BTA15q28-->q29 in a region syntenic to HSA11p11.2. The transcribed ALX4 mRNA encodes a 397-amino-acid protein showing a paired-type homeodomain and a C-terminal stretch of amino acids known as the OAR- or aristaless domain. The predicted protein shares 92.5% identity to human and mouse ALX4 proteins and all three species share almost complete identity in the conserved domains. ALX4 expression was detected by reverse transcriptase polymerase chain reaction in bovine fetal limb bones. The ALX4 gene was evaluated as a candidate gene for bovine syndactyly which has been mapped on the telomeric region of cattle chromosome 15. Sequencing of the four exons with flanking sequences of the bovine ALX4 gene from a panel of 14 affected animals belonging to German Holstein, German Fleckvieh and crossbreds, and 27 unaffected individuals from German Holstein revealed five silent SNPs within the coding region out of eleven SNPs in total. Four SNPs were polymorphic in the affected animals, but in comparison to the genotyped unaffected individuals the genotype distribution showed no evidence for an association to the phenotype. Therefore our data indicate that the ALX4 gene can probably be excluded as candidate gene for bovine syndactyly in the examined animals.

Coordination of murine limb muscle, skeleton and connective tissue development by Lmx1b

The underlying genetic defects of a congenital disease Nail-Patella Syndrome are loss-of-function mutations in the LMX1B gene. Lmx1b encodes a LIM-homeodomain transcription factor that is expressed specifically in the dorsal limb bud mesenchyme. Gain- and loss-of-function experiments suggest that Lmx1b is both necessary and sufficient to specify dorsal limb patterning. However, how Lmx1b coordinates patterning of the dorsal tissues in the limb, including muscle, skeleton and connective tissues, remains unknown. One possibility is that each tissue specifies its own pattern cell-autonomously, i.e., Lmx1b is expressed in tissues in which it functions and different tissues do not communicate with each other. Another possibility is that tissues that express Lmx1b interact with adjacent tissues and provide patterning information thereby directing the development of tissues non-cell-autonomously. Previous results showed that Lmx1b is expressed in limb connective tissue and skeleton, but is not expressed in muscle tissue. Moreover, muscles and muscle connective tissue are closely associated during development. Therefore, we hypothesize that Lmx1b controls limb muscle dorsal-ventral (DV) patterning through muscle connective tissue, but regulates skeleton and tendon/ligament development cell-autonomously. ^ To test this hypothesis, we first examined when and where the limb dorsal-ventral asymmetry is established during development. Subsequently, conditional knockout and overexpression experiments were performed to delete or activate Lmx1b in different tissues within the limb. Our results show that deletion of Lmx1b from whole limb mesenchyme results in all dorsal tissues, including muscle, tendon/ligament and skeleton, transforming into ventral structures. Skeleton-specific knockout of Lmx1b led to the dorsal duplication of distal sesamoid and metacarpal bones, but did not affect the pattern formation of other tissues, suggesting that Lmx1b controls skeleton development cell-autonomously. In addition, this skeleton-specific pattern alteration only occurs in distal limb tissues, not proximal limb tissues, indicating different regulatory mechanisms operate along the limb proximal-distal axis. Moreover, skeleton-specific ectopic expression of Lmx1b reveals a complementary skeletal-specific dorsalized phenotype. This result supports a cell-autonomous role for Lmx1b in dorsal-ventral skeletal patterning. This study enriched our understanding of limb development, and the insights from this research may also be applicable for the development of other organs. ^