# 2017 – 18

Speaker: **Rafel J. Prohens
**

Affiliation: Universitat de les Illes Balears (UIB), Math. Dept.

Title: **New lower bounds for the Hilbert numbers using reversible centers**

Date: Monday, January 15th, 2017, 14:00 – 15:00

Place: Aula 2A, Edifici de Postgrau (Mossèn Antoni Maria Alcover Sureda) at UIB

Summary: In this seminar we present the work (preprint), of the same title, of the authors R. Prohens and J. Torregrosa (Universitat Autònoma de Barcelona). In this work we provide the best lower bounds, that are known up to now, for the Hilbert number of a polynomial vector field of degree N, H(N), for small values of N. These limit cycles appear bifurcating from new symmetric Darboux reversible centers with very high simultaneous cyclicity. The considered systems have, at least, three centers, one on the reversibility straight line and two symmetric about it. More concretely, the limit cycles are in a three nests configuration and the total number of limit cycles writes in the form 2n+m, for some values of n and m. The new lower bounds are obtained using simultaneous degenerate Hopf bifurcations. In particular, H(4)≥28, H(5)≥37, H(6)≥53, H(7)≥74, H(8)≥96, H(9)≥120, and H(10)≥142..

# 2016 – 17

Speaker: **Peter De Maesschalck**

Affiliation: Hasselt University, Campus Diepenbeek, Agoralaan, gebouw D, 3590 Diepenbeek, BELGICA

Title: **Normal forms of saddle connections in the plane**

Date: Wednesday, June 21th, 2017, 11:30 – 12:30

Place: Anselm Turmeda, Sala de Reunions

Summary: When studying limit cycles in the plane near singular points the local behaviour around the singular point becomes important, and for degenerate singular points a lot of open questions remain. In order to deal with the questions we propose to find the most simple local expressions at those singular points by blowing up first, identifying saddle connections close to the blow up locus and extending normal form theory to a nonlocal theory near these saddle connections.

Speaker: ** Antonio E Teruel **

Affiliation: University of Balearic Islands

Title: ** Global phase portrait of a 3D integrable flow **

Date: Wednesday, May 24th, 2017, 15:00 – 16:00

Place: Anselm Turmeda, A128

Summary: In many cases, the knowledge about the integrablility of a flow is synonymous with knowing its behavior. However, there is still an long way to travel for going from integrability to the global description of the phase portrait. Although the necessary tools are elementary, this trip is not always easy to do. In this work, and starting from a system of three-dimensional and integrable differential equations, a description of its overall phase portrait is presented.

Speaker: ** Joan Duran **

Affiliation: University of Balearic Islands

Title: ** Nonlocal Regularizing Constraints in Variational Optical Flow **

Date: Wednesday, March 22th, 2017, 15:00 – 16:00

Place: Anselm Turmeda, A128

Summary: Optical flow methods try to estimate a dense correspondence field describing the motion of the objects between an image pair. We introduce novel nonlocal regularizing constraints for variational optical flow computation. While the use of similarity weights has been restricted to the regularization term so far, the proposed data terms permit to implicitly use the image geometry in order to regularize the flow and better locate motion discontinuities. We show some preliminar results comparing the classical brightness constancy assumption with the proposed data constraints.

Speaker: ** Julia Navarro **

Affiliation: University of Balearic Islands

Title: ** Cost adaptive window for local stereo matching **

Date: Tuesday, February 7th, 2017, 14:00 to 15:00

Place: Mateu Orfila, Room A16

Summary: The goal of stereovision is to estimate the depth of the scene from at least two images taken from different viewpoints. For an epipolar rectified image pair, the problem is reduced to estimate the displacement (disparity) between both views. Local stereo methods estimate disparity by means of a block-matching approach. These methods assume that disparity is constant within the matching window. We present a novel stereo block-matching algorithm which uses adaptive windows. The shape of the window is selected to minimize the matching cost. Such a window might be the less distorted by the disparity function and thus the optimal one for matching. Moreover, we use a coarse-to-fine strategy to limit the number of ambiguous matches and reduce the computational cost. The proposed approach performs as state of the art local matching methods.

Speaker: ** Antonio E. Teruel **

Affiliation: University of Balearic Islands

Title: ** Electrophysiology of Neurons: the Hodgkin-Huxley model **

Date: Tuesday, January 10th, 2017, 14:00 to 15:00

Place: Mateu Orfila, Room A16

Summary: In this introductory talk we will introduce step by step the well known neural model of the giant squid due to Hodgkin and Huxley. The action potential exhibited by the model will be simulated by using numerical tools.

Speaker: ** Bartomeu Coll **

Affiliation: University of Balearic Islands

Title: ** An introduction to optimal transport applied to image fusion **

Date: Tuesday, December 13th, 2016, 14:00 to 15:00.

Place: Mateu Orfila, Room A16

Summary: In this talk, we shall give an introduction to the optimal transport theory which defines a natural and useful geometry to compare measures supported on metric probability spaces as the Wasserstein distance. Based on the optimal transport theory, we propose an image fusion application for the hyperspectral and multispectral images by merging the spectral information of the hyperspectral image with the relevant spatial one contained in the multispectral image. The diffusion process consists in minimizing the sum of the weighted Wasserstein distances.

Speaker: ** Catalina Vich Llompart.**

Affiliation: University of Balearic Islands.

Title: **Inverse methods to estimate synaptic conductances – Two compartment neurons**

Date: Tuesday, November 15th, 2016, 14:00 to 15:00.

Place: Seminari del grup de Matemàtica Aplicada.

Summary: Un dels problemes de neuro-ciència en que s’està treballant avui en dia és el problema d’estimació de conductàncies, important per a entendre la connectivitat cerebral. Les tècniques d’estimació han anat millorant els darrers anys per a fer estimacions en zones de poca activitat neuronal. Ara bé, tots aquests mètodes suposen que la neurona és un únic punt enlloc de tota una estructura morfològica. En aquest seminari es vol donar a conèixer un problema obert d’estimació de conductàncies, el qual involucre models neuronals que combinen equacions diferencials ordinàries i parcials, a més de possible renou.

# 2015-16

Speaker: **Dr. Ana Belén Petro Balaguer.**

Affiliation: University of Balearic Islands.

Title: **The online Retinex benchmark for contrast improvement of Earth observation images.**

Date: Thursday, May 12th, 2016, 13:00 to 14:00.

Place: 3A at the ground floor of Antoni Maria Alcover i Sureda building.

Summary: We present the results of the ranking demo for the evaluation of 13 contrast enhancements of three sorts (retinex, tone mapping, gradient based methods) that were implemented, analyzed and compared on line on Pleiades images by experts at CNES and two universities.This led to the selection of four algorithms, all performing a local enhancement, three of them belonging to the multiscale retinex family. In a second inquiry, we reviewed 9 algorithms that have been proposed in the recent rush toward High dynamic Range. After a careful analysis of these algorithms and of their implementations, we have decided to include five of them in a new online benchmark.

Our conclusion is that a new benchmark must be immediately performed to classify our four preceding winners and match them to our five new ones. We organize this benchmark following the considerations: 1) to make all users rate the same pieces of images; 2) to divide the evaluation into two marks, as the experts will be invited to rate separately the overall contrast, but also to pay a special attention to the loss of contrast in bright areas, resulting in an apparent saturation; 3) to completely divide the evaluation of the grey level process from the evaluation of the color process.

Speaker: **Cati Vich.**

Affiliation: University of Balearic Islands.

Title:** A strategy for non-linear estimation of synaptic conductances**

Date: Thursday, Match 3th, 2016, 13:00 to 14:00.

Place: 3A at the ground floor of Antoni Maria Alcover i Sureda building.

Summary Estimating the synaptic conductances impinging on a single neuron directly from a single trace of its membrane potential is an important problem in order to understand the flow of information in the brain. Despite the existence of some methods, they all present the inconvenience that the estimation is done in subthreshold activity regime assuming the absence of subthreshold ionic currents.

Recently we have proved how these methods cannot estimated correctly the conductances under the presence of subthreshold activated currents; presenting a deterministic model, which provides significant improvements when results are compared with the ones obtained by the usual linear model.

However, experimental recordings are noisy, and stochastic models are called for. In this work we extend the procedure to estimate synaptic conductances based on a quadratization of a stochastic model: a stochastic version of the Quadratic Integrate-and-Fire model. Given the membrane potential trace, we estimate the time course of the model coefficients by approximate maximum likelihood, which allows to estimate the time-varying excitatory and inhibitory conductances. This new procedure has been applied both to simulated data from computational models and using real data, obtaining good estimates in the first case, and reasonable results in the real data.

Speaker: **Dr. María Jesús Álvarez Torres.**

Affiliation: University of Balearic Islands.

Title: **Centers and limit cycles for a family of Abel equations**.

Date: Thursday, February 25th, 2016, 13:00 to 14:00.

Place: 3A at the ground floor of Antoni Maria Alcover i Sureda building.

Summary: We consider the class of generalized Abel equations of the form $x’=(a_1A_1(t)+a_2A_2(t))x^m+(a_3A_3(t)+a_4A_4(t))x^n$, where $n,m>1$ are natural numbers, $a_1,a_2,a_3,a_4\in\mathbb{R}$ and $A_1,A_2,A_3,A_4$ are trigonometric monomials. In this talk we give a brief summary of the known results concerning the existence and non-existence of limit cycles of the previous differential equation, paying special attention to the new theorems, that apply to the case for which $A_1,A_3$ have the same signs as $\sin t$, and $A_2,A_4$ the same signs as $\cos t$. More concretely, these new results prove that the center variety for the previous equation (i.e. the set of values $a_1,a_2,a_3,a_4$ such that the Abel equation has a center) has codimension one or two. Moreover, it is one if and only if $A_1=A_3$ and $A_2=A_4$ and it is two if and only if the family has non-trivial limit cycles (different from $x(t)\equiv 0$) for some values of the parameters.

Speaker: **Dr. Catalina Sbert Juan**

Affiliation: University of Balearic Islands

Title: **A white-patch model for color cast detection in natural images.
**

Date: Friday, November 20th, 2015, 12:30 to 13:30

Place: 2B at the ground floor of Antoni Maria Alcover i Sureda building

Summary: We introduce the image formation model and discuss in a mathematical framework the classical hypotheses for reflectance and illuminant estimation. We show the different role that reflectance and illuminance play for distinguish between dominant color images and color cast images. Then we discuss some methods for color cast detection in natural images.

Speaker: **Dr. Antoni Buades Capó**

Affiliation: University of Balearic Islands

Title: **Video denoising and applications to satellite multi-image restoration**

Date: Friday, October 30th, 2015, 12:30 to 13:30

Place: 2B at the ground floor of Antoni Maria Alcover i Sureda building

Summary: A novel image sequence denoising algorithm is presented. The proposed approach takes advantage of self similarity and redundancy of adjacent frames. The algorithm is inspired by fusion algorithms and, as the number of frames increases, it tends to a pure temporal average. The use of motion compensation by regularized optical flow methods permits robust patch comparison in a spatiotemporal volume. The use of Principal Component Analysis ensures the correct preservation of fine texture and details. An extensive comparison with state of the art methods illustrates the superior performance of the proposed approach, with improved texture and detail reconstruction. The proposed algorithm is applied for the restoration of series of satellite images acquired nearly simultaneously, including compression and aliasing artefact removal.

# 2014-15

Speaker: **Michael Möller.**

Affiliation: Technical University of Munich

Title: **Nonlinear multiscale methods for image and signal analysis**

Date: Thursday, July 16, 2015

Place: AT-D108

Summary: Classical strategies for image and signal analysis apply an orthonormal linear transform, filter the resulting coefficients, and apply the inverse transform. If the filtering itself is also linear, one typically interprets the whole process with the help of the singular value decomposition of the linear operator. This talk discusses the possibility to introduce similar concepts for nonlinear transformations arising from the solution of variational, scale space, or inverse scale space methods. We introduce the notion of generalized frequency and wavelength representations as well as filterings of the corresponding coefficients. First numerical results demonstrate interesting properties of the resulting image decomposition.

Speaker: **Antoni Manel Ferragut**

Affiliation: Universitat Jaume I.

Títol: **Aproximació algebraica de separatrius de sella d’un sistema de Lotka-Volterra**

Date: Tuesday, July 14, 2015

Place: AT-D108

Summary: Given a differential system of Lotka-Volterra type with three positive parameters, consider the families that have a saddle in the first quadrant. Once characterized these families, we focus on the cases where the separatrix joining the origin with the saddle is algebraic, and characterize these subfamilies.

For the remainder subfamilies (where the separatrix not algebraic) we give lower and upper bounds of the separatrix by algebraic functions. From them we can calculate the probability that, as an initial condition in the first quadrant, extinguished species or the other.

Speaker:** Edoardo Provenzi**

Title: **Second order stationarity and tensor product spatiochromatic features of natural images**

Date: Thursday, Juny 4, 2015

Place: AT-D108

Summary: In the seminar, it will be shown that two simple assumptions on the covariance matrices ofcolor images yield eigenvectors made by the Kronecker product of Fourier features times the triadgiven by luminance plus color opponent channels. The first of these assumptions is second orderstationarity while the second one is commutativity between color correlation matrices. The validity ofthese assumptions and the predicted shape of the PCA components of color images are experimentally observed on two large image databases. As a by-product of this experimental study, we will provide novel data to support an exponential decay law of the spatiochromatic covariancebetween pairs of pixels as a function of their spatial distance.

Speaker:** Joan Torregrosa**

Affiliation: Department of Mathematics, Universitat de les Illes Balears

Title: **Center, weak-focus and cyclicity problems for planar systems**

Date: Thursday, May 14, 2015

Place: AT-D108

Summary: The center-focus problem consists in distinguishing whether a monodromic singular point is a center or a focus. For singular points with imaginary eigen-

values, usually called nondegenerate singular points, this problem was already

solved by Poincar ́ and Lyapunov, see [3]. The solution consists in computing several quantities called commonly the Poincar ́–Lyapunov constants, and study whether they are zero or not.

Despite the existence of many methods, the solution of the center-focus problem for simple families, like for instance the complete cubic systems or the quartic systems with homogeneous nonlinearities, has resisted all the attempts. For this reason, we propose to push on this question in another direction. We study this problem for a natural family of differential systems with few free parameters but arbitrary degree. We consider planar systems with a linear center at the origin that in complex coordinates the nonlinearity terms are formed by the sum of few monomials. For some families in this class, we study the center problem, the maximum order of a weak-focus and the cyclicity problem. Several

centers inside this family are done. The list includes a new class of Darboux centers that are also persistent centers. We study if the given list is exhaustive or not.

For small degrees we provide explicit systems with weak foci or high-order centers that, after perturbation, give new lower bounds for the number of limit cycles surrounding a single critical point. These lower bounds are higher than the corresponding Hilbert number known until now for these degrees. The talk will be a review of the results [1,2].

*References*

1. A. Gasull, J. Gin ́, and J. Torregrosa. Center problem for systems with two monomial nonlinearities. Preprint. 2014.

2. H. Liang and J. Torregrosa. Some new results on the order and the cyclicity of weak focus of planar polynomial system. Preprint. 2015.

3. A. M. Lyapunov. The general problem of the stability of motion. Taylor & Francis, Ltd., London, 1992. Translated from Edouard Davaux’s French translation (1907) of the 1892 Russian original and edited by A. T. Fuller. Reprint of Internat. J. Control 55 (1992), no. 3, 521–790.

Speaker: **Antonio E. Teruel**

**Folded singularities in piecewise linear systems**

Speaker: **M.J. Álvarez**

Title: **Period Function of degenerate systems**

Date: Thursday, April 16, 2015

Place: AT-D108

Summary: In this work we study the period function of a system that is the sum of two quasi-homogeneous hamiltonian vector fields, none of them being linear. We prove that, when the period annulus is global, the period function is monotone.

We also give a result in the homogeneous case when the period annulus is nonglobal and we get that,

for an unbounded region of the parameters (quasi-degrees of the vector fields) the maximum number of critical periods is one, and this bound is sharp. We also pay a particular attention to the case of a quasi-homogeneous vector field.

Speaker: **Toni Buades **

Title**: Self-similarity and spectral correlation adaptive algorithm for color demosaicking**

Date: Thursday, March 12, 2015

Place: AT-D108

Summary:** **Most common cameras use a CCD sensor device measuring a single color per pixel. The other two color values of each pixel must be interpolated from the neighboring pixels in the so-called demosaicking process. State-of-the-art demosaicking algorithms take advantage of inter-channel correlation locally selecting the best interpolation direction. These methods give impressive results except when local geometry cannot be inferred from neighboring pixels or channel correlation is low. In these cases, they create interpolation artifacts.

We introduce a new algorithm involving non-local image self-similarity in order to reduce interpolation artifacts when local geometry is ambiguous. The proposed algorithm introduces a clear and intuitive manner of balancing how much channel-correlation must be taken advantage of.

Comparison shows that the proposed algorithm gives state-of-the-art methods in several image bases.

Speaker: **Rafel Prohens**

Title: **Asymptotic behaviours in difference equations having equilibrium parabolic points**

Date: Thursday, February 26, 2015

Place: AT-D108

Summary: Some classes of non-linear difference equations present a parabolic equilibrium point and a natural question is to find the asymptotic behaviour of the solutions tending to it. Roughly speaking, we are mainly interested in to face some questions concerning known asymptotic behaviours appearing in the literature such as the problem to bound (upper and lower) the solutions tending to the equilibrium point. More concretely, we explain why these bounds are guessed. We prove why these guesses are used as natural bounds for the solutions, by proving that they are not only bounds (upper and lower) but the natural bounds of the difference equation itself. These bounds are obtained when looking for the asymptotic behaviour of the solutions in the invariant manifold of the parabolic point. Even more, for some families of difference equations we prove that this invariant manifold is, actually, the graph of a Lipschitz function, i.e. a one dimensional variety in R^n. Since this is a work in progress, we aim to address the more interesting question that is to prove that the solutions bounding the referred invariant manifold are, not only bounds but that they (asymptotically) are the solution itself.

Speaker: **Joan Duran Grimalt**

Title:** A Novel Framework for Vectorial Total Variation and Applications to Image De-noising**

**
**Date: Monday, January 19, 2015

Place: AT-D108Summary: Even after over twenty years of research, the total variation (Tv) introduced by Rudin, Osher and Fatemi remains one of the most popular regularization criteria for image restoration. It has proved to be quite efficient for regularizing images without over-smoothing the boundaries of the objects. By observing that the discrete gradient of a colour image can be considered as a 3D matrix with dimensions corresponding to the pixels, the (local or non-local) derivatives and the colour channels, we propose several vector-valued TV regularizations that penalize the “mixed matrix norms” of the data structure, that is, $\ell^{p,q,r}$ or $(S^p,\ell^q)$ norms.

In this talk, we will discuss the implementation of different local and non-local colour TV regularizations by using the primal-dual hybrid gradient method. We will first analyse the de-noising properties of each mixed matrix norm for suppressing colour artefacts while preserving image features and aligning edges. We will describe then in detail how to compute the proximity operators – a generalization of the projection mapping – of the proposed norms. Finally, a performance comparison between several mixed matrix TV methods on image de-noising will be provided.

Speaker: **Marko Miladinovic**

Title:

Date: Monday, November 3, 2014

Place: AT-D108

Summary: Two topics are going to be covered by the following talk: **Numerical optimization and Computations of generalized inverses of matrices.**

abstract: In optimization gradient methods and quasi-Newton methods play very important role. We will see how original gradient method can be improved by introducing some acceleration parameters. Also two point step size methods will be consider as quasi-Newton methods. The improvement of the original BB method given in terms of scalar correction will be present.

Generalized inverses of matrices are very important tool in mathematics and can be applied on different fields such as: linear statistical models, optimization, singular linear difference equations, probability theory and statistics, financial and insurance mathematics. Successive Matrix Squaring (SMS) algorithm for computing generalized inverses of Toeplitz matrices will be explained. Also, some interesting representations of {2,4} and {2,3}-inverses will be present. Additionally, some of the gradient methods will be apply in order to determine some generalized inverses or to solve the system of linear equations.

Speaker: **Antonio E. Teruel **

Title:** MMOs induced by PWL canards.**

Date: Monday, October 27, 2014

Place: AT-D124

Summary: Mixed mode oscillations (MMOs) are periodic behaviours which are characterized by a sequence of small amplitude oscillations followed by large amplitude oscillations. In the literature, slow-fast differential system-based models are proposed. In this talk we present minimal models based on piecewise linear slow-fast systems. We will show that these systems reproduce qualitative and quantitatively the MMOs exhibited by the differential systems.

# 2013-14

- Speaker:
**Xavier Jarque**

Affiliation: Department of Mathematics, Universitat de Barcelona

Title: **Sobre la connectivitat del conjut de Julia per funcions polinomials, racionals i transcendents**

Date: Thursday, March 27, 2014

Place: Room ATA09

Time: 11:00

Summary : The talk will be about the connectivity of the Julia set, that is the region of the complex plane in which the family of iterates of a holomorphic function is not a normal family. I will present first the polynomial case (the dichotomy of the quadratic family) and some other results for hyperbolic rational maps. Another interesting approach follows Shishikura’s Theorem which state that if the Julia set of a rational map is disconnected then the map has at least two weakly repelling fixed points. In the last part of the seminar I will present some work done by the Barcelona’s research group in holomorphic dynamics about the connectivity of the Julia set for transcendental meromorphic maps. In particular I will state the extension of Shishikura’s Theorem for this case which is a joint work with K. Baranski, N. Fagella and B. Karpinska involving the solution of an old conjecture on the existence of absorbing domains for Baker domains.

- Speaker:
**Rafel Prohens**

Affiliation: Department of Mathematics, Universitat de les Illes Balears

Title: **Alien limit cycles in Liénard equations**

Date: Tuesday, January 21, 2014

Place: Room ATA09

Time: 14:00

Summary : In this work we prove the existence of alien limit cycles in the polynomial Liénard differential equations by constructing an example of a family of polynomial differential equations Liénard exhibiting a limit cycle limit cycle of type alien. This limit cycle is perturbed from a 2-saddle cycle in the boundary of an annulus of

periodic orbits given by a Hamiltonian vector field. The Hamiltonian represents a truncated pendulum of degree $4$. In comparison to a former polynomial example, not only the equations are simpler but a lot of tedious calculations can be avoided, making the example also interesting with respect to simplicity in treatment.

- Speaker:
**Antoni Guillamon**

Affiliation: Department of Applied Mathematics, University (Catalonia)

Title: **A survey on estimation of synaptic conductance**

Date: Wednesday, October 23, 2013

Place: Room ATA09

Time: 13:30

Summary : Brain connectivity is Currently one of the hottest topics in neuroscience . Among all sub Emerging from it , the estimation of conductance from intracellular recordings , Intended to Obtain information about the local synaptic inputs to neuron That receiver , you have a special relevance . Despite of the existing literature , no satisfactory methods have yet BEEN GIVEN , the main obstacle BEING That it is an inverse problem analysis and sound data it requires to have a good Supporting behind in order to estimate model parameters . Associated to Different computational modelling strategies have BEEN proposed alternatives , but none of Them have reached a consensus . However , They constitute , altogether , a nice compendium of mathematical resources to handle a specific practical problem . In this informal talk we will present and discuss some Attempt That BOTH use statistical methods , tools from dynamical systems , master equations for processes or stochastic filtering methods . We will try to shed light on the drawback of EACH approach ( sometimes of practical nature , sometimes of mathematical nature ) Hoping to Encourage discussion to stimulate fresh opinions on the problem .

- Speaker:
**Paolo Massobrio**

Affiliation: Neuroengineering and Bio-nanoTechnology Lab (NBT), Department of Informatics, Bioengineering, Robotics and System Engineering (DIBRIS), University of Genova (Italy)

Title: **Playing with living and in silico neuronal assemblies: the neuroengineering approach**

Date: Wednesday, October 16, 2013

Place: Room ATA04B

Time: 12:30

Summary : In this seminar, I will first introduce the basic concepts of neuronal networks both from a physiological and computational point of view. Issues like how to describe the interactions among neurons (i.e., synaptic transmission), how to estimate, infer and model network connectivity (i.e., functional, effective, morphological) in large-scale networks will be described and discussed.

The second part of the seminar is devoted to present some examples of engineered neuronal networks recreated in vitro over Micro-Electrode Arrays (MEAs) by “playing” on parameters like connectivity, cell density and composition, network organization. By exploiting the MEA technology it is possible to record extracellularly the activity of neuronal assemblies from several electrodes, creating a neuro-electronic interface. Experimental and in silico results regarding the interplay between network connectivity and dynamics, as well as the interactions between heterogeneous neuronal populations (cortical-thalamic circuits) will be presented and discussed. Finally, possible perspectives for the development of hybrid networks where living neuronal assemblies are dynamically interfaced with in silico ones will be discussed as possible proof of concepts for the design of innovative neuroprostheses.

Keywords: Neuronal networks modelling, in vitro neuronal networks, network dynamics, interacting assemblies, neuroengineering.