Activation of neuronal ensembles via controlled synchronization

"In this contribution we present the activation of neuronal ensembles of Hindmarsh-Rose neurons by controlled synchronization. The main problem consists in to impose a particular spiking-bursting behavior in all the neurons of the network. We consider a network where the neurons are in its resting state, it is desired that the neurons change their resting state to a particular behavior of activation, dictated by a neuron called the reference neuron. The goal is reached by controlling some neurons in the network controlling only the membrane potential (electrical synapse). The key feature of the present contribution is that by controlling a small number of neurons in the network a desired behavior is induced in all the neurons in the network despite its network topology. The important parameters are the control gain and the coupling strength, thus the activation of the network lays down on a compromise between the control gain and the coupling strength.

On the emergence of chaos in dynamical networks

"We investigate how changes of specific topological features result on transitions among different bounded behaviours in dynamical networks. In particular, we focus on networks with identical dynamical systems, synchronised to a common equilibrium point, then a transition into chaotic behaviour is observed as the number of nodes and the strength of their coupling changes. We analyse the network's transverse Lyapunov exponents (tLes) to derive conditions for the emergence of bounded complex behaviour on different basic network models. We find that, for networks with a given number of nodes, chaotic behaviour emerges when the coupling strength is within a specific bounded interval; this interval is reduced as the number of nodes increases. Furthermore, the endpoints the emergence interval depend on the coupling structure of network. We also find that networks with homogeneous connectivity, such as regular lattices and small-world networks are more conducive to the emergence of chaos than networks with heterogeneous connectivity like scale-free and star-connected graphs. Our results are illustrated with numerical simulations of the chaotic benchmark Lorenz systems, and to underline their potential applicability to real-world systems, our results are used to establish conditions for the chaotic activation of a network of electrically coupled pancreatic β-cell models.

On the controllability of networks with nonidentical linear nodes

"The controllability of dynamical networks depends on both network structure and node dynamics. For networks of linearly coupled linear dynamical systems the controllability of the network can be determined using the well-known Kalman rank criterion. In the case of identical nodes the problem can be decomposed in local and structural contributions. However, for strictly different nodes an alternative approach is needed. We decomposed the controllability matrix into a structural component, which only depends on the networks structure and a dynamical component which includes the dynamical description of the nodes in the network. Using this approach we show that controllability of dynamical networks with strictly different linear nodes is dominated by the dynamical component. Therefore even a structurally uncontrollable network of different nn dimensional nodes becomes controllable if the dynamics of its nodes are properly chosen. Conversely, a structurally controllable network becomes uncontrollable for a given choice of the node’s dynamics. Furthermore, as nodes are not identical, we can have nodes that are uncontrollable in isolation, while the entire network is controllable, in this sense the node’s controllability is overwritten by the network even if the structure is uncontrollable. We illustrate our results using single-controller networks and extend our findings to conventional networks with large number of nodes.

Lie algebra on synchronization of different systems: a generalized function for Hodgkin-Huxley neurons

"In this contribution two results are taken: (1) The synchronization of noiseless Hodgkin- Huxley (HH) neurons is possible from robust feedback based on Lie algebra approaches and (2) the fact that, from Lie algebra of vector fields, the generalized synchronization of different (triangular form) chaotic systems can be used to derive an explicit synchronization function. Both results are extended to derive the synchronization function in HH neurons despite this systems are not in triangular form. Thus, the Lie algebra of vectors fields permits to establish a theoretical framework for finding the synchroniza- tion function in chaotic systems in face they have different model.

Conditions for synchronization and chaos in networks of β-cells

"The insulin producing and releasing pancreatic β-cells play a key role in glucose homeostasis. The proper function of these cells is characterized by a spiking-bursting activity in their membrane potential. It is known that these cells are coupled with each other and that this coupling can induce synchronization. This fact maybe relevant in the mechanisms of blood glucose regulation. However, not much is known about conditions of synchronization, nor about the topology of real-world β-cell network. Also about the conditions for the emergence of chaotic bursting in networks of β-cells not much is known. We investigate these phenomena in terms of complex network theory as model of β-cells networks, in which synchronization and emergence of chaotic bursting occurs.

Sincronización generalizada entre sistemas Rössler y Lorenz con dinámica Interna

"Se investiga la sincronización generalizada (SG) entre los osciladores caóticos Rössler y Lorenz en una configuración maestro-esclavo. Para lograr la sincronización se diseña un controlador escalar por retroalimentación obtenido a partir de una transformación de coordenadas basada enAlgebra de Lie. La SG se logra en términos de los cambios de coordenadas al estabilizar el error de sincronización en las variables transformadas. En particular, se muestra la existencia de una SG parcial entre Rössler y Lorenz cuando el grado relativo de los sistemas es menor a su dimensión.

Ciencia Odontológica 2.0

Libro que muestra avances de la Investigación Odontológica en MéxicoEs para los integrantes de la Red de Investigación en Estomatología (RIE) una enorme alegría presentar el segundo de una serie de 6 libros sobre casos clínicos, revisiones de la literatura e investigaciones. La RIE está integrada por cuerpos académicos de la UAEH, UAEM, UAC y UdeG

Robust synchronization of a class of uncertain complex networks via discontinuous control

"We propose robust controller designs to synchronize networks with uncertainties in their node dynamics and their connections. We consider two situations: in the first, we assume that the effect of uncertainties vanishes as synchronization is achieved. In the second, disturbances are assume nonvanishing but bounded. To achieve robust synchronization on these situations, we design local feedback controllers, which are smooth in the first case, and discontinuous in the latter. These designs allow us to establish synchronization criteria for this class of uncertain dynamical networks. We use numerical experiments to illustrate our results.

Synchronization in complex networks under structural evolution

"We investigate the effects of structural evolution on the stability of synchronized behavior in complex networks. By structural evolution we mean processes that change the topology of the network. In particular, we consider structural evolution as two simultaneous processes: on one hand, the topology changes according to an arbitrary switching law among a set of admissible patterns of connection; on the other hand, the strength of connection evolves according to an adaptive law. Our results show that by constraining the admissible patterns of connection, and using an adaptive law based on the difference between the nodes, we can guarantee the stability of the synchronized solution of the network despite structural changes. Additionally, we extend our results by considering alternative structural evolution processes, namely, a node-based adaptive strategy and a resetting switching law. We illustrate our results with numerical simulation.