Sincronización de la actividad eléctrica neuronal, utilizando el modelo de Hodgkin-Huxley y el circuito RCLSJ

We simulated the neuronal electrical activity using the Hodgkin-Huxley model (HH) and a superconductor circuit, containing Josephson junctions. These HH model make possible simulate the main neuronal dynamics characteristics such as action potentials, firing threshold and refractory period. The purpose of the manuscript is show a method to syncronize a RCLshunted Josephson junction to a neuronal dynamics represented by the HH model. Thus the RCLSJ circuit is able to mimics the behavior of the HH neuron. We controlated the RCLSJ circuit, using and improved adaptative track scheme, that with the improved Lyapunov functions and the two controllable gain coefficients allowing synchronization of two neuronal models. Results will provide the path to follow forward the understanding neuronal networks synchronization about, generating the intrinsic brain behavior.Simulamos la actividad eléctrica neuronal mediante el modelo de Hodgkin-Huxley (HH) y un circuito superconductor, que contiene uniones Josephson. El modelo HH simulan las características principales de la dinámica neuronal tales como potenciales de acción, umbrales de disparo y el períodos refractarios. El propósito del manuscrito es mostrar un método para sincronizar un circuito con union Josephson RCLSJ a una dinámica neuronal representado por el modelo HH. Así, el circuito RCLSJ es capaz de imitar el comportamiento de la neurona HH. Controlamos el circuito RCLSJ, utilizando un esquema de control adaptativo, que con funciones de Lyapunov y dos coeficientes de ganancia controlables nos permiten la sincronización de los dos modelos neuronales. Los resultados proporcionan una ruta a seguir adelante en el entendimiento de la sincronización de redes neuronales, generadas por el comportamiento intrinseco del cerebro

Lyapunov-based synchronization of two coupled chaotic Hindmarsh-Rose neurons

This paper addresses the synchronization of coupled chaotic Hindmarsh-Rose neurons. A sufficient condition for self-synchronization is first attained by using Lyapunov method. Also, to achieve the synchronization between two coupled Hindmarsh-Rose neurons when the self-synchronization condition not satisfied, a Lyapunov-based nonlinear control law is proposed and its asymptotic stability is proved. To verify the effectiveness of the proposed method, numerical simulations are performed

Synchronization of Coupled Different Chaotic FitzHugh-Nagumo Neurons with Unknown Parameters under Communication-Direction-Dependent Coupling

This paper investigates the chaotic behavior and synchronization of two different coupled chaotic FitzHugh-Nagumo (FHN) neurons with unknown parameters under external electrical stimulation (EES). The coupled FHN neurons of different parameters admit unidirectional and bidirectional gap junctions in the medium between them. Dynamical properties, such as the increase in synchronization error as a consequence of the deviation of neuronal parameters for unlike neurons, the effect of difference in coupling strengths caused by the unidirectional gap junctions, and the impact of large time-delay due to separation of neurons, are studied in exploring the behavior of the coupled system. A novel integral-based nonlinear adaptive control scheme, to cope with the infeasibility of the recovery variable, for synchronization of two coupled delayed chaotic FHN neurons of different and unknown parameters under uncertain EES is derived. Further, to guarantee robust synchronization of different neurons against disturbances, the proposed control methodology is modified to achieve the uniformly ultimately bounded synchronization. The parametric estimation errors can be reduced by selecting suitable control parameters. The effectiveness of the proposed control scheme is illustrated via numerical simulations

Nonlinear Dynamics, Synchronisation and Chaos in Coupled FHN Cardiac and Neural Cells

Physiological systems are amongst the most challenging systems to investigate from a mathematically based approach. The eld of mathematical biology is a relatively recent one when compared to physics. In this thesis I present an introduction to the physiological aspects needed to gain access to both cardiac and neural systems for a researcher trained in a mathematically based discipline. By using techniques from nonlinear dynamical systems theory I show a number of results that have implications for both neural and cardiac cells. Examining a reduced model of an excitable biological oscillator I show how rich the dynamical behaviour of such systems can be when coupled together. Quantifying the dynamics of coupled cells in terms of synchronisation measures is treated at length. Most notably it is shown that for cells that themselves cannot admit chaotic solutions, communication between cells be it through electrical coupling or synaptic like coupling, can lead to the emergence of chaotic behaviour. I also show that in the presence of emergent chaos one nds great variability in intervals of activity between the constituent cells. This implies that chaos in both cardiac and neural systems can be a direct result of interactions between the constituent cells rather than intrinsic to the cells themselves. Furthermore the ubiquity of chaotic solutions in the coupled systems may be a means of information production and signaling in neural systems