| Neuronal Synchronization has been hypothesized to underlay the emergence of cell assemblies and to provide an important mechanism for the large-scale integration of distributed brain activity[1][2]. In this study we investigate the local and global synchronization of an ensemble of delayed interconnected neurons. We begin investigating a circuit composed of Hodgkin and Huxley [3] cells with reciprocal delayed synaptic connections, modeled as alpha function [4]. The different ways in which a neuronal network can be connected, i.e. the topology of the network, and how it affects the dynamic of the network is a critical issue to be explored. With this purpose, we study the influence of different topologies on the local and global synchronization of the network. Five different types of topologies are analyzed: regular, small-world, random, scale-free and all-to-all [5][6][7]. To gain insight into the effects of the delay in the synchronization properties, we consider two different congurations, one in which the delay of all the connections is the same and another in which the delay is broadly distributed. Since data about axonal distributions of conduction velocities in fibers is limited, specially in the case of humans, we explore a large family of distribution shapes obtained from a general gamma distribution function, ranging from exponential to quasi-delta functions. To characterize the synchronization of the neurons we use an order parameter based on the phase difference between elements [8]. Firts we define the phase of each element doing a linear interpolation between spike events [9], then we compute the local synchronization using the phase difference of all pairs of connected neighbors, and finally we compute the global synchronization using the phase difference between all neurons in the network. We found that synchronized firing activity between one neuron and its neighbors (local synchronization) is achieve in all networks considered while a high randomness in the connections is required for a global synchronized activity. The variation of the delay reveals a resonant-like effect in the synchronization even in the mean field topology. We also observe that in the synchronized state the frequency of all the neurons is locked to a lower value than in the desynchronized state, where the frequency is broadly distributed among higher values. Keywords: Neuronal Networks, Delay Distribution, Synchronization. References [1] Singer W. (1999) Neuronal Synchrony: A Versatile Code for the Definition of Relations Neuron, 24:49-65. [2] Varela F.J., Lachaux J.P., Rodriguez E., Martinerie J. (2001) The brainweb: phase synchronization and large-scale integration Nat. Rev. Neurosci., 2:299-230. [3] Hodgkin, A., and Huxley, A. (1952): A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117: 500-544 [4] Destexhe A., Mainen Z.F., Sejnowski T.J. (1994) An efficient method for computing synaptic conductances based on a kinetic model of receptor binding Neural Comput, 6:14-18. [5] Watts, D.J. and Strogatz, S.H. (1998) Collective dynamics of small-world networks Nature, 393 (6684): 440-442. [6] Barabasi, A.L. and Albert, R. (1999) Science, 286: 509-512. [7] Albert, R. and Barabasi, A.L.(2002) Rev. Mod. Phys., 74,47. [8] Grigori V. Osipov, Arkady S. Pikovsky, Michael G. Rosenblum, and Jürgen Kurths (1995) Phase synchronization effects in a lattice of nonidentical R&oauml;ssler oscillators Phys. Rev. E, 55: 2353-2361 [9] Gammaitoni L., Marchesoni F., and Santucci S. (1995) Stochastic Resonance as a Bona Fide Resonance Phys. Rev. Lett, 74:1052-1055. |
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