Emerging dynamics in neuronal networks of diffusively coupled hard oscillators

Oscillatory networks are a special class of neural networks where each neuron exhibits time periodic behavior. They represent bio-inspired architectures which can be exploited to model biological processes such as the binding problem and selective attention. In this paper we investigate the dynamics of networks whose neurons are hard oscillators, namely they exhibit the coexistence of different stable attractors. We consider a constant external stimulus applied to each neuron, which influences the neuron's own natural frequency. We show that, due to the interaction between different kinds of attractors, as well as between attractors and repellors, new interesting dynamics arises, in the form of synchronous oscillations of various amplitudes. We also show that neurons subject to different stimuli are able to synchronize if their couplings are strong enough.

[1]  Eugene M. Izhikevich,et al.  “Subcritical Elliptic Bursting of Bautin Type ” (Izhikevich (2000b)). The following , 2022 .

[2]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[3]  A. Winfree.  The geometry of biological time , 1991 .

[4]  Fernando Corinto,et al.  Weakly Connected Oscillatory Network Models for Associative and Dynamic Memories , 2007, Int. J. Bifurc. Chaos.

[5]  Kazuyuki Aihara,et al.  Analysis of positive Lyapunov exponents from random time series , 1998 .

[6]  C. Koch,et al.  Attention activates winner-take-all competition among visual filters , 1999, Nature Neuroscience.

[7]  Peter König,et al.  Stimulus-Dependent Assembly Formation of Oscillatory Responses: I. Synchronization , 1991, Neural Computation.

[8]  E. Izhikevich,et al.  Weakly connected neural networks , 1997 .

[9]  Michael J. Tsatsomeros,et al.  Doubly diagonally dominant matrices , 1997 .

[10]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[11]  O. Taussky.  Bounds for characteristic roots of matrices , 1948 .

[12]  Fernando Corinto,et al.  Weakly connected oscillatory networks for information processing , 2011 .

[13]  A. Roskies.  The Binding Problem , 1999, Neuron.

[14]  Gábor Orosz,et al.  Learning of Spatio–Temporal Codes in a Coupled Oscillator System , 2009, IEEE Transactions on Neural Networks.

[15]  Peter König,et al.  Binding by temporal structure in multiple feature domains of an oscillatory neuronal network , 1994, Biological Cybernetics.

[16]  O. Taussky.  A Recurring Theorem on Determinants , 1949 .

[17]  Peter König,et al.  Stimulus-Dependent Assembly Formation of Oscillatory Responses: III. Learning , 1992, Neural Computation.

[18]  E. Izhikevich,et al.  Oscillatory Neurocomputers with Dynamic Connectivity , 1999 .

[19]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[20]  Wen-Chyuan Yueh.  EIGENVALUES OF SEVERAL TRIDIAGONAL MATRICES , 2005 .