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Synchronization Analysis in Models of Coupled Oscillators Guilherme M. Toso, Fabricio A. Breve Guilherme Toso So Paulo State University (UNESP) guilherme.toso@unesp.br July 3, 2020 1 Summary 1. Introduction 2. Phase Synchronization 3.

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Synchronization Analysis in Models of Coupled Oscillators

Guilherme M. Toso, Fabricio A. Breve July 3, 2020

Guilherme Toso

SΓ£o Paulo State University (UNESP) guilherme.toso@unesp.br

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Summary

  • 1. Introduction
  • 2. Phase Synchronization
  • 3. Methodology and Models
  • 4. Results
  • 5. Conclusions
  • 6. Bibliography
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  • Visual Attention is a technique used by biological neural

network systems developed to reduce the large amount

  • f visual information that it is received by natural

sensors [5].

  • In 1981, von der Malsburg [13] suggested that each
  • bject is represented by the temporal correlation of

neural firing activities, which can be described by dynamic models

Introduction

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Introduction

  • A natural way of representing the coding of the temporal

correlation is to use synchronization between oscillators.

  • Objective: Study of synchronization in some biological

neurons’ models which exhibit chaotic behaviors, by using a coupling force between the oscillators as in Breve et. all work [3]

  • The motivation is to use this sync method for visual

selection of objects that represents sync neurons' models, while the rest of the image is unsynced.

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  • The phase synchronization of two oscillators p and q

happens when their phases difference |πœ’π‘ž - πœ’π‘Ÿ | is kept below a certain phase threshold C.

  • So as t β†’ ∞, |πœ’π‘ž - πœ’π‘Ÿ | < C. The phase i at time 𝑒𝑗 is

calculated as following [11]: πœ’π‘— = 2πœŒπ‘™ +

𝑒𝑗 βˆ’ 𝑒𝑙 𝑒𝑙+1 βˆ’ 𝑒𝑙

(1)

  • where k is the number of neural activities prior to time 𝑒𝑗,

and 𝑒𝑙 and 𝑒𝑙+1 are the last and the next times of neural activity, respectively.

Phase Synchronization

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  • So that two oscillators can synchronize with each other, a

coupling term is added to the dynamical system as the following: ሢ π‘¦π‘˜

π‘ž = 𝐺 π‘˜ 𝒀, 𝝂 + π‘™βˆ†π‘ž,π‘Ÿ

(2) ሢ π‘¦π‘˜

π‘Ÿ = 𝐺 π‘˜ 𝒀, 𝝂 + π‘™βˆ†π‘Ÿ,π‘ž

  • Where ሢ

π‘¦π‘˜

π‘ž and ሢ

π‘¦π‘˜

π‘Ÿ are the time evolution of the π‘¦π‘˜ state of

the p and q oscillators. 𝐺

π‘˜ 𝒀, 𝝂 is the behaviour’s rate

and π‘™βˆ†π‘ž,π‘Ÿ is the coupling term, where k is a coupling force and βˆ†π‘ž,π‘Ÿ is the difference between the states: βˆ†π‘ž,π‘Ÿ= π‘¦π‘˜

π‘Ÿ βˆ’ π‘¦π‘˜ π‘ž

(3)

Phase Synchronization

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  • The proposed models for the attention system are a two-

dimensional network of neural models' dynamical systems with coupled terms.

  • Dynamical Systems: Hodgkin-Huxley [8], Hindmarsh-

Rose [7], Integrate-and-Fire [10], Spike-Response- Model [6]. It was used the 4th Order Runge-Kutta numerical method.

  • Discrete Models: Aihara’s [1], Rulkov’s [12], Izhikevic [9]

and Courbage-Nekorkin-Vdovin [10].

  • Search for chaos by varying the parameters values in 𝝂

= (𝜈1, 𝜈2, ..., πœˆπ‘—, ..., πœˆπ‘‚) or adding a white noise at the models.

Methodology and Models

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Methodology and Models

  • Fig. 1: Two Oscillator Problem
  • Fig. 2: Vector of Oscillators Coupled
  • Fig. 3: Grid of Oscillators Coupled
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Coupling Force Variation: some oscillators were strongly coupled and others weakly, so that the first were synchronized and hence clusterized.

Methodology and Models

  • Fig. 4: Grid of Neurons
  • Fig. 5: Grid of Sync

Neurons and Unsync.

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Chaotic and stochastic trajectories to represent different neurons and pixels.

Results

  • Fig. 6: Stochastic Hodgkin-

Huxley

  • Fig. 7: Chaotic Hindmarsh-Rose
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  • Fig. 7: Stochastic Integrate-and-Fire

Results

  • Fig. 8: SRM with different limit times
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  • Fig. 9: Chaotic Aihara

Results

  • Fig. 10: Chaotic Rulkov
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  • Fig. 11: Chaotic Izhikevic

Results

  • Fig. 12: Chaotic CNV
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Trajectories and phases difference of a grid of oscillators with phase threshold at 2𝜌 [2].

Results

(a) Trajectories Difference (b) Phases Difference

  • Fig. 13: Hodgkin-Huxley Model
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(a) Trajectories Difference

Results

(b) Phases Difference

  • Fig. 14: Hindmarsh-Rose Model
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(a) Trajectories Difference

Results

(b) Phases Difference

  • Fig. 15: Integrate-and-Fire Model
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(a) Trajectories Difference

Results

(b) Phases Difference

  • Fig. 16: Spike-Response-Model
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(a) Trajectories Difference

Results

(b) Phases Difference

  • Fig. 17:Aihara’s Model
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(a) Trajectories Difference

Results

(b) Phases Difference

  • Fig. 18: Rulkov’s Model
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(a) Trajectories Difference

Results

(b) Phases Difference

  • Fig. 19: Izhikevic’s Model
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(a) Trajectories Difference

Results

(b) Phases Difference

  • Fig. 20: Courbage-Nekorkin-Vdovin Model
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Trajectories and the phases difference of the models (Hodgkin-Huxley, Hindmarsh-Rose and Integrate-and-Fire) in a grid with sync and unsync oscillators.

Results

(a) Synchronized and desynchronized Trajectories (b) Phases Difference

  • Fig. 21: Hodgkin-Huxley Model
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(a) Synchronized and Desynchronized Trajectories

Results

(b) Phases Difference

  • Fig. 22: Hindmarsh-Rose Model
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(a) Synchronized and Desynchronized Trajectories

Results

(b) Phases Difference

  • Fig. 23: Integrate-and-Fire Model
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  • Discrete time models didn’t synchronizes.

Continuous time models synchronizes.

  • Spike-Response-Model synchronizes without a

coupling force, only considering the arrival time

  • f presynaptic stimuli. But did not show chaos

behavior.

  • The continuous models tested for the

synchronization and desynchronization for a cluster formation depending on the coupling force showed a potential solution for a visual selection mechanism for an attention system.

Conclusions

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  • 1. Aihara, K., Takabe, T., Toyoda, M.: Chaotic neural
  • networks. Physics letters A144(6-7), 333-340 (1990)

2. Breve, F.: Aprendizado de maquina utilizando dinÒmica espaço-temporal em redes complexas. São Carlos: Universidade de São Paulo (Tese de Doutorado) (2010) 3. Breve, F.A., Zhao, L., Quiles, M.G., Macau, E.E.: Chaotic phase synchronization for visual selection. In: Neural Networks, 2009. IJCNN 2009. International Joint Conference on, pp. 383-390. IEEE (2009) 4. Courbage, M., Nekorkin, V., Vdovin, L.: Chaotic

  • scillations in a map-based model of neural activity.

Chaos: An Interdisciplinary Journal of Nonlinear Science 17(4), 043109 (2007)

Bibliography

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  • 5. Desimone, R., Duncan, J.: Neural mechanisms of

selective visual attention. Annual review of neuroscience 18(1), 193-222 (1995)

  • 6. Gerstner, W.: A framework for spiking neuron models:

The spike response model. In: Handbook of Biological Physics, vol. 4, pp. 469-516. Elsevier (2001)

  • 7. Hindmarsh, J.L., Rose, R.: A model of neuronal bursting

using three coupled first order differential equations. Proceedings of the Royal society of London. Series B. Biological sciences 221(1222), 87-102 (1984)

  • 8. Hodgkin, A.L., Huxley, A.F.: A quantitative description of

membrane current and its application to conduction and excitation in nerve. The Journal of physiology 117(4), 500-544 (1952)

Bibliography

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  • 9. Izhikevich, E.M.: Simple model of spiking neurons. IEEE

Transactions on neural networks 14(6), 1569-1572 (2003) 10.Lapicque, L.: Recherches quantitatives sur l'excitation electrique des nerfs traitee comme une polarization. Journal de Physiologie et de Pathologie Generalej 9, 620-635 (1907) 11.Pikovsky, A., Rosenblum, M., Kurths, J., Kurths, J.: Synchronization: a universal concept in nonlinear sciences, vol. 12. Cambridge university press (2003) 12.Rulkov, N.F.: Modeling of spiking-bursting neural behavior using two-dimensional map. Physical Review E 65(4), 041922 (2002) 13.von der Malsburg, C.: The correlation theory of brain

  • function. Tech. rep., MPI (1981)

Bibliography