Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 79-82___________________________________________________________________________________________________________

Selective synchronization of coupled bifurcating neurons
for phase shift of background oscillation
Akihiro Yamaguchi*, Yutaka Yamaguti

Department of Information and Systems Engineering, Fukuoka Institute of Technology,
3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295, Japan
Masao Kubo

Department of Computer Science, National Defense Academy of Japan,
1-10-20 Hashirimizu, Yokosuka, Kanagawa, 239-8686, Japan
E-mail: *aki@fit.ac.jp

Abstract
Synchronization in coupled bifurcating neurons was studied from the view point of the selective formation of cell
assembly. The bifurcating neuron is a simple chaotic neuron that exhibits chaotic inter-spike interval dynamics via
the addition of sinusoidal background oscillation. In this research, we introduced the phase shift of the backgroundoscillation and several types of phase response to the input spike sequence. Sixteen coupled bifurcating neurons withcomplete bidirectional coupling were numerically simulated and their synchronized behaviors were analyzed. As aresult, the formation of chaotically synchronized cell assemblies for the phase shift value were observed.
Keywords: chaotic synchronization, neural coding, bifurcating neuron, binding information

network and coupling model. In Section 4, we describe
the numerical simulation that is performed to examine theselective synchronization for the phase shift of the
background oscillation. We summarize the results in
Section 5.

1. Introduction
Synchronized neural activity has an important role inneural information coding and brain information
processing1. The correlated firing among neurons has
been observed in visual information processing in thebrain2,3. The authors have studied the chaoticsynchronization of the neural spike response and its
application to the segmentation of input images4; featurelinking of the visual image of moving objects5; and thedecomposition of superimposed chaotic spike sequences6.
In order to represent multiple information by chaotic
synchronized cell assemblies, confirming the coexistenceof synchronized assemblies is necessary.
In this study, we constructed the network of thebifurcating neuron7 with complete bidirectional couplingand designed a coupling model to achieve selective
chaotic synchronization based on the phase response. InSections 2 and 3, we explain the bifurcating neuron

2. Network model of the bifurcating neuron
The bifurcating neuron was introduced by Lee andFarhat7. In our previous research6, we defined thebifurcating neuron as a form of the spike response model8
to extend the coupling term based on the phase response.
In the following sections, we explain the extended
bifurcating neuron model and its network. Let thenetwork of bifurcating neurons consist of 𝑁 neurons,
where 𝑛 is the 𝑖-th neuron. The dynamics of the 𝑖-thbifurcating neuron is defined as follows:𝑢𝑡𝑢𝜉𝜂𝑡𝑡𝜉
𝜈 ,𝑡Copyright © 2018, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
79(1)

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 79-82___________________________________________________________________________________________________________

where 𝑢 , 𝑢, 𝜂 , 𝜉 , 𝜉 , and 𝜈 denote theinternal potential of 𝑛 , resting potential, kernel
function of internal dynamics, positive coupling term,
negative coupling term, and external noise, respectively.
The coupling terms are described in the next section.
The kernel function 𝜂 is defined as𝜂𝑡𝛼 𝑡𝜂 𝑡, 𝜙𝜂 𝑡,𝜙𝐴 𝑠𝑖𝑛 2𝜋𝜔 𝑡𝜙 ,𝑡;chaotic dynamics in the inter-spike interval, and thephase of the next firing time is determined by the chaotic
one dimensional map of the phase of the last firing time6,7.
The shape of this chaotic map is determined by the phaseshift value, 𝜙 . Examples of this chaotic map are shownin Fig. 1. Furthermore, the bifurcation diagram andLyapunov exponents of this map are shown in Fig. 2.

(2)

3. Coupling model with phase response(3)Let the set 𝑆𝑡 , 𝑡 , 𝑡 , ⋯ be a set of firing
time of the neuron, 𝑛 , where 𝑡 is the firing time of𝑗-th spike of 𝑛 . The set firing time of the input spikes
to 𝑛 is denoted as 𝛤 . For bidirectional complete
coupling, Γ is defined aswhere 𝛼 is the time constant of the kernel 𝜂 , and 𝑡
is the last firing time of 𝑛 . The internal potential 𝑢 islinearly increasing by the kernel 𝜂 . The neuron 𝑛 isfiring when 𝑢 exceeds the threshold, 𝜃, and then 𝑢 isreset to the initial potential given by the backgroundoscillation, 𝜂 𝑡, 𝜙 . The constants, 𝐴 , 𝜔, and 𝜙 are
the amplitude, frequency, and phase shift value of
background oscillation, respectively.
The dynamics of a single neuron without the coupling
term is the same as with the original bifurcating neuron.
The single neuron without coupling exhibits various𝑆

𝛤

.

(4)

The coupling term 𝜉 and 𝜉 in Eq. (1) are thesummation of the positive and negative phase responses
𝜀 and 𝜀 , respectively, such that
𝜉𝑡𝑠 .

𝜀∈

(5)

,The simple phase response is a constant value, such that
𝜀𝑠𝛽 ,

(6)

where 𝛽 and 𝛽 are non-negative coupling constants.
The next firing time is hastened by the positive responseand delayed by the negative response. We describedcoupling with these simple responses as constant positive
and negative coupling, respectively.
We designed the phase response to strengthen thechaotic synchronization between coupled neurons withthe same phase shift value6. The positive phase response
based on the predicted next firing time is defined as

(a)
(b )
Fig. 1. Examples of the chaotic one dimensional map of thephase of a single bifurcating neuron with the phase shift (a) 𝜙𝜋/2 and (b) 𝜙 3𝜋/2.

0𝜀𝑠𝛽𝑡̂

𝛥𝑡̂

0where 𝑡̂that80𝛥𝑡̂𝑠,(7)𝑠is the predicted next firing time of 𝑛 , such𝑡̂

Fig. 2. The bifurcation diagram and Lyapunov exponents of asingle bifurcating neuron without coupling.

𝑡̂𝑠𝑡

𝜃

𝑢 𝑡.
𝛼(8)

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 79-82___________________________________________________________________________________________________________Selective synchronization of theWe describe this coupling as adaptive positive coupling.
The negative phase response based on the last firing timeis defined as𝜀⎧⎪𝑠⎨⎪
⎩0𝛽𝑠𝑡0𝑡𝑠𝑡Δ .

𝑡Δ(9)𝑠We describe this coupling as adaptive negative coupling.
If time, 𝑠, is within the range, Δ , from the predictednext firing time 𝑡̂, then the phase response is positive,
so that the next firing time is hastened. On the other hand,
if time, 𝑠, is within the range, Δ , from the last firing time,𝑡 , then the phase response is negative to delay the next
firing time. By using the phase responses in Eq. (7) and(9), the decomposition of the superimposed chaotic spikesequences with the different phase shift values ispossible6.

Fig. 3. An example of generated spike sequences for thenetwork with the adaptive positive and negative coupling.

4. Numerical experiments

synchronization ratio for the neurons with the same phaseshift value, such that

4.1. Experimental setting
We numerically simulated a network of 16 bifurcatingneurons to analyze the selective synchronization to thesame phase shift value for complete bidirectional
coupling. We assigned 4 different phase shift values, asfollows:𝜙Δ ⌊𝑖/4⌋ 𝑖0, ⋯ ,15 ,(10)Δ,,𝑆𝑅 𝑆 ; 𝑆𝑀𝑆𝑅,/192.

(13)

4.3. Experimental results
An example of our generated spike sequences from thenumerical simulation are shown in Fig. 3, where thephase response coupling is the adaptive positive andnegative coupling. The generated spikes weresynchronized for each 4 neurons using the same phaseshift value, and selective synchronization for the same
phase shift value was observed. We numericallysimulated the network and evaluated the synchronization
ratio to analyze the effect of coupling, and the mean ratiofor each of the 5 types of coupling that were defined inSection 3. Results are shown in Fig. 4 and Table 1. These
results indicated that selective synchronization wasobserved for coupling types (c) and (e). The mean
synchronization ratios were greater than 90% for theAs an index of synchronization between two spikesequences, the synchronization ratio was defined as𝑡/ #𝑆(12)

and mean synchronization ratio for neurons with thedifferent phase shift values, such that

4.2. Characteristics of synchronization𝑡/48,,

is the difference between the assigned
where 0 Δ
phase shift values. In this setting, each 4 neurons have thesame phase shift value. We examined the selectivesynchronization among each 4 neurons with the samephase shift value. Numerical simulation was performed
using the following settings: 𝛼 100 , 𝜃
30 ,70, 𝐴
21.5, 𝜔 1, 𝛽𝛽
2.1, Δ
𝑢0.05, and Δ
𝜋/2.

𝑆𝑅 𝑆 ; 𝑆#𝑡𝑡 ∈ 𝑆 ,𝑡 ∈ 𝑆

𝑆𝑅 𝑆 ; 𝑆𝑀𝑆𝑅

(11)

where # ⋅ denotes the number of elements and Δ is thetemporal resolution of the firing time. Furthermore, wecharacterized the assembly formation of thesynchronized cluster of neurons by defining the mean81

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 79-82___________________________________________________________________________________________________________

1

10.8

0.8

10.6

10.6

0.8

0.8

0.4

0.4

0.6
0.4

00.6

0.2

0.2

0.2

0.2

00
14
12
1088

neuron ID

664

4
20 02

8

64

1014

12

20 0

108

642

1214

neuron ID

neuron ID(a) Constant Positive(b) Constant Negative
1

1
0.8

0.8

10.6

10.6

0.8

0.8

0.4

0.4

0.6

0.2
00.6

0.2

0.4

14
12108

neuron ID6286410120.7861

0.4766(b) Constant Negative

0.18190.0638(c) Adaptive Positive

0.94140.3694(d) Adaptive Negative(e) Adaptive Positive
and Negative

0.65200.20570.97230.3519Acknowledgements

8640 0(a) Constant Positive

002𝑀𝑆𝑅

0.2

0.2

4𝑀𝑆𝑅

0.4

0141210neuron ID

Coupling type

0.4

0141210neuron ID

Table 1. Mean synchronization ratio for the neurons with the, and for the neurons withsame phase shift value, 𝑀𝑆𝑅
the different phase shift values, 𝑀𝑆𝑅.

1420 042(c) Adaptive Positive

86101214This work was partially supported by MEXT KAKENHI
Grant Number 15H05878 and 16K00409.

neuron ID

neuron ID(d) Adaptive Negative

10.8References

10.6

0.8

0.4

1. Fuji, H., Ito, H., Aihara, K., Ichinose, N., and Tsukada, M.,
Dynamical cell assembly hypothesis – Theoreticalpossibility of spatio-temporal coding in the cortex,
Neural Networks, Vol.9, No.8, 1996, pp.1303-1350．
2. Gray, C.M., Koenig, P., Engel, A. K., and Singer, W.,
Oscillatory responses in cat visual cortex exhibit intercolumnar synchronization which reflects global stimulus
properties, Nature, 338, 1989, pp.334-337.
3. Eckhorn, R., Reitboeck, H.J., Arndt, M., Dicke, P., Featurelinking via stimulus-evoked oscillations: experimental
results from cat visual cortex and functional implications
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Neural Networks, 1989, pp. 723-730. .
4. M. Fujiwara, A. Yamaguchi, and M. Kubo, Synchronized
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Neural Network, J. Robotics, Networking and Artificial
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using Synchronized Responses in Chaotic Cellar NeuralNetworks for Visual Stimulus of Moving Objects,
J. Robotics, Networking and Artificial Life, Vol.2, No.4,
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6. A. Yamaguchi, Y. Yamaguti, and M. Kubo,
Decomposition of superimposed chaotic spike sequencesby using the bifurcating neuron, to appear in Advances incognitive neurodynamics (VI), Springer, 2018.
7. Lee G., and Farhat, N.H., The Bifurcating Neuron
Network 1, Neural Networks, Vol. 14, pp. 115-131, 2001.
8. Gerstner, W., and Kistler, W., Spiking Neuron Models:Single Neurons Populations Plasticity, Cambridge
University Press, 2002.

0.60.20.4

00.20141210neuron ID

8642002468101214

neuron ID(e) Adaptive Positive and Negative (both)Fig. 4. The synchronization ratio between two neurons for (a)constant positive, (b) constant negative, (c) adaptive positive,(d) adaptive negative, and (e) both adaptive positive andnegative coupling types.

neurons with the same phase shift value, and less than
40% for the neurons with the different phase shift values.
5. Conclusion
We have constructed a bifurcating neuron network, andassessed the formation of chaotically synchronized cell
assemblies for several types of coupling model. As aresult, the chaotically synchronized cell assemblies wereobserved for adaptive positive coupling of the phase
response, where neurons that have the same phase shift
value were selectively synchronized. This result indicatesthe possibility of binding information based on the phaseshift value of the background oscillation. Our future work
will seek to analyze the effect of adaptive coupling using
a different chaotic neuron model, which will further
elucidate such information coding.

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