Journal of Robotics, Networking and Artificial Life, Vol. 2, No. 4 (March 2016), 230-233

Feature Linking using Synchronized Responses in Chaotic Cellar Neural Networks
for Visual Stimulus of Moving Objects
Akihiro Yamaguchi *, Satoshi Arakane

Department of Information and Systems Engineering, Fukuoka Institute of Technology,
3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0116, JAPANMasao Kubo

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

Abstract
A feature-linking mechanism using the synchronized responses of neural assemblies was studied for chaotic cellar
neural networks (Chaotic-CNN). Chaotic-CNNs consist of chaotic spike response neurons that show chaotic interspike-interval dynamics. In our scheme of feature linking, the features of the target objects are linked by thesynchronized spike responses characterized by the temporal chaotic pattern of the spike sequence. In this paper, weanalyze the synchronized spike responses invoked by the visual stimulus of moving bars. Consequently, theresulting neural assemblies have higher correlation for the visual stimulus of two bars moving in the same direction
than in the opposite direction. We also discuss the possibility of feature linking using the chaotic synchronizedresponses.
Keywords: chaotic synchronization, neural coding, spike response model, feature linking

1. Introduction
In the brain system, feature linking is one of theprincipal functions used to realize recognition of visualobjects. From physiological experiments, correlated
firing among neurons is regarded as a candidate
mechanism for such feature linking1,2. In neural coding
schemes that use the synchronization of neurons,
feature-linking information is represented bysynchronized firing among the corresponding neurons
such that the features detected by neurons are linked if
their firing patterns are synchronized.
On the other hand, chaotic systems are well-known
for their complex behavior. Comparing with a periodic
spike pattern, a chaotic spike pattern is expected to havean advantage in terms of the variety of represented
information. Therefore, there is a possibility ofincreases the performance of information representationby using chaotic synchronization of spike responses3,4.
From the view point of neural coding, the authors
have studied the formation of chaotic cell assemblies inchaotic cellular neural networks (Chaotic-CNNs), which
are two-dimensional coupled networks of chaotic
neurons5-7. Chaotic neurons are modeled by the chaotic
spike response model (Chaotic-SRM), which is anextended spike response model (SRM), to show chaotic
inter-spike intervals by adding background sinusoidal
oscillation5-7.
In our previous study, chaotic cell assemblies wereformed in Chaotic-CNNs for visual stimuli when astationary image was inputted6,7. In this research, weanalyze the formation of chaotic cell assemblies for thevisual stimulus of moving objects and their featurelinking property.

Published by Atlantis Press
Copyright: the authors
230

Akihiro Yamaguchi, Satoshi Arakane, Masao Kubo

2. Chaotic Cellular Neural Networks
Chaotic-CNNs are defined as two-dimensional coupledsystems of Chaotic-SRM5-7. In the following section, thedefinitions of the neuron model and the network aredescribed, respectively.
2.1. Neuron Model
As a neuron model, we use a Chaotic-SRM, which is anextended spike response model (SRM). The SRM wasintroduced by Gerstner and Kistler8. The definition of aChaotic-SRM is as follows.
The membrane potential 𝑢(𝑡) of the neuron at time 𝑡 isdefined as𝑢(𝑡) = 𝑢𝑟𝑟𝑟𝑟 + 𝜂(𝑡 − 𝑡 ∗ ) + 𝛽,(1)∗where 𝑢𝑟𝑟𝑟𝑟 , 𝑡 , and 𝛽 denote the resting potential, thelast firing time, and the external input, respectively. Thekernel function 𝜂 is defined as𝑡 − 𝑡∗𝜂(𝑡 − 𝑡 ∗ ) = −𝜂𝑖𝑖𝑖𝑖 𝑒𝑒𝑒 �𝜏𝜂0� Θ(𝑡 − 𝑡 ∗ ),(2)

where 𝜏𝜂0 is the time constant of the spike response and𝛩 is a step function such that 𝛩(𝑠) is 1 for 𝑠 ≥ 0 and 0for other values.
In this model, when the membrane potential
exceeds the threshold value 𝜃, the neuron fires and themembrane potential is reset by the update of the last
firing time 𝑡 ∗ . The term −𝜂𝑖𝑖𝑖𝑖 is the initial value of thekernel function 𝜂 after firing.
We extended the original SRM to show the chaotic
response by adding a background sinusoidal oscillationin the same way as for a bifurcating neuron9 and achaotic pulse-coupled neural network4. In our model,
background oscillation is added to the term 𝜂𝑖𝑖𝑖𝑖 suchthat𝜂𝑖𝑖𝑖𝑖 = 𝜂0 − 𝐴𝜂0 𝑠𝑠𝑠�2𝜋𝜔𝜂0 𝑡 ∗ + 𝜙�,(3)where 𝐴𝜂0 , 𝜔𝜂0 , and 𝜙 denote the amplitude, thefrequency, and the phase shift of the backgroundoscillation, and 𝜂0 is constant. The Chaotic-SRM shows
various chaotic behaviors depending on the parameters𝐴𝜂0 and 𝛽.
2.2. Definition of a Network

In a Chaotic-CNN, each neuron is located in an 𝑁 × 𝑀
lattice and connected to neighbors. Let 𝑛𝑥,𝑦 be a neuron
located at the position (𝑥, 𝑦) where 𝑥 ∈ {0,1, ⋯ , 𝑁 − 1}and 𝑦 ∈ {0,1, ⋯ , 𝑀 − 1}. The membrane potential 𝑢𝑥,𝑦of the neuron 𝑛𝑥,𝑦 is defined as∗𝑢𝑥,𝑦 (𝑡) = 𝑢𝑟𝑟𝑟𝑟 + 𝜂�𝑡 − 𝑡𝑥,𝑦� + 𝛽𝑥,𝑦+𝜉 ×

�𝑛𝑥′,𝑦′ ∈𝐵(𝑥,𝑦;𝑟)𝑜𝑥 ′ ,𝑦′ (𝑡)(4)∗where 𝑢𝑟𝑟𝑟𝑟 , 𝑡𝑥,𝑦, 𝛽𝑥,𝑦 , 𝜉 , and 𝐵(𝑥, 𝑦; 𝑟) denote theresting potential, the last firing time, the external input,
the coupling weight, and the set of connected neurons,respectively. A set of connected neurons is defined as

𝐵(𝑥, 𝑦; 𝑟) = �𝑛𝑥 ′ ,𝑦′ � (𝑥 ′ , 𝑦 ′ ) ≠ (𝑥, 𝑦),max(|𝑥 − 𝑥 ′ |, |𝑦 − 𝑦 ′ |) ≤ 𝑟 ,0 ≤ 𝑥 ′ ≤ 𝑁 − 1, 0 ≤ 𝑦 ′ ≤ 𝑀 − 1 }.

(5)

The function 𝑜𝑥 ′ ,𝑦′ is the output from the connected
neuron 𝑛𝑥 ′ ,𝑦′ and is defined as𝑜𝑥 ′ ,𝑦′ (𝑡) =(𝑘)�(𝑘)∗ <𝑡𝑡𝑥,𝑦+Δε<𝑡𝑥′ ,𝑦′(𝑘)𝜀(𝑡 − 𝑡𝑥 ′,𝑦′ − Δ𝜀)(6)where 𝑡𝑥 ′ ,𝑦′ denotes the 𝑘th firing time of the neuron𝑛𝑥 ′ ,𝑦′ and Δ𝜀 is the time delay of the synaptic
connection. The kernel function 𝜀 describes theresponse of the synaptic connection. The definition ofthe kernel function 𝜀 is𝜀(𝑠) =𝑠𝜏𝜀𝑠𝑒𝑒𝑒 �− � Θ(𝑠),𝜏𝜀(7)where 𝜏𝜀 is the time constant of the synaptic connection.
2.3. Gradient Field of a Phase Shift

In this research, to emphasize the synchronized response
in one direction, a gradient field of a phase shift isintroduced in Eq. (3). The gradient field of a phase shift
is defined as,
𝜙𝑥,𝑦 (𝛼) = 𝐷(cos(𝛼) 𝑥 + sin(𝛼) 𝑦) 𝑚𝑚𝑚 2𝜋,(8)where 𝐷 is the constant gradient and 𝛼 is the directionof the gradient. Therefore, 𝜙𝑥,𝑦 is the same value in theorthogonal direction to 𝛼 in a Chaotic-CNN.

3. Numerical Analysis of Two Coupled Neurons

As a preliminary numerical experiment, we examined
the synchronization property of two coupled neurons
corresponding to a 2 × 1 Chaotic-CNN. In this case, theneurons 𝑛0,0 and 𝑛1,0 were coupled. In the numericalsimulation, the parameter values of the Chaotic-SRM
were set as follows: 𝑢𝑟𝑟𝑟𝑟 = −70 mv , 𝜃 = −35 mv ,0.75, 𝛢𝜂0 = 10.9 , 𝜏𝜀 = 1.5𝜂0 = 55 , 𝜏𝜂 = 10 , 𝜔𝜂0 =2𝜋
and 𝛥𝛥 = 0.1 . The external input was chosen as

Published by Atlantis Press
Copyright: the authors
231

Feature Linking using Synchronized

𝛽0,0 = 𝛽1,0 = 52.5 such that a single neuron exhibits achaotic spike response.
As an index of synchronization, the crosscorrelation between two spike sequences was analyzed.
Let 𝑆𝑥,𝑦 be the set of the firing times of the neuron 𝑛𝑥,𝑦 .
The cross-correlation between 𝑆𝑥,𝑦 and 𝑆𝑥 ′ ,𝑦′ with thetime shift Δ𝑡 is defined as
(𝑘)

(𝑙)

𝐶𝐶�𝑆𝑥,𝑦 , 𝑆𝑥′,𝑦′ ; Δ𝑡� = #({𝑡𝑥,𝑦 ∈ 𝑆𝑥,𝑦 |∃𝑡𝑥′,𝑦′ ∈ 𝑆𝑥′ ,𝑦′ ,(𝑘)(𝑙)|𝑡𝑥,𝑦 − 𝑡𝑥′,𝑦′ − Δ𝑡| ≤ 𝛥𝛥}) / #(𝑆𝑥,𝑦 )(9)where #(𝑋) denotes the number of elements of 𝑋 and𝛥𝛥 is the time resolution of coincident firing. In thisresearch, 𝛥𝛥 was set to 0.5 ms. The auto-correlation wasdefined as 𝐴𝐴�𝑆𝑥,𝑦 ; 𝛥𝛥� = 𝐶𝐶�𝑆𝑥,𝑦 , 𝑆𝑥,𝑦 ; 𝛥𝛥� . In thecase of 𝛥𝛥 = 0, the cross-correlation 𝐶𝐶�𝑆𝑥,𝑦 , 𝑆𝑥′,𝑦′ ; 0�
corresponds to the ratio of synchronized spikes.
We simulated these two coupled neurons and calculated

the cross-correlation 𝐶𝐶�𝑆0,0 , 𝑆1,0 ; 0� for the generated
spike sequences. Results are shown in Figs. 1 and 2. Fig.
1 shows the dependency of cross-correlation on thecoupling weight 𝜉. In the case of equivalent phase shifts
for the two neurons (𝜙0,0 = 𝜙1,0 ) , the complete
synchronization was such that 𝐶𝐶�𝑆0,0 , 𝑆1,0 ; 0� ≃ 1
was achieved by increasing the coupling weight 𝜉. Fig.
2 shows the dependency of cross-correlation on thephase shift difference (𝜙1,0 − 𝜙0,0 ) . The crosscorrelation exhibits a selective preference for the casewhere the difference of phase shifts is 0. In this research,
we applied this selective preference of the coupledsystem to feature linking.
4. Numerical Experiment4.1. Experimental Model
In order to examine the feature-linking property usingsynchronized responses, we constructed a simulation
model of a 40 × 40 Chaotic-CNN and the motion
picture of moving two bars was inputted as a visualstimulus as in physiological experiments investigating acat’s visual cortex1. In this research, we also used twomotion patterns. One consisted of two bars moving inthe same direction (Fig. 3(a)), and the other was twobars moving in opposite directions (Fig. 3(b)).
The image of a bar was represented by a rectangleof 5 × 12 pixels. The velocity of each bar was0.07 pixels/ms. The external input 𝛽𝑥,𝑦 was 52.5 for theneurons inputting the image of the rectangle and 0 forothers. The connection parameter 𝑟 was 2. The gradient
field of the phase shift was set as 𝐷 = 0.353 and 𝛼 = 0.
4.2. Cross-Correlation Analysis

Fig. 1. Cross-correlation vs. coupling weight 𝜉 for two
coupled neurons.

Fig. 2. Cross-correlation vs. phase shift difference (𝜙1,0 −𝜙0,0 ) for two coupled neurons.

Numerical simulations were performed for spike
responses over 1,000 ms. The cross-correlation was thencalculated for the neuron 𝑛20,10 and others. The neuron
𝑛20,10 was located at the center of the path of bar 1 inFig. 3.
The cross-correlation for the two bars moving in thesame direction (Fig. 3(a)) is shown in Fig. 4(a), whichdepicts the formation of neural assemblies in which
neurons are synchronized. Furthermore, the neuralassembly corresponding to bar 1 had high crosscorrelation with the assembly for bar 2. On the other
hand, the cross-correlation for the two bars moving in
opposite directions (Fig. 3(b)) is shown in Fig. 4(b).
Although Fig. 4(b) also shows the formation of a neuralassembly, the cross-correlation between the assemblyfor bar 1 and that for bar 2 was low.

Published by Atlantis Press
Copyright: the authors
232

Akihiro Yamaguchi, Satoshi Arakane, Masao Kubo

5. Discussion
For the visual stimulus of moving objects, the formationof neural assemblies was observed in the Chaotic-CNN.
By cross-correlation analysis, the invoked neuralassemblies were found to have high correlation for
objects moving in the same direction and low
correlation for objects moving in opposite directions.

These results indicate the possibility of feature linking
using synchronized spike responses in the Chaotic-CNN
in terms of cross-correlation. In our model, this linking
property might be achieved by selective preference ofthe phase shift for the coupled system, as mentioned inSection 3. Analysis of the dynamics of invoked spikesequences along a moving assembly and theirapplication to real images are topics for our future work.
Acknowledgements
This work was supported by JSPS KAKENHI Grant
Number 24650120.
References(a) Same direction(b) Opposite direction

Fig. 3. Visual stimuli for the numerical experiment. The
motion of two bars moving in the same direction (a) and in
opposite directions (b).

(a) Same direction(b) Opposite direction

1. 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, pp. 334–337, 1989.
2. Eckhorn, R., Reitboeck, H.J., Arndt, M., Dicke, P.,
Feature linking via stimulus-evoked oscillations:experimental results from cat visual cortex and functional
implications from a network model, International Joint
Conference on Neural Networks, pp. 723–730, 1989.
3. Yamaguchi, A., Okumura, N., Chaki, H., and Wada, M.,
Chaotic synchronized cluster in the network of spikeresponse neurons, IEICE Tech. Rep., Vol. 99, No. 685,
NC99-119, pp. 15–20, Mar. 2000 (in Japanese).
4. Yamaguti, Y., Ishimura, K., and Wada M., Chaotic
synchronized assembly in Pulse Coupled NeuralNetworks, IEICE Tech. Rep., Vol. 101, No. 615,
NC2001-98, pp.127–134, Jan. 2002 (in Japanese).
5. Yamaguchi, A., On a chaotic synchronization of one-way
coupled two spike response neurons, Fukuoka Institute ofTechnology Reports of the Computer Science
Laboratory, Vol. 24, pp. 1–6, 2013 (in Japanese).
6. Yamaguchi, A., On an information coding using
localized synchronization in the two dimensional coupledsystem of chaotic spike response neurons, Fukuoka
Institute of Technology Reports of the Computer ScienceLaboratory, Vol. 25, pp. 1–6, 2014 (in Japanese).
7. Fujiwara, M., Yamaguchi, A., and Kubo M.,
Synchronized Response to Grayscale Image Inputs inChaotic Cellular Neural Network, Journal of Robotics,
Networking and Artificial Life, Vol. 2, No. 1, pp. 26–29,
2016.
8. Gerstner, W., and Kistler, W., Spiking Neuron Models:Single Neurons Populations Plasticity, Cambridge
University Press, 2002.
9. Lee G., and Farhat, N.H., The Bifurcating Neuron
Network 1, Neural Networks, Vol. 14, pp. 115–131,
2001.

Fig. 4. Cross-correlation between neuron 𝑛20,10 and others
when the visual stimuli shown in Fig. 3 were inputted.

Published by Atlantis Press
Copyright: the authors
233