2024 IEEE International Symposium on Circuits and Systems (ISCAS) | 979-8-3503-3099-1/24/$31.00 ©2024 IEEE | DOI: 10.1109/ISCAS58744.2024.10557836

Feature Extraction of Neuronal Activity
by Attractor Reconstruction in Neural Networks
with Delayed Couplings
Yoko Uwate† , Marie Engelene J. Obien‡ , Urs Frey‡ and Yoshifumi Nishio††

Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp‡
MaxWell Biosystems
Albisriederstrasse 253, 8047 Zurich, Switzerland
Email: {marie.obien, urs.frey}@mxwbio.com

Abstract—We investigate neural activity of the neural network
including delayed couplings by using attractor reconstruction
images. The paper presents the effects on neural activity whenthe value and distribution of delayed coupling is changed. In
particular, we focus on the complexity of attractor images
and evaluate the relationship between delayed coupling andcomplexity.

I. I NTRODUCTION
In the field of neuroscience, several types of neuronal activity measurement devices equipped with high-density electrodearray have been developed [1]- [3]. With the developmentof these technologies, it has become possible to obtain moreaccurate data on neural activity in brains of Wistar rats andother animals [4]. Typically, neural activity data from the brain
is obtained in the form of a raster plot, a graph that represents
the activity patterns of neurons. Standard analysis methods
include focusing on areas where many neurons are spiking
simultaneously, calculating the synchronization rate of spikes,
and investigating the distribution between spikes [5]. These
analytical methods are important for evaluating the state ofneural activity in the brain.
There have been many investigations of neural activity in
networks using mathematical models as a modeling of brain
network function. In Ref. [6], the authors have reported thatthe interaction of delay and STDP causes spiking neurons tospontaneously self-organize into groups, generating a patternof canonical polysynchronous activity. Another study investigated the relationship between neuronal synchronization andexcitatory and inhibitory conduction in a neuronal network
consisting of adaptive integrating firing neurons [7].
As a feature extraction and visualization of neuronal activity, a method that converts raster plots obtained from Wistar rat
brain to time series data and using attractor reconstraction has
been proposed [8]. In addition, it has been shown that attractor
reconstruction can also be used for neuronal activity obtained
from the coupled system of the Izikevich neuronal model tocharacterize the network [9], [10]. The analysis methods forneural activity patterns so far have mostly been linear, however

the proposed approach here is a new attempt using nonlinear
methods. Since neural activity patterns exhibit highly complex
features, it is believed that non-linear analysis is effective.
In this study, we investigate how attractor reconstruction
is affected when delayed coupling is added to the Izikevich
neuron coupling system. By using the computer simulations,
we found that the pattern of neuronal activity changes with thesize and distribution of the delayed couplings. In particular, thevariety of delayed couplings is important for the complexityof neuronal activity.
II. N EURON AND N ETWORK M ODELSThe Izikevich neuron model which is described by a secondorder differential equation Eq. (1) is used.
(v̇ = 0.04v 2 + 5v + 140 − u + Iexu̇ = a(bv − u)(1)

if v > 30 mV, then v ← c and u ← u + d.
Where v corresponds the membrane potential of the neuron,
u corresponds a slow membrane recovery variable, accounting
for the activation of K + ion currents and inactivation of N a+ion currents. Iex denotes the excitatory input current. The
parameters of the excitatory neurons (Regular Spike type) are
fixed with a=0.02, b=0.2, c=-55 and d=8. The parameters of
the inhibitory neurons (Frast Spike type) are fixed with a=0.1,
b=0.2, c=-55 and d=2. In this case, 1,000 neurons are coupled
randomly with coupling probability p = 0.1. The ratio of
excitability and inhibitory neuron is 0.8 and 0.2, respectively.
III. ATTRACTOR R ECONSTRUCTION
The attractor of dynamical systems can be reconstructed
topologically in the embedding space from Takens’ theorem [11]. The state vectors in the reconstructed m-dimensional
embedding space are defined byy(t) = x(t), x(t + τ ), ..., x(t + (m − 1)τ )(2)

where x(t) means a scalar time series and τ is the delay time.

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IV. S IMULATION R ESULTS
A. Without Delayed Couplings
First, we investigate neuronal activity in the neural network
without delayed couplings. Figure 1 shows the simulation
results of the raster plot and spike rate during 1,000 [ms].
It was observed that the pattern of many neurons firing
simultaneously at intervals of 100 to 200 [ms]. The time seriesof spike rates shows that the peaks appear irregularly and theirpeak values are not constant.

three neurons are nearly synchronized (Fig. 5(a)). While, thecase of delay is 5 [ms], Fig. 4(b) shows that the percentageof neurons that fire simultaneously is small. Three neurons
are not synchronized as shown in Fig. 5(b). As the delay
time becomes even longer, the spike timing becomes more
periodic (Fig. 4(c) and (d)). Neuron activities are also nearly
synchronous as shown in Figs. 5 (c) and (d).
Figure 6 shows the reconstructed attractors depending on
the delay time. When the delay is 1 [ms], the shape of theattractor is similar to that without delay. The size of theattractor increases, and the complexity seems to be slightly
weaker. When the delay is 5 ms, the attractor size is smaller
and the complexity is stronger. If the delay is 10 or 12 [ms],the attractor orbit becomes nearly periodic. The size of theattractor is larger with a delay of 12 [ms]. In other words,
many neurons fire at the same time.

Fig. 1. Raster plot and spike rate without delayed couplings.

Action potentials of three randomly selected neurons areshown in Fig. 2. Two neurons behave in regular spikes,
however one neuron produces bursting at some intervals.
The attractor obtained by applying the attractor reconstruction to the spike rate time series is shown in Fig. 3.
The attractor trajectories are not random, but have a certain
structure and complexity for small values.

(a) Delay time: 1 [ms].

(b) Delay time: 5 [ms].
Fig. 2. Examples of three neuronal activities without delayed couplings.

"re_att_d=0ms.dat"

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Fig. 3. Reconstracted attractor without delayed couplings (τ =10).
(d) Delay time: 12 [ms].

B. Constant Delayed Couplings
Simulation results for the case all excitatory coupling delays
were set to the same value are shown in Figs. 4 and 5. In thiscase, all delays of inhibitory coupling were fixed at 3 [ms].
When the delay is 1 [ms], Fig. 4(a) shows that the percentage of neurons that fire simultaneously is large. And theinterval of their firing timing is longer than when there isno delay. The action potentials of the neurons show that theFig. 4. Raster plot and spike rate with constant delayed couplings.

Figure 7 shows the average spike rates when the delayed
coupling time is changed from 0 to 13 [ms]. This result isthe average of 10 different networks with different coupling
patterns. The average spike rate is larger once when there is
a 1 [ms] delay than without delay, however the average spikerate becomes smaller when the delay is longer than 2 [ms].

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C. Distributed Delayed Couplings(a) Delay time: 1 [ms].

(b) Delay time: 5 [ms].

(c) Delay time: 10 [ms].

(d) Delay time: 12 [ms].

Fig. 5. Examples of three neuronal activities with constant delayed couplings.

25000

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Number of couplings"re_att_d=5ms.dat"

Number of couplings"re_att_d=1ms.dat"

Next, we investigate the case of four distributions of delyed
couplings. Four distributions is shown in Fig. 8. The horizontal
axis represents the time delay and the vertical axis represents
the number of couplings. Distribution A is a uniform distribution. Distribution B is a normal distribution. Distribution
C is a distribution with many couplings of short and long
delays. Distribution D has the highest number of couplings
for short delays, and the number of couplings decreases forlonger delays.

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Ave. firing spike rate [Hz]Ave. firing spike rate
Min value of firing spike rate
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Delay time [ms]Fig. 7. Average of spike rate without delay couplings.

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Delay time [ms]1011(c) Distribution C.

(d) Delay time: 12 [ms].

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Then, when the delay is more than 9 [ms], the average spikerate is again larger than without delay. This may be due toan increase in the proportion of synchronous neurons as thedelay time becomes longer.

0

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Fig. 6. Reconstracted attractors with constant delay couplings.

06(a) Distribution A.

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Fig. 8. Four patterns of distributed delayed couplings.

The simulation results of raster plot and spike rate depending on the coupling distributions are shown in Figs. 9. In
the case of Distribution A, the spike firing peaks are observed
at intervals of about 200 [ms]. In the case of Distribution B,
similar to A, spike firing peaks are observed at about 200[ms] intervals, and occasionally the spike rate peaks are high.
In cases C and D, the peak period of spike rates is broken
compared to the case without delay.
Figure 10 shows the neuronal activities of three neurons
selected randamly. In all cases, we see that there are regularand burst spiking neurons.
Figure 11 shows the reconstructed attractors depending on
the delay distributions. It can be seen that case of Distribution
B has the largest attractor size. In all other cases, the attractor
size is about the same. With regarding complexity, the complexity is stronger in cases Distributions A, C, and D comparedto without delay, and it is weaker in case of Distribution B.
The average spike rate with four different distributed delayed couplings are summarized in Table I. This result isthe average of 10 different networks with different coupling
patterns. It can be seen that the delayed coupling reduces themean spike rates for all distributions compared to without
delay case.

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"re_att_d=A.dat""re_att_d=B.dat"

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Fig. 11. Reconstracted attractors with distributed delay couplings.

V. C ONCLUSION(c) Distribution C.

(d) Distribution D.
Fig. 9. Raster plot and spike rate with distributed delayed couplings.

(a) Distribution A.

(b) Distribution B.

(c) Distribution C.

(d) Distribution D.

Fig. 10. Examples of three neuronal activities with distributed delayed
couplings.

TABLE IAVERAGE SPIKE RATE WITH DISTRIBUTED DELAY COUPLINGS
Distribution typeA
B
C
D
Without Delay

Ave.
35.80
37.83
36.14
38.55
46.97

Min
30.77
30.12
31.21
31.90
35.05

Max
40.77
46.90
40.15
44.05
60.41

We investigated how delayed coupling affects the activity
patterns of neurons using images of reconstructed attractors.
We showed the simulation results of raster plot, spike rate
and neuronal activities. Then we apply attractor reconstruction
method to spike rate time series. In particular, the evaluation
focused on attractor size and complexity compared to without
delay. First, in the case of that delay is constant, the attractor
size becomes larger when delay time is 1 [ms]. The complexityof attractor becomes stronger when delay time is 5 [ms]. The
complexity becomes weak when the delay time is longer than
10 [ms]. Next, in the case of the four distributed delayed coupling, attractor size became smaller in all cases. In complexity,
we found that complexity holds in all three cases except for
the normal distribution. These findings indicate that delayed
coupling has a significant effect on neuronal activity. We also
confirmed that the diversity of delayed couplings is importantfor maintaining complexity.
As the part of future work, complexity must first be quantitatively evaluated. One possible way is that we use Poincare
cross sections. Also, since we did not focus on synchronizationof neurons in this study, the effect of delayed coupling on
synchronization should be investigated. Furthermore, while
this study applied nonlinear analysis to the neural activity
patterns of mathematical models, in the future, we aim to extract new features from actual neuron signals from a nonlinear
perspective.
R EFERENCES[1] U. Frey, J. Sedivy, F. Heer, R. Pedron, M. Ballini, J. Mueller, A.
Hierlemann, “Switch-Matrix-Based High- Density Microelectrode Arrayin CMOS Technology,” IEEE Journal of Solid-State Circuits, 45(2),467482. 2010.
[2] J. Muller, M. Ballini, P. Livi, Y. Chen, M. Radivojevic, A. Shadmani,
V. Viswam, I. L. Jones, M. Fiscella, R. Diggelmann, Alexander Stettler,
U. Frey, D. J. Bakkum and A. Hierlemanna, “High-resolution CMOS

Authorized licensed use limited to: Yoshifumi Nishio. Downloaded on August 08,2024 at 08:35:16 UTC from IEEE Xplore. Restrictions apply.

MEA Platform to Study Neurons at Subcellular, Cellular, and Network
Levels,” Lab Chip 15, 2767-2780, https://doi.org/10.1039/C5LC00133A,
2015.
[3] H. Shin, S. Jeong, J. H. Lee, W. Sun, N. Choi and I. J. Cho, “3DHigh-Density Microelectrode Array with Optical Stimulation and Drugdelivery for Investigating Neural Circuit Dynamics,” Nature Communications, vol. 12, Article number: 492, 2021.
[4] D. J. Bakkum U. Frey, M. Radivojevic, T. L. Russell, J. Mller, M.
Fiscella, A. Hierlemann, “Tracking Axonal Action Potential Propagation
on a High-Density Microelectrode Array Across Hun- dreds of Sites,”Nature Communications, 4, 2181. http://doi.org/10.1038/ncomms3181,
2013.
[5] D. J. Bakkum, M. Radivojevic, U. Frey, F. Franke, A. Hierlemann and H.
Takahashi, “Parameters for burst detection,” iFront. Comput. Neurosci.,
https://doi.org/10.3389/fncom.2013.00193, 13 January 2014.
[6] E. M. Izhikevich , “Polychronization: computation with spikes,” NeuralComput. 18(2):245-82 Feb. 2006.
[7] P. R. Protachevicz, F. S. Borges, K. C. Iarosz, M. S. Baptista, E. L.
Lameu, M. Hansen, I, L. Caldas, J. D. Szezech Jr., A. M. Batista, J.
Kurths, “Influence of Delayed Conductance on Neuronal Synchronization,” Front. Physiol., 3, https://doi.org/10.3389/fphys.2020.01053, Sep.
2020.
[8] Y. Uwate, M. E. J. Obien, U. Frey and Y. Nishio, “Time Series Analysis
of Neurons and Visualization of Network Characteristics,” Proceedingsof RISP International Workshop on Nonlinear Circuits, Communications
and Signal Processing (NCSP’19), pp. 450-453, Mar. 2019.
[9] Y. Uwate, Y. Nishio, M. E. J. Obien and U. Frey, “Visualization of
Neuron Data Using Nonlinear Technic,” Proceedings of International
SoC Design Conference (ISOCC’19), pp. 142–143, Oct. 2019.
[10] Y. Uwate, Y. Nishio, M. E. J. Obien and U. Frey, “Nonlinear Time Series
Analysis of Spike Data of Izhikevich Neuron Model,” Proceedings ofInternational SoC Design Conference (ISOCC’20), pp. 203–204, Oct.
2020.
[11] F. Takens, “Detecting Strange Attractors in Turbulence,” In D.A. Rand
and B.-S. Young, editors, Dynamical Systems and Turbulence, vol. 898
of Lecture Notes in Mathematics, pp. 366–381, Warwick 1980, 1981.

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