Local and Global Activities of Izhikevich Neuron
Model in Networks
Yoko Uwate and Yoshifumi Nishio

Marie Engelene J. Obien and Urs Frey

Dept. of E. E. Eng., Tokushima University
2-1 Minami-Josanjima Tokushima
770-8506 Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp

MaxWell Biosystems
Albisriederstrasse 253, Zurich,
8047 Switzerland
Email: {marie.obien, urs.frey}@mxwbio.com

Abstract— It is important to investigate the phenomena innetworks composed of neuron models. In our previous work, wehave proposed a method to explain the phenomena in the wholenetwork by nonlinear time-series analysis. In this study, we will
clarify the relationship between the phenomena at the constituent
nodes of the network and the overall phenomena.

II.

IZHIKEVICH NEURON MODEL

Izhikevich neuron model efficiently produces a wide varietyof neuron spiking and bursting dynamics. The Izhikevich neuronmodel is described by the following equations.

Keywords; neuron data, nonlinear time-series analysis

I.

INTRODUCTION

Uncovering the evolutionary process of the brain is one of
the most important steps in understanding the brain systems that
perform higher-order information processing. In order to makeclear the mechanism of the brain, burst detection and burst
pattern analyzes methods are developed, which are used invarious fields [1]. Nonlinear time series analysis is a different
approach from linear time series analysis because it can capture
the characteristics of a time series in a nonlinear manner [2]. We
consider that this nonlinear time series analysis is effective foranalyzing the activity patterns of the brain.

Where v represents the membrane potential of the neuron, urepresents a slow membrane recovery variable, accounting forthe activation of K+ ion currents and inactivation of Na+ ion
currents. Iex denotes the excitatory input current. In the computer
simulations, 20 neurons were coupled randomly with thecoupling probability p. The ratio of excitability and inhibitory
neuron is 0.8 and 0.2, respectively.
Figure 1 shows the two types of network models for thisstudy. It is confirmed that the Network B has more connections
than the Network A.

Our research group has proposed a method to extract featuresother than burst information from brain activity patterns byvisualizing features of the entire network using attractor
embedding . In this study, spike firing rates were calculated fromraster plots obtained from 1024 electrodes, and attractor
embedding, a nonlinear analysis method, was applied to the timeseries. We applied nonlinear time-series analysis to three sets oftime-series data from a neuronal culture measured at days invitro (DIV) 15, 20 and 30, respectively [3]. By using computer
simulations, we confirmed that the characteristics of theseneuronal networks change across different DIV and that theproposed visualization methods are suitable to capture theessence of these changes [4].

RS
NL

J

JM
SBI

BKC

M

DCNPDIHOGAL

GQ

EO

K

However, the relationship between the results obtained fromthe spike firing rate of an entire neuron region and what it
represents for the neuron information here is unclear.
Additionally, we consider that it is required to construct a modelusing mathematical neuron model producing burst patterns,
because it is difficult to obtain real biological neuron data.

FPQF
A

H

(a) Network A (p=0.1).

(b) Network B (p=0.3).

Fig. 1 Network models (N=20) depending on the coupling
probability p.

In this study, we consider the relationship between local andglobal activities of the networks which is composed ofIzhikevich neuron model. By using attractor reconstruction, we
confirmed that differences in local and global activity status.

978-1-6654-0174-6/21/$31.00 ©2021 IEEERTTE219

ISOCC 2021

III.

SIMULATION RESULTS

delay τ = 10. From this figure, we confirm that two networks
exhibit a clear structure, because the orbit draws in certain
range and does not move about randomly. Furthermore, in thecase of the Network B, the size of the attractor is larger andthe behavior of the orbit is periodicity.

A. Local Activity in the Networks
At the first step, we investigate local neuron activity in thenetwork. Figures 2 and 3 show the simulation results ofattractor reconstruction of the spiking activity of a typicalnode in Network A and B.
In the case of Fig. 2, nodes Q and S behave active and theother nodes show periodicity. While in the case of Fig. 3, allnodes behave very active.

(a) Network A.
40Fig. 4 Raster plot.

40

20200-200-20-40-40-60-80-60-80-100-100
40

40

20

200-100(b) Network B.

-80

0-20-60-100-40-40-20-80-20-60-60
020-40-40-80
40 -100(a) node Q.

-20

140
120
100
80
60
40
200-60020-8040 -100(b) node R.

140
120

200
180
160
140
120
100
80
60
40
200

100
040

40

20

200-200-20-40-40-60-80-60-80-100-80

40
20-100-40-40-20-80

20-40-40-80
40 -100-20

40

020-8040 -100

200-20-40-40-60-80-10040

40202000-20-100-40-40-20-80-20-60-60020-40-40-80
40 -100(a) node Q.

-20-60020-8040 -100(b) node R.

40140

204006080 100
120 140160 180

40200 0608020(b) Network B.

200-20-40-40-60-80-60-80-100

CONCLUSIONSFor the future works, we would like to perform a quantitativeevaluation of the obtained attractor reconstruction. In addition,
since we have investigated one specific network model, morestatistical analysis is needed.

40200-20-100
404020200-100020In this study, we investigated the local and global activities
of Izhikevich neuron model using attractor reconstruction. In anetwork with few connections, there are active and inactivenodes, whereas a network with many connections is composedof active nodes overall. The investigation of overall neuronal
activity shows more complex behavior in networks with fewer
connections, and more periodicity in networks with moreconnections.

-60-80-100-60

120IV.

40200-20-80

10040Fig. 5 Attractor reconstruction of whole network.

-60(c) node S.
(d) node T.
Fig. 2 Attractor reconstruction of Network A.

-100

6080-20-60-600600-20-60200
1801601401201008040(a) Network A.

-100
40
200-100

20-80-20-60-40-40-20-60020-80
40 -100

REFERENCES

0-100-80-20-60-40-40-20-60
020-8040 -100[1](c) node S.
(d) node T.
Fig. 3 Attractor reconstruction of Network B.

[2][3]B. Global Activity in the Network
Next, global activity in the network is investigated. Figure4 shows the raster plot obtained from the Networks A and B.
After that, we calculate the spike rate at time bins, then timeseries data of the whole network is obtained.
Figure 5 shows the simulation results when neuronaltimeseries data is embedded in 3-dimensional space with time978-1-6654-0174-6/21/$31.00 ©2021 IEEE

[4]

220

D. J. Bakkum, M. Radivojevic, U. Frey, F. Franke, A. Hierlemann and H.
Takahashi, “Parameters for Burst Detection,” Frontiers in Computational
Neuroscience, doi: 10.3389/fncom.2013.00193, Jan. 2014.
E. Bradley and H. Kantz, “Nonlinear Time-Series Analysis Revisited,”
Chaos, vol. 25, doi: 10.1063/1.4917289, 2015.
M. E. J. Obien, K. Deligkaris, T. Bullmann, D. J. Bakkum, and U. Frey,“Revealing Neuronal Function through Microelectrode Array
Recordings,”
Frontiers
in
Neuroscience,
8(423).
http://doi.org/10.3389/fnins.2014.00423, 2015.
Y. Uwate, M. E. J. Obien, U. Frey and Y. Nishio, “Modeling of Bursting
Neurons and Its Characteristic using Nonlinear Time Series Analysis”,
Proc. of NCSP’20, March, 2020.

ISOCC 2021