Nonlinear Time Series Analysis of Coupled Bursting
Neuron Model Depending on Coupling Strength
Yoko Uwate and Yoshifumi Nishio
Dept. of E. E. Eng., Tokushima University
2-1 Minami-Josanjima Tokushima 770-8506 Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp

simulation results, we confirm that the attractor has different
shape depending on the coupling strength.

Abstract— It is well known that burst patterns within neurons
may have some important role for operating information
processing in a brain. However, the analysis of judgement ofsynchronization and correlation between burst patterns is not soeasy. Therefore, we have proposed a visualization method of
bursting patterns of whole neural network using nonlinear timeseries analysis. Additionally, we consider that it is required toconstruct a model using mathematical neuron model producing
burst patterns, because it is difficult to obtain real biologicalneuron data. In this study, we propose a visualization method of
network characteristics of FitzHu-Nagumo (FHN) neuron modelusing nonlinear time-series analysis.
Keywords;
nonlineartime-seris
analysis,reconstruction, coupled bursint neuron model

I.

II.

The model of gap junction coupled FHN neurons [4], [5] is
described by the following equation.

(1)

attractor

where x, y are sate variables, provided that orbit is closeenough to the basin of attraction, g denotes the couplingstrength of gap junction, and Io is the external stimulation
expressed by cosine function. Sn is set of nodes which areconnected to i-th neurons. In the coupled neuron system, small
frequency error is added to every neuron.

INTRODUCTION

Burst patterns within neurons may have some importantrole for operating information processing in a brain. Therefore,
to develop detecting and analyze burst pattern are investigatedin various fields [1]. Although it is important to study burst
pattern in order to make clear correlation and communication
process between neurons, unveiling a structure of the whole
neuronal network is also required. While, nonlinear time-series
analysis is a useful tool for characterizing about the dynamics
behind observed time-series data [2]. The neuronal dataobtained from living neurons should be high-dimensional anddynamic data. Then nonlinear time-series analysis matches
well to characterize the neuronal data. Therefore, we haveproposed a visualization method of bursting patterns of whole
neural network using nonlinear time series analysis [3].

Figure 1 shows the bursting patterns of all neurons when
the number of neurons is set to N=10. The coupling strength isfixed with g=0.1.

However, it is difficult obtain amount of real biologicalneuron data. We consider that it is required to construct amodel using mathematical neuron model producing burst
patterns. In this study, we propose a visualization method ofnetwork characteristics of burst patterns obtained from FitzHuNagumo (FHN) neuron model using nonlinear time series
analysis. As a first step, we build up the system model which isthat ten neurons are coupled randomly. In order to mimic a realsystem, the small mismatch of oscillation frequency is added to
all neurons. We calculate the spike timing of the bursting
patterns and obtain the raster plot using the threshold. Next, wecalculate the spike rate at a certain range, then time-series data
is obtained. The attractor reconstruction of nonlinear timeseries is applied to sum of time-series data. From the978-1-7281-2478-0/19/$31.00 ©2019 IEEE

COUPLED BURSING NEURON MODEL

Fig. 1. Bursting patterns of all neurons (N=10).

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III.

At my presentation, the other attractor reconstruction
results are shown, when the coupling strength in the network
are changed.

SIMULATION RESULTS

A. Raster PlotFigure 2 shows the raster plot obtained from all neurons, bychecking the peak points of the bursting spike data. However, it
is difficult to understand the network characteristics from thisfigure. Next, we calculate the spike rate at a certain range, thentime-series data is obtained as shown in Fig. 3. From thisfigure, we can see that the time series has periodicity.

Fig. 4. Simulation result of attractor reconstruction.

IV.

Fig. 2. Raster plot of coupled 10 neurons.

CONCLUSIONSIn this study, we have proposed a visualization method ofnetwork characteristics of neuron data using mathematicalmodel. By calculating the computer simulations, we obtained
the attractors in 3-dimensioanl space. From observed results,
we can see that the proposed system has possibility to show thenetwork characteristics of biological neuron data.
In the future works, we would like to use larger networks,
and different types of network structures. Applying theproposed system to real biological neuron data is also ourfuture work.

REFERENCES
Fig. 3. Time-series data.

[1]

B. Attractor Reconstruction
The attractor of dynamical systems can be reconstructed
topologically in the embedding space from Takens' theorem[6]. The state vectors in the reconstructed m-dimensional
embedding space are defined by[2]where x(t) means a scalar time series and tau is the delay time.

[5]

[3]

[4]

Figure 4 shows the simulation results when neuronal timeseries data (Fig. 3) is embedded in 3-dimensional space bysetting the delay time tau=30. From the figure, we confirm thatthe neuron has some sort of structure because the orbit draws incertain range, does not move about randomly.

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[6]

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