Nonlinear Time Series Analysis of Spike Data ofIzhikevich Neuron Model
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 well known that burst patterns of neuronal
networks may play an important role in information processingin the brain. We consider that it is advantageous to construct amodel using mathematical neuronal models producing burst
patterns, because it is such models are easier to study and moreaccessible as compared to real biological neuronal data. In thisstudy, we use the Izhikevich neuron model to produce burst
patterns and apply a recurrence plot density entropy to theIzhikevich neuron data.

In this study, we apply recurrence plot density entropy tospike burst data generated from Izikevich neuron model andinvestigate the characteristics of spike burst data.
II. IZHIKEVICH NEURON MODEL
Izhikevich neuron model efficiently produces a wide
variety of neuron spiking and bursting dynamics. TheIzhikevich neuron model is described by the followingequations.

Keywords; neuron data, nonlinear time-series analysis

I.
INTRODUCTION
Burst patterns within neurons may play an important role in
information processing in the brain. Therefore, burst detection
and burst pattern analyzes methods are developed, which areused in various fields [1]. Although it is important to study
burst patterns, as these can relate to network syn- chronicity
and synaptic connectivity, unveiling the structure of the wholeneuronal network is also required. Nonlinear time-series
analysis is a useful tool for characterizing the dynamics behindobserved time-series data [2]. Typically, neuronal data obtained
from living neurons is high-dimensional and dynamic data,
therefore nonlinear time-series analysis is suitable tocharacterize neuronal data. Previously, we pro- posed avisualization method for bursting patterns of whole neuralnetworks using nonlinear time series analysis [3]. We applied
nonlinear time-series analysis to three sets of time- series datafrom a neuronal culture measured at days in vitro (DIV) 15, 20
and 30, respectively. By using computer simulations, weconfirmed that the characteristics of these neuronal networks
change across different DIV and that the proposed visualization
methods are suitable to capture the essence of these changes.

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 thecomputer simulations, 1,000 neurons were coupled randomly.
The ratio of excitability and inhibitory neuron is 0.8 and 0.2,respectively. The parameters of the Izhikevich neuron model
are fixed with a=0.02, b=0.2, c=-50 and d=8. By using theseparameters, the Izhikevich neuron model produces tonic
spiking as shown in Fig. 1.

However, it is difficult to extract the ground-truth of theunderlying network morphology for data obtained from real
biological neuronal networks. To relate these time-series
analysis results to the underlying network morphology, it is required to construct a model using mathematical neuron modelsproducing such burst patterns. In our previous study, wepropose a visualization method of network characteristics ofburst patterns obtained from Izhikevich neuron model using
nonlinear time-series analysis. From the simulation results, we
confirm that the Izhikevich neuron model shows similar
development stages as biological neurons, by controlling theparameters of the coupling probability and strength [4].

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Fig. 1. Tonic spiking of the neuron network model.
The coupling probability (p) of the network and thecoupling strength (g) of excitability neurons are importantparameters for modeling the development of neurons. Figure 2
shows the network activity of 1,000 neurons, the firing rateduring a fixed bin, for different values of the coupling
probability and strength.

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B. Simulation Results
Figure 4 shows the simulation results of the RPDE of 10
average. The spike rate of Fig. 3 (b) has highest RPDE. From
this result, it was found that the characteristics of the neuron
spike influence the coupling strength and coupling probability
connection rate of the network. When the coupling strength orthe coupling probability are small, the periodicity of the neuron
activities becomes high (Fig. 4 (a) and (c)).

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time (ms)(b) p=0.08, g=2.2 [mV]
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Fig. 4 Results of RPDE (Hnorm).

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IV. CONCLUSIONSIn this study, we investigated the characteristics of neuronspike data of Izhikevich neuron model using recurrence plot
density entropy. By using the computer simulations, weconfirmed that the spike data have more periodicity when thenetwork parameters are set with strong coupling probability
and weak coupling strength or weak coupling probability andstrong coupling strength.

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(c) p=0.03, g=2.5[mV]Fig. 3 Time series data of the simulated neuronal network.
III.

NONLINEAR TIME SERIES ANALYSISREFERENCES

A. Recurrence Plot Density Entropy (RPDE)
Recurrence time is calculated as the distance between thepoints on the vertical line of the recurrence plot (RP) [5], [6].
Little et al. have proposed the Recurrence Period Density
Entropy method (RPDE). In this method, the iterations of eachpoint to the neighborhood ε are tracked, and the resulting timeintervals are used to build a histogram of iteration times. Thishistogram is used to calculate the recurrence period density
function. The normalized entropy of this density is defined asfollowing equation [7].

[1]

[2]
[3]

[4]

[5]

[6]

The value of Hnorm changes in the range from zero to one. For
the periodic signals, Hnorm=0, while the uniform white noise,
Hnorm=1.

[7]

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