Wavelet Correlation of Neural Activity Bursts Generating Spikes

S. V. Bozhokin, I. B. Suslova and D. A. Tarakanov

Peter the Great Saint-Petersburg Polytechnic University, Polytechnicheskaya 29, Saint-Petersburg, Russia

Keywords: Continuous Wavelet Transform, Wavelet Correlation Function, Synchronization of Neurons.

Abstract: We study neural activity synchronization on the basis of instantaneous wavelet correlation function and

simple mathematical model of brain bursts carrying several spikes. The approach allows us to obtain

analytical solution for two neurons generating a given number of spikes and estimate the coupled behavior

of neurons at different time moments. Neural activity is simulated as a superposition of elementary

nonstationary Gaussian signals with some given parameters. Time-frequency properties of neural signals are

studied by continuous wavelet transform with adaptive Morlet mother wavelet function.

1 INTRODUCTION

Synchronization of individual neurons and complex

neuron ensembles plays an important role in the

central nervous system functioning (Hramov et al.,

2015). A neuron as a pacemaker can generate

rhythms of various duration and frequencies, regular

or chaotic. The electric neural activity shows some

bursts consisting of short pulses (spikes) and low-

frequency oscillations associated with slow changes

in the membrane potential. In accordance with the

spike model (Izhikevich, 2006, Kislev, 2016,

Gerstner et al., 2002) the main carriers of

information are the number, duration and the

moments of occurrence of spikes. Synchronization

of neural activity as a way of information transfer

has been considered in numerous experimental and

theoretical works (Izhikevich, 2006, Kislev, 2016,

Gerstner et al, 2002, Xiaojuan et al., 2011).

A large number of mathematical models of

neurons, simulating the dynamics of their electrical

activity, can be divided into several classes. The first

class includes the models based on kinetic equations

simulating the excitation of nerve impulse. Such

models use a detailed description of ion channels

kinetics. The classical Hodgkin-Huxley model and

its various generalizations (Zhou and Kurths, 2003)

belong to this class of models. The requirements for

these models consist in accurate reproduction of the

electrical activity of the mathematical neuron, whose

electrical activity must correspond to a single pulse

or bursts of pulses of the real biological neuron.

The second class of models includes conceptual

models of neurons (Hramov et al., 2015, Izhikevich,

2003). These phenomenological models describe the

effect of ion currents with identical characteristic

scales by using a single variable. In this case, a few

ordinary differential equations are sufficient to

describe the electrical activity of a single neuron.

The third class includes threshold models of neurons

(Nekorkin, 2008, Tuchwell, 1988). In this case, the

system accumulates threshold signals. Their

combined action results in membrane potential

reaching the threshold value. This fixed value is

treated as a spike. Thereafter, the value of the

membrane potential returns to the initial state. The

process of synchronization is described by using

mathematical models of synaptic connections

(Hramov et al., 2015). It should be noted that both

the models of functioning of a neuron ensemble and

synapses connecting the neurons require large

computing powers.

In this paper, we use the simplest model

describing a separate spike in the form of the

Gaussian signal, which has a certain duration. In this

model, we assume that all spikes produced by

neurons have the same shape. The formation of

individual bursts both for a single neuron and for

two coupled neurons is given phenomenologically in

the form of a sequence of individual spikes created

by the neurons. Such a simple model has the

advantage of allowing us to solve analytically the

synchronization problem for two neurons.

It is assumed that the coding of information in

the brain is carried out through spike frequencies for

a single neuron and a group of several neurons

Bozhokin, S., Suslova, I. and Tarakanov, D.

Wavelet Correlation of Neural Activity Bursts Generating Spikes.

DOI: 10.5220/0006635102010207

In Proceedings of the 11th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2018) - Volume 4: BIOSIGNALS, pages 201-207

ISBN: 978-989-758-279-0

201

(Izhikevich and Gally, 2004). A certain stimulus of

electrical activity causes excitation of a specific

group of neurons. In this case, the relative times for

the production of spikes by different neurons in the

group are strictly fixed. The evidence is that

temporal coding serves to represent and process

information in the cerebral cortex (Lowet et al.,

2016). The same set of neurons can encode a large

number of different stimuli. Each stimulus is

characterized by a unique order of spikes emitted by

different neurons. The same is the case with the

spacing between the spikes. The activation of neural

groups, caused by the presence of any stimulus,

assumes the process of creating spikes with strictly

fixed delays between them. This procedure of

creating spikes is observed for all or almost all

participants of the neural groups.

Coherent analysis, instantaneous phase-locking,

entropy transfer, and nonlinear dynamics methods

have been applied to assess the degree of

synchronization of neurons (Lowet et al., 2016,

Mizuno-Matsumoto et al., 2005). Spectral coherence

method based on the Fourier transform (cross-

spectra) and assuming stationarity of signals has

long been considered the main method for

evaluating the interaction of signals related to

rhythmic brain activity. It should be noted that

instantaneous phase synchronization methods are

recognized as more informative.

The study (Quyen et al., 2001) on the

synchronization of neurons at different frequencies

compares the efficiency of Hilbert transform and

wavelet transform. At present, it is possible to

highlight two main directions in the research on the

synchronization of neurons and neural networks: the

construction of more complex and more realistic

models of neurons and their interactions, and the

development of methods that take into account the

nonstationary and nonlinear nature of neural

rhythmic activity.

The purpose of this work is to develop a model

of impulse activity of neurons generating spikes; to

derive analytical wavelet transform that determines

the time-frequency properties of spikes; to calculate

wavelet correlation function analytically for

comprehensive analysis of the synchronization

processes.

2 METHODS

2.1 Mathematical Model of Neural

Rhythmic Activity

In contrast to the complicated models (Hramov et

al., 2015, Izhikevich, 2003, Nekorkin, 2008, Zhou

and Kurths, 2003), which consider in detail the

operation of ion channels, we propose a simple

mathematical model of the coupling activity of two

neurons



and



, and formulate the criteria for

their synchronization. Let the signal

 



electrical activity of the first



- neuron, be a

superposition of



elementary Gaussian signals

 

ttztZ 









)(

, (1)

 





















exp

ttb

ttz

. (2)

The simulation (1) assumes the first



- neuron

involving a number of spikes. We determine the

time of each spike occurrence either

phenomenologically or as a solution of more

complex models of neuron functioning. Each spike

 

;;

L L L

L b t

occurs at a time moment

tt 

, has

its own amplitude

and duration



. In addition

to the spikes with



much smaller than the time

interval between spikes, this model comprises some

bursts. Each burst has its own center

LВ

and its

characteristic duration



(





). Various

bursts of



- neuron do not overlap (



is much

smaller than the distance between bursts), and the

number of spikes in each burst can vary. The

proposed model makes it possible to take into

account the long-term changes in the work of an

individual neuron. Various types of effects on the

neuron (medicinal and light effects) can evoke such

perturbations. We represent the signal of second

neuron

)(tZ



as a superposition

 

ttztZ 









)(

(3)

of Gaussian peaks

 

ttz 

characterized by a set

of parameters

 

KKK

tbK  ;;

Let us consider the expression for the correlation

function,

BIOSIGNALS 2018 - 11th International Conference on Bio-inspired Systems and Signal Processing

202

     

tdttZtZtССF



















, (4)

which shows the coupling between two signals

()Zt



and

)(tZ



at different time points. We can

obtain the analytical expression for the cross-

correlation function of two Gaussian signals

 

exp

t t t

ССF t















  



   



. (5)

It is convenient to use the normalized cross-

correlation function

 

()n

ССF t

with the maximum

value of

 

()n

ССF t

equal to one at

t t t

Taking into account (1), (3), the correlation function

(4) takes the form

   

()

CCF t CCF t















(6)

We find out that function (6) does not change if

we reverse the direction of time, and replace all the

times of spikes’ and neurons’ occurrences by the

opposite values

tt

Hereinafter, we assume that all spikes of neurons



and



are equal

    

, where



=0.001s,

and their maximum amplitudes

bb   

are

equal to one. Suppose that for



-neuron we have



=4 of spikes

 

0 1 2 3

; ; ;

     

localized at

 

0.01 ;0.03;0.06;0.1

ts

. For



-neuron we

have



=4,

 

0 1 2 3

; ; ;

     

with

 

0.71;0.73;0.79;0.89

ts

. Fig. 1 shows the signal

 



of the first



-neuron. We highlight in bold

four spikes of this neuron, given by the Gaussian

peaks. Four spikes forming the signal

 



associated with



-neuron are not highlighted.

The first peak of

 

CCF t



=0.61 s is due to

the coupling (correlation) of spikes

( ; )

. The

second peak at 0.88 s corresponds to the coupling of

spikes

( ; )

. The peak having a doubled

amplitude at

=0.70 s is due to the synchronization

of two pairs of spikes

);(



and

);(



. All

other peaks are related to the difference in

localization times of other spikes and neurons (Fig.

1).

Figure 1: The dependence of time

,ts

for



-neuron

signal

 



(in bold) и



-neuron signal.

Fig.2 shows instantaneous correlation function

 

ССF t



(4) for signals

 



and

 



Figure 2: The dependence of time

,ts

for

 

CCF t



2.2 Continuous Wavelet Transform of

Neural Signal

Time-frequency properties of



and



neurons

vary with time. We can successfully process such

nonstationary signals by Continuous Wavelet

Transform – CWT. This type of integral transform

maps nonstationary signal

)(tZ

to the plane of time

(s) and frequency



(Hz) (Bozhokin, 2010,

Bozhokin and Suslova, 2015) by the formula

      

tdtttZtV



















, (7)

where

 

x

is the mother wavelet function, symbol

* means complex conjugation. We use here a new

adaptive Morlet wavelet function (AMW) with the

control parameter

(Bozhokin and Suslova,

2016):

Wavelet Correlation of Neural Activity Bursts Generating Spikes

203

   

 

 

.exp2exp

exp

Dx 















(8)

In (8) we have the parameter

m  

, and

constant

defined from the condition

 

 x

The properties of the mother wavelet (8) are given in

(Bozhokin and Suslova, 2016). In (Bozhokin and

Suslova, 2016) the value of

acts as a control

parameter of AMW. We can change temporal

resolution



and frequency resolution



signals under study by varying

. The

characteristic moments of time



, which make the

main contribution to the integral (7), satisfy the

relation

t t t



       

. The adaptive

mother wavelet (8) acts as a varying window, which

depends on control parameter

. The window width

automatically becomes large for small frequencies



and small for large ones.

To calculate (8), we use the Fourier transform of

all functions in (7):

     

, exp 2

V t Z f ift df







   









, (9)

where

 

and

 

/f

are the Fourier

components of

 

and

 

x

. The Fourier

component

 

F

equals:

   

 

exp 1

1 exp 2 .





    







  



(10)

The application of the AMW makes it possible to

improve the results for spectral and time resolutions

)(tZ

in comparison with the application of the

conventional Morlet wavelet. We can illustrate the

fact by considering an infinite harmonic signal

( ) cos(2 )Z t f t

with frequency

. We can derive

the analytical expression for continuous wavelet

transform

 

,Vt

(7) using AMW (8). The maximum

 

,Vt

is located at point

f

. The full width

at half maximum of

 

,Vt

given by

2 2ln2 /( )

FWHM

fm  

decreases with the

increase in parameter

If we present the signal

 



as a superposition

(1) of elementary signals (2), then

 

,Vt





also

should be the superposition of wavelet images

 

Vt

corresponding to the signals (2). The

wavelet images

 

Vt

for our model of neural

signals can be derived analytically:

 

,exp

exp

exp),(

2222









































































amx

(11)

where

 

0 L

x t t  

2 2 2

1 2 /

am   

3 RESULTS

3.1 Wavelet Analysis of Model Neural

Signals

Using the simulation (1), we can calculate

analytically the modulus of wavelet transform

),( tV 



for signal

)(tZ



with four spikes

 

0.01 ;0.03;0.06;0.1

ts

presented in Fig.1.

Fig. 3 shows the ridges of two-dimensional surface

),( tV 



Figure 3: Modulus of wavelet transform

),( tV 



depending on frequency



, Hz and time

,ts

Fig. 4 displays the skeleton of the wavelet

transform, which shows the location of extremal

ridges on time-frequency plane.

Figure 4: Skeleton of Modulus of wavelet transform

),( tV 



depending on frequency



,Hz and time

st,

BIOSIGNALS 2018 - 11th International Conference on Bio-inspired Systems and Signal Processing

204

3.2 Wavelet Correlation Behavior

We introduce wavelet correlation function

),( tWCF 



by the formula

   

tdttVtVtWCF



















,,),(

, (12)

which, in contrast to (4), shows the correlation

between CWT of two signals

),( tV 



and

),( tV 



taken at different time moments. According to the

principle of superposition wavelet correlation

),( tWCF 



(10) can be represented as a double sum,

which includes

( , )

WCF t

calculated for

elementary Gaussian signals. Note that under this

approach, we can also derive

( , )

WCF t

analytically.

Hereinafter, we will need to know the

normalized function

 

( ) 2

, 8 exp 2

exp 2 2exp 1 .

LK m

WCF t m e A

iB iB





      





   





   



   



   



(13)

calculated as

 

( ) (max)

( , ) , /

LK LK LK

WCF t WCF t WCF  

with

maximal value equal to unit. In (13) we have

e 

2.72 and

 

2 2 2

   



, (14)

 

(

t t t

  



, (15)

 

t t t

  







. (16)

Function

( , )

WCF t

reaches its maximal value

(max)

WCF

t t t

. Under the condition

    

, the value of

(max)

WCF

has its

maximum at the point

 

max

1 1/ (16 /(4 )m    

, which for



=0.001 s approximately equals to 80 Hz.

To achieve the best time resolution of spikes, we

use here the control parameter

1m 

Fig.5 shows

),( tWCF 



for two neurons



and



Figure 5: Dependence of

),( tWCF 



on frequency

,Hz

and time

st,

The analysis of Fig. 5 shows that at large

frequencies

1/(4 )  

(



80 Hz) the time

behaviour of wavelet correlation modulus

( , )WCF t





corresponds exactly to the classical

correlation function

 

ССF t



. The doubled peak at

=0.70 s at these frequencies also appears to be due

to the synchronization of two pairs of spikes

);(



and

);(



. The special features of low-

frequency behaviour of

),( tWCF 



are associated

with the characteristic intervals of peaks sequence in

the process of neurons synchronization.

4 CONCLUSION

We propose a simple mathematical model of neural

signals, which allows us to obtain analytical

expressions for wavelet correlation functions

( , )WCF t





.The neural signal as a sequence of bursts

containing a certain number of spikes is simulated

by the superposition of elementary Gaussian signals

characterized by several parameters such as

amplitude

, duration



and time of occurrence

tt

The study of the maximal value of

( , )WCF t





depending on time gives the opportunity to detect

the synchronization of spikes between various

neurons at different time moments. The dependence

),( tWCF 



on frequency



provides additional

information on the correlation of spikes. In the

limiting case

1/(4 )  

, the time behavior of

),( tWCF 



is identical to that of the classical

correlation function

 

ССF t



Wavelet Correlation of Neural Activity Bursts Generating Spikes

205

The task of determining neuronal correlations is

particularly important in the development of neuro-

interfaces, which are multi-electrode arrays that

exchange information between the neuronal

population and the outside world (Bursáki et al.,

2012,). Such neuro-interfaces allow both stimulation

and synchronous probing of dozens of neurons at the

cellular level. Our method can be used to determine

individual spikes of neurons in the patch-clamp

method (Suk et al., 2017), as well as in studying the

functioning of mirror neurons (Hou et al., 2017).

The wavelet-correlation function introduced in

this paper can be used as a tool to study rapidly

changing burst processes in radio-physics, plasma

physics and astrophysics, as well as the stability of

quantum frequency standards.

ACKNOWLEDGMENTS

The work has been supported by the Russian

Science Foundation (Grant of the RSF 17-12-

01085).

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