Chaos, Solitons and Fractals 123 (2019) 35–42

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals

Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Temperature dependence of phase and spike synchronization of

neural networks

R.C. Budzinski a , B.R.R. Boaretto

a , T.L. Prado

a , b , S.R. Lopes a , ∗

a Departamento de Física, Universidade Federal do Paraná, Curitiba 81531-980, PR, Brazil b Instituto de Engenharia, Ciência e Tecnologia, Universidade Federal dos Vales do Jequitinhonha e Mucuri, Janaúba 39440–0 0 0, MG, Brazil

a r t i c l e i n f o

Article history:

Received 2 January 2019

Revised 26 March 2019

Accepted 29 March 2019

Keywords:

Neural network

Synchronization

Nonlinear dynamics,

a b s t r a c t

We simulate a small-world neural network composed of 20 0 0 thermally sensitive identical Hodgkin–

Huxley type neurons investigating the synchronization characteristics as a function of the coupling

strength and the temperature of the neurons. The Kuramoto order parameter computed over individ-

ual neuron membrane potential signals, and recurrence analysis evaluated from the mean field of the

network are used to identify the non-monotonous behavior of the synchronization level as a function of

the coupling parameter. We show that moderated high temperatures induce a low variability of the inter-

burst intervals of neurons leading to phase synchronization and further increases of temperature result

in a low variability of inter-spike intervals leading the network to display spike synchronization.

© 2019 Elsevier Ltd. All rights reserved.

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. Introduction

Due to advances of computational methods, real neural net-

orks can now be satisfactorily studied by using mathematical

eural models coupled through complex networks. In fact, be-

ond neural networks, chemical, biological, physical and even so-

ial problems can be analyzed using complex networks models and

oupled oscillators [1–5] . Neural networks are composed of inter-

onnected neural cells and can be found in small systems of hun-

red of neurons as the C. Elegans [6] neural network as well as

orming much more complex problems such as the human brain

etwork [7] . To simulate mathematically the real neural network,

eurons are considered as nodes possessing their own dynamics

nd linked to others through a connection matrix. Usually, random,

cale–free and small–world topologies are used to simulated the

onnection schemes of neural networks [8–11] since such topolo-

ies find support in real situations [6,12,13] . For all these connec-

ion schemes, a transition from unsynchronized to synchronized

tates is observed.

An important characteristic of complex systems consists of the

mergent collective behavior, where global phenomena are, in gen-

ral, different and richer than the simple sum of individual be-

avior of system elements [3] . A common collective behavior of

neural network is the presence of synchronized states. In spe-

ial, a large number of networks exhibit a quite complex transi-

∗ Corresponding author.

E-mail address: lopes@fisica.ufpr.br (S.R. Lopes).

H

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ttps://doi.org/10.1016/j.chaos.2019.03.039

960-0779/© 2019 Elsevier Ltd. All rights reserved.

ion to synchronized states, characterized by a non-monotonic de-

endence on the coupling parameter [14–17] leading the network

o an intermittent and nonstationary transition to synchronization

9,14] . Neural networks composed of bursting neurons display syn-

hronization in two time scales [18] : a fast one, related to spike

ynchronization, and a slow one, associated to phase (bursting)

ynchronization. These dynamical properties are very useful to the

eural system understanding since there are neural diseases asso-

iated to an excess of (sometimes anomalous) synchronization, as

bserved in [19,20] and neurological disorders related to the inter-

ittent (nonstationary) behavior [21] of the network dynamics. A

articularly interesting point consists of the effect of the tempera-

ure on the neural system since it is known that temperature in-

reases can lead the brain to undergo an epilepsy scenario [22,23] .

ther studies suggest that high body temperature can be associ-

ted with an acute crisis in Parkinson disease [24] . On the other

and, a theoretical approach of dynamical properties as a function

f the temperature of the neural system is given in [25] , where the

emperature increases result in a decrease of individual neuron be-

avior variability.

Here 20 0 0 thermally sensitive identical bursting neurons in a

mall-world topology are simulated. Synchronization characteris-

ics are studied as a function of the temperature and the cou-

ling parameter. The model of Braun et. al. [26] is used to simulate

he neuron behavior, a modified version of the original Hodgkin–

uxley model [27] . Following many real neural systems [28] , the

eural behavior considered is the bursting regime, which consists

f a sequence of chaotic spikes related to the fast time scale of the

36 R.C. Budzinski, B.R.R. Boaretto and T.L. Prado et al. / Chaos, Solitons and Fractals 123 (2019) 35–42

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dynamics, followed by a resting period that is related to a slow

time scale [29] .

We use the Kuramoto order parameter [30] , recurrence quantifi-

cation analysis, particularly the determinism [31] and standard de-

viation of membrane potential signal [17] to evaluate unequivocally

the synchronization characteristics. The order parameter is able to

quantify the phase synchronization level through the individual

signal of each neuron which is associated with a geometric phase

of the bursting activity. On the other hand, the determinism mea-

sures the ratio of recurrent points belonging to diagonal structures

of the recurrence plot and is evaluated from the potential mean

field of the neural network as proposed in [14] . At last, the stan-

dard deviation of the membrane potential of the neurons is able to

bring information about synchronization in slow (phase synchro-

nization) and fast (spike synchronization) time scales and is used

to corroborates the results. Through these quantifiers, it is possible

to investigate the synchronization level in a general way, where

is observed that the neural system can display non–monotonous

synchronization phenomena as a function of coupling strength and

temperature changes, including synchronization in two time scales,

namely phase synchronization related to the bursting time scale,

and spike synchronization associated to the time scale of the neu-

ron spikes. We show that exist a lower coupling regime where

phase and spike synchronization can be closely related to the in-

dividual dynamics of the neurons. For such regimes, the collective

dynamics is strongly related to the small variability of the neu-

rons and the synchronization frequency is closely related to the

frequency of the neurons itself. For strong coupling, the neuron

variability is not a limiter and phase synchronization can be ob-

tained for a larger interval of frequencies.

This paper is organized as follows. In Section 2 the neural

model and topology characteristics are discussed, in Section 3 the

synchronization quantifiers are defined and in Section 4 the results

and discussions are shown to support the conclusion in Section 5 .

At last, the acknowledgment and references are presented.

2. Neural model and connection architecture

The model of Braun et. al. [26] is a variation for the original

Hodgkin–Huxley model [27] , where the temperature dependence

and new ionic fluxes are considered. The original work of Braun

et al. was developed in order to investigate the temperature de-

pendence and its effects in the crayfish cold receptors [26] . How-

ever, many authors have developed adaptations to the model to

mimic different biological scenarios related to the temperature in-

fluence in the neural system [17,32,33] . Recently, Burek et al. have

investigated the temperature influence on neuronal firing rates and

bursting transitions [34] by using the model of Braun et al. The re-

sults have shown the temperature influence on the neural activity

and the ability of the model to reproduce some experimentally ob-

served phenomena such as the increase of the depolarization time

for low temperatures [35] .

The model is able to reproduce bursting neural activity and the

main equation is based on capacitor modeling

m

dV i

dt = −J i, Na − J i, K − J i, sd − J i, sa − J i, L + J i, coupling , (1)

where the membrane capacitance is given by C m

= 1.0 μF/cm

2 ,

J i,υ is the ionic flux (mA/cm

2 ) over the i th neuron membrane and

related to Sodium (Na), Potassium (K) and leak (L) currents. Addi-

tionally, two slow fluxes due to Calcium, (sd) and (sa) are consid-

ered. The flux due to coupling to other neurons is represented by

J i ,coupling . The parameters of the model were chosen following [36] .

The fluxes related to each ion are given by their specific maxi-

mum conductances,

J i, Na = 1 . 5 ραi, Na (V i − 50) , (2)

i, K = 2 . 0 ραi, K (V i + 90) , (3)

i, sd = 0 . 25 ραi, sd (V i − 50) , (4)

i, sa = 0 . 4 ραi, sa (V i + 90) , (5)

i, L = 0 . 1(V i + 60) . (6)

The Nernst potential associated with each ion is supposed as

efined in [36] . The first temperature factor is given by

= 1 . 3

T−50 10 , (7)

here T is the temperature of the neural system, T ∈ [37, 40], mea-

ured in Celsius degrees. The activation channel variables α have

ynamical evolution given by

dαi, Na

dt =

φ

0 . 05

(αi, Na , ∞

− αi, Na ) , (8)

dαi, K

dt =

φ

2 . 0

(αi, K , ∞

− αi, K ) , (9)

dαi, sd

dt =

φ

10

(αi, sd , ∞

− αi, sd ) , (10)

dαi, sa

dt =

−φ

20

(0 . 012 J i, sd + 0 . 17 αi , sa ) , (11)

here

= 3 . 0

T−50 10 (12)

s the second temperature dependence. The membrane potential

ependences of the activation variable are given by

i, Na , ∞

=

1

1 + exp [ −0 . 25(V i + 25)] , (13)

i, K , ∞

=

1

1 + exp [ −0 . 25(V i + 25)] , (14)

i, sd , ∞

=

1

1 + exp [ −0 . 09(V i + 40)] . (15)

The coupling term is modeled by excitatory chemical synapses

37] through a small-world (Newman–Watts route [38] ) binary ma-

rix, e i,j , given by

i, coupling =

ε

χ

N ∑

j=1

e i, j r j (V syn − V i ) , (16)

here ε is the coupling strength (mS/cm

2 ), χ = 4 . 8 is the aver-

ge number of connections, V syn = 20 mV is the synaptic reversal

otential, N = 20 0 0 is the number of neurons and r j is the kinetic

erm representing the fraction of receptors available to receive con-

ections [37]

dr i dt

=

(1

τr − 1

τd

)1 − r i

1 + exp [ −s 0 (V i − V 0 )] − r i

τd

, (17)

here s 0 = 1 (mV) −1 , V 0 = −20 mV, τr = 0 . 5 ms and τd = 8 ms are

onstants.

Fig. 1 depicts the typical dynamics of a neuron, where the black

ine depicts bursting behavior and the magenta line is the tempo-

al behavior of the variable defined as U ≡ 1/ αsa , which assumes a

aximum every time a fast sequence of spikes starts (the burst),

aking possible a geometric phase association to each slow time

cale bursting activity. For all coupling and temperature values

sed here, the bursting behavior is obtained and the phase associ-

tion is valid. Here, we simulate the slow bursting frequencies cen-

ered around 1 Hz and the spike frequencies are centered around

.05 Hz. Such values reflect biologically acceptable values but the

R.C. Budzinski, B.R.R. Boaretto and T.L. Prado et al. / Chaos, Solitons and Fractals 123 (2019) 35–42 37

Fig. 1. Dynamical behavior of the membrane potential V ( t ) for a neuron (black line)

and the variable U ≡ 1/ αsa that assumes a maximum at each burst starting (magenta

line). (For interpretation of the references to color in this figure legend, the reader

is referred to the web version of this article.)

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odel is general since Eqs. (1) – (17) can be re-scaled in time and

ther suitable sets of constants leads to the same dynamical be-

avior.

The total number of connection of the network is given by

= 4 N ︸︷︷︸ local

+ N(N − 5) p ︸ ︷︷ ︸ nonlocal

, (18)

here p is the probability to add a nonlocal connection. p → 1

eads to a globally connected network. Here, p = 0 . 0 0 04 is used

esulting in 1590 nonlocal connection added by 80 0 0 local ones

esulting in a small–world network with 9590 connections.

A 12th Adams’ predictor-corrector method with an absolute tol-

rance of 10 −8 [39] is used to integrate the differential equations.

he initial conditions are randomly distributed in the intervals

iven by V i ∈ [ −65 , 0] mV, αυ,i ∈ [0 . 1 , 1] , where υ represent Na,

, sd, sa, L and r i = 0 . 1 , avoiding any synchronization trend and/or

umerical divergence.

. Synchronization quantifiers

The Kuramoto order parameter can constitute a quantifier to

hase synchronization since the variable U assumes a maximum

very time a burst starts, as shown in Fig. 1 , such that we can as-

ociate a geometric phase θ i to each neuron, increasing it by 2 πvery maximum of U . To obtain a continuous variation of phase

e use simple linear interpolation [40,41]

i (t) = 2 πk i + 2 πt − t k,i

t k +1 ,i − t k,i

, t k,i < t < t k +1 ,i , (19)

here t k,i is the time when the k th burst occurs in the i th neuron.

To obtain the Kuramoto order parameter of the entire network

t is necessary to evaluate the phase of all neurons through [30]

(t) =

∣∣∣∣∣1

N

N ∑

j=1

e iθ j (t)

∣∣∣∣∣. (20)

If the system is (not) in a phase synchronized state, R → 1

R → 0).

Better visualization of the synchronization as a function of the

oupling parameter or the temperature is obtained from the mean

alue of the Kuramoto order parameter, defined as

R 〉 =

1

A

t f ∑

t= t 0 R (t) , (21)

here t f is the total simulation time, t 0 is the transient time and

= (t f − t 0 ) /h, where h = 0 . 01 ms.

Recurrence analysis is an additional tool to evaluate phase syn-

hronization [14,16] . Since it does not use information of neuron

hases, it can be computed over the mean field of the network

omputing a recurrence matrix defined as [42]

ab (δ) = �(δ − || x a − x b || ) , x a ∈ R , a, b = 1 , 2 , . . . , S, (22)

here x in our case are the successive values in the time of the

ean field of the network. � is the Heaviside function, δ is the

ecurrence threshold and S is the size of time series analyzed. If a

tate x a is (not) recurrent to x b , R ab receives (zero) one.

The determinism is a recurrence quantifier that measures the

atio of recurrent points belonging to diagonal structures over all

ecurrent points in the recurrence matrix [31] , and distinguishes

he level of synchronization [9]

(� min , δ) =

S ∑

� = � min

�P (�, δ)

S ∑

� =1

�P (�, δ)

, (23)

here � are the diagonal sizes, � min is the minimum diagonal size

onsidered and P ( � , δ) is the probability distribution function (PDF)

f diagonal lines computed over the mean field of the neural net-

ork

(t) =

1

N

N ∑

i =1

V i (t) . (24)

In a similar approach as used to Kuramoto order parameter, we

efine the mean value of the determinism as

〉 =

1

A

t f ∑

t= t 0 (t) . (25)

Here, the determinism ( ) of the mean field potential is eval-

ated through Eq. (23) using an overlapped moving window.

hrough this procedure a single value of for each window is

btained. Considering the total time of simulation, a time series

or the determinism is obtained and a mean value in time is com-

uted, 〈 〉 . The recurrence threshold is chosen using the maximum sensi-

iveness of to capture small changes in the dynamics as pro-

osed in [43] . Using this procedure, phase synchronized network

esults in a mean field that leads to 〈 〉 ∼ 1. Due to the spike na-

ure of the mean field signal, spike synchronization results in small

alues of 〈 〉 associated with 〈 R 〉 ∼ 1. The standard deviation as a

unction of time computed over the neuron membrane potential

ignals

(t) =

√ √ √ √

N ∑

i =1

(V i (t) − V (t)) 2

N

. (26)

s used to corroborates the results of and R [17] , since the tem-

oral coincidence of all membrane potentials leads to σ → 0 quan-

ifying a complete synchronization. On the other hand, higher val-

es of σ indicate unsynchronized states.

. Results and discussions

Fig. 2 (a) and (b) are a summary of our findings and depict color

oded mean values of the Kuramoto order parameter and the de-

erminism in the parameter space ε × T . The results are based on

ime series of t f = 150 s and the transient time is given by t 0 =20 s. Here, are considered 15 different initializations of the sys-

em. The recurrence threshold parameter δ = 0 . 11 is chosen from

he condition that d [ ( δ)]/ d δ assumes a maximum [43] , which

esults in the highest sensitiveness of the quantifier. The mini-

um diagonal size is set to � min = 35 ms to avoid small diagonals

9,44] and the determinism is evaluated using a moving window

f 10 s (10,0 0 0 points) using a overlapping of 9.99 s.

38 R.C. Budzinski, B.R.R. Boaretto and T.L. Prado et al. / Chaos, Solitons and Fractals 123 (2019) 35–42

Fig. 2. Mean values of the Kuramoto order parameter (panel (a)) and the determinism (panel (b)) described by Eqs. (21) and (25) , respectively. Here, 〈 R 〉 and 〈 〉 are

computed as a function of coupling parameter ( ε) and neurons temperature ( T ). The network is highly sensitive to T for small coupling regime.

Fig. 3. Mean values of the Kuramoto order parameter ( 〈 R 〉 ) (black line) and the de-

terminism ( 〈 〉 ) (red line), for ε = 0 . 005 as function of temperature ( T ). Increases

on temperature induce synchronization to neural system. A decrease of 〈 〉 associ-

ated to an increase of 〈 R 〉 is a diagnostic of spike synchronization (SS). (For inter-

pretation of the references to color in this figure legend, the reader is referred to

the web version of this article.)

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In general the system is not sensitive to temperature for cou-

pling strength larger than 0.025 where a (partially) phase syn-

chronized regime (PS2) is observed for all range of T . In this case,

the collective behavior imposed by the coupling overcomes the in-

fluences of the temperature over the local dynamics of the neu-

rons. On the other hand, for small coupling strength the increase

of the temperature results in a very complex dynamical behavior

due to the interplay between temperature and coupling effects. For

T < 38.0 ◦C and coupling smaller than 0.01, the neural network dis-

plays unsynchronized state (dark areas in both panels in Fig. 2 ) fol-

lowed by a monotonic transition to phase synchronization as the

coupling strength increases. For temperatures higher than 38.0 ◦C

and coupling strength 0.005 < ε < 0.015, a non-monotonic synchro-

nization is observed where a local maximum is noticed in both

panels for 〈 R 〉 and 〈 〉 denouncing some level of phase synchro-

nization even for low coupling strengths (PS1). This first phase

synchronization regime is strongly related to the small interval of

frequencies experienced by the neurons, resulting in a small vari-

ability of the neuron frequencies, as we will going to show later on

in this text. In this region of the parameter space, a rising coupling

strength makes the system to lose synchronization characteristic.

Similar behavior was reported as an anomalous phase synchroniza-

tion regime [14,16,17,25,45] . The unsynchronized regime extends

further for higher values of the temperature and coupling parame-

ter.

For temperatures above 39 ◦C and small coupling strengths

0.001 < ε < 0.01, the neural system displays high levels of synchro-

nization, indicating that higher temperatures can even induce spike

synchronization (SS). Such the regime of SS is related to an ex-

treme small variability of neurons resulting in a spike synchroniza-

tion of the neurons. PS1, SS and PS2 regimes will be investigated

in details along this text.

Differently from other situations where high values of the de-

terminism and the Kuramoto order parameter are observed simul-

taneously for the synchronized regime, what in fact, characterizes

phase synchronization [14] , for T > 39.0 ◦C and for 0.001 < ε < 0.01

it is noticed that 〈 R 〉 displays a local maximum but 〈 〉 shows a

local minimum, denouncing a distinct synchronization regime in

this region, depicted as “SS” in Fig. 2 .

A better view of this region is depicted in Fig. 3 where mean

values of the Kuramoto order parameter 〈 R 〉 and the determin-

ism 〈 〉 are plotted as a function of the temperature for a fixed

coupling parameter ε = 0 . 005 . An initial increase of the tempera-

ture makes the neural system to transit from unsynchronized to

phase synchronized states, however, for T > 39.0 ◦C, 〈 R 〉 reaches al-

c

ost 1 at the same time that 〈 〉 decreases. So, we say that the

imultaneous observations of 〈 R 〉 and 〈 〉 make possible the dis-

inction among two distinct synchronization regimes, namely the

hase and spike synchronizations. It is possible since differently

rom the order parameter, 〈 〉 is computed based on the network

ean field and it shows to be sensitive enough to capture distinct

ehaviors of the mean field of the membrane potential of the neu-

al network ( Eq. (24) ) of both regimes.

Fig. 4 (black lines) depicts details of V (t) ( Eq. (24) ) and its

tandard deviation σ ( t ) (magenta lines) of the neuron membrane

otential ( Eq. (26) ) for unsynchronized regime occurring for

= 0 . 005 and T = 37 . 0 ◦C (a), phase synchronized regime

or ε = 0 . 0040 and T = 40 . 0 ◦C (b) and spike synchronized regime

or ε = 0 . 005 and T = 40 . 0 ◦C (c). For panel (a), the amplitude of

scillation of V (t) is almost null and σ ( t ) depicts just random

uctuations around a large value, corroborating the idea of an un-

ynchronized network for these coupling and temperature values.

anels (b) and (c) depict higher synchronized regimes and for both

ases, the Kuramoto order parameter shows high and similar mean

alues (see Fig. 3 ), however for the panel (b) case, the determinism

ollows the Kuramoto order parameter trend, while for panel (c)

he Kuramoto order parameter presents a local maximum and the

eterminism a local minimum. The mean field helps to understand

he real situation since for both cases there are a high amplitude

f both quantifiers indicating the existence of synchronization, but

he case (c) shows two time scales, a first one related to phase syn-

hronization (slow time scale) and a second one related to spike

R.C. Budzinski, B.R.R. Boaretto and T.L. Prado et al. / Chaos, Solitons and Fractals 123 (2019) 35–42 39

Fig. 4. Mean field of the neuron membrane potential ( Eq. (24) ) (black lines) and

its standard deviation ( Eq. (26) ) (magenta lines) as function of time for the neural

network for (a) ε = 0 . 005 and T = 37 . 0 ◦C, (b) ε = 0 . 0040 and T = 40 . 0 ◦C, and (c)

ε = 0 . 005 and T = 40 . 0 ◦C. (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article.)

s

[

o

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d

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〈

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(

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ynchronization (fast time scale). A similar scenario is found in

18] where the spike synchronization regime is reached after the

ccurrence of a burst regime. This fact explains the antagonistic

ehavior of 〈 R 〉 and 〈 〉 since the determinism is evaluated from

ean field signal. Corroborating this scenario, σ ( t ) in panel (b)

epicts large slow time oscillations related to the phase syn-

hronization process. When the network is in spike regime, σ ( t )

isplays values larger than those expected for an unsynchronized

egime while smaller values are expected for the quiescent one.

ccordingly σ ( t ) decays when temporal coherence of the mem-

rane potential is observed. Therefore, panel (c) depicts smaller

scillations of σ ( t ) resulted of a more synchronized spike regime

ccurring in the fast time scale. Even slower values of σ ( t ) is

bserved for the quiescent regime pointing out for a very (spike)

ynchronized network. In fact, an absolute level of synchronization

s obtained by the mean value of σ ( t )

σ 〉 =

1

T

∫ T 0

σ (t ) dt . (27)

esulting in 〈 σ 〉 = 10 . 44 for unsynchronized case of Fig. 4 (a), 〈 σ 〉 = . 25 for phase synchronized case of Fig. 4 (b) and 〈 σ 〉 = 5 . 94 for

pike synchronized case of Fig. 4 (c).

Fig. 5 (a), (b) and (c) depict examples of recurrence plots eval-

ated through Eq. (22) from the mean field potential time se-

ies, as described by Eq. (24) for unsynchronized states (panel (a))

here ε = 0 . 005 and T = 37 . 0 ◦C, phase synchronized states (panel

b)) where ε = 0 . 040 and T = 40 . 0 ◦C and spike synchronized states

ig. 5. Recurrence plots obtained from Eq. (22) evaluated using data from the mean field

he unsynchronized, phase synchronized and spike synchronized states, respectively.

panel (c)) where ε = 0 . 005 and T = 40 . 0 ◦C. The difference be-

ween the synchronized and unsynchronized case is very clear

ince in the panels (b) and (c) it is possible to observe diagonal

tructures while in the panel (a) no structure is noticed [9,44] . On

he other hand, the differences between burst and spike synchro-

ization are subtler. Panels (b) and (c) depict similar diagonal lines,

owever, as before mentioned the phase synchronization is related

o the slow time scale while the spike synchronization to the fast

ne, resulting in smaller structures in the fast time scale of the

anel (c), which denounces the partial spike synchronization.

Fig. 6 shows a qualitatively way to observe distinct network

ynchronization phenomena induced by temperature increases,

amely the Poincarè surface at V = −20 mV for all neurons in the

etwork. Panel (a) represents the case where ε = 0 . 005 and T =7 . 0 ◦C, panel (b) ε = 0 . 040 and T = 40 . 0 ◦C and panel (c) ε = 0 . 005

nd T = 40 . 0 ◦C. For the first panel, there is not spatial-temporal

oherence of membrane potential and it is representative of an

nsynchronized state. Panels (b) and (c) depicts spatial patterns

bserved for synchronized states, however, panel (c) displays syn-

hronization even for the fast time scale denouncing spike syn-

hronization. The synchronization on the slow time scale or phase

ynchronization is very similar in both panels (b,c), which is ex-

ected since the 〈 R 〉 is almost the same value for each case. On

he other hand, spike synchronization should be not expected to

ccur due to small values of the coupling parameter. In this case,

e conclude that spike synchronization is induced just by the in-

rease of the temperature.

In some cases, the global behavior of a network synchronization

an be associated to the individual neuron variability [25] . To in-

estigate the correlation between individual variability of the neu-

ons and the global synchronization states of the network, Fig. 7

epicts bifurcation diagrams for the inter-spike (ISI) and inter-burst

IBI) intervals for an uncoupled neuron, panels (a) and (d); for a

andom chosen neuron of the network, panels (b) and (e), for a

oupling strength of ε = 0 . 0050 as an example of weak coupling

egime; and for a random chosen neuron of the network, panels

c) and (f) for a coupling strength of ε = 0 . 040 , a strong coupling

egime. For ISI, we compute just time intervals belonging to the

ame burst, discharging the time interval between the last spike of

ne burst and the first spike of the next.

For 37.5 ◦ � T � 40 ◦ the ISI for an uncoupled neuron suffers a

ouble period bifurcation process, going from a great variably

haotic ISI to a single value of ISI diminishing drastically its

ariability as T grows as observed in Fig. 7 (a). Despite the in-

ividual behavior of the uncoupled neurons, a weakly network

oupling imposes an almost constant variability of ISI of the cou-

led neurons for the interval 37 ◦ � T � 38.5 ◦ as observed in panels

b). We conclude that even a weak coupling regime results to be

trong enough to destroy the stability of all sequence of periodic

potential of the neural network as depicted in Fig. 4 . Panels (a), (b) and (c) depict

40 R.C. Budzinski, B.R.R. Boaretto and T.L. Prado et al. / Chaos, Solitons and Fractals 123 (2019) 35–42

Fig. 6. The Poincare surface in V = −20 mV for the neural network. Panel (a) depicts an unsynchronized state, where ε = 0 . 005 and T = 37 . 0 ◦C. Panel (b) depicts a phase

synchronized state, where ε = 0 . 040 and T = 40 . 0 ◦C and panel (c) depicts a synchronization in the two time scales, the fast and the slow ones, which suggests a burst and

spike synchronization level ( ε = 0 . 005 and T = 40 . 0 ◦C).

Fig. 7. Inter-spike intervals (ISI) and inter-bursting intervals (IBI) as a function of

the temperature of the system ( T ). Panels (a,d) depict the ISI and IBI for an uncou-

pled neuron, respectively, panels (b,e) depict the ISI and IBI for a weakly coupled

neuron where ε = 0 . 0050 . Panels (c,f) display the ISI and IBI for a strongly coupled

neuron where ε = 0 . 040 . The increase of the temperature results in a decrease of

the variability of a single neuron, however, the variability of the weakly coupled

neuron is almost constant for large intervals suffering subtle changes related to the

bifurcations occurring in the individual dynamics of the neurons (panels (b,e). For

strong coupling (panels (c,f) the coupling destroys all influence of the individual

dynamics of the neurons.

Fig. 8. Power spectrum of a representative neuron and T = 38 . 5 ◦C for (a) An un-

coupled neuron ε = 0 . 0 , showing an intrinsic (natural) period width of 53.9 ms. (b)

A weakly coupled neuron, ε = 0 . 006 and period width slightly increased to 65.8 ms.

(c) A strong coupled neuron, ε = 0 . 040 and period width increased to 141.2 ms as

the effect of the strong coupling.

t

a

w

r

w

a

3

s

t

(

r

i

3

r

i

c

c

f

t

a

(

l

behavior depicted by ISI. Nevertheless, the strong stability of the

period “one” orbit of the individual dynamics of the (uncoupled)

neurons, starting in T ≈ 39.25 ◦C influences heavily the network

neuron behavior, resulting in a broader ISI but centered on its for-

mer uncoupled behavior. For greater temperature, the progressive

increasing stability of the period “one” orbit for the uncoupled

neuron makes the variability of the coupled neurons to diminish

as seen in Fig. 7 (b). Based on Figs. 2 and 3 , such a behavior can

be associated with the appearance of spike synchronization of the

network occurring for a weak coupling regime.

On the other hand, for the strong coupling as observed in panel

(c), the coupling effect destroys all influences of the individual be-

havior of the neurons, producing a large variability for all intervals

of the temperature. So, for strong coupling, the influence of the

natural spike frequencies of the neurons is lost and spike synchro-

nization is not induced, leading the network to display only phase

synchronization as shown in Figs. 2 and 3 and detailed next.

Fig. 7 (d), (e) and (f) depict the behavior of the intervals be-

ween consecutive bursts (IBI) as a function of the temperature for

n uncoupled (d) and a weakly (e) and strongly (f) coupled net-

ork neurons. Again, the variability of IBI for an uncoupled neu-

on decreases as the temperature increases, panel (d). The net-

ork coupling makes the variability of IBI of a network neuron

lmost constant for a large part of the period doubling cascade

7 ◦ < T < 38 ◦C, and again, it seems to be not strong enough to de-

troy completely the stability properties of the period “one” orbit

hat starts at T ≈ 37.9 ◦C, and even high temperature period “two”

T � 38.5 ◦C, resulting in a drastic diminishing of the network neu-

on variability, panel (e). The result of this small variability in IBI

s the transition to phase synchronization observed in Figs. 2 and

even for weak coupling regime. So the phase synchronization

egimes occurring for weak couplings are strongly related to the

ndividual dynamics of the neurons, in contrast to the phase syn-

hronization occurring for larger values of the coupling. For strong

oupling, panel (f), the presence of phase synchronization is not a

unction of the individual dynamics of the neurons. In this case,

he strong coupling imposes a collective dynamics and large vari-

bility neurons can also be synchronized.

A view of the neuron dynamics for T = 38 ◦ in the period space

P ( ζ ) × ζ ) is plotted in Fig. 8 . Panel (a) depicts results for an iso-

ated neuron. It is characterized by a discrete spike periods for

R.C. Budzinski, B.R.R. Boaretto and T.L. Prado et al. / Chaos, Solitons and Fractals 123 (2019) 35–42 41

ζ

1

a

r

n

d

f

b

n

s

S

c

n

a

F

c

s

p

s

o

c

t

5

t

d

t

m

a

c

p

n

t

a

g

c

w

T

a

r

a

t

i

o

o

r

s

o

r

d

a

a

s

n

c

s

D

A

f

n

s

-

m

b

(

R

[

[

[

[

[

[

< 100 ms (cyan dashed area) and a clear burst period around

0 0 0 ms with a variability period band of 53.9 ms (orange dashed

rea). Fig 8 (b) depicts results for a representative network neu-

on for the same temperature and considering a phase synchro-

ized state occurring for a weak coupling regime ε = 0 . 006 . The

iscrete nature of the spike periods is lost due to the coupling ef-

ect as suggested by the broader band of periods for ζ < 200 ms

ut due to the weak influence of the coupling in the burst dy-

amics, as pointed in Fig. 7 (e), the burst period around 10 0 0 ms is

till present and just the variability is slightly increased to 65.8 ms.

o we conclude that the phase synchronization occurring for weak

oupling has a strong influence of the individual dynamics of the

eurons, the network will display a synchronous behavior almost

t the same bursting frequency depicted by the neurons. Finally

ig. 8 (c) shows results for a phase synchronized state and a strong

oupling regime ( ε = 0 . 04 ) for the same temperature. Now we ob-

erve a large broad band in the spike period regime, and the burst

eriod regime is characterized by a broader peak around 1200 ms

howing a variability of 141.2 ms. The large variability corroborates

ur results depicted in Fig. 7 (f) and suggests that the phase syn-

hronization occurring for strong coupling destroys (at least par-

ially) the influence of any particular neuron dynamics.

. Conclusions

We have shown that a neural network composed of 20 0 0

hermally sensitive Hodgkin–Huxley like neurons can present rich

ynamical characteristics as a function of the temperature of

he neurons and coupling strength. The network displays a non-

onotonous synchronization scenario for weak coupling regime

nd temperature greater than 38 ◦C, where phase and spike syn-

hronizations are observed. For these synchronization regimes, the

resence of a small coupling parameter results in a highly synchro-

ized network. Further increases of the coupling parameter destroy

he phase and spike synchronizations and after a recovery region,

s the coupling parameter is continuously increased, an asymptotic

lobally stable phase synchronization is acquired.

Corroborating reference [25] , we have shown that for the weak

oupling regime, the phase synchronization observed in the net-

ork is strongly related to the individual variability of the neuron.

he increase of the temperature promotes a decrease of the vari-

bility of the neurons and even weak couplings between the neu-

ons lead the network to display slow time scale synchronization,

phase synchronization regime. Further increases of the tempera-

ure promote also a decrease in the variability of the spike time

ntervals (a fast time scale regime). The result of this reduction

f variability is the emergence of a spike synchronization regime

ccurring for weak coupling parameters. For both synchronization

egimes, further increases of the coupling parameter destroy the

ynchronization regimes.

It is known that neural diseases can be associated to an excess

f synchronization [19,20] and that high body temperature can be

elated to epilepsy phenomenon [22,23,46] or even to Parkinson’s

iseases [24] . Here, it is shown that the neural system can present

high level (excess) of synchronization just increasing the temper-

ture of the system. We have shown that such an increase of the

ynchronization level is associated to individual variability of the

eurons and the correct understanding of this dependence relation

an improve the understanding of the scenario of phase and spike

ynchronization associated to important deceases.

eclaration of Interest Statement

None.

cknowledgments

This study was financed in part by the Coordenação de Aper-

eiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Fi-

ance Code 001. The authors acknowledge the support of Con-

elho Nacional de Desenvolvimento Científico e Tecnológico, CNPq

Brazil, grant number 302785/2017-5 , Coordenação de Aperfeiçoa-

ento de pessoal de Nível Superior, CAPES , trough project num-

ers 88881.119252/2016-01 and Financiadora de Estudos e Projetos

CTINFRA-FINEP).

eferences

[1] Strogatz SH . Exploring complex networks. Nature 2001;410:268–76 .

[2] Bar-Yam Y . Dynamics of complex systems. Reading, MA: Addison-Wesley; 1997 .[3] Boccara N . Modeling complex systems. Springer Science & Business Media;

2010 .

[4] Rubido N , Cabeza C , Kahan S , Ávila GR , Marti AC . Synchronization regionsof two pulse-coupled electronic piecewise linear oscillators. Eur Phys J D

2011;62:51–6 . [5] Etémé AS , Tabi CB , Mohamadou A . Synchronized nonlinear patterns in electri-

cally coupled hindmarsh–rose neural networks with long-range diffusive inter-actions. Chaos Solitons Fractals 2017;104:813–26 .

[6] Varshney LR , Chen BL , Paniagua E , Hall DH , Chklovskii DB . Structural prop-

erties of the caenorhabditis elegans neuronal network. PLOS Comput Biol2011;7:1–21 .

[7] Hilgetag CC , Kaiser M . Lectures in supercomputational neuroscience: dynamicsin complex brain networks. Berlin: Springer; 2008 .

[8] Eguiluz VM , Chialvo DR , Cecchi GA , Baliki M , Apkarian AV . Scale-free brainfunctional networks. Phys Rev Lett 2005;94:018102 .

[9] Budzinski RC , Boaretto BRR , Rossi KL , Prado TL , Kurths J , Lopes SR . Nonsta-tionary transition to phase synchronization of neural networks induced by the

coupling architecture. Physica A 2018;507:321–34 .

[10] Zhou W , Yang J , Zhou L , Tong D . Stability and synchronization control ofstochastic neural networks. Berlin: Springer; 2015 .

[11] Bassett DS , Bullmore ED . Small-world brain networks. Neuroscientist2006;12:512–23 .

[12] He Y , Chen ZJ , Evans AC . Small-world anatomical networks in the human brainrevealed by cortical thickness from MRI. Cereb Cortex 2007;17:2407–19 .

[13] Stam CJ , Straaten ECWV . The organization of physiological brain networks. Clin

Neurophysiol 2012;123:1067–87 . [14] Budzinski RC , Boaretto BRR , Prado TL , Lopes SR . Detection of nonstationary

transition to synchronized states of a neural network using recurrence anal-yses. Phys Rev E 2017;96:012320 .

[15] Xu K , Maidana JP , Castro S , Orio P . Synchronization transition in neuronal net-works composed of chaotic or non-chaotic oscillators. Sci Rep 2018;8:8370 .

[16] Boaretto BRR , Budzinski RC , Prado TL , Kurths J , Lopes SR . Suppression of

anomalous synchronization and nonstationary behavior of neural network un-der small-world topology. Physica A 2018;497:126–38 .

[17] Prado TL , Lopes SR , Batista CAS , Kurths J , Viana RL . Synchronization ofbursting hodgkin-huxley-type neurons in clustered networks. Phys Rev E

2014;90:032818 . [18] Dhamala M , Jirsa VK , Ding M . Transitions to synchrony in coupled bursting

neurons. Phys Rev Lett 2004;92:028101 .

[19] Dinstein I , Pierce K , Eyler L , Solso S , Malach R , Behrmann M , Courch-esne E . Disrupted neural synchronization in toddlers with autism. Neuron

2011;70:1218–25 . 20] Galvan A , Wichmann T . Pathophysiology of parkinsonism. Clin Neurophysiol

2008;119:1459–74 . [21] Iasemidis LD , Sackellares JC . Review: Chaos theory and epilepsy. Neuroscientist

1996;2:118–26 .

22] Morimoto T , Hayakawa T , Sugie H , Awaya Y , Fukuyama Y . Epileptic seizuresprecipitated by constant light, movement in daily life, and hot water immer-

sion. Epilepsia 1985;26:237–42 . 23] Satishchandra P , Shivaramakrishana A , Kaliaperumal VG , Schoenberg BS .

Hot-water epilepsy: a variant of reflex epilepsy in southern india. Epilepsia1988;29:52–6 .

[24] Brugger F , Erro R , Balint B , Kägi G , Barone P , Bhatia KP . Why is there motor

deterioration in parkinsonâ disease during systemic infections-a hypotheticalview. NPJ Parkinson’s Dis 2015;1:15014 .

25] Boaretto BRR , Budzinski RC , Prado TL , Kurths J , Lopes SR . Neuron dynamicsvariability and anomalous phase synchronization of neural networks. Chaos

2018;28:106304 . 26] Braun HA , Huber MT , Dewald M , Schafer K , Voigt K . Computer simulations of

neuronal signal transduction: the role of nonlinear dynamics and noise. Int JBifurc Chaos 1998;8:881–9 .

[27] Hodgkin AL , Huxley AF . A quantitative description of membrane cur-

rent and itsc application to conduction and excitation in nerve. J Physiol1952;117:500–44 .

28] Schäfer K , Braun HA , Peters RC , Bretschneider F . Periodic firing pattern in af-ferent discharges from electroreceptor organs of catfish. Pflügers Arch Eur J

Physiol 1995;429:378–85 .

42 R.C. Budzinski, B.R.R. Boaretto and T.L. Prado et al. / Chaos, Solitons and Fractals 123 (2019) 35–42

[

[29] Coombes S , Bressloff PC . Bursting: the genesis of rhythm in the nervous sys-tem. World Scientific; 2005 .

[30] Kuramoto Y . Chemical oscillations, waves, and turbulence, 19. Springer Science& Business Media; 2012 .

[31] Marwan N , Romano MC , Thiel M , Kurths J . Recurrence plots for the analysis ofcomplex systems. Phys Rep 2007;438:237–329 .

[32] Feudel U , Neiman A , Pei X , Wojtenek W , Braun HA , Huber M , Moss F . Homo-clinic bifurcation in a hodgkin–huxley model of thermally sensitive neurons.

Chaos 20 0 0;10:231–9 .

[33] Olivares E , Salgado S , Maidana JP , Herrera G , Campos M , Madrid R , Orio P .Trpm8-dependent dynamic response in a mathematical model of cold ther-

moreceptor. PloS one 2015;10:e0139314 . [34] Burek M , Follmann R , Rosa E . Temperature effects on neuronal firing rates and

tonic-to-bursting transitions. Biosystems 2019;180:1–6 . [35] Thompson SM , Masukawa LM , Prince DA . Temperature dependence of intrinsic

membrane properties and synaptic potentials in hippocampal ca1 neurons in

vitro. J Neurosci 1985;5:817–24 . [36] Braun W , Eckhardt B , Braun HA , Huber M . Phase-space structure of a ther-

moreceptor. Phys Rev E 20 0 0;62:6352–60 . [37] Destexhe A , Mainen ZF , Sejnowski TJ . An efficient method for computing

synaptic conductances based on a kinetic model of receptor binding. NeuralComput 1994;6:14–18 .

[38] Newman MEJ , Watts DJ . Scaling and percolation in the small-world networkmodel. Phys Rev E 1999;60:7332–42 .

[39] Cohen SD , Hindmarsh AC , Dubois PF , et al. Cvode, a stiff/nonstiff ode solver inc. Comput Phys 1996;10:138–43 .

[40] Ivanchenko MV , Osipov GV , Shalfeev VD , Kurths J . Phase synchronization inensembles of bursting oscillators. Phys Rev Lett 2004;93:134101 .

[41] Boccaletti S , Kurths J , Osipov G , Valladares D , Zhou C . The synchronization ofchaotic systems. Phys Rep 2002;366:1–101 .

[42] Eckmann JP , Kamphorst SO , Ruelle D . Recurrence plots of dynamical systems.

EPL (Europhys Lett) 1987;4:973 . [43] Prado TL , Lima GZS , Lobão Soares B , Nascimento GCd , Corso G , Fontenele-A-

raujo J , Kurths J , Lopes SR . Optimizing the detection of nonstationary signalsby using recurrence analysis. Chaos 2018;28:085703 .

44] Budzinski RC , Boaretto BRR , Prado TL , Lopes SR . Phase synchronization andintermittent behavior in healthy and alzheimer-affected human-brain-based

neural network. Phys Rev E 2019;99:022402 .

[45] Xie Y , Gong Y , Hao Y , Ma X . Synchronization transitions on com-plex thermo-sensitive neuron networks with time delays. Biophys Chem

2010;146:126–32 . [46] Mormann F , Lehnertz K , David P , Elger CE . Mean phase coherence as a measure

for phase synchronization and its application to the eeg of epilepsy patients.Physica D 20 0 0;144:358–69 .