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37
Нелинейная
^^^^^^^^ динамика и нейронаука
Известия высших учебных заведений. Прикладная нелинейная динамика. 2021. Т. 29, № 3 Izvestiya Vysshikh Uchebnykh Zavedeniy. Applied Nonlinear Dynamics. 2021;29(3)
Article
DOI: 10.18500/0869-6632-2021-29-3-440-448
Calcium concentration in astrocytes: Emergence of complicated spontaneous oscillations and their cessation
M.S. Sinitsinaim, S. Yu. Gordleeva1'2, V.B. Kazantsev1'2'^, E. V. Pankratova1
1Lobachevsky State University of Nizhni Novgorod, Russia 2Innopolis University, Russia 3 Samara State Medical University, Russia E-mail: [email protected], [email protected], [email protected], [email protected] Received 17.11.2020, accepted 27.01.2021, published 31.05.2021
Abstract. The purpose of this work is to show the mechanisms of transitions between different dynamic modes of spontaneous astrocytic calcium activity. With this aim, dynamics of recently introduced Lavrentovich-Hemkin mathematical model was examined by both analytical and numerical techniques. Methods. In order to obtain the conditions for the oscillations cessation, the linear stability analysis for the equilibrium point was carried out. Complicated dynamics was studied numerically by calculations of time traces and bifurcation diagrams. Results. The mechanisms of oscillatory mode development with the increase of the maximal calcium flux out of the SERCA pump in the presence of low and high level of extracellular calcium concentration were demonstrated. It was shown that emergence of oscillations occurs via supercritical Andronov-Hopf bifurcation, and the properties of the oscillatory mode with further increase of the maximal calcium flux out of the SERCA pump are highly dependent on the value of extracellular calcium concentration. Notably, emergence of chaotic spontaneous calcium oscillations for specific level of calcium ions outside the cell was revealed. Conclusion. Based on the analysis of various dynamical modes of spontaneous astrocytic chemical activity, the peculiarities in astrocyte-neuron interaction in complex multicellular systems can be further investigated.
Keywords: mathematical modeling, calcium concentration in astrocytes, oscillatory mode, stationary mode.
Acknowledgements. This work was supported by grant of the President of the Russian Federation for state support of leading scientific schools No. NSh-2653.2020.2. SG work was supported by the RFBR grants No. 20-32-70081, 18-29-10068. This study was supported by the Ministry of Science and Higher Education of the Russian Federation (project No. 0729-2020-0061).
For citation: Sinitsina MS, Gordleeva SYu, Kazantsev VB, Pankratova EV. Calcium concentration in astrocytes: Emergence of complicated spontaneous oscillations and their cessation. Izvestiya VUZ. Applied Nonlinear Dynamics. 2021;29(3):440-448. DOI: 10.18500/0869-6632-2021-29-3-440-448
This is an open access article distributed under the terms of Creative Commons Attribution License (CC-BY 4.0).
440
© Синицина М.С., Гордлеева С. Ю., Казанцев В. Б., Панкратова Е. В., 2021
Научная статья УДК 530.182
DOI: 10.18500/0869-6632-2021-29-3-440-448
Концентрация кальция в астроцитах: возникновение сложных спонтанных колебаний и их затухание
М. С. Синицина1И, С.Ю. Гордлеева 1'2, В. Б. Казанцев1'2'3, Е.В. Панкратова1
1 Нижегородский государственный университет им. Н. И. Лобачевского, Россия 2Университет Иннополис, Россия 3Самарский государственный медицинский университет, Россия E-mail: [email protected], [email protected], [email protected], [email protected] Поступила в редакцию 17.11.2020, принята к публикации 27.01.2021, опубликована 31.05.2021
Аннотация. Цель данной работы - показать механизмы переходов между различными динамическими режимами спонтанной кальциевой активности астроцитов. С этой целью динамика недавно представленной математической модели Лаврентовича-Хемкина была исследована как аналитическими, так и численными методами. Методы. Для получения условий прекращения колебаний был проведен линейный анализ устойчивости точки равновесия. Сложная динамика изучалась численно в рамках построения реализаций и бифуркационных диаграмм. Результаты. Продемонстрированы механизмы развития колебательных режимов с увеличением максимального потока кальция через SERCA-насос при низком и высоком уровне концентрации внеклеточного кальция. Было показано, что возникновение колебаний происходит в результате суперкритической бифуркации Андронова-Хопфа, а свойства колебательного режима при дальнейшем увеличении максимального потока кальция через SERCA-насос сильно зависят от величины концентрации внеклеточного кальция. В частности, показано, что при определенном уровне концентрации ионов кальция снаружи клетки наблюдается возникновение хаотических спонтанных кальциевых колебаний. Заключение. На основе анализа различных динамических режимов спонтанной химической активности астроцитов возможно дальнейшее изучение особенностей взаимодействия астроцитов и нейронов в сложных многоклеточных системах.
Ключевые слова: математическое моделирование, концентрация кальция в астроцитах, колебательный режим, стационарный режим.
Благодарности. Работа выполнена при поддержке гранта Президента Российской Федерации на государственную поддержку ведущих научных школ № НШ-2653.2020.2. Работа С.Г. выполнена при поддержке гранта РФФИ № 20-32-70081, 18-29-10068. Работа поддержана Министерством науки и высшего образования Российской Федерации (проект № 0729-2020-0061).
Для цитирования: Синицина М. С., Гордлеева С.Ю., Казанцев В. Б., Панкратова Е.В. Концентрация кальция в астроцитах: возникновение сложных спонтанных колебаний и их затухание // Известия вузов. ПНД. 2021. T. 29, № 3. С. 440-448. DOI: 10.18500/0869-6632-2021-29-3-440-448
Статья опубликована на условиях лицензии Creative Commons Attribution License (CC-BY 4.0).
Introduction
Neural networks are highly complicated collection of various interconnected structural units. Few decades ago, it was believed that only neurons define the information processing in the brain. Such representation of real neural networks allowed to obtain a lot of interesting and important results concerning the coherence and synchronization of neural elements [1-8]. At the same time, recent studies shown that a lot of factors can significantly modify the neuronal response [9-14]. Particularly, various mathematical models for neuron-astrocyte interaction were introduced and examined [15-19].
Astrocytes are glial cells of the central nervous system that are capable to generate impulses of chemical activity. This kind of chemical activity was experimentally detected by various methods
of fluorescence and confocal laser microscopy, that allowed revealing the changes in concentration of free cytosolic calcium in course of time. Extensive experimental study shown that astrocytes are characterized by both spontaneous changes in calcium concentration [10,20,21], and calcium signals caused by neuronal activity [22-24]. These astrocytic calcium concentration changes can be localized in synapses or spread along the astrocytic network. Peculiarity of such chemical signal is in the presence of oscillations with different amplitude in calcium concentration: the so-called blips and puffs [25]. Blips are short and weak peaks that correspond to the opening of one IP3R channel (or one tetramer of an IP3R channel), while the puffs are longer and higher peaks resulting from the coordinated opening of a group of neighboring IP3R channels (or their tetramers) through the calcium-induced calcium release principle (CICR). Recently introduced Lavrentovich-Hemkin mathematical model [26] gives clear description for these peculiarities of astrocytic response. This model allows taking into account various physiological parameters: the flow of calcium from the extracellular space into the astrocyte's cytosol, IP3R-mediated flow of calcium from the endoplasmic reticulum (ER) to the cytosol and so on. Since the appearance of various spatiotemporal changes in calcium concentration is associated with various network functions, the analysis of the dynamic modes observed with the change of parameters in corresponding mathematical model is of particular significance.
In this work, the role of the maximal calcium flux out of the SERCA pump* in changes of spontaneous chemical activity in astrocytes is examined.
1. Description of the mathematical model
In accordance with Lavrentovich-Hemkin model [26], calcium concentration changes in the cytosol of astrocytes and in its endoplasmic reticulum, and Ca2+-dependent dynamics of inositol-1,4,5-triphosphate (IP3) concentration are governed by the following equations:
d[Cad2+]cyt = Jin - fcout[Ca2+]cyt + JcicR - Jserca + kf ([Ca2+]ER - [Ca2+]cyt),
= Jserca - Jcicr + kf ([Ca2+]cyt - [Ca2+]ER), (1)
d[Ca2+]ER — T T------ ^ . _ ГП.2+
dt
[IPajcyt
dt = JPLC - fcdeg([IP3]cyt),
where the expressions for Jserca, JCICR and JPLC are:
[Ca2+]2
J _ ( [Ca2+]2yt \
Jserca _VM \ [Ca2+]2yt + k2j '
[Ca2+]c2yt +
Jo,or_4vm■((^2+]»^^]^J([iP^+W([Ca2+] (2)
( [Ca2+] cyt
[Ca 2
Jplo _ V.
\([Ca2+]c2yt + kl)f
Here, we consider the following set of parameters: vm3 = 40 s-1, vp = 0.05 ^M/s, ^ = 0.1 p,M,
fc0aA _ 0.15 цМ, fc0ai _ 0.15 цМ, fcip3 _ 0.1 цМ, kp _ 0.3 цМ, kdeg _ 0.08 s 1, kout _ 0.5 s 1
*The sarco(endo)plasmic reticulum calcium transport ATPase (SERCA) is a pump that transports calcium ions from the cytoplasm into the ER [27,28].
kf = 0.5 s-1, n = 2.02, m = 2.2. In [26,29], for these parameters and vm2 = 15 [xM/s, the existence of chaotic attractor was shown. Here, we focus on role of the parameter vm2 in both emergence of irregular calcium activity and cessation of spontaneous calcium oscillations in the presence of various level of extracellular calcium Jin.
2. Results
0.7
0.6
гм
си
О 0.5
С
Е 0.4
и + 0.3
гм
ГО
и 0.2
X
го
Е 0.1
0.0
J п = 0.0:
/ Jin = 0.05
/ Jir = 0.04
/
2.1. Emergence of periodic oscillations of calcium concentration. In order to obtain the threshold for the oscillations birth, we calculate the difference between the maximal and minimal values of [Ca2+]cyt with the change of parameter vm2. Fig. 1 shows the curves obtained for three levels of extracellular calcium. For Jin = 0.03 [xM/s, slowly growing dependence for the amplitude of oscillations is observed. The increase of Jin leads to the shift of the "quiescent-oscillatory" threshold: for larger values of Jin, transition to oscillatory mode for larger values of vm2 occurs. Moreover, for Jin = 0.04 [xM/s, the growth of the amplitude is much faster then in the previous case. With further increase of Jin, sudden jump in amplitude is observed. It should be noted that, for all considered cases, transition to oscillatory mode occurs through the Andronov-Hopf bifurcation. To show this, the linear stability
analysis for the equilibrium point was carried out. Namely, the roots of the characteristic equation were calculated. The results obtained for Jin = 0.03 [xM/s and Jin = 0.05 [M/s are presented in Fig. 2, a and Fig. 2, b, respectively. As seen from these figures, for Jin = 0.03 [xM/s, the real part of the complex root (red curve in the figure) becomes equal to zero for vm2 ~ 2.5 [xM/s while for Jin = 0.05 [xM/s the Andronov-Hopf bifurcation is observed for vm2 ~ 11 [M/s.
0.0
2.5
5.0
12.5 15.0
7.5 10.0
Vm, jjM/s
Fig. 1. Difference between the maximal and minimal values of [Ca2+]cyt oscillations with the change of parameter vm2 for three values of Jin Jin = 0.05 p,M/s - red, Jin = 0.04 p,M/s -green and Jin = 0.03 ^M/s - blue curves. Zero value of the difference corresponds to the case of stable equilibrium state
2.0
1.5
1.0
0.5
0.0
-0.5
Im s*
"Re
а
6 8 10 Vm, JxM/s
12
14
3 2 1
c^ 0 -1
-2
16
Ir ri
/ке
9 10 11 12 L3 14
b
Vm, (iM/s
Fig. 2. Roots of the characteristic equation for two values of Jin: Jin = 0.03 ^M/s (a) and Jin = 0.05 ^M/s (b). Maximal values of the real roots are shown by blue color, red and green curves correspond to change of real and imaginary parts of complex roots, respectively
2.2. Emergence of irregular oscillations of calcium concentration. To examine the changes in oscillatory dynamics of the calcium concentration in the cytosol, we depict the values of its local maxima on diagram with the change of vM2 for two values of Jin. In Fig. 3, a, the diagram obtained for Jin = 0.03 ^M/s is shown. These data demonstrate the appearance of additional branch for vm2 > 11.7 ^M/s. This branch is due to the one-blip (weak peak after the high main peak) emergence in calcium concentration output. To show this, in Fig. 3, b and Fig. 3, c, the time traces for [Ca2+]cyt when vM2 = 10 ^M/s and vm2 = 20 ^M/s are presented. As seen from these traces, the increase of vM2 leads to both the increase of puff's amplitude and the increase of the interpuffs intervals, i.e. the time between subsequent high peaks of calcium concentration.
Similar calculation of [Ca2+]cyt local maxima for Jin = 0.04 ^M/s (not shown) demonstrates the successive transition from one-blip oscillations to two-, three-, two-, and again one-blip oscillatory mode with the increase of vM2. As for Jin = 0.03 ^M/s, the amplitude of puffs smoothly increases with the increase of the maximal flux of the calcium ions from the cytosol to ER of the cell.
The bifurcation diagram obtained for higher level of extracellular calcium, Jin = 0.05 ^M/s, looks differently. Fig. 4, a shows that oscillatory mode with high peak of calcium concentration (puff) appears suddenly for vM2 « 14.82 ^M/s. To show this, in Fig. 4, b and Fig. 4, c, the time traces for [Ca2+]cyt when vM2 = 14.8177 ^M/s and vM2 = 14.8179 ^M/s are presented. It is seen that both these oscillatory modes are chaotic. It can be shown that emergence of small-amplitude chaos occurs via a period doubling cascade with the increase of vM2. For vM2 > 15 ^M/s, successive transition from the regime with large number of blips (six-blips to five-blips etc.) to one-blip oscillatory mode can be observed.
Finally note that, for higher levels of extracellular calcium, the chaotic dynamics of cytosolic calcium concentration is not observed.
1 -,
0.8 -
5-0.6 -
x
™ 0.2 -
10
20
Vm, (iM/s
0.6 —1 "0.40.2 -0
5000
I0'8! =¡-0.6-
i_& 0.4 -
i 0.2-^ 0-
5000
5200
5400
5600
5800
5200
5400
5600
5800
/, S
t, s
Fig. 3. a - Bifurcation diagram obtained for maximal values of [Ca2+]cyt with the change of vm2 for Jm = 0.03 ^M/s. Time traces for [Ca2+]cyt(i) when vm2 = 10 ^M/s and vm2 = 20 ^M/s are shown in (b) and (c), respectively
1 -,
Vm, цМ/s
и
0.24 -0.20 -:0.16-0.12 ■ 0.08 -0.04 -0 -
5000
5 0,8 3-0.6-
r-i 0.4-+
To 0.2 H
U " 0
5000
5200
5200
5400
5600
5800
f, S
5400
5600
5800
f, S
Fig. 4. а -part of the in (b) and
Bifurcation diagrams obtained for maximal values of [Ca2+]cyt with the change of vm2. In the inset, the enlarged diagram is shown. Time traces for [Ca2+]cyt(i) when vm2 = 14.8177 ^M/s and vm2 = 14.8179 ^M/s are shown (c), respectively. Jin = 0.05 ^M/s
b
a
с
b
a
с
Conclusion
In this work, the dynamics of the astrocytic calcium concentration have been studied within the framework of the Lavrentovich-Hemkin model. The impact of the maximal flux of the calcium ions from the cytosol to endoplasmic reticulum of the cell has been analysed. The peculiarities of transition to the quiescent mode have been studied. Emergence of chaotic spontaneous calcium oscillations for specific level of calcium ions outside the cell has been revealed.
In [26], it was shown that chaos is possible in dynamics of spontaneous calcium activity. In this study, we have demonstrated that even small variation in the maximal calcium flux out of the SERCA can lead to stabilization of the spontaneous calcium dynamics: either small-amplitude periodic oscillations (for less values of vm2) or periodic oscillations with puffs (for larger values of vm2) are observed.
Since the changes in calcium concentration are able to modify the neuronal response, the conditions for emergence of spontaneous calcium oscillations, that can be chaotic, particularly, is of significance. Based on the analysis of various dynamical modes of spontaneous astrocytic chemical activity, the peculiarities in astrocyte-neuron interaction in complex multicellular systems can be further investigated. Nowadays biophysical investigations of the subcellular mechanisms of calcium signals emergence in astrocytes is required due to recently identified roles of astrocytic signalling in synaptic, neural network, and memory functions [30,31]. Understanding the complex dynamic mechanisms of intracellular Ca2+ activity has remained a major challenge and will open a new therapeutic opportunities to fight against pathological and aging-induced impairments [32-34].
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Синицина Мария Сергеевна - родилась в Нижнем Новгороде (1999). В настоящее время проходит обучение по направлению «Фундаментальная информатика и информационные технологии» Института информационных технологий, математики и механики Нижегородского государственного университета им. Н. И. Лобачевского. Победитель конкурса проектов Летней школы Intel по компьютерному зрению «Computer Vision Summer Camp». Научные интересы: математическое моделирование, нелинейная динамика, хаос. По результатам научных исследований опубликованы 3 работы в сборниках материалов международных конференций.
Россия, 603950, Нижний Новгород, пр. Гагарина, 23 Институт информационных технологий, математики и механики, Нижегородский государственный университет им. Н. И. Лобачевского E-mail: [email protected] ORCID: 0000-0002-0687-0598
Гордлеева Сусанна Юрьевна - кандидат физико-математических наук, доцент кафедры нейротехнологий ННГУ им Н. И. Лобачевского. Родилась в 1987 году. Выпускница кафедры общей физики радиофизического факультета ННГУ. В 2015 году защитила кандидатскую диссертацию по теме «Эффекты мультистабильной динамики в системах взаимодействующих биологических осцилляторов». Область научных интересов: нейронаука, биофизика, нелинейная динамика, математические модели нейронов и нейрон-астроцитарных сетей, анализ ЭЭГ, нейроинтерфейсы.
Россия, 603950, Нижний Новгород, пр. Гагарина, 23 Институт биологии и биомедицины,
Нижегородский государственный университет им. Н. И. Лобачевского Россия, 420500 Иннополис, Университетская, 1
Центр технологий компонентов робототехники и мехатроники, Университет Иннополис E-mail: [email protected] ORCID: 0000-0002-7687-3065
Казанцев Виктор Борисович - доктор физико-математических наук, доцент. Родился в 1973 году. Выпускник кафедры теории колебаний и автоматического регулирования радиофизического факультета ННГУ, основанной академиком А. А. Андроновым. В 1999 году защитил кандидатскую диссертацию по теме «Структуры, волны и их взаимодействие в многослойных активных решетках». В 2005 году стал доктором физико-математических наук, защитив докторскую диссертацию «Кооперативные эффекты нелинейной динамики активных многоэлементных систем: структуры, волны, хаос, управление». Область научных интересов: нейронаука, математические модели нейронов и нейронных сетей, нейроги-бридные и нейроморфные системы, нейроинтерфейсы, нейрон-глиальные взаимодействия, колебания и волны в нейродинамике. С 2005 года В.Б. Казанцев работает заведующим кафедрой нейротехнологий биологического факультета ННГУ им. Н.И. Лобачевского (Нижний Новгород), в 2020 году получил позицию профессора университета Иннополис (Казань), а также возглавляет лабораторию нейромоделирования НИИ Нейронаук СамГМУ (Самара). В.Б. Казанцев является соавтором более 150 научных статей в российских и зарубежных реферируемых изданиях, нескольких глав в книгах и монографиях, множества патентов и учебно-методических разработок. Руководитель ведущей научной школы «Нелинейная динамика сетевых нейросистем: фундаментальные аспекты и приложения» в рамках гранта Президента РФ 2020-2021. Под руководством В.Б. Казанцева защитилось 6 кандидатов наук.
Россия, 603950, Нижний Новгород, пр. Гагарина, 23 Институт биологии и биомедицины,
Нижегородский государственный университет им. Н. И. Лобачевского Россия, 420500 Иннополис, Университетская, 1
Центр технологий компонентов робототехники и мехатроники, Университет Иннополис Россия, 443079 Российская Федерация, Самара, ул. Гагарина, 18 Самарский государственный медицинский университет E-mail: [email protected]
Панкратова Евгения Валерьевна - родилась в Нижнем Новгороде (1981). В 2004 году окончила радиофизический факультет Нижегородского государственного университета им. Н. И. Лобачевского. В 2008 году защитила диссертацию на соискание ученой степени кандидата физико-математических наук на тему «Синхронизация регулярных и хаотических колебаний в нейродинамических системах» по специальности «Радиофизика». С 2015 года работает на кафедре прикладной математики Института информационных технологий, математики и механики ННГУ в должности доцента. Научные интересы: математическое моделирование, нелинейная динамика, синхронизация, стохастические процессы. Автор более 30 научных работ в ведущих отечественных и зарубежных журналах, а также сборниках материалов международных конференций.
Россия, 603950, Нижний Новгород, пр. Гагарина, 23 Институт информационных технологий, математики и механики, Нижегородский государственный университет им. Н. И. Лобачевского E-mail: [email protected]
