Synaptic transmission as a cooperative phenomenon in confined systems
In this review paper, the theory of synaptic transmission (ST) was developed and discussed. We used the hypothesis of isomorphism between: (a) the cooperative behavior of mediators — acetylcholine molecules (ACh) and cholinoreceptors in a synaptic cleft with binding into mediator-receptor (AChR) co...
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irk-123456789-1569682019-06-20T01:25:30Z Synaptic transmission as a cooperative phenomenon in confined systems Chalyi, A.V. Vasilev, A.N. Zaitseva, E.V. In this review paper, the theory of synaptic transmission (ST) was developed and discussed. We used the hypothesis of isomorphism between: (a) the cooperative behavior of mediators — acetylcholine molecules (ACh) and cholinoreceptors in a synaptic cleft with binding into mediator-receptor (AChR) complexes, (b) the critical phenomena in confined binary liquid mixtures. The systems of two (or three) nonlinear differential equations were proposed to find the change of concentrations of ACh, AChR complexes, and ferment acetylcholinesterase. The main findings of our study: the linear size of the activation zone was evaluated; the process of postsynaptic membrane activation was described as a cooperative process; different approximations of ACh synchronous release were examined; stationary states and types of singular points were studied for the proposed models of ST; the nonlinear kinetic model with three order parameters demonstrated a strange-attractor behavior. У цьому оглядi була розроблена i обговорена теорiя синаптичної передачi (СП). Ми застосували гiпотезу iзоморфiзму мiж: (а) кооперативною поведiнкою медiаторiв — молекул ацетилхолiну (ACh) i холiнорецепторiв в синаптичнiй щiлинi з утворенням медiатор-рецепторних (AChR) комплексiв, (б) критичними явищами в просторово обмежених бiнарних рiдких сумiшах. Були запропонованi системи двох (або трьох) нелiнiйних диференцiальних рiвнянь для опису змiни концентрацiй ACh, комплексiв AChR i ферменту ацетилхолiнестерази. Основнi результати нашого дослiдження: оцiнено лiнiйний розмiр зони активацiї; описано процес активацiї постсинаптичної мембрани як кооперативний процес; розглянуто рiзнi апроксимацiйнi функцiї синхронного вивiльнення ACh; дослiджено стацiонарнi стани i типи особливих точок для запропонованих моделей СП; показано, що нелiнiйна кiнетична модель з трьома параметрами порядку демонструє поведiнку дивного атрактора. 2017 Article Synaptic transmission as a cooperative phenomenon in confined systems / A.V. Chalyi, A.N. Vasilev, E.V. Zaitseva // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13804: 1–12. — Бібліогр.: 44 назв. — англ. 1607-324X PACS: 82.40.Ck, 87.10.Ed DOI:10.5488/CMP.20.13804 arXiv:1703.10410 http://dspace.nbuv.gov.ua/handle/123456789/156968 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
In this review paper, the theory of synaptic transmission (ST) was developed and discussed. We used the hypothesis of isomorphism between: (a) the cooperative behavior of mediators — acetylcholine molecules (ACh)
and cholinoreceptors in a synaptic cleft with binding into mediator-receptor (AChR) complexes, (b) the critical
phenomena in confined binary liquid mixtures. The systems of two (or three) nonlinear differential equations
were proposed to find the change of concentrations of ACh, AChR complexes, and ferment acetylcholinesterase.
The main findings of our study: the linear size of the activation zone was evaluated; the process of postsynaptic
membrane activation was described as a cooperative process; different approximations of ACh synchronous
release were examined; stationary states and types of singular points were studied for the proposed models of
ST; the nonlinear kinetic model with three order parameters demonstrated a strange-attractor behavior. |
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Article |
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Chalyi, A.V. Vasilev, A.N. Zaitseva, E.V. |
| spellingShingle |
Chalyi, A.V. Vasilev, A.N. Zaitseva, E.V. Synaptic transmission as a cooperative phenomenon in confined systems Condensed Matter Physics |
| author_facet |
Chalyi, A.V. Vasilev, A.N. Zaitseva, E.V. |
| author_sort |
Chalyi, A.V. |
| title |
Synaptic transmission as a cooperative phenomenon in confined systems |
| title_short |
Synaptic transmission as a cooperative phenomenon in confined systems |
| title_full |
Synaptic transmission as a cooperative phenomenon in confined systems |
| title_fullStr |
Synaptic transmission as a cooperative phenomenon in confined systems |
| title_full_unstemmed |
Synaptic transmission as a cooperative phenomenon in confined systems |
| title_sort |
synaptic transmission as a cooperative phenomenon in confined systems |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/156968 |
| citation_txt |
Synaptic transmission as a cooperative phenomenon in confined systems / A.V. Chalyi, A.N. Vasilev, E.V. Zaitseva // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13804: 1–12. — Бібліогр.: 44 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT chalyiav synaptictransmissionasacooperativephenomenoninconfinedsystems AT vasilevan synaptictransmissionasacooperativephenomenoninconfinedsystems AT zaitsevaev synaptictransmissionasacooperativephenomenoninconfinedsystems |
| first_indexed |
2025-07-14T09:19:41Z |
| last_indexed |
2025-07-14T09:19:41Z |
| _version_ |
1837613482979098624 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 1, 13804: 1–12
DOI: 10.5488/CMP.20.13804
http://www.icmp.lviv.ua/journal
Synaptic transmission as a cooperative phenomenon
in confined systems
A.V. Chalyi1∗, A.N. Vasilev2†, E.V. Zaitseva1
1 Department of Medical and Biological Physics, Bogomolets National Medical University,
13 Shevchenko Blvd., 01601 Kyiv, Ukraine
2 Department of Theoretical Physics, Taras Shenchenko Kyiv National University,
6 Acad. Glushkov Ave., 03022 Kyiv, Ukraine
Received January 29, 2017, in final form March 12, 2017
In this review paper, the theory of synaptic transmission (ST) was developed and discussed. We used the hy-
pothesis of isomorphism between: (a) the cooperative behavior of mediators — acetylcholine molecules (ACh)
and cholinoreceptors in a synaptic cleft with binding into mediator-receptor (AChR) complexes, (b) the critical
phenomena in confined binary liquid mixtures. The systems of two (or three) nonlinear differential equations
were proposed to find the change of concentrations of ACh, AChR complexes, and ferment acetylcholinesterase.
The main findings of our study: the linear size of the activation zone was evaluated; the process of postsynaptic
membrane activation was described as a cooperative process; different approximations of ACh synchronous
release were examined; stationary states and types of singular points were studied for the proposed models of
ST; the nonlinear kinetic model with three order parameters demonstrated a strange-attractor behavior.
Key words: cell-to-cell communication, synaptic transmission, phase transition, binary liquid mixture, critical
mixing point, finite-size (confined) systems
PACS: 82.40.Ck, 87.10.Ed
“Biophysics is a Tower of Babel”
P.G. de Gennes
The Nobel Laureate in Physics (1991)
1. Introduction
The problems of evolution of living (organic) and non-living (inorganic) nature still attract a great
interest and agitate inquisitive human mind. Among these problems, synaptic transmission (ST) and cell-
to-cell communication play an absolutely outstanding role. Actually, cell-to-cell communication is of the
same fundamental importance in animate nature as intermolecular interaction in non-living nature. It
makes it possible to understand the basic principles of a decisive problem, i.e., how our brain works and,
in particular, how our nonlinear brain system organizes its thinking function [1–4].
The understanding of the processes of cell origin (their typical sizes are about 10−6 m) came from the
study of evolution processes in the living nature at distances intermediate relative to the farthest bound-
ary of the Universe, i.e., more than 10 billion light years or 1025 m, and at the smallest distances inside the
atomic nucleus, i.e., less than 10−15 m. The cells emerged due to the creation of plasmatic membranes,
i.e., the process concerned with the so-called self-assembly of amphiphilic molecules of phospholipids in
aquatic environment, which was absolutely determined (see appendix B).
Generation and propagation of electric potentials is the most important phenomenon in living cells
and tissues based on excitation of cells, regulation of intracellular processes, muscular contraction, and
∗E-mail: avchal@nmu.ua
†E-mail: vasilev@univ.kiev.ua
© A.V. Chalyi, A.N. Vasilev, E.V. Zaitseva, 2017 13804-1
https://doi.org/10.5488/CMP.20.13804
http://www.icmp.lviv.ua/journal
A.V. Chalyi, A.N. Vasilev, E.V. Zaitseva
nerve system functioning. The origin of electric membrane potentials of two types (in particular, the rest-
ing potential and the action potential) may be explained by a specific feature of distribution and diffusion
of ions considering their permeability through a membrane [5–10].
In 1949–1952 Hodgkin and Huxley [1] experimentally and theoretically studied the generation and
propagation of the action potential. In 1963 they were awarded, together with Eccles, a Nobel Prize in
medicine or physiology “. . . for their discoveries concerning the ionic mechanisms involved in excitation
and inhibition in the peripheric and central portions of the cell membrane”.
The problems of ST and cell-to-cell communication have actually two important aspects: electrical and
chemical. The electrical intercellular interaction aspect is associated with the process of the action poten-
tial propagation along the nerve fibers (axons). The chemical aspect is connected with chemical reactions
between neurotransmitters and receptors in synaptic clefts (SC), or synapses separating dendrites of one
neuron from other neurons, and in neuro-muscular junctions [1–4].
It is known that the body (soma) of a nervous cell (neuron), similarly to any other cells, consists
of mitochondria, ribosomes and many other organelles: nucleus with its genetic information; dendrites
emitting synaptic information from their own soma and receiving this information from other nervous
cells; axons transmitting information to dendrites of other neurons.
Consider some important characteristics and properties of the cell-to-cell communication and the ST
[1–11]: 1) size of a human nervous cell (neuron) — from 10 to 100 µm (the largest size is typical of Betz
cells— giant pyramidal cells discovered in 1874 by a Ukrainian anatomist Volodymyr Betz, who worked
at the medical faculty of St. Volodymyr Kyiv University [11]); 2) size of SC (thickness of synapse) — from
10 to 200 nm; 3) diameter of ion channels in the membrane — up to 10 nm; 4) axon length — up to 1 m;
width (diameter) of axons — up to 0.5−1 mm for a giant squid axon; 5) the number of neurons in a
human brain— about 1011 (a more precise number is 86 billions); 6) the number of contacts (synapses)
between dendrites of one neuron with the other neurons — up to 1.5 · 104 ; 7) the number of synapses
in a human brain — about 1014−1015; 8) time of a signal exchange — about 10−3 s; 9) the number of
channels in the membrane surface— about 102−103 per µm2; 10) the number of impulses generated by
a neuron per 1 second — from 1 to 103; 11) the number of neurotransmitters, say, acetylcholine (ACh)
molecules contained in 1 vesicle— about 104; 12) the number of ACh molecules released simultaneously
from the axonal presynaptic terminal with approximately 103 vesicles, under the action of one nerve
impulse (action potential)— about 107.
In this contribution, we propose a new theoretical approach to the problem of ST, based on the non-
linear kinetic models and the hypothesis of isomorphism between (a) the cooperative behavior of a
neurotransmitter-receptor system in the synaptic cleft (SC), (b) the critical phenomena in confined bi-
nary mixture near its critical mixing point. This hypothesis was originally proposed in 1989 and was
published in [12]. Further, it was developed in the lecture “Critical phenomena in finite-size systems and
synaptic transmission”, presented in 1991 at the ARWNATO “Dynamic phenomena at interfaces, surfaces
and membranes” and was published in the Les Houches Series in [13]. The preface to this book was writ-
ten by the Nobel Laureate P.G. de Gennes (1991), with its first phrase being used here as an epigraph.
As an application of the above-mentioned hypothesis to cholinergic synapses, the systems of two (or
three) nonlinear differential equations were proposed to examine the change of order parameters such
as concentrations of ACh, acetylcholine-receptor (R) complexes (AChR), and acetylcholinesterase (AChE),
being a ferment which removes ACh from the R-location area. The main findings of our study were as
follows: the linear size of the activation zone was evaluated as the correlation length (CL) for a mediator-
receptor mixture (section 2); the process of the postsynaptic membrane activation was described as a
cooperative process (section 3); the nonlinear kinetic model of ST with three order parameters ACh, AChR
and AChE demonstrated a strange-attractor behavior, different approximations for the function of ACh
synchronous release were examined, the stationary states and types of singular points were studied for
the proposed models of ST (section 4).
2. Model of ST as phase transition in “transmitter-receptor” system
The chemical reactions near the bifurcation (critical) points are of a great importance for the ST
mechanisms of nerve impulses. The phenomena occurring near the points (lines) of phase transitions
13804-2
Synaptic transmission as a cooperative phenomenon in confined systems
or boundaries of stability demonstrate a universal behavior in physical, chemical, biological and other
systems [14–32]. The reason is related to cooperative (synergetic) nature of such events, defined by in-
teraction of fluctuations of the characteristic order parameters, their correlations at large spatial and
temporal intervals.
To understand this reason more deeply, let us consider the analogy between the equations of motion
for 1) systems described by the fluctuation Hamiltonians with two interacting order parameters [14–
23, 26–28], and 2) nonlinear reaction-diffusion kinetic models [6, 13, 19, 33–36]. To study such systems
undergoing phase transitions and critical phenomena, one usually uses the fluctuation Hamiltonian [14–
19]
H =
∑
i
HLG[ϕi ]+Hint[ϕ1,ϕ2]. (2.1)
Here,
HLG[ϕi ]=
∫[
1
2
aiϕ
2
i +
1
4
biϕ
4
i +
1
2
ci (∇∇∇ϕi )2
]
dV (2.2)
is the Landau-Ginzburg Hamiltonian, while the second term in (2.1) corresponds to the Hamiltonian of
interaction between two order parameters ϕ1
i
and ϕ2
i
. Taking into account Hamiltonians (2.1) and (2.2),
one can write kinetic equations, or equations of motion for order parameters
∂ϕi
∂t
=−Γi
δH
δϕi
=−Γi
(
aiϕ+biϕ
3
i −ci∆ϕi +
δHint
δϕi
)
, (2.3)
where Γi are the Onsager coefficients. Actually, such equations (2.3) resemble the kinetic equations of
reaction-diffusion kinetic models containing both the diffusion term, being proportional to ∆ϕi with the
diffusion coefficient Di = Γi ci , and the non-linearity, connected with theϕ3
i
and corresponding nonlinear
terms in the functional derivative δHint/δϕi .
It is known that in the vicinity of the critical state, Hamiltonian-like systems demonstrate a universal
behavior in the sense that their characteristics do not depend on the particular form of a short-range po-
tential of interparticle interaction. And in the critical state, these systems demonstrate long-range effects
that also do not depend on the particular form of interpartical potential. Concerning the ST problem, our
principal position, which is based on experimental data, is that the main characteristics of the system
do not qualitatively depend on the parameters of the local interaction, and the system demonstrates a
cooperative long-range behavior at the regime of its ordinal functioning that could be interpreted as a
critical-like state (see additional considerations in the end of section 3).
Convenient theories of cell-to-cell communication and ST are commonly based on the ideas regarding
the chemical intermediates (transmitter agents) securing interaction between two neurons in the synap-
tic cleft or between the motor neuron and muscle fibre in the neuro-muscular junction.
Here, we shall consider the cholinergic synapse and the corresponding neurotransmitter — ACh. The
sequence of major events in cholinergic synapse are as follows: ACh is synthesized by cholineacetyltrans-
ferase and stored in spheroid vesicles in the presynaptic membrane, then ACh molecules are released
and, after their diffusion through the synaptic cleft, they react with R— specific ACh receptors. The bind-
ing ACh and R∗ into ACh–R∗ complexes produces conformal changes in the postsynaptic membrane and,
therefore, the change in a membrane electric potential. Finally, ACh is either inactivated by AChE or is
removed by diffusion.
Such a scheme of chemical reactions corresponds to the processes in the cholinergic synapse:
ACh+R
k1⇐⇒ACh−R∗ k3=⇒ P+R∗ k2=⇒R . (2.4)
Here, R and R∗ are receptors in non-active and active states, respectively; P is a product of ACh destruction
by the action of AChE.
The kinetic equations describing, in accordance with (2.4), the temporal evolution of the receptor
concentration R∗ in the active state (variable x) and the concentration of ACh–R∗ complexes (variable y),
can be written in the following form [12]:
dx
dt
= k3 y −k2x, (2.5)
13804-3
A.V. Chalyi, A.N. Vasilev, E.V. Zaitseva
dy
dt
= k1[ACh]([R]0 − y)− (k1 +k3)y, (2.6)
where [ACh] is concentration of ACh molecules, [R]0 is initial concentration of receptors in non-active
state, ki are coefficients of reactions velocities.
It must be stressed that the process of ACh release from vesicles in the presynaptic membrane into
the synaptic cleft is a cooperative phenomenon: about 107 ACh molecules are simultaneously released
under the action of one nerve impulse (spike). Such a synchronous activation of a large zone of receptors
by ACh molecules can be treated in detail as the process which is isomorphic to the critical phenomena
in finite-size binary liquid mixtures near the critical mixing point [13, 14, 19, 20].
Here, we begin to study the ST process starting with the simplest model [12] with two order parame-
ters and two kinetic equations (2.5), (2.6) and then we continue to correspondingly examine more com-
plicated models with three order parameters (and kinetic equations) in section 4. It is worthy to mention
that the 3rd order parameter associated with the concentration of ferment AChE appears at the last stage
of ST process when ACh is inactivated by AChE. Actually, the hypothesis of isomorphism corresponds only
to the models of ST process in sections 2, 3. This problem was investigated in detail in chapter 5 of [33]
and was explicitly examined in [34–36].
Correlation function and correlation length in ACh–R∗ system in SC. The expression for the pair
correlation function (CF) G2(r ) of the Ornstein-Zernike approximation G2(r ) = A exp(−r /ξ)/r , is valid
for 3-dimensional systems with zero boundary conditions and system linear dimension L0 ≫ ξ, where ξ
is the correlation length of order parameter fluctuations. As it is known, the concentration is an order
parameter of binary liquid mixture near the critical mixing point (see, e.g., [19, 27]). The CF G2(r ) → 0 if
r = |r1 − r2| →∞. The anomalous growth of CL ξ and the long-range behavior of the CF G2(r )∼ r−1 takes
place at the phase transition or critical points only for spatially infinite systems.
To study the correlation properties of ACh–R∗ system, a fundamentally important point is the fact
that the SC is a finite-size (confined) system. For obvious reasons, it is clear that the CL ξ in the synaptic
cleft cannot exceed the linear size L0 of a system. Therefore, an actual problem is to find the correlation
properties [CF G2(r ) and CL ξ] of ACh–R∗ system inside the spatially confined synaptic cleft.
Such a problem was first solved for a plane-parallel geometry of the synaptic cleft in [16] and then in
detail in [28, 29, 33–37]. The main contribution to the CF is given by the following expression:
G2(ρ, z) =
1
2πL0
K0
[
ρ
(
κ2 +
n2π2
4L2
0
)1/2]
cos
(
πz
2L0
)
, (2.7)
where ρ = (x2 + y2)1/2, K0(u) is the cylindrical Macdonald function, and κ= ξ−1 is the reverse CL of spa-
tially infinite systems. Correlation function (2.7) demonstrates an oscillatory behavior in the z direction
of spatial limitation for plane-parallel layers confirming the theoretical results and computer-simulation
studies (see e.g. [30]) for the radial distribution function g (r ) in liquids in restricted geometry.
As is seen from (2.7), the pair correlation function G2(ρ, z) of ACh–R∗ system in confined geometry has
a non-exponential shape. Therefore, it is natural to determine CL ξ of the order parameter fluctuations
in such bounded liquids according to the following relation:
ξ=
√
M2 , M2 =
∫
G2(r )r 2dr
∫
G2(r )dr
, (2.8)
where M2 is the second normalized spatial moment of the pair correlation function.
In the SC with a plane-parallel geometry with thickness H = 2L0, taking into account the formu-
la (2.7) for the CF G2(r ), expression for the Macdonald function Kν(u) and gamma-function Γ(u) [34]
∫∞
0 xµKν(ax)dx = 2µ−1a−µ−1
Γ
(
(1+µ+ν)/2
)
Γ
(
(1+µ−ν)/2
)
, one can get such a formula for the CL ξ of
concentration fluctuations in the SC:
ξ=
[(
κ2 +
π2
4L2
0
)−1
+
(
1−
8
π2
)(
L0
2
)2
]1/2
.
An anisotropic behavior demonstrated by CL ξ is determined by two contributions ξ= (ξ2
x y+ξ2
z )1/2, where
ξx y = 1/(κ2 +π2/4L2
0)1/2 is the CL in the x y plane and ξz = (L0/2)(1−8/π2)1/2 is the CL in the z direction,
13804-4
Synaptic transmission as a cooperative phenomenon in confined systems
respectively, parallel (perpendicular) to presynaptic (postsynaptic) membrane. The analysis of the ex-
pression for ξ shows that the maximum value of the CL in the x y plane equals (ξx y )max = 4L0/π in the
mixing point where κ= 0. Usually, the thickness of chemical synapses is about several dozens of nanome-
ters, so we may assume for definiteness that the membrane thickness H = 2L0 ≈ 20 nm. Thus, in the case
of chemical synapses, the maximum value of the CL in the x y plane equals: (ξx y )max ≈ 6.4 nm, while the
CL in z direction equals: ξz ≈ 2.2 nm. In a neuro-muscular junction, the thickness H is higher, reaching
100 nm, i.e., L0 ≈ 50 nm. In this case, taking the width of the neuro-muscular junction 2L0 ≈ 10−7 m, one
has a maximum value of the CL ξx y : (ξx y )max ≈ 32 nm, while the CL ξz ≈ 11 nm.
It is known that at the end plate of the postsynaptic membrane, about several million receptor mole-
cules are placedwith a high packing at a density which is equal approximately to ρsurf. density ≈ 104 molec-
ular receptor per 1 µm2. It makes it possible to evaluate the number NACh–R∗ of correlated ACh–R∗ com-
plexes in the postsynaptic membrane. For this purpose, let us find the surface area Sact = πd2
act/4 of
the activation zone in the postsynaptic membrane of the chemical synapse with taking into account its
maximum linear size — the diameter dact ≈ (ξx y )max ≈ 6.4 nm. As a result, one has Sact ≈ π(ξ2
x y )max/4 ≈
32.2 nm2 = 32.2 ·10−6 µm2. Then, the number NACh–R∗ of correlated receptors or ACh–R∗ complexes in
the postsynaptic membrane of the chemical synapse equals the product of ρsurf. density and Sact , i.e., one
has NACh–R∗ = ρsurf. density ×Sact ≈ 104 receptors/µm2 ×32.2 ·10−6 µm2 ≈ 32.2 ·10−2 receptors. This result
definitely shows that there are no correlated receptors in a chemical synapse. Thus, receptors of chemical
synapses work independently, without any correlations with each other.
Another situation is realized in the neuro-muscular junctions. Really, due to a larger maximum value
of the CL (ξx y )max ≈ 32 nm, the surface area Sact of the activation zone in the neuro-muscular junc-
tion is 25 times larger and equals Sact ≈ π(ξ2
x y )max/4 ≈ 805 nm2 = 8.05 · 10−4 µm2. Then, the number
NACh–R∗ of correlated complexes equals: NACh–R∗ = ρsurf. density×Sact ≈ 104 receptors/µm2×8.05·10−4 µm2
≈ 8, i.e., approximately 8 receptors mutually correlate in the neuro-muscular junction.
3. Model of postsynaptic membrane activation as cooperative process
Here, we consider the model which describes the process of the postsynaptic membrane activation
as a cooperative process [32, 38–40]. Our model accounts for the interaction of ACh with R on the post-
synaptic membrane. Due to this interaction, the receptors on the postsynaptic membrane transfer to the
activated state which means generating a signal by the postsynaptic membrane neuron. Basing on this
model, we calculate the temporal evolution of the amount of the R∗ on the postsynaptic membrane. In
particular, we analyze how the system responds to the ACh pulse and also how R upon the postsynaptic
membrane interacts with the constant amount of ACh. The most notable result of our modelling is that
the key factor which determines the activation process is the total amount of ACh injected into the cleft.
Basic equation. The key process in passing the signal through a SC is the biochemical reaction that
activates the postsynaptic membrane. This provides a pulse for generating a signal at the adjacent neu-
ron. In a simplified form, we can treat the activation process as the binding of ACh with the R on the
postsynaptic membrane. Thus, within this approach, we have some R on the postsynaptic membrane
that are ready to interact with ACh, and also we have some R∗ (it means that they have already interacted
with ACh). If we denote x to be the concentration of the R, and y to be the concentration of ACh, then we
can write the following equations to determine the dynamics of ACh and R:
dx
dt
= k1(x0 − x)−k2x y, (3.1)
dy
dt
= f (t)−k2x y. (3.2)
Here, the x0 stands for the total concentration of R on the postsynaptic membrane, the coefficients k1
and k2 are phenomenological parameters of the model. The function f (t) determines the intensity of
ACh injection into the SC. We also neglect here the diffusion processes within the synapse (but in general
case, these effects should also be taken into account). The system must be supplemented with the initial
conditions x(0) = x0 and y(0) = 0 which means that at the beginning, all the receptors are in the non-
activated state and there is no ACh in the SC.
13804-5
A.V. Chalyi, A.N. Vasilev, E.V. Zaitseva
After making substitutions t = k1x0t , x = x0u and y = x0v , we get the following system of differential
equations:
du
dt
= k(1−u)−uv, (3.3)
dv
dt
= j (t)−uv, (3.4)
where k = k2/(k1x0), and j (t) = f (t x−1
0 k−1
1 )/(k1x2
0) is the dimensionless function that determines the
intensity of the ACh injection into the SC. Initial conditions in this case are of the form u(0) = 1 and
v(0) = 0.
The system of the nonlinear equations (3.3)–(3.4) can be solved numerically if we know the function
j (t) (and the value of the parameter k). Next, we will consider some special situations.
Acetylcholine in synaptic cleft. First of all, it is interesting to clear out what happens if ACh appears
in the SC. We mean that the function j (t) is zeroth, but the initial concentration V of the ACh is nonzero,
that is v(0) =V . In this case, the concentration of R decreases at first and then it goes up to the initial unit
value, meanwhile the concentration of ACh goes from the initial value down to zero. The concentration
of the R∗ changes opposite to the concentration of the R: at first, the concentration of the R∗ increases
and then it goes down to the zeroth value. This behavior is totally expectable since the presence of ACh
causes the interaction with the receptors, so the amount of R∗ increase. Due to the deactivation process,
the amount of the R∗ decreases to the initial zeroth value.
More realistic situation is observed when some portion (i.e., an impulse) of ACh arrives at the post-
synaptic membrane. This process can be modulated with the nonzero function j (t) in equation (3.4).
Unfortunately, we do not know the precise expression for the function j (t). Nevertheless, theoretical
and experimental data testify that this function (in the case when a single ACh impulse arrives) should
be strictly localized in time. Thus, as the probe function we take the expression of the following form:
j (t) = A exp[−a(t − t0)2], where the A constant determines the “strength” of the ACh impulse, the a pa-
rameter determines the localization of the impulse and the t0 is the time of the impulse arrival. We will
consider this case. In other words, we consider the j (t) function and the initial condition for the amount
of the ACh concentration being zero, so we take v(0) = 0. Increasing the parameter A, leads to an increase
of the R∗ amount. This is a natural effect since increasing the parameter A (when the other parameters
are fixed) means increasing the total amount of ACh which is injected into the SC. The amount of the R∗
also depends on the value of the parameter a. Aswe can see, increasing the parameter a causes a decrease
of the amount of the R∗ due to a decrease of the total amount of ACh injected into the cleft. Thus, we can
expectedly state that varying the parameters A and a causes the change of the amount of the R∗. Never-
theless, it is notable that if A ∼
p
a, then changing the parameter a changes the value of the parameter A,
but the total amount of ACh injected into the cleft is still the same. Numerical calculations show that this
synchronic change of the parameters A and a does not significantly affect the profile of the R∗ curve.
Figure 1 demonstrates the dependence of the concentration of the activated receptors on time. It is
obvious that this behavior is universal in the sense that it does not qualitatively depend on the param-
eter A =
p
a and, thus, it could be an argument for the initial preposition regarding the isomorphism
hypothesis.
This feature can be important for analyzing the ST. Indeed, when studying ACh spreading from the
presynaptic membrane through the SC to the postsynaptic membrane, it is important to know the pecu-
liarities of the transportation process and the initial distribution of ACh in the SC. Thus, the question of
stability is relative to the ACh distribution profile that could play a crucial role for the ST theory.
4. Models of ST with M- and N-receptors and effects of diffusion
Here, in this section, we shall study the kinetic models of ST with three order parameters and dif-
ferent types of receptors taken into account, specifically, (a) the muscarinergic (M) receptors and (b) the
nicotinergic (N) receptors (see appendix B).
The 1st kinetic model of ST with 3 order parameters and M-receptors takes into account the func-
tion of ACh release depending on time, being approximated by using function cos(Ωt), where parame-
ter Ω characterizes a slight deviation of the release speed of ACh from the constant value in accordance
13804-6
Synaptic transmission as a cooperative phenomenon in confined systems
Figure 1. The concentration of the activated receptors as the function of time. It is assumed that k = 1,
t0 = 2, A =
p
a and the solid line is for the value a = 10, the dashed line is for the value a = 50 and the
dotted line is for the value a = 75.
with the experimental data [41–43]. Since the process of release of ACh in the SC is quite fast, in order
to take account of this fact and the gradual slowdown supply of ACh molecules in the SC, trigonometric
function was expanded in Taylor series preserving the quadratic term in (4.1). The corresponding system
of differential equations is as follows:
dx
dt
= M
(
1−
Ω
2t 2
2
)
−k1x(a − y)−k2xz −k3x, (4.1)
dy
dt
= k1x(a − y)−k4x y z, (4.2)
dz
dt
= k5z −k2xz −k4x y z. (4.3)
The first term in (4.1) describes the speed of release of free ACh molecules into SC, the second term
describes the rate of binding of ACh with R (a — total number of M-receptors on the postsynaptic mem-
brane), the third shows the rate of ACh molecules decay at the presence of AChE, the fourth shows the
withdrawal of ACh from the synaptic cleft by diffusion and reuptake (return of mediator molecules back
to the presynaptic membrane). The following processes are present in (4.2): the first term defines a pos-
itive contribution to the rate of change of the concentration of ACh–R∗ complexes by binding the ACh
molecules with R, and the second term describes the speed of the decay of ACh–R∗ complexes at the pres-
ence of AChE. In (4.3), the first term describes the speed of release of free AChE at the decay of ACh–R∗
complexes, the second and third terms have the same meaning as similar terms in (4.1) and (4.2). The
stationary system of equations (4.1)–(4.3) has the following solutions:
a) x0 = 0, y0 = g , z0 = 0 (g < a). This solution is classified as a strange attractor, because the roots of
the characteristic equation are as follows: λ1 = −ζ, λ2 = 0, λ3 = k5, where ζ = k1a +k2g +k3 and k5 is
the rate of release of AChE. The emergence of this solution can be explained by the fact that the release
function of ACh in the SC is considered as a short, small perturbation;
b) x0 = 0, y0 = 0, z0 = h, where h is the concentration of the enzyme AChE. This solution satisfies
system (4.1)–(4.3) provided that k5 = 0. Indeed, for long periods of time, all ACh molecules and ACh–
R∗ complexes are split and the AChE release rate will be equal to zero. The roots of the corresponding
characteristic equation are as follows: λ1 =−ζ, λ2 = 0, λ3 = 0. In this case, since λ1 < 0, λ2 = 0, λ3 > 0, this
singular point is classified as stable torus.
The 2nd model with 3 order parameters and N-receptors was created to study the mechanism of
ST, which involved N-cholinergic receptors (see appendix B). Let us present the processes that occur in
the synaptic cleft, with the following scheme of biochemical reactions:
ACh+R
k1⇐⇒AChR, (4.4)
13804-7
A.V. Chalyi, A.N. Vasilev, E.V. Zaitseva
AChR+ACh
k4⇐⇒ACh2R
k4⇐⇒ACh2R
∗ k5=⇒R. (4.5)
In other words, at secretion of AChmolecule into the SC being bound to the R (4.4), this complex bindswith
another AChmolecule, then the mediator-receptor complex enters an excited state, and then it returns to
the initial state (4.5). As in previous models, the reverse processes are not taken into account. Since the
activation of N-cholinergic receptors requires two molecules of the ACh, there is an intermediate inactive
ACh–R complex with one ACh molecule. To adequately describe such a system, one should increase the
number of variables which leads to complications in kinetic equations and increases their numbers. To
avoid this, we assume the concentration of the enzyme AChE to be constant. Thus, we will have the
following three nonlinear differential equations for the concentration of ACh molecules (A), receptors in
the unexcited state (X1) and excited mediator-receptor complexes (X2):
Ȧ = Mδ(t)−k1 AX1 −k2 AE −k3 A−k4 A[X0 − (X1 +X2)]+k5X2 +k6[X0 − (X1 +X2)], (4.6)
Ẋ1 = k5X2 −k1 AX1 +k6[X0 − (X1 +X2)], (4.7)
Ẋ2 = k4 A[X0 − (X1 +X2)]−k5 X2 . (4.8)
In the first equation, it is accepted that the concentration E of the enzyme AChE is constant, and the
release of ACh in the SC is described by Dirac δ-function [see the first term of (4.6)]. The second and
fifth terms describe the binding of the ACh with free R and inactive neurotransmitter-receptor complex,
respectively, the third term describes the hydrolysis of ACh in the presence of AChE, the fourth term
shows the removal of ACh from the cleft by diffusion and reuptake, and the sixth and seventh terms
show disconnection of ACh molecules from the active and inactive neurotransmitter-receptor complex,
respectively. Terms, which are included in (4.7): the first and the third mean the same as the sixth and
the seventh terms in (4.6), respectively, and a free ACh binding with R is described by the second term.
In (4.8), the first term describes the transition of an inactive neurotransmitter-receptor complex in the
excited state, and the second shows the collapse of neurotransmitter-receptor complex.
The stationary system of equations has a solution: a = A/A0 = 0, x1 = X1/X0 = 1, x2 = X2/X0 = 0
(A0 is the total concentration of ACh molecules at the time of release, X0 sets the total concentration of
cholinergic receptors in the SC), which corresponds to the real situation when there are no molecules of
ACh in the SC, and no mediator-receptor complexes, and all receptors are free. This solution is classified
as a stable node because numerical calculation of the roots of the characteristic equation with using
experimental data [41–43] gives the following result: λ1 ≈ −5 ms−1, λ2 ≈ −3082 ms−1, λ3 ≈ −37 ms−1.
The corresponding relaxation times are equal to: τ1 ≈ 0.2 ms, τ2 ≈ 3 ·10−4 ms, τ3 ≈ 2.7 ·10−2 ms. Since
these relaxation times differ quite significantly (τ1/τ2 ≈ 670, τ1/τ3 ≈ 7.5), the main contribution to the
kinetics of the system approach to equilibrium gives the slowest relaxation process with time 0.2 ms.
Kinetic models of ST with 3 order parameters and diffusion effects. Here, we have also consid-
ered non-linear kinetic models of synaptic transmission in the synapses withM-cholinergic receptors and
N-cholinergic receptors, where the impact of diffusion processes was considered more consistently. For
both models, it is believed that the process of synaptic transmission has the following common features:
1) molecules of ACh are released in the space of SC very quickly, so the release of ACh function is
approximated using the Dirac δ-function;
2) ACh molecules then diffuse to the postsynaptic membrane, where they bind with the receptor
molecules, which causes excitation of nerve impulses in the next cell;
3) then, the mediator-receptor complexes decompose.
In the 3rd kineticmodel, the processes occurring in the cleft of the synapse withM-cholinergic recep-
tors are taken into account. It is assumed that one ACh molecule activates the receptor, and AChE partici-
pates in the destruction of neurotransmitter-receptor complexes. The corresponding system of nonlinear
differential equations for concentrations of molecules of ACh (x), for the ACh–R∗ complexes (y) and for
the enzyme AChE (z) is as follows:
ẋ = Kδ(t)−k1x(a − y)−k2xz −k3x +D1∆x, (4.9)
ẏ = k1x(a − y)−k4x y z, (4.10)
ż = k5z −k2xz −k4x y z −D3∆z. (4.11)
13804-8
Synaptic transmission as a cooperative phenomenon in confined systems
The first term on the right-hand side of the equation (4.9) describes the rate of ACh release in the SC,
the latter shows the removal of neurotransmitter molecules from the cleft by diffusion. The first term in
the third equation depending on the sign will have a double meaning: for k5 Ê 0 — a rate of release of
the molecules of the enzyme AChE; for k5 < 0, it is a speed of binding of the enzyme AChE molecules with
OPhC that suppresses the speed of release of enzymemolecules. The last term of equation (4.11) describes
the movement of molecules of the enzyme in the region with high concentration of acetylcholine through
lateral diffusion. All other terms in the equations (4.9)–(4.11) have the same meaning as the correspond-
ing equations in (4.1)–(4.3).
Singular points of the system (4.9)–(4.11):
(a) x0 = 0, y0 = g , z0 = 0, where g is some concentration of R∗, it is clear that g É a;
(b) x0 = m, y0 = a, z0 = 0, m is some free concentration of ACh;
(c) x0 = 0, y0 = g , z0 = h, h is certain concentration of AChE.
Depending on the sign of the parameter k5 that plays the role of a control parameter of the model,
points (a), (b) and (c) have the Lyapunov classification (to assess their respective solutions of the charac-
teristic equation, experimental data from the articles [41–43] were used):
• for k5 Ê 0 point (a) is classified as a strange attractor, point (b) may be a stable node and a saddle,
and point (c) is classified as a strange attractor;
• for k5 < 0 point (a) is classified as a limiting cycle, point (b) is classified as a stable node, and point
(c) is classified as a limiting cycle.
It should be noted that lateral diffusion of molecules of the enzyme has less impact on the process of
synaptic transmission than the diffusion of mediator molecules.
In the last 4th model, the influence of the diffusion of mediator molecules on the synaptic trans-
mission was analyzed in the cleft with N-cholinergic receptors. Therefore, a proper system of nonlinear
differential equations of the model is substantially similar to the system of equations (4.6)–(4.8) of the
second model, but differs from it by the presence of diffusion term ∆A in the first equation.
Stationary system of equations of the model has the same solution as the 2nd model: a = 0, x1 = 1,
x2 = 0, which corresponds to the real situation when there are nomolecules of acetylcholine in the synap-
tic cleft, and no mediator-receptor complexes, and the receptors are all free.
An analysis of the solutions of characteristic equation shows that at any values of the diffusion coef-
ficient D , a singular point is classified as a stable node (the numerical evaluation of experimental data
was taken from [41–43]) because the roots of the characteristic equation are equal to: λ1 ≈ −14 ms−1,
λ2 ≈ −3717 ms−1, λ3 ≈ −37 ms−1. The diffusion process does not change the type of a singular point,
although it affects the relaxation time values: τ1 = 0.07 ms, τ2 = 2.6 ·10−4 ms, τ3 = 2.7 ·10−2 ms. Having
compared the estimates given in the relevant 2nd model, we see that at a consistent account of the pro-
cesses of diffusion, the relaxation times τ1 and τ2 decreased approximately 2.8 and 1.2 times, while τ3
did not change.
5. Conclusions
We have created and studied the non-linear kinetic models of ST and intercellular interaction based
on actual processes that occur in the synapse of chemical type and isomorphism of coherent aspects of
ST and critical phenomena in liquid systems with restricted geometry. The main results and conclusions
can be summarized as follows.
1. The linear size of the activation zone was evaluated as the CL ξ for a binary mediator-receptor
mixture. It was found that approximately 8 receptors mutually correlate in the neuro-muscular junction
with its thickness of 100 nm.
2. The process of the postsynaptic membrane activation was described as a cooperative process. The
system of nonlinear equations for concentrations of R and ACh was numerically solved to find its tem-
poral evolution in different specific conditions. The question of stability relative to the ACh distribution
profile could play a crucial role for the ST theory.
13804-9
A.V. Chalyi, A.N. Vasilev, E.V. Zaitseva
3. The analysis of ST models with three order parameters and diffusion effects taken into account
showed that if the control parameter changed its sign from positive to negative, the type of singular
points also changed — they all got stability, i.e., the bifurcation occurred: a strange attractor turned into
the limiting cycle.
4. Based on the experimental data and numerical calculations for kinetic models with N-receptors,
the relaxation times of three order parameters were equal to: τ1 = 0.2 ms, τ2 = 3 · 10−4 ms, τ3 = 2.7
× 10−2 ms. Thus, the approach of the system to an equilibrium state was characterized by the slowest
relaxation process with time 0.2 ms. Diffusion processes did not change the types of singular points, but
affected the relaxation times τ1 = 0.07 ms, τ2 = 2.6 ·10−4 ms, while τ3 did not change.
Acknowledgements
The authors wish to thank Professor Yu.V. Holovatch, a famous scientist in the field of nanobiophysics,
critical phenomena and physics of complex systems, one of the best representatives of the widely-known
scientific school by Academician I.R. Yukhnovskii.
A. Abbreviations
ST — synaptic transmission; ACh — acytineholine; R — non-activated receptors; R∗
— activated re-
ceptors; SC — synaptic cleft; AChR — mediator-receptor complexes; AChE — acetylcholinesterase; CF —
correlation function; CL — correlation length; N — nicotinergic (receptors); M — muscarinergic (recep-
tors); OPhC — organophosphate compounds
B. Glossary
Amphiphilicmolecule possesses both hydrophilic and hydrophobic properties. This sort of molecules
can form bilayers andmicelles in water, so that their polar hydrophilic heads turn to the water molecules,
while their hydrophobic tails hide inside the formation.
Phospholipids are a sort of amphiphilic molecules.
Mediator is a biologically active chemical substance transmitting the nerve impulse from one cell to
another.
Receptor provides transformation of impacts of environment and inner mediuminto nerve impulse.
There are a lot of types of receptors in the human body.
Acetylcholine is a neuromediator providing nerve-to-muscle transmission of the nerve impulse. With-
out going into biophysical details of intercellular interactions [1–10], let us briefly note that there are two
types of ACh receptors: 1) only onemediator molecule is sufficient for activation receptors of the first type
(so-calledmuscarinergic receptors, orM-receptors), while 2) activation of receptors of the second type (so-
called nicotinergic receptors, or N-receptors) requires two mediator molecules. Here, we shall consider
several nonlinear kinetic models with M- and N-receptors and the effects of diffusion [40, 44].
Acetylcholinesterase is a hydrolytic ferment catalizing the hydrolysis of acetylcholine which lets the
cell change to the rest state.
Organophosphor compounds inhibit the action of acetylcholinesterase which leads to the effects of
strong poisoning and even to the death of a human body.
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https://doi.org/10.3367/UFNr.0173.200302c.0175
https://doi.org/10.1070%2FPU2003v046n02ABEH001077
https://doi.org/10.1016/S0167-7322(99)00187-7
https://doi.org/10.1016/0167-7322(93)80066-5
https://doi.org/10.1134/S0006350910040147
https://doi.org/10.1007/978-1-4020-5872-1_26
https://doi.org/10.1007/BF02145692
A.V. Chalyi, A.N. Vasilev, E.V. Zaitseva
Синаптична передача як кооперативне явище в просторово
обмежених системах
О.В. Чалий1, О.М. Васильєв2, О.В. Зайцева1
1 Кафедра медичної та бiологiчної фiзики, Нацiональний медичний унiверситет iменi О.О. Богомольця,
бульв. Шевченка, 13, 01601 Київ, Україна
2 Кафедра теоретичної фiзики, Київський нацiональний унiверситет iменi Тараса Шевченка,
пр. Глушкова, 6, 03022 Київ, Україна
У цьому оглядi була розроблена i обговорена теорiя синаптичної передачi (СП). Ми застосували гiпотезу
iзоморфiзму мiж: (а) кооперативною поведiнкою медiаторiв — молекул ацетилхолiну (ACh) i холiнореце-
пторiв в синаптичнiй щiлинi з утворенням медiатор-рецепторних (AChR) комплексiв, (б) критичними яви-
щами в просторово обмежених бiнарних рiдких сумiшах. Були запропонованi системи двох (або трьох)
нелiнiйних диференцiальних рiвнянь для опису змiни концентрацiй ACh, комплексiв AChR i ферменту
ацетилхолiнестерази. Основнi результати нашого дослiдження: оцiнено лiнiйний розмiр зони активацiї;
описано процес активацiї постсинаптичної мембрани як кооперативний процес; розглянуто рiзнi апро-
ксимацiйнi функцiї синхронного вивiльнення ACh; дослiджено стацiонарнi стани i типи особливих точок
для запропонованих моделей СП; показано, що нелiнiйна кiнетична модель з трьома параметрами по-
рядку демонструє поведiнку дивного атрактора.
Ключовi слова: мiжклiтинна взаємодiя, синаптична передача, фазовий перехiд, рiдка бiнарна сумiш,
критична точка змiшування-розшарування, просторово обмеженi системи
13804-12
Introduction
Model of ST as phase transition in ``transmitter-receptor'' system
Model of postsynaptic membrane activation as cooperative process
Models of ST with M- and N-receptors and effects of diffusion
Conclusions
Abbreviations
Glossary
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