Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model
We studied bursting patterns underlied by bifurcation phenomena and chaotic spiking in a computational leech heartbeat model. We observed the gradient physical properties of the ISI trains and amplitude (shift of the membrane potential) when the parameter gleak was mildly changed and found differ...
Saved in:
| Date: | 2014 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут фізіології ім. О.О. Богомольця НАН України
2014
|
| Series: | Нейрофизиология |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148274 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model / Y.Zh. Guan, Sh.Q. Liu, Y.J. Zeng // Нейрофизиология. — 2014. — Т. 46, № 2. — С. 121-128. — Бібліогр.: 22 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-148274 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1482742025-02-23T18:28:10Z Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model Градієнтні тригерні механізми, пов’язані з режимами бістабільності, в моделі контролю серцевих скорочень у п’явки Guan, Y.Zh Liu, Sh.Q. Zeng, Y.J. We studied bursting patterns underlied by bifurcation phenomena and chaotic spiking in a computational leech heartbeat model. We observed the gradient physical properties of the ISI trains and amplitude (shift of the membrane potential) when the parameter gleak was mildly changed and found different bistable areas. The resulting computation implies that (i) classification of the intensity of the input information is feasible in this regime, (ii) a neuron’s working level can be marked by its range in a typical bifurcation, and (iii) there are invisible triggers underlying subtle mechanisms in the model. Ми досліджували пачкові імпульсні патерни, що формувалися на основі феноменів біфуркації, та хаотичну імпульсну активність у комп’ютерній моделі керування серцевими скороченнями у п’явки. Ми спостерігали градієнтність фізичних властивостей, що визначали характеристики послідовностей імпульсів та амплітуду (зміщення мембранного потенціалу), при невеликих змінах параметра gleak (провідності витоку). Було також виявилено, що існують різні зони бістабільності. Результати комп’ютерного моделювання вказують на те, що, по-перше, в такому режимі може забезпечуватися класифікація інтенсивності вхідної інформації; по-друге, робочий рівень для нейрона визначаеться його положенням у типовій біфуркації, і, по-третє, існують «невидимі» тригери, на яких базуються тонкі механізми моделі. This work was supported by the National Natural Science Foundation o f China (Grant No. 11172103). 2014 Article Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model / Y.Zh. Guan, Sh.Q. Liu, Y.J. Zeng // Нейрофизиология. — 2014. — Т. 46, № 2. — С. 121-128. — Бібліогр.: 22 назв. — англ. 0028-2561 https://nasplib.isofts.kiev.ua/handle/123456789/148274 612.172:004.94 en Нейрофизиология application/pdf Інститут фізіології ім. О.О. Богомольця НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We studied bursting patterns underlied by bifurcation phenomena and chaotic spiking in a
computational leech heartbeat model. We observed the gradient physical properties of the
ISI trains and amplitude (shift of the membrane potential) when the parameter gleak was
mildly changed and found different bistable areas. The resulting computation implies that (i)
classification of the intensity of the input information is feasible in this regime, (ii) a neuron’s
working level can be marked by its range in a typical bifurcation, and (iii) there are invisible
triggers underlying subtle mechanisms in the model. |
| format |
Article |
| author |
Guan, Y.Zh Liu, Sh.Q. Zeng, Y.J. |
| spellingShingle |
Guan, Y.Zh Liu, Sh.Q. Zeng, Y.J. Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model Нейрофизиология |
| author_facet |
Guan, Y.Zh Liu, Sh.Q. Zeng, Y.J. |
| author_sort |
Guan, Y.Zh |
| title |
Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model |
| title_short |
Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model |
| title_full |
Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model |
| title_fullStr |
Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model |
| title_full_unstemmed |
Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model |
| title_sort |
gradient trigger mechanisms related to bistability regimes in a leech heartbeat model |
| publisher |
Інститут фізіології ім. О.О. Богомольця НАН України |
| publishDate |
2014 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148274 |
| citation_txt |
Gradient Trigger Mechanisms Related to Bistability Regimes in a Leech Heartbeat Model / Y.Zh. Guan, Sh.Q. Liu, Y.J. Zeng // Нейрофизиология. — 2014. — Т. 46, № 2. — С. 121-128. — Бібліогр.: 22 назв. — англ. |
| series |
Нейрофизиология |
| work_keys_str_mv |
AT guanyzh gradienttriggermechanismsrelatedtobistabilityregimesinaleechheartbeatmodel AT liushq gradienttriggermechanismsrelatedtobistabilityregimesinaleechheartbeatmodel AT zengyj gradienttriggermechanismsrelatedtobistabilityregimesinaleechheartbeatmodel AT guanyzh gradíêntnítrigernímehanízmipovâzanízrežimamibístabílʹnostívmodelíkontrolûsercevihskoročenʹupâvki AT liushq gradíêntnítrigernímehanízmipovâzanízrežimamibístabílʹnostívmodelíkontrolûsercevihskoročenʹupâvki AT zengyj gradíêntnítrigernímehanízmipovâzanízrežimamibístabílʹnostívmodelíkontrolûsercevihskoročenʹupâvki |
| first_indexed |
2025-11-24T10:26:49Z |
| last_indexed |
2025-11-24T10:26:49Z |
| _version_ |
1849667102877679616 |
| fulltext |
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2014.—T. 46, № 2 121
UDC 612.172:004.94
Y. ZH. GUAN1, SH. Q. LIU1, and Y. J. ZENG2
GRADIENT TRIGGER MECHANISMS RELATED TO BISTABILITY REGIMES IN A
LEECH HEARTBEAT MODEL
Received May 4, 2013
We studied bursting patterns underlied by bifurcation phenomena and chaotic spiking in a
computational leech heartbeat model. We observed the gradient physical properties of the
ISI trains and amplitude (shift of the membrane potential) when the parameter gleak was
mildly changed and found different bistable areas. The resulting computation implies that (i)
classification of the intensity of the input information is feasible in this regime, (ii) a neuron’s
working level can be marked by its range in a typical bifurcation, and (iii) there are invisible
triggers underlying subtle mechanisms in the model.
Keywords: leech heartbeat model, leech heart interneuron, bifurcation and chaos, leak
current, injected current, interspike intervals (ISIs).
1 Department of Mathematics, South China University of Technology,
Guangzhou, China.
2 Biomedical Engineering Center, Beijing University of Technology, Beijing,
China.
Correspondence should be addressed to
Y. Zh. Guan (e-mail: yzhguan@scutedu.cn) or
Y. J. Zeng (e-mail: yjzeng@bipu.edu.cn)
INTRODUCTION
Some general considerations on the mechanisms that
provide shifting of excitable cells among different
modes of multistability have been proposed, but
details of such mechanisms are unknown in most
cases. If we master these mechanisms, we will be able
to apply the respective techniques to the treatments
using biofeedback stimulation in the cases of serious
illnesses. For example, we might prevent the onset of
pathological regimes of seizures by returning cells of
the involved networks back to the normal regime with
a switch induced by stimulation treatment [1, 2].
In many organisms, there are central pattern
generators (CPGs) comprised of neuronal circuits that
generate and organize repetitive motor patterns in a
few regimes. The network controlling leech heartbeat
is one of the simplest examples of such networks.
Control of the leech heartbeat pattern has attracted
significant attention and advanced discussion. Two
segmental oscillators were found in the neuronal
network pacing heartbeat in the leech. They are
located in the third and fourth ganglia of the ventral
nerve chain [3, 4]. In order to simulate the respective
regimes, Hill et al. modeled a segmental oscillator
pacing heartbeat in the leech using Hodgkin–Huxley
equations [5]. This model was constructed as a
network of six heart interneurons localized within a
single ganglion [6]. The network comprised the basic
rhythm generator and demonstrated properties more
similar to those of the biological prototype than the
previous model [7].
Cymbalyuk et al. [7] provided diagrams for the
activity of single interneurons of this model. There
were three separate areas in these diagrams, (i) a
bursting regime coexisting with silence, (ii) bursting
coexisting with tonic spiking, and (iii) tonic spiking
coexisting with silence. Shilnikov et al. [8] described
the connection between a codimension of bifurcation
and a transition from tonic spiking behavior to
bursting behavior. Malashchenko et al. [9] pointed out
the existence of a saddle periodic orbit as the main
factor providing coexistence of the bursting and silent
regimes.
In our study, as a further step towards understanding
the relation between a single cell regime and network
functions, we explore more physical properties
of the leech heartbeating model, paying special
attention to interspike intervals (ISIs) as one of the
notable characteristics in the respective models. We
investigated bursting patterns under conditions of
application of electric current (DC) to the canonical
leech heart interneuron model, and we also analyzed
bifurcation diagrams for ISIs and amplitudes (shifts of
the membrane potential) characterizing the transitions
between different firing patterns with variation of the
magnitude of DC fields. Equations were integrated
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2014.—T. 46, № 2122
Y. ZH. GUAN, SH. Q. LIU, and Y. J. ZENG
using the fourth-order Runge-Kutta method. Values
of the ISI(s) are a comprehensive feature for many
computational results; so, these parameters may
reveal some regular features beyond the biological
experiments; at the same time, they may also miss
some details of the pattern. Thus, we extended our
investigation to the results derived from the leech
heartbeat computational model and found out that
mild changes in the amplitude of the above-mentioned
influence (DC electric field) may determine specific
diagrams of bifurcation and chaos characteristics of
the firing patterns. We noticed that ISIs regularly
exhibit bifurcation and chaos patterns when the
model was subjected to weak electrical stimulation.
This uncommon feature allowed us to deduce certain
relations between properties of single neurons,
neuronal behavior in the CPG network in general, that
in the leech special structure in particular, and some
biological phenomena (motor activity).
DESCRIPTION OF THE MODEL
Previous works [10, 11] revealed that there are three
classes of central neurons controlling heartbeat in the
leech kinetic system. The compartment of the network
is formed by 14 heart interneurons (HN cells); it
generates the heartbeat rhythm and controls the
activity of excitatory heart motor neurons (HE cells)
[12]. Furthermore, heart accessory neurons (HA cells)
also receive inputs from HNs [13]. In general, HN
cells are “leading” elements of the system. Figure 1
describes the synaptic connectivity in the heartbeat
CPG.
A single heart interneuron was modeled [6] based on
Hodgkin–Huxley equations [5] including intrinsic and
synaptic conductances of the HN cells. The respective
oscillations (changes in the amplitude, i.e., in values
of the membrane potential), especially those typical
of the bursting pattern, are similar in their form to
oscillations observed in the biological system. [7]. In
Fig. 1, the locations of interneurons in this model are
shown.
The model involves the following currents: fast
calcium current, slow calcium current, fast sodium
current, delayed-rectifier potassium current, persistent
potassium current, fast transient potassium current,
hyperpolarization-activated cationic current, persistent
sodium current, and leak current. These currents are
described by a system of 14 differential equations.
.
(1)
The values corresponding to the model parameters
in Eq. (1) have been adopted from the communication
by Hill et al. [6]. All parameters including Eleak were
set to the canonical values; only the gleak was set from
11.3 to 12.9 nS in the following analysis.
SIMULATION RESULTS RELATED TO
BIFURCATION AND CHAOS PHENOMENA
Sequences of action potentials (APs) in bifurcation
and chaos modes. The bursting regime and its
relation to physical parameters of the neurons are of
high physiological significance in the processing of
information by neuronal networks [14, 15]. We do not
discuss the special problem of how the bursting AP
organization influences the target cells, as compared
with the effects of regular rhythmic spiking; this
aspect is beyond the topic of our communication. The
complexity of dynamical regimes of spiking interferes
with clear understanding of the respective mechanisms
responsible for one type or another of biological
behavior. The fast/slow analysis [16–19] became an
extensively accepted method for recognizing several
typical bursting patterns in neuronal networks. In a
A B
1
2
3 3
4
5
6
3
4
5
6
4
1
HN
HE HE 2
3
4
12 12
3434
F i g. 1. Scheme of neurons and synapses in the heartbeat central
pattern generator. A) Network of synaptic connections between
HNs and HEs. Circles represent neurons, lines represent their
connections, and filled small circles denote inhibitory chemical
synapses. B) Basic oscillator built by neurons and inhibitory
synapses among HN cells.
Р и с. 1. Схема нейронів і синапсів у центральному генераторі,
що контролює скорочення серця.
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2014.—T. 46, № 2 123
GRADIENT TRIGGER MECHANISMS RELATED TO BISTABILITY REGIMES
single neuron, different currents interplay and provide
the coexistence of different types of firing activity
(known as regular spiking, bursting, silence, and
oscillations) [20–22].
We firstly exhibited some typical sequences of
APs for characterization of HN neurons in our model
and focused on the bursting patterns underlied by
bifurcation phenomena and chaos spiking. Bursting
patterns produced by the model neuron are shown in
Fig. 2. In this case, the gleak was constant and equal
to 12.7 nS, while the injected direct current Iinj was
changed (A–F).
As is well known, the frequency, period, and duty
cycle of bursting are important physical properties
for firing patterns of neurons in fast/slow systems.
Undoubtedly, we can use these indices to summarize
5 5
5
5 5
5
mV mV
mV
mV mV
mV
A
C
E F
C
B
sec sec
sec
sec sec
sec
70 70
70
70 70
70
75 75
75
75 75
75
80 80
80
80 80
80
85 85
85
85 85
85
90 90
90
90 90
90
–15 –15
–15
–15 –15
–15
–35 –35
–35
–35 –35
–35
–55 –55
–55
–55 –55
–55
F i g. 2. Patterns of spiking produced by the model neuron. A) Tonic spiking, B and C) bursts consisting of two spikes, D) bursting with a
varied number of spikes and unstable interspike intervals (chaos), E and F) bursting with three and four spikes, respectively. gleak = 12.7 nS;
injected direct current (Iinj) = 0.017, 0.0159, 0.0154, 0.015, 0.0145, and 0.0126 nA in A–F, respectively.
Р и с. 2. Патерни імпульсної активності, генерованої модельним нейроном.
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2014.—T. 46, № 2124
Y. ZH. GUAN, SH. Q. LIU, and Y. J. ZENG
information on the pattern from biological results
to physical features, allowing us to correlate firing
behavior of the neuron with transition behavior of the
network. Although it is difficult to explain directly
the animal’s biological patterns by description of
the respective micromechanisms till now, it is a
promising branch for understanding the regimes
of neuronal activity. The bursting patterns in Fig. 2
demonstrate that the ISI groupings vary from one to
two, two to three, and so on. This phenomenon means
that bifurcation occurs. We explore more physical
properties to reveal the biological process, such as
ISI sequences and amplitudes (shifts of the membrane
potential).
Gradient mechanisms across bistability regimes.
The previous study on the leech heartbeat HN
neuron model showed that the bistability of bursting
and silence is associated with the Andronov-Hopf
bifurcation [7]. This implies that the barrier separating
two attractors, bursting and silence, is the stable
manifold of a saddle periodic orbit [9]. Below, we
show preview diagrams (scatter plots) demonstrating
the interrelation between ISIs and values of the
amplitude and gleak.
Hill et al. [6] found that an increase in the maximum
conductance for the leak current (gleak) leads to
decreases in the cycle period and spike frequency.
Cymbalyuk et al. [7] showed bifurcation diagrams for
a single cell and pointed out three coexisting areas,
bursting and silence, bursting and tonic spiking,
and tonic spiking and silence. Shilnikov et al. [8]
demonstrated that bifurcation of a codimension can
explain the transition between tonic spiking behavior
and bursting behavior. Malashchenko et al. [9] showed
that the main factor for a regime of coexistence of
bursting and silence is a saddle periodic orbit.
0
0
–55
–55
12,5
12,2 12,2
12,512,6
12,3 12,3
12,612,7
12,4 12,4
12,712,8
12,5 12,5
12,812,9
12,6 12,6
12,913,0
12,7 12,7
13,013,1
12,8 12,8
13,113,2
12,9 12,9
13,2
1
1
–50
–50
2
2
–45
–45
3
3
–40
–40
sec
sec mV
mV
nS
nS nS
nS
A
C D
B
F i g. 3. Scatter plots for the relations between ISIs, sec, and gleak , nS (A and C) and between the amplitude, mV, and gleak , nS, (B and D). In
A and B, gleak = 12.5 to 13.2 nS, Iinj = 0.0147 nA; in C and D, gleak = 12.2 to 12.9 nS, Iinj = 0.0121 nA; Eleak = –0.061 mV in all cases.
Р и с. 3. Діаграми розсіяння для відношень між тривалістю міжімпульсних інтервалів, с, та gleak , нСм, (А та С) і між амплітудою,
мВ, та gleak , нСм (B та D відповідно).
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2014.—T. 46, № 2 125
GRADIENT TRIGGER MECHANISMS RELATED TO BISTABILITY REGIMES
appears almost along a horizontal line. We believe
that this is an uncommon physical characteristic
in the neuronal models. When we compare Fig. 3 A
and C vs. B and D in groups, we find that different
injection current intensities underlie the gradient
structure variation. This implies that (i) the neuron can
“recognize” the intensity of the input “information”,
(ii) the neuron can “inform” other units with respect to
its working level by its range in bifurcation, and (iii)
there are invisible triggers existing in the mechanisms
of the bistability regimes for this model. It might help
us to understand why some phenomenon appears or
disappears unpredictably.
ISI and amplitude bifurcation reveals trigger
details in the gradient mechanisms . Since the
Lapicque formula approximates well the strength-
duration physical law under the action of square pulse
stimulation current [9], we believe that we can reveal
some physical properties related to the biological
features in this computation model. As before, we
recognized bifurcation in Fig 2; when we increase or
decrease the gleak variable, bifurcation appears as the
fitting of the complete section of the injected current.
We illustrate the ISI, Iinj, and gleak interdependence by
diagrams (scatter plots) in Fig 4. The settings of gleak
varied between 12.3 and 12.7 nS (A–C).
It is interesting that bifurcation in A changes
from two to four, but that from three to six follows.
In B, bifurcation changes from two to four, three to
six follows, and four to eight occurs finally. In C,
bifurcation occurs as in B and finally from five to
ten. As we increase the variable, the law determining
bifurcation develops as a linear and double-interlaced
increase.
The diagram of the amplitude shows that the peak
of the amplitude is located along the horizontal line,
and a trend toward the bottom is not clear. Once we
amplify the nadir of the amplitude only in our plane
rectangular coordinate system, bifurcation emerges.
We illustrate the respective three diagrams in Fig 5.
The gleak settings are the same as in Fig 4, i.e., 12.3,
12.5, and 12.7 nS (A–C, respectively).
Comparison of Figs. 4 and 5 shows that these panels
display similar patterns as those that we predicted
before, including the law described in the Discussion
and location of the bifurcation beginning in the next
Fig. 6. When we explored the detailed transition
modes in Fig. 5, we were surprised that, instead of
the “divergence” property of ISIs, the nadir of the
amplitude displays some “convergence” properties.
In Fig. 4, when we follow the tracks of the outward
C
A
B
0
0
0
7,20•10–3
1,20•10–2
9,00•10–3
8,18•10–3
1,30•10–2
1,01•10–2
9,16•10–3
1,39•10–2
1,12•10–2
1,01•10–2
1,49•10–2
1,23•10–2
1,11•10–2
1,58•10–2
1,34•10–2
1,21•10–2
1,68•10–2
1,45•10–2
1
1
1
2
2
2
3
3
3
s
s
s
nA
nA
nA
F i g. 4. Scatter plots for the dependence between ISIs, sec, and
Iinj , nA. In A – C, gleak = 12.3, 12.5, and 12.7 nS, respectively.
Р и с. 4. Діаграми розсіяння для залежності між тривалістю
міжімпульсних інтервалів (с) та Iinj (нА).
We discovered in more detail the influence of the
gleak values across different areas [7] including gradient
physical properties for ISIs, in particular a nadir of
the amplitude. In particular, the peak of the amplitude
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2014.—T. 46, № 2126
Y. ZH. GUAN, SH. Q. LIU, and Y. J. ZENG
bifurcation, all points are surely to be localized inside
the domain.
Relation between the injected current intensity and
trigger mechanisms. We notice that the location of the
bifurcation moves to opposite directions, while the
stimulation influence is manifested in a monotonic
increase or decrease. The trail of bifurcation looks like
a tree in Figs. 4 and 5. So, we choose the “root” value
to mark the location of bifurcation. This means that we
used the value of the current mostly at the right side
in Figs. 4 and 5. We computed nine points where the
gleak changed from 11.3 to 12.9 nS. We used MATLAB
to plot the points, and the points are connected by a
broken line. The result is shown in Fig. 6.
It is obvious that there is a linear relationship
between the gleak and the location. We developed a
formula for fitting [Eq. (2)]. As gleak increases, the
injected current must be increased so that ISI and
amplitude bifurcation can be found in a region shifted
toward the less intense current. The linear regression
formula looks like:
Iinj = 0.0121 · gleak – 0.137 (2)
DISCUSSION
We focused on the bursting patterns underlied by the
bifurcation phenomenon or chaotic impulsation in
the model used (Fig. 2). As was demonstrated, ISI
groupings vary from one to two, two to three, and so
A
7,20•10–3 8,18•10–3 9,16•10–3 1,01•10–2 1,11•10–2 1,21•10–2
mA
nA
B
9,00•10–3 1,01•10–2 1,12•10–2 1,23•10–2 1,34•10–2 1,45•10–2
mA
nA–55
–55
–50
–50
–45
–45
–40
–40
C
1,20•10–2 1,30•10–2 1,39•10–2 1,49•10–2 1,58•10–2 1,68•10–2
mV
nA–55
nA
nS–5
0
11,2 11,4 11,6 11,8 12 12,2 12,4 12,6 12,8 13
5
10
15
20
–50
–45
–40
F i g. 5. Scatter plots for the dependence between the amplitudes
(changes in the membrane potential), mV, and Iinj , nA. In A–C,
gleak = 12.3, 12.5, and 12.7 nS, respectively.
Р и с. 5. Діаграми розсіяння для залежності між амплітудою
(зміною мембранного потенціалу, мВ) та Iinj (нА).
F i g. 6. Graph of the dependence between gleak and Iinj for the
bifurcation position.
Р и с. 6. Графік залежності між gleak та Iinj, що визначає положення
біфуркації.
bifurcation, we cannot be sure whether the point is
inside the range or not. The situation means that if
we can build a mathematical model for outward
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2014.—T. 46, № 2 127
GRADIENT TRIGGER MECHANISMS RELATED TO BISTABILITY REGIMES
on. We recognized that this phenomenon means the
existence of bifurcations. Thus, we explored more
model physical properties to reveal the biological
parameters, such as ISI durations and amplitude (in
Fig. 3). We discovered more details during changing
the gleak across different areas [7] for the gradient
physical properties for ISIs (in Fig. 4) and the nadir
of the amplitude (in Fig. 5). We ignore the fact that
the peak of the amplitude appears almost along the
horizontal line, though it is an unexpected physical
characteristic in the model of the neuron. The result
of computation implies that (i) the neuron can
“recognize” the intensity of the input “information,”
(ii) the neuron can “tell” others about its working
level by its range for bifurcation, and (iii) there are
invisible triggers that exist in the mechanisms of
bistability regimes for this model. These findings
might explain why some phenomenon appears or
disappears unpredictably. In addition, we provided
the locations where the bifurcation begins and found a
linear dependence between the injected current and the
examined parameter, independently of the coefficients
in Eq. (1).
Acknowledgment.This work was supported by the Nati-
onal Natural Science Foundat ion of China (Grant
No. 11172103).
The authors, Y. Zh. Guan, Sh. Q. Liu, and Y. J. Zeng,
confirm that they have no conflict of interest.
Ї. Ж. Гуан1, Ш. К. Лю1, Ї. Дж. Зенг2
ГРАДІЄНТНІ ТРИГЕРНІ МЕХАНІЗМИ, ПОВ’ЯЗАНІ З РЕ-
ЖИМАМИ БІСТАБІЛЬНОСТІ, В МОДЕЛІ КОНТРОЛЮ
СЕРЦЕВИХ СКОРОЧЕНЬ У П’ЯВКИ
1 Південно-китайський технологічний університет, Гуанч-
жоу (Китай).
2 Пекінський технологічний університет (Китай).
Р е з ю м е
Ми досліджували пачкові імпульсні патерни, що формува-
лися на основі феноменів біфуркації, та хаотичну імпульс-
ну активність у комп’ютерній моделі керування серцеви-
ми скороченнями у п’явки. Ми спостерігали градієнтність
фізичних властивостей, що визначали характеристики по-
слідовностей імпульсів та амплітуду (зміщення мембранно-
го потенціалу), при невеликих змінах параметра gleak (про-
відності витоку). Було також виявилено, що існують різні
зони бістабільності. Результати комп’ютерного моделюван-
ня вказують на те, що, по-перше, в такому режимі може за-
безпечуватися класифікація інтенсивності вхідної інформа-
ції; по-друге, робочий рівень для нейрона визначаеться його
положенням у типовій біфуркації, і, по-третє, існують «не-
видимі» тригери, на яких базуються тонкі механізми моделі.
REFERENCES
1. P. Suffczynski, S. Kalitzin, and F. H. Lopes da Silva,
“Dynamics of non-convulsive epileptic phenomena modeled
by a bistable neuronal network,” Neuroscience, 126, No. 2,
467-484 (2004).
2. P. A. Tass and C. Hauptmann, “Therapeutic modulation of
synaptic connectivity with desynchronizing brain stimulation,”
Int. J. Psychophysiol., 64, No. 1, 53-61 (2007).
3. E. L. Peterson, “Generation and coordination of heartbeat
timing oscillation in the medicinal leech. I. Oscillation in
isolated ganglia,” J. Neurophysiol., 49, No. 3, 611–626 (1983).
4. E. L. Peterson, “Generation and coordination of heartbeat
timing oscillation in the medicinal leech. II. Intersegmental
coordination,” J. Neurophysiol., 49, No. 3, 627–638 (1983).
5. A. L. Hodgkin and A. F. Huxley, “A quantitative description
of membrane current and its application to conduction and
excitation in nerve,” J. Physiol., 117, No. 4, 500-544 (1952).
6. A. A. Hill, J. Lu, M. A. Masino, et al., “A model of a segmental
oscillator in the leech heartbeat neuronal network,” J. Comput.
Neurosci., 10, No. 3, 281-302 (2001).
7. G. S. Cymbalyuk, Q. Gaudry, M. A. Masino, and R. L. Ca-
labrese, “Bursting in leech heart interneurons: cell autonomous
and network based mechanisms,” J. Neurosci., 22/24, 10580-
10592 (2002).
8. A. Shilnikov, R. L. Calabrese, and G. Cymbalyuk, “Mechanism
of bistability: Tonic spiking and bursting in a neuron model,”
Phys. Rev. E, 71, 056214 (2005).
9. T. Malashchenko, A. Shilnikov, and G. Cymbalyuk, “Bistability
of bursting and silence regimes in a model of a leech heart
interneuron,” Phys. Rev. E, 84, 041910 (2011).
10. E. A. Arbas and R. L. Calabrese, “Ionic conductances
underlying the activity of interneurons that control heartbeat
in the medicinal leech,” J. Neurosci., 7, No. 12, 3945-3952
(1987).
11. E. A. Arbas and R. L. Calabrese, “Slow oscillations of
membrane potential in interneurons that control heartbeat in
the medicinal leech,” J. Neurosci., 7, 3953-3960 (1987).
12. W. J. Thompson and G. S. Stent, “Neuronal control of heartbeat
in the medicinal leech. II. Intersegmental coordination of
heart motor neuron activity by heart interneurons,” J. Comp.
Physiol., 111, 281-307 (1976)
13. R. L. Calabrese and A. R. Maranto, “Neural control of the
heart in the leech, Hirudo medicinalis. III. Regulation of
myogenicity by heart accessory neurons,” J. Comp. Physiol.,
154, 393-406 (1984).
14. J. E. Lisman, “Bursts as a unit of neural information: Making
unreliable synapses reliable,” Trends Neurosci., 20, No. 1, 38-
43 (1997).
15. E. M. Izhikevich, “Neural excitability, spiking and bursting,”
Int. J. Bifurc. Chaos, 10, No. 6,: 1171-1266 (2000).
16. J. Rinzel and Y. S. Lee, “Dissection of a model for neuronal
parabolic bursting,” J. Math. Biol., 25, No. 6, 653-675 (1987).
17. A. Sherman and J. Rinzel, “Rhythmogenic effects of weak
electrotonic coupling in neuronal models,” Proc. Natl. Acad.
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2014.—T. 46, № 2128
Y. ZH. GUAN, SH. Q. LIU, and Y. J. ZENG
Sci. USA, 89, No. 6, 2471-2474 (1992).
18. E. Av-Ron, H. Parnas, and L. A. Segel, “A basic biophysical
model for bursting neurons,” Biol. Cybern., 69, No. 1, 87-95
(1993).
19. M. E. Rush and J. Rinzel, “Analysis of bursting in a thalamic
neuron model.” Biol. Cybern., 71, No. 3, 281-291(1994).
20. J. Hounsgaard and O. Kiehn, “Serotonin-induced bistability of
turtle motoneurones caused by a nifedipine-sensitive calcium
plateau potential,” J. Physiol., 414, 265-282 (1989).
21. P. Fuentealba, I. Timofeev, M. Bazhenov, et al., “Membrane
bistability in thalamic reticular neurons during spindle
oscillations, “ J. Neurophysiol., 93, No. 1, 294-304 (2005).
22. Y. Loewenstein, S. Mahon, P. Chadderton, et al., “Bistability
of cerebellar Purkinje cells modulated by sensory stimulation.”
Nat. Neurosci., 8, 202-211(2005).
|