Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study
Simple spikes and complex spikes are two distinguishing features in neurons of the cerebellar cortex; the motor learning and memory processes are dependent on these firing patterns. In our research, the detailed firing behaviors of Purkinje cells were investigated using a computer compartmental n...
Gespeichert in:
| Veröffentlicht in: | Нейрофизиология |
|---|---|
| Datum: | 2015 |
| Hauptverfasser: | , , , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізіології ім. О.О. Богомольця НАН України
2015
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148138 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study / X. Zhang, S. Q. Liu, H. X. Ren, Y. Wen, Y. J. Zeng // Нейрофизиология. — 2015. — Т. 47, № 1. — С. 4-13. — Бібліогр.: 17 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-148138 |
|---|---|
| record_format |
dspace |
| spelling |
Zhang, X. Liu, Q. Ren, H. X. Wen, Y. Zeng, Y.J. 2019-02-17T10:43:13Z 2019-02-17T10:43:13Z 2015 Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study / X. Zhang, S. Q. Liu, H. X. Ren, Y. Wen, Y. J. Zeng // Нейрофизиология. — 2015. — Т. 47, № 1. — С. 4-13. — Бібліогр.: 17 назв. — англ. 0028-2561 https://nasplib.isofts.kiev.ua/handle/123456789/148138 577.352.5:612.827 Simple spikes and complex spikes are two distinguishing features in neurons of the cerebellar cortex; the motor learning and memory processes are dependent on these firing patterns. In our research, the detailed firing behaviors of Purkinje cells were investigated using a computer compartmental neuronal model. By means of application of numerical stimuli, the abundant dynamical properties involved in the multifarious firing patterns, such as the Max-Min potentials of each spike and period-adding/period-doubling bifurcations, appeared. Neuronal interspike interval (ISI) diagrams, frequency diagrams, and current-voltage diagrams for different ions were plotted. Finally, Poincare mapping was used as a theoretical method to strongly distinguish timing of the above firing patterns. Our simulation results indicated that firing of Purkinje cells changes dynamically depending on different electrophysiological parameters of these neurons, and the respective properties may play significant roles in the formation of the mentioned characteristics of dynamical firings in the coding strategy for information processing and learning. Генерація простих та складних потенціалів дії є специфіч- ною властивістю нейронів мозочкової кори; моторне навчання і проце си формування пам’яті за лежать від генерації даних патернів розряду. В нашій роботі ми провелидетальне до слідження проце сів генерації імпульсної активності клітинами Пуркін’є з використанням компартментної (включаючи сому) моделі нейрона. В умовах прикладання оцифрованих стимулів у модельованого нейрона проявлявся багатий набір динамічних властивостей, що зумовлювало генерацію різноманітних розрядних патернів; це відбивало сь у відповідних діаграмах максимальних/мінімальнихпотенціалів для кожного піку та появі біфуркацій із феноменами додання або подвоєння періодів. Були побудовані діаграми міжімпульсних інтервалів, значень частоти та залежностей струм–потенціал для різних іонів. Нарешті, побудова мап Пуанкаре була використана як теоретичний метод для переконливої диференціації часових характеристик зазначених вище розрядних патернів. Як показали результати нашого моделювання, розрядна активність клітин Пуркін’є динамічно змінюється залежно від варіації електрофізіологічних параметрів цих нейронів, і відповідні властивості можуть відігравати істотну роль у формації згаданих вище характеристик динамічних розрядів, що мають відношення до стратегії кодування в перебігу обробки інформації та процесів навчання. The authors would like to acknowledge the generous support by the National Undergraduates Innovating Experimentation Project of China, No. 111056144. en Інститут фізіології ім. О.О. Богомольця НАН України Нейрофизиология Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study Динамічні властивості клітин Пуркін’є, що мають різні електрофізіологічні параметри: модельне дослідження Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study |
| spellingShingle |
Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study Zhang, X. Liu, Q. Ren, H. X. Wen, Y. Zeng, Y.J. |
| title_short |
Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study |
| title_full |
Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study |
| title_fullStr |
Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study |
| title_full_unstemmed |
Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study |
| title_sort |
dynamic properties of purkinje cells having different electrophysiological parameters: a model study |
| author |
Zhang, X. Liu, Q. Ren, H. X. Wen, Y. Zeng, Y.J. |
| author_facet |
Zhang, X. Liu, Q. Ren, H. X. Wen, Y. Zeng, Y.J. |
| publishDate |
2015 |
| language |
English |
| container_title |
Нейрофизиология |
| publisher |
Інститут фізіології ім. О.О. Богомольця НАН України |
| format |
Article |
| title_alt |
Динамічні властивості клітин Пуркін’є, що мають різні електрофізіологічні параметри: модельне дослідження |
| description |
Simple spikes and complex spikes are two distinguishing features in neurons of the cerebellar
cortex; the motor learning and memory processes are dependent on these firing patterns.
In our research, the detailed firing behaviors of Purkinje cells were investigated using a
computer compartmental neuronal model. By means of application of numerical stimuli,
the abundant dynamical properties involved in the multifarious firing patterns, such as the
Max-Min potentials of each spike and period-adding/period-doubling bifurcations, appeared.
Neuronal interspike interval (ISI) diagrams, frequency diagrams, and current-voltage diagrams
for different ions were plotted. Finally, Poincare mapping was used as a theoretical method
to strongly distinguish timing of the above firing patterns. Our simulation results indicated
that firing of Purkinje cells changes dynamically depending on different electrophysiological
parameters of these neurons, and the respective properties may play significant roles in the
formation of the mentioned characteristics of dynamical firings in the coding strategy for
information processing and learning.
Генерація простих та складних потенціалів дії є специфіч-
ною властивістю нейронів мозочкової кори; моторне
навчання і проце си формування пам’яті за лежать
від генерації даних патернів розряду. В нашій роботі
ми провелидетальне до слідження проце сів генерації
імпульсної активності клітинами Пуркін’є з використанням
компартментної (включаючи сому) моделі нейрона. В
умовах прикладання оцифрованих стимулів у модельованого
нейрона проявлявся багатий набір динамічних
властивостей, що зумовлювало генерацію різноманітних
розрядних патернів; це відбивало сь у відповідних
діаграмах максимальних/мінімальнихпотенціалів для
кожного піку та появі біфуркацій із феноменами додання
або подвоєння періодів. Були побудовані діаграми
міжімпульсних інтервалів, значень частоти та залежностей
струм–потенціал для різних іонів. Нарешті, побудова
мап Пуанкаре була використана як теоретичний метод
для переконливої диференціації часових характеристик
зазначених вище розрядних патернів. Як показали
результати нашого моделювання, розрядна активність
клітин Пуркін’є динамічно змінюється залежно від варіації
електрофізіологічних параметрів цих нейронів, і відповідні
властивості можуть відігравати істотну роль у формації
згаданих вище характеристик динамічних розрядів, що
мають відношення до стратегії кодування в перебігу
обробки інформації та процесів навчання.
|
| issn |
0028-2561 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148138 |
| citation_txt |
Dynamic Properties of Purkinje Cells Having Different Electrophysiological Parameters: A Model Study / X. Zhang, S. Q. Liu, H. X. Ren, Y. Wen, Y. J. Zeng // Нейрофизиология. — 2015. — Т. 47, № 1. — С. 4-13. — Бібліогр.: 17 назв. — англ. |
| work_keys_str_mv |
AT zhangx dynamicpropertiesofpurkinjecellshavingdifferentelectrophysiologicalparametersamodelstudy AT liuq dynamicpropertiesofpurkinjecellshavingdifferentelectrophysiologicalparametersamodelstudy AT renhx dynamicpropertiesofpurkinjecellshavingdifferentelectrophysiologicalparametersamodelstudy AT weny dynamicpropertiesofpurkinjecellshavingdifferentelectrophysiologicalparametersamodelstudy AT zengyj dynamicpropertiesofpurkinjecellshavingdifferentelectrophysiologicalparametersamodelstudy AT zhangx dinamíčnívlastivostíklítinpurkínêŝomaûtʹrízníelektrofízíologíčníparametrimodelʹnedoslídžennâ AT liuq dinamíčnívlastivostíklítinpurkínêŝomaûtʹrízníelektrofízíologíčníparametrimodelʹnedoslídžennâ AT renhx dinamíčnívlastivostíklítinpurkínêŝomaûtʹrízníelektrofízíologíčníparametrimodelʹnedoslídžennâ AT weny dinamíčnívlastivostíklítinpurkínêŝomaûtʹrízníelektrofízíologíčníparametrimodelʹnedoslídžennâ AT zengyj dinamíčnívlastivostíklítinpurkínêŝomaûtʹrízníelektrofízíologíčníparametrimodelʹnedoslídžennâ |
| first_indexed |
2025-11-25T12:51:38Z |
| last_indexed |
2025-11-25T12:51:38Z |
| _version_ |
1850514895208972288 |
| fulltext |
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 14
UDC 577.352.5:612.827
X. ZHANG,1 S. Q. LIU,1 H. X. REN,1Y. WEN,1 and Y. J. ZENG2
DYNAMIC PROPERTIES OF PURKINJE CELLS HAVING DIFFERENT
ELECTROPHYSIOLOGICAL PARAMETERS: A MODEL STUDY
Received December 2, 2013
Simple spikes and complex spikes are two distinguishing features in neurons of the cerebellar
cortex; the motor learning and memory processes are dependent on these firing patterns.
In our research, the detailed firing behaviors of Purkinje cells were investigated using a
computer compartmental neuronal model. By means of application of numerical stimuli,
the abundant dynamical properties involved in the multifarious firing patterns, such as the
Max-Min potentials of each spike and period-adding/period-doubling bifurcations, appeared.
Neuronal interspike interval (ISI) diagrams, frequency diagrams, and current-voltage diagrams
for different ions were plotted. Finally, Poincare mapping was used as a theoretical method
to strongly distinguish timing of the above firing patterns. Our simulation results indicated
that firing of Purkinje cells changes dynamically depending on different electrophysiological
parameters of these neurons, and the respective properties may play significant roles in the
formation of the mentioned characteristics of dynamical firings in the coding strategy for
information processing and learning.
Keywords: Purkinje cell, computer compartmental model, Max-Min potential, interspike
intervals (ISIs), current-voltage diagram, frequency, Poincare mapping.
1 South China University of Technology, Department of Mathematics,
Guangzhou, China.
2 Beijing University of Technology, Biomedical Engineering Center, Beijing,
China.
Correspondence should be addressed to S. Q. Liu or Y. J. Zeng
(e-mail: yjzeng@bjut.edu.cn)
INTRODUCTION
Purkinje cells of the cerebellar cortex are some of the
largest neurons in the brains of mammals (including
humans) [1]. These cells have an intricately elaborated
dendritic arborization characterized by an enormous
number of dendritic spines. Purkinje cells receive
two main excitatory inputs, from climbing fibers
and parallel fibers, and two inhibitory inputs, from
basket cells and stellate cells [2, 3]. The literature
on the Purkinje cell models and simulation of their
spiking behaviors is very rich. In particular, it was
found in detailed models of these cells that the Ca2+
dynamics are effectively controlled by Ca2+-activated
K+ channels, and that a compensating mechanism
largely eliminates the effect by removing diffusion
in a model on the Ca2+ dynamics over multiple time
scales [4]. Transition from bursting to high-frequency
single spikes in a reduced mathematical model of a
cerebellar Purkinje cell was studied; some results on
the influence of tetrodotoxin (TTX) and cAMP on the
activity of Purkinje cells were also described [5, 6].
Ion currents underlying generation of spontaneous
action potentials (APs) were examined by Raman [7].
There are two distinct forms of electrophysiological
activity of Purkinje cells. Simple spikes occur at
rates of 17 to 50 sec–1 [7], while complex spikes are
generated much more rarely, with a 1-3 sec–1 frequency.
The latter spikes are characterized by an initial
prolonged high-amplitude component followed by a
high-frequency bursting of smaller-amplitude APs. The
two electrophysiological activities are closely related
to the functioning of sodium and calcium channels. It
was recently shown that activation of climbing fibers
terminating on the Purkinje cell can shift its activity
from a quiet state to a spontaneously active state and,
vice versa, can serve as a type of the toggle switch [7].
However, all the papers mentioned above were
based on examination of a complete Purkinje cell, and
the impact of definite separate electrophysiological
parameters has not been studied in these cases. Based
on the anatomical data, we constructed a computer
single somatic compartmental model allowing us to
separately study the influences of external stimuli, ion
conductances, and temperature on firing patterns of the
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 1 5
DYNAMIC PROPERTIES OF PURKINJE CELLS
Purkinje cell. Then we analyzed the Max-Min potential,
neuronal ISIs observed under different situations,
changes in the spiking frequency, and current-voltage
diagrams for different ions. Finally, we exploited the
Poincare mapping to further distinguish different firing
patterns generated by the model of the Purkinje cell used.
METHODS
To deeply examine the complex spike phenomenon,
we in this study employed some earlier papers to
describe the necessary ion channels and equations [8-
10]. The somatic compartment includes eight channel
types, namely two Na+ channels, three Ca2+ channels,
two K+ channels, and leak channels; the details are
shown in Fig. 1.
According to the classic Hodgkin-Huxley model
[11], the description of the membrane potential in this
neuron is performed using the following equation:
1
2( )
M
dV
ext NaF NaP CaP CaT CaE KD Kh Ldt C I I I I I I I I I= − − − − − − − −
where V represents the membrane potential, the
membrane capacitance CM is 0.8 μF/cm
2, Iext and IL
represent the external current and the leak current, and
other components separately represent all kinds of ion
currents. The entire ion conductance can be depicted
by classic Hodgkin-Huxley formalism:
1 2
1 1
3 3
1 1 2 2 2 3 2
4 4 5 5 6 6
7
1 1
( ), ( ), ( ),
( ), ( ), ( ),
( ), ( );
( )[ ( )(1 ) ( ) ],
NaF NaF NaF NaP NaP NaP CaP CaP CaP
CaT CaT CaT CaE CaE CaE KD KD KD
Kh Kh Kh L L L
dm dm
m mdt dt
I g m h V V I g m V V I g m V V
I g m h V V I g m h V V I g m h V V
I g m V V I g V V
T V m V mφ α β φ
= − = − = −
= − = − = −
= − = −
= − − =
2 2
3 4
3 3 4 4
5 6
5 5 6 6
2 2
3 3 4 4
5 5 6 6
7 7 7 7
( )[ ( )(1 ) ( ) ],
( )[ ( )(1 ) ( ) ], ( )[ ( )(1 ) ( ) ],
( )[ ( )(1 ) ( ) ], ( )[ ( )(1 ) ( ) ],
0.8*(m + mexp*(minf-m )+0.2*(m + n
m m
dm dm
m m m mdt dt
dm dm
m m m mdt dt
T V m V m
T V m V m T V m V m
T V m V m T V m V m
m
α β
φ α β φ α β
φ α β φ α β
− −
= − − = − −
= − − = − −
=
1 4
1 1 4 4
5 6
5 5 6 6
1 2
7
1 1 4 4
5 5 6 6
exp*(minf-m )),
( )[ ( )(1 ) ( ) ], ( )[ ( )(1 ) ( ) ],
( )[ ( )(1 ) ( ) ], ( )[ ( )(1 ) ( ) ],
( ) 35/exp((v+5)/(-10)), ( )= 200/(1+exp((v-
dh dh
h h h hdt dt
dh dh
h h h hdt dt
m m
T V h V h T V h V h
T V h V h T V h V h
V V
φ α β φ α β
φ α β φ α β
α α
= − − = − −
= − − = − −
=
3 4
5 6
1 2
3
18)/(-16))),
( ) 8.5/(1+exp((v-8)/(-12.5))), = 2.6/(1+exp((v+21)/(-8))),
2.6/(1+exp((v+7)/(-8))), 8.5/(1+exp((v+17)/(-12.5))),
( ) 7/exp((v+65)/20), ( ) 25/(1+exp((v+58)/8)),
( ) 35/(1
m m
m m
m m
m
V
V V
V
α α
α α
β β
β
=
= =
= =
=
4
5 6
1 4
5 6
+exp((v+74)/14.5)), = 0.18/(1+exp((v+40)/4)),
= 0.18/(1+exp((v+26)/4)), = 35/(1+exp((v+99)/14.5)),
( ) 0.225/(1+exp((v+80)/10)), 0.0025/(1+exp((v+40)/8)),
0.0025/(1+exp((v+32)/8)),
m
m m
h h
h h
V
β
β β
α α
α α
= =
=
1 4
5 6
0.0015/(1+exp((v+89)/8)),
( ) 7.5/exp((v-3)/(-18)), = 0.19/(1+exp((v+50)/(-10))),
= 0.19/(1+exp((v+42)/(-10))), = 0.0055/(1+exp((v+83)/(-8))),
mexp 1 - exp(tinc/38),minf = 1/(1+exp((v+78)/7
h h
h h
Vβ β
β β
=
=
= )), nexp = 1 - exp(tinc/319),
tinc -dt* ( ), ( ) 3exp((celsius - 37)/10).T Tφ φ= =
where mn1 and hn2 (n1 = 1~7, n2 = 1, 4, 5, 6) are
gating variables representing the activation and
inactivation of different ion
4433
and,,,
nnnn mmmm βαβα
channels, (n3 = 1~6, n4 = 1, 4, 5, 6) are some necessary
parameters of the V. The equilibrium potentials for
each ion channel and leak potential are VNaF, VNaP, VCaP2,
VCaE, VCaT, VKD, VKh, and VLeak. Varied ion conductances
are gNaF, gNaP, gCaP2, gCaE, gCaT, gKD, gKh, and gLeak; Φ(T)
is the temperature factor. Some parameters shown
above are the following: VNaF = 45 mV, VNaP = 45 mV,
VCaP2 = 135 mV, VCaT = 135 mV, VCaE = 135 mV,
VKD = –85 mV, VKh = –30 mV, and Vleak = –30 mV,
gNaF = 7.5, gNaP = 0.01, gCaP2 = 0.0045, gCaT = 0.0005,
gCaT = 0.0005, gKD = 0.0045, gKh = –30, and
gleak = 0.0003 msec/cm
2.
All simulations were performed using NEURON
software, and we used MATLAB software to process
the data.
RESULTS
Impact of Different External Stimuli. As was
described in the earlier paper [5, 12], when the soma is
separately stimulated with direct current, alternating
current, and square-wave current, the firing patterns
generated by the model are different (Fig. 2A). The
differences appear in the potential amplitude and the
spike frequency.
Furthermore, when we stimulated the soma with
0.29 nA direct current, the following was found
(Fig. 2B). When the firing patterns in the soma were
modified, the Ca2+ concentration changed, and all the
ion currents changed synchronously. The difference
F i g. 1. Scheme of the model of the Purkinje cell somatic
compartment; inside, the electrical diagram of this compartment is
shown.
Р и с. 1. Схема моделі соматичного компартмента клітини
Пуркін’є (всередині наведено електричну діаграму цього
компартмента).
Iinj Vm
Kh
KD
CaTC gNaF gCaT gKD gI
Leak
CaP2
CaE
NaF
NaP
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 16
X. ZHANG, S. Q. LIU, H. X. REN, et al.
40
40
40
20
20
20
0
0
0
0 100 200 300 400 500 600 700 800 900
msec
1000
–20
–20
–20
–40
–40
–40
–60
–60
–60
–80
–80
–80
mV
Potential
Current
CaE
Khh
NaF CaP
[Ca2+]
A
1 1
2
2
3
3
B
mV
mV
F i g. 2. Effects of changes in the stimulus intensity. A) Firing patterns in the soma at application of different stimuli: 0.29 nA (1),
0.29·(t%100>85) nA (2), and 0.29+0.29sin(0.02 t) nA (3), duration 1000 msec. B) Changes in the membrane potential (1), Ca2+ concentration
(2), and all stimulated ions currents (3).
Р и с. 2. Впливи змін інтенсивності стимуляції.
is that changes of the K+ current were positive, while
those of Ca2+ and Na+ currents were negative. We
should notice that the CaP current always remains at
a 0 nA level. The simulation results are similar to the
published experimental ones [5]. The firing pattern
changes significantly when the stimulus intensity
increases. When the stimulus was around 3 nA, a huge
transition appeared. After this, some oscillations were
observed, and the amplitude of the latter decreased.
The ISI diagram (Fig. 4A) shows that the global
change includes a slow fall and a fast increase. The
Max-Min potential diagram, however, demonstrates
the clear period-adding and period-doubling
bifurcation phenomena are manifested with increase
in the stimulus intensity. When the stimulus was
varied between 3.0 and 3.3 nA, the clear period-
doubling bifurcation phenomenon could be observed.
When the above parameter varied between 2.6 and
3.0 nA, the firing sequences in the soma showed
the period-doubling bifurcation and inverse period-
doubling bifurcation. The period-adding bifurcation
phenomenon was also initiated when the stimulus
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 1 7
DYNAMIC PROPERTIES OF PURKINJE CELLS
varied from 2.2 to 2.6 nA. In this case, the firing
sequence changed from period-2 spiking doubling
to period-4 spiking doubling. Moreover, when the
stimulus intensity increased, the frequency of spikes
increased from 0 to more than 50 sec-1; then, after a
slow increase, it dropped to 0.
In Fig. 4B, the same approach was used to analyze
behavior of a granule cell (compared to A). The ISI
diagram shows the distinct period-adding and period-
doubling bifurcation phenomena both from global and
local aspects, but the Max-Min potential diagram is
very complex.
The aforementioned analysis shows that the external
stimuli exert a powerful influence on somatic spike
activities. The observed rich diverse spike patterns
and some dynamic properties help us to further
understand the generation and conduction of neuronal
AP sequences in the cell under study.
Impact of Different CaT Conductances. Calcium
ions have been proved to play a very important role
in the functioning of cerebellar networks [4, 12,
13]. In particular, some analysis of the work of P/Q-
Type calcium channels related to calcium-activated
potassium channels was carried out [14-16]. The main
ion channels working in our neuronal compartmental
model were calcium, sodium, and potassium ones. For
different ions, the integral conductances reflect the
densities of different ion channels, and variations of
this conductance show the extent of opening of ion
channels and the processes of ion accumulation. These
factors affect the generation of APs; so, the detailed
research of the CaT conductance is expedient.
As we can see in Fig. 5A, 1, the ISI diagram
demonstrated an unclear period-doubling bifurcation;
at the same time, the Max-Min diagram (B, 1) shows
an exquisite period-adding phenomenon where the
CaT conductance changes from 0.46 to 0.67 msec/cm2.
The period-doubling and inverse period-doubling
bifurcation phenomena appear with increase in the
CaT conductance from 0.22 to 0.46 msec/cm2; and
this is followed by a period-adding phenomenon when
the CaT conductance varied from 0 to 0.22 msec/cm2.
During this process, the firing sequence in the soma
changes from period-2 spiking doubling to period-4
spiking doubling (sometimes, even to period-8 spiking
doubling). Besides this, we analyzed the frequency of
spiking related to different CaT conductances. The
general tendency was almost the same as that when
the stimulus intensity increased, but the conspicuous
difference is that the spike frequency is not zero when
the CaT conductance is 0 msec/cm2.
Considering the results of the above analysis,
it can be concluded that the influence of the CaT
conductance on neuronal activities is extraordinarily
important. Abundant spike patterns were observed
with variations of the CaT conductance. These facts
confirm that the state of the Ca2+ system is extremely
important the Purkinje cells.
40
20
0
0 050 50100 100150 150200 200250 250300 300350 350400 400450 450
msecmsec
500 500
–20
–40
–60
–80
mV mV
20
0
–20
–40
–60
–80
F i g. 3. Oscillations
in the soma. A-D) The
stimuli are 2.99, 3.16,
3.24, and 3.27 nA,
respectively; duration
500 msec.
Р и с. 3. Осциляції в
сомі.
A
C
B
D
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 18
X. ZHANG, S. Q. LIU, H. X. REN, et al.
20 –20
25 –10
30 0
35 10
40 20
45 30
50 40
5 –50
10 –40
15 –30
00 3.5 2.2 3.3
mA
mV
nA nA
msec
msec sec–1
1
2
–60
40
50
60
70
80
90
10
20
30
0
0 0
5
10
15
0 3.2
40
60
80
100
120
20
0
0 010 10 1020 2030 30 3040 4050 50 5060 6070 70 7080 8090 90 90100 100 100
–20
–10
0
10
20
30
–50
–40
–30
–60
–70
F i g. 4. ISI bifurcation diagrams, bifurcation diagrams of Max-Min potentials, and frequency diagrams (A-C, respectively); 1 and 2) effects
of changes in the stimulation intensity applied to the Purkinje cell soma (1) and to the granule cell (2).
Р и с. 4. Діаграми біфуркацій міжімпульсних інтервалів, біфуркацій максимальних/мінімальних потенціалів та змін частоти
(А–С відповідно) в умовах змін інтенсивності стимуляції соми клітини Пуркін'є (1) та гранулярної клітини (2).
Impact of Different NaF Conductances. It was
shown that the parameters of electrophysiological
activity of Purkinje cells are both sodium- and
calcium-dependent [6]. Based on the model approach,
Scutter and Bower [12] got some differential equations
and relevant parameters. Later on, Miyasho et al. [13]
subjected to further studies impacts of the some ion
conductances for Ca2+and K+ on the generation of APs
by comparing different firing patterns observed in
the soma and dendrites. Here, we mainly analyze the
impact of the NaF conductance.
In Fig. 5B, the ISI graph shows that there is a similar
period-2 doubling bifurcation phenomenon. According
to the Max-Min diagram, the firing sequence in the
soma shows a noticeable period-adding phenomenon
with increase in the NaF conductance from 5 to
31 msec/cm2. When the NaF conductance changes
from 26.7 to 27.1 msec/cm2, both the maximum and
minimum potentials change suddenly. This event is not
a gradually developing process; the potentials change
at once, and we still could not find the meticulous
process in the detailed analysis. The frequency
change is the same as that when the stimulus intensity
increased. This parameter rises from 0 to around
70 sec–1 and then decreases gradually to zero when the
NaF conductance reaches 31 msec/cm2.
The NaF conductance is very important for the
formation of firing patterns and transmission processes
of neuronal potential sequences. The approaches used
in the analysis (bifurcation diagrams of the Max-Min
potential and ISI diagrams) give us some inspirations
to deeply acquire the conduction of neuronal potential.
Impact of Different Temperature. Temperature
is a very important factor influencing firing patterns
generated by all kinds of neurons; however,
information on the temperature effects remains
relatively limited [17]. In our model, we estimated the
temperature impact on the Hodgkin-Huxley model of
a Purkinje cell.
In Fig. 5C, graph A shows the ISI diagram, while
graph B is the Max-Min diagram when the temperature
increases from 22 to 63°C. The firing sequences in
the soma show in this case obvious period-adding
and period-doubling phenomena. The firing is
transformed into the chaotic one, in particular, when
the temperature varied from 22 to around 40°C. Within
this interval, the firing sequence in the soma changed
from period-2 spiking doubling to period-4 spiking and
to period-16 spiking in a nonperiodic manner. In graph
C, the frequency of spikes gradually increased when
A B C
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 1 9
DYNAMIC PROPERTIES OF PURKINJE CELLS
the temperature changed from 0 to 50°C and rapidly
decreased to zero when the temperature changed from
50 to 60°C. Thus, there was no spiking under the
latter extreme conditions, where the model leaves the
adequate temperature range.
In any case, the temperature is an especially pivotal
factor with respect to the Purkinje cell (the prototype
of our model), and, with no doubts, it would also
crucially influence spike activity of other neuronal
types. Estimation of different firing sequences, the
bifurcation phenomenon of Max-Min potential, and
other results of simulation, would help us to study the
functioning of different neuronal species in subsequent
research.
Analysis of Current-Voltage Diagrams for Different
Ions. As is generally known according to the basic
Ohm’s law formula, the electric resistance is equal to a
voltage/current ratio. We can introduce this formula to
gain the “electric resistance” relationship in our model.
According to the former analyses, potassium channel
could be considered an analog of the inductance
circuit in some their characteristics; so, the electric
resistance can transformed into the classical current-
voltage mode [16]. We used this approach to plot the
current-voltage graphs. When we estimated the impact
of temperature (see above), the bifurcation diagrams
changed from chaotic to periodical modes, and we
used the temperature factor in the analysis of changes
in the electric resistance.
As can be seen in Fig. 6, the current-voltage
diagrams were significantly modified with the
temperature changes. As is obvious from the top
diagram, the firing pattern of the soma is disordered,
and the current-voltage diagrams for three ions are
50
–50
100
0
150
50
msec mV
0
–100
0 0 00.07 0.67 0.66
–40
–20
0
20
40
60
–80
–60
50
60
70
80
90
40
10
20
30
0
50
100
60
120
70
140
80
160
40
80
10
20
20
40
30
60
3031
30
0
0
55
5
–30
–20
–10
0
10
20
30
–50
–40
25
25
25
30
30
35
35
40
20
20
20
5
5
5
10
10
10
15
15
15
0
0
0
0
0 0
30
100
°C °C °C
100
Fig. 5. ISI bifurcation diagrams, bifurcation diagrams of Max-Min potentials, and frequency diagrams (A-C, respectively); from 1 to 3,
changes in the CaT conductance (1), NaF conductance (2), and temperature (3).
Р и с. 5. Діаграми біфуркацій міжімпульсних інтервалів, біфуркацій максимальних/мінімальних потенціалів та змін частоти
(А–С відповідно) в умовах змін CaT- і NaF-провідності, а також температури (1–3 відповідно).
1
2
3
A B C
sec–1
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 110
X. ZHANG, S. Q. LIU, H. X. REN, et al.
–10
0
10
20
150 200 250 300 350 400 450 500
–40
–30
–20
–50
–0.34
20
0
–20
–40
–60 100
200
300
400
500
–0.32
–0.30
–0.28
–0.26
–0.24
–0.22
–0.5
–1.0
–1.5
–2.0
1.4
1.2
1.0
0.8
0.6
0.4
20
0
0
–20
–40
–60 100
200
300
400
500 20
0
–20
–40
–60 100
200
300
400
500
–30
–25
–33
–28
–32
–27
–31
–26
–34
–29
150 200 250 300 350 400 450 500
–0.260
–0.255
–0.250
–0.245
–0.240
–0.235
–26
–28
–30
–32
–34 100
200
300
400
500
–26
–28
–30
–32
–34 100
200
300
400
500 –26
–28
–30
–32
–34 100
200
300
400
500
0.65
0.70
0.75
0.80
–0.44
–0.46
–0.42
–0.40
–0.38
–0.36
–0.34
F i g. 6. Current-voltage diagrams for different ions at different temperatures. A and B, at 36 and 63 °C; 1) for Ca2+; 2) for K+; 3) for Na+.
Р и с. 6. Діаграми струм–потенціал для різних іонів при змінах температури.
A
B
1
1
2
2
3
3
4
4
not regular. However, all the bottom diagrams were
transformed into irregular ellipses, and a periodical
mode appears. It seems that some bioelectrical
characteristics could be obtained easily, and the
electric resistances for different ion channels might be
associated with changes in the potential of the neuron
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 1 11
DYNAMIC PROPERTIES OF PURKINJE CELLS
examined.
By comparison of the effects of two different
temperatures, it can be found that the current-voltage
diagram shows certain specified changes in time.
Estimation of different shapes of current-voltage
diagrams (different electric resistances) provides a
convenient method to study the periodicity in electrical
activity of the cell under study.
Analysis of Poincare Mapping. The so-called
Poincare map is the intersection of a periodic orbit in
the given space of an uninterrupted dynamical system
with a certain lower-dimensional subspace (called the
Poincare surface of section), transversal to the flow of
the system.
According to the point of intersection between the
periodic orbit and Poincare surface of section, the
periodic motion in the phase space can be discovered.
If the point of intersection includes only one stationary
point or a few discrete points, the motion is periodic.
Conversely, if the point of intersection is dense and
unordered, the motion is chaotic. In addition, if the
point of intersection looks as a closed curve, the
motion is quasiperiodic. In our study, we used Poincare
mapping to explain some special cases of the firing
periodicity. Here, we mainly analyzed the impact of
temperature, while other parameters were left stable.
In Fig. 7B (the black area), there are several points
of intersection on the Poincare surface in response to
A. This corresponds to the functioning of the system
with several period-doubling spiking. In contrast, there
is only one stable point of intersection on the Poincare
surface in D (responses to C), which represents the
system with one period-doubling spiking.
Poincare mapping is rather practical to resolve some
problems related to the periodicity.
DISCUSSION
By studying the mechanisms of the formation of firing
patterns in the somatic compartmental model, one can
gain perceptions into a series of firing modes and some
dynamic behaviors. Such interpretations can help
researchers to understand what factors might impact
the above firing patterns. Based on a single somatic
compartmental model, we performed the dynamic
analyzes of behavior of a cortical Purkinje cell that
can generate complex spikes. This allowed us to gain
several theoretical results summarized below:
External s t imuli and changes in the NaF
conductance, CaT conductance, and temperature
would significantly influence the firing pattern in the
soma. The respective changes include modifications
of the distributions of interspike intervals (ISIs),
–40
–28
–30
–27
–20
–26
–10
–25
0
0
–10
–3
0
1.5
1.0
0.5
0
1.0
0.8
0.8
0.6
0.4
0.2
0
0.6
0.4
0.2
0 –20
–4 –2
2 4
–10
10 20 30
1
0.8
0.6
–1 0
10
20
A
D
B C
FE
–50
–29
–30
–31
–32
–33
–34
150 200 250 300 350 400 450 500
150 200 250 300 350 400 450 500
F i g. 7. Graphs of Poincare mapping. A, D) Firing patterns of the Purkinje cell at 36 and 63°C, respectively; B, E) Poincare surfaces of
section responds to two different systems; C, F) two different orbits in the given space. (axis X is the potential, axis Y is mNaF, and axis Z
is mNaP)
Р и с. 7. Результати побудови мап Пуанкаре.
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 112
X. ZHANG, S. Q. LIU, H. X. REN, et al.
Max-Min potential diagrams, frequency graphs, and
current-voltage diagrams for different ions. Changes
of the electrophysiological parameters listed above
induce period-doubling and period-adding bifurcation
phenomena. The frequency diagrams show that
spiking of the cell examined increases firstly and
then decreases. With increases of all kinds of the
electrophysiological parameters, the current-voltage
diagrams become limited into a regular area, and
the electric resistance tends to manifest periodical
oscillations.
Poincare mapping was applied to research the
behavior of a chaotic system, a periodic system,
and transformations between these two states. By
comparison of two different firing patterns and
points on the Poincare surface of section, it becomes
obvious that these two systems can be rather easily
differentiated, and some “periodic” problems can be
resolved.
From the aforementioned analyses, it is essentially
important to take into account that the repertoire of
firing patterns of the Purkinje cell is really abundant,
and some important information can be obtained
according to the results of alteration of different
factors in the model used. As is well known, a problem
of neural coding is one of the scientific frontier and
hot topics at present. So, the research of specificities
of generation of AP sequences and varying modes
among different firing patterns are becoming more
and more necessary and expedient. However, since
the whole neuron code researches are too complex
in bio-experiments, the model computation results
in mathematics will support the research a lot in the
future.
Acknowledgments. The authors would like to acknowledge
the generous support by the National Undergraduates Innovating
Experimentation Project of China, No. 111056144.
Appendix A. Supporting information. Supplementary data
associated with this article can be found in the online version at
http://neuromorpho.org.
This study was not associated with any experiments on
animals or tests involving human subjects; therefore, confir-
mation of compliance with existing ethical standards is not re-
quired from this aspect.
The authors of this study, X. Zhang, S. Q. Liu, H. X. Ren,
Y. Wen, and Y. J. Zeng, confirm that the research and publication
of the results were not associated with any conflicts regarding
commercial or financial relations, relations with organizations
and/or individuals who may have been related to the study, and
interrelations betveen co-authors of the article.
Кс. Жанг1, Ш. К. Лю1, Х. Кс. Рен1, Ю. Вен1, Я. Дж. Зенг2
ДИНАМІЧНІ ВЛАСТИВОСТІ КЛІТИН ПУРКІН’Є, ЩО
МАЮТЬ РІЗНІ ЕЛЕКТРОФІЗІОЛОГІЧНІ ПАРАМЕТРИ:
МОДЕЛЬНЕ ДОСЛІДЖЕННЯ
1 Південно-Китайський еехнологічній університет, Гуанжоу
(Китай).
2 Пекінський технологічний університет, Центр
біомедичного інженирінга, Пекін (Китай).
Р е з ю м е
Генерація простих та складних потенціалів дії є специфіч-
ною властивістю нейронів мозочкової кори; моторне
навчання і процеси формування пам’яті залежать
від генерації даних патернів розряду. В нашій роботі
ми провелидетальне дослідження процесів генерації
імпульсної активності клітинами Пуркін’є з використанням
компартментної (включаючи сому) моделі нейрона. В
умовах прикладання оцифрованих стимулів у модельованого
нейрона проявлявся багатий наб ір динамічних
властивостей, що зумовлювало генерацію різноманітних
розрядних патернів; це відбивалось у відповідних
діаграмах максимальних/мінімальнихпотенціалів для
кожного піку та появі біфуркацій із феноменами додання
або подвоєння періодів. Були побудовані діаграми
міжімпульсних інтервалів, значень частоти та залежностей
струм–потенціал для різних іонів. Нарешті, побудова
мап Пуанкаре була використана як теоретичний метод
для переконливої диференціації часових характеристик
зазначених вище розрядних патернів. Як показали
результати нашого моделювання, розрядна активність
клітин Пуркін’є динамічно змінюється залежно від варіації
електрофізіологічних параметрів цих нейронів, і відповідні
властивості можуть відігравати істотну роль у формації
згаданих вище характеристик динамічних розрядів, що
мають відношення до стратегії кодування в перебігу
обробки інформації та процесів навчання.
REFERENCE
1. D. Purves, G. J. Augustine, D. Fitzpatrick, et al., Neuroscience
(4th edition), Sinauer Ass., Sunderland (MA) (2008).
2. R. D Traub, S. J. Middleton, T. Knöpfel, and M. A. Whittington,
“Model of very fast (>75 Hz) network oscillations generated by
electrical coupling between the proximal axons of cerebellar
Purkinje cells,” Eur. J. Neurosci., 28, No. 8, 1603-1616 (2008).
3. J. I. Wadiche and C. E. Jahr, “Multivesicular release at
climbing fiber-Purkinje cell synapses,” Neuron, 32, No. 2,
301-313 (2001).
4. H. Anwar, S. Hong, and E. D. Schutter, “Controlling Ca2+-
activated K+ channels with models of Ca2+ buffering in Purkinje
cells,” Cerebellum, 11, No. 3, 681-693 (2012).
5. M. A. Kramer, R. D. Traub, and N. J. Kopell, “New dynamics
in cerebellar Purkinje cells: torus canards,” Phys. Rev. Lett.,
101, No. 6, 068103 (2008).
NEUROPHYSIOLOGY / НЕЙРОФИЗИОЛОГИЯ.—2015.—T. 47, № 1 13
DYNAMIC PROPERTIES OF PURKINJE CELLS
6. A. M. Swensen and B. P. Bean, “Ionic mechanisms of burst
firing in dissociated Purkinje neurons,” J. Neurosci., 23,
No. 29, 9650-9663 (2003).
7. I. M. Raman and B. P. Bean, “Ionic currents underlying
spontaneous action potentials in isolated cerebellar Purkinje
neurons,” J. Neurosci., 19, No. 5, 1663-1674 (1999).
8. A. Roth and M. Häusser, “Compartmental models of rat
cerebellar Purkinje cells based on simultaneous somatic and
dendritic patch-clamp recordings,” J. Physiol., 535, 445-472
(2001).
9. W. Akemann and T. Knopfel, “Interaction of Kv3 potassium
channels and resurgent sodium current influences the rate of
spontaneous firing of Purkinje neurons,” J. Neurosci., 26,
No. 17, 4602-4612 (2006).
10. W. M. Yamada, C. Koch, and P. R. Adams, Methods in
Neuronal Modeling, Cambridge, MIT Press, London (1987).
11. A. L. Hodgkin, A. F. Huxley, “Currents carried by sodium and
potassium ions through the membrane of the giant axon of
Loligo,” J. Physiol., 116, 449-472 (1952).
12. E. D. Schutter and J. M. Bower, “An active membrane model
of the cerebellar Purkinje cell I. Simulation of current clamps
in slice,” J. Neurophysiol., 71, No. 1. 375-400 (1994).
13. T. Miyasho, H. Takagi, H. Suzuki, et al., “Low-threshold
potassium channels and a low-threshold calcium channel
regulate Ca2+ spike firing in the dendrites of cerebellar Purkinje
neurons: A modeling study,” Brain Res., 891, 106-115 (2001)
14. U. Wolf, M. J. Rapoport, and T. A. Schweizer, “Evaluating
the affective component of the cerebellar cognitive affective
syndrome,” J. Neuropsychiat. Clin. Neurosci., 21, No. 3, 245-
53 (2009).
15. M. D. Womack, C. Chevez, and K. Khodakhah, “Calcium-
activated potassium channels are selectively coupled to P/Q-
type calcium channels in cerebellar Purkinje neurons,” J.
Neurosci., 24, No. 40, 8818-8822 (2004).
16. P. Wang, Q. Н. Song, Q. M Zeng, et al., “The research about
the inductance characteristics of potassium channel,” Beijing
Biomed. Eng., 22, No. 1, 77-79 (2003).
17. Y. Loewenstein, S. Mahon, P. Chadderton, et al., “Bistability
of cerebellar Purkinje cells modulated by sensory stimulation,”
Nat. Neurosci., 8, 202-211 (2005).
|