Neurocomputing 44–46 (2002) 103 – 108
www.elsevier.com/locate/neucom
Modeling the layer V cortical pyramidal neurons
showing theta-rhythmic ring in the presence
of muscarine
Tomoki Fukaia; b; ∗ , Katsunori Kitanoa , Toshio Aoyagic; b ,
Youngnam Kangb; d
a Department
of Information-Communication Engineering, Tamagawa University,
Tamagawagakuen, 6-1-1 Machida, Tokyo 194-8610, Japan
b CREST, JST, Japan Science and Technology Corporation, Japan
c Department of Applied Analysis and Complex Dynamical Systems, Kyoto University,
Sakyoku, Kyoto 606-8501, Japan
d Division of Brain Science, Health Science Center, Hokkaido Health Science University,
Sapporo, Japan
Abstract
In this study, we model a type of layer V pyramidal neurons in the prefrontal cortex of rats.
In the presence of muscarine, the pyramidal neurons exhibit a self-sustained rhythmic ring
in the theta frequency range (4 –7 Hz) in response to a brief depolarizing step current. Since
muscarine is known to activate the IP3 receptors of the intracellular Ca2+ store, the sustained
ring seems to occur due to the activation of a certain Ca2+ -dependent current, presumably
the Ca2+ -activated nonselective cationic current, during the IP3 -induced Ca2+ release. Here we
present a two-compartmental model neuron that shows the observed patterns of neuronal rec 2002 Elsevier Science B.V. All rights reserved.
sponses. Keywords: Theta oscillation; Compartment neuron model; Pyramidal neuron; Muscarinic modulation
The brain activity of behaving animals including rats, monkeys and humans often
shows theta oscillations. For instance, in the retention period of some working memory
tasks, the activity of prefrontal cortex and that of parietal cortex show an enhanced
coherence in the theta frequency range [10]. Thus, the oscillations are likely to play
∗ Corresponding author. Tel=fax.: +81-42-739-8434. Department of Information-Communication Engineering, Tamagawa University, Tamagawagakuen, 6-1-1 Machida, Tokyo 194-8610, Japan.
E-mail address: [email protected] (T. Fukai).
c 2002 Elsevier Science B.V. All rights reserved.
0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 2 ) 0 0 3 6 9 - 7
104
T. Fukai et al. / Neurocomputing 44–46 (2002) 103 – 108
Fig. 1. Responses of the layer-V pyramidal neurons to a brief step current (Kishi, Takahashi, Kang, unpublished observation). (a) The responses in the control condition and in the presence of muscarine. The
neuron was slightly hyperpolarized such that the step current elicited a single spike. (b) The sustained
theta-frequency ring triggered by a brief step current.
some cognitive roles. In the hippocampus, networks of mutually inhibiting interneurons, in which the GABAergic synaptic currents are mediated by fast and slow GABAA
receptors, were proposed as the source of the mixed theta–gamma oscillations [11]. Interneurons, however, do not have long-range synaptic connections and therefore are
unlikely to underlie the coherent theta oscillation between distant cortical areas. Therefore, the subtype of the layer-V pyramidal neurons modeled in this study is a likely
candidate for the pacemaker of the cortical theta rhythm.
In in vitro recording studies [4], the response prole of the pyramidal neuron to
a brief step current was obtained as shown in Fig. 1a. The response shows afterhyperpolarization (AHP) that immediately follows an action potential and lasts for
about 200 –300 ms. In the presence of muscarine, AHP is followed by DAP that often
lasts for more than 10 s. The DAP showed its peak in 200 –400 ms after an action potential. In the control condition without muscarine, the DAP disappears. The response
pattern with a single spike usually appeared when the neuron was slightly hyperpolarized by the current injection. When the neuron was in or slightly depolarized from the
resting state, self-sustained ring at 4 –5 Hz (Fig. 1b) was triggered by a step current
in the presence of muscarine. This theta rhythmic ring is considered to occur through
the alternate induction of AHP and DAP in the neuron.
In order to explain the response patterns of the neuron, we consider a twocompartment model neuron shown in Fig. 2. The somatic compartment involves the
spike generating sodium and potassium currents, the high-voltage-activated (HVA)
Ca2+ current, and Ca2+ -activated potassium (SK) current. On the other hand, the dendritic compartment involves the spike generating sodium and potassium currents, the
HVA Ca2+ current, Ca2+ -activated cationic current, and the intracellular Ca2+ store
that causes the IP3 (inositol 1,4,5-trisphosphate)-induced Ca2+ release (IICR). As explained later, the SK channel should not be activated by the Ca2+ released from the
store. The condition may be fullled if the store and the SK channel are spatially
separated in the pyramidal neurons. In the present model, they are located in dierent
compartments.
An action potential generated at the soma back-propagates into the dendrite and
depolarizes the membrane potentials of both compartments. The depolarization of the
membrane potentials leads to the Ca2+ entry through the HVA Ca2+ channels. In
T. Fukai et al. / Neurocomputing 44–46 (2002) 103 – 108
SOMA
DENDRITE
Na
Na
pump
+
HVA- Ca
Musc-R
2+
gc
IP3R
pump
store
2+
HVA- Ca 2+
pump
+
Na /Ca exchanger
105
2+
Na /Ca exchanger
leak
DK
SK (AHP)
DK
2+
Ca -dep. cation (ADP)
Fig. 2. Our two-compartment model of the pyramidal neuron. The solid arrows directing the inward (outward)
of the compartments represent the sodium (potassium) currents, while the dashed arrows indicate the calcium
ows.
the soma, the Ca2+ entry activates the SK channel and generates AHP. In real neurons,
the Ca2+ entry into the dendrites may lead to IICR in the presence of muscarine through
the following cascade of reactions. Muscarine activates the metabotropic muscarinic
receptors on the cell membrane and the activated muscarinic receptors generate G
proteins. The G proteins and the Ca2+ -sensitive activity of phospholipase C lead to
the synthesis of IP3 [8]. Thus produced IP3 and the Ca2+ entering through the HVA
channel activate the IP3 receptors on the Ca2+ store to release a large amount of Ca2+
(∼ several micro-moles) into the intracellular space. Thus, in the pyramidal neurons,
the IP3 synthesis may be time-locked to the action potential generation. Note that
the released Ca2+ gives a positive feedback to the IP3 synthesis. In this model, the
IP3 synthesis is not modeled explicitly: [IP3 ], which is 0:2 M at the base line level,
instantaneously rises up to 2 M in 20 ms after every action potential and decays back
to the base line level with a time constant of 4 s.
The open probability of the IP3 -induced Ca2+ release shows a bell-shaped dependence on the calcium concentration, giving a peak probability at [Ca2+ ] = 0:2 − 0:3 M.
In this model, we follow Li-Rinzel re-formulation [7] of De Young and Keizer model
[3], but ignore the Ca2+ -dependence of the reaction time for the slow IP3 -receptor
inactivation to avoid Ca2+ oscillations. An improved model involving more precise
descriptions of the IP3 -receptor kinetics and the IP3 synthesis is under construction and
will be reported elsewhere.
The Ca2+ -dependent cationic current and the SK current are described as follows in
terms of the Michaelis–Menten-type activation functions:
Icat = gcat
[Ca2+ ]
(V − Ecat );
[Ca2+ ] + Kd; cat
ISK = gSK
[Ca2+ ]4
(V − EK );
[Ca2+ ]4 + Kd;4 SK
where Ecat = −42 mV, Kd; cat = 10 M and Kd; SK = 0:4 M. The Ca2+ pumping into
the store is described by the Michaelis–Menten equation with Hill coecient 2 and
T. Fukai et al. / Neurocomputing 44–46 (2002) 103 – 108
0
-40
IP3 synthesis
-60
-80
0
2
40
20
0
-20
-40
-60
-80
(c)
8
4
2
0
10
0
2
(b)
40
20
0
[Ca 2+] (μM)
Membrane potential (mV)
(a)
4
6
Time [sec]
6
2+
control
Frequency (Hz)
-20
8
[Ca ] (µM)
Membrane potential (mV)
106
1
2
Time (sec)
3
4
5
(d)
8
10
6
5
4
3
2
0
4
6
Time (sec)
0 _ 10s
30_ 40s
0.04 0.06 0.08 0.1
g cat (mS/cm 2 )
Fig. 3. The single-spike responses of the model neuron to a brief step current. (a) The temporal proles
of the membrane potential are shown in the control condition and in the presence of the IP3 synthesis.
(b) A large amount of Ca2+ is released from the store in the presence of the IP3 synthesis. (c) The theta
rhythmic ring of the model neurons. The temporal prole of the membrane potential is displayed by a
gray thin curve, while the prole of [Ca2+ ] in the dendritic compartment is indicated by a thick curve. (d)
The average rates of the sustained ring as a function of the maximum conductance of the Ca2+ -dependent
cationic current.
the half-activation concentration of 0:2 M [2]. The Na+ =Ca2+ exchanger at the cell
membrane obeys the Michaelis–Menten equation with Hill coecient 1 and the halfactivation concentration of 5:0 M [5]. The densities of the pumps and exchangers are
determined such that the model may produce the experimentally observed temporal
prole of IICR. Ca2+ buers with the dissociation constants of 1 M and 1 mM are
included in the cytosol and the store, respectively. It is assumed that Ca2+ ions do not
diuse between dierent compartments.
In Fig. 3a, we show a typical example of the single-spike response of the model
neuron to a brief step current (10 ms duration) to the soma. The time evolution of
the calcium concentration in the dendritic compartment is also displayed (Fig. 3b).
The temporal prole of [Ca2+ ] resembles the experimentally observed prole in the
hippocampal CA1 pyramidal neurons [9]. If the SK channel was inuenced by the
released Ca2+ , AHP would appear not only immediately after an action potential, but
also several seconds later when [Ca2+ ] again becomes low in the dendrite. In this
model, the dendritic Ca2+ concentration easily oscillates for shorter time constants
(typically 10 s) of the Ca2+ -dependent IP3 -receptor inactivation. This is because the
store re-releases Ca2+ when Ca2+ returns to low concentrations. In some cases, the
period of the oscillations can be more than 20 s.
Fig. 3c displays the self-sustained ring induced by a brief step current in the model
neuron. In the present model, it is somewhat tricky to achieve a stable self-sustained
T. Fukai et al. / Neurocomputing 44–46 (2002) 103 – 108
107
ring, while in real neurons it often lasts for more than minutes. Because the IP3 receptors almost completely close at [Ca2+ ] of 10 M [4], the dendritic [Ca2+ ] must be
kept within a narrow range (a few micro-moles) during the sustained ring. Otherwise
the sustained ring ceases in several seconds. It is interesting to ask as to how real
neurons manage to overcome this situation. One possibility is that the Ca2+ -dependent
inactivation of IICR is suppressed by the generation of an excessive amount of intracellular IP3 (∼ 10 M or more) [6]. Whether such a large amount of IP3 is available
in realistic physiological conditions, however, remains to be examined. The rates of
the sustained ring are shown in Fig. 3d as a function of the maximum conductance
of the cationic current. The average rate over the initial 10 s is in general higher than
that over later 10 s, since [Ca2+ ] is much elevated for an initial few seconds. The rates
are in the range of theta frequency.
We have already proposed a model of chattering cells [1], a class of pyramidal cells
that exhibit fast rhythmic bursting at the gamma frequency range (30 –70 Hz). In the
model of chattering cells, high-frequency bursting (¿ 40 Hz) was achieved by the interplay between the Ca2+ -dependent cationic current and the SK current in the absence
of IICR. The present model, which involves IICR, suggests that the Ca2+ -dependent
cationic current provides the ionic mechanisms for both gamma and theta frequency
oscillations in the cerebral cortex.
Acknowledgements
K. Kitano was supported by Japanese Society for Promotion of Science.
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