星形胶质细胞与前馈回路神经元网络的随机共振效应

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"该文研究了星形胶质细胞如何影响神经元网络中的随机共振现象，特别是在三元前馈环(FFL)结构中的表现。通过对神经元采用霍奇金-赫克斯利模型，星形胶质细胞用扩展的Li-Rinzel模型来模拟，并考虑了扩散效应。研究发现，星形胶质细胞的存在可以显著增强网络对微弱信号的处理能力，同时其效果对噪声水平和神经元-星形胶质细胞耦合强度有钟形依赖性。" 文章高亮点： 1. **神经元网络基序的建模**：研究采用了星形胶质细胞场中的前馈环网络模型，考虑了神经元与星形胶质细胞之间的相互作用。 2. **随机共振的研究**：探讨了网络模式在不同噪声条件下的随机行为，特别是在星形胶质细胞参与的情况下的表现。 3. **星形胶质细胞的角色**：星形胶质细胞通过随机共振机制提升了神经元网络的信号处理性能，这表明它们可能是神经系统的内在噪声源，有助于信号检测和处理。 4. **功能讨论**：文章讨论了星形胶质细胞在神经网络动态过程中的功能角色，强调了它们对噪声诱导动力学的贡献。关键词： - 神经元：神经系统的组成单元，负责信息的传递和处理。 - 基序：神经网络中的基本连接模式，揭示了网络动力学的基础。 - 星形胶质细胞：大脑中的非神经元细胞，与神经元相互作用，参与信号处理和调节。 - 钙：在神经元和星形胶质细胞中起关键信号转导作用的离子。 - 噪声：在神经网络中，噪声可以影响信号传输并可能导致随机共振现象。文章历史： - 收稿日期：2013年4月16日 - 修订版接收日期：2013年7月2日 - 接受日期：2013年7月7日 - 在线发布日期：2013年7月16日总结：这篇文章揭示了星形胶质细胞在神经网络中的重要性，尤其是在随机共振现象中的作用。通过模拟研究，作者指出星形胶质细胞不仅可以适应噪声环境，还能通过增强信号处理能力来优化神经网络的功能，这为理解和利用大脑的复杂动力学提供了新的视角。

The model equations are expressed as follows (Gerstner and

Kistler, 2002):

=dt ¼g

ðV

V

Þg

ðV

V

Þg

ðV

V

þI

ext

þ I

syn

þ I

noise

þ γ

astro

;

=dt ¼ α

ðV

Þð1n

Þβ

ðV

Þn

;

=dt ¼ α

ðV

Þð1m

Þβ

ðV

Þm

;

=dt ¼ α

ðV

Þð1h

Þβ

ðV

Þh

; ð1Þ

where i ¼1; 2; 3 denote the neuron index, V

represent the mem-

brane potentials for each neuron i in the motif, g

, g

, and g

stand for the maximal potassium, sodium, and leakage conduc-

tances per unit area, and V

, V

, and V

denote their correspond-

ing reversal potentials. Without loss of generality, the maximum

conductance of the potassium, solidum, and leakage channels are

set as g

¼36 ms/cm

, g

¼120 ms/cm

and g

¼0:3 ms/cm

respectively, and their corresponding reversal potentials are set as

¼12 mV, V

¼115 mV and V

¼10:6 mV, respectively. The

membrane capacitance is set as C

¼1 μ F/cm

. The parameters m

, n

represent the saturation values of the gating variables with

those terms in (1) governed by

¼0:01ðV

þ 50Þ=½1expð0:1ðV

þ 50ÞÞ;

¼0:125 expð0:125ðV

þ 60ÞÞ;

¼0:1ðV

þ 35Þ=½1expð0:1ðV

þ 35ÞÞ;

¼4:0 expð0:0556ðV

þ 60ÞÞ;

¼0:07 expð0:05ðV

þ 60ÞÞ;

¼1=½1 þ expð0:1ðV

þ 30ÞÞ:

In Eq. (1), the term I

ext

denotes the externally applied current.

The term I

syn

denotes the synaptic coupling current among the

neurons, which is modeled by the linear sum of the synaptic

current onto neuron i coming from all the pre-synaptic neurons,

i.e. I

syn

¼∑

ðEV

Þ, where g

describes the coupling strength of

the synapse from neuron j to neuron i. For simplicity, we assume

the coupling strength is identical for all the connections, denoted

as g

¼g for short. The reversal potential E¼0 mV if neuron j is

excitatory and E ¼80 mV if neuron j is inhibitory. When a

neuron ﬁres an action potential, the synaptic conductance r

of its postsynaptic target neuron is increased according to

¼r

þ 0:5ð1r

Þ, and in other time dr

=dt ¼r

=τ

, where the

refractory time τ

is set as 15 ms. The term I

noise

denotes the noisy

synaptic current representing the external or internal ﬂuctuations,

which is modeled by the Ornstein–Uhlenbeck (OU) process

(Destexhe and Rudolph, 2012)

noise

=dt ¼I

noise

ﬃﬃﬃﬃﬃﬃﬃ

ηðtÞ; ð2Þ

where ηðtÞ is a Gaussian white noise with zero mean and unit

variance. For simplicity, here we assume the noise intensity for

each neuron is the same, denoted as D. The constant τ

denotes the

relaxation time of the synaptic noise, which is ﬁxed at τ

¼2 ms.

The synaptic inward current I

astro

in (1) denotes the depolar-

ization current feedbacked from astrocyte to neuron i with a

coupling strength γ

. The value of I

astro

is related to the calcium

concentration in a two-dimensional ﬁeld. More details are given in

Section 2.2.

2.2. Astrocyte ﬁeld model

In literature, several models have been suggested to describe

the dynamics of the intracellular Ca

2þ

concentration (Li and Rinzel,

1994; Atri et al., 1993; Amiria et al., 2012; Höfer et al., 2002; Sneyd

et al., 1995). In the study of dressed neuron (a neuron coupled

by an astrocyte), the Li–Rinzel model is typically used, which

resembles the Hodgkin–Huxley mode for electrically excitable

membranes by replacing the transmembrane potential by Ca

2þ

concentration. In this model, the calcium only changes in the time

domain, and the signaling way from astrocyte to neuron is directly

modeled by a depolarization current determined by the time-

dependent Ca

2þ

.InSotero and Cancino (2010), a dynamical mean

ﬁeld was proposed to model the behavior of an ensemble of

interacting neurons and astrocytes, which gives a simple and

efﬁcient description of neural-glial mass. Yet, there still exist

rooms for modeling improvement. Firstly, the astrocytes are

spatially extended objects. The Ca

2þ

can be diffused spatially from

astrocyte to astrocyte through gap junctions, and thus the Ca

2þ

concentration in different spatial domains may be different

(Allegrini et al., 2009; Liu and Li, 2013). The original Li–Rinzel

model is unable to represent the Ca

2þ

concentration in spatial

domain and thus cannot give a more accurate evaluation of the

depolarization current I

astro

, as it is position-dependent. Secondly,

for a network motif coupled by several astrocytes, it is difﬁcult to

model the precise relations between astrocytes and neurons if

only the time domain is considered, since each astrocyte may

interact with several neurons in its neighborhood and meanwhile

each neuron may be affected by more than one astrocyte

(Newman, 2003). Considering these two aspects, it is more

reasonable to model the astrocytes as a two-dimensional ﬁeld,

and then the depolarization current from astrocyte to neuron can

be directly determined by Ca

2þ

in the spatial domain (Allegrini

et al., 2009; Liu and Li, 2013).

Following this motivation, in this paper, the Li–Rinzel model is

extended to a two-dimensional ﬁeld, and the diffusion effects of

gliotransmitters, IP

and Ca

2þ

from astrocyte to astrocyte are

introduced. Mathematically, the model equations can be expressed

as follows (Ullah et al., 2006):

∂½IP

=∂t ¼D

ð∂

½IP

=∂

x þ ∂

½IP

=∂

yÞþ1=τ

ð½IP



½IP

Þ

þ½v

ð½Ca

2þ

þð1αÞk

Þ=ðk

þ½Ca

2þ

Þ þ r

ΘðV

25mVÞ

ð3aÞ

∂½Ca

2þ

=∂t ¼ D

ð∂

½Ca

2þ

=∂

x þ ∂

½Ca

2þ

=∂

yÞJ

flux

J

leak

J

pump

ð3bÞ

dq=dt ¼ k

=ðk

þ½Ca

2þ



Þq: ð3cÞ

The ﬁrst terms in right hand side (RHS) of (3a) and (3b) reﬂect

the diffusion of IP

and Ca

2þ

, moving from astrocyte to astrocyte

via gap junction with diffusion coefﬁcients D

¼300ðμmÞ

1

and

¼20ðμmÞ

1

, respectively (Allegrini et al., 2009; Atri et al.,

1993). In the dynamical equation (3a) for IP

, the second term

describes the degradation of IP

with a degradation rate of 1=τ

and an equilibrium point ð½IP

Þ

, which accounts for a refractory

time in calcium responses of astrocytes; the third term describes

the production of IP

caused by Ca

2þ

, where the constants are

¼0:13 μMs

1

, k

¼1:1 μM, and α ¼0:8; the last term reﬂects the

glutamate-induced production of the cytosolic IP

with the pro-

duction rate r

via the Heaviside function Θ. To be speciﬁc, it is

activated when the membrane potential of the neuron j,

≥25 mV. It is also worth pointing out that as the inhibitory

neurons cannot release much excitatory neurotransmitter, gluta-

mate, which is necessary for establishing neuron–astrocyte inter-

action, the signaling way from neuron to astrocyte is not

considered for the inhibitory neurons, i.e. r

¼0 if neuron j is

inhibitory.

In the dynamical equation (3b) for Ca

2þ

concentration, J

flux

describes the rate of calcium released from ER into the cytoplasm

through the IP

receptors, which designates a positive feedback

controlled by the slow inactivation gate q, J

pump

denotes the rate of

calcium ﬂux pumping from the cytoplasm into ER and into the

extracellular space; J

leak

denotes the leakage ﬂux from the ER to

Y. Liu, C. Li / Journal of Theoretical Biology 335 (2013) 265–275 267

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