ISNN 2010

Functional modeling of astrocytes in epilepsy:a feedback system perspective

Mahmood Amiri • Fariba Bahrami •

Mahyar Janahmadi

Received: 3 March 2010 / Accepted: 19 October 2010 / Published online: 9 November 2010

� Springer-Verlag London Limited 2010

Abstract Astrocytes, a subtype of glial cells, in the brain

provide structural and metabolic supports to the nervous

system. They are also active partners in synaptic trans-

mission and neuronal activities. In the present study, a

biologically plausible thalamocortical neural population

model (TCM) originally proposed by Suffczynski et al.

(Neuroscience 126(2):467–484, 2004) is extended by

integrating the functional role of astrocytes in the regula-

tion of synaptic transmission. Therefore, the original TCM

is modified to consider neuron-astrocyte interactions.

Using the modified model, it is demonstrated that the

healthy astrocytes are capable to compensate the variation

of cortical excitatory input by increasing their firing fre-

quency. In this way, they can preserve the attractor corre-

sponding to the normal activity. Furthermore, the

performance of the pathological astrocytes is also investi-

gated. It is hypothesized that one of the plausible causes of

seizures is the malfunction of astrocytes in the regulatory

feedback loop. That is, pathologic astrocytes are not any

more able to regulate and/or compensate the excessive

increase of the cortical input. Therefore, pathologic astro-

cytes lead to the emergence of paroxysmal attractor.

Results demonstrate that disruption of the homeostatic or

signaling function of astrocytes can initiate the synchro-

nous firing of neurons, suggesting that astrocytes might be

one of the potential targets for the treatment of epilepsy.

Keywords Astrocyte � Epilepsy �Thalamocortical model � Functional modeling

1 Introduction

Epilepsy is one of the most prevalent serious neurological

disorders, which affects more than 50 million people

worldwide. It is characterized by a sudden and intermittent

occurrence of synchronous activity which is termed seizure

or ictal event that impairs the normal function of the brain

[1]. These seizures are transient signs of hypersynchronous

neuronal activity in the brain. Several experimental and

clinical findings conducted over the past decade have

revealed several underlying mechanisms involved for epi-

lepsy [1, 2]. However, many issues are still under investi-

gation. There are now increasing evidences that an

improved understanding of the epileptic process can be

achieved through analysis of bidirectional interactions

between astrocytes and neuronal cells [3, 4]. Astrocytes are

the most abundant type and the best studied glial cells

which are connected together by gap junctions forming a

large functional syncytium. In different sites of brain, each

astrocyte can contact 100,000 synapses. Theses synapses

belong to, on average, six neurons and form ‘‘synaptic

islands’’ [3, 5]. Astrocytes regulate neurotransmitter

release, modulate synaptic transmission and control extra-

cellular fluid content and ion homeostasis [6]. Although

astrocytes cannot generate action potentials, they respond

to neuronal activities with an elevation of their intracellular

M. Amiri (&)

School of Electrical and Computer Engineering,

College of Engineering, University of Tehran, Tehran, Iran

e-mail: ma.amiri@ece.ut.ac.ir

F. Bahrami

CIPCE, School of Electrical and Computer Engineering,

College of Engineering, University of Tehran, Tehran, Iran

e-mail: fbahrami@ut.ac.ir

M. Janahmadi

Neuroscience Research Center and Department of Physiology,

Medical School, Shahid Beheshti Medical Sciences University,

Tehran, Iran

123

Neural Comput & Applic (2011) 20:1131–1139

DOI 10.1007/s00521-010-0479-0

calcium levels. In this way, not only astrocytes can sense

neuronal transmission but also their calcium elevation

leads to release of gliotransmitters including glutamate or

ATP (adenosine triphosphate), thereby can control synaptic

strength of neighboring neurons [7, 8]. This fact led to the

concept of the ‘‘tripartite synapse’’ [9–11] in which the

astrocyte, a third active element of the synapse, ‘‘listens

and responds’’ to the synapse [12, 13].

Computational models with biological plausibility are

interesting due to their capability to describe neurological

phenomena at various levels of complexity from cellular to

neural population level [14]. They can be utilized to

understand the basic mechanisms of neuron-astrocyte

interactions and for analyzing the effects of varying neu-

rological parameters on the dynamical characteristics of

model. All this will facilitate to get more thorough under-

standing of the behavior of the original system. In addition,

computational models are helpful to design experiments to

validate the resulted novel insights [15]. An increasing

number of models have been proposed to describe the loop

of information exchange between astrocytes and neurons.

Nadkarni and Jung proposed ‘‘dressed neuron’’ model and

provided a mathematical framework for the synaptic inter-

actions between neurons and astrocytes in the tripartite

synapse [16–18]. A general and nondimensional model for

the tripartite synapse is proposed by Postnov and colleagues

[19]. Recently this model was modified in order to apply it

to the spatially extended neuron-astrocyte network [20].

Garbo [21] proposed a minimal model consisting of a

pyramidal neuron, an interneuron and an astrocyte and

studied how the presence of ATP and interneuron affect the

overall neural activities. Ullah et al. considered the gap

junction mediated calcium waves and investigated coordi-

nation of cytosolic calcium oscillations in two coupled

astrocytes [22]. Finally, some of the other researchers

studied a single glial cell dynamics and characterized its

different calcium-signaling responses [23, 24].

Recent physiological findings show how astrocytes

regulate the synaptic strength and synaptic transmission via

uptake of neurotransmitters or release of gliotransmitters

[6–13]. In the light of these findings, in the present study, a

thalamocortical neural population model (TCM) while was

proposed originally by Suffczynski et al. [25] is modified

by integrating the role of astrocytes in the model under

normal and pathological conditions. The modified TCM

(MTCM) is then used to understand one of the basic

functional mechanisms that cause epileptic seizure. The

TCM describes the neuro-electrophysiology of cortical and

thalamic neural populations and was constructed at an

intermediate level. In this way, instead of simulating the

explicit behavior of individual neurons, a network of

interacting populations was considered. This approach

allows investigating the dynamical properties of system

and the role played by different mechanisms in normal and

pathological conditions more easily. The feedback mech-

anism organized by astrocytes in the structure of the

MTCM, ensures normal asynchronous behavior in spite of

being subjected to those abnormal perturbations inducing

synchronous activities. However, disturbing the mecha-

nism of synaptic transmission can lead to epilepsy. Con-

sequently, dysfunction in the feedback actions generated by

astrocytes can also produce abnormal hypersynchronized

oscillations and seizure-like activities.

The rest of the paper is organized as follows: in Sect. 2,

the mathematical description of the astrocyte model is

explained. The original TCM and its modified version i.e.

MTCM which integrates the role of astrocytes in regulation

of synaptic strengths are presented in Sect. 3. The results of

some simulations are discussed in Sect. 4. Finally, Sect. 5

concludes the paper.

2 Astrocyte model

Astrocytes are vital for normal neuronal functioning and

survival. They have a large number of receptors that are

used to get information about synaptic activity. Although

astrocytes do not have adequate voltage-gated sodium

channels to generate action potential, they are excitable

with respect to intracellular calcium [3, 8].

At the cellular level, the main mechanisms underlying

the tripartite synapse are as follows: neurotransmitters

(such as glutamate) released from the presynaptic neuron

are partially bound to the metabotrobic glutamate receptors

(mGluR) of the synaptic astrocytes. This leads to produc-

tion of inositol (1,4,5)-trisphosphate (IP3) which, in turn,

lead to the release of Ca2? into the astrocytic cytoplasm

from endoplasmic reticulum (ER). These calcium eleva-

tions propagate into nearby astrocytes as intercellular cal-

cium waves with the passage of second messengers through

gap junctions [6, 26]. As a consequence of the increased

intracellular Ca2? concentration, astrocyte releases glio-

transmitters including glutamate and ATP into the extra-

cellular space. The released gliotransmitters feedback onto

the pre and postsynaptic terminals, which implies that

astrocytes regulate synaptic information transfer [10–12].

To model the dynamics of the intracellular Ca2? waves

produced by astrocytes, a recently introduced dynamic

model of the astrocyte [19, 20] is used. This is a general

and simplified mathematical model for a small neuron-

astrocyte ensemble and considers the main pathways of

neuron-astrocyte interactions. The model will be useful to

study the main types of astrocyte responses and the

resulting dynamical patterns. This will allow us to predict

the changes in dynamical patterns with varying control

parameters, which will be introduced later in this section.

1132 Neural Comput & Applic (2011) 20:1131–1139

123

This model is explained with the following set of equations

[19, 20]:

scdc

dt¼ �c� c4f ðc; ceÞ þ ðr þ bSmÞ ð1Þ

ec scdce

dt¼ f ðc; ceÞ ð2Þ

f ðc; ceÞ ¼ c1

c2

1þ c2� c2

e

1þ c2e

� �c4

c42 þ c4

� �� c3ce ð3Þ

sSm

dSm

dt¼ 1þ tan h sSmðz� hSmÞ½ �ð Þ � 1� Smð Þ � Sm

dSm

ð4Þ

sGm

dGm

dt¼ 1þ tan h sGmðc� hGmÞ½ �ð Þ � 1� Gmð Þ � Gm

dGm

ð5Þ

where c is the calcium concentration in the astrocytic cyto-

plasm. ce denotes the calcium concentration within the

endoplasmic reticulum. The parameters ec and sc together

define the characteristic time for calcium oscillations. The

calcium influx from the extracellular space is sensitive to the

production of secondary messenger Sm (IP3), which is con-

trolled by the factor b. The initial state of the calcium oscil-

lation is controlled by the parameter r. The calcium exchange

between the cytoplasm and the endoplasmic reticulum is

defined by the nonlinear function f (c, ce). We set the control

parameters sSm; sGm; sSm; sGm; hSm; hGm; dSm; dGm to the

values used by [19]. An increase in the cytoplasmic calcium

concentration causes release of Gm, an astrocytic mediator.

In terms of the thalamocortical network model, Gm is the

output of the astrocyte. The interaction between astrocyte

and subpopulations is denoted with the parameter z that

shows the synaptic activity of the subpopulations of PY

(pyramidal) and IN (interneuron).

3 Modified thalamocortical model

In the present paper, we consider a lumped neural popu-

lation model based on the thalamocortical model (TCM)

proposed by Suffczynski et al. [25]. The TCM considers

the basic components involved in absence seizures and

therefore consists of two main modules: the thalamic

module and the cortical. Each individual module includes

two mutually interconnected neuronal subpopulations. This

model helps to explore the role of astrocytes in epilepsy

from a macroscopic point of view.

The thalamic module consists of RE (reticular thalamic)

and TC (thalamocortical) subpopulations and the cortical

module consists of PY (pyramidal cell) and IN (inter-

neuron) subpopulations. The interaction between these

subpopulations is facilitated via AMPA (a-amino-3-

hydroxy-5-methyl-4isoxazolepropionic acid) mediated the

fast excitatory synapses and GABAA (c-aminobutric acid)

and GABAB mediated both the fast and slow inhibitory

synapses. The schematic diagram of the TCM is shown in

Fig. 1a. Each subpopulation in the model denotes a lump of

neurons that share the same membrane voltage in the cor-

responding area. It is described by two variables: membrane

potential and firing rate. A sigmoid transfer function is used

to convert the mean membrane potential of the subpopu-

lation to its firing rate to make the output of that population.

In the thalamic subpopulation, for transformation between

mean membrane potential and firing density, the low-

threshold calcium current, IT, that underlies burst firing in

the thalamic cells is also considered. This current plays an

important role in the pathophysiology of epileptic seizures

in animals and humans [27, 28]. The intra-TCM inter-

actions, i.e., the interactions between the PY and the IN

subpopulations, are modeled by constant coupling gains.

The network has three external inputs. The pyramidal

population receives excitatory (glutamergic) input from

other cortical pyramidal neurons not included in the popu-

lation. The TC subpopulation receives excitatory input that

represents sensory inputs from the ascending afferents. RE

cells receives inhibitory inputs from the neighboring RE

cells because they are interconnected by mutual inhibitory

synapses. Cortical input PcxðtÞ is modeled by non-zero

mean white Gaussian noise which was mentioned before is

the output of other cortical pyramidal cells not incorporated

in the model. The output of the TCM is the mean membrane

potential of the pyramidal cells which simulates experi-

mental recordings of the local field potentials. Selecting

suitable mean value of the input signal PcxðtÞh ið Þ, the TCM

demonstrates normal activity. For further details, readers

are referred to [25, 27].

As previously was discussed, astrocytes monitor activ-

ity-dependent changes in the chemical environment of the

extracellular space shared with neurons and play a

dynamic role in regulation of synaptic transmission. It is

well established that transmitters released from neurons

can stimulate astrocyte, leading to release of glutamate,

ATP and other neuroactive substances from the astrocyte.

These gliotransmitters can affect neuronal excitability by

facilitating or by suppressing synaptic transmissions

between neurons [6, 7, 12]. To show the astrocyte-

dependent regulation of neural activity, the TCM is

extended and a new block that represents astrocytes is

added to the original TCM. This block indicates a popu-

lation of astrocytes that regulate internal synapses between

pyramidal and interneuron subpopulations. The modified

TCM (called MTCM) is shown in Fig. 1b. It should be

mentioned that all model parameters such as synaptic

kinetics, sigmoid parameters are similar to the original

TCM [25].

Neural Comput & Applic (2011) 20:1131–1139 1133

123

To monitor the synaptic activity, firing rate of the mean

pyramidal neuronal subpopulation FPYðtÞh i is estimated

from the actual firing rate FPYðtÞ as follows [14, 29]:

d FpyðtÞh idt

¼ a FPYðtÞ � FpyðtÞh i� �

ð6Þ

Based on the results of [14, 29], a time constant of 2 s

(a = 0.5) is used. To remove the DC level of the input

signal of the astrocytes, the estimated mean firing rate

FPYðtÞh ið Þ is subtracted from the actual firing rate FPYðtÞð Þ.The obtained signal is multiplied by the gain Ks and

constitutes the input signal of the astrocytes. This, in fact,

is equivalent to monitor the synaptic transmission by the

astrocytes in the nervous system. The astrocyte outputs

should control and regulate the synaptic activities of the

pyramidal and interneuron cells [11, 12]. This is modeled

by adjusting the strength of interneuron to pyramidal

neuron subpopulations inhibitory gain ðCIN!PYÞ and the

strength of pyramidal neuron to interneuron subpopulations

excitatory gain ðCPY!INÞ as follows:

CIN!PYðtÞ ¼ C�IN!PY þ KiðAs OutðtÞÞ ð7Þ

CPY!INðtÞ ¼ C�PY!IN � KeðAs OutðtÞÞ ð8Þ

where As OutðtÞis the output of the astrocytes and C�IN!PY

and C�PY!IN have the same nominal values as in the original

TCM. In other words, in the original TCM, the model

parameters including CIN!PYðtÞ and CPY!INðtÞ are constant

(they are set in nominal values) and the pyramidal neuron

subpopulation output changes with PCxh i; thereby the ori-

ginal TCM is called static: whereas, in the MTCM these

synaptic strengths are dynamically regulated by the astro-

cytes. In the feedback structure of the MTCM, the astro-

cytes control the balance between the excitation and

inhibition levels so that the pyramidal cell firing rate is

maintained in a normal range and accordingly compensates

the increase or decrease of the excitation and inhibition

levels.

4 Simulation results and discussions

In this section the result of the simulations are presented to

explore the role of normal and pathological astrocytes. It

should be pointed out that all the simulations were imple-

mented in Simulink on a personal computer with Intel(R)

Core(TM) 2 Duo CPU 2.66 GHz.

The TCM is a computational model of absence seizures.

Such seizures are paroxysmal losses of consciousness that

start and end suddenly and are accompanied by bilaterally

synchronous rhythmic spike and wave (SW) discharges

that can be recorded on the EEG. Based on the model

simulations and system analysis in [25], it was verified that

the TCM exhibits bistable dynamics, i.e., the normal and

paroxysmal state coexist for the same set of parameters.

Noise present in the network may switch the system

between these two types of behavior. Paroxysmal dis-

charges characterized by large amplitude oscillations take

place and finish spontaneously due to changes in the

intrinsic system parameters of the population [27].

Results from re-simulations of the static TCM

(CIN!PYðtÞ ¼ C�IN!PY and CPY!INðtÞ ¼ C�PY!IN) are shown

in Fig. 2a. In the static TCM, the firing-rate of pyramidal

neuron subpopulation increases and/or decreases with

cortical input PCxh i. Figure 2a clearly shows that changing

PCxh i alters the model output significantly since it has no

alternative to model the neural homeostasis which is

essential for the normal operation of the brain. However,

synaptic transmission and intrinsic properties must be

tuned through release or/and uptake of neurotransmitters

and ions in order to maintain the stable ongoing activity in

Fig. 1 a The original TCM

proposed by Suffczynski et al.

[25]. b The modified TCM

(MTCM) in which astrocytes

dynamically regulate the

synaptic strengths between

pyramidal and interneuron

subpopulations

1134 Neural Comput & Applic (2011) 20:1131–1139

123

neural circuits. Recent studies on communications between

astrocytes and neurons revealed that glutamate release

from single astrocyte may control simultaneously the

excitability of several neighboring pyramidal cells [30].

Figure 2b illustrates the output of MTCM with dynamic

gains CIN!PYðtÞ and CPY!INðtÞ for the changing input

PCxh i. It is observed that the astrocytes modify the synaptic

strengths appropriately and maintain the balance of exci-

tation and inhibition levels in order to track the changes of

PCxh i. This will prevent from the occurrence of spontane-

ous paroxysmal episodes. For the simulations shown in

Fig. 2b the value of the parameters are as follows:

Ks ¼ 3:5, Ki ¼ 3 and Ke ¼ 3:1. For this case, the output of

the astrocytes and the concentration of the intracellular

calcium are shown in Fig. 2c. This figure demonstrates that

by changing the value of PCxh i, the astrocytes regulate and

control the synaptic strengths properly by changing their

firing frequency. Increasing/decreasing PCxh i leads to the

increase/decrease of the firing rate of the astrocyte output.

These simulation data are compatible with experimental

findings showing that astrocytes modulate and regulate the

synaptic activities [6, 11, 12].

In addition to the physiological role of astrocytes in the

control of neuronal excitability and synaptic transmission,

they can contribute to the disorders of nervous system

[4, 30]. Experimental findings demonstrated that patho-

logical astrocytes undergo structural alterations which are

accompanied by variation in their functions [31]. These

findings imply that alterations in astrocyte properties,

specifically glutamate, K? and Ca2? homeostasis, could be

part of the pathophysiology of neuronal hyperexcitability,

which is a characteristic of epilepsy [4, 32]. Hence, here a

pathological condition is considered in which astrocytes

cannot carry out their responsibilities properly. One way to

simulate pathological astrocytes is simply reducing the

capability of astrocytes in monitoring the synaptic activi-

ties. This is physiologically corresponding to deficit in the

structure of astrocyte receptors or scarcity in bindings of

neurotransmitters with astrocyte receptors. This is simu-

lated by decreasing the value of Ks from 3.5 to 1.5 in the

MTCM with the same changing input PCxh i as in Fig. 2b.

The simulation for pathological astrocytes is shown in

Fig. 2d. It is hypothesized that one of the plausible cause of

seizures is pathology in the feedback structure generated by

the astrocytes. In this case, astrocytes cannot regulate and/

or compensate excessive increase of the cortical input

through release of gliotransmitters and/or uptake of neu-

rotransmitters and ions. This is in accordance with the

physiological findings showing that paroxysmal depolar-

ization can be induced by pathological astrocytes in neu-

rons through calcium signaling pathway [3, 30, 33]. Indeed,

before the development of seizures and in the absence of

other pathological changes, hypertrophy of astrocytes

during the process of epileptogenesis has been reported

[34].

To obtain more insights into the role of astrocytes in the

active control of neuronal activity and synaptic neuro-

transmission, the excitation level is plotted versus inhibi-

tion level of pyramidal cells for (a) the MTCM with

healthy astrocytes and (b) the MTCM with pathological

astrocytes in Fig. 3. The top (bottom) panels are related to

the time interval 50–60 s (120–130 s) of the corresponding

model in Fig. 2. In the context of epilepsy, there is a par-

ticular interest to the balance between excitation and

inhibition levels. In fact, numerous researches have shown

that ictal spike discharges initiate in an area of the cortex

that is excessively excitable. This increased excitability is

due to increased excitation, reduced inhibition or both [35,

36]. As a consequence, increasing the ratio of excitation to

inhibition influences the dynamics of the modeled popu-

lation of neurons which can produce sustained discharges

of spikes. In the MTCM, the excitation and inhibition

levels for the cortical module are described by AMPA

mediated excitatory synapses and GABAA and GABAB-

mediated inhibitory synapses. It should be mentioned that

in Fig. 3, for simplicity, only the AMPA excitatory syn-

apses are plotted against the GABAA inhibitory synapses.

As can be seen in Fig. 3a, in spite of abnormal increase

in cortical excitatory input, healthy astrocytes maintain

the balance between excitation and inhibition through

regulation of excitatory and inhibitory neurotransmitter

concentrations in the synaptic cleft. This is achieved via

mechanisms that are responsible for release of gliotrans-

mitters and uptake of neurotransmitters. However, Fig. 3b

shows considerable changes in the level of excitation and

inhibition for the case of pathological astrocytes. Signifi-

cant decrease in the inhibition and to some extent increase

in the excitation levels are the result of the pathological

condition in astrocytes. This is consistent with experi-

mental findings where increased extracellular levels of

glutamate have been found in epileptogenic foci [37].

Hence, it is logical that several drugs impacting

GABAergic synaptic transmission are usually used in the

therapy of epilepsy disorders. In this issue, during gluta-

matergic neurotransmission, glutamate is released by neu-

rons into the synaptic cleft. This glutamate is mainly

exported and taken up into astrocytes by the glutamate

transporters where it is converted into glutamine. Gluta-

mine is then released from astrocytes and in turn is shuttled

back into neurons, where it is converted back to glutamate

[4, 32, 38]. Therefore, due to pathology in astrocytes,

glutamate to glutamine cycling is impaired and this could

contribute to seizures in the epileptic brain. Furthermore,

vesicular GABA levels partially dependent on glutamine

transport into inhibitory neurons. Down-regulation of

glutamine synthetase could partially reduce inhibitory

Neural Comput & Applic (2011) 20:1131–1139 1135

123

synaptic terminals of GABA and as a result impair GAB-

Aergic inhibition [32]. In conclusion, normal asynchro-

nizing behavior is maintained by applying appropriate

feedback action, while pathology in the feedback mecha-

nism caused by pathological astrocytes gives rise to sei-

zure-like activity.

Finally, applying principles of the dynamical system

theory on the MTCM, performance of the modified model is

analyzed and a dynamical scenario of transition between

normal and pathological states is presented. Fig. 4 shows the

phase portraits of the system for the MTCM with healthy (a)

and pathological (b) astrocytes. The phase portraits are in

line with Fig. 2 and are created by plotting the mean

membrane potential of the population of pyramidal cells

(VPY) versus that of the interneurons population (VIN) for

two different time intervals. The top panel is for the time

interval 50–70 s and the middle panel is corresponding to

the time interval 120–140 s. The bottom panel shows

simultaneous plotting of the top and middle panels. Inter-

estingly, different kinds of system’s behavior are observed

depending on the state of astrocytes. It is obvious that

healthy astrocytes can preserve the attractor corresponding

Fig. 2 a The effect of altering the value of PCxh i in TCM leads to

changing the occurrence of spontaneous paroxysmal episodes,

dramatically. b Population output with normal astrocyte in MTCM.

In this simulation, astrocytes effectively monitor the synaptic activity

and modify the synaptic strengths to prevent from the occurrence of

spontaneous paroxysmal episodes. c Astrocyte output of simulation in

b. In order to compensate the excessive increase of cortical input,

firing rate is accordingly increased. d The output of the MTCM with

pathological astrocyte. Due to infection of astrocytes, the paroxysmal

discharges emerge. Vout(t) is the output of the MTCM/TCM and is the

mean membrane potential of the pyramidal cells. As_Out(t) is the

output of the astrocytes in the MTCM

1136 Neural Comput & Applic (2011) 20:1131–1139

123

to the normal activity. Therefore, in spite of being subjected

to abnormal increase of cortical excitatory input, the normal

ongoing activity is continued. However, due to the astro-

cytic pathology, paroxysmal attractor is emerged which

implies that the neuronal network has a bistable dynamic.

This means that, in addition to an attractor for normal

activity, there might exist another attractor for the hyper-

synchronous behavior. Therefore, due to abnormal fluctua-

tions of some variables (in the present simulations,

abnormal increase of cortical input), the transition between

normal ongoing and the seizure activities can occur. This

concept is in agreement with the general theoretical

framework of the dynamics of the brain activity which

considers epilepsy as a dynamical disease of brain system

[39, 40]. To illustrates the transition between normal and

seizure attractors, the phase-planes of model outputs for the

time interval 45–75 s (top panel) and 95–125 s (bottom

panel) for healthy and pathological astrocytes are shown in

Fig. 5a and b, respectively. In Fig. 5, the phase-planes are

obtained by plotting the value of Vout(t ? 1) versus Vout(t) of

Fig. 2b and d. The emergence of seizure attractor due to

pathology in astrocytes is again obvious. Consequently, in

Fig. 3 Plot of excitation level versus inhibition level for a the MTCM with healthy astrocytes b the MTCM with pathological astrocytes.

The top panel is related to the time interval 50–60 s and the bottom panel is associated with the time interval 120–130 s of Fig. 2

Fig. 4 The phase portrait of a the MTCM with healthy astrocytes

b the MTCM with pathological astrocytes. The top panel is

corresponding to the time interval 50–70 s and the middle panel is

related to the time interval 120–140 s of Fig. 2. The bottom panel

shows the simultaneous plotting of the top and the middle panels. By

plotting the mean membrane potential of the population of pyramidal

cells (VPY) versus that of the interneurons population (VIN), the phase

portraits are created

Neural Comput & Applic (2011) 20:1131–1139 1137

123

the pathological case, changes in the cortical excitatory

input alter the trajectory of the system such that it can pro-

gress on the road leading to the seizure attractor. To sum-

marize, pathology in the feedback structure generated by the

abnormal astrocytes can be one of the plausible reasons that

can cause hypersynchronous discharges.

5 Conclusion

During the last decade, application of variety of new tools

and recording of a large amount of experimental data con-

firmed that glial cells are active partners of neurons in

synaptic integration and information processing. Astrocytes,

a subtype of glial cell, through Ca2?-dependent release of

neuroactive chemicals such as glutamate or ATP, are able to

affect neuronal excitability and thereby modulate, regulate

and optimize the synaptic transmission between neurons

[12, 18, 38]. The vast bulk of researches on epilepsy have

been focused mainly on neurons. Whereas, it has been

reported that a variety of changes in neuronal properties can

induce seizure activities in human or experimental epilepsy,

recent findings indicate that astrocytes are involved in epi-

leptogensis. In fact, functional alterations of astrocyte

receptors, transporters and membrane channels have been

discovered in several neurological disorders including epi-

lepsy [4, 30, 37]. Therefore, advances in the understanding

of cellular mechanisms by which astrocytes modulate

excitability holds promise for the development of improved

pharmacological treatments of epilepsy [30].

In this paper, a biologically plausible lumped neural

population model reported previously in the literature was

extended in an attempt to understand the functional role of

astrocytes in regulating and controlling the synaptic activ-

ities. In the feedback structure of the MTCM, astrocytes

dynamically regulate the synaptic strengths between pyra-

midal and interneuron subpopulations. It was shown that the

healthy (normal) astrocytes which monitor the synaptic

activities efficiently, could apply the appropriate feedback

action. They change their firing rate in order to compensate

the abnormal changes of cortical excitatory input. This

study explained how a very apparent neuronal synchroni-

zation could be successfully inhibited by appropriate neu-

ron-astrocyte interactions. Furthermore, the performances

of the pathological astrocytes were also investigated. We

provided theoretical supports that pathology in astrocytes

while diminishing neuron-astrocyte cross-talk, might lead

to epileptic seizure in the MTCM. This was characterized

by self-sustained oscillations in the population output.

In addition to the systems-level interpretation of epi-

leptic seizures, the proposed modeling approach is helpful

for the development and testing of new strategies to control

seizures using electrical stimulation. This issue will be

addressed in our future studies. Moreover, the results of

this paper have allowed us to explain some of the diverse

dynamic roles of astrocytes. Definitely, these results should

be evaluated in the light of experimental studies on animal

models or clinical trials. We hope that this study contribute

to the development of new clinical methods of prevention

and intervention in epileptic patients.

Fig. 5 The phase-plane representing of the attractors for a healthy

astrocytes b pathological astrocytes is obtained by plotting the value

of Vout(t ? 1) versus Vout(t). The top panel is corresponding to the

time interval 45–75 s and the bottom panel is related to the time

interval 95–125 s of Fig. 2b and d, respectively

1138 Neural Comput & Applic (2011) 20:1131–1139

123

References

1. Lehnertz K, Bialonski S, Horstmann MT, Krug D, Rothkegel A,

Staniek M, Wagner T (2009) Synchronization phenomena in

human epileptic brain networks. J Neurosci Meth 183(1):42–48

2. Demont-Guignard S, Benquet P, Gerber U, Wendling F (2009)

Analysis of intracerebral EEG recordings of epileptic spikes:

insights from a neural network model. IEEE Trans Biomed Eng

56(12):2782–2795

3. Ricci G, Volpi L, Pasquali L, Petrozzi L, Siciliano G (2009)

Astrocyte-neuron interactions in neurological disorders. J Biol

Phys 35:317–336

4. Seifert G, Carmignoto G, Steinhauser C (in press) Astrocyte

dysfunction in epilepsy. Brian Res Rev

5. Gertrudis P, Araque A (2005) Properties of synaptically evoked

astrocyte calcium signal reveal synaptic information processing

by astrocytes. J Neurosci 25(9):2192–2203

6. Newman EA (2003) New roles for astrocytes: regulation of

synaptic transmission. Trends Neurosci 26:536–542

7. Voltarra A, Steinhauser C (2004) Glial modulation of synaptic

transmission in the hippocampus. GLIA 47:249–257

8. Hertz L, Zielke HR (2004) Astrocytic control of glutamatergic

activity: astrocytes as stars of the show. Trends Neurosci 27(12):

735–743

9. Araque A, Parpura V, Sanzgiri RP, Haydon PG (1999) Tripartite

synapses: glia, the unacknowledged partner. Trends Neurosci

22:208–215

10. Halassa MM, Fellin T, Haydon PG (2007) The tripartite synapse:

roles for gliotransmission in health and disease. Trends Mol Med

13(2):54–63

11. Halassa MM, Fellin T, Haydon PG (2009) Tripartite synapses:

roles for astrocytic purines in the control of synaptic physiology

and behavior. Neuropharmacol 57(4):343–346

12. Fellin T, Pascual O, Haydon PG (2006) Astrocytes coordinate

synaptic networks: balanced excitation and inhibition. Physiology

21:208–215

13. Fellin T, Carmignoto G (2004) Neuron-to-astrocyte signaling in

the brain represents a distinct multifunctional unit. J Physiol 559:

3–15

14. Chakravarthy N, Tsakalis K, Sabesan S, Iasemidis LD (2009)

Homeostasis of brain dynamics in epilepsy: a feedback control

systems perspective of seizures. Ann Biomed Eng 37(3):565–585

15. Suffczynski P, Wendling F, Bellanger JJ, Lopes da Silva FH

(2006) Some insights into computational models of (patho)

physiological brain activity. Proc IEEE 94(4):784–804

16. Nadkarni S, Jung P (2004) Dressed neurons: modeling neural-

glial interactions. Phys Biol 1:35–41

17. Nadkarni S, Jung P (2007) Modeling synaptic transmission of the

tripartite synapse. Phys Biol 4:1–9

18. Nadkarni S, Jung P, Levine H (2008) Astrocytes optimize the

synaptic transmission of information. PLoS Comput Biol 4(5):

1–11

19. Postnove DE, Ryazanova LS, Sosnovtseva OV (2007) Functional

modeling of neural-glial interaction. BioSystems 89:84–91

20. Postnov DE, Koreshkov RN, Brazhe NA, Brazhe AR, Sos-

novtseva OV (2009) Dynamical patterns of calcium signaling in a

functional model of neuron-astrocyte networks. J Biol Phys

35:425–445

21. Garbo AD (2009) Dynamics of a minimal neural model con-

sisting of an astrocyte, a neuron, and an interneuron. J Biol Phys

35:361–382

22. Ullah G, Jung P, Cornell-Bell AH (2006) Anti-phase calcium

oscillations in astrocytes via inositol (1, 4, 5)-trisphosphate

regeneration. Cell Calcium 39:197–208

23. Garbo AD, Barbi M, Chillemi S, Alloisio S, Nobile M (2007)

Calcium signaling in astrocytes and modulation of neural activity.

BioSystems 89:74–83

24. Lavrentovich M, Hemkin SA (2008) Mathematical model of

spontaneous calcium (II) oscillations in astrocytes. J Theoretic

Biol 251(4):553–560

25. Suffczynski P, Kalitzin S, Lopes da Silva FH (2004) Dynamics of

non-convulsive epileptic phenomena modeled by a bistable

neuronal network. Neuroscience 126(2):467–484

26. Hung J, Colicos MA (2008) Astrocytic Ca2? waves guide CNS

growth cones to remote regions of neuronal activity. PLoS ONE

3(11):e3692

27. Suffczynski P, Kalitzin S, Lopes da Silva FH (2008) A neuronal

network model of corticothalamic oscillations: the emergence of

epileptiform absence seizures. In: Soltesz I, Staley K (eds)

Computational neuroscience in epilepsy. Elsevier, Amsterdam,

pp 403–418

28. Destexhe A (2008) Corticothalamic feedback: a key to explain

absence seizures. In: Soltesz I, Staley K (eds) Computational

neuroscience in epilepsy. Elsevier, Amsterdam, pp 184–211

29. Chakravarthy N, Sabesan S, Iasemidis LD, Tsakalis K (2007)

Modeling and controlling synchronization in a neuron level

population model. Int J Neural Syst 17(2):123–138

30. D’Ambrosio R (2004) The role of glial membrane ion channels in

seizures and epileptogenesis. Pharmacol Therapeut 103:95–108

31. Silchenko AN, Tass PA (2008) Computational modeling of par-

oxysmal depolarization shifts in neurons induced by the gluta-

mate release from astrocytes. Biol Cybern 98:61–74

32. Wetherington J, Serrano G, Dingledine R (2008) Astrocytes in

the epileptic brain. Neuron 58:168–178

33. De Keyser J, Mostert JP, Koch MW (2008) Dysfunctional

astrocytes as key players in the pathogenesis of central nervous

system disorders. J Neurol Sci 267:3–16

34. Rogawski MA (2005) Astrocytes get in the act in epilepsy. Nat

Med 11(9):919–920

35. Wendling F, Bellanger JJ, Bartolomei F, Chauvel P (2000) Rel-

evance of nonlinear lumped-parameter models in the analysis of

depth-EEG epileptic signals. Biol Cybern 83(4):367–378

36. Wendling F, Chauvel P (2008) Transition to ictal activity in

temporal lobe epilepsy: insights from macroscopic models. In:

Soltesz I, Staley K (eds) Computational neuroscience in epilepsy.

Elsevier, Amsterdam, pp 356–386

37. Binder DK, Steinhauser C (2009) Role of astrocytes in epilepsy.

astrocytes in (Patho)physiology of the nervous system. Springer,

Berlin, pp 649–671

38. Santello M, Volterra A (2009) Synaptic modulation by astrocytes

via Ca2?-dependent glutamate release. Neuroscience 158(1):

253–259

39. Lopes da Silva FH, Blanes W, Kalitzin SN, Parra J, Suffczynski

P, Velis DN (2003) Epilepsies as dynamical diseases of brain

systems: basic models of the transition between normal and

epileptic activity. Epilepsia 44(12):72–83

40. Lopes da Silva FH (2008) The impact of EEG/MEG signal pro-

cessing and modeling in the diagnostic and management of epi-

lepsy. IEEE Rev Biomed Eng 1:143–156

Neural Comput & Applic (2011) 20:1131–1139 1139

123