ORIGINAL RESEARCH ARTICLEpublished: 04 July 2013

doi: 10.3389/fncom.2013.00081

Implementing the cellular mechanisms of synaptictransmission in a neural mass model of thethalamo-cortical circuitryBasabdatta S. Bhattacharya*

Engineering Hub, School of Engineering, University of Lincoln, Lincoln, UK

Edited by:

Peter Robinson, University ofSydney, Australia

Reviewed by:

Peter Robinson, University ofSydney, AustraliaPulin Gong, University of Sydney,Australia

*Correspondence:

Basabdatta S. Bhattacharya, Schoolof Engineering, University ofLincoln, Engineering Hub, BrayfordPool, Lincoln LN6 7TS, Lincolnshire,UKe-mail: basab@ieee.org

A novel direction to existing neural mass modeling technique is proposed wherethe commonly used “alpha function” for representing synaptic transmission isreplaced by a kinetic framework of neurotransmitter and receptor dynamics.The aim is to underpin neuro-transmission dynamics associated with abnormalbrain rhythms commonly observed in neurological and psychiatric disorders. Anexisting thalamocortical neural mass model is modified by using the kineticframework for modeling synaptic transmission mediated by glutamatergic and GABA(gamma-aminobutyric-acid)-ergic receptors. The model output is compared qualitativelywith existing literature on in vitro experimental studies of ferret thalamic slices, aswell as on single-neuron-level model based studies of neuro-receptor and transmitterdynamics in the thalamocortical tissue. The results are consistent with these studies:the activation of ligand-gated GABA receptors is essential for generation of spindlewaves in the model, while blocking this pathway leads to low-frequency synchronizedoscillations such as observed in slow-wave sleep; the frequency of spindle oscillationsincrease with increased levels of post-synaptic membrane conductance for AMPA(alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionic-acid) receptors, and blocking thispathway effects a quiescent model output. In terms of computational efficiency, thesimulation time is improved by a factor of 10 compared to a similar neural mass modelbased on alpha functions. This implies a dramatic improvement in computational resourcesfor large-scale network simulation using this model. Thus, the model provides a platformfor correlating high-level brain oscillatory activity with low-level synaptic attributes, andmakes a significant contribution toward advancements in current neural mass modelingparadigm as a potential computational tool to better the understanding of brain oscillationsin sickness and in health.

Keywords: neural mass model, thalamocortical circuitry, kinetic framework, brain oscillations, AMPA, GABA

1. INTRODUCTIONNeural mass computational models mimicking synchronousbehavior in populations of thalamocortical neurons are oftenused to study brain oscillations (David and Friston, 2003;Suffczynski et al., 2004; Breakspear et al., 2006; Sotero et al.,2007; Deco et al., 2008; Izhikevich and Edelman, 2008; Ponset al., 2010; Robinson et al., 2011; de Haan et al., 2012).The term “neural mass” was coined by Freeman (1975), whilethe neural mass modeling paradigm is based on the math-ematical framework proposed by Wilson and Cowan (1973);each cell population in a neural mass model represents aneuronal “ensemble” of mesoscopic-scale (104–107), which aredensely packed in space and work at the same temporal-scale,so that for all practical purposes, they can be mathemati-cally treated as a single entity (Liljenström, 2012), whence“mass”. In a seminal work, da Silva et al. (1974) used a neu-ral mass model of a simple thalamocortical circuitry to sim-ulate EEG (Electroencephalography) alpha rhythms (8–13 Hz).

Subsequently, this model has been the basis of several research(Zetterberg et al., 1978; Stam et al., 1999; Suffczynski, 2000;Bhattacharya et al., 2011a), albeit with modifications andenhancements; of special mention is the modification introducedby Jansen and Rit (1995) where the model is expressed as a setof ordinary differential equations (ODE). This modification, inturn, has been the basis of many significant research (Wendlinget al., 2002; Grimbert and Faugeras, 2006; Ursino et al., 2010).However, the computational basis of the models remain thesame—the conversion from firing rate to membrane potentialby excitatory and inhibitory neurotransmitters is simulated byconvolution of the input from a pre-synaptic neuronal masswith an exponential function, commonly known as the “alphafunction”, proposed by Rall (1967). Although the alpha func-tion is a fair estimate of the synaptic process (Bernard et al.,1994), it does not allow an insight into the underlying cellularmechanisms of synaptic transmission associated with abnormalbrain oscillations—an aspect emphasized to be crucial as an

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COMPUTATIONAL NEUROSCIENCE

Bhattacharya Synaptic transmission in neural mass models

aid to research in brain disorders (McCormick, 1992; Basar andGuntekin, 2008). The importance of understanding the neuro-transmission mechanisms in slow wave synchronized as well asspindle oscillations is also discussed in several relevant experi-mental studies (Steriade et al., 1993; von Krosigk et al., 1993).Moreover, correlating synaptic kinetics with brain oscillatoryactivity has the potential to aid neuropharmacological advancesin treating the diseased brain (Aradi and Erdi, 2006). Alongthese lines, Destexhe et al. (1998) argue that the alpha func-tion is inappropriate for representing post synaptic events otherthan the originally proposed post-synaptic potential in spikingneural networks; they propose a kinetic framework as a morebiologically plausible method of modeling synaptic transmis-sion compared to the alpha function (Destexhe et al., 2002).The ability of such a modeling framework to capture the phys-iological properties of synaptic transmission was demonstratedby fitting the model outputs to experimental data from hip-pocampal slices. Moreover, kinetic modeling is reported to becomputationally efficient (Destexhe et al., 1994), a vital pre-requisite in large-scale computational models. Subsequently,the kinetic models of neurotransmission was used in severalsingle-neuronal-level model-based studies—to investigate thala-mic oscillations (Destexhe et al., 1996) and corticothalamic influ-ence on brain oscillatory activity (Destexhe, 2008); to investigatenetwork synchrony (Breakspear et al., 2003); to simulate syn-chronous behavior observed during in vitro experimental studieson ferret thalamic slice by Wang and Rinzel (1992), Golomb et al.(1994, 1996) and Wang et al. (1995).

A significant modification to current neural mass model-ing framework was proposed by Suffczynski et al. (2004) byapplying single-neuronal-level model based techniques. Towardthis, they proposed an “ensemble” representation of the mem-brane conductance and post-synaptic current in a neuronal massmodel of the thalamocortical circuitry; an integrator is usedto generate the “ensemble” post-synaptic membrane potential.In the work presented here, a similar approach is adopted toimplement the kinetic framework of synaptic transmission inneural mass models—each post-synaptic attribute is assumed tobe an “ensemble” representation corresponding to a “neuronalmass”. For brevity, only two-state (“open” and “closed”) ion-channels (Destexhe et al., 1998) are considered, the desensitizedstate is ignored. While two-state models are a significant sim-plification of the very complex nature of ion channel dynamicsin biology, they have shown a remarkable fit to biological datacompared to more-than-two-state models (Destexhe et al., 1998,2002). This work aims to interface an abstraction of the ion chan-nel dynamics, such as the two-state ion channel kinetic models,with an abstraction of the population level neuronal behavior,such as neural mass models. The goal is to enable the correlationof higher-level brain dynamics observed in EEG with cellular-leveldynamics.

The work is presented thus: first, the kinetic frame-work for modeling AMPA (α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic-acid) and GABA (γ-amino-butyric-acid)receptor mediated synapses is introduced in an existing thala-mocortical neural mass model (section 2); second, a qualitativecomparison of the model behavior with experimental studies on

ferret thalamocortical tissue reported in von Krosigk et al. (1993)as well as to single-neuronal-level model based observationsreported in Golomb et al. (1996; section 3) is presented; the lack ofa quantitative study is mainly to avoid erratic conclusions as dif-ference in model structure and simulation techniques are boundto induce mismatch in numerical results. The model behavior isobserved to be consistent with these studies (von Krosigk et al.,1993; Golomb et al., 1996)—The post synaptic membrane con-ductance in both the thalamocortical relay (TCR) and thalamicreticular nucleus (TRN) cell population plays a role in effecting abifurcation in model behavior from spindling mode [oscillationswith the characteristic waxing-and-waning pattern seen in earlystages of sleep (Steriade et al., 1993; Hughes et al., 2004) as well asin alpha rhythmic oscillations during resting brain state (da Silvaet al., 1973)] to a limit-cycle mode (synchronized oscillations asseen in later stages of sleep or during absence seizures). The post-synaptic membrane conductance for both AMPA and GABA inthe TRN cell population is responsible for sustaining and mod-ulating spindle oscillations in the model output. Blocking theGABA-ergic synapses in the self-inhibitory loop of the TRN cellpopulation effects a low-frequency synchronized oscillation inthe model; this is aided by the secondary-messenger-gated GABAsynapses in the TCR cell population. In addition, the reverse rateof transmitter binding plays a role in increasing or decreasing thefrequency of synchronized oscillations, besides functioning as abifurcation parameter, an observation that has not been reportedin experimental studies. A comparison of the simulation time ofthe model with previous research using neural mass models basedon alpha functions show a factor of 10 improvement in simulationtime. This is a dramatic improvement on computational effi-ciency and emphasizes the appropriateness of the model proposedherein toward building large-scale software models for investi-gating neuronal disorders. The observations from this study aswell as issues related to the modeling approach are discussed insection 4.

2. MATERIALS AND METHODS2.1. FROM ALPHA FUNCTION TO KINETIC MODEL: A BRIEF OUTLINEA single neuronal mass structure as used commonly in neu-ral mass models is shown in Figure 1 and is defined inEquations (1–5):

hw(t) = Hw

τwtexp(−t/τw) (1)

yN(t) =∑

hw(t) ⊗ ENw (t) (2)

yN(t) = Hw

τwEN

w (t) − 2

τwyN(t) − 1

τ2w

yN(t) (3)

VP(t) =∑

N ∈ {1, 2, 3,...n}CN .yN(t) (4)

EPw(t) = S(VP) = 2e0

1 + eν(s0 − VP)(5)

where w ∈ {e, i} represents pre-synaptic neuronal populationswhich make excitatory (e) and inhibitory (i) synapses on apost-synaptic neuronal population; τw is the time constant and

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Bhattacharya Synaptic transmission in neural mass models

FIGURE 1 | Block diagram of a single “neuronal mass” in current state-of-the-art neural mass models.

Hw is the amplitude of the synapse; ENw (t), N ∈ {1, 2 . . . n} is the

firing frequency of an extrinsic or intrinsic cell population that ispre-synaptic to the population P; CN is a percentage of the totalnumber of synapses from all afferents to P; VP is the “ensemblepost-synaptic membrane potential”; EP

w is the “ensemble firingrate” of P and is defined by a sigmoid function where 2e0 is themaximum firing rate of the population, s0 is the threshold poten-tial at which the neurons spike and ν is the sigmoid steepnessparameter.

2.1.1. A modified neural mass representationIn a recent work, Suffczynski et al. (2004) modified the neu-ral mass representation of a cell population and introducedpost-synaptic current mediated by the ligand-gated glutamater-gic receptors AMPA, and the ligand- and secondary-messenger-gated GABA-ergic receptors GABAA and GABAB, respectively.The input EN

ξ(t), ξ ∈ {AMPA, GABAA, GABAB}, is the firing rate

of an excitatory (AMPA) or inhibitory (GABAA and GABAB)pre-synaptic neuronal population N ∈ {1, 2 . . . , n}. The model(Figure 2A) is defined in Equations (6–11):

hξ(t) = Hξ

(exp

(−t/τa

ξ

)− exp

(−t/τb

ξ

)), τb

ξ> τa

ξ(6)

gNξ

(t) =∑

hξ(t) ⊗ ENξ

(t) (7)

gNξ

(t) = 1

τaξτbξ

[Hξ

(τaξ− τb

ξ

)EN

ξ(t) −

(τaξ+ τb

ξ

)gNξ

(t)

− gNξ

(t)]

(8)

INξ

(t) = gNξ

(t)(

VP(t) − Vξ

)(9)

κmVP(t) = −∑

N ∈ {1,2,3,...,n}CN .IN

ξ(t) − Iλ(t) (10)

Iλ(t) = gλ

(VP(t) − Vλ

)(11)

where hξ(t) is the synaptic transmission function with τaξ

and

τbξ

as the rise and decay times, respectively; gNξ

denote the post-

synaptic “ensemble” membrane conductance; Vξ is the reversal

potential for the synapse mediated by ξ; VP is the ensemble

post synaptic membrane potential of the population P due toPSC from all pre-synaptic cell populations N ∈ {1, 2 . . . , n}; κm

is the ensemble membrane capacitance; CN is the synaptic con-nectivity parameter; Iλ, gλ and Vλ are the ensemble leakagecurrent, conductance and reversal potential, respectively for P.The ensemble firing rate EP

ξ(t) is as defined in Equation (5)

and is the pre-synaptic firing rate input to other neuronalpopulations.

2.1.2. Introducing kinetic model of synapses in a neural massrepresentation

The single neuronal mass structure presented in Figures 1, 2A ismodified by replacing the alpha function with kinetic models ofAMPA, GABAA, and GABAB synapses; the enhanced representa-tion (Figure 2B) is defined in Equations (12–19):

[T]χ(Vχ) = Tmax

1 + e− Vχ − θsσs

(12)

drξ1χ (t)

dt= αξ1 [T]χ

(1 − rξ1

χ (t))

− βξ1 rξ1χ (t) (13)

dRξ2χ (t)

dt= αξ2 [T]χ

(1 − Rξ2

χ (t))

− βξ2 Rξ2χ (t) (14)

d[X](t)

dt= αξ2 Rξ2

χ (t) − βξ2 [X](t) (15)

rξ2χ ((t)) = [X]n(t)

[X]n(t) + Kd(16)

Iξχ(t) = g ξrξ

χ(t)(

VP(t) − V ξ)

(17)

κmdVP(t)

dt= −

∑χ∈{1,2}

Iξχ(t).Cχ − Iλ

P(t) (18)

IλP(t) = gλ

P

(VP(t) − Vλ

P

)(19)

Let Vχ, χ ∈ {1, 2} be the “ensemble” membrane potential of twopre-synaptic neuronal population that are afferent to the post-synaptic population P such that the synapses made by χ = 1 ismediated by a ligand-gated receptor ξ1 ∈ {AMPA, GABAA} whilethat made by χ = 2 is mediated by a secondary-messenger-gatedreceptor ξ2 ∈ {GABAB}. The concentration of neurotransmitters

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Bhattacharya Synaptic transmission in neural mass models

FIGURE 2 | Block diagram of (A) Suffczynski et al. (2004)’s modification

of the structure in Figure 1 by introducing “ensemble” representation of

post-synaptic membrane conductance and current. (B) Neuronal massstructure with the kinetic framework implemented for modeling synaptic

transmission as an alternative to the alpha function (hw (t) in Figure 1). Adiagrammatic representation of the ion-channel kinetics during synaptictransmission is presented in Figures 3A,B. (C) A thalamocortical circuitryimplementing the modified neuronal mass representation in (B).

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Bhattacharya Synaptic transmission in neural mass models

FIGURE 3 | The state transition diagrams for (A) AMPA and GABAA

neuro-receptor dynamics defined in Equation (13); α and β are rate of

transitions between the two states and α is a function of the

transmitter concentration in the synaptic cleft [T ] defined in

Equation (12); (B) GABAB neuro-transmission as defined in

Equations (14–16)—the neurotransmitter T binds to the inactivated

receptor R0; a fraction of activated receptors R act as a catalyst to

transform the G-protein from an inactivated form X0 to an activated

form X , which binds at n independent sites to open a fraction of the

ion channels. The desensitized state of the ion-channels are ignored in thiswork for brevity (see Destexhe et al., 1998 for a detailed comparison ofkinetic models with more than two-states).

[T]χ in the synaptic cleft is defined as a function of Vχ and isexpressed by a sigmoid function (Equation 12) where Tmax isthe maximum neuronal concentration in the synaptic cleft andis well approximated by 1 mM (Destexhe et al., 1998), θs rep-resents the threshold at which [T] = 0.5Tmax and σs denote thesteepness of the sigmoid. The proportion of open ion-channelsdue to the bound receptors ξ1 on the ensemble membrane ofthe post-synaptic cell population corresponding to the synapsemade by the population χ = 1 is defined in Equation (13) where

αξ1 and βξ1 are the forward and backward rate constants, respec-tively for transmitter binding. The transition diagram is shownin Figure 3A. However, GABAB mediated synapses, unlike AMPAand GABAA synapses, activate G-proteins which in turn act as the“secondary messengers” and initiate the opening of ion channels.

The process is defined in Equations (14–16) where Rξ2χ is the frac-

tion of activated ξ2 receptors, which acts as a catalyst in activatingthe secondary-messenger G-protein (guanine nucleotideUbind-ing proteins); [X] is the concentration of the activated G-protein;

rξ2χ is the fraction of open ion channels caused by binding of

X with independent binding sites; αξ2 and βξ2 are the bindingrate constants; n is the number of bound receptor sites and Kd

is the dissociation constant of binding of X with the ion chan-nels. The transition diagram of this process is shown in Figure 3B.The resulting ensemble PSC mediated by the receptor ξ ≡ ξ1 ∪ ξ2

due to a synapse from the pre-synaptic population χ is defined in

Equation (17) where g ξ and V ξ are the maximum conductanceand reverse potential, respectively corresponding to ξ mediatedsynapse. VP (Equation 18) is the ensemble post-synaptic poten-tial (PSP) of P, where κm is the ensemble membrane capacitanceof P, Cχ, χ ∈ {1, 2} is the synaptic connectivity parameter. Iλ

P(Equation 19) is the ensemble leak current of the post-synapticmembrane, where gλ

P and VλP are conductance and reverse poten-

tial, respectively, corresponding to “non-specific” leak (Golombet al., 1996; Suffczynski et al., 2004) in the ensemble membraneof the post synaptic cell population. In the following section,we implement this framework in a neural mass model of thethalamocortical circuitry.

2.2. NEURAL MASS MODEL OF A THALAMOCORTICAL CIRCUITRYWITH KINETIC SYNAPSES

The thalamocortical circuitry is shown in Figure 2C and con-sists of the two thalamic cell populations that communicatewith the cortex viz. the TCR and TRN. The third group ofcells viz. the Interneurons (IN) participate in intra-thalamiccommunications and are ignored here for brevity. The synap-tic structure and connectivity are informed from experimentaldata based on the dorsal thalamic Lateral Geniculate Nucleus(LGNd) (Horn et al., 2000). The input to the model is assumedto be the ensemble membrane potential of pre-synaptic reti-nal cells (Vret) in a resting state with no sensory input and issimulated using a Gaussian white noise (da Silva et al., 1973).The TCR cells make AMPA receptor mediated glutamatergicsynapses on the TRN cells (other types of glutamatergic recep-tors are ignored in this work for brevity); the TRN cells makeGABA-ergic synapses on the TCR cells mediated by both theligand-gated GABAA and the secondary-messenger-gated GABAB

receptors. Furthermore, the TRN cells make GABAA receptormediated synapses within the population. The model is definedin Equations (20–27); all variables and parameters in the modelare assumed to be the ensemble representation corresponding to aneural mass:

[T]� (V�(t)) = Tmax

1 + exp(− V� (t)−θs

σs

) (20)

drη1

�(t)

dt= αη1 [T]�

(1 − rη1

�(t)

)− βη1 rη1

�(t) (21)

dRη2

�(t)

dt= α

η21 [T]�

(1 − Rη2

�(t)

)− β

η21 Rη2

�(t) (22)

d[X](t)

dt= α

η22 Rη2

�(t) − β

η22 [X](t) (23)

rη2

�((t)) = [X]n(t)

[X]n(t) + Kd, (24)

Iη

�(t) = g ηrη

�(t)

(Vϒ (t) − V η

)(25)

κmdVϒ (t)

dt= −

∑

� ∈ {ret, trn, trn}Iη

�(t).Cuvw − Iλ

ϒ(t), (26)

Iλϒ

(t) = gλϒ

(Vϒ (t) − Vλϒ

), (27)

where � ∈ {ret, trn, trn} represent the afferent cell popula-tions; ϒ = {trn, trn} represent the efferent cell populations; η1 ∈{AMPA, GABAA}, η2 ∈ {GABAB}, η ≡ η1 ∪ η2; Cuvw are con-nectivity parameters where u ∈ {t, n} and v ∈ {r, t, n, s} denotethe post-synaptic and pre-synaptic cell populations, respectivelyof the retina (r), TCR (t), TRN (n), while s denote an intra-population afferent; w ∈ {e, i} represent an excitatory (e) or aninhibitory (i) synapse. All other parameter nomenclatures are asdefined in section 2.1. The initial parameter values are mentionedin Table 1.

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Bhattacharya Synaptic transmission in neural mass models

Table 1 | Initial values of the parameters defined in Equations (21–27).

Neuroreceptors → AMPA GABAA GABAB

Units↓

(A) NEUROTRANSMISSION PARAMETERS

mM.msec−1 αη1 = 2 αη1 = 2α

η21 = 0.02

αη22 = 0.03

msec−1 βη1 = 0.1 βη1 = 0.08β

η21 = 0.05

βη22 = 0.01

mS gη = 0.1gη

TRN to TCR = 0.1gη = 0.06

gη

TRN to TRN = 0.2

mV Vη = 0Vη

TRN to TCR = −85Vη = −100

Vη

TRN to TRN = −75

Kd = 100

n = 4

(B) CELL MEMBRANE PARAMETERS

TCR TRN

gλϒ

(mS) 0.01 0.01

Vλϒ

(mV) −55 −72.5

Vrest (mV) −61 −84

(C) CONNECTIVITY PARAMETERS

Efferents →TCR

TRNRetinal

Afferents ↓ GABAA GABAB

TCR XCa

tni Cbtni Ctre

34 of 30.9 1

4 of 30.9 7.1

TRNCnte Cnsi X X35 20

Data in (A) and (B) are as in Golomb et al. (1996) and Suffczynski et al. (2004).

In Equation (20), both θs and σs act as bifurcation parameters in the model

(see Bhattacharya et al., 2012). However, the emphasis here is on post-synaptic

membrane attributes as in von Krosigk et al. (1993) and Golomb et al. (1996).

Thus these parameters (θs = −35 and σs = 2) are set by trial and error at values

just before the model undergoes bifurcation from a “point-attractor” mode to

a “limit-cycle” mode, based on a recent study where we observed rich model

dynamics and power spectral behavior around the bifurcation point (Bhattacharya

et al., 2013); Tmax = 1 mM (Destexhe et al., 1994). The input noise mean

μ = −45 mV and standard deviation ϕ = 20 mV2 are set by trial and error and

represents the resting state membrane potential fluctuations in retinal cells.

While the total number of GABA-ergic synaptic count on TCR cells is reported

as 30.9%, specific data on GABAA and GABAB are not available in literature

to the best of our knowledge. Thus, values for Catni , Cb

tni and Cnsi in (C) are

selected, within the reported biological range [see Bhattacharya et al. (2011b) for

details], when the model output showed an increased frequency content within

the theta (4–7 Hz) and alpha (8–13 Hz) bands. Ctre and Cnte are as in Bhattacharya

et al. (2011b). All variables in the ODEs are initialized to an arbitrarily small value

0.0002.

3. RESULTSThe ODEs are solved using the 4th/5th order Runge-Kutta-Fehlberg method (RKF45) in Matlab for a total duration of 600 s(10 min) at a resolution of 1 ms. The output voltage time series is

averaged over 20 simulations, each simulation run with differentseed for the noisy input. For frequency analysis, an epoch from100–599 s of the output signal is sampled every 4 ms (250 Hz) andbandpass filtered between 3.5–14 Hz with a Butterworth filter oforder 10. Short Time Fourier Transform (STFT) is done with aHamming window of duration 10 s and overlap of 50%.

The model displays a point-attractor mode behavior (initialtransient oscillations before settling down to a low amplitudenoisy output, which reflects the noisy input of the model) cor-responding to initial parameter values (Figure 4A). There is abehavioral transition in the model to a limit cycle mode withincreasing values of βampa, which correlates with a decrease inthe fraction of open ion channels in the post-synaptic ensem-ble membrane (Figures 4B,C). Varying αampa, on the other hand,does not affect the model behaviour (Figures 4D,E). A transi-tion from the limit cycle mode to a spindling mode is effectedin the model by increasing gampa, the post-synaptic membraneconductance for AMPA mediated synapses in both TCR and TRNcell population, and shown in Figures 4F,G. STFT of the outputtime series indicates the non-stationary behavior of the model(Figures 4H–K). A decrease and increase, respectively of the thetaand alpha band components imply an overall increase in fre-quency with increasing values of gampa ≡ {gampa

TCR , gampaTRN }, where

gampaTCR and g

ampaTRN correspond to the incoming signal from the

retina (to the TCR) and TCR (to the TRN), respectively in themodel. These observations are consistent with similar reports of atransition in the state of the model output with increasing valuesof gampa in Golomb et al. (1996; pp. 756–757), accompanied byan abrupt increase in the ratio of the frequency of oscillation ofthe TCR and the TRN cell populations; we have not studied thelatter aspect in this work. A more detailed study on the model pre-sented herein where g

ampaTCR and g

ampaTRN are varied separately specify

the gampaTCR as the control parameter that causes a bifurcation in

the model output from a limit cycle mode to the spindling modewith an increase in its value. On the other hand, the g

ampaTRN does

not effect any behavioral change in the model output, rather, it iseffective in increasing the inter-spindle frequency with an increasein its value when the model is in a spindling mode. This observa-tion implies that a change in AMPA receptor related attributesin the TRN plays a role in modulating thalamocortical spin-dle oscillations, which finds strong support in the experimentalstudy by von Krosigk et al. (1993), where “activation of AMPA-kainate receptors on the PGN” (Peri-geniculate nucleus—the partof the TRN associated with the LGNd) is described as “criticalto the generation of spindle waves”. Furthermore, this observa-tion is in line with the TRN being widely implicated as beingthe key “ingredient” in the generation of thalamocortical spindleoscillations (McCormick, 1992; Steriade et al., 1993).

Varying the GABA-ergic synaptic attributes when the modelis in a point-attractor mode does not show any change in modelbehavior. When the model is in a spindling mode (Figure 5A),increased synchronization within the limit cycle mode with

increasing values of ggabaATRN to TCR (Figures 5B,C) is observed. An

increase in the parameter βgabaA affects the output only whenthe model is in a limit-cycle mode and counters the effect

of increase in ggabaATRN to TCR (Figure 5D). However, varying αgabaA

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Bhattacharya Synaptic transmission in neural mass models

0 100 200 300 400 500 600

−60

−50

−40

−30

A

mod

el o

utpu

t (m

V)

240 300

0 100 200 300 400 500 600

−60

−50

−40

−30

B

240 300

0 100 200 300 400 500 600−70

−60

−50

−40

−30

C

240 300

0 100 200 300 400 500 600

−60

−50

−40

−30

D

240 300

αampa=10 βampa=1 gampa=0.1αampa=2 βampa=1 gampa=0.1αampa=2, βampa=0.1, gampa=0.1 αampa=2 βampa=0.2 gampa=0.1

0 100 200 300 400 500 600−65

−60

−55

−50

−45

−40

−35

−30

E

mod

el o

utpu

t (m

V)

240 300

200 300 400 500 600−45

−44.5

−44

−43.5

−43

−42.5

−42

F

200 300 400 500 600−43.2

−43

−42.8

−42.6

−42.4

G

αampa=20 βampa=1 gampa=0.2αampa=20 βampa=1 gampa=0.1 αampa=20 βampa=1 gampa=0.3

J

10 20 30 408

9

10

11

12

13

H

Freq

uenc

y (H

z)

10 20 30 40

4

5

6

7

I

10 20 30 40

4

5

6

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FIGURE 4 | The model output time series with (A) all parameters at their

initial values as in Table 1. (B) The model displays a bifurcation in outputbehavior when βampa is increased to 0.2; (C) a further increase in theparameter shows sustained oscillations with increased magnitude anddecreased frequency. The frequency behavior may be observed qualitativelyin the embedded line plots displaying the respective time-series in each plotfor an arbitrarily selected period of 60 s. (D) An increase in αampa has areverse effect resulting in reduced magnitude and increased frequency in thelimit-cycle mode; (E) further increase in the parameter do not show anysignificant effect on the magnitude or the limit-cycle behavior while there is a

slight increase in frequency. (F) Maintaining these modified values ofαampa = 20 and βampa = 1, an increase in gampa brings a bifurcation in modelbehavior from a limit-cycle mode to a “spindling” mode. A “zoomed-in” plotfrom the 3rd minute to the 9th minute is shown; the initial transientoscillations are neglected. (G) The frequency of spindle oscillations increasewith increasing values of gampa. This is indicated by a distinct (H,I) decrease(more blue pixels) in theta band components and (J, K) increase (more red,orange and yellow pixels) in alpha band components in the correspondingoutput time series plots. The abscissa in the figures denote (A–G) time(seconds) (H–K) time windows (seconds).

does not affect the model output. For ggabaATRN to TCR � 0.5, which

is the approximate bifurcation point (Figure 5B), increasing

ggabaATRN to TRN causes the model to revert back to the spindling

mode; the frequency of the inter-spindle oscillations increasewith increasing values of the parameter (Figures 5E,F). Thisis also indicated by a decrease (Figures 5H,I) and increase(Figures 5J,K) of theta and alpha band components respectivelyin the STFT of the output time series. In other words, decreas-

ing values of the parameter ggabaATRN to TRN causes increased syn-

chronization within the spindling mode behavior of the modelalong with a decrease in the inter-spindle frequency. However,

blocking ggabaATRN to TRN effects a switch in the model behavior to

a very low-frequency oscillatory state (Figure 5G). These resultsare consistent with experimental findings (von Krosigk et al.,1993) where application of GABAA inhibitor either “abolishedspindle waves or decreased within-spindle frequency,” whichcorrespond to the condition of either blocking or decreasing,

respectively of ggabaATRN to TRN in our model. Thus, the model impli-

cate the intra-TRN synaptic activity to be a key factor in sus-taining spindle oscillations in the thalamocortical circuitry, anobservation which conforms to those made in Golomb et al.

(1996; p. 755). Furthermore, a “frequency jump” with increas-ing ggabaA , and associated transition in model behavior is alsoreported in Golomb et al. (1996; see Figure 7) as a comparativestudy between the TCR and TRN cells. This is similar with theincrease in frequency of the spindle oscillations corresponding to

increasing ggabaATRN to TRN in the present model, although we have not

done a comparative study with the TRN cell population behavior.However, the current study implicate the increased post-synapticconductance for GABAA receptors in the TCR cell population

(ggabaATRN to TCR) to play a significant role in effecting state-transition

between spindle and slow-wave oscillations, an observation thatis yet to find support from experimental or model-based studies.

A quiescent state is observed corresponding to blocking

either AMPA (gampa = 0) or both GABAA (ggabaATRN to TCR = 0) and

GABAB (ggabaBTRN to TCR = 0) mediated synapses in the TRN to TCR

pathway (not shown). This is consistent with both experimen-tal (von Krosigk et al., 1993) and model-based (Golomb et al.,1996) studies. The role of the synaptic parameters in the GABAB

pathway in our model was minimal—to sustain a non-quiescentmodel behavior with blockage of GABAA; to sustain a highamplitude of limit-cycle oscillations in the model. Again, this

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Bhattacharya Synaptic transmission in neural mass models

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FIGURE 5 | (A) The model output (corresponding to the “zoomed-in” plot inFigure 4G) when αampa = 20, βampa = 1, gampa = 0.3). Retaining theseparameter values, (B) increasing ggabaA

n2r , where “n2r” denotes the TRN toTCR pathway, from its initial value effects a bifurcation in model behaviorfrom a spindling mode to a limit-cycle mode, indicating highly synchronizedoscillations in the thalamocortical circuitry. (C) Synchronization increases withincrease in ggabaA

n2r , indicated by increased magnitude and decreased

frequency of oscillation. For ggabaAn2r � 0.5 [approximate point of bifurcation,

shown in (B)], (D) increasing βgabaA effects an increase in frequency within the

limit-cycle mode, while (E) increasing ggabaAn2n , where “n2n” denotes the self

inhibitory pathway of the TRN, effects a transition from the limit-cycle modeto the spindling mode. (F) Further increase in ggabaA

n2n causes an increase in thefrequency of the spindle oscillations, indicated by (H,I) a decrease (more bluepixels) in theta band components and (J,K) increase (more red, orange andyellow pixels) in alpha band components in the corresponding output timeseries plots. (G) Blocking of ggabaA

n2n shows a very low-frequency (≈0.03 Hz)synchronized oscillation whose magnitude decreases with time. The abscissain the figures denote (A–G) time (seconds) (H–K) time windows (seconds).

is in agreement with experimental studies (von Krosigk et al.,1993), where activation of GABAB receptors are reported as“not essential” for generating synchronized oscillations, whileapplication of GABAB antagonist abolished “evoked or sponta-neous slowed oscillations.” The model in Golomb et al. (1996; seein Discussion p. 763) is also mentioned as being consistent withthese experimental results.

In a recent work (Bhattacharya et al., 2012), a simple neuralmass model implementing kinetic modeling for synaptic trans-mission is presented; the synaptic connectivity parameters inthe model correlate directly to that of an alpha function basedneural mass model [modified Alpha Rhythm model (modARm)from Bhattacharya et al. (2011a)]. The model behavior is studiedcorresponding to changes in the synaptic connectivity parame-ters as well as transmitter concentration related parameters, anda relevant comparison is made with the modARm. However,the model presented in this work has a larger set of synapticconnectivity parameters; model behavior corresponding to thisparameter space and its usefulness in understanding neurologicaldisorders will be the topic of a future work.

4. DISCUSSIONThe work presented here explores a novel approach toward corre-lating current neural mass model based studies with underlying

cellular mechanisms during synaptic transmission. The aim isto underpin the synaptic correlates of abnormal brain oscilla-tions in neurological and psychiatric disorders such as observedin Electroencephalogram (EEG). A kinetic framework for mod-eling AMPA and GABA receptor mediated synapses is imple-mented in an existing thalamocortical neural mass model con-sisting of an excitatory and an inhibitory neural mass, repre-senting cell populations of the thalamocortical relay (TCR) andthe thalamic reticular nucleus (TRN), respectively. Parametersin the model are assumed to be “ensemble” representationsof the corresponding attributes in a single neuron. A prelim-inary observation is made on the model behavior by varyingthe parameters corresponding to the post-synaptic membraneconductance of the cell populations as well as the forward andreverse rates of synaptic reaction; of specific interest is thetransition of the model behavior between the spindle oscilla-tory mode and the limit-cycle mode, the latter resembling theslow-wave (high-amplitude, low-frequency) synchronized oscil-lations that are signatures of absence seizures as well as slow-wave sleep. Furthermore, only the alpha (8–13 Hz) and theta(4–7 Hz) frequency bands of the output power spectra are stud-ied here, as EEG alpha and theta bands are believed to havea strong correlation with thalamocortical oscillations (Hugheset al., 2004).

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Bhattacharya Synaptic transmission in neural mass models

The results indicate that: (1) The post synaptic membrane con-ductance for both AMPA and GABAA receptors in the TRN cellpopulation play a role in sustaining spindle oscillations of theTCR cell population (the model output). (2) Blocking the GABAA

mediated synapses in the self-inhibitory feedback pathway of theTRN cell population effects synchronized oscillations with highamplitude and increased time-period of oscillation (≈0.03 Hz).(3) The post-synaptic membrane conductance for GABAB in theTCR cell population does not play any role in generating orsustaining spindle oscillations, but is responsible for sustainingthe slow-wave oscillations in the model associated with blockingof the intra-TRN GABAA synapses. (4) Blocking both GABAA

and GABAB or only the AMPA mediated synapses in the TCRcell population results in a quiescent model output. These find-ings are consistent with in vitro studies based on multiple unitrecordings from ferret thalamic slices (von Krosigk et al., 1993)as well as single-neuron-level model based studies (Golomb et al.,1996). In addition, this study identifies—(a) the reverse rate oftransmitter binding as an important attribute in effecting thala-mocortical synchronized oscillations that can be induced in themodel by increasing (decreasing) the fraction of open channelsdue to GABAA (AMPA) mediated synapses in the TCR (TRN)cell populations; (b) the post-synaptic membrane conductancefor GABAA in the TCR cell population as a control parameter foreffecting a behavioral transition in the model.

It may be noted that the above-mentioned observations areonly a qualitative comparison with single-neuron-level model-based (Golomb et al., 1996) and experimental (von Krosigk et al.,1993) studies; a drawback of the current work is a lack of quan-titative comparison with these studies. The neural mass modelpresented in this work is at a mesoscopic scale, representationof a population of ≈104−107 neurons, unlike that in Golombet al. (1996), which is at single-neuronal-level. Similarly, the mul-tiple unit recording based study in von Krosigk et al. (1993)observes neuronal behavior of either a single neuron or a popula-tion of <102 neurons. In addition, the modeling and simulationmethods in the current work and that in Golomb et al. (1996)are not similar. Thus, a quantitative comparison of the cur-rent work with these studies may lead to erroneous conclusions.However, model validation with experimental data is a crucialcriteria when investigating brain disorders. Along these lines, anongoing work is investigating ways to validate the model pre-sented herein with EEG data, and will be the topic of a futurestudy.

The model structure in the current work is a consider-ably simplified representation of the thalamocortical circuitry.The role of the thalamocortical circuitry in generating slowwave brain oscillations is discussed at length in Steriade et al.(1993), based on in vivo and in vitro studies. More recently,three parameters in the thalamo-cortico-thalamic loop viz. thecortico-thalamic, thalamo-cortical and intra-thalamic pathwaysare specified in Breakspear et al. (2006) for generating insta-bilities in the thalamocortical circuitry, leading to synchronizedoscillations such as seen during absence seizures. Furthermore, anon-linear dynamical analysis of the model is shown to predictseizure onset by validating with patient EEG data. In a previ-ous research (Bhattacharya et al., 2011b), we have proposed a

more elaborate alpha-function based neural mass model that haveconsidered these vital pathways in the thalamocorticothalamicloop. Also, we have performed a non-linear dynamical analy-sis of a simple thalamocortical model based on alpha functionsin Bhattacharya et al. (2013) to understand EEG power spectraabnormalities associated with several neurological disorders. Suchresearch directions will be considered as an extended work basedon the model presented herein.

It is worth mentioning here that biologically plausible param-eterizations has been a major constraint in neural mass modelingof brain dynamics. This is largely due to insufficient experimentaldata, published or otherwise, as well as to a lack of “homogene-ity” of published data from different experimental laboratories.The trend thus far has been to use biologically plausible dataif and when available; otherwise, i.e., for parameter values thatcannot be availed from experimental data, the models are tunedto estimated parameter values which provide a desirable out-put in context to the objectives of the research [the reader mayrefer to Robinson et al. (2004) for a model parameterizationsrelated work and discussion]. Thus, the model in Breakspear et al.(2006) was based on neurophysiological parameters obtainedfrom Robinson et al. (2002), which in turn are based on inverseparameterizations during model validation with EEG data frompatients of epileptic seizures. The parameterizations of the modelpresented in this work is largely based on neurophysiologicalparameters obtained from experimental studies: the cellular-levelparameters, including those of the synaptic kinetics, are based onin vitro studies and model-based studies of thalamocortical tissueby von Krosigk et al. (1993) and Golomb et al. (1996), respec-tively; the model connectivity parameters are based on experi-mental studies of the cat and rat thalamus obtained from Hornet al. (2000); Sherman and Guillery (2001). On the other hand,the extrinsic (retinal) input and neuro-transmitter concentrationparameter values are adjusted to maintain a “dynamically active”model behavior (this is as opposed to a continuous “quiescent”state of the model corresponding to certain parameter values, anddoes not conform to biology). However, technological advancesin the field of neuro-imaging during the last decade such as func-tional Magnetic Resonance Imaging (fMRI), Diffusion TensorImaging (DTI) and Transcanial Magnetic Stimulation (TMS) arepaving the way for biologically-realistic mapping of parametervalues in computational models; for example as in Izhikevich andEdelman (2008).

The observations made herein support the motivation towardthis preliminary work, which is to correlate higher-level braindynamics with underlying cellular-level synaptic mechanisms. Itmay be noted that in all our previous works using alpha functionbased neural mass models, the emphasis has been on studyingthe model behavior with varying values of synaptic connec-tivity parameters toward a meaningful mapping to Alzheimerdisease-related EEG anomalies. However, such “synaptic parame-ter variation only” studies are highly constrained and do not makemuch sense when trying to understand generic brain-state con-ditions e.g., the sleep-awake cycle, or several other neurologicaland psychiatric disorders e.g., absence seizures, which rely heav-ily on various aspects of cellular dynamics in the thalamocorticalcircuitry. Rather, the emphasis of this work is on laying the

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ground-work for a more elaborate, and yet computationally effi-cient scheme, whereby large-scale computational models may besimulated to mimic brain rhythms, which can then be correlatedto model parameters emulating cellular dynamics. The synaptictransmission kinetics and subsequent post-synaptic membraneparameters are some of the key constituents of brain signaling,and are affected significantly in various brain diseases. Clearly,the alpha-function based neural mass models are inadequate indealing with research directions where model parameters can bemapped in a biologically plausible manner to synaptic attributes.In terms of computational efficiency, the time for simulating

20 trials with the model presented in this work takes 60 s; thismay be contrasted with 600 s for simulating a similar model [themodified Alpha Rhythm model in Bhattacharya et al. (2011a)]based on alpha functions. This is a dramatic improvement incomputational efficiency and highlight the plausibility of usingthe kinetic-model based neural mass modeling framework insimulating large-scale computational models toward mimickingreal-time EEG signals. This in turn will provide a powerful toolfor specifying cellular pathways that need be targeted for symp-tomatic alleviation of anomalous brain rhythms as well as toinform effective neuropharmacological research directions.

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Conflict of Interest Statement: Theauthors declare that the researchwas conducted in the absence of anycommercial or financial relationshipsthat could be construed as a potentialconflict of interest.

Received: 29 November 2012; paperpending published: 03 February 2013;accepted: 06 June 2013; published online:04 July 2013.Citation: Bhattacharya BS (2013)Implementing the cellular mechanismsof synaptic transmission in a neural massmodel of the thalamo-cortical circuitry.Front. Comput. Neurosci. 7:81. doi:10.3389/fncom.2013.00081Copyright © 2013 Bhattacharya. This isan open-access article distributed underthe terms of the Creative CommonsAttribution License, which permits use,distribution and reproduction in otherforums, provided the original authorsand source are credited and subject to anycopyright notices concerning any third-party graphics etc.

Frontiers in Computational Neuroscience www.frontiersin.org July 2013 | Volume 7 | Article 81 | 11