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In this article under material and methods I need you to recreate section 2.1 only in matlab and do not worry about I_syn. I need the matlab code and proof that it worked on your end. You can use this...

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Gamma Oscillation by Synaptic Inhibition in a Hippocampal
Interneuronal Network Model
Xiao-Jing Wang1 and György Buzsáki2
1Physics Department and Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02254, and
2Center for Molecular and Behavioral Neuroscience, Rutgers University, Newark, New Jersey 07102
Fast neuronal oscillations (gamma, 20–80 Hz) have been
observed in the neocortex and hippocampus during behav-
ioral arousal. Using computer simulations, we investigated
the hypothesis that such rhythmic activity can emerge in a
andom network of interconnected GABAergic fast-spiking
interneurons. Specific conditions for the population synchro-
nization, on properties of single cells and the circuit, were
identified. These include the following: (1) that the amplitude
of spike afterhyperpolarization be above the GABAA synaptic
eversal potential; (2) that the ratio between the synaptic
decay time constant and the oscillation period be sufficiently
large; (3) that the effects of heterogeneities be modest be-
cause of a steep frequency–cu
ent relationship of fast-
spiking neurons. Furthermore, using a population coherence
measure, based on coincident firings of neural pairs, it is
demonstrated that large-scale network synchronization re-
quires a critical (minimal) average number of synaptic con-
tacts per cell, which is not sensitive to the network size.
By changing the GABAA synaptic maximal conductance,
synaptic decay time constant, or the mean external excitatory
drive to the network, the neuronal firing frequencies were grad-
ually and monotonically varied. By contrast, the network syn-
chronization was found to be high only within a frequency band
coinciding with the gamma (20–80 Hz) range. We conclude that
the GABAA synaptic transmission provides a suitable mecha-
nism for synchronized gamma oscillations in a sparsely con-
nected network of fast-spiking interneurons. In turn, the inter-
neuronal network can presumably maintain subthreshold
oscillations in principal cell populations and serve to synchro-
nize discharges of spatially distributed neurons.
Key words: gamma rhythm; hippocampus; interneurons;
GABAA; synchronization; computer model
Although fast gamma cortical oscillation has been the subject of
active investigation in recent years (cf. Singer and Gray, 1995), its
underlying neuronal mechanisms remain elusive. Two major is-
sues are the cellular origin of rhythmicity (Llinás et al., 1991;
McCormick et al., 1993; Wang, 1993) and the mechanism(s) of
large-scale population synchronicity (Freeman, 1975; Bush and
Douglas 1991; Engel et al., 1991; Hansel and Sompolinsky, 1996).
Traditionally, recu
ent excitation between principal (pyramidal)
neurons is viewed as a major source of rhythmogenesis as well as
neuronal synchronization. However, in model studies in which
quantitative data about the synaptic time course were incorpo-
ated, it was found that glutamatergic synaptic excitation of the
AMPA type usually desynchronizes rather than synchronizes re-
petitive spike firings of mutually coupled neurons (Hansel et al.,
1995; van Vreeswijk et al., XXXXXXXXXXTherefore, recu
ent connec-
tions between pyramidal cells alone do not seem to account fo
the network coherence during cortical gamma oscillations. It was
suggested that pyramidal cell populations may be entrained by
synchronous rhythmic inhibition originating from fast-spiking in-
terneurons (Buzsáki et al., 1983; Lytton and Sejnowski, 1991).
During field gamma oscillations, intracellular recordings from
pyramidal cells revealed both EPSPs and IPSPs phase-locked to
the field oscillation frequencies (Jagadeesh et al., 1992; Chen and
Fetz, 1993; Soltész and Deschênes, 1993).
In this paper, we address the question whether, in the hip-
pocampus, an interneuronal network can generate a coherent
oscillatory output to the pyramidal neurons, thereby providing a
substrate for the synaptic organization of coherent gamma popu-
lation oscillations. In the behaving rat, physiologically identified
interneurons were shown to fire spikes in the gamma frequency
ange and phase-locked to the local field waves (Bragin et al.,
1995). Intracellular studies and immunochemical staining demon-
strated that these interneurons are interconnected via GABAergic
synapses (Lacaille et al., 1987; Sik et al., 1995; Gulyás et al., 1996).
Theoretical studies suggest that these GABAergic interconnec-
tions may synchronize an interneuronal network when appropri-
ate conditions on the time course of synaptic transmission are
satisfied (Wang and Rinzel, 1992, 1993; van Vreeswijk et al.,
1995). Moreover, in a recent in vitro experiment (Whittington et
al., 1995; Traub et al., 1996), the excitatory glutamate AMPA and
NMDA synaptic transmissions were blocked in the hippocampal
slice. When metabotropic glutamate receptors were activated,
transient oscillatory IPSPs in the 40 Hz frequency range were
observed in pyramidal cells. These IPSPs were assumed to origi-
nate from the firing activities of fast-spiking interneurons synchro-
nized by their interconnections. Computer simulations (Whitting-
ton et al., 1995; Traub et al., 1996) lend further support to this
To assess whether an interneuronal network can subserve an
adequate basis for the gamma frequency population rhythm in the
hippocampus, it is necessary to identify its specific requirements
Received May 5, 1996; revised June 25, 1996; accepted July 31, 1996.
This work was supported by the National Institute of Mental Health (MH53717-
01), Office of Naval Research (N XXXXXXXXXX), and the Sloan Foundation to
X.J.W.; and HFSP and the National Institute of Neurological Disease and Stroke
(NS34994) to G.B. and X.J.W. Simulations were partly performed at the Pittsburgh
Supercomputing Center. We thank D. Golomb, D. Hansel, J.-C. Lacaille, and C.
McBain for discussions, A. Sik for preparing Figure 2, and L. A
ott, J. Lisman, and
R. Traub for carefully reading this manuscript.
espondence should be addressed to Xiao-Jing Wang, Center for Complex
Systems, Brandeis University, Waltham, MA 02254.
Copyright q 1996 Society for Neuroscience XXXXXXXXXX/96/ XXXXXXXXXX$05.00/0
The Journal of Neuroscience, October 15, 1996, 16(20):6402–6413
on the cellular properties and network connectivities, as well as to
determine whether these conditions are satisfied by particula
interneuronal subtypes. The present study is devoted to investi-
gate such requirements using computer simulations. We found
that synaptic transmission via GABAA receptors in a sparsely
connected network of model interneurons can provide a mecha-
nism for gamma frequency oscillations, and we compared the
modeling results with the anatomical and electrophysiological
data from hippocampal fast spiking interneurons.
Model neuron. Each interneuron is described by a single compartment
and obeys the cu
ent balance equation:
5 2INa 2 IK 2 IL 2 Isyn 1 Iapp , (2.1)
where Cm 5 1 mF/cm
2 and Iapp is the injected cu
ent (in mA/cm
2). The
leak cu
ent IL 5 gL(V 2 EL) has a conductance gL 5 0.1 mS/cm
2, so that
the passive time constant t0 5 Cm/gL 5 10 msec; EL 5 265 mV.
The spike-generating Na1 and K1 voltage-dependent ion cu
ents (INa
and IK) are of the Hodgkin–Huxley type (Hodgkin and Huxley, 1952).
The transient sodium cu
ent INa 5 gNam`
3 h(V 2 ENa), where the acti-
vation variable m is assumed fast and substituted by its steady-state
function m` 5 am/(am 1 bm); am(V XXXXXXXXXXV 1 35)/(exp(20.1(V 1
XXXXXXXXXX), bm(V ) 5 4exp(2(V 1 60)/18). The inactivation variable h
obeys a first-order kinetics:
5 f~ah~1 2 h! 2 bhh! (2.2)
where ah(V XXXXXXXXXXexp(2(V 1 58)/20) and bh(V ) 5 1/(exp(20.1(V 1
XXXXXXXXXXgNa 5 35 mS/cm
2; ENa 5 55 mV, f 5 5.
The delayed rectifier IK 5 gKn
4 (V 2 EK), where the activation
variable n obeys the following equation:
5 f~an~1 2 n! 2 bnn! (2.3)
with an(V XXXXXXXXXXV1 34)/(exp(20.1(V XXXXXXXXXXand bn(V XXXXXXXXXX
exp(2(V 1 44)/80); gK 5 9 mS/cm
2, and EK 5 290 mV.
These kinetics and maximal conductances are modified from Hodgkin
and Huxley (1952), so that our neuron model displays two salient prop-
erties of hippocampal and neocortical fast-spiking interneurons. First, the
action potential in these cells is followed by a
ief afterhyperpolarization
(AHP) of about 215 mV measured from the spike threshold of approx-
imately 255 mV (McCormick et al., 1985; Lacaille and Williams, 1990;
Morin et al., 1995; Zhang and McBain, XXXXXXXXXXThus, during the spike
epolarization the mem
ane potential reaches a minimum of about 270
mV, rather than being close to the reversal potential of the K1 cu
EK 5 290 mV. This is accomplished in the model by relatively small
maximal conductance gK and fast gating process of IK so that it deacti-
vates quickly during spike repolarization.
Second, these interneurons have the ability to fire repetitive spikes at
high frequencies (with the frequency–cu
ent slope up to 200–400 Hz
nA) (McCormick et al., 1985; Lacaille and Williams, 1990; Zhang and
McBain, XXXXXXXXXXWith fast kinetics of the inactivation (h) of INa, the
activation (n) of IK, and the relatively high threshold of IK, the model
interneuron displays a large range of repetitive spiking frequencies in
esponse to a constant injected cu
ent Iapp (Fig. 1A, left). It has a small
ent threshold (the rheobase Iapp . 0.2 mA/cm2), and the firing rate is
as high as 400 Hz for Iapp . 20 mA/cm2. Similar to cortical interneurons
(McCormick et al., 1985; Lacaille and Williams, 1990), the whole fre-
ent curve is not linear, and the frequency–cu
ent slope is
larger at smaller Iapp values (lower frequencies) (Fig. 1A, right). As a
consequence, the neural population is more sensitive to input heteroge-
neities at smaller Iapp values. This is demonstrated in Figure 1B, where a
Gaussian distribution of Iapp is applied to a population of uncoupled
neurons (N 5 100), with a mean Im and standard deviation Is. Given a
fixed and small Is 5 0.03, the mean drive Im is varied systematically, and
the resulting dispersion in the neuronal firing frequencies, fs /fm (standard
deviation of the firing rate/mean firing rate) is shown as function of Im
(Fig. 1B, top). When plotted versus fm, it is evident that with the same
amount of dispersion in applied cu
ent (Is) the dispersion in firing rates
fs /fm is dramatically increased for fm , 20 Hz (Fig. 1B, bottom). This
feature has important implications for the frequency-dependent network
ehaviors (see Results).
Model synapse. The synaptic cu
ent Isyn 5 gsyns(V 2 Esyn), where gsyn
is the maximal synaptic conductance and Esyn is the reversal potential.
Typically, we set gsyn 5 0.1 mS/cm
2 and Esyn 5 275 mV (Buhl et al.,
1995). The gating variable s represents the fraction of open synaptic ion
channels. We assume that s obeys a first-order kinetics (Perkel et al.,
1981; Wang and Rinzel 1993):
5 aF~Vpre!~1 2 s! 2 bs, (2.4)
where the normalized concentration of the postsynaptic transmitter-
eceptor complex, F(Vpre), is assumed to be an instantaneous and sigmoid
function of the presynaptic mem
ane potential, F(Vpre) 5 1/(1 1
exp(2(Vpre 2 usyn)/2)), where usyn (set to 0 mV) is high enough so that the
transmitter release occurs only when the presynaptic cell emits a spike.
Figure 1. Model of single neuron and synapse. A, Left, Firing frequency
versus applied cu
ent intensity ( f 2 Iapp curve) of the model neuron. The
firing rate can be as high as 400 Hz. Right, The derivative df/dIapp shows
that the f/Iapp slope is much larger at smaller Iapp (lower f ) values. B,
Dispersion in firing rates caused by heterogeneity in input cu
ent. A
Gaussian distribution for input cu
ents, with standard deviation Is 5 0.03,
is applied to a population of uncoupled neurons. The dispersion in firing
ates was computed as the ratio between the standard deviation and the
mean of firing rates ( fs /fm). This ratio is much larger for smaller mean
ent amplitude Im (top). Plotting fs /fm versus fm shows that the disper-
sion in firing rates is dramatically increased for fm , 20 Hz (bottom). C,
ief cu
ent pulse applied to a presynaptic cell generates a single
action potential, which elicits an inhibitory postsynaptic cu
ent (Isyn) and
ane potential change in a postsynaptic cell (gsyn 5 0.1 mS/cm
Wang and Buzsáki • Gamma Rhythm in an Interneuronal Network J. Neurosci., October 15, 1996, 16(20):6402– XXXXXXXXXX
The channel opening rate a 5 12 msec21 assures a fast rise of the Isyn, and
the channel closing rate b is the inverse of the decay time constant of the
Isyn; typically, we set b 5 0.1 msec
21 (tsyn 5 10 msec). An example of Isyn
and IPSP elicited by a single presynaptic spike is illustrated in Figure 1C.
Random network connectivity. The network model consists of N cells.
The coupling between neurons is randomly assigned, with a fixed average
number of synaptic inputs per neuron,Msyn. The probability that a pair of
neurons are connected in either direction is p 5 Msyn/N. For comparison,
we also used fully coupled (all-to-all) connectivity (Msyn 5 N ). In the
model, the maximal synaptic conductance gsyn is divided by Msyn, so that
when the number of synapses Msyn is varied, the total synaptic drive pe
cell in average remains the same.
Msyn is the convergence/divergence factor of the neural coupling in the
network. Experimentally, an estimate of this important quantity has been
obtained for an interneuronal network of the CA1 hippocampus (Sik et
al., XXXXXXXXXXA parvalbumin-positive (PV1) basket interneuron was stained
intracellularly by biocytin in vivo. Its axonal a
orization was largely
confined in the striatum pyramidale (Fig. 2A). Other PV1 interneurons
were stained immunochemically, and the contacts made by the biocytin-
filled cell with other PV1 cells were counted (Sik et al., XXXXXXXXXXIt was
concluded that a single PV1 basket cell makes synaptic contacts with at
least 60
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