Coherent and intermittent ensemble oscillations …physics.wustl.edu/nd/fac/wessel/2016HoseiniWesselCoherentAnd...Coherent and intermittent ensemble oscillations emerge ... from spatially

Coherent and intermittent ensemble oscillations emerge from networks ofirregular spiking neurons

Mahmood S. Hoseini and Ralf WesselDepartment of Physics, Washington University, St. Louis, Missouri

Submitted 10 June 2015; accepted in final form 4 November 2015

Hoseini MS, Wessel R. Coherent and intermittent ensemble oscil-lations emerge from networks of irregular spiking neurons. J Neuro-physiol 115: 457–469, 2016. First published November 11, 2015;doi:10.1152/jn.00578.2015.—Local field potential (LFP) recordingsfrom spatially distant cortical circuits reveal episodes of coherent gammaoscillations that are intermittent, and of variable peak frequency andduration. Concurrently, single neuron spiking remains largely irregularand of low rate. The underlying potential mechanisms of this emergentnetwork activity have long been debated. Here we reproduce suchintermittent ensemble oscillations in a model network, consisting ofexcitatory and inhibitory model neurons with the characteristics of reg-ular-spiking (RS) pyramidal neurons, and fast-spiking (FS) and low-threshold spiking (LTS) interneurons. We find that fluctuations in theexternal inputs trigger reciprocally connected and irregularly spiking RSand FS neurons in episodes of ensemble oscillations, which are termi-nated by the recruitment of the LTS population with concurrent accumu-lation of inhibitory conductance in both RS and FS neurons. The modelqualitatively reproduces experimentally observed phase drift, oscillationepisode duration distributions, variation in the peak frequency, and theconcurrent irregular single-neuron spiking at low rate. Furthermore,consistent with previous experimental studies using optogenetic manip-ulation, periodic activation of FS, but not RS, model neurons causesenhancement of gamma oscillations. In addition, increasing the couplingbetween two model networks from low to high reveals a transition fromindependent intermittent oscillations to coherent intermittent oscillations.In conclusion, the model network suggests biologically plausible mech-anisms for the generation of episodes of coherent intermittent ensembleoscillations with irregular spiking neurons in cortical circuits.

�-band oscillation; intermittent; coherence; cortex

LOCAL FIELD POTENTIAL (LFP) recordings from cortical tissuehave long revealed short epochs of periodic voltage fluctua-tions (Buzsáki et al. 1992; Csicsvari et al. 1999). Such recordedvoltage fluctuations reflect the superposition of synchronizedoscillatory local extracellular currents from spiking neuronsand/or synaptic inputs within the recording volume determinedby the LFP reach (Buzsáki et al. 2012; Linden et al. 2011).

The LFP oscillations are ubiquitous across different corticalareas, species, and behavioral and cognitive context (Buzsákiet al. 2013; Fries et al. 2007; Jensen et al. 2007; Prechtl 1994).Despite this diversity in origin, cortical LFP oscillations tend toshare four commonly observed features. First, for the popula-tion of neurons located within the LFP recording volume of anelectrode, single neuron spiking is sparse (i.e., firing rate issmaller than oscillation frequency) (Lehky et al. 2005; Rolls etal. 2004, 2006; Treves et al. 1999) and irregular, both duringepochs of oscillations and in the absence of oscillations (Fries etal. 2001). Second, throughout an epoch of elevated spectral

power, the phase of the oscillation is not conserved, i.e., notautocoherent (Burns et al. 2010, 2011). Third, the duration and thepower spectral density (PSD) distribution of LFP oscillations varyfrom epoch to epoch (Csicsvari et al. 1999; Xing et al. 2012), aswell as for different conditions of sensory stimulation (Barbieri etal. 2014; Engel et al. 1990; Jia et al. 2011). Fourth, the fluctuatingaspects of LFP oscillations covary across spatially separate corti-cal recording sites (Roberts et al. 2013).

The ubiquity of LFP oscillations raises the question as towhat these mesoscopic emergent phenomena reveal about theunderlying cortical microcircuits. In other words, what combi-nation of biophysical ingredients of a model network cangenerate the four common features of emergent cortical oscil-lations? Numerous network models of varying levels of com-plexity have reproduced subsets of the listed four features ofcortical LFP oscillations (Bartos et al. 2007; Buzsáki andWang 2012; Traub and Whittington 2010). For instance, de-tailed model networks of irregularly and sparsely spikingneurons generated oscillations (Brunel 2000; Brunel and Ha-kim 1999; Brunel and Wang 2003; Geisler et al. 2005).However, these important investigations did not address inter-mittency and autocoherence. At a higher level of abstraction, arecurrent network consisting of one excitatory and one inhib-itory node generated intermittent oscillations of variable dura-tions and peak frequencies (Xing et al. 2012), but, by design ofa rate model, these investigations did not speak to the nature ofneuronal spiking. To the best of our knowledge, despite thevast literature on the mechanisms of cortical oscillations(Wang 2010), no computational model has captured simulta-neously the four important features of observed cortical oscil-lations, i.e., irregular and sparse spiking, phase drift, epoch-to-epoch variations, and coherence across multiple networks.

Here, we propose underlying mechanisms for cortical oscil-lations that explain all observed features simultaneously. Tothis end, we investigated a model network consisting of regu-lar-spiking (RS) pyramidal neurons, fast-spiking (FS) interneu-rons, low-threshold spiking (LTS) interneurons, and stochasticexternal inputs. With the use of biologically plausible param-eters, the model network spontaneously generated intermittentepochs of activity oscillations with evolving phase, whilemodel neurons spiked sparsely and irregularly. Durations andpeak frequencies varied from epoch to epoch, but covaried fortwo networks with sufficient coupling.

MATERIALS AND METHODS

Model Network

We consider a model network of 2,000 RS pyramidal neurons, 250FS interneurons, 250 LTS interneurons and external inputs (Fig. 1A).

Address for reprint requests and other correspondence: M. S. Hoseini, Dept.of Physics, Campus Box 1105, Washington Univ., St. Louis, MO 63130-4899(e-mail: [email protected]).

J Neurophysiol 115: 457–469, 2016.First published November 11, 2015; doi:10.1152/jn.00578.2015.

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The network structure resembles cortical feedback triads (Caudill etal. 2009). Specifically, each RS and LTS neuron can connect to allother neurons. In contrast, FS neurons can connect to other RS and FSneurons, but not to LTS neurons (Banks et al. 2000). The maindifferences between the two groups of interneurons, besides project-ing to different targets, are 1) LTS neurons operate on much longertime scales (Banks et al. 2000; Goldberg and Yuste 2005; Silberbergand Markram 2007), and 2) LTS neurons tend not to spike asfrequently as FS neurons, since they need coordinated input from RScells (Beierlein et al. 2003; Kapfer et al. 2007). We implemented thesedistinctions through different synaptic time constants and leak con-ductances. Of the possible connections, constrained by the networkstructure, a subset of connections is chosen with probability, Pc, thusgenerating a network with sparse and random connectivity. As aresult, each model neuron obtains on average 400 excitatory and 100inhibitory connections from within the network. In addition, eachneuron has 800 excitatory connections from private external inputswith a given total firing rate (Brunel and Wang 2003). Membranepotentials and action potentials of model neurons of each type aresimulated as leaky integrate-and-fire neurons, with the subthresholddynamics of the membrane potential Vi of neuron i given by C[dVi(t)/dt] � �Iext � Ileak � Isyn, where C is the membrane capacitance,Ileak � gleak(V � Vleak) is the leak current (gleak is leak conductance),and Isyn � gsynS(t)(V � Vsyn) is the synaptic current due to excitatoryand inhibitory inputs from within the network (gsyn is synapticconductance and S is the gating variable) (Table 1). The externalexcitatory input, Iext, is conductance based and has the samedynamics as the excitatory synapses of connections withinthe network. The synaptic gating variable S(t) � [�m/(�d ��r)](exp{�[(t � �l)/�d]} � exp{�[(t � �l)/�r]}) describes the timecourse of synaptic conductance change (Fig. 1B). The normaliza-tion constant ensures that the time integral of the synaptic gatingvariable equals the membrane time constant, and thus varying thesynaptic time constant does not affect the time integral of thesynaptic current. This normalization ensures the balance of exci-tation and inhibition. When the membrane potential of a neuronexceeds a threshold, the neuron fires an action potential, and themembrane potential resets and stays at the rest value for durationof the refractory period. The spike train of neuron i is given by thetime series �i(t). The numerical values in Table 2 are matched withexperimental values for synaptic time scales (Angulo et al. 1999;Bartos et al. 2001; Gupta et al. 2000; Xiang et al. 1998; Zhou andHablitz 1998) and conductances (Buhl et al. 1997; Markram et al.1997). A model neuron receives external inputs with total firingrate of 6 and 8 kHz to RS and FS/LTS neurons. This external inputcan be delivered using a different number of synaptic connections.

Considering 800 excitatory synaptic inputs makes each synapse todeliver an independent Poisson pulse train with mean rate of 7.5Hz for RS and 10 Hz for FS and LTS neurons. For simplicity, thePoisson to Binomial distribution approximation is used. Synapticconductance for external inputs is 1.75 nS for RS and 2.0 nS for FSand LTS neurons. All model parameters are defined, and theirvalues are given in Tables 1 and 2. Simulations were performedusing Euler’s method with a time step of 0.05 ms. Simulationswere carried out for 4 s in real time unless stated otherwise.

Analysis of Simulation Results

Single-neuron spike trains. The sparseness of spiking is evaluatedby comparing the distribution of single-neuron mean firing rates withthe dominant frequency of the oscillation. The irregularity of spikingis evaluated via the interspike interval (ISI) distribution of the spiketrains �i(t). For a Poisson pulse train we expect an exponential decayof the ISI distribution (for ISIs larger than the refractory period) anda coefficient of variation (CV) with a value near one. The CV isdefined as the SD divided by the mean ISI. For a bursting spike train,the CV can be larger than one because of two completely differenttime scales present in the spike train. In contrast, a regular spike trainhas a CV value near zero.

A B

Fig. 1. Model network with recurrent connec-tions and synaptic dynamics. A: schematic rep-resentation of network structure. Each regular-spiking (RS) and low-threshold spiking (LTS)neuron can connect to all other neurons. Incontrast, fast-spiking (FS) neurons can inhibitRS and FS but not LTS neurons. B: gatingvariable dynamics for three selected synapses.They are normalized in such a way that thearea under the curve equals the membrane timeconstant.

Table 1. Neuron and network properties

Parameter Description Value

Vleak, mV Leakage reversal potential �70Vthr, mV Threshold membrane potential �59Vreset, mV Reset membrane potential �52VE, mV Excitatory reversal potential 0Vl, mV Inhibitory reversal potential �80�ref

E , ms RS refractory time 2�ref

I FS and LTS refractory time 1�l, ms Delay time constant 0.5�m

E , ms Excitatory membrane time constant 20�m

I , ms Inhibitory membrane time constant 10CE, nF Excitatory membrane capacitance 0.2CI, nF Inhibitory membrane capacitance 0.1dt, ms Simulation time step 0.05t, ms Time bin size 0.5NRS Number of RS pyramidals 2,000NFS Number of FS interneurons 250NLTS Number of LTS interneurons 250Pc Connection probability 0.2

RS, regular spiking; FS, fast spiking; LTS, low-threshold spiking.

458 NETWORK MECHANISMS OF ENSEMBLE OSCILLATIONS

J Neurophysiol • doi:10.1152/jn.00578.2015 • www.jn.org

Synchrony. To quantify the level of synchrony among the spiketrains �i(t) within the model network we bin the time, �t � 0.5 ms.From the spike train of neuron i, we get the instantaneous firing rateri(t), i.e., the number of spikes within a time bin divided by the binsize. From this, the network instantaneous firing rate is r(t) � �i�1

N

ri(t). Operationally, this quantity is derived by summing the number ofspikes from all neurons within a time bin. We then compare thevariance var[r(t)/N] of the network instantaneous firing rate normal-ized by the network size against the population-averaged variance 1/N

�i�1N var[ri(t)] of the variance of instantaneous firing rates from

individual spike trains (Ginzburg and Sompolinsky 1994; Hansel andSompolinsky 1992). Thus, we define the synchrony measure

�2�N� � �var�r�t� ⁄ N�� ⁄ 1

N�i�1N var�ri�t��.

For fully synchronous spike trains this measure equals one and isindependent of network size. In contrast, for asynchronous spiketrains, �(N) takes on values between zero and one and, importantly,varies linearly with the inverse of the square root of the network size,

�(N) � �(�) � �/�N where � is a constant (Golomb et al. 2006).Oscillations. We evaluate the level of oscillations in the network

activity from the PSD via Fourier transform of the networkinstantaneous firing rate. Calculations are performed using themultitaper method (Henri and Shapley 2005; Mitra and Pesaran1999). In general, time bins, �t, must be much larger than thesimulation time steps and much smaller than the period of theoscillations in network activity. Results are robust for time binsbetween 0.5 and 2 ms. To identify an episode of oscillation innetwork instantaneous firing rate signal, we first took the points inthe time-frequency space that have at least one-half of the maxi-mum power at any frequency between 20 and 100 Hz. Amongthese, points are accepted that are above three times the SD of a5-Hz frequency band around it. Now neighboring points form anepoch. This procedure allows us to estimate the time duration andpeak frequency of an oscillation episode.

Autocoherence. Complementary to evaluating the amplitude of afrequency component, it is informative to evaluate its residual phase,which is the difference between the phase of the analyzed frequencycomponent and the phase of a pure sinusoidal function of the samefrequency. To obtain the phase of the analyzed frequency component,the network instantaneous firing rate r(t) is convolved with a Gaborfilter

��t;�, f� �1

�2exp��t � ��2 ⁄ �22��exp�� j2ft�

and the resulting function is Fourier transformed (j is the square rootof �1 not an index)

G��, f� � ��

�ds r�s��s � t;�, f� � R��,f�exp� j ��, f��.

This “continuous Gabor transform” (Mallat 1999) yields the ampli-tude R(�, f) and the phase (�, f) of the network instantaneous firingrate r(t) in each time-frequency point. The circular variation

CiV� f� � 1 � �� R��, f�exp�j r��, f�� R��, f�

where phase residual r(�, f) is the difference between phase of thesignal and phase of a pure sinusoidal signal with that frequency, r(�,f) � (�, f) � sinusoidal(�, f) where sinusoidal(�, f) � 2f mod[�,(1/f)]. As an example, complex Gabor transform of a pure sinusoidal(autocoherent) signal results in a circle trajectory phase portrait. Timeevolution of the signal manifest itself in the circular counterclockwiserotation of the state on that circle around the origin. Subtracting sinusoidal, which corresponds to a clockwise rotation in the referencecoordinate, will result in one single point in the phase space asexpected, since we started with an autocoherent signal. Circularvariation (CiV) quantifies the localization of the residual phase in thephase portrait and is a criterion for the degree of autocoherence of thatfrequency component (Mardia 1972). The values of CiV are boundedby zero and one, with zero for the most coherent oscillation (puresinusoidal) and one for random signals (Burns et al. 2010).

RESULTS

Overview

To gain insight into what combination of cellular and/orcircuit mechanisms can generate cortical oscillations, we in-vestigated a model network motivated by cortical neurons,synapses, and circuits. Here we focus on four important fea-tures of experimentally observed cortical oscillations: irregularand sparse spiking, intermittent network oscillations withphase drift, epoch-to-epoch variations in network oscillations,and coherence across multiple networks. The model networkconsists of 2,000 RS excitatory neurons, 250 FS interneurons,250 LTS interneurons, and excitatory external inputs (Fig. 1A).Connectivity is sparse and random (see MATERIALS AND METH-ODS) such that each model neuron receives on average 400excitatory and 100 inhibitory connections from within thenetwork, plus 800 excitatory connections from private externalinputs. Each external input is simulated as an independentPoisson pulse train with mean rate of 7.5 Hz to pyramidalneurons and 10 Hz to interneurons.

Single-Neuron Spiking is Sparse, Irregular, andAsynchronous

In response to the continuing external stochastic inputs andthe inputs from the resulting spiking activity of other neuronsfrom within the network, all neurons spike sparsely and irreg-ularly (Figs. 2 and 3A). Time-averaged firing rates vary fromneuron to neuron. Firing rates are distributed within the rangeof �1 and 20 Hz (mean � 6 Hz; maximum � 25 Hz) for RSexcitatory neurons, whereas firing rates for inhibitory neuronsrange from �10 to 50 Hz (mean � 23 Hz for LTS and 34 Hzfor FS; maximum � 50 Hz) (Fig. 2A). The CV values aredistributed around slightly below one (Fig. 2B). The interspikeinterval distributions for all neurons decay approximately ex-ponentially (Fig. 2C), as expected for Poisson pulse trains, thusindicating irregular spiking. To evaluate the level of correlationamong the spike trains, we use the synchrony measure �(N)

Table 2. Synaptic parameters

Pre/Post RS FS LTS

Synaptic conductance, nSRS 0.25 3.8 3.8FS 0.30 4.00 4.00LTS 0.30 0 4.00

Rise time constant, msRS 1.00 0.50 5.0FS 0.20 0.50 5.0LTS 0.20 5.0

Decay time constant, msRS 5 5 50FS 1 5 50LTS 1 50

Leakage conductance, nS 10 12.5 20

459NETWORK MECHANISMS OF ENSEMBLE OSCILLATIONS


(see MATERIALS AND METHODS). This measure is defined as theratio of the variance of the population-averaged instantaneousfiring rate and the average variance of the instantaneous firingrates from individual spike trains. Simulated spike trains in ourmodel yield values of �(N) �1 and, importantly, scale withnetwork size according to �(N) ��(�) � (�/�N) (Fig. 2D).Both observations are characteristics of asynchronous spik-ing. This conclusion is further corroborated by the similarityof the results for network spike trains and shuffled spiketrains. It is important to note, however, that the synchronymeasure �(N) is a time-averaged (total simulation time)measure and thus does not speak to transient correlationsamong spike trains.

Network Oscillations Are Intermittent

To evaluate the possibility of transient epochs of oscilla-tions, we looked beyond the individual spike trains �i(t) (Fig.3A) and analyzed the network instantaneous firing rate. Unlikethe individual spike trains, this measure of network activityreveals significant oscillatory temporal structure (Fig. 3B). Atime-resolved Fourier transform (see MATERIALS AND METHODS)reveals elevated power in narrow frequency bands for shortperiods of time (Fig. 3C). Importantly, however, spiking re-mains sparse. Even at the highest peak in network instanta-neous firing rate near 70 Hz, �5% of the neurons spiked during

a given time bin (0.5 ms). Thus, visual inspection of a subsetof spike trains (Fig. 3A) offers little information about epochsof network oscillations.

In contrast, the network instantaneous firing rates for the RSand FS populations of neurons (fast time scale), and thedynamics of inhibitory synaptic conductance (slow time scale),are informative about the underlying mechanisms of the pop-ulation dynamics (Fig. 4). Qualitatively, the intermittent oscil-lations of network activity arise through the following se-quence of biophysical interactions. First, stochastic fluctu-ations in the spike occurrence of the external inputs to theRS and FS neurons activate these groups of neurons. Sec-ond, from the thus transiently increased RS/FS activity andthe recurrent interaction between RS/FS neurons, an oscil-lation in network activity emerges. Third, the increasedsynchrony of RS excitatory inputs to LTS neurons causes anincreased activity of this group of inhibitory neurons duringthe oscillation. Fourth, the long synaptic decay time (50 ms)of LTS inhibitory synaptic conductances (Fig. 1B) causes agradual build-up of inhibitory synaptic conductances in allneurons. Fifth, the increasing inhibitory synaptic conduc-tance has two biophysical effects: 1) all membrane poten-tials drift closer to the inhibitory reversal potential of �80mV and away from the spiking threshold of �59 mV; and 2)all effective membrane time constants decrease. Sixth, both

A B

C D

Fig. 2. Spiking is irregular and asynchronous onaverage. A: distribution of time-averaged firingrates for excitatory (red) and inhibitory (blue)neurons. For clarity, the distributions of inhibi-tory LTS and FS neurons are merged. Averagefiring rates are 6 Hz for RS, 23 Hz for LTS, and34 Hz for FS neurons. B: distribution of coeffi-cients of variation (CVs) for excitatory (red) andinhibitory (red) neurons. The mean CVs are 0.80for RS and 0.81 for LTS and FS neurons com-bined. C: interspike interval (ISI) distributionsfor a few excitatory (red) and inhibitory (blue)neurons on the right tail of time-averaged meanrate distribution in A. D: synchrony measure�(N) as a function of network size N. The linearincrease with the inverse of the square root of thenetwork size indicates the asynchrony of individ-ual spike trains.



effects reduce the spiking probabilities in all neurons andthus terminate the epoch of network oscillation.

Consistent with this description of mechanisms, the oscilla-tion peak frequency decreases with increasing FS synaptic riseor decay time constant (Fig. 5, A and B), whereas LTS synapticrise and decay time constants do not impact the oscillation peakfrequency (Fig. 5, C and D). However, LTS biophysicalparameters determine the duration of oscillatory epochs(Fig. 5, E and F). LTS decay time impacts the duration ofepisodes in the following way. Synaptic current is defined

by Isyn � gsynS(t)(V � Vsyn), where the gating variable, S(t)(see MATERIALS AND METHODS), includes the normalization con-stant, �m/(�d � �r), which is chosen so that the time integral ofthe gating variable is equal to the membrane time constant(Brunel and Wang 2003). This normalization was adopted tokeep the balance between excitation and inhibition. Varyingthe synaptic time constants does not affect the time integral ofa postsynaptic current but reduces the peak postsynaptic cur-rent. Thus, increasing synaptic time constants of LTS neurons,either �r or �d, reduces the LTS peak postsynaptic current.

A

B

C

Fig. 3. Irregular asynchronous spiking is consistent with intermittent oscillations of network activity. A: raster plots of spikes for 160 RS, 20 FS, and 20 LTSneurons, respectively. Irregular spiking for all three types of neurons is apparent by visual inspection. B: network instantaneous firing rate r(t) � �i�1

N ri(t) revealsoscillations in this continuous variable. An episode of oscillation starts at around 1.75 s and lasts for about 100 ms. C: power spectrum of the networkinstantaneous firing rate shows an episode of elevated power between 1.75 and 1.85 s in time and around 70 Hz in frequency.



Therefore, more spikes in LTS are needed to terminate oscil-lations. Thus increasing synaptic time constants of LTS neu-rons leads to longer-lasting oscillations.

The Phase Drifts Within an Epoch of Oscillation

The stochastic nature of the generation of network oscilla-tions in this model raises the question whether a resultingepoch of network oscillation resembles a sinusoid with a fixedphase. To address this question, we employed the continuousGabor transform (see MATERIALS AND METHODS) to evaluate thedifference between the phase of the analyzed frequency com-ponent and the phase of a pure sinusoidal function of the samefrequency. We use CiV to quantify the localization of residualphase in the phase portrait. The CiV is a measure for the degreeof phase drift of that frequency component, with zero for themost coherent oscillation (pure sinusoidal) and one for signalswith random phase. For all epochs of network oscillationstested, phase portrait trajectories for given frequency bandsbetween 20 and 100 Hz fill out the space with a significantnonzero CiV (Fig. 6). This observation indicates that, over thetime window of a given epoch of network oscillation, the phaseis not constant. The network oscillation is not autocoherent.

Epochs of Oscillations Are Variable

To what extent do the features of network oscillations varyfrom epoch to epoch? This question too is motivated by thestochastic nature of network oscillation generation. To address

this question, we detected epochs of oscillations in a longsimulation (60 s) and characterized each epoch in terms of itsduration and peak frequency (see MATERIALS AND METHODS). Forthe model parameters chosen, the epoch durations are normallydistributed with a mean of 74 ms (Fig. 7A). The durationdistribution depends on LTS biophysical parameters (Fig. 5, Eand F). The peak frequencies are normally distributed arounda mean of 94 Hz (Fig. 7B). As described above (Fig. 5, A andB), peak frequencies depend on the FS synaptic rise or decaytime constants. Epoch duration and peak frequency of networkoscillations are not correlated (Fig. 7C).

The Role of FS Neurons in Rhythmogenesis

The recurrent interaction between RS and FS neurons haslong been thought to be the core mechanism for the generationof network oscillations (Borgers and Kopell 2005; Mann andPaulsen 2007; Wang and Rinzel 1992; Wang et al. 1995). Toevaluate the role of each neuron type in this model (Fig. 1A),we stimulated RS or FS neurons through a 40% decrease intheir leak conductance (pulses with 0.5- and 5-ms rise anddecay time) repeated at regular intervals of 8 or 40 Hz fre-quency (Fig. 8A). First, we verified that this level of leakconductance modulation is sufficient to cause a correspondingincrease in the network instantaneous firing rate in isolated RSand FS neurons (Fig. 8A). We then evaluated the impact of theimposed leak conductance modulation on network activity inthe network with all possible connections intact (Fig. 8B).Modulating RS leak conductance at 8 Hz caused a small, butsignificant (P � 0.002; 2-sample t-test distribution), increase innetwork activity in the 6- to 10-Hz frequency band (Fig. 8C).Modulation of RS leak conductance at 40 Hz had no significanteffect (P � 0.23) on network activity in the 38- to 42-Hzfrequency band (Fig. 8D). Qualitatively, the modulation-in-duced correlated RS activity triggers LTS spiking, which, inturn, produces extended inhibition (50-ms decay time) of bothRS and FS neurons. As a result, the LTS inhibition largelysuppresses the drive-evoked RS spiking response. ModulatingFS leak conductance, in contrast, evokes a different set ofbiophysical mechanisms. Periodic modulation of FS leak con-ductance at 8 Hz has little effect (P � 0.91) on the networkactivity in the 6- to 10-Hz range (Fig. 8E). With the short decaytime (5 ms) of FS inhibition, the impact of FS inhibition on RSspiking is much shorter than the period of modulation (120ms), thus leaving most of the RS spiking to be dominated bystochastic external input. The situation changes for 40-Hzmodulation of FS leak conductance. This rate of modulationsignificantly (P �� 0.01) increased the power of the networkactivity in the 36- to 42-Hz range (Fig. 8F). Qualitatively, at 40Hz modulation and with the decay time (5 ms) of FS inhibition,the external input-evoked RS spiking undergoes sinusoidalfluctuations. This observation highlights the impact of periodicFS activity on network rhythmogenesis.

Coherent Oscillations Emerge Dynamically in CoupledNetworks

As described above, the stochastic nature of rhythmogenesiscauses fluctuations in occurrence, phase, duration, and fre-quency content of oscillation episodes. Such fluctuations raisethe question to what extent intermittent oscillations in twonetworks covary. To address this question, we constructed two

Fig. 4. Dynamics of intermittent oscillations represented in three continuousmodel variables. A fluctuation in external inputs (data not shown) increasesthe RS and FS population-averaged instantaneous firing rates (dark green). Therecurrent interaction between RS and FS neurons mediates oscillations. Thecoordinated activity of RS neurons activates LTS neurons. Because of the longdecay of LTS synaptic conductances in other neurons, the population-averagedconductance accumulates. This accumulation causes a decline in RS and FSactivity, which in turn terminates the oscillatory RS and FS activity. Thevanishing coordinated activity in RS neurons causes a decline in LTS activity(light green). In this parametric plot the state of the network at a certain timeis represented by a dot within 3-dimensional space spanned by the 150-Hzlow-pass-filtered RS and FS instantaneous firing rates and the population-averaged LTS conductance in all neurons. Dots are plotted for 500 ms withtime increments of �t � 0.5 ms. Time increases from dark green to light green.



identical networks but with independent external inputs andvaried the coupling between the two networks (Fig. 9A).

Within each network, of all possible connections, con-strained by the network structure (Fig. 1A), a subset of con-nections is chosen with probability, Pc � 0.2. This proceduregenerates sparse and random connectivity within each network.Parameters are tuned (Fig. 5, A and B) so that each networkgenerates intermittent oscillations at frequencies around 100Hz. Coupling between the two networks is constrained by thesame rules of the network structure. A subset of connectionsbetween the two networks is chosen from the possible connec-tions with internetwork probability that ranges from 0 to 0.2.

The coupling between the two networks, quantified as the ratioof inter- to intranetwork connection probability, ranges fromzero (two independent networks of 2,500 neurons each) to one(one network of 5,000 neurons).

As expected, in uncoupled networks local activity fluctua-tions within each network trigger independent episodes ofensemble oscillations resulting in vanishing coherence values,which were calculated over the 4-s simulation time (Fig. 9B).However, a small increase in the coupling between the twonetworks causes a sharp rise in the coherence of the twonetwork activities in the frequency range around 100 Hz. Thisis despite the fact that the two networks receive independent

A B

C D

E F

Fig. 5. Dependence of oscillations on synaptictime constants. A and B: peak frequency de-creases with increasing FS rise time and withincreasing decay time. For time constants usedin this work see Table 2, except �rise

FS � 1.5 msin B. C and D: LTS synaptic time constants donot impact peak frequency. Parameters arebased on Table 2, except �rise

FS � 1.5 ms. E andF: average duration of oscillation epochs in-creases with increasing LTS rise time and decaytime.



external inputs. In conclusion, coherent oscillations emergedynamically in two networks with modest reciprocal coupling.

Above some intermediate coupling, the coherence decreaseswith increasing coupling. This is because, in our model, inhib-itory (FS and LTS) neurons have a larger time-averaged firingrate than excitatory (RS) neurons (Fig. 2A). Thus the couplingbetween the two networks decreases the overall excitation-to-

inhibition ratio, which in turn decreases the overall networkactivity (Fig. 9C) and thus reduces coherence.

DISCUSSION

In this model investigation we have shown that intermittentensemble oscillations can arise from the interaction of excit-

Fig. 6. The phase is not constant within an epoch of oscillation. Displayed are “phase portraits” of one oscillation epoch for frequencies in the gamma range(20–100 Hz). The oscillation epoch had a peak frequency of 70 Hz and a duration of 200 ms. A phase portrait for a given frequency value is a parametric plotof the residual phase (represented by the angle) and the amplitude (radius) at consecutive points in time (parameter). Circular variation (CiV) is a measure forthe degree of phase drift of that frequency component. The CiV values and the given frequency are shown above each graph. For comparison, a perfect sine waveat the given frequency would result in a point on this graph and a vanishing CiV value. In contrast, signals with a random phase result in a CiV value of 1.



atory RS and inhibitory FS and LTS neurons, while singleneuron spiking remains largely irregular and sparse. The re-current interaction between the RS and FS neurons provides agenerative mechanism of oscillation. Periodic activation of FS,but not RS, model neurons causes enhancement of gammaoscillations. The emerging correlated RS spiking during anoscillation activates LTS neurons. The long LTS synapticdecay time causes an accumulation of inhibitory synapticconductances in all neurons, which eventually terminates theensemble oscillation. The stochastic nature of ensemble oscil-lation causes a phase drift during an epoch of oscillation, anda large variability in durations and peak frequencies from

epoch to epoch. Importantly, however, oscillations largelycovary for two networks with sufficient coupling.

Discrete and Continuous Variables Interact in the Network

The dichotomy between discrete irregular single-neuronspiking and continuous network activity lies at the core ofcortical oscillations (Brunel 2000; Brunel and Hakim 1999;Brunel and Wang 2003). First, the firing rate of individualneurons is typically much smaller than the dominant oscillationfrequency (Csicsvari et al. 1999, 2003; Destexhe et al. 1999;Engel et al. 1990). Second, discrete spike events are trans-formed into continuous variables, such as the synaptic gating

A B C

Fig. 7. Epochs of oscillations vary in duration and peak frequency. A: epoch durations are distributed around 74 51 ms (mean SD; the mean is indicatedby the vertical yellow broken line). B: epoch peak frequencies are distributed around 94 11 Hz (mean SD). C: epoch peak frequency and duration areunrelated. Values for a total number of 200 epochs are shown.

A B C D

E F

Fig. 8. The role of RS/FS neurons in network rhythmogenesis. A: alpha function leak conductance modulation pulses with 0.5- and 5-ms rise and decay timesmodulate network spontaneous firing rate of uncoupled networks. B: schematic representation of the periodic leak conductance modulation of RS and FS neurons.C: low-frequency modulation of RS cells causes a significant relative power increase (P � 0.002) in a 4-Hz frequency band around stimulation frequency, 8 Hz.To compute the power we use multitaper power spectral density with 3 tapers, 500-ms time window sliding by 50 ms. Relative power in the given frequencyis defined as the ratio of power in a 4-Hz frequency range around the given frequency to the total power from 0 to 100 Hz. Black bar is for no modulation case(baseline), and green is in the presence of periodic modulation. Error bars are calculated over 25 repetitions of the simulation. D: 40 Hz modulation of RS neuronsdoes not have a significant (P � 0.23) impact of the relative power at high frequencies centered around 40 Hz. E: low-frequency leak conductance modulationof FS interneurons does not cause a significant (P � 0.91) change in relative power around 8 Hz. F: stimulating FS neurons at gamma-band frequenciessignificantly (P � � 0.01) increases relative power in gamma-band frequencies. �Significant difference between two bars or values.



variable and the membrane potential. Third, the continuousvariables determine the probability of single-neuron spiking.Fourth, in a large population of neurons, the probability ofsingle-neuron spiking translates into network instantaneousfiring rates. In conclusion, at the level of continuous variables,the network dynamic is fully described within the three-dimensional state space spanned by the instantaneous firingrates of RS, FS neurons, and LTS-mediated slow inhibitoryconductances in all neurons (Fig. 4).

Fast Negative Feedback Generates Oscillations

The biophysical mechanisms of sparse pyramidal neuronspiking during gamma oscillation have been explored in recentstudies emphasizing the role of spike frequency adaptation andglobal inhibition (Kilpatrick and Ermentrout 2011) as well asshunting inhibition (Kotani et al. 2014; Krupa et al. 2014).Pharmacological blockade of inhibition disrupts oscillations(Borgers and Kopell 2005; McMahon et al. 1998; Mann andPaulsen 2007; Traub et al. 2004; Wang and Rinzel 1992; Wang

et al. 1995). Our model results (Fig. 8F) corroborate the needfor inhibition in rhythmogenesis and further predict how the FSsynaptic rise and decay time constants impact the oscillationfrequency (Fig. 5, A and B). The oscillation of the continuousvariables is, however, the result of the convolution of discreteand stochastic spike events. The stochastic nature of rhythmo-genesis results in phase drifts during an epoch of oscillation(Fig. 6). In other words, the phase of an oscillation is notconserved during the elevated power of an oscillation. Thislack of autocoherence has been observed in LFP recordingsfrom primary visual cortex of monkeys (Burns et al. 2010;Xing et al. 2012). In addition, the stochastic nature of rhyth-mogenesis results in a large epoch-to-epoch variability in peakfrequencies (Fig. 7B), largely similar to the variability exam-ined in primary visual cortex (Xing et al. 2012).

Optogenetic manipulation of barrel cortex in vivo showedthat light-driven periodic (40-Hz) activation of FS neuronsamplifies oscillations in the gamma range, whereas similaractivation of RS had no such effect (Cardin et al. 2009). Our

A

B

C

Fig. 9. Oscillations covary in networks withmodest coupling. A: schematic representationof two networks with all possible connec-tions and internetwork coupling. Internet-work coupling is similar to intranetwork con-nection, but with different connection prob-ability. B: coherence plot between twonetworks as a function of frequency and theratio of inter- to intranetwork probabilitiesfor a 4-s simulation. C: maximum amplitudeof network instantaneous firing rate as afunction of the ratio of inter- to intranetworkprobabilities.



model reproduces this experimental observation (Fig. 8, D andF) and thus, consistent with investigations of a more complexmodel (Vierling-Claassen et al. 2010), emphasizes the role ofFS neurons in rhythmogenesis.

Slow Negative Feedback Terminates Oscillations

Synchronous RS activity during an oscillation recruits LTSneurons, and their slowly accumulating inhibition eventuallyterminates the oscillation (Fig. 4). Because of the underlyingstochastic spiking, accumulation of activity too is stochastic,which translates into a distribution of oscillation epoch dura-tions (Fig. 7A). Such duration variability is qualitatively similarto what has been observed in primary visual cortex (Xing et al.2012). On average, the duration of an oscillation epoch in-creases with increasing LTS synaptic time constants (Fig. 5, Eand F). With biologically plausible LTS synaptic time con-stants, the model generated oscillation durations of up to a fewhundred milliseconds (Fig. 7A). Because the network stopsfulfilling all four features for very large LTS synaptic timeconstants, additional mechanisms would have to be exploredfor longer durations. Importantly, our model makes the exper-imentally testable prediction that optogenetic activation of LTSneurons after the onset of an oscillation will shorten theduration of that oscillation epoch.

Finally, our model investigation raises an important questionabout the FS and LTS connectivity (Fig. 1A). Our simulationsshow that, in the presence of FS-to-LTS connections of suffi-cient strength, spontaneously started oscillations continue in-definitely, while the peak frequency distribution remainslargely unchanged (data not shown). The qualitative change innetwork behavior from intermittent to continuous as a functionof FS-to-LTS synaptic conductance occurs within a narrowrange between 4.0 nS (intermittent oscillations) and 4.7 nS(continuous oscillations).

The key mechanisms of oscillation termination are the ac-cumulation of activity and the resulting negative feedback. TheLTS neurons are but one biophysical implementation of thetermination mechanism. Alternatively, oscillation terminationcan be accomplished by nonlinear transfer functions (Mem-mesheimer 2010), possibly implemented by the nonlinearproperties of dendrites and synapses (Gasparini and Magee2006; Nevian et al. 2007). Memmesheimer (Memmesheimer2010) has shown that incorporating supralinear dendritic en-hancement of synchronous inputs leads to the generation ofintermittent sharp-wave ripples (200 Hz). This poses the ques-tion for future studies whether coexistence of these two mech-anisms, slow inhibition and nonlinear transfer function, rendersintermittent oscillations more robust. The model presented hereshows that intermittency can arise due to the network structurerather than single-neuron property. However, the model is notrobust. Intermittent oscillations arise from small volumeswithin the multidimensional parameter space.

For completeness, we discuss two important features ofnetwork models with nonlinear components, robustness anddegeneracy. First, an exhaustive scan of the �25-dimensionalparameter space is computationally extremely expensive and isbeyond the scope of the present paper. Nevertheless, networkbehavior is robust with respect to small variations of connec-tion probability (from 30% up to 50%) and synaptic conduc-tances for AMPA and GABA channels (�10% around the

values given in Table 2). Second, the fact that, in networkswith nonlinear elements multiple combinations of parameterscan give rise to the same output, is well established in networktheory and nonlinear dynamics, was introduced in neurosci-ence with a detailed model simulation (Prinz et al. 2004), andreceived further intellectual support in the analytic investiga-tion of a simple model (Caudill et al. 2009). An exhaustivescan of the parameter space in search for degeneracy is beyondthe scope of this manuscript.

Oscillations Covary for Coupled Networks

The stochastic nature of rhythmogenesis and the resultingvariability in phase, peak frequency, and duration raise ques-tions as to the potential coordination of oscillations acrossmultiple networks. Regardless of the observed variability andthe fact that phase does not unfold linearly with extended time,it is thought crucial for two networks to be able to offsetdifferences in oscillation frequencies. In this manner, networkscan initiate and maintain oscillations as communication meansbetween distant neuronal groups (Fries 2005; Miller andBuschman 2013; Nikolic et al. 2013; Roberts et al. 2013;Salinas and Sejnowski 2001). We have shown that oscillationsbetween distant regions remain robustly coordinated despitesignificant variations in their internal dynamics. Our resultsindicate that a wide range of frequencies could be exploited asmechanisms for information transmission between two net-works with recurrent connections during perceptual and cog-nitive processing.

Simulating Important Features of Cortical Oscillations

Numerous previous theoretical investigations have offeredphysical intuition about individual aspects of rhythmogenesisin cortical circuits (Bathellier et al. 2008; Gerstner and vanHemmen 1993; Geisler et al. 2005; van Vreeswijk et al. 1994;Vierling-Claassen et al. 2010; Wang and Rinzel 1992). Herewe highlight selected examples. 1) A spiking model wasdesigned to generate ensemble oscillations in the presence ofirregular and sparse spiking, but did not reproduce the inter-mittency of rhythms (Brunel and Wang 2003). 2) Anotherspiking model included spike timing-dependent plasticity anddisplayed the transition between different frequency bands(Izhikevich 2006). However, the degree of synchronization ofspikes during network oscillations was inconsistent with theexperimentally observed irregular and sparse spiking. 3) Clus-tered connections have been proposed to generate slow dynam-ics and high variability in a network of spiking neurons(Litwin-Kumar and Doiron 2012). This model, however, doesnot reproduce intermittent ensemble oscillations with variablepeak frequencies. 4) electroencephalogram (EEG) model net-works (not spiking) generate intermittent oscillations (Good-fellow et al. 2011; Jia et al. 2013; Lopes Da Silva et al. 1974,1975; Wendling et al. 2002; Xing et al. 2012). However, bydesign (not spiking), these EEG models do not speak toirregular and sparse spiking.

In contrast, the spiking model proposed here, for the firsttime, reproduces five important features of observed corticaloscillations: 1) irregular and sparse spiking, 2) phase driftduring an epoch of oscillation, 3) intermittent oscillations, 4)epoch-to-epoch variations in peak frequency and duration, and5) coherence across multiple networks.



In addition, the model makes two testable predictions. First,our model predicts a lack of connections from FS to LTSneurons. Experimental observations in hippocampus (Banks etal. 2000; Miles et al. 1996) support this prediction. In cortex,the connectivity from FS to LTS neurons could be testedexperimentally in the following two ways: 1) search for phys-iological connections via dual whole cell recordings in a sliceof cortex and 2) search for anatomical connections via electronmicroscopic survey (connectomics) of a piece of cortical tis-sue. Second, the model predicts that optogenetic hyperpolar-ization of LTS neurons would transform oscillations fromintermittent to more continuous. This experiment could berefined by triggering the optogenetic manipulation of LTSneurons on a detected oscillation (for online oscillation detec-tion, see, for instance, Rutishauser et al. 2013). The predictionis that oscillation-triggered hyperpolarization of LTS neuronswould increase oscillation episode duration, whereas oscilla-tion-triggered depolarization of LTS neurons would decreaseoscillation episode duration.

ACKNOWLEDGMENTS

We thank members of the Neurophysics Laboratory for useful discussions.

GRANTS

This research was supported by a Whitehall Foundation Grant no.20121221 and National Science Foundation CRCNS Grant no. 1308159 to R.Wessel.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

Author contributions: M.S.H. and R.W. conception and design of research;M.S.H. analyzed data; M.S.H. prepared figures; M.S.H. drafted manuscript;M.S.H. and R.W. edited and revised manuscript; M.S.H. and R.W. approvedfinal version of manuscript.

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Coherent and intermittent ensemble oscillations …physics.wustl.edu/nd/fac/wessel/2016HoseiniWesselCoherentAnd...Coherent and intermittent ensemble oscillations emerge ... from spatially

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