Gamma rhythms and beta rhythms have different synchronization properties N. Kopell�����, G. B. Ermentrout��, M. A. Whittington��, and R. D. Traubi ��Department of Mathematics and Center for BioDynamics, Boston University, Boston MA 02215 ��Department of Mathematics, University of Pittsburgh, Pittsburgh PA 15260 ��School of Biomedical Sciences, Worsley Building, University of Leeds, Leeds LS2 9NL, United Kingdom and iDivision of Neuroscience, The Medical School, University of Birmingham, Birmingham B15 2TT, United Kingdom Contributed by Nancy J. Kopell, December 3, 1999 Experimental and modeling efforts suggest that rhythms in the CA1 region of the hippocampus that are in the beta range (12���29 Hz) have a different dynamical structure than that of gamma (30���70 Hz). We use a simplified model to show that the different rhythms employ different dynamical mechanisms to synchronize, based on different ionic currents. The beta frequency is able to synchronize over long conduction delays (corresponding to signals traveling a significant distance in the brain) that apparently cannot be tolerated by gamma rhythms. The synchronization properties are consistent with data suggesting that gamma rhythms are used for relatively local compu- tations whereas beta rhythms are used for higher level interactions involving more distant structures. R(12���30 hythms in the gamma range (30���80 Hz) and the beta range Hz) are found in many parts of the nervous system and are associated with attention, perception, and cognition (1���3). It has been noted in electroencephalogram (EEG) signals that rhythms of different frequencies are found simultaneously (4). Beta oscillationsarereadilyobservableimmediatelyafterevokedgamma oscillations in sensory evoked potential recordings (5). This beta activity has been correlated with the long-range synchronous ac- tivity of neocortical regions during visuomotor reflex activation (6). This paper concerns the correlation between the frequency band of coherent oscillations and conduction delays between the sites participating in the coherent rhythm. It has been noted (7) in human EEG subjects that gamma rhythms are prevalent in local visual response synchronization, but more distant coherence oc- curring during multimodal integration between parietal and tem- poralcorticesusesrhythmsatfrequenciesof12���20Hz(theso-called beta 1 range). We shall use data from the CA1 region of the hippocampus (8���10) as a paradigm to address the questions of how long-distance synchrony is achieved and why there is a correlation between oscillation frequency and the temporal distances between partici- pating sites. The data available from the rat hippocampus slice preparation give clues about details of dynamics that are important to the synchronization process. The work builds on earlier work (11���12) describing and analyzing the role of doublet spikes in interneurons in producing synchrony when there are significant conduction delays. Earlier work (13) using rate models showed, via simulations, that longer conduction delays could be tolerated and still produce synchrony if the carrier rhythm had lower frequencies. However, a rate model is not consistent with the situation in which excitatory cells fire at most one spike per cycle, and with high precision in phase. An alternative solutionwassuggestedbydataandlarge-scalemodelsofthegamma rhythm in the hippocampus (8, 9). In both data and models, the ability to synchronize happened in those parameter regimes in which interneurons produced a spike doublet in many of the cycles. This mechanism was analyzed by Ermentrout and Kopell (11), where it was shown how the doublet provides a feedback mecha- nism for the timing. The analysis given there predicted that, for long conduction delays (above 8���10 ms, depending on network param- eters), synchronization in the gamma frequency band is not robust. Although conduction delays in the neocortex are variable, there is evidence that the delays between association areas could be sig- nificantly larger than 10 ms (see Discussion). In this paper, we show that the beta rhythm observed in the hippocampal slices is not merely a slower version of gamma, but has a distinct dynamical structure and makes use of intrinsic membrane currents not expressed during gamma. Furthermore, the beta rhythm is much better adapted to synchronization in the presence of long conduction delays. Via a very reduced model, we analyze why this is so. Predictions from the analysis are shown to hold in the large-scale models. Background on production of beta and gamma rhythms in the hippocampal slice can be found in ref. 3. In the tetanic stimulation paradigm (9, 10), with sufficiently strong stimuli, the hippocampal slice produces both gamma and beta, with a transition between them. In intracellular recordings of beta in pyramidal cells, gamma- frequencyoscillationscontinuebetweenbeta-frequencypopulation spikes, suggesting that the interneuron network continues to oscil- late at gamma frequency, which the pyramidal cells cannot follow (Fig. 1A). Two system parameters alter in time before and during the transition to beta: the strength of recurrent excitatory synapses and the amplitude of one or more slow K conductances. Both of these parameters increase and then level off, and experimental data and large-scale simulations suggest that evolution of both param- eters is necessary for the switch to beta to occur (10, 14). Beta oscillations are synchronized between the two sites when both sites are stimulated together intensely (10, 14). In human EEG, occurring spontaneously or evoked by auditory stimulation by novel sounds, power in the gamma range coexists withbeta,consistentwiththebeat-skippingstructure[C.Haenschel and J. Gruzelier, personal communication also see the work by Tallon-Baudry et al. (15)]. Local Inhibition-Based Rhythms. Thedataandlargescalesimulations cited above all concern the behavior of the network when two sites are intensely stimulated together. To better understand the mech- anism behind the network behavior, we first consider the behavior at one site. We show, via very reduced models, that the transition from gamma to beta can be understood as a consequence of the changes in recurrent excitatory synapses and expressions of K- conductances. Although this had previously been documented in large-scale simulations (14), the ability of the small network to reproduce this creates an excellent model within which to under- stand more deeply the long-distance synchronization properties of beta and gamma. We use models that are much reduced from the large scale simulations in two ways. The network is pared down to a minimal number of cells and connections. We work with a local network of two pyramidal cells (excitatory, or E-cells) and two interneurons (inhibitory, or I-cells). All cells are coupled to one another, except Abbreviation: EEG, electroencephalogram AHP, after-hyperpolarization. ���To whom reprint requests should be addressed. E-mail: nk@bu.edu. The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked ���advertisement��� in accordance with 18 U.S.C. ��1734 solely to indicate this fact. PNAS u February 15, 2000 u vol. 97 u no. 4 u 1867���1872 NEUROBIOLOGY

possiblyforcouplingbetweentheexcitatorycell(dottedlinesinFig. 2A). E-E coupling, although sparse in the CA1 (16), turns out to be important for the beta rhythm (10, 14) gamma rhythms, however, canbesimulatedwithoutE-Ecoupling(11,17).Thegammarhythm corresponds to one in which all of the cells fire synchronously at 30���70 Hz whereas in the beta rhythm, the I-cells synchronize at a gamma frequency and the E-cells synchronize at a frequency half as fast. Compared with the detailed biophysical models (14), the cells are also simple: they are modeled as single compartment cells with fast spiking currents for the gamma rhythm for the beta rhythm, an extra after-hyperpolarization (AHP) current (slow K conductance) is added to the E-cells (see Appendix). Although we use a specific current (M-current) in the simulations, the analysis in the mathematics given below will work for any AHP current with the appropriate decay time. Increases in K-conductances, plus increases in the strength of synapses between excitatory cells, can transform the output of the network of E and I cells from gamma to beta. This transformation was documented in the detailed biophysical model (14) with two connected sites. In Fig. 2B, we show that the transition is repro- duced in the reduced model, even without the synaptic connections betweenthesites.ThefirstpartofFig.2B displaysthevoltagetraces of the two E-cells and one of the I-cells (they are synchronous) with parameters that elicit a gamma rhythm. In the middle section, a slowerK-conductancehasbeenaddedtothemodelE-cells nowthe E-cells, slowed down by the K-conductance, each fire on half of the gamma cycles. Note that they fire on opposite cycles. For the third part of Fig. 2B, the parameters were further changed by adding synaptic (AMPA-mediated) connections between the E-cells the network is now as in Fig. 2A with the dotted lines. Now the E-cells still fire every other cycle, but this time on the same cycle that is, they produce beta. TounderstandwhytheE-cellsmissoppositecyclesintheabsence of the E-E coupling, we note that the firing of one E-cell effectively silences the other in a given cycle through feedback inhibition, unless the lagging cell is so close that it fires before the onset of the feedback inhibition. A major effect of the mutual excitatory con- nections is to increase the range of initial conditions under which the second cell can fire before receiving inhibition in a manner graded with the size of the excitatory conductance, the excitation advances the firing of the second E-cell, preventing suppression in that cycle. With some E-E coupling, there can be other initial conditions for the same parameters for which the E-cells do fire on opposite cycles throughout the trajectory. However, if the E-E coupling is suffi- ciently large, that solution does not stably exist. With enough excitation from the cell that fires in a given cycle, the other cell is forced to fire in the same cycle, ruling out the solution in which cells fire on opposite cycles. We also note that there are many different ways to change parameters to produce the gamma-to-beta transition. In addition to the new excitatory connections, the essential change is to lower the excitability of the E-cells relative to the I-cells, by changing relative drives or intrinsic conductances. As we will see in the next section, synaptic input from distant sources can also change the balance of excitability. Long-Distance Synchronization in Inhibition-Based Rhythms. Strategy and basic dynamical properties. The different dynamical structures and currents associated with gamma and beta lead to different results when these rhythms are used to coordinate dynamical activity of loci at a distance from one another. To show this, our strategy is to look at the dynamics near the gamma or beta rhythm and create a map (a function relating the timing of one cycle to that of the next) containing information about whether, and in what parameter ranges, that rhythm is dynamically stable. There are two principles that govern the behavior of the maps. The first is that E-cells are able to fire when inhibition, either synaptic or intrinsic (from AHP currents), has worn off sufficiently. This situation obtains when the effective membrane time constant of the excitatory cells is small compared with the decay time of the synaptic current andyor the AHP current the voltage then tracks the time course of the synapses or AHP currents. Second, the I-cells have an extra property, associated with relative refractory period. Suppose a cell fires at t 5 0 and receives Fig. 1. Gamma and beta oscillations in vitro. Intracellular recordings of gamma and beta oscillations in a CA1 hippocampal pyramidal neuron. Oscillations were induced by brief tetanic stimulation [see Whittington et al. (9) for methods]. The initial posttetanic response is a gamma oscillation with action potentials (fre- quency 38 Hz) separated by a period of hyperpolarization made up of both AHP and inhibitory synaptic activity. After the transition to beta activity, the under- lying gamma membrane potential oscillation is still apparent (frequency 42 Hz), but spiking occurs on every second or third period (frequency 18 Hz). Action potentialsareseparatedbytheinitialAHPyIPSPhyperpolarizationandadditional IPSPs. (Bar 5 1 mV, 100 ms.) Fig. 2. (A) Minimal network for investigating local synchronization of gamma and beta rhythms. For the gamma rhythms, the E-E connections are absent for the beta rhythms, they are a necessary part of the circuit. (B) Gamma-to-beta transition of local rhythms occurs as the AHP is turned on and the local E-E connectionsarestrengthened.Parametersareasintheappendix.Forthegamma rhythm, gee 5 0andgm 5 0.Atthefirstarrow,gm issetto1,switchingtherhythm from gamma to a rhythm in which the E-cells miss beats and fires nonsynchro- nously. At t 5 400, gee 5 0.15, and the network quickly suppresses the nonsyn- chronous solution, leaving only the synchronous local state. Throughout the transitions, the I cells shown below exhibit only minor changes, slowing down slightly because of the decreased excitation (excitation every other cycle). 1868 u www.pnas.org Kopell et al.