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Inhibition can sometimes synchronize networks
Synchronization can occur in strictly inhibitory networks, and produce rhythms in the gamma frequency range (30-80 Hz). Motivated by the work of Traub, Whittington and Jefferys on rhythms in hippocampal slices, White, Chow, Ritt, Soto-Trevino and Kopell wrote a series of papers on networks of spiking cells, with emphasis on the time scales of the synaptic interactions. This work analyzed local interactions of inhibitory cells (given metabotropic excitation) and showed that, in the presence of even mild heterogeneity, stable synchronization is possible only in the parameter range in which the network frequency is determined mainly by the decay time of the inhibition.
Rhythms in the gamma frequency range also can occur in networks of excitatory and inhibitory cells, also dependent on decay time of inhibition. In current work, Borgers and Kopell showed that noise in the system constrains the frequency range over which this can occur to the normal gamma range. They also showed that this E/I network gamma is much more robust to heterogeneity, and can form a coherent rhythm in the presence of very sparse coupling; a mathematical explanation is given.
Gap junctional coupling can help synchronization - or not
Electrical coupling between inhibitory interneurons exists in the hippocampus and neocortex. Chow and Kopell showed that the electrical coupling need not be synchronizing, and that the shapes of spikes and the frequency of the system are significant in determining whether a system coupled electrically does synchronize. Since chemical synapses are less effective at very high frequencies, electrical coupling may be especially important for very high frequency synchronization.
Hippocampal CA1 slices can produce inhibition-based gamma rhythms (all excitatory synapses are blocked). When gap junctions are blocked, the synchronization between cells at a significant distance in the slice is degraded, showing that gap junctions help the inhibitory coupling to perform synchronization. Detailed models by Traub et al. show that even very small electrical conductances, added to the much larger inhibitory conductances, can accomplish tight synchronization across the slice, in spite of the fact that the electrical coupling is much more local than the chemical synaptic coupling. Kopell and Ermentrout are currently working on mathematics to explain the behavior of this distributed network with both chemical and electrical coupling.
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C. Chow and N. Kopell, "Dynamics of spiking neurons with electrical coupling", Neural Computation, 12 (2000) 1643-1678. R.D. Traub, N. Kopell, A. Bibbig, E.H. Buhl, F.E.N. LeBeau, and M.A. Whittington, "Gap junctions between interneuron dendrites can enhance synchrony of gamma oscillations in distributed networks", J. Neurosci., 21 (2001) 9478-9486. R.D. Traub, S. Schmitz, J.G.R. Jefferys and A. Draguhnm "High-frequency population oscillations are predicted to occur in hippocampa neuronal network interconnected by axo-axonal gap junctions", Neuroscience 92 (1999) 407-426. T.J. Lewis and J. Rinzel, "Dynamics of spiking neurons connected by both inhibitory and electrical coupling", to appear in J.Comput.Neurosci.(2002). | |
Here are MPEG movies of a 49 cell inhibitory network, without and with electrical coupling. The cells are shown as a 2-D lattice (which does not reflect coupling), with membrane potential plotted on the vertical axis. Note that even a small degree of electrical coupling greatly enhances network coherence.
Gamma and beta rhythms are inhibition-based rhythms
In hippocampal slices (CA1), Whittington et al. have shown that gamma and beta (12-30Hz) rhythms can be produced by tetanic stimulation: weak stimulation produces only gamma rhythms, while somewhat stronger stimulation produces gamma, followed by a spontaneous switch to beta. In this beta rhythm, there is an underlying inhibitory gamma rhythm, with pyramidal cells skipping cycles, and firing on the same cycles enough to create a population rhythm at the lower beta frequency. The gamma rhythm and beta rhythms have been obtained in vitro using many different forms of stimulation. The similarities and differences have been described in Whittington et al, 2000.
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M.A. Whittington, R.D. Traub, N. Kopell, G.B. Ermentrout and E.H. Buhl, "Inhibition-based rhythms: Experimental and mathematical observation on network dynamics", Int. J. of Psychophysiology 38 (2000) 315-336 R.D. Traub, M.A. Whittington. E.H. Buhl, J.G.R. Jefferys an H. Faulkner, "On the mechanism of the gamma-beta frequency shift in neuronal oscillations induced in rat hippocampal slices by tetanic stimulation", J. Neurosci. 19 (1999) 1088-1105. N. Kopell , G.B. Ermentrout, M. Whittington and R.D. Traub, "Gamma rhythms and beta rhythms have different synchronization properties", Proc. Nat. Acad. Sci. USA, 97 (2000) 1867-1872. | |
The gamma rhythm can act as a preprocessor for the beta rhythm
As described above, there is a spontaneous transition from the gamma rhythm to the beta rhythm in CA1 in vitro. (This has also been seen in vivo in the auditory cortex.) In the presence of heterogeneous inputs to model pyramidal cells, the gamma rhythm is able to create a partition, with only cells having sufficient drive participating. When the network is in the beta parameter range, the same heterogeneous set of tonic inputs used during gamma leads to a population behavior in which all cells participate, but at different rates. But if some connections among the pyramidal cells are plastically changed during the gamma rhythm preceding the beta rhythm, the network can display well-defined cell assemblies during the beta portion of the rhythm, with a separation in timing of cells previously firing during the gamma portion and those cells not participating in the gamma rhythm. Experimental corroboration of the separation is presented. This work shows the ability of gamma to act as a preprocessor for beta, and thus helps to explain why the gamma-beta combination may be used by the nervous system.
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M. Olufsen, MA. Whittington, M. Camperi and N. Kopell, "New functions for the gamma rhythm: population tuning and preprocessing" J. Comput. Neurosci., in press 2002. C. Haenschel, T. Baldewih, R. Croft, M.A. Whittington and J. Gruzelier, "Gamma and beta frequency cortical oscillation in response to novel auditory stimuli: A comparison of human electroencephalogram (EEG) data with in vitro models", Proc. Nat. Acad. Sci USA 97 (2000) 7645-7650. | |
Benzodiazepines produce beta rhythms in the motor cortex
Benzodiazepines such as diapam produce a beta rhythm in S1 (primary somatosensory cortex). A model of the mechanism of this beta-frequency coherence was done by Jensen et al., using properties of beat-skipping beta described above.
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O. Jensen, M. Pohja, P. Goel, G.B. Ermentrout, N. Kopell and R. Hari, "On the physiological basis of the 15-30 Hz motor-cortex rhythm", Proc. of 13th Int. Conf. On Biomagnetism, 2002. | |
Intrinsic currents can determine if excitation is synchronizing
The excitatory stellate cells of the entorhinal cortex provide the main input to the hippocampus from the cortex. These cells are individually capable of a creating a theta-frequency rhythm (4-12 Hz) , and synchronize to form a population the rhythm in that frequency range. Acker et al. have shown that slow currents displayed by those cells are responsible for the ability to form the coherent rhythms when there is excitatory coupling. The techniques use "spike time response curves” (STRCs), a variation of "phase response curves”, and can account for nonlinear effects that come with large forcing. The techniques show that even very small coupling conductances can be "effectively large” in the sense that firing times depend nonlinearly on synaptic strength for small inputs. One dramatic nonlinear effect of such inputs is to make a periodically firing neuron skip beats, which is shown to increase the rate of synchronization. Electrical membrane noise makes cycle skipping more prevalent.
The techniques of the modeling can be carried out in real stellate cells. Netoff et al. have experimentally computed STRCs for stellate cells in vitro. They also created a hybrid network with one real and one model stellate cell. With dynamic clamp techniques, they could connect these cell by inhibition or excitation. As expected, the cells synchronized with mutual excitation, but not with mutual inhibition. When the cells were replaced by rules for updating spike times using the STRCs, the results were qualitatively and quantitatively the same as the full simulations.
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Acker, C. "Synchronization of strongly coupled excitatory neurons: relating biophysics to network behavior", Masters of Science, Boston University (2000). G.B. Ermentrout, B.Gutkin and M. Pascal, "The effects of spike frequency adaptation on the synchronization of neural oscillators", Neural Computation, in press. C. Acker, N. Kopell and J. White, "Synchronization of strongly coupled excitatory neurons: relating network behavior to biophysics", preprint 2002. T. Netoff, A. Dorval, C. Acker, J. Haas, N. Kopell and J. White , "Synchrony in excitatory and inhibitory hybrid neuronal networks in the entorhinal cortex" , preprint 2002. | |
Fast and slow GABA_A kinetics give mixed gamma/theta rhythm
M. Banks and R. Pearce have discovered a GABA_A receptor pharmacologically similar to the standard GABA_A but 5-6 times slower in its kinetics. In CA1, there is a population of "slow" GABA_A cells that innervates a population of "fast" GABA_A cells, with the kinetics dependent on the pre-synaptic cell. White and colleagues have produced models showing that such populations can give rise to nested theta/gamma rhythms. Furthermore, even in parameter ranges in which the two populations do not produce the nested rhythms (or theta), phase-dispersed input from the entorhinal cortex can lead to sharply focussed theta in CA1, nested with gamma. Kopell and Serenevy are currently examining the mechanism by which phase-dispersed input leads to a higher spectral input.
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M. I. Banks, J. A. White and R. Pearce, "Interactions between distinct GABA_A circuits in hippocampus", Neuron, 25 (2000) 449-457. M. Banks, T. Li and R. Pearce, "The synaptic basis of GABA_A, Slow", J. Neurosci. 18 (1998) 1305-17. J. White, M. Banks, R. Pearce and N. Kopell, "Networks of interneurons with fast and slow GABA_A kinetics provide substrate for mixed gamma-theta rhythm", Proc. Nat. Acad. Sci. USA 97 (2000) 8128-8133. Addendum containing numerical methods. | |
Review: Mechanisms of synchronization
Kopell and Ermentrout reviewed a variety of mechanisms by which phase relationships between a pair of oscillators are determined. The review focuses on different mathematical techniques for determining if the oscillators stably synchronize.
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N. Kopell and G.B. Ermentrout "Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators”, for Handbook on Dynamical Systems, vol. 2 Toward applications . Ed. B. Fiedler, Elsevier, 2002, pp 3-54. | |
Modulatory changes during theta oscillations
Papers by Fox et al. and Skaggs et al. show phasic changes in interneuron activity during theta which could result in phasic changes in influences at both GABA_A and GABA_B receptors. During the period of strong GABA receptor activation, synaptic transmission could be weaker due to activation of presynaptic GABA_B receptors. Data supporting this has been obtained by Wyble, Linster and Hasselmo, and Molyneaux and Hasselmo. In computational models of sequence encoding in hippocampus, phases of weak synaptic transmission allow more effective encoding of new sequences (Wallenstein and Hasselmo, 1997) and more effective retrieval of specific previously encoded sequences (Sohal and Hasselmo, 1998a, 1998b).
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S.E. Fox, S. Wolfson and JB Ranck, "Hippocampal theta rhythm and the firing of neurons in walking and urethane anesthetized rats", Exp Brain Res 62 (1986) 495-508. W.E. Skaggs, B.L. McNaughton, M.A. Wilson and C.A. Barnes, "Theta phase precession in hippocampal neuronal populations and the compression of temporal sequences", Hippocampus 6 (1996) 149-172. V.S. Sohal and M.E. Hasselmo, "Changes in GABA_B modulation during a theta cycle may be analogous to the fall of temperature during annealing", Neural Comput., 10 (1998a) 869-902. V.S. Sohal and M.E. Hasselmo, "GABA_B modulation improves sequence disambiguation in computational models of hippocampal region CA3", Hippocampus 8 (1998b) 171-193. G.V. Wallenstein, and M.E. Hasselmo, "GABAergic modulation of hippocampal activity: Sequence learning, place field development, and the phase precession effect", J. Neurophysiology, 78 (1997) 393-408. | |
Cholinergic modulation of feedback in waking and slow-wave sleep
Cortical acetylcholine levels are dramatically lower during slow-wave sleep compared to waking (Marrosu et al., 1995). This change means a dramatic increase in strength of excitatory feedback from hippocampus to neocortex, as acetylcholine normally suppresses this excitatory feedback (Hasselmo and Schnell, 1994; Hasselmo et al., 1995, Hasselmo, 1999). This increase in strength of excitatory feedback could contribute to initiation and propagation of sharp waves from hippocampus to neocortex (Chrobak and Buzsaki, 1994). Propagation of sharp waves could be a major factor in consolidation - the formation of associations in neocortex on the basis of previously encoded associations in hippocampus (Buzsaki, 1989). Models demonstrate how encoded attractors could be reactivated with random activity in hippocampus and then spread back to neocortex (Hasselmo et al., 1996).
Atropine-resistant theta in an inhibitory network is switched to gamma by AMPA-mediated excitation
Gillies et al. have shown that an atropine-resistent theta can be formed in CA1 in vitro using mGluR agonists in the presence of blockade of AMPA-mediated excitation. If the AMPA is not blocked, the network produces a gamma rhythm instead. The theta rhythm is dependent on the current I_h, which is displayed in the OLM interneuron, whose IPSPs are at least twice as long as the fast-firing interneurons. The OLM cells have currents similar to those in the stellate cells, and are therefore not expected to synchronize by inhibition. Models by Rotstein et al. show that the ability of those cells to synchronize involves a balance between the amount of inhibitory conductance and the conductance of I_h. Even if the OLM cells cannot form a coherent theta by themselves, addition of fast-firing interneurons to the network can synchronize the OLM cells. Addition of pyramidal cells (AMPA not blocked) switches the network to a gamma rhythm.
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M.J. Gillies, R.D. Traub, F.E.N. LeBeau, C,H. Davies, T. Glovelli, E.H. Buhl and M.A. Whittington, "A model of atropine-resistent theta oscillations in rat hippocampal area CA1" Physiology in Press; published online July 19, 2002 as 10. 1113/physiol. 20002.024588 . H. Rotstein, M. Whittington, M. Gillies, E.H. Buhl and N. Kopell, "Atropine-resistant theta in an inhibitory network is switched to gamma by AMPA-mediated excitation: a model", in prep. 2002. | |
The same excitatory-inhibitory network can display alpha, beta or gamma rhythms
The model of alpha given here is motivated by the “augmenting response” of neocortical neurons: because of the inhibition activated current I_h and the inhibition de-inactivated low threshold calcium current, inhibition creates a rebound whose time constant makes the resulting network display a rhythm in the alpha-frequency (9-11 Hz) range. Modulation of the drive to the excitatory cell, mimicking the excitatory action of ACh, changes alpha to gamma or theta, depending on the amount of excitatory drive. The model contains slow potassium currents, needed to produce the beat-skipping beta rhythm.
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D. Pinto, S. Jones, T. Kaper and N. Kopell, "Analysis of state-dependent transitions in frequency and long-distance coordination in a model oscillatory cortical circuit", to appear in J. Comput. Neurosci. | |
Changes in ionic currents change network behavior
Work by Kopell and LeMasson concerns the ability of networks with the ubiquitous h-current to be able to regulate the fraction of cells participating in a rhythm while keeping the frequency constant. The paper by Terman et al. shows how a small change in the intrinsic properties of a class of thalamic cells can lead to effective rewiring of the circuit and different network behavior. The work in Acker et al. includes descriptions about how synchrony relies on the size of the I_h conductance and the conductance of a slow K-current.
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N. Kopell and G. LeMasson, "Rhythmogenesis, amplitude modulation and multiplexing in a cortical architecture", Proc. Nat. Acad. Sci. USA 91 (1994) 10586-10590. D. Terman, A. Bose and N. Kopell, "Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms", Proc. Nat. Acad. Sci. USA 93 (1996), 15417-15422. C. Acker, N. Kopell and J. White, "Synchronization of strongly coupled excitatory neurons: relating network behavior to biophysics", preprint 2002 | |
Long-distance synchronization in gamma and beta rhythms are facilitated by fine structure in interneuronal firing
Traub, Jefferys and Whittington reported that spike doublets in the interneurons occur when there is synchronization in the gamma frequency range between local circuits that are separated. If there are significant conduction delays, the mechanisms described above in the work of White et al. and Chow et al. work poorly. Kopell and G.B. Ermentrout produced a different mechanism to account for synchronization in the presence of delays, using information encoded in timing between the doublets. The work focuses on the roles of the E(xcitatory) to I(nhibitory) and I to E connections in the network, and shows that the mechanism can be captured in a one-dimensional map describing the time between separated E-cells in successive cycles. A generalization of that work from two local circuits to large networks with spatial disorder has been done by N. Kopell and J. Karbowski.
Another generalization done recently by McMillen and Kopell considers separated pairs of large E/I networks, each with heterogeneity in drive. They show that networks need not synchronize if the percentage of I-cells firing on the second spike of the doublet varies considerably from cycle to cycle; this is also captured by a low-dimensional map. Addition of noise to the network restabilizes synchrony by evening out the fraction of I cell firing on different cycles.
Fuchs et al. used transgenic methods to construct a strain of mice in which the time constant of the EPSPs onto interneurons are increased by approximately 10 %. Data show that the local synchronization of hippocampal CA1 slices is not affected, but synchronization between separated sites is significantly decreased. Detailed biophysical models reproduce this effect, which is explained in terms the doublet mechanism described above.
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G.B. Ermentrout and N. Kopell, "Fine structure of neural spiking and synchronization in the presence of conduction delays", Proc. Nat. Acad. Sci. USA 95 (1998) 1259-1264. J. Karbowski and N. Kopell, "Multispikes and synchronization in a large-scale neural network with delays", Neural Computation, 12 (2000) 1573-1606. R.D. Traub, M.A. Whittington, I.M. Stanford, J.G.R. Jefferys, "A mechanisms for generation of long-range synchronous fast oscillations in the neocortex", Nature 383 (1996) 621-624. D. McMillen and N.Kopell, "Noise-stabilized synchronization in populations of model neurons", preprint 2002. E. C. Fuchs, H. Doheny, H. Faulkner, A. Caputi, R.D. Traub, A. Bibbig, N. Kopell, M. Whittington, and H. Monyer, "Genetically altered AMPA-type glutamate receptor kinetics in interneurons disrupt long-range synchrony of gamma neurons", Proc. Nat. Acad. Sci USA 98 (2001) 3571-3576. | |
Networks displaying the beta rhythm can synchronize with longer conduction delays
Hippocampal slices that receive high intensity stimulation display a spontaneous change in frequency from the gamma frequency (30-80 Hz) to the beta frequency (12-30 Hz). The beta rhythm has a different dynamical structure than the gamma, and is dependent on different ionic currents. During the beta phase, there are functional connections (between pyramidal cells) that are absent during the gamma phase. Kopell, Ermentrout, Traub and Whittington have elucidated the mechanisms of the gamma to beta transition, and shown that the beta rhythm can synchronize precisely in the presence of much larger conduction delays than can gamma. This helps to explain why synchronization across parts of the brain separated by long conduction delays appears to happen with rhythms in the beta frequency range.
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R.D. Traub, M.A. Whittington,E.H. Buhl, J.G.R. Jefferys and H.J. Faulkner, "On the mechanism of the gamma-beta shift in neuronal oscillations induced in rat hippocampal slices by tetanic stimulation", J. Neurosci. 19 (1999) 1088-1105. N. Kopell , G.B. Ermentrout, M. Whittington and R.D. Traub, "Gamma rhythms and beta rhythms have different synchronization properties." Proc. Nat. Acad. Sci. USA 97 (2000) 1867-1872. | |
Some alpha rhythms do not synchronize over long distances
Using ionic currents known to exist in layer V of the neocortex, where at least some alpha frequency rhythms (9-11 Hz) are believed to originate, Jones et al. constructed a model of a circuit, containing pyramidal cells and interneurons, that displays an alpha rhythm. Using spike timing maps as in the above work on gamma and beta, they showed two such networks interacting with conduction delays, and using the "doublet" mechanism as above, do not synchronize. This demonstrates that the ability of the beta rhythm to synchronize over longer conduction delays than the gamma rhythm is not due simply to the longer period of beta (since alpha has a period that is still longer). Non-zero phase lags have been measured between distant circuits when the latter are displaying alpha (e.g. Roelfsema et al. 1997).
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S.R. Jones, D. Pinto, T. Kaper and N. Kopell, "Alpha-frequency rhythms desynchronize over long cortical distances: a modelling study", J. Comput. Neurosci. 9 (2000) 271-291. P.R. Roelfsema, A.K. Engel, P. Konig and W. Singer, "Visuomotor integration is associated with zero time-lag synchronization cortical areas", Nature 385 (1997) | |
C.E. Wayne and D. Pinto are studying integro-differential equations that describe the behavior of neural activity as it propagates through synaptically coupled sheets of excitatory and inhibitory neurons. Numerically, this system has been employed extensively, and quite successfully, to understand such phenomena as cortical epilepsy, migraine, and spatial perception. However, there has been very little work attempting to understand the underlying mathematics. At present, the work is focused on developing a set of novel analytic techniques to explore how spatially structured activity in this system is initiated and maintained.
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