Binding by synchrony
|Wolf Singer (2007), Scholarpedia, 2(12):1657.||doi:10.4249/scholarpedia.1657||revision #124403 [link to/cite this article]|
The brain is a highly distributed system in which numerous operations are executed in parallel and that lacks a single coordinating center. This raises the questions of i) how the computations occurring simultaneously in spatially segregated processing areas are coordinated and bound together to give rise to coherent percepts and actions, ii) how signals are selected and routed from sensory to executive structures without being confounded, and finally iii) how information about the relatedness of contents is encoded. One of the coordinating mechanisms appears to be the synchronization of neuronal activity by phase locking of self-generated network oscillations.
In 1986 experiments have been performed on awake behaving kittens that had multiple electrodes implanted in the visual cortex. The goal of these experiments was to follow the time course of experience dependent changes in the receptive field properties of cortical neurons following short periods of monocular deprivation and to study the stability of orientation maps during reverse monocular occlusion (Mioche and Singer, 1989). During these experiments it was noted that groups of simultaneously recorded, spatially segregated neurons engaged in synchronous oscillatory activity when activated by visual stimuli consisting of drifting square wave gratings. The frequency of these oscillations was in the range of 40 Hz and thus differed from the periodic activation induced by the grating and also from the line noise artifact, suggesting that the oscillations and their synchronization were due to internal neuronal interactions. A few months later, Dr. Charles Gray arrived from the States in order to work with us as a postdoc on developmental questions. However, because the oscillatory patterning of neuronal activity and, in particular, the synchronization of the oscillatory responses appeared as a first indication for the possibility that the cerebral cortex might exploit the time domain for coding, it was decided to invest in the further investigation of these phenomena. This led to a series of early studies which confirmed the robustness of these oscillations and to the demonstration that the synchronization of the discharges of spatially segregated neurons depended critically on the configuration of the applied light stimuli. It soon became clear that neurons preferentially synchronized their responses if activated by contours that were either continuous or moved coherently with the same speed in the same direction. This led to the hypothesis that the cerebral cortex might exploit the option to synchronize the discharges of neurons with millisecond precision in order to bind these responses for further joint processing, i.e. to exploit temporal synchrony to encode relations. Purely theoretical formulations of the binding-by-synchrony hypothesis were proposed earlier by Milner (1974), Grossberg (1976), and von der Malsburg (1981), but the Singer lab was the first to obtain experimental evidence supporting the potential role of synchrony as a relational code.
The reason for this conjecture was that synchronization probability appeared to depend on the Gestalt criteria that the visual system applies in order to accomplish scene segmentation and perceptual grouping (Gray and Singer, 1989; Gray et al., 1989). Once we were confident about the robustness of these early findings, we presented them in 1987 at the in-house symposium of the Max Planck Institute for Brain Research to which colleagues from the region were invited, among them Reinhard Eckhorn from Marburg University. The Marburg lab immediately repeated our experiments, confirmed the main results and rapidly published them in Biological Cybernetices without informing us about these actions. This answers the frequently posed question why the discovery of synchronous gamma oscillations had occurred virtually simultaneously in two labs segregated by less than a two hours drive. In the following years, studies in the motion sensitive area MT of the visual cortex of awake monkeys (Kreiter and Singer, 1996), the optic tectum of pigeons (Neuenschwander et al., 1996), other cortical areas in the cat (Engel et al., 1991(a)) and the retina (Neuenschwander et al., 1996) provided evidence that the oscillatory patterning of neuronal responses and the synchronization of these oscillations is an ubiquitous and stimulus dependent phenomenon that is highly sensitive to context. Combining multi-site recordings with the sectioning of the corpus callosum (Engel et al., 1991(b)) and with developmental studies (König et al., 1993; Löwel and Singer, 1992) indicated that the long distance synchronization of these oscillatory responses was mediated by the network of reciprocal cortico-cortical connections. Multi-site recordings also provided evidence that synchronization could occur between widely distributed structures, such as the primary visual cortex, the optic tectum and the suprasylvian cortex in cats (Brecht et al., 1998).
At that time (in the early Nineties) heated controversies arose as to whether these oscillatory patterns did exist and whether the precise synchronization of discharges of spatially segregated neurons was functionally relevant. A summary of these discussions can be found in a special issue of Neuron (e. g. Gray, 1999; Singer, 1999).
In order to test whether synchrony might also play a role in the coordination of widely distributed functions as is required e.g. in sensory-motor processing, multi-site recordings were performed in awake behaviorally trained cats. These indicated that synchronized oscillatory activity is not only stimulus driven but does occur also across widely distributed networks of interconnected cortical areas in anticipation of an attention demanding discrimination task. This observation led to the hypothesis that self-generated oscillatory activity in the beta and gamma frequency range could be a correlate of focused attention and serve both modality specific selection of stimuli and the coordination of sensory and executive subsystems required for the execution of the anticipated task (Roelfsema et al., 1997). This then led to a series of experiments in which a relation between arousal, activated cortical states and the occurrence of gamma oscillations was investigated. These experiments showed that gamma oscillations are associated with activated cortical states and require for their expression activation of the reticular arousal system and activation of muscarinic receptors in the cortex (Herculano-Houzel et al., 1999; Munk et al., 1996). Motivated by the idea that synchronization could be used to raise the impact of synchronized responses and to use this mechanism for response selection in the context of attentional processes and perhaps even to gate access to conscious processing, experiments have been performed on binocular rivalry in order to see whether there was a relation between stimulus selection and response synchronization. These results provided clear evidence that stimuli which are consciously perceived are associated with strong synchronization of oscillatory responses in primary visual cortex of awake cats while stimuli that are suppressed and excluded from being perceived led to responses that are much less well synchronized (Fries et al., 1997; 2002). Because the modulation of discharge rates of neurons in primary visual cortex did not reflect shifts in eye dominance, it was concluded that synchronization rather than discharge rate was used as a mechanism for response selection. The notion that the saliency of responses can be enhanced in a complementary way either by increases of discharge rate or by synchronization has later received direct support from experiments on apparent brightness perception (Biederlack et al., 2006). In-vitro studies (Volgushev et al., 1998) and multi-site recordings in the visual cortex of cats (Fries et al., 2001) then provided evidence, that the oscillatory patterning of neuronal activity is an efficient mechanism to adjust the precise timing of spikes and is potentially a versatile mechanism to convert rate coded input to cells into a temporal code defined by the time of occurrence of spikes relative to the oscillation cycle (see also Fries et al., 2007).
The expansion of the research field
Motivated by the discovery of the synchronization phenomena more and more labs joined into the search for relations between cognitive and executive functions and the synchronization of oscillatory activity. While labs focusing on the analysis of single cell activity continued to obtain controversial results, partly because it is difficult to detect transient, non-stationary oscillatory patterns in the responses of single units due to undersampling, laboratories applying EEG- and MEG-recording methods provided rapidly growing evidence for a close relation between synchronous oscillatory activity in the beta- and gamma frequency range and a variety of cognitive functions such as perceptual grouping, focused attention, maintenance of contents in short term memory, poly-sensory integration, formation of associative memories and sensory motor coordination (for review see Singer, 2004).
Already in the first studies on synchronous gamma oscillations in the visual cortex it had been noted that this phenomenon is best seen in local field potentials which reflect the synchronous activity of local groups of neurons. This led to a revival of studies exploiting the potential of field potential recordings for the detection of synchronized activity and these studies, performed by an increasing number of laboratories, provided independent evidence from a variety of species (monkeys, cats, ferrets, rats, mice and birds) that cognitive and executive functions are often associated with the oscillatory patterning and the synchronization of the responses of neuronal groups and can occur over widely distributed networks. Often, the occurrence of these patterns is associated with expectancy and response preparation, with focused attention and with the anticipated need to coordinate functions distributed across several cortical areas or to retrieve information stored in working or long term memory. Most recently, evidence has become available that large scale synchronizations of cortical activity in the beta and gamma frequency range and, in particular, the precise phase synchronization of this activity, is a prerequisite for sensory information to have access to conscious perception (Melloni et al., 2007). And there is also growing evidence that disturbed cognitive functions, such as occur in patients suffering from schizophrenia, go along with abnormalities in the ability to synchronize high frequency oscillations across spatially distributed cortical areas (Uhlhaas et al., 2006). Abnormalities in the ability of neuronal networks to synchronize and to engage in high frequency oscillations have also been described in a variety of other diseases such as Alzheimer Disease, Autism, Parkinson and, obviously, Epilepsy (for review see Uhlhaas and Singer, 2006, Uhlhaas and Singer, 2010 and Uhlhaas et al., 2010).
Despite of this meanwhile abundant evidence for close correlations between synchronized oscillatory activity and cognitive as well as executive functions we are still far from understanding the full implications of the dynamics expressed in oscillatory activity in various frequency bands and the related synchronization phenomena. Also, most of the evidence supporting the hypothesis that oscillations and synchrony are the backbone of temporal coding strategies is still correlative in nature. It is only very recently that manipulations of gamma oscillations with optogenetic methods provided direct evidence for a causal role of these oscillations in stimulus encoding and cognition (Cardin et al., 2009).
The discovery of synchronized oscillations has motivated a large number of in-vitro studies searching for the mechanisms that would generate these oscillations and this led to a re-evaluation of the functional role of inhibitory inter-neurons. Classically, the network of inhibitory inter-neurons has been considered as a mechanism for gain control and improvement of signal to noise ratios. In-vitro investigations of oscillating networks led to the conclusion that it is the network of inhibitory interneurons that is responsible for the rhythmic pacing of neuronal activity. Thus, inhibitory interneurons appear to play a crucial role not only in controlling response amplitudes but also in adjusting the precise timing of discharges of excitatory neurons. Through the latter effect they assume a pivotal function in the temporal structuring and coordination of neuronal responses (for review see Whittington et al., 2001, Roopun et al., 2008).
The discovery of synchronous oscillatory activity in the cerebral cortex has also motivated a very large number of theoretical studies investigating the functional properties of networks capable of engaging in oscillations and stimulus dependent synchronization patterns. These studies provided deep insights into both the mechanisms that sustain oscillations and their synchronization as well as the putative functions of the temporal coding strategies that can be implemented in such networks with essentially non-linear dynamics (for review see e.g. Traub et al., 2001, Buzsaki and Draguhn, 2004 and Buzsaki, 2006).
From the now-abundant literature on oscillation and synchrony, it appears that the oscillatory patterning of neuronal activity and the associated synchronization of discharges serve important functions for the computations performed by neuronal networks. The main effect of the oscillatory modulation of the membrane potential is that it constrains the time interval during which cells are susceptible to excitatory input and can emit action potentials themselves. The oscillation cycle can be subdivided into two phases: the depolarizing and the hyperpolarizing phase. During the first phase EPSPs can effectively contribute to the excitation of the cells and trigger spikes. The highest probability for generating spikes is at the peak of the depolarizing cycle but if excitatory drive is strong, spikes can also occur during the rising phase. In this case, the timing of spikes relative to the phase of the oscillation cycle depends on the strength of the excitatory drive. The stronger the drive, the earlier the discharges. In this way, the amplitude of excitatory drive can be converted into spike timing whereby the phase precession of discharges relative to the depolarizing peak of the oscillations is a direct measure of input intensity. This relation makes it possible to convert rate coded amplitude values into a temporal code of spike timing. The second phase of the oscillation cycle is hyperpolarizing and dominated by a massive barrage of IPSPs. During this phase cells are very unlikely to respond to excitatory drive because both the shunting and the hyperpolarizing effect of IPSPs prevent effective summation of EPSPs.
These effects of an oscillatory modulation of cell excitability can be exploited in many different ways in order to encode information and to define relations between the activity of spatially distributed neuron groups. When neuronal groups become entrained in synchronous oscillations, they will tend to emit spikes in synchrony and this enhances the impact that these output signals will have on target cells. Synchronization can thus be used to select signals for further joint processing and to accelerate the propagation of the signals across distributed networks.
Entrainment into coherent oscillations can also be used to route activity selectively to particular target structures. If discharges from sending groups of neurons are delivered in packages that are synchronized to the oscillation period of the sending group of neurons and if the target groups are also oscillating, only those target groups in the network will be able to effectively integrate the arriving volleys and to convert them into output signals that are phase synchronized to the sending group and receive the afferent volleys during the depolarizing phase of the oscillation cycle. Cell groups oscillating in anti-phase or at different frequencies will be unable to convert the broadcast information into output signals (Womelsdorf et al., 2007). This option for selective routing is not an all-or-none process. By adjusting oscillation frequencies, phase angles and exploiting variable conduction times of transmission pathways, coupling can be gated over a wide dynamic range and if oscillation frequencies of sender and receiver exhibit m:n-relations, the routing of signals can be multiplexed in distinct time frames to different targets. Thus, entrainment of cell groups into oscillations can be used to prepare hand shaking between distributed cortical processing areas in those computations that require selective cooperation among different cortical processors. Because of the distributed organization of cortical functions, this is required both for low level intramodal processes such as feature binding and figure-ground segregation as well as for attention dependent selection and binding of contents across different sensory modalities (subsystem integration).
Part of these binding operations can of course be accomplished by generating conjunction specific neurons through fixed anatomical connections and this option is exploited by the cerebral cortex. However, this binding strategy must be complemented by a more versatile mechanism, especially when higher functions are considered such as polymodal integration, sensory motor-coordination and temporary binding of ever changing constellations of input signals in working memory. In these cases, the computational results obtained in parallel in a large number of cortical processing areas need to be bound selectively and transiently, processed jointly in rapidly changing combinations and these temporally bound results need in turn be routed selectively to the networks dealing with executive functions. Such versatility can be obtained by exploiting the temporal dimension as coding space. Here, oscillatory networks appear as particularly effective. They offer the option to use phase relations of spike timing for the selection, gating and routing of signals. By adjusting oscillation frequencies, phase relations of oscillations and exploiting a variable spectrum of conduction velocities, binding and routing of signals can be achieved in an extremely flexible way.
Encoding information in the relative timing of spikes has the further advantage that relations between distributed activity patterns can be defined with very high temporal precision. Cortical neurons are highly sensitive coincidence detectors because of a number of specific features such as frequency adaptation of synaptic release, adaptation of postsynaptic receptors, active dendritic conductances, the shortness of synaptic potentials in vivo and the inverse relation between membrane potential slope and spike threshold (Azouz and Gray, 2003). This already high sensitivity to coincident input is further enhanced if the cells are entrained in oscillations because the oscillation cycle further limits the responsiveness of cells to narrow temporal windows and reduces the probability that cells respond to temporally dispersed inputs. The sensitivity of oscillating networks to timing relations depends critically on the oscillation frequency. The width of the window over which afferent activity can be integrated becomes narrower with increasing oscillation frequency. Thus, the higher the oscillation frequency, the higher the precision with which the timing of spikes can be adjusted and the higher the sensitivity to precise timing. Cells oscillating at low frequencies are able to integrate over longer time windows and this does in principle allow for the nesting of relations: Slowly oscillating cell groups can integrate activity from fast oscillating cell groups even if these are oscillating at different frequencies. Thus, while the features represented by fast oscillating groups remain segregated, they may be bound together by more slowly oscillating groups. In principle, this allows for the encoding of hierarchically structured relational graphs and the encoding of nested relations.
Encoding information in the timing of individual spikes rather than in their rates can in principle also allow for a drastic acceleration of processing speed. If, e.g the intensity of a stimulus is converted into a latency value or a phase angle between a spike and the peak of a population oscillation, the brightness distribution of a visual scene could be read out from the latency distribution of response onsets across the matrix of stimulated cells within a few tens of milliseconds, i.e. within the range of the latency scatter of responses or within one gamma cycle (for a more detailed discussion of this aspect see Van Rullen and Thorpe, 2007, Fries et al., 2007).
Last but not least, precise adjustment of the timing of action potentials relative to oscillation cycles is likely to also serve as a relation defining code in learning processes. Use dependent synaptic modifications are exquisitely sensitive to the precise timing relations between pre- and post-synaptic activation (spike timing dependent plasticity, STDP). This predicts that in oscillating networks phase relations between the timing of spikes relative to the oscillation cycle will determine the polarity of use dependent synaptic gain changes. Evidence that this is indeed the case has been obtained in a recent in vitro study on long term potentiation and long term depression in oscillating cells (Wespatat et al., 2004). Thus, it appears as if both in signal processing and learning, temporal relations among individual spiking events are used as tag of relatedness, whereby these relations are defined with a temporal precision in the millisecond range. That the same signature is used for the definition of relations both in signal processing and learning is of course not surprising because otherwise, the risk would be high that synaptic modifications establish false conjunctions. It is thus possible to reverse the line of argumentation: If in learning processes relations are defined by the precise temporal coincidence between pre- and post-synaptic spiking activity, then relations need to be defined in exactly the same way in signal processing.
Thus, two independent lines of evidence indicate that precise timing of spiking activity matters for information processing in the nervous system and available evidence strongly suggests that it is the oscillatory patterning of neuronal activity that serves the adjustment of precise spike timing and sets the time frame for integrating distributed activity.
Azouz, R. and Gray, C. M. Adaptive coincidence detection and dynamic gain control in visual cortical neurons in vivo. Neuron. 2003; 37:513-523.
Biederlack, J., Castelo-Branco, M., Neuenschwander, S., Wheeler, D. W., Singer, W., and Nikolic, D. Brightness induction: Rate enhancement and neuronal synchronization as complementary codes. Neuron. 2006; 52: 1073-1083.
Brecht, M., Singer, W., and Engel, A. K. Correlation analysis of corticotectal interactions in the cat visual system. J. Neurophysiol. 1998; 79: 2394-2407.
Buzsaki, G. Rhythms of the Brain. Oxford University Press. 2006.
Cardin, J. A., Carlén, M., Meletis, K., Knoblich, U. Zhang, F., Deisseroth, K., Tsai, L.-H., and Moore, C. I. Driving fast-spiking cells induces gamma rhythm and controls sensory responses. Nature. 2009; 459: 663-667.
Engel, A. K., Kreiter, A. K., König, P., and Singer, W. Synchronization of oscillatory neuronal responses between striate and extrastriate visual cortical areas of the cat. Proc. Natl. Acad. Sci. USA. 1991(a); 88: 6048-6052.
Engel, A. K., König, P., Kreiter, A. K., and Singer, W. Interhemispheric synchronization of oscillatory neuronal responses in cat visual cortex. Science. 1991(b); 252: 1177-1179.
Fries, P., Neuenschwander, S., Engel, A. K., Goebel, R., and Singer, W. Rapid feature selective neuronal synchronization through correlated latency shifting. Nature Neurosci. 2001; 4(2): 194-200.
Fries, P., Roelfsema, P. R., Engel, A. K., König, P., and Singer, W. Synchronization of oscillatory responses in visual cortex correlates with perception in interocular rivalry. Proc. Natl. Acad. Sci. USA. 1997; 94: 12699-12704.
Fries, P., Schröder, J.-H., Roelfsma, P. R., Singer, W., and Engel, A. K. Oscillatory neuronal synchronization in primary visual cortex as a correlate of stimulus selection. J. Neurosci. 2002; 22(9): 3739-3754.
Fries, P., Nikolic, D., and Singer, W. The gamma cycle. Trends Neurosci. 2007; 30(7): 309-316.
Gray, C. M. The temporal correlation hypothesis of visual feature integration: Still alive and well. Neuron. 1999; 24: 31-47.
Gray, C. M., König, P., Engel, A. K., and Singer, W. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature. 1989; 338: 334-337.
Gray, C. M. and Singer, W. Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc. Natl. Acad. Sci. USA. 1989; 86: 1698-1702.
Grossberg, S. Adaptive pattern classification and universal recoding, II: Feedback, expectations, olfaction, and illusions. Biol. Cybernetics, 187-202.
Herculano-Houzel, S., Munk, M. H. J., Neuenschwander, S., and Singer, W. Precisely synchronized oscillatory firing patterns require electroencephalographic activation. J. Neurosci. 1999; 19(10): 3992-4010.
König, P., Engel, A. K., Löwel, S., and Singer, W. Squint affects synchronization of oscillatory responses in cat visual cortex. Eur. J. Neurosci. 1993; 5: 501-508.
Kreiter, A. K. and Singer, W. Stimulus-dependent synchronization of neuronal responses in the visual cortex of the awake macaque monkey. J. Neurosci. 1996; 16(7): 2381-2396.
Löwel, S. and Singer, W. Selection of intrinsic horizontal connections in the visual cortex by correlated neuronal activity. Science. 1992; 255: 209-212.
Melloni, L., Molina, C., Pena, M., Torres, D., Singer, W., and Rodriguez, E. Synchronization of neural activity across cortical areas correlates with conscious perception. J. Neurosci. 2007; in press.
Milner, P. M. A model for visual shape recognition. Psychological Review. 1974; 81(6), 521-535.
Mioche, L. and Singer, W. Chronic recordings from single sites of kitten striate cortex during experience-dependent modifications of receptive field properties. J. Neurophysiol. 1989; 62: 185-197.
Munk, M. H. J., Roelfsema, P. R., König, P., Engel, A. K., and Singer, W. Role of reticular activation in the modulation of intracortical synchronization. Science. 1996; 272: 271-274.
Neuenschwander, S., Engel, A. K., König, P., Singer, W., and Varela, F. J. Synchronization of neuronal responses in the optic tectum of awake pigeons. Vis. Neurosci. 1996; 13: 575-584.
Roelfsema, P. R., Engel, A. K., König, P., and Singer, W. Visuomotor integration is associated with zero time-lag synchronization among cortical areas. Nature. 1997; 385: 157-161.
Roopun, A. K., Kramer, M. A., Carracedo, L. M., Kaiser, M., Davies, C. H., Traub, R. D., Kopell, N., and Whittington, M. A. Temporal interactions between cortical rhythms. Front. Neurosci. 2008; 2, 2: 145-154.
Singer, W. Neuronal synchrony: A versatile code for the definition of relations? Neuron. 1999; 24: 49-65.
Singer, W. Synchrony, oscillations, and relational codes. Chalupa, L. M. and Werner, J. S. The Visual Neurosciences . Cambridge, Massachusetts: The MIT Press, A Bradford Book; 2004; pp. 1665-1681.
Traub, R. D., Kopell, N., Bibbig, A., Buhl, E. H., LeBeau, F. E. N., and Whittington, M. A. Gap junctions between interneuron dendrites can enhance synchrony of gamma oscillations in distributed networks. J. Neurosci. 2001; 21(23): 9478-9486.
Uhlhaas, P. J., Linden, D. E. J., Singer, W., Haenschel, C., Lindner, M., Maurer, K., and Rodriguez, E. Dysfunctional long-range coordination of neural activity during Gestalt perception in schizophrenia. J. Neurosci. 2006; 26(31): 8168-8175.
Uhlhaas, P. J. and Singer, W. Neural synchrony in brain disorders: Relevance for cognitive dysfunctions and pathophysiology. Neuron. 2006; 52: 155-168.
Uhhaas, P. J. and Singer, W. Abnormal neural oscillations and synchrony in schizophrenia. Nat. Rev. Neurosci. 2010; 11: 100-113.
Uhlhaas, P. J., Roux, F., Rotarska-Jagiela, A. Rodriguez, E., and Singer, W. Neural synchrony and the development of cortical networks. Trends Cogn. Sci. 2010; 14: 72-80.
Van Rullen, R. and Thorpe, S. J. Rate coding versus temporal order coding: What the retinal ganglion cells tell the visual cortex. Neural Computation. 2001; 13:1255-1283.
Volgushev, M., Chistiakova, M., and Singer, W. Modification of discharge patterns of neocortical neurons by induced oscillations of the membrane potential. Neurosci. 1998; 83(1): 15-25.
von der Malsburg, C. The correlation theory of brain function. 1981; Internal Report 81-2, Dept. of Neurobiology, Max-Planck-Institute for Biophysical Chemistry, Göttingen, Germany. Reprinted in: Domany E, van Hemmen JL, Schulten K (eds) Models of neural networks II (1994). Springer, Berlin.
Wespatat, V., Tennigkeit, F., and Singer, W. Phase sensitivity of synaptic modifications in oscillating cells of rat visual cortex. J. Neurosci. 2004; 24(41): 9067-9075.
Whittington, M. A., Doheny, H. C., Traub, R. D., LeBeau, F. E. N., and Buhl, E. H. Differential expression of synaptic and nonsynaptic mechanisms underlying stimulus-induced gamma oscillations in vitro. J. Neurosci. 2001; 21(5): 1727-1738.
Womelsdorf, T., Schoeffelen, J.-M., Oostenveld, R., Singer W., Desimone, R., Engel, A. K., and Fries, P. Modulation of neuronal interactions through neuronal synchronization. Science. 2007; 316:1609-1612.
- Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Roger D. Traub (2006) Fast oscillations. Scholarpedia, 1(12):1764.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- John Dowling (2007) Retina. Scholarpedia, 2(12):3487.
- Don H. Johnson (2006) Signal-to-noise ratio. Scholarpedia, 1(12):2088.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
- Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.