Workshop 4: Rhythms and Oscillations (March 18-22, 2013)
Organizers: Carmen Canavier, Bard Ermentrout, Pascal Fries, and Todd Troyer
The rodent hippocampus and cortex contain spatially tuned neurons—such as place, grid, and border cells—that are tuned to fire selectively when a rat visits specific locations in space. Such neurons are commonly assumed to implement a ‘population vector code’ which represents the animal’s position within its environment as a distributed pattern of neural firing rates. However, most spatially tuned neurons exhibit rhythmic modulation of their firing rates by 4-10 Hz theta oscillations, and the phases of such oscillations carry information about the rat’s position. Efforts to explain this observation have given rise to a class of models which posit that spatially tuned neurons may derive their position-dependent firing by detecting synchrony among theta oscillators with frequencies that vary with the animal’s movement velocity, in such a way that their phases (and thus their synchrony with one another) depend strictly upon the animal’s position in its environment. Here, I will outline one version of such a model, which posits that the rodent spatial memory system contains two major classes of neurons: rhythm generators (RGs) and synchrony detectors (SDs). It shall be hypothesized that RGs correspond to ‘theta cells’ that burst rhythmically at velocity-dependent frequencies, whereas SDs correspond to spatially tuned neurons that burst selectively at locations where they detect synchrony among a preferred subset of RGs. I will then describe results from recent experiments that have been carried out in our lab to test two key predictions of this hypothesis: 1) that the burst frequencies of theta cells vary as the cosine of the rat’s movement direction, and 2) that synchrony among pairs of simultaneously recorded theta cells vary with the rat’s position on a 2D sinusoidal grating defined over the surface of the environment.
The synaptic interaction of excitatory and inhibitory neurons (E- and I-cells) can generate oscillations, provided that the drive to the I-cells is low enough, so that they don't get ahead of the E-cells but merely respond to them. If the drive to the I-cells is high, and inhibitory synapses are strong, the E-cells are suppressed altogether. We think about the transition from rhythmicity (low drive to the I-cells) to suppression of the E-cells (high drive to the I-cells). In work with Nancy Kopell several years ago, we suggested that this transition, if it is abrupt, could be exploited to allow the network to toggle between rhythmic activity and suppression, and that this could be useful in attentional selection. In contrast with the earlier work, here we assume that synchronization of the I-cells is always enforced by gap junctions. We find that in this case, the transition from rhythmicity to suppression is much more abrupt when the I-cells have a type 2 phase response (excitation early in the phase retards them) than when they have a type 1 phase response (excitation always accelerates them). We demonstrate this with simulations and explain it using a one-dimensional map.
Electrophysiological recordings from neurons in the hippocampal and entorhinal cortices of freely moving rodents provide detailed information regarding the neural representations of spatial location and orientation, and indicate a functional role for neural coding with respect to the theta rhythm of the local field potential. I will describe some of these experiments and the computational mechanisms they imply. These emphasise the roles of environmental boundaries in self-localization, via boundary vector cell firing, and temporal oscillations in the theta band in path integration, via grid cell firing. Both types of information are combined in the firing of place cells. I will describe the implications of these findings for the mechanisms supporting human spatial memory, and provide examples of electrophysiological and functional neuroimaging experiments designed to test these implications.
Cognitive control is the ability to direct behavior in a goal-directed manner. This ability lies at the core of intelligent behaviors, allowing us to focus our limited cognitive capacity on one task while still maintaining the flexibility to quickly switch to another task as the situation (or our goals) change. I will present results on the role of frontal and parietal cortices in two forms of cognitive control: the control of attention and flexible rule use. By leveraging large-scale, multiple-region electrophysiology in non-human primates we are able to observe synchronization of neural activity within and between brain regions while animals perform these complex behaviors. I will describe results suggesting synchrony supports cognitive behaviors and how its dynamic nature may underlie cognitive flexibility.
Brain rhythms reflect periodically synchronized electrical activity across groups of neurons and are thought to be important for neuronal communication across disparate brain regions. Gamma rhythms are a particular type of rhythm that occurs throughout many regions of the brain and have been linked to functions such as attentional selection and memory. Gamma oscillations vary in frequency (from ~25 Hz to ~ 100 Hz) from one brain region to another and also within a given brain region from one moment to the next. The exact frequency of oscillations is important because different areas will communicate most effectively when their oscillatory timing is the same. In the hippocampus, a brain region critically involved in memory operations, two distinct subtypes of gamma oscillations, slow and fast gamma, occur at different times. During slow gamma (~40 Hz), hippocampal subfield CA1 is coupled with neighboring subfield CA3, an area involved in memory retrieval. During fast gamma (~80 Hz), CA1 is coupled with the entorhinal cortex, a region transmitting information about the current environment. In this lecture, I will present new data supporting the hypothesis that slow and fast gamma rhythms serve different functions, namely that slow gamma facilitates memory retrieval and fast gamma promotes memory encoding.
Brain-wide networks operating at a millisecond timescale are thought to underlie our cognitive functions, but have never been observed directly. Neuroimaging studies based on hemodynamic signals visualized the precise topographies of brain-wide functional networks, but at low temporal resolution. Neurons and areas within these networks likely cooperate through rhythmic synchronization in multiple frequency bands. However, limitations of current recording methods have restricted our ability to detect and investigate these putative brain-wide synchronization networks. Only if extended corticocortical synchronization networks are observed directly and in behaving subjects, will we simultaneously reveal their topographies, frequencies, directions of information flow and cognitive functions, and thereby the relations among those properties. I will present data from large-scale, high-density electrocorticography grids, combining millisecond temporal and millimeter spatial resolution with coverage of large parts of one hemisphere. I will show that a given brain area may simultaneously participate in different networks that synchronize in distinct frequencies and mediate influences in counter-streams. A gamma-band (50 90 Hz) network synchronizes visual-occipital areas and parts of parietal cortex, and gamma-mediated inter-areal influences are bottom-up. A beta-band (peaking at 14 18 Hz) network synchronizes parietal and frontal areas and parts of visual cortex, and beta-mediated inter-areal influences are mostly top-down. Both networks subserve the cognitive function of attention: gamma- and beta-mediated inter-areal influences are enhanced when they mediate behaviorally relevant signals. The direct topographical demonstration of rhythmic synchronization-defined networks constitutes a new quality of brain network investigation and opens an important window onto their function.
A core goal of neuroscience is to determine how sensory inputs map to neural circuits and form functional cortical architectures. Recently, our understanding of the neural representation of space by medial entorhinal cortical neurons has evolved to the point of providing a unique opportunity to determine the mechanisms and function of circuit organization in a region highly associated with spatial navigation and memory. My research has specifically concentrated on the mechanisms underlying the representation of space by medial entorhinal cortex neurons called ‘grid cells’. A strong characteristic of grid cells is their spatial scale, which is organized topographically, increasing progressively from dorsal to ventral medial entorhinal cortex. I have focused on unraveling the potential substrates underlying this topographical expansion of grid scale. In addition, our recent work has highlighted the presence of a topographic gradient in another functionally-defined medial entorhinal cell type; the head direction cell.
It is well known that the brain produces rhythms at many different frequencies and combination of frequencies, and that these rhythms are associated with different cognitive states and tasks. It is still mysterious, however, how the rhythms take part in function. This talk discusses two working hypotheses about this issue.
- Clues to the function of rhythms come from the physiology underlying the rhythms (Function Follows Physiology). That is, it is not just the frequency that matters, but also the mechanistic bases of rhythms that matter in their functional applications. In particular, cortical layers are critical.
- The ways in which rhythms support function are complex; they are not apt to be obvious from data, and require modeling to form hypotheses.
These themes are illustrated with examples involving bottom-up and top-down processing, including cortical and thalamic rhythms.
It is clear that we need mathematical models and analyses of them to understand brain dynamics. However, in building our mathematical models of neuronal networks, it is not clear what data, how much data, and what level(s) of detail are most appropriate to use. Furthermore, it is clear that cellular characteristics need to be considered in our models as experiments continue to show cellular specificity in affecting network output. However, network size and connectivity also have to be taken into consideration. In this talk, I will present our work examining the generation of gamma rhythms in models of fast-spiking inhibitory cell networks. These models are based on CA1 hippocampus considering an ING-type mechanism and using the experimental context of a whole hippocampus preparation exhibiting spontaneous population activities. Consideration of these aspects together leads to the requirement of weak inhibitory connections between fast-spiking cells and strong excitatory drives to them for coherent gamma rhythms to emerge. Most interestingly, a sharp transition in gamma coherence is found for small changes in excitatory drive, thus suggesting a potential design property underlying theta/gamma rhythms. It is important to note that without the experimental context and constraints, this does not arise. Our work thus illustrates how a mathematical mechanism coupled with a well-defined experimental context could lead to biological insights.
Neuronal communication between cortical areas heavily relies on oscillatory, periodic mechanisms whose precise timing critically determines the flow of information. Yet little is known about the perceptual and psychological consequences of such periodic neuronal dynamics at the rapid time scale of the oscillatory cycle: what perceptual changes accompany the drastic changes of neuronal activity observed between opposite phases of the cycle? I will show several experimental examples of these perceptual consequences in the visual domain. To summarize, visual perception and attention seem to wax and wane intermittently at frequencies in the theta (~7Hz) and alpha (~10Hz) range, possibly reflecting the underlying periodic neuronal processes. Based on spiking neural network simulations, I will argue that similar perceptual cycles can also exist at higher frequencies (gamma range), and that our perceptual experience may be the result of cross-frequency interactions between these different rhythms.
Posters
Circadian rhythms – oscillations with a robust ~24 h period – coordinate many cellular processes in neurons, including firing rate, signaling and long-term potentiation. Their relevance for cognitive processes is less clear, however, growing evidence suggests an interplay between circadian oscillations in the expression of clock genes and molecular networks related to learning (cAMP/MAPK/CREB pathway).
Here, we explore the functional implications of circadian oscillations in the context of hippocampal-dependent memory formation. Based on recent hypotheses, neurons are recruited to the memory trace according to the phase of their circadian cycle and, subsequently, the responsive neurons suffer a phase shift in this cycle, eventually leading to long-term memory formation. Interestingly, neurons in the hippocampus show highly asynchronous circadian oscillations.
We propose a minimal model based on a population of uncoupled phase-oscillators to investigate the functional implications of the aforementioned selection mechanism – a phase-resetting stimulus acting on a population of desynchronized circadian oscillators with a phase-dependent response. This model generates a uniform distribution of phases due to the diversity in oscillator periods, representing circadian asynchrony in the absence of stimulus. At the onset of the phase-resetting stimulus, only oscillators within a phase range respond and synchronize, gradually losing phase coherence over time. We show that even such a simple system provides sparsity in memory encoding, enhances the memory of periodic events, allows multiple memories to coexist by phase separation (multiplexing), and creates a temporal tagging of the events.
In conclusion, we argue that the dynamics of the phase-coherence of oscillations in the microhertz frequency range – provocatively named "tau-waves" – could perform previously unrecognized functions in memory processes.
Synchronization of ongoing neural oscillations is argued to be important for gating information flow, perceptual binding, attentional modulation, and other cognitive processes. To study the organization of synchronization in space, frequency, and (in event-related paradigms) time, neurophysiological recordings are made using a variety of tasks across different subject populations. However, identifying synchronization differences is difficult in high-dimensional datasets, a problem that becomes especially acute when looking at cross-frequency and inter-regional coupling. Here, traditional methods of statistical testing are inadequate, while more advanced techniques (such as cluster-based statistics) have not been applied in such high-dimensional contexts and may identify hard-to-interpret differences.
The 'tensor factorization' technique can be employed in order to simplify and analyze such data. Tensor factorization is a generalization of matrix factorization methods like singular value decomposition (SVD) to higher-dimensional matrices, or tensors. It produces a set of factors, with each factor having a set of loadings on dimensions such as space, frequency, time, and condition. Factor loadings can be visualized and interpreted in the context of known or predicted patterns of oscillatory activity. Furthermore, the goodness-of-fit of a particular factorization can be statistically tested by non-parametric methods.
To demonstrate the potential of this technique in studying differences in oscillatory synchronization, we generate synthetic 32-electrode EEG data displaying theta and gamma oscillations and varying levels of theta-gamma cross-frequency coupling, and we then show that tensor factorization uncovers differences in the organization of this oscillatory activity. In addition, we apply our method to a preliminary analysis of EEG data from a perceptual decision-making task.
Dopamine modulates cortical circuit activity, in part, through its actions on GABAergic interneurons, including increasing the excitability of fast spiking interneurons by suppressing a voltage-independent leak K+ current via D1/D5 receptor mechanism (Gorelova et al., 2002). Though such effects have been demonstrated in single cells, there are no studies to our knowledge that examine how such mechanisms may lead to the effects of DA at a neural network level. With this motivation, we investigated the effects of DA on synchronization in a simulated neural network, composed of 50 excitatory (XX) and 20 fast-spiking inhibitory (Wang-Buzsaki) cells coupled with all-to-all connectivity. The effects of DA were implemented through varying potassium leak conductance of in the fast spiking interneurons. The network synchronization was analyzed through examining gamma band (~40 Hz) power in the local field potential. Parametrically varying the potassium leak conductance revealed an inverted-U shaped relationship, that is, with low gamma band power at both low and high conductance levels, with optimal synchronization occurring at intermediate conductance levels. Our results show that the physiologic effects of DA on single fast-spiking interneurons can give rise to a non-monotonic relationship between cortical gamma band synchrony and DA levels. These findings suggest that the effects of DA at the single neuron level may give rise to more complex behavior at the network level, consistent with literature describing inverted U-shaped cortical activity as a function of DA activity.
Gamma oscillation of the local field potential plays many functional roles for cognition and/or attention. To analyze a mathematical relationship between individual inhibitory neurons and macroscopic oscillations in mathematical and physiological manners, we derived a model, namely “modified theta model” and analyzed their collective dynamics by phase reduction and bifurcation analyses.
Differential equations play a major role in modeling natural phenomena such as heat conduction, population growth, and the spread of disease. However, it is often difficult, if not impossible, to find solutions to complex models. Despite this fact, it is sometimes possible to prove the existence of solutions to some differential equations and to classify them.
The study of C^(n)-almost periodic (and more generally, C^(n)-almost automorphic) solutions to differential equations is one of the most interesting topics in qualitative theory of differential equations due to their applications. Almost automorphy generalizes almost periodicity, which generalizes continuous periodic functions. Extensions of these areas include the study of existence of such solutions to some of differential equations in abstract spaces such as Banach and Hilbert spaces.
In this presentation, we introduce and examine a new concept called C^(n)-pseudo-almost automorphy, which generalizes both the notions of C^(n)-almost periodicity and that of C^(n)-almost automorphy. Properties of such a new concept will be presented. We then prove the existence of C^(n)-pseudo-almost automorphic solutions to some n-order differential equations. Current work includes extending these results to the unbounded case. These results represent joint work with Toka Diagana, Ph.D.
Electroencephalogram (EEG) records non-invasive electrophysiological brain signals. These signals display periodic rhythmicity, observed across multiple subject populations while resting or performing a variety of perceptual and cognitive tasks. In this project we will explore oscillatory brain dynamics associated with awareness of the visual world and the percepts prompt to a decision using measures derived from time-frequency analysis such as event-related spectral perturbation, phase synchrony, and cross-frequency coupling.
In order to achieve this, we will perform a series of studies using different tasks, paradigms, and subject populations. First, we will use resting-state EEG data from individuals diagnosed with schizophrenia and healthy controls, comparing a visual input condition (i.e. eyes open resting-state) against a condition with no visual input (i.e. eyes closed resting state) during cognitive task (i.e. serial subtraction) and no cognitive task (i.e. resting).
Second, we will use the perception of apparent motion as a paradigm for brain activation using impoverished stimuli and pictures of human faces. It has been suggested that simple motion displays, are capable of inducing effective perception of animacy through the capture of high-order information embedded in the moving patterns, despite low-level commonalities of the stimuli used.
Finally, a parametric perceptual discrimination task will be used to specifically measure the influence of task demands and perceptual processes in brain dynamics correlating with perceptual decision-making.
Thus, non-invasively recorded oscillatory brain dynamics as described by time-frequency analysis may provide an appropriate measurement of the building blocks for the emergence of visual awareness, and that directly contribute to cognitive processes such as perceptual decision-making. The current project targets the measurement and characterization of signatures of the interplay between bottom-up and top-down influences across multiple paradigms, tasks, and populations; ensuring that the results are not an artifact dependent of a particular task but indeed represent robust markers of the underlying substrates of visual perception.
This work is based on recent experimental results using optogenetic tools to stimulate both pyramidal cells (PYR) and parvalbumin-immunoreative interneurons (INT) in the hippocampus of freely-behaving rodents. 'In vitro', PYR exhibit theta range subthreshold (membrane potential) resonance. Whether this translates to spiking resonance in behaving animals is unknown. 'In vivo', Individual directly stimulated PYR cells exhibited narrow-band spiking centered on a wide range of frequencies rather than spiking predominantly at theta. In contrast, 'in vivo' INT photostimulation indirectly induced theta band-limited spiking in pyramidal cells accompanied by post-inhibitory rebound spiking. We present a minimal, biophysical (conductance-based) model of a CA1 hippocampal network that qualitatively reproduces the experimental results. This model includes three cell types: PYR, INT and OLM (oriens-lacunosum moleculare) cells. The presence of subthreshold resonance in isolated PYR cells is not enough to generate robust theta-band spiking resonance in PYR cells embeded in this network. Theta-band spiking resonance was especially robust when the model included a timing mechanism, implemented by either a network-mediated time inhibition provided by the OLM cell or synaptic depression of the INT synapses.
Repeated injections of the psychostimulant D-Amphetamine (AMPH) constitute a standard way to model addiction. We record LFP from prefrontal cortex and hippocampus of awake, behaving rats during this behavioral sensitization protocol. We study how neural activity become transiently phase locked and unlocked in the theta frequency band. Short but frequent desynchronized events dominate synchronized dynamics in each examined animal. After the first drug injection, only increases in the relative prevalence of short desynchronization episodes (but not in average synchrony strength) were significant. Throughout sensitization both strength and the fine temporal structure of synchrony (measured as relative prevalence of short desynchronizations) were similarly altered with AMPH injections, with each measure decreasing in the pre-injection epoch and increasing after injection. Decoupling between locomotor activity and synchrony was observed in AMPH, but not saline, animals. The increase in numerous short desynchronizations (as opposed to infrequent, but long desynchronizations) in AMPH treated animals may indicate that synchrony is easy to form yet easy to break. These data yield insight into how synchrony is dynamically altered in cortical networks by AMPH and identify neurophysiological changes that may be important to understand the behavioral pathologies of addiction.
We examine a two population neural-field model with temporal periodic forcing and a piecewise linear firing rate. We perform computer simulations and observe subharmonic, quarter-harmonic and harmonic bifurcations with non-zero wave-number. We then use the piecewise linear nature of the firing rate to construct analytically the underlying spatially homogeneous periodic orbit. Performing linear stability analysis allows us to predict the position of the bifurcation. Finally we formulate the amplitude equations for this model.
Synchrony within and between brain regions has been implicated in many aspects of cognition, movement, and disease. In order to explore general principles of synchronization between synaptically coupled neurons, we constructed hybrid circuits of one model and one biological neuron using the dynamic clamp. The synchronization tendencies of each isolated neuron were characterized by measuring how much a single input from the other neuron transiently shortened or lengthened the cycle period of the oscillation. We used this information to construct a map to predict the cycle by cycle dynamics of the hybrid circuits. Most experiments exhibited transitions between phase locking and phase walkthrough. We hypothesize that these transitions result from bifurcations in which fixed points of the map emerged or disappeared. We tested the ability of three noise models to account for these transitions. A model in which the intrinsic period of the biological neuron evolved as a slow random process best explained the data. Our main result is that the both the bifurcation structure and the form of the dynamical noise are critical for understanding episodes of phase-locked activity in neural systems.
Augmented breaths, or sighs, increase air intake and a lack of sighing is associated with sudden infant death syndrome. In rodents, both sighs and regular (eupnic) breaths are generated within the area of the brain called the pre-Bötzinger complex (preBötC). This region contains approximately 500 neurons, which continue to generate eupnic-like and sigh-like patterns even when they are isolated in a brain slice preparation. The mechanism of sigh generation is currently unknown but it depends on connections between the neurons in the preBötC. Although sighs and eupnic breaths are generated within the same brain area, it is still debated if they are produced by the same network of neurons or by distinct sub-networks. We developed two mathematical models to determine the mechanism for sigh generation. One model proposes that sighs are generated by two distinct neural sub-networks, while the second model suggests that sighs are generated within the same network. By analyzing predictions of models, we concluded that model of sigh generation by separate sub-network is more consistent with in-vitro experimental data.
We identify and describe the principal bifurcations of bursting rhythms in multi-functional central pattern generators (CPG) composed of three neurons connected by fast inhibitory or excitatory synapses.
We develop a set of computational tools that reduce high-order dynamics in biologically relevant CPG models to low-dimensional return mappings that measure the phase lags between cells. We examine bifurcations of fixed points and invariant curves in such mappings as coupling properties of the synapses are varied. These bifurcations correspond to changes in the availability of the network's phase locked rhythmic activities such as periodic and aperiodic bursting patterns. As such, our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of, and switching between, motor patterns generated by various animals.
