A model for the origin and properties of flicker-induced geometric phosphenes

Spatiotemporal resonance in mouse primary visual cortex

Human primary visual cortex (V1) responds more strongly, or resonates, when exposed to ∼10, ∼15-20, and ∼40-50 Hz rhythmic flickering light. Full-field flicker also evokes the perception of hallucinatory geometric patterns, which mathematical models explain as standing-wave formations emerging from periodic forcing at resonant frequencies of the simulated neural network. However, empirical evidence for such flicker-induced standing waves in the visual cortex was missing. We recorded cortical responses to flicker in awake mice using high-spatial-resolution widefield imaging in combination with high-temporal-resolution glutamate-sensing fluorescent reporter (iGluSnFR). The temporal frequency tuning curves in the mouse V1 were similar to those observed in humans, showing a banded structure with multiple resonance peaks (8, 15, and 33 Hz). Spatially, all flicker frequencies evoked responses in V1 corresponding to retinotopic stimulus location, but some evoked additional peaks. These flicker-induced cortical patterns displayed standing-wave characteristics and matched linear wave equation solutions in an area restricted to the visual cortex. Taken together, the interaction of periodic traveling waves with cortical area boundaries leads to spatiotemporal activity patterns that may affect perception.

Visual hallucinations induced by Ganzflicker and Ganzfeld differ in frequency, complexity, and content

Visual hallucinations can be phenomenologically divided into those of a simple or complex nature. Both simple and complex hallucinations can occur in pathological and non-pathological states, and can also be induced experimentally by visual stimulation or deprivation-for example using a high-frequency, eyes-open flicker (Ganzflicker) and perceptual deprivation (Ganzfeld). Here we leverage the differences in visual stimulation that these two techniques involve to investigate the role of bottom-up and top-down processes in shifting the complexity of visual hallucinations, and to assess whether these techniques involve a shared underlying hallucinatory mechanism despite their differences. For each technique, we measured the frequency and complexity of the hallucinations produced, utilising button presses, retrospective drawing, interviews, and questionnaires. For both experimental techniques, simple hallucinations were more common than complex hallucinations. Crucially, we found that Ganzflicker was more effective than Ganzfeld at eliciting simple hallucinations, while complex hallucinations remained equivalent across the two conditions. As a result, the likelihood that an experienced hallucination was complex was higher during Ganzfeld. Despite these differences, we found a correlation between the frequency and total time spent hallucinating in Ganzflicker and Ganzfeld conditions, suggesting some shared mechanisms between the two methodologies. We attribute the tendency to experience frequent simple hallucinations in both conditions to a shared low-level core hallucinatory mechanism, such as excitability of visual cortex, potentially amplified in Ganzflicker compared to Ganzfeld due to heightened bottom-up input. The tendency to experience complex hallucinations, in contrast, may be related to top-down processes less affected by visual stimulation.

Pattern Formation in Mesic Savannas

We analyze a spatially extended version of a well-known model of forest-savanna dynamics, which presents as a system of nonlinear partial integro-differential equations, and study necessary conditions for pattern-forming bifurcations. Homogeneous solutions dominate the dynamics of the standard forest-savanna model, regardless of the length scales of the various spatial processes considered. However, several different pattern-forming scenarios are possible upon including spatial resource limitation, such as competition for water, soil nutrients, or herbivory effects. Using numerical simulations and continuation, we study the nature of the resulting patterns as a function of system parameters and length scales, uncovering subcritical pattern-forming bifurcations and observing significant regions of multistability for realistic parameter regimes. Finally, we discuss our results in the context of extant savanna-forest modeling efforts and highlight ongoing challenges in building a unifying mathematical model for savannas across different rainfall levels.

Effect of frequency and rhythmicity on flicker light-induced hallucinatory phenomena

Flicker light stimulation (FLS) uses stroboscopic light on closed eyes to induce transient visual hallucinatory phenomena, such as the perception of geometric patterns, motion, and colours. It remains an open question where the neural correlates of these hallucinatory experiences emerge along the visual pathway. To allow future testing of suggested underlying mechanisms (e.g., changes in functional connectivity, neural entrainment), we sought to systematically characterise the effects of frequency (3 Hz, 8 Hz, 10 Hz and 18 Hz) and rhythmicity (rhythmic and arrhythmic conditions) on flicker-induced subjective experiences. Using a novel questionnaire, we found that flicker frequency and rhythmicity significantly influenced the degree to which participants experienced simple visual hallucinations, particularly the perception of Klüver forms and dynamics (e.g., motion). Participants reported their experience of geometric patterns and dynamics was at highest intensity during 10 Hz rhythmic stimulation. Further, we found that frequency-matched arrhythmic FLS strongly reduced these subjective effects compared to equivalent rhythmic stimulation. Together, these results provide evidence that flicker rhythmicity critically contributes to the effects of FLS beyond the effects of frequency alone, indicating that neural entrainment may drive the induced phenomenal experience.

Evolution of the Wilson-Cowan equations

The Wilson-Cowan equations were developed to provide a simplified yet powerful description of neural network dynamics. As such, they embraced nonlinear dynamics, but in an interpretable form. Most importantly, it was the first mathematical formulation to emphasize the significance of interactions between excitatory and inhibitory neural populations, thereby incorporating both cooperation and competition. Subsequent research by many has documented the Wilson-Cowan significance in such diverse fields as visual hallucinations, memory, binocular rivalry, and epilepsy. The fact that these equations are still being used to elucidate a wide range of phenomena attests to their validity as a dynamical approximation to more detailed descriptions of complex neural computations.

Functional harmonics reveal multi-dimensional basis functions underlying cortical organization

The human brain consists of specialized areas that flexibly interact to form a multitude of functional networks. Complementary to this notion of modular organization, brain function has been shown to vary along a smooth continuum across the whole cortex. We demonstrate a mathematical framework that accounts for both of these perspectives: harmonic modes. We calculate the harmonic modes of the brain's functional connectivity graph, called "functional harmonics," revealing a multi-dimensional, frequency-ordered set of basis functions. Functional harmonics link characteristics of cortical organization across several spatial scales, capturing aspects of intra-areal organizational features (retinotopy, somatotopy), delineating brain areas, and explaining macroscopic functional networks as well as global cortical gradients. Furthermore, we show how the activity patterns elicited by seven different tasks are reconstructed from a very small subset of functional harmonics. Our results suggest that the principle of harmonicity, ubiquitous in nature, also underlies functional cortical organization in the human brain.

Altered states phenomena induced by visual flicker light stimulation

Flicker light stimulation can induce short-term alterations in consciousness including hallucinatory color perception and geometric patterns. In the study at hand, the subjective experiences during 3 Hz and 10 Hz stroboscopic light stimulation of the closed eyes were assessed. In a within-subjects design (N = 24), we applied the Positive and Negative Affect Schedule (mood state), time perception ratings, the Altered State of Consciousness Rating Scale, and the Phenomenology of Consciousness Inventory. Furthermore, we tested for effects of personality traits (NEO Five-Factor Inventory-2 and Tellegen Absorption Scale) on subjective experiences. Such systematic quantification improves replicability, facilitates comparisons between pharmacological and non-pharmacological techniques to induce altered states of consciousness, and is the prerequisite to study their underlying neuronal mechanisms. The resulting data showed that flicker light stimulation-induced states were characterized by vivid visual hallucinations of simple types, with effects strongest in the 10 Hz condition. Additionally, participants' personality trait of Absorption scores highly correlated with the experienced alterations in consciousness. Our data demonstrate that flicker light stimulation is capable of inducing visual effects with an intensity rated to be similar in strength to effects induced by psychedelic substances and thereby support the investigation of potentially shared underlying neuronal mechanisms.

Migraine Visual Aura and Cortical Spreading Depression-Linking Mathematical Models to Empirical Evidence

This review describes the subjective experience of visual aura in migraine, outlines theoretical models of this phenomenon, and explores how these may be linked to neurochemical, electrophysiological, and psychophysical differences in sensory processing that have been reported in migraine with aura. Reaction-diffusion models have been used to model the hallucinations thought to arise from cortical spreading depolarisation and depression in migraine aura. One aim of this review is to make the underlying principles of these models accessible to a general readership. Cortical spreading depolarisation and depression in these models depends on the balance of the diffusion rate between excitation and inhibition and the occurrence of a large spike in activity to initiate spontaneous pattern formation. We review experimental evidence, including recordings of brain activity made during the aura and attack phase, self-reported triggers of migraine, and psychophysical studies of visual processing in migraine with aura, and how these might relate to mechanisms of excitability that make some people susceptible to aura. Increased cortical excitability, increased neural noise, and fluctuations in oscillatory activity across the migraine cycle are all factors that are likely to contribute to the occurrence of migraine aura. There remain many outstanding questions relating to the current limitations of both models and experimental evidence. Nevertheless, reaction-diffusion models, by providing an integrative theoretical framework, support the generation of testable experimental hypotheses to guide future research.

Hallucinations on demand: the utility of experimentally induced phenomena in hallucination research

Despite the desire to delve deeper into hallucinations of all types, methodological obstacles have frustrated development of more rigorous quantitative experimental techniques, thereby hampering research progress. Here, we discuss these obstacles and, with reference to visual phenomena, argue that experimentally induced phenomena (e.g. hallucinations induced by flickering light and classical conditioning) can bring hallucinations within reach of more objective behavioural and neural measurement. Expanding the scope of hallucination research raises questions about which phenomena qualify as hallucinations, and how to identify phenomena suitable for use as laboratory models of hallucination. Due to the ambiguity inherent in current hallucination definitions, we suggest that the utility of phenomena for use as laboratory hallucination models should be represented on a continuous spectrum, where suitability varies with the degree to which external sensory information constrains conscious experience. We suggest that existing strategies that group pathological hallucinations into meaningful subtypes based on hallucination characteristics (including phenomenology, disorder and neural activity) can guide extrapolation from hallucination models to other hallucinatory phenomena. Using a spectrum of phenomena to guide scientific hallucination research should help unite the historically separate fields of psychophysics, cognitive neuroscience and clinical research to better understand and treat hallucinations, and inform models of consciousness. This article is part of the theme issue 'Offline perception: voluntary and spontaneous perceptual experiences without matching external stimulation'.

Neural field models for latent state inference: Application to large-scale neuronal recordings

Large-scale neural recording methods now allow us to observe large populations of identified single neurons simultaneously, opening a window into neural population dynamics in living organisms. However, distilling such large-scale recordings to build theories of emergent collective dynamics remains a fundamental statistical challenge. The neural field models of Wilson, Cowan, and colleagues remain the mainstay of mathematical population modeling owing to their interpretable, mechanistic parameters and amenability to mathematical analysis. Inspired by recent advances in biochemical modeling, we develop a method based on moment closure to interpret neural field models as latent state-space point-process models, making them amenable to statistical inference. With this approach we can infer the intrinsic states of neurons, such as active and refractory, solely from spiking activity in large populations. After validating this approach with synthetic data, we apply it to high-density recordings of spiking activity in the developing mouse retina. This confirms the essential role of a long lasting refractory state in shaping spatiotemporal properties of neonatal retinal waves. This conceptual and methodological advance opens up new theoretical connections between mathematical theory and point-process state-space models in neural data analysis.

Developing statistical tools to connect single-neuron activity to emergent collective dynamics is vital for building interpretable models of neural activity. Neural field models relate single-neuron activity to emergent collective dynamics in neural populations, but integrating them with data remains challenging. Recently, latent state-space models have emerged as a powerful tool for constructing phenomenological models of neural population activity. The advent of high-density multi-electrode array recordings now enables us to examine large-scale collective neural activity. We show that classical neural field approaches can yield latent state-space equations and demonstrate that this enables inference of the intrinsic states of neurons from recorded spike trains in large populations.

Network mechanisms underlying the role of oscillations in cognitive tasks

Oscillatory activity robustly correlates with task demands during many cognitive tasks. However, not only are the network mechanisms underlying the generation of these rhythms poorly understood, but it is also still unknown to what extent they may play a functional role, as opposed to being a mere epiphenomenon. Here we study the mechanisms underlying the influence of oscillatory drive on network dynamics related to cognitive processing in simple working memory (WM), and memory recall tasks. Specifically, we investigate how the frequency of oscillatory input interacts with the intrinsic dynamics in networks of recurrently coupled spiking neurons to cause changes of state: the neuronal correlates of the corresponding cognitive process. We find that slow oscillations, in the delta and theta band, are effective in activating network states associated with memory recall. On the other hand, faster oscillations, in the beta range, can serve to clear memory states by resonantly driving transient bouts of spike synchrony which destabilize the activity. We leverage a recently derived set of exact mean-field equations for networks of quadratic integrate-and-fire neurons to systematically study the bifurcation structure in the periodically forced spiking network. Interestingly, we find that the oscillatory signals which are most effective in allowing flexible switching between network states are not smooth, pure sinusoids, but rather burst-like, with a sharp onset. We show that such periodic bursts themselves readily arise spontaneously in networks of excitatory and inhibitory neurons, and that the burst frequency can be tuned via changes in tonic drive. Finally, we show that oscillations in the gamma range can actually stabilize WM states which otherwise would not persist.

Phase reorganization leads to transient β-LFP spatial wave patterns in motor cortex during steady-state movement preparation

Previous studies on the origin and properties of spatial patterns in motor cortex β-local field potential (β-LFP) oscillations have focused on planar traveling waves. However, it is unclear 1) whether β-LFP waves are limited to plane waves, or even 2) whether they are propagating waves of excito-excitatory activity, i.e., primarily traveling waves in excitable media; they could reflect, instead, reorganization in the relative phases of transient oscillations at different spatial sites. We addressed these two problems in β-LFPs recorded via microelectrode arrays implanted in three adjacent motor cortex areas of nonhuman primates during steady-state movement preparation. Our findings are fourfold: 1) β-LFP wave patterns emerged as transient events, despite stable firing rates of single neurons concurrently recorded during the same periods. 2) β-LFP waves showed a richer variety of spatial dynamics, including rotating and complex waves. 3) β-LFP wave patterns showed no characteristic wavelength, presenting instead a range of scales with global zero-lag phase synchrony as a limiting case, features surprising for purely excito-excitatory waves but consistent with waves in coupled oscillator systems. 4) Furthermore, excito-excitatory traveling waves induced by optogenetic stimulation in motor cortex showed, in contrast, a characteristic wavelength and reduced phase synchrony. Overall, β-LFP wave statistics differed from those of induced traveling waves in excitable media recorded under the same microelectrode array setup. Our findings suggest phase reorganization in neural coupled oscillators contribute significantly to the origin of transient β-LFP spatial dynamics during preparatory steady states and outline important constraints for spatially extended models of β-LFP dynamics in motor cortex. NEW & NOTEWORTHY We show that a rich variety of transient β-local field potential (β-LFP) wave patterns emerge in motor cortex during preparatory steady states, despite stable neuronal firing rates. Furthermore, unlike optogenetically induced traveling waves, β-LFP waves showed no characteristic wavelength, presenting instead a range of scales with global phase synchrony as a limiting case. Overall, our statistical analyses suggest that transient phase reorganization in neural coupled oscillators, beyond purely excito-excitatory traveling waves, contribute significantly to the origin of motor cortex β-LFP wave patterns.

Serotonergic Hallucinogen-Induced Visual Perceptual Alterations

Serotonergic hallucinogens, such as lysergic acid diethylamide (LSD), psilocybin, and N,N-dimethyltryptamine (DMT), are famous for their capacity to temporally and profoundly alter an individual's visual experiences. These visual alterations show consistent attributes despite large inter- and intra-individual variances. Many reports document a common perception of colors as more saturated, with increased brightness and contrast in the environment ("Visual Intensifications"). Environmental objects might be altered in size ("Visual illusions") or take on a modified and special meaning for the subject ("Altered self-reference"). Subjects may perceive light flashes or geometrical figures containing recurrent patterns ("Elementary imagery and hallucinations") influenced by auditory stimuli ("Audiovisual synesthesia"), or they may envision images of people, animals, or landscapes ("Complex imagery and hallucinations") without any physical stimuli supporting their percepts. This wide assortment of visual phenomena suggests that one single neuropsychopharmacological mechanism is unlikely to explain such vast phenomenological diversity. Starting with mechanisms that act at the cellular level, the key role of 5-HT2A receptor activation and the subsequent increased cortical excitation will be considered. Next, it will be shown that area specific anatomical and dynamical features link increased excitation to the specific visual contents of hallucinations. The decrease of alpha oscillations by hallucinogens will then be introduced as a systemic mechanism for amplifying internal-driven excitation that overwhelms stimulus-induced excitations. Finally, the hallucinogen-induced parallel decrease of the N170 visual evoked potential and increased medial P1 potential will be discussed as key mechanisms for inducing a dysbalance between global integration and early visual gain that may explain several hallucinogen-induced visual experiences, including visual hallucinations, illusions, and intensifications.

Harmonic Brain Modes: A Unifying Framework for Linking Space and Time in Brain Dynamics

A fundamental characteristic of spontaneous brain activity is coherent oscillations covering a wide range of frequencies. Interestingly, these temporal oscillations are highly correlated among spatially distributed cortical areas forming structured correlation patterns known as the resting state networks, although the brain is never truly at "rest." Here, we introduce the concept of harmonic brain modes-fundamental building blocks of complex spatiotemporal patterns of neural activity. We define these elementary harmonic brain modes as harmonic modes of structural connectivity; that is, connectome harmonics, yielding fully synchronous neural activity patterns with different frequency oscillations emerging on and constrained by the particular structure of the brain. Hence, this particular definition implicitly links the hitherto poorly understood dimensions of space and time in brain dynamics and its underlying anatomy. Further we show how harmonic brain modes can explain the relationship between neurophysiological, temporal, and network-level changes in the brain across different mental states ( wakefulness, sleep, anesthesia, psychedelic). Notably, when decoded as activation of connectome harmonics, spatial and temporal characteristics of neural activity naturally emerge from the interplay between excitation and inhibition and this critical relation fits the spatial, temporal, and neurophysiological changes associated with different mental states. Thus, the introduced framework of harmonic brain modes not only establishes a relation between the spatial structure of correlation patterns and temporal oscillations (linking space and time in brain dynamics), but also enables a new dimension of tools for understanding fundamental principles underlying brain dynamics in different states of consciousness.

Sensory dynamics of visual hallucinations in the normal population

Hallucinations occur in both normal and clinical populations. Due to their unpredictability and complexity, the mechanisms underlying hallucinations remain largely untested. Here we show that visual hallucinations can be induced in the normal population by visual flicker, limited to an annulus that constricts content complexity to simple moving grey blobs, allowing objective mechanistic investigation. Hallucination strength peaked at ~11 Hz flicker and was dependent on cortical processing. Hallucinated motion speed increased with flicker rate, when mapped onto visual cortex it was independent of eccentricity, underwent local sensory adaptation and showed the same bistable and mnemonic dynamics as sensory perception. A neural field model with motion selectivity provides a mechanism for both hallucinations and perception. Our results demonstrate that hallucinations can be studied objectively, and they share multiple mechanisms with sensory perception. We anticipate that this assay will be critical to test theories of human consciousness and clinical models of hallucination.

Pattern formation in oscillatory media without lateral inhibition

Spontaneous symmetry breaking instabilities are the most common mechanism for how biological, chemical, and physical systems produce spatial patterns. Beginning with Turing's original paper, so-called lateral inhibition-in which negative feedback has greater spread than positive feedback-has been the underlying mechanism for pattern formation in biological models. Despite this, there are many biological systems that exhibit pattern formation but do not have lateral inhibition. In this paper, we present an example of such a system that is able to generate robust patterns emerging from a spatially homogeneous state. In fact, patterns can arise when there is only spatial spread of the activator. Unlike classic Turing pattern formation, these patterns arise from a spatially homogeneous oscillation rather than from a constant steady state.

Visual Contextual Effects of Orientation, Contrast, Flicker, and Luminance: All Are Affected by Normal Aging

The perception of a visual stimulus can be markedly altered by spatial interactions between the stimulus and its surround. For example, a grating stimulus appears lower in contrast when surrounded by a similar pattern of higher contrast: a phenomenon known as surround suppression of perceived contrast. Such center–surround interactions in visual perception are numerous and arise from both cortical and pre-cortical neural circuitry. For example, perceptual surround suppression of luminance and flicker are predominantly mediated pre-cortically, whereas contrast and orientation suppression have strong cortical contributions. Here, we compare the perception of older and younger observers on a battery of tasks designed to assess such visual contextual effects. For all visual dimensions tested (luminance, flicker, contrast, and orientation), on average the older adults showed greater suppression of central targets than the younger adult group. The increase in suppression was consistent in magnitude across all tasks, suggesting that normal aging produces a generalized, non-specific alteration to contextual processing in vision.

Human brain networks function in connectome-specific harmonic waves

A key characteristic of human brain activity is coherent, spatially distributed oscillations forming behaviour-dependent brain networks. However, a fundamental principle underlying these networks remains unknown. Here we report that functional networks of the human brain are predicted by harmonic patterns, ubiquitous throughout nature, steered by the anatomy of the human cerebral cortex, the human connectome. We introduce a new technique extending the Fourier basis to the human connectome. In this new frequency-specific representation of cortical activity, that we call 'connectome harmonics', oscillatory networks of the human brain at rest match harmonic wave patterns of certain frequencies. We demonstrate a neural mechanism behind the self-organization of connectome harmonics with a continuous neural field model of excitatory-inhibitory interactions on the connectome. Remarkably, the critical relation between the neural field patterns and the delicate excitation-inhibition balance fits the neurophysiological changes observed during the loss and recovery of consciousness.

Periodic Forcing of Inhibition-Stabilized Networks: Nonlinear Resonances and Phase-Amplitude Coupling

Inhibition-stabilized networks (ISNs) are neural architectures with strong positive feedback among pyramidal neurons balanced by strong negative feedback from inhibitory interneurons, a circuit element found in the hippocampus and the primary visual cortex. In their working regime, ISNs produce damped oscillations in the [Formula: see text]-range in response to inputs to the inhibitory population. In order to understand the properties of interconnected ISNs, we investigated periodic forcing of ISNs. We show that ISNs can be excited over a range of frequencies and derive properties of the resonance peaks. In particular, we studied the phase-locked solutions, the torus solutions, and the resonance peaks. Periodically forced ISNs respond with (possibly multistable) phase-locked activity, whereas networks with sustained intrinsic oscillations respond more dynamically to periodic inputs with tori. Hence, the dynamics are surprisingly rich, and phase effects alone do not adequately describe the network response. This strengthens the importance of phase-amplitude coupling as opposed to phase-phase coupling in providing multiple frequencies for multiplexing and routing information.

A bidirectional link between brain oscillations and geometric patterns

Like hallucinogenic drugs, full-field flickering visual stimulation produces regular, geometric hallucinations such as radial or spiral patterns. Computational and theoretical models have revealed that the geometry of these hallucinations can be related to functional neuro-anatomy. However, while experimental evidence links both visual flicker and hallucinogenic drugs to upward and downward modulations of brain oscillatory activity, the exact relation between brain oscillations and geometric hallucinations remains a mystery. Here we demonstrate that, in human observers, this link is bidirectional. The same flicker frequencies that preferentially induced radial (<10 Hz) or spiral (10-20 Hz) hallucinations in a behavioral experiment involving full-field uniform flicker without any actual shape displayed, also showed selective oscillatory EEG enhancement when observers viewed a genuine static image of a radial or spiral pattern without any flicker. This bidirectional property constrains the possible neuronal events at the origin of visual hallucinations, and further suggests that brain oscillations, which are strictly temporal in nature, could nonetheless act as preferential channels for spatial information.

The flickering wheel illusion: when α rhythms make a static wheel flicker

α oscillations (8-14 Hz) greatly influence brain activity, yet we generally do not experience them consciously: the world does not appear to oscillate. Dedicated strategies must exist in the brain to prevent these oscillations from disrupting normal processing. Could suitable stimuli fool these strategies and lead to the conscious experience of our own brain oscillations? We describe and explore a novel illusion in which the center of a static wheel stimulus (with 30-40 spokes) is experienced as flickering when viewed in the visual periphery. The key feature of this illusion is that the stimulus fluctuations are experienced as a regular and consistent flicker, which our human observers estimated at ~9 Hz during a psychophysical matching task. Correspondingly, the occipital α rhythm of the EEG was the only oscillation that showed a time course compatible with the reported illusion: when α amplitude was strong, the probability of reporting illusory flicker increased. The peak oscillatory frequency for these flicker-induced modulations was significantly correlated, on a subject-by-subject basis, with the individual α frequency measured during rest, in the absence of visual stimulation. Finally, although the effect is strongest during eye movements, we showed that stimulus motion relative to the retina is not necessary to perceive the illusion: the flicker can also be perceived on the afterimage of the wheel, yet by definition this afterimage is stationary on the retina. We conclude that this new flickering illusion is a unique way to experience the α rhythms that constantly occur in the brain but normally remain unnoticed.

Perceptual echoes at 10 Hz in the human brain

The occipital alpha rhythm (∼10 Hz) is the most prominent electrophysiological activity in the awake human brain, yet its functional role and relation to visual perception are little understood. Transient stimuli normally elicit a short series of positive and negative deflections lasting between 300 and 500 ms: the visual-evoked potential (VEP). Alpha oscillations, on the other hand, are generally suppressed by transient visual input; they only augment in response to periodic ("steady-state") inputs around 10 Hz. Here, we applied reverse-correlation techniques to the visual presentation of random, nonperiodic dynamic stimulation sequences and found that the brain response to each stimulus transient was not merely a short-lived VEP but also included a strong ∼10 Hz oscillation that lasted for more than 1 s. In other words, the alpha rhythm implements an "echo" or reverberation of the input sequence. These echoes are correlated in magnitude and frequency with the observer's occipital alpha rhythm, are enhanced by visual attention, and can be rendered perceptually apparent in the form of ∼10 Hz flicker. These findings suggest a role for the alpha rhythm in the maintenance of sensory representations over time.

Response of traveling waves to transient inputs in neural fields

We analyze the effects of transient stimulation on traveling waves in neural field equations. Neural fields are modeled as integro-differential equations whose convolution term represents the synaptic connections of a spatially extended neuronal network. The adjoint of the linearized wave equation can be used to identify how a particular input will shift the location of a traveling wave. This wave response function is analogous to the phase response curve of limit cycle oscillators. For traveling fronts in an excitatory network, the sign of the shift depends solely on the sign of the transient input. A complementary estimate of the effective shift is derived using an equation for the time-dependent speed of the perturbed front. Traveling pulses are analyzed in an asymmetric lateral inhibitory network and they can be advanced or delayed, depending on the position of spatially localized transient inputs. We also develop bounds on the amplitude of transient input necessary to terminate traveling pulses, based on the global bifurcation structure of the neural field.