An amplitude equation approach to contextual effects in visual cortex

Firing rate models for gamma oscillations in I-I and E-I networks

Firing rate models for describing the mean-field activities of neuronal ensembles can be used effectively to study network function and dynamics, including synchronization and rhythmicity of excitatory-inhibitory populations. However, traditional Wilson-Cowan-like models, even when extended to include an explicit dynamic synaptic activation variable, are found unable to capture some dynamics such as Interneuronal Network Gamma oscillations (ING). Use of an explicit delay is helpful in simulations at the expense of complicating mathematical analysis. We resolve this issue by introducing a dynamic variable, u, that acts as an effective delay in the negative feedback loop between firing rate (r) and synaptic gating of inhibition (s). In effect, u endows synaptic activation with second order dynamics. With linear stability analysis, numerical branch-tracking and simulations, we show that our r-u-s rate model captures some key qualitative features of spiking network models for ING. We also propose an alternative formulation, a v-u-s model, in which mean membrane potential v satisfies an averaged current-balance equation. Furthermore, we extend the framework to E-I networks. With our six-variable v-u-s model, we demonstrate in firing rate models the transition from Pyramidal-Interneuronal Network Gamma (PING) to ING by increasing the external drive to the inhibitory population without adjusting synaptic weights. Having PING and ING available in a single network, without invoking synaptic blockers, is plausible and natural for explaining the emergence and transition of two different types of gamma oscillations.

A Cortical-Inspired Sub-Riemannian Model for Poggendorff-Type Visual Illusions

We consider Wilson-Cowan-type models for the mathematical description of orientation-dependent Poggendorff-like illusions. Our modelling improves two previously proposed cortical-inspired approaches, embedding the sub-Riemannian heat kernel into the neuronal interaction term, in agreement with the intrinsically anisotropic functional architecture of V1 based on both local and lateral connections. For the numerical realisation of both models, we consider standard gradient descent algorithms combined with Fourier-based approaches for the efficient computation of the sub-Laplacian evolution. Our numerical results show that the use of the sub-Riemannian kernel allows us to reproduce numerically visual misperceptions and inpainting-type biases in a stronger way in comparison with the previous approaches.

Stochastic neural fields as gradient dynamical systems

Continuous attractor neural networks are used extensively to model a variety of experimentally observed coherent brain states, ranging from cortical waves of activity to stationary activity bumps. The latter are thought to play an important role in various forms of neural information processing, including population coding in primary visual cortex (V1) and working memory in prefrontal cortex. However, one limitation of continuous attractor networks is that the location of the peak of an activity bump (or wave) can diffuse due to intrinsic network noise. This reflects marginal stability of bump solutions with respect to the action of an underlying continuous symmetry group. Previous studies have used perturbation theory to derive an approximate stochastic differential equation for the location of the peak (phase) of the bump. Although this method captures the diffusive wandering of a bump solution, it ignores fluctuations in the amplitude of the bump. In this paper, we show how amplitude fluctuations can be analyzed by reducing the underlying stochastic neural field equation to a finite-dimensional stochastic gradient dynamical system that tracks the stochastic motion of both the amplitude and phase of bump solutions. This allows us to derive exact expressions for the steady-state probability density and its moments, which are then used to investigate two major issues: (i) the input-dependent suppression of neural variability and (ii) noise-induced transitions to bump extinction. We develop the theory by considering the particular example of a ring attractor network with SO(2) symmetry, which is the most common architecture used in attractor models of working memory and population tuning in V1. However, we also extend the analysis to a higher-dimensional spherical attractor network with SO(3) symmetry which has previously been proposed as a model of orientation and spatial frequency tuning in V1. We thus establish how a combination of stochastic analysis and group theoretic methods provides a powerful tool for investigating the effects of noise in continuous attractor networks.

Stochastic neural field model of stimulus-dependent variability in cortical neurons

We use stochastic neural field theory to analyze the stimulus-dependent tuning of neural variability in ring attractor networks. We apply perturbation methods to show how the neural field equations can be reduced to a pair of stochastic nonlinear phase equations describing the stochastic wandering of spontaneously formed tuning curves or bump solutions. These equations are analyzed using a modified version of the bivariate von Mises distribution, which is well-known in the theory of circular statistics. We first consider a single ring network and derive a simple mathematical expression that accounts for the experimentally observed bimodal (or M-shaped) tuning of neural variability. We then explore the effects of inter-network coupling on stimulus-dependent variability in a pair of ring networks. These could represent populations of cells in two different layers of a cortical hypercolumn linked via vertical synaptic connections, or two different cortical hypercolumns linked by horizontal patchy connections within the same layer. We find that neural variability can be suppressed or facilitated, depending on whether the inter-network coupling is excitatory or inhibitory, and on the relative strengths and biases of the external stimuli to the two networks. These results are consistent with the general observation that increasing the mean firing rate via external stimuli or modulating drives tends to reduce neural variability.

Circuits and Mechanisms for Surround Modulation in Visual Cortex

Surround modulation (SM) is a fundamental property of sensory neurons in many species and sensory modalities. SM is the ability of stimuli in the surround of a neuron's receptive field (RF) to modulate (typically suppress) the neuron's response to stimuli simultaneously presented inside the RF, a property thought to underlie optimal coding of sensory information and important perceptual functions. Understanding the circuit and mechanisms for SM can reveal fundamental principles of computations in sensory cortices, from mouse to human. Current debate is centered over whether feedforward or intracortical circuits generate SM, and whether this results from increased inhibition or reduced excitation. Here we present a working hypothesis, based on theoretical and experimental evidence, that SM results from feedforward, horizontal, and feedback interactions with local recurrent connections, via synaptic mechanisms involving both increased inhibition and reduced recurrent excitation. In particular, strong and balanced recurrent excitatory and inhibitory circuits play a crucial role in the computation of SM.

Stochastic Network Models in Neuroscience: A Festschrift for Jack Cowan. Introduction to the Special Issue

Jack Cowan's remarkable career has spanned, and molded, the development of neuroscience as a quantitative and mathematical discipline combining deep theoretical contributions, rigorous mathematical work and groundbreaking biological insights. The Banff International Research Station hosted a workshop in his honor, on Stochastic Network Models of Neocortex, July 17-24, 2014. This accompanying Festschrift celebrates Cowan's contributions by assembling current research in stochastic phenomena in neural networks. It combines historical perspectives with new results including applications to epilepsy, path-integral methods, stochastic synchronization, higher-order correlation analysis, and pattern formation in visual cortex.

Laminar Neural Field Model of Laterally Propagating Waves of Orientation Selectivity

We construct a laminar neural-field model of primary visual cortex (V1) consisting of a superficial layer of neurons that encode the spatial location and orientation of a local visual stimulus coupled to a deep layer of neurons that only encode spatial location. The spatially-structured connections in the deep layer support the propagation of a traveling front, which then drives propagating orientation-dependent activity in the superficial layer. Using a combination of mathematical analysis and numerical simulations, we establish that the existence of a coherent orientation-selective wave relies on the presence of weak, long-range connections in the superficial layer that couple cells of similar orientation preference. Moreover, the wave persists in the presence of feedback from the superficial layer to the deep layer. Our results are consistent with recent experimental studies that indicate that deep and superficial layers work in tandem to determine the patterns of cortical activity observed in vivo.

Short term synaptic depression improves information transfer in perceptual multistability

Competitive neural networks are often used to model the dynamics of perceptual bistability. Switching between percepts can occur through fluctuations and/or a slow adaptive process. Here, we analyze switching statistics in competitive networks with short term synaptic depression and noise. We start by analyzing a ring model that yields spatially structured solutions and complement this with a study of a space-free network whose populations are coupled with mutual inhibition. Dominance times arising from depression driven switching can be approximated using a separation of timescales in the ring and space-free model. For purely noise-driven switching, we derive approximate energy functions to justify how dominance times are exponentially related to input strength. We also show that a combination of depression and noise generates realistic distributions of dominance times. Unimodal functions of dominance times are more easily told apart by sampling, so switches induced by synaptic depression induced provide more information about stimuli than noise-driven switching. Finally, we analyze a competitive network model of perceptual tristability, showing depression generates a history-dependence in dominance switching.

Strong recurrent networks compute the orientation tuning of surround modulation in the primate primary visual cortex

In macaque primary visual cortex (V1), neuronal responses to stimuli inside the receptive field (RF) are modulated by stimuli in the RF surround. This modulation is orientation specific. Previous studies suggested that, for some cells, this specificity may not be fixed but changes with the stimulus orientation presented to the RF. We demonstrate, in recording studies, that this tuning behavior is instead highly prevalent in V1 and, in theoretical work, that it arises only if V1 operates in a regime of strong local recurrence. Strongest surround suppression occurs when the stimuli in the RF and the surround are iso-oriented, and strongest facilitation when the stimuli are cross-oriented. This is the case even when the RF is suboptimally activated by a stimulus of nonpreferred orientation but only if this stimulus can activate the cell when presented alone. This tuning behavior emerges from the interaction of lateral inhibition (via the surround pathways), which is tuned to the preferred orientation of the RF, with weakly tuned, but strong, local recurrent connections, causing maximal withdrawal of recurrent excitation at the feedforward input orientation. Thus, horizontal and feedback modulation of strong recurrent circuits allows the tuning of contextual effects to change with changing feedforward inputs.

Evolutionary constraints on visual cortex architecture from the dynamics of hallucinations

In the cat or primate primary visual cortex (V1), normal vision corresponds to a state where neural excitation patterns are driven by external visual stimuli. A spectacular failure mode of V1 occurs when such patterns are overwhelmed by spontaneously generated spatially self-organized patterns of neural excitation. These are experienced as geometric visual hallucinations. The problem of identifying the mechanisms by which V1 avoids this failure is made acute by recent advances in the statistical mechanics of pattern formation, which suggest that the hallucinatory state should be very robust. Here, we report how incorporating physiologically realistic long-range connections between inhibitory neurons changes the behavior of a model of V1. We find that the sparsity of long-range inhibition in V1 plays a previously unrecognized but key functional role in preserving the normal vision state. Surprisingly, it also contributes to the observed regularity of geometric visual hallucinations. Our results provide an explanation for the observed sparsity of long-range inhibition in V1--this generic architectural feature is an evolutionary adaptation that tunes V1 to the normal vision state. In addition, it has been shown that exactly the same long-range connections play a key role in the development of orientation preference maps. Thus V1's most striking long-range features--patchy excitatory connections and sparse inhibitory connections--are strongly constrained by two requirements: the need for the visual state to be robust and the developmental requirements of the orientational preference map.

Neural field model of binocular rivalry waves

We present a neural field model of binocular rivalry waves in visual cortex. For each eye we consider a one-dimensional network of neurons that respond maximally to a particular feature of the corresponding image such as the orientation of a grating stimulus. Recurrent connections within each one-dimensional network are assumed to be excitatory, whereas connections between the two networks are inhibitory (cross-inhibition). Slow adaptation is incorporated into the model by taking the network connections to exhibit synaptic depression. We derive an analytical expression for the speed of a binocular rivalry wave as a function of various neurophysiological parameters, and show how properties of the wave are consistent with the wave-like propagation of perceptual dominance observed in recent psychophysical experiments. In addition to providing an analytical framework for studying binocular rivalry waves, we show how neural field methods provide insights into the mechanisms underlying the generation of the waves. In particular, we highlight the important role of slow adaptation in providing a "symmetry breaking mechanism" that allows waves to propagate.

Investigating the mechanisms of hallucinogen-induced visions using 3,4-methylenedioxyamphetamine (MDA): a randomized controlled trial in humans

BACKGROUND

The mechanisms of drug-induced visions are poorly understood. Very few serotonergic hallucinogens have been studied in humans in decades, despite widespread use of these drugs and potential relevance of their mechanisms to hallucinations occurring in psychiatric and neurological disorders.

METHODOLOGY/PRINCIPAL FINDINGS

We investigated the mechanisms of hallucinogen-induced visions by measuring the visual and perceptual effects of the hallucinogenic serotonin 5-HT2AR receptor agonist and monoamine releaser, 3,4-methylenedioxyamphetamine (MDA), in a double-blind placebo-controlled study. We found that MDA increased self-report measures of mystical-type experience and other hallucinogen-like effects, including reported visual alterations. MDA produced a significant increase in closed-eye visions (CEVs), with considerable individual variation. Magnitude of CEVs after MDA was associated with lower performance on measures of contour integration and object recognition.

CONCLUSIONS/SIGNIFICANCE

Drug-induced visions may have greater intensity in people with poor sensory or perceptual processing, suggesting common mechanisms with other hallucinatory syndromes. MDA is a potential tool to investigate mystical experiences and visual perception.

TRIAL REGISTRATION

Clinicaltrials.gov NCT00823407.

Steps toward numerical mode analysis of organizing systems

A well established method to analyze dynamical systems described by coupled nonlinear differential equations is to determine their normal modes and reduce the dynamics, by adiabatic elimination of stable modes, to a much smaller system for the amplitudes of unstable modes and their nonlinear interactions. So far, this analysis is possible only for idealized symmetric model systems. We aim to build a framework in which realistic systems with less symmetry can be analyzed automatically. In this paper we present a first example of mode analysis with the assistance of numerical computation. Our method is illustrated using a model system for the ontogenesis of retinotopy, and the results reproduce those from theoretical analysis precisely. Aspects of organization generalized from this model system are discussed.

Nonlocal Ginzburg-Landau equation for cortical pattern formation

We show how a nonlocal version of the real Ginzburg-Landau (GL) equation arises in a large-scale recurrent network model of primary visual cortex. We treat cortex as a continuous two-dimensional sheet of cells that signal both the position and orientation of a local visual stimulus. The recurrent circuitry is decomposed into a local part, which contributes primarily to the orientation tuning properties of the cells, and a long-range part that introduces spatial correlations. We assume that (a) the local network exists in a balanced state such that it operates close to a point of instability and (b) the long-range connections are weak and scale with the bifurcation parameter of the dynamical instability generated by the local circuitry. Carrying out a perturbation expansion with respect to the long-range coupling strength then generates a nonlocal coupling term in the GL amplitude equation. We use the nonlocal GL equation to analyze how axonal propagation delays arising from the slow conduction velocities of the long-range connections affect spontaneous pattern formation.

Tuned solutions in dynamic neural fields as building blocks for extended EEG models

The most prominent functional property of cortical neurons in sensory areas are their tuned receptive fields which provide specific responses of the neurons to external stimuli. Tuned neural firing indeed reflects the most basic and best worked out level of cognitive representations. Tuning properties can be dynamic on a short time-scale of fractions of a second. Such dynamic effects have been modeled by localised solutions (also called "bumps" or "peaks") in dynamic neural fields. In the present work we develop an approximation method to reduce the dynamics of localised activation peaks in systems of n coupled nonlinear d-dimensional neural fields with transmission delays to a small set of delay differential equations for the peak amplitudes and widths only. The method considerably simplifies the analysis of peaked solutions as demonstrated for a two-dimensional example model of neural feature selectivity in the brain. The reduced equations describe the effective interaction between pools of local neurons of several (n) classes that participate in shaping the dynamic receptive field responses. To lowest order they resemble neural mass models as they often form the base of EEG-models. Thereby they provide a link between functional small-scale receptive field models and more coarse-grained EEG-models. More specifically, they connect the dynamics in feature-selective cortical microcircuits to the more abstract local elements used in coarse-grained models. However, beside amplitudes the reduced equations also reflect the sharpness of tuning of the activity in a d-dimensional feature space in response to localised stimuli.

Space and time in visual context

No sensory stimulus is an island unto itself; rather, it can only properly be interpreted in light of the stimuli that surround it in space and time. This can result in entertaining illusions and puzzling results in psychological and neurophysiological experiments. We concentrate on perhaps the best studied test case, namely orientation or tilt, which gives rise to the notorious tilt illusion and the adaptation tilt after-effect. We review the empirical literature and discuss the computational and statistical ideas that are battling to explain these conundrums, and thereby gain favour as more general accounts of cortical processing.

The role of feedback in shaping the extra-classical receptive field of cortical neurons: a recurrent network model

The responses of neurons in sensory cortices are affected by the spatial context within which stimuli are embedded. In the primary visual cortex (V1), orientation-selective responses to stimuli in the receptive field (RF) center are suppressed by similarly oriented stimuli in the RF surround. Surround suppression, a likely neural correlate of perceptual figure-ground segregation, is traditionally thought to be generated within V1 by long-range horizontal connections. Recently however, it has been shown that these connections are too short and too slow to mediate fast suppression from distant regions of the RF surround. We use an anatomically and physiologically constrained recurrent network model of macaque V1 to show how interareal feedback connections, which are faster and longer-range than horizontal connections, can generate "far" surround suppression. We provide a novel solution to the puzzle of how surround suppression can arise from excitatory feedback axons contacting predominantly excitatory neurons in V1. The basic mechanism involves divergent feedback connections from the far surround targeting pyramidal neurons sending monosynaptic horizontal connections to excitatory and inhibitory neurons in the RF center. One of several predictions of our model is that the "suppressive far surround" is not always suppressive, but can facilitate the response of the RF center, depending on the amount of excitatory drive to the local inhibitors. Our model provides a general mechanism of how top-down feedback signals directly contribute to generating cortical neuron responses to simple sensory stimuli.

The "silent" surround of V1 receptive fields: theory and experiments

The spiking response of a primary visual cortical cell to a stimulus placed within its receptive field can be up- and down-regulated by the simultaneous presentation of objects or scenes placed in the "silent" regions which surround the receptive field. We here review recent progresses that have been made both at the experimental and theoretical levels in the description of these so-called "Center/Surround" modulations and in the understanding of their neural basis. Without denying the role of a modulatory feedback from higher cortical areas, recent results support the view that some of these phenomena result from the dynamic interplay between feedforward projections and horizontal intracortical connectivity in V1. Uncovering the functional role of the contextual periphery of cortical receptive fields has become an area of active investigation. The detailed comparison of electrophysiological and psychophysical data reveals strong correlations between the integrative behavior of V1 cells and some aspects of "low-level" and "mid-level" conscious perception. These suggest that as early as the V1 stage, the visual system is able to make use of contextual cues to recover local visual scene properties or correct their interpretation. Promising ideas have emerged on the importance of such a strategy for the coding of visual scenes, and the processing of static and moving objects.

The functional geometry of local and horizontal connections in a model of V1

A mathematical model of interacting hypercolumns in primary visual cortex (V1) is presented that incorporates details concerning the geometry of local and long-range horizontal connections. Each hypercolumn is modeled as a network of interacting excitatory and inhibitory neural populations with orientation and spatial frequency preferences organized around a pair of pinwheels. The pinwheels are arranged on a planar lattice, reflecting the crystalline-like structure of cortex. Local interactions within a hypercolumn generate orientation and spatial frequency tuning curves, which are modulated by horizontal connections between different hypercolumns on the lattice. The symmetry properties of the local and long-range connections play an important role in determining the types of spontaneous activity patterns that can arise in cortex.

A spherical model for orientation and spatial-frequency tuning in a cortical hypercolumn

A theory is presented of the way in which the hypercolumns in primary visual cortex (V1) are organized to detect important features of visual images, namely local orientation and spatial-frequency. Given the existence in V1 of dual maps for these features, both organized around orientation pinwheels, we constructed a model of a hypercolumn in which orientation and spatial-frequency preferences are represented by the two angular coordinates of a sphere. The two poles of this sphere are taken to correspond, respectively, to high and low spatial-frequency preferences. In Part I of the paper, we use mean-field methods to derive exact solutions for localized activity states on the sphere. We show how cortical amplification through recurrent interactions generates a sharply tuned, contrast-invariant population response to both local orientation and local spatial frequency, even in the case of a weakly biased input from the lateral geniculate nucleus (LGN). A major prediction of our model is that this response is non-separable with respect to the local orientation and spatial frequency of a stimulus. That is, orientation tuning is weaker around the pinwheels, and there is a shift in spatial-frequency tuning towards that of the closest pinwheel at non-optimal orientations. In Part II of the paper, we demonstrate that a simple feed-forward model of spatial-frequency preference, unlike that for orientation preference, does not generate a faithful representation when amplified by recurrent interactions in V1. We then introduce the idea that cortico-geniculate feedback modulates LGN activity to generate a faithful representation, thus providing a new functional interpretation of the role of this feedback pathway. Using linear filter theory, we show that if the feedback from a cortical cell is taken to be approximately equal to the reciprocal of the corresponding feed-forward receptive field (in the two-dimensional Fourier domain), then the mismatch between the feed-forward and cortical frequency representations is eliminated. We therefore predict that cortico-geniculate feedback connections innervate the LGN in a pattern determined by the orientation and spatial-frequency biases of feed-forward receptive fields. Finally, we show how recurrent cortical interactions can generate cross-orientation suppression.