This paper is concerned with a striking visual experience: that of seeing geometric visual hallucinations. Hallucinatory images were classi¢ed by Klu« ver into four groups called form constants comprising (i) gratings, lattices, fretworks, ¢ligrees, honeycombs and chequer-boards, (ii) cobwebs, (iii) tunnels, funnels, alleys, cones and vessels, and (iv) spirals. This paper describes a mathematical investigation of their origin based on the assumption that the patterns of connection between retina and striate cortex (henceforth referred to as V1)öthe retinocortical mapöand of neuronal circuits in V1, both local and lateral, determine their geometry.In the ¢rst part of the paper we show that form constants, when viewed in V1 coordinates, essentially correspond to combinations of plane waves, the wavelengths of which are integral multiples of the width of a human Hubel^Wiesel hypercolumn, ca. 1.33^2 mm. We next introduce a mathematical description of the large-scale dynamics of V1 in terms of the continuum limit of a lattice of interconnected hypercolumns, each of which itself comprises a number of interconnected iso-orientation columns. We then show that the patterns of interconnection in V1 exhibit a very interesting symmetry, i.e. they are invariant under the action of the planar Euclidean group E(2)öthe group of rigid motions in the planeö rotations, re£ections and translations. What is novel is that the lateral connectivity of V1 is such that a new group action is needed to represent its properties: by virtue of its anisotropy it is invariant with respect to certain shifts and twists of the plane. It is this shift^twist invariance that generates new representations of E(2). Assuming that the strength of lateral connections is weak compared with that of local connections, we next calculate the eigenvalues and eigenfunctions of the cortical dynamics, using Rayleigh^Schro« dinger perturbation theory. The result is that in the absence of lateral connections, the eigenfunctions are degenerate, comprising both even and odd combinations of sinusoids in , the cortical label for orientation preference, and plane waves in r, the cortical position coordinate.`Switching-on' the lateral interactions breaks the degeneracy and either even or else odd eigenfunctions are selected. These results can be shown to follow directly from the Euclidean symmetry we have imposed.In the second part of the paper we study the nature of various even and odd combinations of eigenfunctions or planforms, the symmetries of which are such that they remain invariant under the particular action of E(2) we have imposed. These symmetries correspond to certain subgroups of E(2), the so-called axial subgroups. Axial subgroups are important in that the equivariant branching lemma indicates that when a symmetrical dynamical system becomes unstable, new solutions emerge which have symmetries corresponding to the axial subgroups of the underlying symmetry group. This is precisely the case studied in this paper. Thus we study the various planforms that emer...

Many observers see geometric visual hallucinations after taking hallucinogens such as LSD, cannabis, mescaline or psilocybin; on viewing bright flickering lights; on waking up or falling asleep; in "near-death" experiences; and in many other syndromes. Klüver organized the images into four groups called form constants: (I) tunnels and funnels, (II) spirals, (III) lattices, including honeycombs and triangles, and (IV) cobwebs. In most cases, the images are seen in both eyes and move with them. We interpret this to mean that they are generated in the brain. Here, we sum- nectivity between hypercolumns exhibits symmetries, rendering it invariant under the action of the Euclidean group E(2), composed of reflections and translations in the plane, and a (novel) shift-twist action. Using this symmetry, we show that the various patterns of activity that spontaneously emerge when V1's spatially uniform resting state becomes unstable correspond to the form constants when transformed to the visual field using the retino-cortical map. The results are sensitive to the detailed specification of the lateral connectivity and suggest that the cortical mechanisms that generate geometric visual hallucinations are closely related to those used to process edges, contours, surfaces, and textures.

When a cortical neuron is repeatedly injected with the same fluctuating current stimulus (frozen noise) the timing of the spikes is highly precise from trial to trial and the spike pattern appears to be unique. We show here that the same repeated stimulus can produce more than one reliable temporal pattern of spikes. A new method is introduced to find these patterns in raw multitrial data and is tested on surrogate data sets. Using it, multiple coexisting spike patterns were discovered in pyramidal cells recorded from rat prefrontal cortex in vitro, in data obtained in vivo from the middle temporal area of the monkey (Buracas et al., 1998) and from the cat lateral geniculate nucleus (Reinagel and Reid, 2002). The spike patterns lasted from a few tens of milliseconds in vitro to several seconds in vivo. We conclude that the prestimulus history of a neuron may influence the precise timing of the spikes in response to a stimulus over a wide range of time scales.

The spike timing reliability of Aplysia motoneurons stimulated by repeated presentation of periodic or aperiodic input currents is investigated. Two properties of the input are varied, the frequency content and the relative amplitude of the fluctuations to the mean (expressed as the coefficient of variation: CV). It is shown that, for small relative amplitude fluctuations (CV approximately 0.05-0.15), the reliability of spike timing is enhanced if the input contains a resonant frequency equal to the firing rate of the neuron in response to the DC component of the input. This resonance-related enhancement in reliability decreases as the relative amplitude of the fluctuations increases (CV-->1). Similar results were obtained for a leaky integrate-and-fire neuronal model, suggesting that these effects are a general property of encoders that combine a threshold with a leaky integrator. These observations suggest that, when the magnitude of input fluctuations is small, changes in the power spectrum of the current fluctuations or in the spike discharge rate can have a pronounced effect on the ability of the neuron to encode a time-varying input with reliably timed spikes.

Oscillations and noise are ubiquitous in physical and biological systems. When oscillations arise from a deterministic limit cycle, entrainment and synchronization may be analyzed in terms of the asymptotic phase function. In the presence of noise, the asymptotic phase is no longer well defined. We introduce a new definition of asymptotic phase in terms of the slowest decaying modes of the Kolmogorov backward operator. Our stochastic asymptotic phase is well defined for noisy oscillators, even when the oscillations are noise dependent. It reduces to the classical asymptotic phase in the limit of vanishing noise. The phase can be obtained either by solving an eigenvalue problem, or by empirical observation of an oscillating density's approach to its steady state.Introduction. Limit cycles (LC) appear in deterministic models of nonlinear oscillators such as spiking nerve cells [1], central pattern generators [2], and nonlinear circuits [3]. The reduction of LC systems to onedimensional "phase" variables is an indispensable tool for understanding entrainment and synchronization of weakly coupled oscillators [4,5]. Within the deterministic framework, all initial points converge to the LC, on which we can define a phase that progresses at a constant rate (θ = ω LC = 2π/T LC ). The phase θ(x 0 ) of any point x 0 is then defined by the asymptotic convergence of the trajectory to that phase on the LC. However, stochastic oscillations are ubiquitous, for example in biological systems [6], and in this setting the classical definition of the phase breaks down. For a noisy dynamics, all initial densities will converge to the same stationary density. Thus the large-t asymptotic behavior no longer disambiguates initial conditions, and the classical asymptotic phase is not well defined.Schwabedal and Pikovsky attacked this problem by defining the phase for a stochastic oscillator in terms of the mean first passage times (MFPT) between surfaces analogous to the isochrons (level curves of the phase function θ(x)) of deterministic LC [7][8][9]. Here we formulate an alternative definition that is tied directly to the asymptotic behavior of the density, rather than the first passage time, and is grounded in the analysis of the forward and backward operators governing the evolution of system densities. Our operator approach leads to two distinct notions of "phase" for stochastic systems. As we argue below, the phase associated with the backward or adjoint operator is closely related to the classical asymptotic phase.General framework. Consider the conditional density ρ(y, t|x, s), for times t > s, evolving according to the forward and backward equations