Billock and Tsou. 10.1073/pnas.0610813104. |
Fig. 6. Static (single frame) versions of a few of the dynamic fractal patterns used in Exp. 3. Each fractal has a spatial amplitude spectra (fs-b). From top to bottom, the b values are 0.4, 0.8, 1.2, 1.6, and 2.0. The dynamic sequences we used also have power law temporal spectra (ft-a), and as a consequence the pixels appear to be in constant motion. If these noise patterns are viewed through concentric circular, spiral, or fan-shaped transparencies, various illusory percepts can occur. For low a and b (white noise) the fractal noise appears to stream in directions orthogonal to the geometry of the overlaying pattern (a variation of the MacKay effect). If b is increased, actual patterns form; large shadowy but distinct illusory radial spokes appear to be seen rotating over the noise and physical concentric circles; noisy pulsating circles form on the physical fan shapes. The rotation/pulsation is faster for low a than high.
SI Text
Producing Classroom Demonstrations of Photopically Induced Hallucinations.
It is easy to produce flicker-induced hallucinations and to replicate the findings of Exp. 1 for classroom demonstrations. Obtain a stroboscope like those sold in RadioShack or Spencer's Gifts for about $20.00. The best strobes for this purpose have a smooth mirror behind the flash bulb and a clear plastic safety shield in front of the bulb.
To Demonstrate Random Hallucinations.
Hold the strobe close to the subject's face and have the subject watch the strobe flash through closed eyelids (or flicker illuminate any relatively uniform surface; the matte hemispheres used in ERG experiments work beautifully for this, but so does a movie screen or a flickering CRT). After the subject reports shapes and colors, ask him/her to place one hand over one eye. The subject will often report that the hallucinatory pattern changed as well (generally it becomes simpler). [Other arrangements are possible: Purkinje discovered these hallucinations in 1819 by fanning his spread fingers between his closed eyes and a gas light (35). Brewster reported similar effects in 1834, which he found by dashing past a high sunlight fence with his eyes closed (36), an experiment hard to understand but easy to replicate by passengers in cars driving on tree-lined streets.]
To Produce Flicker-Induced MacKay Effects.
Enlarge the fan-shaped and concentric patterns of Fig. 1, or, better yet, use one of MacKay's higher spatial frequency patterns from ref. 22 (higher spatial frequencies work better for this particular effect). In a dark room illuminate the figures with a stroboscope. The physical patterns will appear to be overlaid by illusory patterns roughly orthogonal to the physical patterns. MacKay suggested that the physical pattern is seen in the flash of the strobe and the orthogonal illusion is an opponent aftereffect seen in the dark period after the flash, an idea reinforced by MacKay's work with afterimages (22, 23). MacKay's notion of geometric opponency via sequential contrast formed the basis for understanding the simultaneous contrast effects seen in Exp 1.
To Produce Biased Hallucinations Like Those Seen in Exp 1.
Photocopy the fan shape and concentric circular patterns from Fig. 1 onto ordinary white paper. Use the magnify/minify feature of the photocopier to size the stimuli: good results are obtained from stimuli that are 1/10th to one-third of the flash area. Use the strobe to back-illuminate the stimuli and surrounding field. Good results are obtained in a dark room with this stimulus. Have the subjects hold the strobe about an arm's length away and use a flicker rate of 10-15 Hz. Some subjects will prefer a different distance or flicker rate; cheap strobes often do not generate flicker over 10 Hz. If the central figure on the flickering field is a set of concentric circles, subjects will generally report an illusory fan shape rotating in the flickering area. If the central figure is a fan shape, subjects will generally report pulsing or wobbling illusory circles. Illusory fans are generally more salient than illusory circles. For this stimulus configuration, the hallucination may extend through the central stimulus. To stop this from happening, place an opaque disk the size of the central figure behind it to block the light from the strobe. Then raise the room light so that the central area can be seen and flicker the surrounding area as before. Of course, if a CRT is used instead of a strobe, it is straightforward to not flicker the biasing area. CRTs also enable other possibilities which have pedagogical uses. For example, one can create a stimulus that simultaneously demonstrates Fechner-Benham subjective colors and induced hallucinations by flickering the biasing field out-of-phase with the background field. This not only prevents the induced hallucination from extending through the biased area, but also results in the biasing field becoming colored; the illusory color of the biasing field can be changed by manipulating the phase lag between the two flickering fields (the same principle was once used to present color images on black-and-white TVs). For this demonstration, the flicker rate used is a compromise between the optimum flicker rate for Fechner-Benham colors (about 4-10 Hz) and the optimum flicker rates for induced hallucinations (about 8-15 Hz).
To Produce the Classic Noise-Induced MacKay Effect.
Create a large fan-shaped or concentric circular pattern [perhaps by magnifying the patterns in Fig. 1, or use Mackay's (22, 23) or generate one of your own with any desired spatial frequency] and print as a transparency. Tape the transparency over the screen of a television and find a channel containing only static. MacKay found that the static seen through the open areas of the transparency would appear to move perpendicular to the opaque features of the transparency and sometimes define an orthogonal shape. In informal experiments using this method we found that the orthogonal motion effect was more apparent on some empty TV channels than others and that no illusory shapes formed, suggesting that perhaps the statistics/contrast of the static were important. We also saw cases in which the apparent motion was parallel to features of the transparency, so that the noise appeared to flow coherently in the channels defined by the open areas of the transparency. [As a matter of historical interest, MacKay experimented with generating different noise statistics but was limited by the technology of his day; e.g., using a moving sandpaper belt (or a movie of the same) and varying the coarseness of the sandpaper.]
To Demonstrate Hallucinations Seen in Fractal Noise.
A computer program for demonstrating the phenomena described in Exp. 3 is available on request from V.A.B. The user can create an array of small dynamic fractals of various spatial and temporal statistics, each covered by a virtual transparency and determine by inspection which noise generates the strongest pattern formation.
Background on Hallucinatory Pattern Formation
Although some mental conditions and intoxicative states lead to cognitively interesting hallucinations (faces, landscapes, etc.), here we deal only with simple geometric hallucinations. Such hallucinations may be a simple as a spiral or as complicated as an oriental carpet. Geometric hallucinations have been recorded for the early phases of LSD, mescaline, and THC intoxication, as well as in hypoglycemia, migraine, epileptic seizure, and synesthesia (11, 13, 14, 15, 18, 37). They can be induced by transcranial magnetic stimulation (TMS), bilateral pressure on the eyes, and by flickering light (11, 12, 35-50). They can also be induced by alternating current through the eye or cortex; the former emulates the effects of flicker, and the latter is functionally equivalent to TMS (an alternating magnetic field induces an electric current). Klüver (37) summarized the extensive literature on some of these conditions and pointed out that many similar hallucinatory patterns are produced across conditions. Among these were lattices (including rhomboidal and hexagonal forms), cobwebs, polar symmetric shapes (tunnels, funnels, cones, fan shapes, and concentric circles) and spirals. There are also specific hallucinations that we associate with particular states but can be experienced in other states. For example, the fortification aura that often accompanies migraine headaches is marked by a moving scotoma outlined by bright flashing line segments that are set at sharp angles to their nearest neighbors, resembling a particular style of medieval fortification architecture (13, 18, 51). The size of the line segments scales beautifully with what we would expect from the cortical magnification factor and can in fact be used to estimate retinocortical scaling (52,53). However, similar jagged spatial structures in both geometry and scaling are sometimes seen in flicker-induced hallucinations, minus the bright flashes of the segments. Moreover, migraine sufferers do report geometric hallucinations characteristic of Klüver's system (18), just not as often as they report fortifications. Hypoglycemia sufferers also report both Klüver's characteristic shapes, as well as fortification illusions. A variety of evidence points to a cortical origin for these hallucinations, including the ability to induce geometric hallucinations in blind subjects whose eyes are nonfunctional or enucleated.
Since the discovery of flicker-induced hallucinations (35, 36), the phenomenon - often called Purkinje's colors in the older literature - has been studied sporadically (38-50), using a variety of experimental techniques, including rotating patterns (38-39) and strobes, either viewed through closed eyelids or on a Ganzfeld. Most such studies suggest that flicker-induced hallucinations are similar to (but generally not as striking or as stable as) the better studied drug hallucinations. Both migraine- and mescaline-induced hallucinations are often described as being exceptionally bright, with fine lines that are described as being like glowing filaments (18, 37). Color is more often reported in drug-induced hallucinations than in any other type of hallucination, and the colors are often described as being exceptionally saturated. In flicker-induced hallucinations, the patterns that are produced are often like the superposition of simpler drug-induced states and the colors are often desaturated, both conditions that would be expected from a more multistable system [see Williams et al. (54) on this effect in Brewster's illusory colors]. Flicker-induced hallucinations often seem to be the superposition of many simpler patterns; somewhat simpler patterns can be achieved by flickering one eye instead of using binocular stimulation. However, in monocular vision, both Brown and Gebhard (40) and Smythies (41-44) note an alternation between hallucinations typical of Klüver's system and less geometric patterns, such as colored and swirling particulate patterns, that seem to be seen in the nonstimulated eye. Because of a similar alteration between the stimulated and unstimulated eye in binocular rivalry experiments, these nongeometric patterns are sometimes called "dark phase patterns." In our experiments, the effects we reported occurred for both monocular and binocular stimulation, which were conducted with binocular stimulation and thus the dark phase effect was not a factor; however, it may be worth noting that, in Exp. 2, the swirling textures and storm-like patterns seen in the dark square defined by the apparent motion are reminiscent of the dark phase patterns found by Brown and Gebhard (40) and Smythies (41-44).
A Brief Tutorial on Pattern Formation in Physical and Biological Systems
There are many physical and biological systems that form patterns autonomously (55). They may require outside stimulation, but the stimulation can be spatially uniform, and the pattern that forms is thus more a property of the system rather than of the stimulus. Two classic examples are pattern formation in fluid dynamics and on animal coats. The former system is particularly simple.
A Physical Example: Patterned Convection in Fluid Dynamics.
Suppose we have a pan of oil that is heated from below on a stove and cooled by the air above. Physical intuition tells us that the hot oil will rise and the cold oil will sink. If the temperature difference is small, this will happen at random throughout the pan. However, if the temperature difference is big enough, the convection currents will start reinforcing one another, until a stable pattern emerges. It turns out that two kinds of patterns are relatively stable. (i) The currents can organize themselves into rotating cylinders of oil. For example, one cylinder of oil will be rising from the right and falling toward the left (counterclockwise). The cylinder to its left will form parallel to the first and its currents will rise from the left and fall to the right (clockwise, so the two streams of falling liquid reinforce one another). Seen from above, the rotating cylinders look like stripes. Once they form, they are relatively stable, but nothing usually dictates what orientation they form in; it is a matter of chance that depends both on the initial currents that arise and the ways that they happen to reinforce or cancel one another. The transition from non-pattern-forming to pattern-forming is called a bifurcation, and the heat difference it occurs at is called a critical value. (ii) Although cylinder orientations will compete with one another for domination, it is possible under some conditions (e.g., high surface tension) for three cylinder patterns at 60° angles to superimpose. Seen from above, this is a hexagonal pattern of cells, with the fluid rising in the center of each cell and falling along its edges. The individual hexagons are known as convection cells and are important in the theory of meteorology. Similar patterns using related mechanisms have been described in laser modes, geophysical systems, cellular automata and neural networks (21, 55).
A Biological Example: Turing's Model of Pattern Formation on Animal Coats.
Turing (56) wondered what governed the formation of stripes and spots on the coats of animals. He found that if at least two chemical species which could affect pigment production were allowed to spatially interact via diffusion, a great variety of spot and stripe patterns could be produced that depend on initial conditions. The simplest such system requires an "activator" species and an "inhibitor" species, which interact asymmetrically and which diffuse at different rates (the so-called Turing instability).
∂A/∂t = F(A,B) + DA▼2A [1]
∂B/∂t = G(A,B) + DB▼2B
Where A and B are the concentrations of the activator and inhibitor species, respectively, with diffusion rates DA and DB; F(A,B) and G(A,B) are (generally) nonlinear functions that describe reaction kinetics (asymmetric activator-inhibitor interactions); and ▼2[A or B] is the Laplacian (second spatial derivative) operator; e.g., ¶2A/¶x2 + ¶2A/¶y2. The primary condition for pattern formation is that DA¹ DB. Murray (57), applying Turing's reaction-diffusion theory, illustrates how variations in initial conditions lead to the various patterns seen in animal coats. Similar models explain patterns that form on sea shells (58, 59). In such models the rate of diffusion is controlled by the product of the diffusion constant (a property of the system) and the concentration gradient (as quantified by the second spatial derivative). Something like a Nth spatial derivative is found in every pattern-forming reaction-diffusion-like system, but it is not always apparent. For example, in neural network versions, the Nth order derivative can be implicit in the interplay between excitation and inhibition; a "Mexican hat' or difference of Gaussians spatial weighting function is functionally equivalent to a second derivative of a Gaussian, while the receptive fields of simple cells can be modeled by higher-order derivatives. Recently Billock et al. (60) used reaction-diffusion theory to model gradients of illusory forbidden colors (reddish greens and greenish reds) seen in retinally stabilized equiluminous images. With some slight modifications it can be used to model spatial patchiness in interacting species or in disease spread (57). For example, in predator-prey interactions, there are two kinds of asymmetries that can lead to species patchiness: (i) the prey feeds the predators while the predators do nothing directly to benefit the prey, and (ii) the predator and the prey have different territorial ranges. The versatile but confusing concept of pattern formation via the Turing instability can be made clear using Murray's parable of the sweaty grasshopper (57):
"Consider a field of dry grass in which there is a large number of grasshoppers which can generate a lot of moisture by sweating if they get warm. Now suppose the grass is set alight at some point and a flame front begins to propagate. We can think of the grasshopper as an inhibitor and the fire as an activator. If there was no moisture to quench the flames the fire would simply spread over the whole field which would result in a uniform charred area. Suppose, however, that when the grasshoppers get warm enough they can generate enough moisture to dampen the grass so that when the flames reach such a premoistened area, the grass will not burn. The scenario for spatial pattern is then as follows. The fire starts to spread - it is one of the 'reactants', the activator, with a 'diffusion' coefficient, DF say. When the grasshoppers, the inhibitor 'reactant', ahead of the flame front feel it coming they move quickly well ahead of it - that is they have a 'diffusion' coefficient DG say, which is much larger than DF. The grasshoppers then sweat profusely and generate enough moisture and thus prevent the fire from spreading into the moistened area. In this way the charred area is restricted to a finite domain which depends on the 'diffusion' coefficients of the reactants - fire and grasshoppers - and various 'reaction' parameters. If, instead of a single initial fire, there was a random scattering of them we can see how this process would result in a final spatially inhomogeneous steady state of charred and uncharred regions in the field, since around each fire the above scenario would take place. It is clear that if the grasshoppers and the flame front 'diffused' at the same speed no such spatial pattern could evolve."
In a conservative system (a system with no outside inputs) pattern formation generally reaches a final steady state, although this does not have to happen quickly. For example, in some chemical systems of three or more interacting reactants, the patterns can evolve in a multistable fashion for as long as there are reactants left to interact. In nonconservative systems, in which there are outside resources being introduced into the system (a category that includes neural systems), there is no steady state requirement, and the patterns may evolve continuously. This is especially relevant to flicker-induced hallucinations, in which every flash is a perturbation and the percept is likely an evolving superposition of previously and currently elicited states. Indeed, flicker-induced hallucinations are generally far more unstable and complicated than their drug-induced counterparts, despite evidence (below) for a common mechanism. However, it does not mean that the system cannot be stabilized or biased to a single simple pattern under some conditions. A good example is biased pattern formation in convection (see below).
The Ermentrout and Cowan Model and Related Ideas.
As discussed above, Klüver (37), comparing mescaline illusions to illusions produced under many different conditions (e.g., epilepsy, hypoglycemia, and migraine), found that some patterns were particularly common. These include patterns with radial symmetry (e.g., fan shapes), patterns with circular symmetry (e.g., bull's-eyes), spirals, and lattices, either square, rhomboidal or hexagonal. Similar patterns have also been demonstrated for transcranial magnetic stimulation, ocular pressure (applied simultaneously to both eyes), and flicker. Ermentrout and Cowan (14) realized that that after compensating for the nonlinear retinocortical mapping (a complex logarithmic conformal mapping; see refs. 53 and 61 for a detailed analysis), fan shapes, concentric circles, and spirals all correspond to gratings of activity on V1, each differing only by orientation (e.g., Fig. 1). For example, a bull's-eye pattern and a fan-shaped pattern on the retina map to cortex as mutually orthogonal gratings (for direct imaging evidence of this specific mapping see ref. 62), as do clockwise and counterclockwise twisted spirals. This means that if a grating of activity could be produced on cortex, then it would look like a fan pattern, or a spiral or a bull's-eye shape, depending on the orientation it happened to have. Moreover, lattice patterns could be modeled by superimposing two or three grating patterns at 90° or 60° angles (compare to the convection example above).
What remains then is to show that neural networks could generate stripe patterns of various geometries. One model that generates such stripes is the Ermentrout and Cowan (14) neural network, based on Wilson and Cowan's models of interacting excitatory and inhibitory cortical neurons. The model is a set of coupled integro-differential equations; each pair of which represent on particular location in cortex.
dE/dt = -E + SE{aWEE*E - bWIE*I + Sensory Input} [2]
dI/dt = -I + SI{cWEI*E - dWII*I}
Where SE and SI are sigmoid compressive nonlinearities and the W*E and W*I terms indicate spatial convolutions with lateral inhibitory and excitatory spatial weighting functions [e.g., W(x,y) *E(x,y) = òòW(t1,t2)E(x-t1, y-t2)dt1dt2, where t1 and t2 are dummy variables of integration]. If the level of spatially uniform excitation is sufficiently high, Ermentrout and Cowan find that these networks give rise to parallel stripes of neural activity and inactivity that can form at any orientation. Note that there are two potential routes to raising the overall level of spatially uniform excitation. The first, used by Ermentrout and Cowan and some of their successors to model mescaline hallucinations and migraine aura, involves increasing the value of a (and perhaps c) in Eq. 2 (the options and possibilities for this are discussed further below). The other route to increasing spatially uniform excitation is to provide a spatially uniform input, via the Input term in Eq. 2. Most visual neurons have little response to temporally static spatially uniform stimuli, which provides a possible basis for understanding why many stimuli that produce hallucinatory pattern formation are temporally phasic (e.g., flicker, pulsed transcranial magnetic stimulation, alternating electric current stimulation of the retina). The interplay between increasing excitation via the coupling constant a and the Input term in Eq. 2 suggests that conditions that drive one of these two terms should interact synergistically with conditions that drive the other. Some evidence for this is adduced below.
Why does the Ermentrout and Cowan model produce stripes? Recall the two pattern forming systems discussed above. In the case of reaction-diffusion, the pattern formation is made possible by the Turing instabilities - the asymmetry between the diffusion rates for the activator and inhibitor species, and the asymmetry in the interactions between activator and inhibitor. Strongly analogous asymmetries occur in the Ermentrout and Cowan system: (i) excitatory cells excite both excitatory and inhibitory cells, while inhibitory cells inhibit both excitatory and inhibitory cells; and (ii) the spatial reach of excitatory and inhibitory interactions is not equal. Note that Eq. 1 has four spatial weighting functions, each defining the effects of neighboring excitatory and inhibitory cells on one another. Ermentrout and Cowan sensibly modeled these weighting functions as Gaussians of different variances. It is possible to treat Eq. 1 as a mathematical version of an electrophysiological experiment and probe its response to spots or lines, just as an electrophysiologist would measure a line-spread or point-spread function. The original Ermentrout and Cowan model (14, 15), which produces stripes of cortical activity, has excitation predominate at both short and long spatial ranges, with inhibition predominating at moderate ranges. Such a network has a line spread function with an excitatory peak, two smaller symmetrically flanking inhibitory troughs, and two still smaller flanking excitatory peaks. Graphically, it looks like a symmetric Gabor function, although a fourth derivative of a Gaussian would be a better fit. Physiologically, it looks like an even-symmetric simple cell receptive field. A line is good stimulus for such a weighting function, but a grating patch is an even better fit. Now imagine such networks being randomly stimulated all over cortex, and interacting with their neighboring cortical networks, much as the rising and falling currents in the convection example do. Some network interactions reinforce: a stripe of excitatory activity next to a stripe of inhibitory activity. In the absence of governing initial conditions or boundary conditions, it is not surprising that the Ermentrout and Cowan network forms gratings of activity on cortex, whose orientation, like in the convection example, cannot be determined in advance. It is also possible for two or more stripe patterns to reinforce one another in a stable fashion. The two most likely combinations are for three orientation patterns at 60° angles from one another, forming a hexagonal lattice and two orientation patterns combining at 90° angles, forming a square or plaid pattern. After compensating for the nonlinear retinocortical mapping, these cortical patterns predict some cobweb- and lattice-like patterns reported during hallucinations. The Ermentrout and Cowan model has limitations, and in its simplest form cannot explain biased pattern formation or some of the more complicated hallucinatory patterns that are seen. The next two sections will address these points in more detail.
Arieli (1) and Ringach (6) comment that some of the most interesting aspects of autonomous pattern formation are the ways that the pattern formation might interact with stimulus-driven perception. This concern drove our own research. There have been very few studies of this in the pattern formation literature, but one such study suggested to us that a physical stimulus could be used to bias the resulting pattern formation. Bestehorn and Haken (63) studied the convection system that we introduced above. As discussed, the cylinders of rising hot and falling cold oil (rolls) can form in any orientation. In simulations of convection currents they found that injecting a single line of current along a single orientation was extremely effective in biasing the formation of all of the surrounding rolls to that orientation (see Fig. 2). This motivated us to explore using small physical stimuli embedded in a large flickering field. The large flickering fields would promote random pattern formation, while the small embedded shapes would function like the current injections in Bestehorn and Haken's study - biasing the emerging pattern. Because perceptual fan shapes and concentric circles map to orthogonal stripe patterns of neural activity on cortex, we expected the physical fan to induce more parallel stripes of activity on cortex and to extend the length of the stripes induced by the physical stimulus. So, for example, we expected that flickering around a small fan shape might induce a larger fan shape, while flickering around a set of concentric circles might induce larger illusory circles to form concentric to the physical circles. Hence, the formation of illusory fan shapes in the flickering field around physical circles (and vice versa) was unexpected.
Elaborations of the Ermentrout and Cowan Model and Related Models.
Variations of the Ermentrout and Cowan model have been made that use a single neural population with mixed excitation and inhibition and still produce similar results. Although interesting, these do not concern us here. Some more relevant modifications and elaborations are considered below.
(i) Tass (64) studied the case where weighting functions WEE and WEI have equal spatial ranges and differ, if at all by a scaling constant. Similarly, WII and WIE equal spatial ranges. The spatial weighting function for the network is then a simple difference-of-Gaussians, which can be modeled as the second spatial derivative of a Gaussian, exactly like the Laplacian term of a reaction-diffusion model. As would be expected from Turing's stripe producing reaction diffusion systems, this reduction from a fourth derivative to a second derivative seems to have little effect on the system's ability to produce patterns.
(ii) Ermentrout and Cowan (14) studied the case where excitation is increased, which can be done in two ways: by increasing a - the weight of WEE (and perhaps c - the weight of WEI as well) or by increasing the value of the Input. The first case, being internal to the system, seems especially suitable for thinking about drug-induced hallucinations and cortical spontaneous activity, while the second seems more suitable for thinking about TMS-induced and photopically induced hallucinations. However, from the near symmetry of the equations, it would seem like a similar result could be obtained by decreasing inhibition, which Tass (64) has used as a model of peit mal epileptic hallucinations. Tass (17) examined pattern production for every combination of increasing or decreasing the four excitatory and inhibitory terms in Eq. 2. He found parameterizations where winner-take-all dynamics reigned and only one hallucinatory state (dictated by random initial conditions) could emerge. He also found parameterizations where as many as four states could coexist, and parameterizations that lead to alternations between states.
(iii) Similarly, Tass (17) exampled the effect of strengthening or weakening the coupling between the excitatory and inhibitory elements of the network. Weaker coupling favor simple and stable geometric hallucinations, while stronger coupling leads to oscillations between competing states (e.g., perception of fan shapes alternating with perception of competing circles). Tass (64) notes that adding a noise input results in concentric circles that pulsate and fans that rotate. This seems relevant to our own study, in which rotating fans were seen in all three experiments, while pulsating circles were occasionally seen in Exp. 1 but were always seen in Exp. 3. In our Exp. 1, the only noise would be internal, but Exp. 3 featured externally supplied noise.
(iv) The spatial weight functions used by Ermentrout and Cowan (14) are spatially symmetric and the stripe formation arises from local reinforcement by neighboring members of the network, with allowable patterns being any doubly periodic system that tiles the visual cortex. However, this kind of model is not theoretically exhaustive; some recent models of neural networks allow for asymmetric directional connectivity on cortex (19, 20, 65-67). Although this sort of network has not yet been used for modeling hallucinatory pattern formation, it would seem to be the kind of network that would be useful in modeling the spatial opponency of our present results. One can imagine a set of Ermentrout-Cowan-like networks, each biased to form stripes in horizontal, vertical or several oblique angles along V1, with competition between these networks for the right to fire. Such a system could produce opponency at a distance, if the vertical and horizontal stripe producing networks had mutual geometrically antagonistic long range connections preferential arranged along particular directions on V1 [much like the connectivity envisioned by Mitchison and Crick (68)]. An anisotropic model would also facilitate modeling why the present experiments favor production of fans and concentric circles over spirals, when isotropic models show no such preference.
(v) A very specific instantiation of an antistrophic pattern production network has recently been considered by Bessloff et al. (19, 20) which serves to illustrate the power of an anisotropic connection approach. In this case, the specificity of the connections is in how the connections are labeled for perception. They note that not all reported hallucinations can be accounted for by the Ermentrout and Cowan model. For example, there are some hallucinations in which a texture is superimposed on a geometry; e.g., a rhomboidal lattice can form on a circular or spiral structure (for a beautiful illustration see ref. 69). It has been shown that such hallucinations can be accommodated by generalizing the Ermentrout and Cowan model for orientation. In each V1 cortical column are cells tuned to particular geometries. If each oriented population belongs to a separate Ermentrout and Cowan network, then the patterns that arise will be textured because all of the activated cells will be labeled for orientation. The orientation of the stripe patterns determines the overall geometry of the illusion, and the orientation labels of the activated cortical cells within the network determine the texture that is attached to the geometry. If populations labeled for particular orientations inhibit one another, or otherwise compete for the ability to fire, the resulting stripes of activity need not have the same orientation label and richly complicated percepts can result.
(vi) Although probably relevant not here, for completeness we should mention that there is another route that produces spiral pattern formation without resorting to oblique stripe formation on cortex. Some systems, like chemical reaction-diffusion systems, and cardiac muscle can produce spreading scroll-like wavefronts that viewed in two dimensions are spirals. Fohlmeister et al. (70) has modeling an Ermentrout and Cowan-like system that produces both cortical stripes and cortical spirals, which can grow from a single point. Often the only way to distinguish a cortical spiral from an oblique cortical grating is that the scroll-wave spiral is an Archimedean spiral, while the oblique cortical grating results in a logarithmic spiral percept. Also, because scroll waves grow out of a single origin, more than one can emerge and they can merge into more complicated patterns. We doubt that a Fohlmeister-like mechanism is important in our own experiment because, as Fohlmeister notes, it results in spirals being favored over all other percepts, while in our studies spirals were the least favored. Schwartz (53) notes another source of highly localized pinwheel like spirals sometimes seen in place of phosphenes during point stimulation of cortex: the local structure of orientation slabs in a ocular dominance column, if activated in sequence by a spreading wavefront, should mimic a rotating highly localized spiral.
(vii) It has recently been shown that reaction-diffusion systems can show stripe-direction opponency strikingly reminiscent of that which we found in Exp. 1. Interestingly, the phenomenon involves anisotropic diffusion that seems quite analogous to the anisotropic neural networks just discussed. Shoji et al. (71) studied two specific cases in which the diffusing activator and inhibitor species have preferred directions of diffusion and the inhibitor has greater spatial range. (1) If both activator and inhibitor have the same preferred direction of diffusion, then stripes form parallel to the diffusion of the activator species, until the activator reaches the limit of its range. At this point, the stripes continue to form, but perpendicular to the first set of stripes, and the switch is abrupt. (2) In the other case, the activator and inhibitor have different directional diffusion preferences. The same behavior results but the switch in the orientation of the stripes is more gradual. The first case, which resembles the neural phenomena of disinhibition, seems closer (in the patterns formed) to our results in Exp. 1 and could potentially be modeled by a system of long-range anisotropic inhibition combined with shorter range anisotropic excitation stimulated by the Input term in Eq. 2. The closely related Ermentrout and Cowan models (14-20) already has the range asymmetry built into it and would only require asymmetric orientation preferences on cortex to function like the Shoji model. [Note: cortical orientation preference in this context means preference of direction of neural connections across physical cortex (65) and does not refer to orientation preference in retinotopic coordinates, as in the Bressloff et al. (19, 20) models.] To model the various cortical grating patterns underlying the different geometric percepts, there would have to be several of these anisotropic Ermentrout and Cowan networks, each with a different cortical orientation preference, and competition between them.
Synergies Between Pattern Formation Mechanisms.
Aside from the ubiquity of the Klüver patterns (37), which argue for a common mechanism for many induced hallucinations, there are other findings suggestive of a common mechanism. Sometimes conditions that lead to pattern formation appear to have additive effects. For example, consider mescaline and flicker-induced hallucinations. Mescaline hallucination, although similar to flicker-induced hallucinations, are reported to be more vivid (37, 44). Smythies reports that subjects given a subclinical dose of mescaline (not enough to hallucinate) immediately experienced mescaline-quality geometric hallucinations when exposed to flickering lights (44). The flicker modulates the Input term in Eq. 2, while Ermentrout and Cowan (14) model mescaline as increasing the a term in Eq. 2. A glance at Eq. 2 shows that the two terms sum (albeit subadditively because of the saturating nonlinearity SE). If there is the critical level of stimulation required for pattern formation, it might be useful to think of mescaline dose as a pedestal of stimulation to which a periodic stimulus is added. If the sum of the stimuli crosses the critical level, the pattern formation can occur. Such a model could also account for the usual temporal instability of flicker-induced hallucinations; every flash acts as a perturbation, whereas stimulation induced by mescaline remains at a steady, dose-dependent level. Similar combination effects have been reported for triggering hallucinatory events: (i) Shevelev et al. (72) recorded from cortex while using flicker to trigger geometric hallucinations through closed eyelids. They report that on trials in which the flicker was synchronized to the patients' alpha rhythm, pattern formation took 2-5 seconds, compared to 10-15 seconds for nonsynchronized trials. (ii) Siegal (10) reports a patient who experienced LSD "flashbacks" triggered by flickering neon lights after heavy nicotine use. (iii) Several studies have shown that migrainers between episodes have reduced thresholds for phosphene induction by transcranial magnetic stimulation (for a review see ref. 73). (iv) Conversely, flicker is a trigger for some patients who experience hallucinatory "auras" with migraine or epileptic seizure. Most of these cases make sense if considered as an interaction between the excitability parameter a and the Input term of Eq. 2.
Effects of Noise: 1/f Noise and Stochastic Resonance in Spatiotemporal Pattern Formation
A Variety of Roles for 1/f Noise in Pattern Formation.
In Exp. 3, replacing white noise with fractal (1/f) noise in the MacKay effect has a dramatic effect: induced geometric hallucinations become much more visible. There are many links between 1/fa noise and pattern formation. (i) Usher et al. (74), studying pattern formation in a neural network, find that the emergent correlations decay over time in a manner consistent with 1/f dynamics. (ii) Kelso (75), reviewing multichannel EEG and dense-array SQUID MEG studies, states that "from a relatively incoherent or rest state (with a 1/f distribution of component frequencies) the brain manifests coherent spatiotemporal patters immediately [when] it is confronted with a meaningful task." (iii) Busch and Kaiser (28, 76) report that 1/f noise is more effective than white noise in driving some kinds of spatiotemporal pattern formation. The first is more descriptive than explanatory, and not directly applicable to Exp. 3. The second could suggest that the pattern is synthesized from the spectral contents of the noise, roughly as Fourier might have suggested. The third is an atypical example of a phenomenon known as stochastic resonance and holds promise for understanding the results of Exp. 3. This section reviews stochastic resonance in sensory systems, pattern formation, and 1/f noise.
Types and Mechanisms of Stochastic Resonance.
Stochastic resonance (SR) is defined as a qualitative change in the behavior of a system with the addition of noise (77, 78). There are three basic kinds of SR: (i) Adding noise can make a signal more detectible. (ii) Adding noise can make a stable system multistable. (iii) Adding noise can make a system generate an autonomous behavior, like forming a pattern (79, 80). Such a pattern can be one-dimensional, like a time series, or multidimensional, e.g., in an excitable medium the patterns can form in all three spatial dimensions and vary over time as well. There are many proposed mechanisms for stochastic resonance. Here we focus on several that are implicated in stochastic resonance of the first kind: (i) adding noise can make a signal more detectable because of a hard threshold, which signal plus noise combine to overcome. This kind of stochastic resonance seems rather prosaic but can have a rather large impact because the system's output signal is tightly correlated with the input signal and has very little noise component. Thus, a true increase in signal-to-noise ratio is obtained and can be quite large. Indeed, in most noise studies, the recognized signature of SR is an increase in signal-to-noise for a limited range of noise strengths (usually measured from the statistical variance or standard deviation of the noise). The reason for this is clear: too little noise does not boost the signal above threshold, while too much can trigger a system response to the noise alone. (ii) Another mechanism that has been identified is a soft threshold, like the logistic response of many psychophysical observers. (iii) In the older literature, stochastic resonance was associated with oscillatory entrainment; e.g., the stochastic signal triggers an oscillation in the system of characteristic frequency and phase. The noise also triggers oscillations, but these couple to the oscillation induced to the signal and become entrained to it, e.g., the oscillations induced by the noise shift their phase and frequency to match the signal-induced oscillations. The synchronized oscillations combine to have a greater amplitude than that induced by signal alone, yet they are indistinguishable by frequency or phase from those induced by signal alone, and so constitute an amplification.
Evidence of Stochastic Resonance in Various Sensory Systems Including Vision.
Behavioral and neural demonstrations of stochastic resonance have been found in a variety of sources. Greenwood et al. (81) found that noise in the electrosensory system of paddlefish enhanced the sensitivity of that sensory system. Collins et al. (82) found an SR effect with the application of vibratory noise on information transmission in rat cutaneous mechanoreceptors, and similar effects have been documented in crayfish and cricket (83, 84). Collins et al. (85, 86) have found evidence of vibratory enhancement in human tactile sensitivity. Similar results have been demonstrated for absolute thresholds in human hearing (87). A model for this enhancement (based on the biophysics of hair cells) has been proposed by Wisenfeld and Jaramillo (78). There are several studies that find indications of stochastic resonance in human vision. Ditzinger et al. (88) used autostereopsis to demonstrate a SR effect of the second kind; perception of harder-to-see percepts in multistable visual perception was enhanced by the addition of noise. Riani and Simonotto (89) modeled stochastic resonance in a bistable visual system. A few studies have looked at signal enhancement. Kitajo et al. (90) used an unusual design in which the brightness of a sinusoidal signal presented in the right eye was crossmodally matched via a handgrip strength measurement. Random noise was presented in the left eye. Inferred brightness of the sinusoid was enhanced for intermediate levels of noise, a SR-like effect, with the interocular noise transfer indicating a central locus for the enhancement. Simonitto et al. (91) added dynamic noises to an image with a spatial frequency gradient and found for intermediate levels of noise that acuity was enhanced. A follow-up experiment (92), using fMRI, measured the increase in activity in V1 during the enhancement process. Yang (93) examined the effect of adding noise to very weak images and found enhancement for intermediate noise strengths.
Effects of "Colored" (1/fb) Noise on Stochastic Resonance.
The early literature on SR presumed that the optimal noise for signal boosting would be "white" (spectrally flat), because all phases and frequencies needed would be present. However, several studies have documented or simulated systems for which 1/f noises are more efficient (94-98). Especially relevant studies were simulations that looked at an array of 200 ´ 200 oscillators meant to simulate a neural network (28, 76). 1/f noise was more efficient in both the spatial and temporal domain in inducing spatiotemporal pattern formation. In the temporal domain, for the nervous system, this can be a consequence of the neural refractory period; 1/f noise is more efficient because it gives greater weight to slower temporal frequencies that are less likely to generate neural events when the neuron has just initiated an action potential and cannot follow up. It is not as clear why spatial 1/f noises are especially potent in spatiotemporal pattern formation, but it would be consistent with evidence that human observers are adapted to the 1/f spatial and temporal spectra of natural images (99,100).
Perceptual Phenomena That May Be Related to Flicker-Induced Pattern Formation
Parvo-Magno Distinctions and Second Order Mechanisms.
Hardage and Tyler (101) discovered an interesting "induced twinkle effect" where viewing dynamic noise in a central area can induce a sense of twinkle in the unstimulated surround, and vice versa. The phenomenon has an induction range of about 10° from each edge of a patch presented near the fovea. In some respects this effect seems analogous to one version of the MacKay effect, where viewing a blank patch after viewing a geometric pattern leads to a perception of noise in the blank patch streaming orthogonal to the orientation of the features of the previous image. It also seems analogous to the center/surround behavior shown by Exp. 1, although a cursory check of flicker-induced hallucinations, using a biasing annulus with a 25° hole, did not show an expected gap of several degrees in the centrally induced hallucination. Tyler and Hardage's (102) follow-up study of the induced-twinkle effect notes that some properties of the induced twinkle effect seem commensurate with the properties of the M cell pathway. Although the twinkle effect is reminiscent of some of the phenomena that we have observed, we doubt that the M cell pathway could be exclusively responsible for our phenomena. One reason is that flicker-induced hallucinations are sometimes colored (indeed, flicker-induced hallucinations are sometimes known as Purkinje colors). Should we then consider a P cell substrate? One related phenomenon, often considered to be P cell-based, is Fechner-Benham subjective color, which is optimal at 6 Hz. Yet flicker-induced hallucinations are best produced at significantly higher rates of 10-20 Hz, which is near or above the flicker fusion rate for chromatic stimuli. A likely explanation for these contradictions is that flicker-induced hallucinations are produced by a second-order mechanism, whose network-based properties may reflect some, but not all, of the properties of the retinogeniculate mechanisms that are afferent to the network.
It is also worth noting that V1 is not the only visual cortex in which pattern formation may occur [see recent evidence for activity in V3 and other extrastriate areas during migraine aura (103)]. Moreover, there are cells in V4 that are preferentially sensitive to radial and concentric circular stimuli (104). Given interactions between V1 and extrastriate areas, complicated interaction between pattern-forming and geometry sensitive systems in V1, V3, and V4 cannot be ruled out. It may be possible to sort out some of these interactions using functional imaging or selective deactivation techniques like TMS.
Opponent Processing of Form.
Our findings are striking examples of geometric form opponency, but they are not the first; MacKay found opponency in the perception of afterimages by circles and fan shapes, as well as noise-streaming opponency in white noise viewed through a geometric mask (22, 23). Vidasgasgar et al. (105) and Francis's laboratory (32, 106) have extended these results in successive contrast experiments. Clifford found a similar opponency between perception of circular and radial symmetry in Glass patterns (30). Other evidence of pattern opponency is found in afterimage opponency of visibility regulation for patterns of various scales, orientations, and direction of motion (31, 32). MacKay's opponent geometric forms can be used as complementary inducing pairs in the McCollough effect (33). Georgeson looked at opponency between motion and orientation mechanisms and between/within spatial frequency and orientation mechanisms, some of which are relevant to MacKay effects (31, 107). Francis has extensively modeled opponency between spatial orientations that arises from a neural network designed to minimize afterimage persistence (32, 106). These models are tantalizing, but they are designed to account for very local successive contrast effects and would need to have a long range spatially opponency to model the phenomena we describe in Exps. 1 and 2.
Functional Imaging Predictions Associated with Photopically Induced Hallucinations
Based on our studies we can distinguish seven classes of photopically induced hallucinations and closely related phenomena, six (four unique to our study) of which produce specific rather than random effects. We also make specific predictions about the patterns of cortical activity that each of these phenomena would produce, if V1 were imaged, if Ermentrout and Cowan's (14) hypothesis of self-organized cortical stripe formation were to be tested.
Class 1: Ordinary Flicker-Induced Hallucinations.
Viewing spatially uniform flickering fields give rise to multistable fine scale colored forms; the scale and multistability of the associated activity would be difficult to resolve or study on V1. Previous studies of similar percepts induced by migraine (51-53) and mescaline intoxication (19, 20) suggest a natural default scale of cortical activation of about 2 mm, which would correspond roughly to the dimensions of an orientation hypercolumn on V1 (but not to hypercolumn scale in extrastriate corticies, which are a factor of 2-4 more course; ref. 108).
Class 2: MacKay Effects.
MacKay (22) found that flicker-illuminating a geometric form leads to perception of an illusory opponent form superimposed on a physical form. The induced radial forms sometimes rotate paradoxically; individual arms can seem stationary, even when the pattern as a whole appears to rotate. If imaged on V1, such MacKay effects should look like two orthogonal stripe patterns arranged as a lattice (for superimposed perceptual circles and fans) or as a plaid (for superimposed spirals). MacKay (23) also found that viewing white noise through a geometrical transparency leads to perception of orthogonal streaming which can define a "complementary afterimage." This result would seem to suggest that dynamic noise activates a geometrical opponency that can be separately manifested in a formless motion pathway. Our Exp. 3 suggests that overt pattern formation can be dependent on the spectrum of noise and on the noise contrast level.
Class 3: Hallucinations in the Empty Flickering Region Around a Flickering Geometrical Target.
If the target and empty region flicker in-phase, the percept is like a MacKay effect that extends through both the target and the flickering region. Unlike the normal MacKay effect, the induced radial hallucinations usually appear to rotate overtly (nonparadoxically). Circles induce fan shapes and vice versa; spirals induce countertwisted spirals. Spatial frequency of the induced patterns is generally similar to the spatial frequency of the inducer, but the dynamic nature of the stimulus makes counting difficult. Induced spirals can rotate; induced circles can wobble or pulsate but often remain stable. If imaged on V1, the combination of the physical pattern and the hallucination would look like a lattice or plaid close to the foveal representation and like a grating in peripheral cortex. Except for the case where circles are induced by fan shapes, the patterns should be in continuous flux because the induced fan shapes and spirals would rotate over the steady inducing shapes, leading to a scrolling of one orthogonal grating over the other near the fovea and a single scrolling grating away from the fovea.
Class 4: Hallucinations in the Empty Flickering Region Around a Steady Geometrical Target.
These do not extend through the target (see Fig. 3 a and b for artist illustrations). Fans generally rotate; spirals may rotate; induced circles may wobble or pulsate. Their cortical representation is similar to that predicted for Class 3 except that the foveal region would contain only one steady grating, orthogonal to the potentially scrolling grating in peripheral cortex.
Class 5: Hallucinations in an Empty Flickering Field Enclosed by a Flickering Geometrically Pattered Annulus.
These extend through the annulus and behave much like those in Class 3. Their predicted cortical representation is similar to Class 3, save that the lattice-like pattern is in peripheral cortex.
Class 6: Hallucinations in an Empty Flickering Field Enclosed by a Steady Geometrically Patterned Annulus.
Similar to Class 5 except that the induced forms do not extend through the annulus. See Fig. 3 c and d for artist depictions. Color textures (generally hexagonally packed) emerge first and then take on an overall geometry. Circular surrounds induce fan-shaped hallucinations (and vice versa), and spiral surrounds induce fleeting reverse twisted spiral hallucinations. Induced circles can resemble a spinning vinyl record or whirling ball of yarn and often wobble; induced fan shapes are often broad four-bladed patterns, although additional blades appear fleetingly. The predicted cortical representation of the various induced patterns are similar to Class 4, save that the hallucinatory grating is in foveal/parafoveal cortex.
Class 7: Apparent Motion-Biased Effects.
As discussed above, apparent motion in the Mayzner configurations (e.g., Fig. 4) is effective in producing an illusion of radial stimuli rotating with the apparent motion of the icons. The number of illusory arms is always an integer multiple of the number of active icons and becomes multistable when visual persistence allows inactive icons to continue to be perceived (see above). If functionally imaged, the cortical pattern should appear like the pattern shown in top-right part of Fig. 1 (with additional outlying spots for the inducing icons) and should appear to scroll as the perceptual pattern rotates. Spirals are also initially evoked by these stimuli but unwrap into fan-shape patterns upon continued inspection. This is interesting from an imaging viewpoint because it would correspond to a rotation as well as a scrolling of the cortical stripe patterns. However, the best opportunity to image the associated cortical phenomena may be the high spatial frequency stationary fan shape discussed above, because current imaging technologies are more limited by temporal resolution than spatial resolution.
A few caveats about potential imaging studies. If the predicted cortical stripes consist of stripes of cortical activity flanked by stripes of inactivity, then those stripes could be as measurable as those imaged by Tootell and associates using deoxyglucose radioactive staining and fMRI (62, 109). If, however, the stripes consist of stripes of activity in excitatory cells, flanked by stripes of activity in inhibitory activity, then unless excitatory and inhibitory cells produce different hemodynamic responses, this experiment will have to await imaging paradigms that can distinguish between excitatory and inhibitory activity, perhaps a neurotransmitter specific fMRI. In the meantime, it may be possible to use techniques like that used by Kenet et al. (3) to correlate cortical activity induced by oriented geometric patterns and similar cortical patterns found during spontaneous cortical activity. In this case, one would need to correlate cortical activity induced by physical geometric patterns and with cortical activity induced by stimuli that are inducing geometrically similar hallucinations.
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