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Conceived and designed the experiments: BE. Performed the experiments: MR MS BE. Analyzed the data: MR MS BE. Contributed reagents/materials/analysis tools: MR BE. Wrote the paper: BE. Designed code for the 2D simulations: MR.

The authors have declared that no competing interests exist.

We present a model for flicker phosphenes, the spontaneous appearance of geometric patterns in the visual field when a subject is exposed to diffuse flickering light. We suggest that the phenomenon results from interaction of cortical lateral inhibition with resonant periodic stimuli. We find that the best temporal frequency for eliciting phosphenes is a multiple of intrinsic (damped) oscillatory rhythms in the cortex. We show how both the quantitative and qualitative aspects of the patterns change with frequency of stimulation and provide an explanation for these differences. We use Floquet theory combined with the theory of pattern formation to derive the parameter regimes where the phosphenes occur. We use symmetric bifurcation theory to show why low frequency flicker should produce hexagonal patterns while high frequency produces pinwheels, targets, and spirals.

When the human visual system is subjected to diffuse flickering light in the range of 5-25 Hz, many subjects report beautiful swirling colorful geometric patterns. In the years since Jan Purkinje first described them, there have been many qualitative and quantitative analyses of the conditions in which they occur. Here, we use a simple excitatory-inhibitory neural network to explain the dynamics of these fascinating patterns. We employ a combination of computational and mathematical methods to show why these patterns arise. We demonstrate that the geometric forms of the patterns are intimately tied to the frequency of the flickering stimulus.

Ever since they were first described by Jan Purkinje in 1819, the swirling geometric visual patterns brought on by diffuse flickering light have fascinated both scientists and artists. Helmholtz described the patterns at the turn of the twentieth century. The invention of the stroboscope enabled investigators to classify conditions in which they occurred, including, the interactions with hallucinogens. In several papers, Smythies

The first attempts to

Based on earlier models of hallucinations

Our goal in this paper is to propose a computational and theoretical model for the spontaneous formation of geometric patterns in the presence of flickering light. We first propose a model for a spatially distributed network of excitatory and inhibitory neurons where each neuron is represented by its firing rate

We utilize a variant of the Wilson-Cowan equations

For simplicity, and to avoid edge effects, in our simulations, the boundary conditions are periodic. For most of the paper, we fix parameters to be

In the last section of the results, we couple two such two-dimensional networks to represent the left and right hemifields of the visual cortex. Coupling is achieved as follows. Let

When

Analysis of the linearized equations about the oscillatory homogeneous state (for

The phosphenes reported by subjects vary tremendously, but among them are the commonly seen so-called

Remole

There is a well-known topographic mapping from retinal coordinates to cortical coordinates (

We begin with simulations of a one-dimensional domain since it is much easier to visualize the spatio-temporal dynamics.

Each point represents a simulation at a fixed value of the amplitude and period for the flashing stimulus. The gray-scale represents the magnitude of the pattern averaged over time. Specifically, the time average of

If

Two dimensional simulations reveal some striking differences.

Top row shows patterns seen with high frequency stimuli. Pairs show the results of different random initial conditions. Bottom row shows patterns seen at lower frequency; each pattern has the same period as the stimulus.

The lower row of

(A) high frequency stimulation (18.2 Hz); (B) low frequency (9.1 Hz) stimulation. Note that in (A) after one temporal cycle of 55 msec, the pattern is shifted by one half of a spatial cycle.

Each square is a simulation of a

In sum, the simulations show (i) high frequency stimulation tends to lead to stripes; (ii) low frequency tends to lead to hexagonal patterns; and within each frequency band, the higher frequencies have coarser spatial structure. We lastly remark that the two different regimes are reminiscent of Remole’s observations that one subject had two resonance regions at periods of 90 msec and 42 msec. Our goal in the remainder of this paper is to better understand the reasons for these observations.

Before turning to the analysis of the spatially distributed domains, we first consider a very reduced system. Suppose that there are two E-I pairs:

We assume similar equations for

(A) Plots of

In sum, even with as few as two units, the overall dynamics is qualitatively similar to the full spatially extended networks. The shape of two-network phase-diagram differs from that of the spatially extended network. This is due to the fact that the full spatial system has an infinite number of eigendirections, compared to just the two for the reduced model and that the ratio of inhibitory to excitatory coupling is slightly different.

We now want to understand the mechanism for these patterns and to better quantify the dependence of the patterns on the stimulus period. To do this, we next show how to compute numerically boundaries for pattern formation as the frequency and amplitude of the flashing light change. The analysis holds in any dimension and in many types of domains as long as certain conditions are met. We describe the approach generally for

For example, if the domain is the circle (that is periodic boundary conditions in one dimension), then

This is a nonlinear periodically forced system, so we are not guaranteed that there is a periodic solution. Let

We now invoke our hypothesis about eigenfunctions. We write

If we plug this into (6), we see that

The way to solve a linear equation with periodic coefficients is to compute the so-called monodromy matrix. Let

For our system,

To compute stability boundaries, we need to find and parameterize the eigenvalues,

For our system, we have used Gaussian spatial interactions with space constants,

(A,B) Values of the determinant (green) and stability conditions for a +1 (black) and −1 (red) Floquet multiplier as the wave-number,

Once we have found an intersection of one of the curves,

We can understand

In

So far, the simulations and stability analyses have all been for equations (1–2) when

Stability boundaries for the homogeneous solution as amplitude and period vary, but the inhibitory population receives input of strength

In order to get pattern formation we have to make several important assumptions on the local circuit dynamics and the coupling. With no coupling, the “space-clamped” system should have a damped return to a stable rest state. Furthermore, the stable equilibrium should lie on the middle branch of the excitatory nullcline (the so-called “inhibition-stabilized” regime

One of the most striking findings of our simulations is that low frequency stimuli mainly lead to hexagons and high frequency generally lead to stripes. There turns out to be a deep theoretical reason for this result that is based on the ideas of symmetric bifurcation theory. We do not discuss the rigorous mathematics that underlies this theory, but rather, summarize the basic ideas. Near the onset of the instability, the pattern will look like a sum of the eigenfunctions,

(A) Bifurcation at low frequencies in equation (8) for the case where

When flicker hallucinations are perceived, they are often seen as whole-field patterns and the patterns are “pure” rather than a mixture of say pinwheels and targets. Thus, a natural question is how can the two halves of the visual cortex “synchronize” their spatial patterns. There is strong anatomical

(A) Period 55 msec without coupling (left two images) and coupled

In this paper, we have suggested a simple mechanism for flicker-induced hallucinations. We suggest that all that is needed is a spatially extended lateral-inhibitory network of excitatory and inhibitory neurons along with some resonance properties such as a damped oscillatory return to the resting state. The lateral inhibition is necessary to produce spatial instabilities as has already been suggested by

Our model, being based on the earlier models for hallucinations

In order to produce a model that is capable of creating these patterns, the cortex has to be in a particular state. Geometrically, we want the excitatory and inhibitory nullclines of the space-clamped system (the local circuitry) to both have positive slopes at the resting state. In a recent combination of theory and experiment,

There are many generalizations of this model which could be considered. Smythies

With very little change in the details of the equations, it should be possible to introduce the “seeding” of patterns into the model. For example, suppose that we are in the low frequency stimulation regime and now add a small bias in the form of say a low contrast target or pinwheel. (In the equations, we would model this as a low contrast grating of the appropriate orientation.) We could then see if the model would produce stripes instead of hexagons as stripes remain a possible pattern. Indeed, the schematic bifurcation diagram in

Many of the phosphenes reported by subjects are not the broad forms shown in

An exciting direction to go in this work is to explore the role of color. The phosphenes themselves are extremely colorful. In addition, the color of the light stimulus can have a strong effect on the pattern

The emergence of patterns in periodically forced spatially distributed systems has a long history, particularly, in the area of fluid mechanics

Flicker stimuli provide an excellent way to probe the intrinsic pattern forming capabilities of the visual cortex since, unlike drug-induced hallucinations, they can be readily controlled. Indeed,