Hallucinations occur in a wide variety of situations such as with migraine headaches, epilepsy, advanced syphilis and, particularly since the 1960's, as a result of external stimulus by drugs such as the extremely dangerous LSD and mescaline (derived from the peyote cactus). General descriptions are given by Oster (1970) and, with much more detail, by Kluver (1967).

Hallucinogenic drugs acquired a certain mystique, since users felt the drugs could alter their perceptions of reality. From extensive studies of drug-induced hallucinations by Kluver (1967), it appears that in the early stages the subject sees a series of simple geometric patterns which can be grouped into four pattern types. These four categories (Kluver 1967) are: (i) lattice, network, grating honeycomb; (ii) cobweb; (iii) spiral; (iv) tunnel, funnel, cone. Figure 12.8 shows typical examples of these pattern types. In Figure 12.8(a) the fretwork type is characterised by regular tesselation of the plane by a repeating unit; that in Figure 12.8(b), the spiderweb, is a kind of distorted Figure 12.8(a).

The hallucinations are independent of peripheral input: for example, experiments showed that LSD could produce visual hallucinations in blind subjects. These experiments, and others such as those in which electrodes in the subcortical regions generated visual experiences, suggest that the hallucinogenic patterns are generated in the visual cortex. Ermentrout and Cowan's (1979) seminal paper is based on the assumption that the hallucinations are cortical in origin and proposed and analysed a neural net model for generating the basic patterns; see also the discussion on large scale nervous activity by Cowan (1982) and the less technical, more physiological, exposition by Cowan (1987). Ermentrout and Cowan (1979) suggest the patterns arise from instabilities in neural activity in the visual cortex; we discuss their model in detail in this section.

A visual image in the retina is projected conformally onto the cortical domain. The retinal image, which is described in polar coordinates (r,0), is distorted in the process of transcription to the cortical image where it is described in (x, y) Cartesian coordinates. It is a mechanism for the creation of these cortical projection patterns that we need to

model. The packing of retinal ganglion cells (the ones that transmit the image via the lateral geniculate nucleus which relays it to the visual cortex) decreases with distance from the centre and so small objects in the centre of the visual field are much bigger when mapped onto the cortical plane. So, a small area dx dy in the cortical plane corresponds to Mr dr dO in the retinal disk, where M is the cortical magnification parameter which is a function of r and O. Cowan (1977) deduced the specific form of the visuo-cortical transformation from physiological measurements; it is defined by x = a ln[pr + (1 + p 2r 2)1/2], y = ap r O(1 + p2r 2)—1/2, (12.26)

where a and p are constants. Close to the centre (the fovea) of the visual field, that is, r small, the transformation is approximately given by x ~ apr, y ~ aprO, r « 1, (12.27)

whereas for r far enough away from the centre (roughly greater than a solid angle of 1°)

Thus, except very close to the fovea, a point on the retina denoted by the complex coordinate z is mapped onto the point with complex coordinate w in the visual cortex according to w = x + iy = a ln[2ßr] + iad = a ln[z], z = 2ßr exp[i0].

This is the ordinary complex logarithmic mapping. It has been specifically discussed in connection with the retino-cortical magnification factor M. Figure 12.9 shows typical patterns in the retinal plane and their corresponding shapes in the cortical plane as a result of the transformation (12.29); see any complex variable book which discusses conformal mappings in the complex plane or simply apply (12.29) to the various shapes such as circles, rectangles and so on. We can thus summarise, from Figure 12.9, the cortical patterns which a mechanism must be able to produce as: (i) cellular patterns of squares and hexagons, and (ii) roll patterns along some constant direction. All of these patterns tessellate the plane and belong to the class of doubly periodic patterns in the plane. In Section 2.4 in Chapter 2 we saw that reaction diffusion mechanisms can generate similar patterns at least near the bifurcation from homogeneity to heterogeneity.

The basic assumption in the model is that the effect of drugs, or any of the other causes of hallucinations, is to cause instabilities in the neural activity in the visual cortex and these instabilities result in the visual patterns experienced by the subject. Ermentrout and Cowan's (1979) model considers the cortical neurons, or nerve cells, to be of two types, excitatory and inhibitory, and assume that they influence each other's activity or firing rate (recall the discussion in Section 12.1). We denote the continuum spatially distributed neural firing rates of the two cell types by e(r, t) and i (r, t) and assume that cells at position r and time t influence themselves and their neighbours in an excitatory and inhibitory way much as we described in the last section with activation and inhibition kernels.

Here the activity at time t strictly depends on the time history of previous activity and so in place of the dependent variables e and i we introduce the time coarse grained activities

where h(t) is a temporal response function which incorporates decay and delay times; h(t) is a decreasing function with time which is typically approximated by a decaying exponential exp[-at] with a > 0.

There is physiological evidence (see, for example, Ermentrout and Cowan 1979) that suggests the activity depends on the self-activation through E and inhibition through I . The activity of E and of I also decay exponentially with time, so the model mechanism can be written as d E

— = -I + Si (aei wei * E - auwu * I), d t where, from physiological evidence, the functions SE and SI are typical threshold functions of their argument, such as the S shown in Figure 12.10(a) and the a's are constants related to the physiology and, for example, drug dosage; note that S is bounded

Figure 12.9. Corresponding patterns under the visuo-cortical transformation. (a) The lattice patterns in Figure 12.8(a), except for distortions, are effectively unchanged. The other visual field hallucination patterns are on the left with their corresponding cortical images on the right: (c) tunnel; (d) funnel; (e) spiral. (After Ermentrout and Cowan 1979)

Figure 12.9. Corresponding patterns under the visuo-cortical transformation. (a) The lattice patterns in Figure 12.8(a), except for distortions, are effectively unchanged. The other visual field hallucination patterns are on the left with their corresponding cortical images on the right: (c) tunnel; (d) funnel; (e) spiral. (After Ermentrout and Cowan 1979)

for all values of its argument with S(0) = 0. The convolutions are taken over the two-dimensional cortical domain and the kernels here are nonnegative, symmetric and decaying with distance, as illustrated in Figure 12.10(b): a symmetric decaying exponential such as exp[-(x2 + y2)] is an example. The argument in the interaction functions, SE, for example, represents the difference between the weighted activation of the local excitation and the local inhibition due to the presence of inhibitors. The inhibitors are enhanced through the argument of the Si function, in the I equation, via the wEI convolution. The inhibitors also inhibit their own production via the wII convolution. There are similarities with the model discussed in the last section except there the activation and inhibition were included in each kernel.

Let us now examine the linear stability of the spatially uniform steady state of (12.31), namely, E = I = 0, that is, the rest state. The nonlinearity in the system is in the functions S so the linearised form of (12.31), where now E and I are small, is d E .

— = -I + Si(0)(aEiWEi * E - anwn * I), d t where, because of the forms in Figure 12.10(a), the derivatives S'E (0) and Si (0) are positive constants. We now look for spatially structured solutions in a similar way to that used in the last two sections except that here we are dealing with a system rather than single equations ((12.5) and (12.23)), by setting

E^r'^ = Vexp[Àt + ik ■ r] = Vexp[A.t + ikix + ik2y], (12.33)

where k is the wave vector with wavenumbers ki and k2 in the (x, y) coordinate directions; X is the growth factor and V the eigenvector. If X > 0 for certain k, these eigenfunctions are linearly unstable in the usual way.

Substituting (12.33) into the linear system (12.32) gives a quadratic equation for

X = X(k), the dispersion relation, where k = | k | = (kf + k|)1/2. For example, with (12.33)

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