Użytkownik "Marek Celiński" <m...@w...pl> napisał w wiadomości
news:dqok9s$osh$1@news.onet.pl...

> znaleźć np. matematyczny model tworzenia się rozkładów pobudzeń kory
> V1 prowadzących do powstawania halucynacji tunelowych na skutek

Cytat (przepraszam za brak symboli matematycznych - zamiast nich
będzie sieczka):

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Geometric Visual Hallucinations
Geometric visual hallucinations are seen by many observers after
taking hallucinogens such as LSD, cannabis, or mescaline; on
viewing bright flickering lights; on waking up or falling asleep; in
"near death" experiences; and in many other syndromes. The Chi-
cago neurologist Klu¨ver organized such images into four groups
called form constants: (I) tunnels and funnels, (II) spirals, (III)
lat-
tices, including honeycombs and triangles, and (IV) cobwebs, all
of which contain repeated geometric structures. Figure 2A shows
their appearance in the visual field. Note in particular the
difference
between the first two noncontoured images, which consist of al-
ternating regions of light and dark, and the contoured nature of the
last two images.
Ermentrout and Cowan (1979) provided a first account of the
generation of visual hallucinations, based on the idea that some
disturbance such as a drug or flickering light can destabilize the
primary visual cortex (V1), inducing spontaneous pattern of cor-
tical activity that reflects the underlying architecture of V1. They
studied interacting populations of excitatory and inhibitory neurons
distributed within a two-dimensional cortical sheet. Modeling the
evolution of the network in terms of a set of Wilson-Cowan equa-
tions, they showed how spatially periodic patterns such as stripes,
squares, and hexagons bifurcate from a low-activity homogeneous
state via a Turing instability. They then noted that there is an or-
derly retinotopic mapping of the visual field onto the surface of
cortex, with the left and right halves of visual field mapped onto
the right and left cortices, respectively. Except close to the fovea
(the center of the visual field), this map can be approximated by a
complex logarithm (Schwartz, 1977) as illustrated in Figure 2B.
Applying the inverse of this retinocortical map, they showed that
when the periodic cortical patterns are mapped back into visual
field coordinates, noncontoured hallucinatory images such as the
form constants (I) and (II) of Figure 2A are reproduced.
Interestingly, the model cannot reproduce the contoured images
(III) and (IV), since there is no information in it regarding the
orientation selectivity of neurons in V1. Recently, a much more
detailed model of the functional and anatomical structure of V1 has
been developed that treats the cortex as a continuum of interacting
hypercolumns, each of which has the internal structure of the ring
model for orientation tuning (Bressloff et al., 2001). In this new
model, both contoured and noncontoured hallucinatory images can
be generated depending on whether each isolated hypercolumn un-
dergoes a local Turing or bulk instability with respect to orientation
(see Figure 1A).
We now consider the theory of cortical pattern formation in more
detail. For simplicity, we describe the earlier version due to Er-
mentrout and Cowan. Let aE(r, t) be the activity of excitatory neu-
rons in a given volume element of a slab of neural tissue located
at r
R2, and let aI(r, t) be the corresponding activity of
inhibitory
neurons; aE and aI can be interpreted as local spatiotemporal av-
erages of the membrane potentials or voltages of the relevant neural
populations. When neuron activation rates are low, they can be
shown to satisfy nonlinear evolution equations of a similar form to
Equation 1:
s
al  al  w . r[a ]  h (10)  lm m l
t mE,l
where w . r now signifies the convolution
(w . r[a])(r)   w(|r  r|)r[a(r, t)]dr (11) R2
with wlm(|rr|) giving the weight per unit volume of all synapses
to the lth population from neurons of the mth population a distance
|rr| away. Note that wlE
 0 and wlI
 0 and the external input
hl is assumed to be constant. An important property of the weight
distributions wlm is that they are invariant under the action of the
planar Euclidean group-the group of rigid motions in the plane
consisting of translations, rotations, and reflections. This symmetry
plays a crucial role in determining the types of pattern that emerge
through a Turing instability.
For a sigmoid firing rate function r, it can be shown that there
exists at least one fixed-point solution a?l of Equation 10:
a? l  W . r[a? ], W  w (r)dr (12)
If the external input h1 is sufficiently small relative to the
threshold
for firing, then this fixed point is unique and stable. There are thus
two ways to increase the excitability of the network and thus de-
stabilize the fixed point: either by increasing the external input or
reducing the threshold. The latter can occur through the action of
drugs on certain brainstem nuclei, which therefore provides a
mechanism for generating geometric visual hallucinations. The lo-
cal stability of the fixed point is found by linearization. Setting
al(r,
t)  a?l
 ul(r)ekt/s leads to the eigenvalue equation
(k  1)ul(r)  l (w . u )(r) (13)  m lm m
mE,I
where ll
r(a?l). This can be diagonalized by introducing Fourier
transforms Wlm(k) and Um(k) and using the convolution theorem.
The result is a matrix dispersion relation for k as a function of q
 |k| given by solutions of the characteristic equation det ([k 
1]I  K(q))  0, where K
lm(q)  lmWlm(|k|) and I is the unit
matrix. One can simplify the formulation by assuming that wEE

wIE and wII
 wEI so that the dispersion relation reduces to k(q)
1  lW(q), where W(q) is the Fourier transform of w(r) 
[lEwEE(r)  lIwII(r)]/l.
It is then relatively straightforward to set up the conditions under
which the homogeneous state undergoes a Turing instability,
namely, that W(q) be bandpass. This can be achieved with the
"Mexican hat" function shown in Figure 2C, representing short-
range excitation and long-range inhibition. It is simple to establish
that k then passes through zero at the critical value lc
 1/W(qc),
signaling the growth of spatially periodic patterns with wave num-
ber qc, where W(qc)maxq{W(q)}. Close to the bifurcation point,
these patterns can be represented as linear combinations of plane
waves:
N
u(r)   [z eikn.r  z*eikn.r] (14) n n
n1
where the sum is over all wave vectors with |kn|  qc. Rotation
symmetry implies that the space of such modes is infinite dimen-
sional. That is, all plane waves with wave vectors on the critical
circle |kn|  qc are allowed (see Figure 3A). However, translation
symmetry means that we can restrict the space of solutions to that
of doubly periodic functions corresponding to regular tilings of the
plane. The symmetry group is then reduced to that of certain crystal
lattices: square, rhomboid, and hexagonal lattices (see Figure 3B).
The sum over n in Equation 14 is now finite with N  2 (square,
rhomboid) or N  3 (hexagonal), and depending on the boundary
conditions, various patterns of stripes or spots can be obtained as
solutions. Amplitude equations for the coefficients zn can then be
obtained by using the perturbation approach described in the dis-
cussion of orientation tuning. Here the rotation and translation sym-
metries introduced above restrict the structure of the amplitude
equations. In the case of a square or rhombic lattice, we can take
k1
 qc(1, 0) and k2
 qc(cos u, sin u) such that
dzn  znl  lc c0|zn|2 2cu |z |2 (15)  m   dt m
n
for n  1, 2, where cu depends on the angle u. In the case of a
hexagonal lattice, we can take kn
 qc(cos un, sin un) with u1

0, u2
 2p/3, and u3
 4p/3 such that
dzn  zn[l  lc c0|zn|2  gzn*1zn*1] dt
 2cu2zn(|zn1|2 |zn1|2) (16)
where n  1, 2, 3 (mod 3).
These ordinary differential equations can then be analyzed to
determine which particular types of pattern are selected and to cal-
culate their stability. The results can be summarized in a bifurcation
diagram as illustrated in Figure 3C for the hexagonal lattice with
g  0 and 2cu2
 c0. (Note that such patterns have also been
observed in fluids in the form of convection rolls and honeycombs
as well as in animal coat markings in the form of stripes and spots.
This indicates that although the physics may be very different, the
interactions in all these phenomena are such that they can all be
represented within the framework of the Turing mechanism.)
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