A Cellular Automaton Model Based on Cortical Physiology

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Complex Systems
1 (1987) 187-202
A Cellular Automaton Model Based on
Cortical Physiology
Martin I.Hofmann
Departments
of
Zoology
and
Physiology,Uni versity
of
Toronto,
Toronto,
Ontario,
M5S l AI,
Canada
Abstract.A model is proposed that
is
based
on certain
aspects
of sensory cortex.The
basic
model is a cellular aut omaton,where
the cells represent small groups of neurons
in
the cor t ex.All cells
ar e locally interconnected with their three nearest neighbours in each
direction,a variable set of weights determines the st rengt h of the
connections.The output of the cell is a fun cti on of the weighted sum
of the inputs.Other variable parameters are the threshold and the
number of states.The model is shown
to
be able to exhibit a wide
range of behaviours analogous to the types of behaviour seen in other
more general cellular automata.A subset of these behaviour types
may be applicable to modelling the funct ions of sensory cortex.
1.
Introduction
Most modelling of the visual system has been done using a hierarchical
linear filter approach (e.g.[1]),where each level filters the output from the
previous level and thus projects to the next higher level.For the low levels
of the visual system this method accurately models many aspects of the
processing of visual images.Even at the level of the primary visual cortex
this approach has given valuable insight into the pr ocessing t hat occurs.
However only about half of all the cells in t he primary visual cortex show
linear or almost linear responses,the rest have varying degrees of non­
linearity.
Primary sensory cortical areas are organized into a regular array of
"hy­
percoIumns"[2,3].Within the hypercolumn there is furt her subdivision into
columnar st ructure:For instance in the primary visual cor tex,each hyp er­
column cont ains orientation and ocular dominanc e columns.The cellular
str uct ure within all columns is basically the same.Ther e are two major
features of note in the connections within t he cortex (see figure 1).First,
most connect ions,except input and output,are local in nature,extending
"Compute r
addre ss:
Ueenet:{allegr a,
i.hnp4
,decvax
,pyru.id}
!utzoo!vision
(C)
1987 Comnlex Sveeeme Publicafi one.Inc.
188
Martin
1.
Hofmann
Deep
pyramidal
neuron
Efferent
output
.(0;'-""'1:::=1
Superficial
pyramidal
n'h;;y'~,,,,,?,~.~n.e u r o n
Afferent
input
Fig ure 1:Simplified circuit diagram of the visual cortex showing
some of the types of connections that ar e found there.The pyrami­
dal cells are considered to be principal neurons and the smaller cells
are local int erneurons.The shaded cells represent inhibitory neurons.
(Adapted from Szentagothai [41 and Shepherd [51.)
no further than t he size of a hypercolumn.Second,many of t hese connec­
t ions can be seen as part of feedback loops.The approximate homogeneity
of a cortical ar ea.and local connections leads naturally t o the idea of mod­
elling us ing cellular au tomata.Models of this sort have been proposed for
t he cerebellar cortex
161
and the visual cortex
[71.
Cellular automata mod ­
els assume homogeneity over the ent ire system and t he results show bulk
properties of the syst em be ing modelled.
In
contrast to this are various
models of parallel processing and learning (some examples may be foun d in
Rumelhart and McLelland
18])
which depend on non-homogeneity
"fd
are
more concerned with det ailed propert ies of t he system.These models may
be more applicable to higher corti cal regions t hat do not show t he same
regularity of structure as the sensory and cerebellar cortices.
Recent advances in the theory of cellular automata [9,10,11],provide
the background for the present model.The model is a cellular automaton
based on some of the known propert ies of cort ical neurons.A relati vely
small number of parameters determine the rule governing the behaviour of
the model.Using t his mod el I shall examine the various types of behaviour
t hat t he system can exhibit and how t he various parameters of t he model
affect that behaviour.Secondly I shall extend t he model to include input
from outside and an asynchronous mode of calculation,which int roduces a
more realistic and stochastic element into the system.
2.Model descrip t ion
A cellular automaton is basically an n-dimens iona l array,s with elements
8 j.
Each element or
"cell"
may t ake on a range of values 0,1...,
k -
1,
the
A Cellular Automaton Model Based on Cortical Physio logy
189
(3.1)
value of the cell being t ermed its state.The states of cells are changed in
discrete t ime steps,t he array of states at some time
t
is called a generation.
The next stat e of a cell depends on the state of a cell and its neighbours.
The rule which determines t he next stat e is t he same for all cells.The range
determines the number of neighbours in some direction t hat can contribute
to the next state.The present model always uses a range r
=
3.
3.Basic model
The basic model is a cellular automat on where each successive generation is
calculated using a two-st ep procedure.The first st ep is t o take a weight ed
sum of of a cell and its neighbours.A function is t hen applied t o thi s value
and the result is the value of the cell at the next generation.This procedure
in one dimension is given by the formula:
3
s:+1
=
fp(
L
Wi S:+i )
i =- 3
where
w
is
the weights vector
(W- 31...,W3).
Similar ly
in
two dimensions:
3 3
s:1
1
=
f p(
L L
WJ;lS:+.l:i+ l)
J;=-3l= - 3
(3.2)
where w
is
the weights matrix
[Wi;].
i,
j
=
{3,...3}.
The weights used may be positive or negative.
The functions
fp
are bas ed on a difference of logistic functions (see figure
2).
I.(x)
=
L
1(x)
-"'(L,( x)
where
L,
is the logist ic function
(3.3)
1
L ·
-
(3.4)
,-
1
+
exp( -(:~:;» )'
The parameters
Xi
and
p
are calculated as a function of
k,
0
J
w.
The
function
fp
is normali zed and integerized so that the output is in the cor­
rect
range
(0,..,
k
-
1).Two condit ions are necessary in this scheme to
res t rict t he possible rules to
"legal"
rul es
110].
First,the weights must be
symmetrical,i.e.in one dimension
W-i
=
Wi
or in 2 dimensions
W-i-i
=
wi -i
=
w- ii
=
wii·
Second,the functions
fp
must sat isfy:
1.(0)
=
o.
These conditions are satisfied in all cases.
(3.5)
(3.6)
(3.7)
190
Martin
1.
Hofmann
1O ~------ -- ----------..,
8
2
__ ___________ -1°.
0
---- i0
5
4
stimul us (input)
Figure 2:The
basic
stimulus-response (SR) functions showing t he
effect of var ying
j.
The value of
'1
is shown to the right of each curve.
4.Forcing input
One ext ension t o t he basic model descri bed above t o apply a constant (time
invariant) forcing input.This means t hat a given cell
will
receive input not
only from its elf and it s neighbours but some external source as well.The
following formul a expresses t his:
,
.;+1
=
/P(
L
w;':+,
+
t ao)
;=-3
(4.1)
where
f
is th e forcing weight,
~
is
th e value of the
Ji h
cell of the forcing
input.
5.Asynchronous calculation
The bas ic model uses synchronous calculation t o determine the next gener­
ation,where th e value of all cells ar e calculated before changing any values.
It
is also possibl e to calculate th e next generat ion asynchronously [12].One
way to do this is t o choose a number of cells at random and change only
their values before continuing.The second extension to the basic model
does t his,t he number of cells to be changed is input as
a
variable parame­
t er,
A.
With t his method of calculation th e notion of generat ion is no longer
well-defined.For purposes of display the next generation was arbitraril y
defined as occurring after the tot al number of changes made equalled or
exceeded the number of cells.
A Cellular Automaton Model Based on Cortical Physiology
6.Behaviour of the model
191
The types of behavi our of t he model and t he effects of t he various param­
eters on
t his
behaviour are shown in figur es 3 through 5.The number of
states and the absolute values of the weights have little qualitati ve effect on
the behaviour of the systems.The sh ape parameter
"t
has its major influ­
ence on the temporal behaviour of the system.When
"1
=
0 (with a positi ve
center weight) the system tends towards stable states,with t he majority of
cells taking a value of either 0 or t he maximum.As
"1
increases the system
goes to oscillatory
~r
chaotic types of behaviour,depending on t he weights
and the t hreshold.Systems that conver ge to st able states generally do so
very rapidly (about 5 generat ions for Den-dimensional syst ems and 10 - 20
for two-dimensional systems).
The weighting function and the threshold have their major influence
on the spat ial structure of t he system.The distribution of excitatory and
inhibitory weights affects the spatial structure of t he states,e.g.alternat ­
ing excitatory and inhibitory weights pr oduce a system wit h alt ernating
zero and non-zero st ates in each generati on.The rate of lat er al growth
of patterns is det ermined by both the weights and the t hreshold.A high
threshold or too much side inhibition inhibits later al spread of patterns.
The r at e of growth is an important factor
in
det ermining whether a pat­
t ern will exhibi t oscillat ory or chaotic behaviour.Class IV like behaviour
occurs in the transition region as
threshold
of a chaotic system is increased
(figure 5).In this model Class IV like behaviour appears to occur when the
tendency of a pat t ern to spread laterally is inh ibited by a high threshold.
7.Extensions
The major effect of us ing an asynchronous mode of calculation ( figur e 6)
is on systems that show oscillatory behaviour under synchronous calcula­
tion.Strict oscillatory behaviour is dependent on synchronous calculation.
Other types of behaviour are affected to a much smaller degree alt hough
the rate of convergence is somewhat slower.The effect of applying a forcing
input (figure 7) is to entrain the system to t he forcing inp ut,alt hough t he
behaviour is not significantly affected otherwise.The behaviour of one­
dimensional systems (figur e 8) is not qualit atively differ ent,except that
oscillatory behaviour appears to be less common,given the general for m of
the weighting functions I have been using.
8.Discussion
The relation of t he model to cortical physiology occurs at two basic levels.
First,the model was constructed based on certain aspects of the nervous
system.Second,t he behaviour of the model should correspond to the
behaviour of the nervous system for appropriate values of
.t he
paramet ers.
192
mnrrnmmm
nrrrnmr In
rrnrr nnnnr
n- 1T,f1T - -­
- -_
..-1--11"-
(U)
.=1-1
-3 241-3-1)
,..0,5,
1ll.
15,10
.,
_
O.
(l.ll
_"{ot
.3 1'
Z.~
.1)
'E O.~.,.1
Inrlll~nlllli
(UI
_ _
(0_\1 4 1_1
OJ
,,,,o.IO.3).,.O~
lilllllIDnmnmnmulllll
nmillllUI
iQllffilffimllnmnlnll limmml llii
i llt n[ lmlliillm ~I;;lillliUr~ f l~ii
- -lIi lh"-------]!fiW
Irrllf---,rnr
(1.1)..
_ {.l l _H -3 1 -1
I
'_0.$,10.1$.,,,",0
Martin
1.
Hofmann
T1TlIlnlnnmmn
nnnnmrrt
-lrnnrrTl rl
T --- ilT -n- ­
r r"--'- ------
(U I
._1_1
-a
24
t·].l )
'_0.5,10.15,20,,"0.6
--­
.1 11
...
1
11111111"(111'.
r -111II1I1111111:1
'nrrml'lRl
(l.t) _ _ {O.I
2 l 2 _1
OJ
,
'"
0,
5,10.15.20,,,_0
(3.6l
w
· IO_l t 4 2 _I O)
,
E
0..."I
(3.0'1_(_12 -3..-31.11
,..0.....I
Fi gure 3:The effects of var ying
"1
and t he t hr eshold,
9,
are shown for
a sample of different weighting functions.For a given set of weights
incr easing
"'I
leads to either oscillatory or chaotic behav iour.
Increas­
ing
threshold
(from top
to
bottom in t he figures) generally reduces
the number of active sites.All
syste ms
started from a random initial
configuration and had t he numb er of possible states,
k
=
8.
A Cellular Automaton Model Based
on Cortical
Physiology
193

I
I
I I I
 I I
I
0 0
a
a
833
a
(4.1) w ={ -I -3 2 4 2 -3 -I }
o
::::
0,
'1
::::
0,0.5,1 initial
condit ions;
isolated
points
'If"1I'''!''''X
10ft
.~~
ft'
l.
a
',1
i
I
~ B~a
~
d
si
.~:
(4.3)
w ~ {-I
-3 24 2 -3 - I }
8
::::0,
"f
::::
1;init ial conditions:
periodic,without and with added
noise
1111111111111111111111111'
11111111111[1111111111111'
rrrrrrrrrrrrr
rrrrrrmrnr
TTTTTTl
TI
rr'TI'TI
Ti
TiT
(4.2) w ={-I -3 2 4 2 -3 - I }
()::::0,
'1
::::
0;initial conditions:
peri odic,without and with added
noise
(4.4) w ={o -122 2 -I O}
(J::::
0,
"7
::::
1;initial conditions:
periodic,without and with added
noise
Figure 4:The behaviour of some systems starting from non-random
initial conditions.Parts 4.2 - 4.4
also
show the effect of adding noise
to the initial config uration.The noise level
in
parts 4.2 and 4.3
is
a
change
of
±1
state value
in
about half of the sites.In part
4.4
a
single
site
near the center was changed and the disturbance can be
seen propagating out from this region.
194
~
!
,
(5.1) w ={O -1 2 2 2 -1
oj
() =O,'1 =l
~
-.
(5.3) w={O -1 2 2 2 -1
o}
() =
5,
'1
=
1
Martin
1.
Hofmann
~~
,
,
",
,)
(5.2) w ={O -1 2 2 2 -1 OJ
0 =
4,
i
=
1
,rrwl'l
i'f
1'1~ ~"i'~ l ~
l
....ill"-
ffT"'l'
~""'"
....
· r t· · ~
I
(5.4) w={O -1 2 2 2 -1
oj
() =
6,8,1,
"1
=
1
Figure 5:The
effect
of increasing threshold on a chaotic system.The
system passes through a region where
Class
IV-like behaviour is ob­
served.
A Cellular A utomaton Model Based on Cortic al Physiology
195
rnmrmrnnflrn
n
rnmmrrrn
mmnmrm
,
rrrnrr rrn­
r
r'Tr'T
"1"1'-
'~'T
(6.1)
w ~ { - I
-3 2 4 2 -3 -I}
() =
0,5,10,15,20,
I
=
0,
asynchronous calculation:A
=
10
(6.3)
w ~{o
-I 2 2 2 -I
oj
(}
=
0,
"1
=
1,asynchronous
calculation:A
=
10
I
or.
~ ·t.J,f(ff'1\IT.l.
t
.£j,
·l..
I
il})1
~
.,
.
(6.2)
w={-I
-3 2
4
2 -3
-I }
o
=
0,
"I
=
1,asynchronous
calculation:A
=
10
(6.4)
w ~{o
-1 2 2 2 -1
o}
6
=
4,
...,
=
1,asynchronous
calcul ation:A
=
10
Figure 6:The effect of asynchronous calcula.tion on the behaviour
of the syst ems.The systems shown correspond to stable,oscillatory,
chaotic and Class IV-like respectively.Compare these with figures
3.1,3.3,5.1 and 5.2.
196
Martin
1.
Hofmann
....
~
..
~ -'-'~r 1"""T
,

#
'.
rrmnm1fflurrn
._-~------------- --- ----_.
rmrrrrrrm
-------------
l~lrlrrl ·l -ll ~r
--- ---- -_._-
(7.1)
w={.l
-3 2 2 2 -3 -1}
9
=
10,
"1
=
0,forcin g funct ion
U
=
2):isolated po ints and
periodic
--
-
..
- -
--,-.-_._._-
(7.3) w ={O -1 2 2 2 -1 O}
o
=
10,
I
=
1,
forci ng
funct ion
(J
=
2):isolated po ints and
periodic
(7.2)
w ~{- l
-3 2 2 2 -3 -1}
o
=
10,
"'1
=
1,
for cing function
(f
=
2):isolated points and
peri odi c
(7.4)
w= {O-1 2 2 2 -1 O}
8
=
10,
"f
=
1,forcin g function
(J
=
2):isolated po ints and
periodic;asynchronous
calculat ion:A
=
10
Figure 7:The addi tion of a forcing input
to
the model The forcing
input
is
shown as a single line line below each system.All initial
configurations were random.
A
Cellular
Automaton
Model Based
on Cortical
Physiology
197
(8.1b) evolution in time
of the
cent er
row of cells
0 0
- 1 - 1 - 1
0 0
0
- 1
-2
-2 - 2
- 1
0
- 1
- 2
4 4 4
- 2
- 1
A;
- 1
- 2
4
8
4
- 2
- 1
- 1
- 2
4 4 4
- 2
- 1
0
-1
- 2 -2 - 2
- 1
0
0 0
- 1 -1
-1
0 0
Figure 8:Some examples of
two-dimensional
systems.The configu­
rations shown are either the final (stable) state or the one-hundredth
generation.
A
cross -section through the centre of the system is also
shown beside the final configuration.The weights mat rix A used are
shown here.
198 Martin
1.
Hofmann
8.1 Physiological basis of the model
The basic elements of t he model are the cells of the array which constitute
t he cellular automaton.The correspondence of these cells is not necessarily
one-t o-one with neurons in the nervous system.The cellular aut omaton
cells may be considered rather to correspond with a small group of neurons
(module) within a hypercolumn of t he cortex.For example,the module
may correspond t o a group of cells such as t hose shown in the basic circuit
di agram in figur e 1.
Most neurons require a minimum level of input before they fire;this is
called the threshold.A second effect of threshold is to shift t he stimulus­
response curve to higher val ues,this effect was incorpor ated int o the pr esent
model.The weights represent t he st rengt h of the connect ions between
neighb ouring modules.These weights are assumed t o be the same for all
modules and to be relatively const ant on a short time scale.Changes in t he
weights over long time scales may be,at least in part,a basis for lear ning.
The stimulus-response function represents the input-output relations of
this module.In t he case where the SR function is based on t he logistic
curve
('"1
= 0),t he module may represent a single neuron or a group of
simi lar ne urons.This type of curve is seen at
different
levels of the nervous
system,from t he level of the synapse [13J,t o t he level of psychophysics
whi ch represents the functioning of many parts of the brain [141.However,
the peaked SR functions
(1
>
0) are unlikely to imp lemented in a single
type of neuron and may correspond to the interactions of both excitatory
and inhibi tory neurons.For example t he sum of an excitat ory neuron
and a higher t hreshold inhibi tory neuron could produce these types of SR
functions.
The extensions to the basic model were des igned to pr oduce a more
realistic model of the cortex.The basic model in 2-dimensions corresponds
to a sheet of cortical tissue,however,the cortex does not exist as an iso­
lated sheet.The input t o a region of t he cortex may come from a distant
region,e.g.input t o t he visual cort ex comes from t he lateral geniculate
nucleus.The addit ion of a forcing function t o t he model is intended to
simulate this input,alt hough t he restrict ion to t ime invar iant input is itself
a simplification.The ext ension to asynchronous mode of calculation was
introduced because the cortex,unlike a computer,does not have a single
clock controlling all cells.The method of asynchronous calculation used
here also introduces a stochast ic element int o the model.
8.2 Rel a tion of t h e model with p hysiology
I have tried to t o show that this model has a reasonable basis in known
physiology,but the acid t est of any model is whether or not it can pr edict
t he behaviour of the system being modelled.Unfortunately,at the present
st at e of t echnology the measurement of t he individual states of a large
group of neurons
is
imposs ible.Nevert heless,some general observations
can be made.
A Cell ular
Automaton
Model Based
on
Cortical Physiology
199
Systems that have a high t hreshold or weights wit h outside inhi bit ion
tend to produce localized structures.The excitation of a region of space
does not propagate to dist ant regions,this is also true of cortical regions un­
der normal condi t ions.Under cer tain pathological conditions (e.g.epilepsy,
hallucinat ions) this is no longer true and uncontrolled spread of excitat ion is
observed.Reduction or elimination of t he inhibitory porti ons of the weights
appears to produce just t his kind of behaviour.Ermentrout and Cowan 1151
have proposed a model of visual ha llucinat ions where t he t riggering insta­
bility is decreased inhibition and increased excitation.Babloyantz,Salazar
and Nicolis [16) have shown that EEG measurements of the brain under
various condit ions give different values for the dimension of t he underlying
dynamics.In particular t he arousal stat e of t he organism
affects
the di­
mensionality.Increasi ng t hreshold reduces t he excitability of the cells and
reduces the entropy or dimensionality.
The cont rol of threshold an d the ratio of excit at ion t o inhibition could
occur in at least two different manners in t he nervous system.First,there
are a number of non-specific fibr es originating in t he brain stem and ter­
minating in a relatively large region of cortex.These fibres may be used to
change t he parameters of cells in a given region of cortex or t he ent ire cor­
tex.Second,there are various chemical fact ors called neuromodulat ors that
can react with receptors on the cells'surface and change t heir behaviour.
Systems that use a simple logist ic based SR funct ion tend t o stabilize
very quickly.The final stable states can be compared to point attractors
in some phase space.The basin of attracti on consists of those initial con­
figurations that closely resemble one ano t her.These types of sys tems seem
to provide a better model of t he normal functions of the br ain in that th e
systems are resistant t o noise and will produce an out put t hat is relat ed to
the input.Chaotic and Class IV systems are,by defini tion,sensit ive to the
initial conditions (input).
9.Conclusions
Cortical physiology provides the basic parameters used t o const ruct t he
model presented here.The model is not intended t o accurat ely model
any real nervous sys tem,but rat her t o examine overall pr operties of a
connected group of discret e neuron-like elements;A relat ively small number
of parameters,similar t o th ose t hat might be found in a nervous syst em,
is used to determine the rule used by the model.Despite the fact that t his
model
is
more rest rict ed in its rules t han many cellular automata,where
an output state can be assigne d to each individual input configurat ion,it
can nevertheless exhibit a wide range of behaviour as the paramet ers are
varied.Some types of behaviour may be applicable t o cort ical funct ion,for
example the localizat ion of excitation and the compression of informat ion
that occur in the systems showing stablizing behaviour.
200
Martin
1.
Hofmann
Acknowledgements
I wish to thank P.Hallett,R.Hansell,A.Jepson and H.Kwan for stimu­
lating discussi on and advice.This work was made possible
by
equipment
obtained by P.Hallett under MRC grant number MT 7673.
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11]
J.J.
Kulikowski,S.Marcelja and P.O.
Bishop,
"Theory of spatial positi on
and spatial frequency relations
in
the receptive fields of simple
cells
in
the
visual cortex"
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Biological Cybernetics,
43 (1982) 187-198.
[2) D.H.Hubel and T.N.Wiesel,"Receptive fields,binocular interaction and
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I
Journal
of
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160 (1962) 106-154.
!3]
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Jones,«Anatomy of
t he cerebral cortex:columnar input-output organi­
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in
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Organization
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Schmidt,
F.G.Worden,G.Adelman and S.G.Dennis,(MIT Press,Cambridge,
1981).
f4}
J.
Szentagothai,
"The
modular architectonic principle of centers",
Review
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Physiology,Biochemist ry
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98 (1983) 11-61.
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G.M.Shepherd,
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