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Andr'es Fraguela Collar, Juan Alberto Escamilla
A MATHEMATICAL MODEL TO STUDY THE ACTIVACION
PROCESS IN THE CELEBRAL CORTEX
Introduction
The first step requires analysis of neuroscientific models on different scales.
Subsequently we will consider a non­linear distributed model on a macroscopic
scale, for the interaction of cortical neurons in between which action potentials
travel with finite speed of propagation, without taking into account synaptic
delays. The model includes several important physiological parameters such
as: density of assosciation fibres, density of synaptic contacts, decreasing
scales of connections, firing threshold, magnitude of excitatory and inhibitory
postsynaptic potentials, etc. The model describes the space­time evolution
of stimulated synaptic activity in the presence of a local process of lateral
inhibition. A non­linear system of hyperbolic equations is obtained for the
excitation--inhibition process, which when applying results from the theory of
perturbations can be reduced into one single equation for a new variable acti­
vation calling. This model can provide results about the existence of rhythms
(stationary waves) and other physiological states (travelling waves), control
of the functional activity of the cerebral cortex (control) and reactions from
the cortex towards certain stimuli from structures within the brain (periodical
solutions).
1. Deduction of the mathematical model
We will begin by stating some basic physiological considerations for the
model's construction. The cerebral cortex has a stratified structure made
up of six layers of neurons, grouped in units known as columns, which in­
teract amongst each other. The first estimation is that cortical neurons can
be divided into two large categories: main neurons which establish cortical
connection or distances measuring between one to 20 centimeters, such as
neuronal pyramids, and interneurons which only have intracortical neurons
and work with a radius of less than a millimeter. The main cells carry out the
excitatory synapses while the interneurons carry out the inhibited synapses on
the main cells. We suppose that the afferent inputs to the cortex, proceeding
from the brain's inferior structures are only excitatory and we will put to one
47

side the synaptic delays which are of no interest for a model of global activity.
To simplify the model we will consider a unidimensional band of the cortex,
which we will represent by the whole R of real numbers. In this way the
model's parameters and variables are the following (signs +, \Gamma correspond to
the excitatory and inhibited activity respectively):
– \Gamma1
\Sigma ­ Longitude scale of connections decrease (cm);
V \Sigma ­ Postsynaptic potentials produced by synapses (mV );
R \Sigma (x; x 0 ; v) ­ Density of corticortical fibers (+) and intracortical fibers (\Gamma)
which connect localizations in x in R and x 0
in R, through which the velocity
of propagation of action potentials is v (seg=cm 3 );
F \Sigma (v) ­ Density of distribution of velocities of propagation of action poten­
tials (seg=cm 3 );
g 0 (x; t) ­ Fraction of total of neurons in localization x that are active in
instant t (adimensional);
h \Sigma (x; t) ­ Lineal density of active synapses in instant t on the neurons
located in x (1=cm);
S \Sigma (x) ­ Lineal density of synapses in localization x (1=cm);
h 0 (x; t) ­ Lineal density of excitatory synaptic afferent input to the cortex
in localization x in instant t (1=cm).
Supposing for simplicity that
F \Sigma (v) = ffi(v \Gamma v \Sigma ) ;
where ffi is the delta of Dirac and v \Sigma are the characteristic velocities of propa­
gation of the action potentials in the excitatory and inhibitory cells and that
besides have the following expression
R \Sigma (x; x 0 ; v) = – \Sigma
2 S \Sigma (x)e \Gamma– \Sigma jx\Gammax 0 j ffi(v \Gamma v \Sigma ) ; (1)
we obtain the system
h+ (x; t) = h 0 (x; t) + –+
2 S+ (x)
Z 1
\Gamma1
e \Gamma– + jx\Gammax 0 j g(x 0 ; t \Gamma jx \Gamma x 0 j
v +
)dx 0 ; (2)
h \Gamma (x; t) = – \Gamma
2 S \Gamma (x)
Z x+''
x\Gamma''
e \Gamma– \Gamma jx\Gammax 0 j g(x 0 ; t \Gamma
jx \Gamma x 0 j
v \Gamma
)dx 0 ; (3)
where the parameter ('' ? 0) corresponds to the hypotheses that inhibitory
activity is short range.
48

2. Reduction and simplification of the system (2), (3)
If in (??) and (??) we divide both sides by S+ (x) and S \Gamma (x) respectively
and we apply the operator @ 2
@x 2 , then for the new unknown functions
H \Sigma (x; t) = h \Sigma (x; t)
S \Sigma (x)
the equivalent system of differential equations is obtained
@ 2 H+
@t 2
+ 2–+ v +
@H+
@t
+ – 2
+ v 2
+ H+ \Gamma v 2
+
@ 2 H+
@x 2
= (4)
F 0 (x; t) + – 2
+ v 2
+ g 0 (x; t) + –+ v +
@g 0
@t
(x; t) ;
@ 2 H \Gamma
@t 2
+ 2– \Gamma v \Gamma
@H \Gamma
@t
+ – 2
\Gamma v 2
\Gamma H \Gamma \Gamma v 2
\Gamma
@ 2 H \Gamma
@x 2
= G 0 (x; t; '') ; (5)
where
F 0 (x; t) = @ 2 H 0
@t 2
+ 2–+ v +
@H 0
@t + – 2
+ v 2
+ H 0 \Gamma v 2
+
@ 2 H 0
@x 2
;
H 0 (x; t) = h 0 (x; t)
S+ (x) ;
G 0 (x; t; '') = – 2
\Gamma v 2
\Gamma g 0 (x; t) + – \Gamma v \Gamma
@g 0
@t
(x; t)
\Gamma 1
2 – \Gamma v \Gamma e \Gamma– \Gamma ''
(
– \Gamma v \Gamma
''
g 0
/
x + ''; t \Gamma ''
v \Gamma
!
+ @g 0
@t
/
x + ''; t \Gamma ''
v \Gamma
!#
+
''
g 0
/
x \Gamma ''; t \Gamma ''
v \Gamma
!
+ @g 0
@t
/
x \Gamma ''; t \Gamma ''
v \Gamma
!#)
\Gamma
1
2
– \Gamma v 2
\Gamma e \Gamma– \Gamma ''
(
@g 0
@x
/
x + ''; t \Gamma ''
v \Gamma
!
\Gamma @g 0
@x
/
x \Gamma ''; t \Gamma ''
v \Gamma
!)
:
If we suppose that g 0 and grad(g 0 ) are continuous functions, then G 0 (x; t; '')
converge when '' ! 0, uniformly at zero on compacts sets of R 2 . From this
it can be concluded that the solution of Cauchy's problem associated to the
equation (??) converges uniformly on compacts sets of R 2 to solution the
corresponding Cauchy problem when (??) is substituted G 0 (x; t; '') by zero.
With this in mind and moving on to adimensional variables
Ü = –+ v + t ; y = –+x
49

starting from (??) and (??) we obtain the following simplified model
@ 2 u
@Ü 2
+ 2 @u
@Ü
+ u \Gamma @ 2 u
@y 2
= V+
''
g(y; u) + @g
@u
(y; u)
@u
@Ü
#
(6)
+f 0 (y; Ü ) + C 0 (y; Ü )
for the new variable
u(y; Ü ) = V+H+ (y; Ü ) \Gamma V \Gamma H \Gamma (y; Ü );
which we will call the activation variable. In (??) we have considered that
g 0 (x; t) = g(y; u) where function g, which is associated to neuronal activity
for a determined physiological state is for each y fixed a sigmoid function of
the activation variable. Besides the term
f 0 (y; Ü ) = V+
– 2
+ v 2
+
F 0
/
y
–+
;
Ü
–+ v +
!
corresponds to the excitatory input produced by the brain's interior structures
and
C 0 (y; Ü ) = \GammaV \Gamma
2
4
@
@Ü 2
+ 2 @
@Ü
+ 1 \Gamma @ 2
@y 2
3
5 H \Gamma
/
y
–+
;
Ü
–+ v +
!
is the control produced by the inhibitory activity. It must be noted that on
considering
G 0 (x; Ü; '') j 0
on the right hand side of (??) , inhibitory control C 0 (y; Ü ) depends uniquely
on the initial state of inhibition.
3. Some results obtained from model (??)
Firstly we will suppose that f 0 (y; Ü ) +C 0 (y; Ü ) j 0 ; g(y; u) = g(u) in (??)
and then study the existence of travelling waves
u(y; Ü ) = '(y \Gamma cÜ )
in the resulting model and that they travel at speed c. In this case placing
y \Gamma cÜ = ¸ ; ' 0 (¸) = /(¸) ;
equation (??) is reduced to system
d
dÜ
0
@
'
/
1
A =
0
B B B @
0 1
1
1\Gammac 2
\Gamma2c
1\Gammac 2
1
C C C A
0
@
'
/
1
A + V+
1 \Gamma c 2
0
B B B @
0
cg 0 (')/ \Gamma g(')
1
C C C A
: (7)
50

The existence of travelling waves of (??) is equivalent to the existence of
bounded solutions of (??) differing to the stationary solution.
Theorem 1. We consider that in (??) is
g = g(u) =
i
1 + e \Gammaffu+fi j \Gamma1
; ff ; : fi ? 0;
and also
f 0 (y; Ü ) + C 0 (y; Ü ) j 0:
We suppose that 2V+ ff ! 1 and also in this case ' 0 ? 0, the only constant
solution of the equation (??). Then for each ae ? max(' 0 ; 1
2ff
) exist numbers
0 ! c 1 (ae) ! c 2 (ae) ! 1 so that c 1 (ae) ! 0 and c 2 (ae) ! 1 when ae ! 1 and
satisfy the following properties: if 0 ! c ! c 1 (ae) the only bounded solution of
(??) contained in
j'j 2 + j/j 2 Ÿ ae 2
is the stationary solution
0
@
' 0
0
1
A , and if
c 1 (ae) Ÿ c Ÿ c 2 (ae) ;
then homoclinical solutions exist of (??) contained in
j'j 2 + j/j 2 Ÿ ae 2
with asymptotic value ' 0 , when '' ! \Sigma1 ; and that correspond to travelling
waves of (??) with speed c.
Proof. If G(Ü; c) denotes the main Green principle that corresponds to the
matrix 0
B
B
B
@
0 1
1
1\Gammac 2
\Gamma 2c
1\Gammac 2
1
C
C
C
A
;
then the bounded solutions of (??) coincide [1, page 81] with the solutions of
the non­linear integral equation
0
B B B
@
'(¸)
/(¸)
1
C C C
A
= V+
1 \Gamma c 2
Z 1
\Gamma1
G(¸ \Gamma Ü; c)
0
B B B
@
0
cg 0 ('(Ü ))/(Ü ) \Gamma g('(Ü ))
1
C C C
A
dÜ: (8)
If
V+
1 \Gamma c 2
jcg 0 (' 1 )/ 1 \Gamma g(/ 1 )j Ÿ M
51

and
V+
1 \Gamma c 2
j(cg 0 (' 1 )/ 1 \Gamma g(' 1 )) \Gamma (cg 0 (' 2 )/ 2 \Gamma g(' 2 )))j Ÿ
q(j' 1 \Gamma ' 2 j 2 + j/ 1 \Gamma / 2 j 2 ) 1=2
for
j' i j 2 + j/ i j 2 Ÿ ae 2 :; i = 1 ; 2;
and besides
k G(Ü; c) kŸ Ne \Gammaš jÜ j ;
then the condition for the integral operator in (??) is contracting in
B ae =
8
? ? ? !
? ? ? :
0
B B B @
'(¸)
/(¸)
1
C C C A
: R ! R 2 continues j'(¸)j 2 + j/(¸)j 2 Ÿ ae 2
9
? ? ? =
? ? ? ;
is that satisfies the inequalities
M Ÿ aeš
2N ; q !
š
2N : (9)
The inequations system (??) are satisfied for 0 ! c ! c 1 (ae) and in this case
the only bounded solution of (??) is the stationary solution
0
@
' 0
0
1
A . To prove
the second part of the theorem we denote by K
0
@
'
/ ; c
1
A the integral operator
in (??). Then, there is c 2 (ae) ? c 1 (ae) such that for 0 ! c Ÿ c 2 (ae), K is an
operator of B ae in B ae , and their differential
D i '
/
j K
i i ' 0
0
j
; c
j
has a norm less than 1 when
c 1 (ae) Ÿ c Ÿ c 2 (ae):
Applying the theorem of the implicit function to operator K in B ae it can be
obtained for each ~
c in a neighborhood of c
(c 1 (ae) Ÿ c Ÿ c 2 (ae));
that there is a bounded solution
0
B B B @
'(¸; ~
c)
/(¸; ~ c)
1
C C C A
52

of (??) different to
0
@
' 0
0
1
A , which corresponds to a homoclinical solution of
(??) and therefore, defines a travelling wave of (??) with speed ~ c. We will
consider in what follows a small portion of cortex where we can suppose that
there is no spatial variation of the activation function. Then from (??) we
obtain for u = u(Ü ) the following Lienard equation
d 2 u
dÜ 2
+ (2 \Gamma V+ g 0 (u)) du
dÜ
+ (u \Gamma V+ g(u)) = f 0 (Ü ) + c 0 (Ü ): (10)
Note that on eliminating the dependence of the spatial variable y, one has to
c 0 (Ü ) = \GammaV \Gamma
/
– \Gamma v \Gamma
–+v+
\Gamma 1
!
e \Gamma – \Gamma v \Gamma
– + v +
Ü
''/
a
–+v+
Ü + b
! /
– \Gamma v \Gamma
–+v+
\Gamma 1
!
\Gamma
2a
–+v+
#
; (11)
where a and b are parameters of control that depend on the initial conditions
of inhibition.
Lemma 1. The solution of the equation (??) which satisfies the initial
conditions u(0) = u 0 , u 0 (0) = u 0
0 , coincides with the solution of the equation
of first order
du
dÜ
+ u \Gamma V+ g(u) = (f(Ü ) + c(Ü ) +K)e \GammaÜ ; (12)
which satisfies the initial condition u(0) = u 0 where
f(Ü ) =
Z Ü
0
f 0 (t)e t dt ; c(Ü ) =
Z Ü
0
c 0 (t)e t dt ;
K = u 0
0 + u 0 \Gamma V+ g(u 0 ) :
The previous lemma allows us to prove the following theorem.
Theorem 2. If (f(Ü ) + c(Ü ) + k)e \GammaÜ is a periodic function (something
which occurs if the afferent input f 0 (Ü ) is asymptotically periodic) and also
V+ g 0 (u) ! 1 : (u 2 R) :; then for any value K 2 R there is a unique periodic
solution of (??) with the same previous period whose initial conditions satisfy
the equality
K = u 0
0 + u 0 \Gamma V+g(u 0 ):
Proof. Equation (??) is proved to have bounded solutions for which
Banach's theorem of fixed point is used whose application requires condition
V+ g 0 (u) ! 1. Therefore from the periodicity of (f(Ü ) + c(Ü ) + k)e \GammaÜ [3, page
116] the existence of and unicity of periodic solution is deduced for (??) and
from lemma 1 the affirmation of the theorem is concluded.
53

Theorem 3. Equation (??) is controllable regards parameters a and b,
that is to say for any instant T ? 0 , initial conditions u 0 :; : u 0
0
and final
conditions u T and u 0
T
, exist parameters a and b :
a = a(u 0 ; u 0
0 ; u T ; u 0
T
; T ) ; b = b(u 0 ; u 0
0 ; u T ; u 0
T
; T )
for which the solution u(Ü ) of (??) which
u(0) = u 0 and u 0 (0) = u 0
0
executes also satisfies
u(T ) = u T and u 0 (T ) = u 0
T
:
Proof. If the solution of (??) satisfying initial conditions
u(0) = u 0 ; u 0 (0) = u 0
0
is denoted for u(Ü; a; b), and we consider the transformation
U : R 2 ! R 2 ;
U
i a
b
j
=
0
B B B @
u(T ; a; b)
du
dÜ
(T ; a; b)
1
C C C A
;
then the Jacobean matrix DU has a determinant different from the zero of
the whole R 2 . From this we can conclude that the image of U is a non­empty
open subset R 2 . On the other hand, if the sucesion
0
B B B @
u(T ; a n ; b n )
du
dÜ
(T ; a n ; b n )
1
C C C A
is convergent of the continuous dependency of the solution of (??) regards
the parameters a and b we can conclude that the limit belongs to the image
of U . With this we conclude that the image of U is whole R 2 , from which we
obtain equation (??) which is controllable.
54

4. Conclusions
Theorem 1 corresponds to the existence of physiological states which prop­
agates itself on the cortex with a velocity lower than that of the propagation
of the excitatory action potentials, just as seems to occur with rhythms of
cortical activity [5], [6].
Theorem 2 tells us that cortical activity of a piece of cortex can join with
that of the sustained periodic afferent input as has been obtained experimen­
tally ([3], [4]). In [2] we also conclude the controlling role, of the inhibitory
activity in the cortex, which is confirmed in theorem 3.
References
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[2] Dickson C.T., Alonso A. Muscarinic induction of Synchronous popu­
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[3] Hale J. and Kocak H. Dynamics and Bifurcations. Springer Verlag, New
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[4] Lukatch H.S. and Mac Iver B. Physiology, Pharmacology and Topog­
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[5] Nu~nez, P. L. and Katznelson, R. D. Electric Fields on the Brain, the
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55