Adv Syst Sci Appl 2023; 03:177–190
Published online at https://ijassa.ipu.ru.

On Bi-Laminar Neural Field Models of Electrical Activity
in the Primary Visual Cortex

Evgenii Burlakov1,2*, Ivan Malkov1

1Tyumen State University, Tyumen, Russian Federation
2 Derzhavin Tambov State University, Tambov, Russian Federation

Abstract: We investigate the modelling framework for studying electrical activity in the primary
visual cortex of the brain based on a bi-laminar neural field equation. The deep layer of the neural
field models the orientation-independent electrical activity, whereas the orientation-dependent
superficial layer captures the selectivity to spatially oriented stimuli of the orientation columns in
the primary visual cortex. We verify the solvability of a Cauchy problem for the bi-laminar neural
field equation with both sigmoidal and Heaviside-type neuronal activation. We also construct
connections between the solutions that correspond to these types of neuronal activation, which
justifies the use of the Heaviside-type neuronal activation functions that is crucial in the problems
of computer simulations involving vast ensembles on neurons. We prove the possibility of a
correct approximation of the bi-laminar neural field model with a two-layer neuronal network.
We also highlight some perspectives opened by the results of the present research related to the
studies of travelling waves of evoked electrical activity in the visual cortex as well as the neural
activity control problems in the framework of the neurofeedback paradigm.

Keywords: mathematical models of primary visual cortex, bi-laminar neural field models, two-
layer neuronal nerwork models, well-posedness, Heaviside activation function

1. INTRODUCTION

Mathematical models of macro- and mesoscopic neuronal activity of the human brain cortex
involve the description of electrical activity of vast ensembles of neuronal elements, which
can be registered using electro- and magnetoencefalography (EEG and MEG) [1, 2] and
indirectly observed in fMRI recordings [3], are usually presented in the form of neural field
equations (see e.g. the pioneering work [4] and the review [5]). The most well-known neural
field model is the Amari neural field equation (see [4])

∂tu(t, x) = −τu(t, x) +
∫
Ω

ω(x, y)g(u(t, y))dy. (1.1)

Here u(t, x) represents the level of electrical activity in the neural field Ω at time t and position
x, the so-called connectivity function ω defines the strengths of interneuronal connections in
the neural field, the value g(u) determines the probability of activation (firing) of a neuron
with electrical activity level u. In the mathematical neuroscience community, the connectivity
ω is typically assumed to be an exponentially decaying function symmetric with respect to
the vertical axis or a sum of such functions, and the activation function f is taken to be a
continuous sigmoidal-shaped function.

∗Corresponding author: eb @bk.ru



178 E. BURLAKOV, I. MALKOV

Several extensions of (1.1) relying on the fact that the neural media is not spatially
uniform formalize the heterogeneity of the brain cortex in general forms (see e.g. [6–8]). Such
mathematical models are rather well-studied, including the issues of well-posedness [8, 9],
construction and justification of the schemes for numerical simulations [8–10], studies of
special types of solutions that are physiologically relevant [6, 7, 11, 12]. However, none of
these neural field models could capture the characteristical features of primary visual cortex,
whose elements of microstructure are selective with respect to perceptions of visual stimuli
of certain directions, before the introduction of the following model (see [13]):

∂tud(t, x) = −τdud(t, x) +
∫
Ω

ωd(x, y)gd(u(t, y))dy

+νd

π
2∫

−π
2

gs(us(t, x, ψ))dψ,

∂tus(t, x, φ) = −τsus(t, x, φ) +
∫
Ω

π
2∫

−π
2

ωs(x, φ, y, ψ)gs(u(t, y, ψ))dψdy

+νsgd(ud(t, x)).

(1.2)

Here ud(t, x) defines the level of orientation-independent activity in the deep layer and
us(t, x, φ) is the orientation-dependent activity in the superficial layer. The connectivities in
the two layers are denoted by ωd and ωs, respectively, and the corresponding time constants
are given by τd and τs. We also include vertical inputs from the deep to the superficial layer
with the strength νd and back from the superficial to the deep layer with the strength νs
(the latter is averaged with respect to the orientation preference of neurons in the superficial
layer). Functions gd, gs are probabilistic functions of firing (neuronal activation functions) for
the deep layer and the superficial layer, respectively. We refer the reader to the work [13] for
more details on the biophysical justification of the modeling framework (1.2).

To the best of our knowledge, the work [13] is the only published study capturing
the orientation selectivity in the visual cortex in the framework of the neural field setting.
However, several important mathematical issues were overlooked in this study. The aim of the
present paper is to fill in these mathematical blank spots in the justification of the modelling
framework (1.2). Namely, in Section 2 we verify the solvability of a Cauchy problem for (1.2)
and prove the possibility of correct spatially discretized approximation of (1.2). Section 3
deals with the solvability problem for (1.2) in the case when the probabilistic activation
functions are replaced with the instant-activation Heaviside-type functions. In Section 4 we
establish connections between the modelling approaches of the two previous sections. In
Section 5 we highlight some close perspectives opened by the results of the present research.

2. BI-LAMINAR NEURAL FIELD MODEL WITH CONTINUOUS ACTIVATION
FUNCTIONS

For convenience of the presentation of the forthcoming material, we introduce the following
notations. For any metric space Λ, any λ0 ∈ Λ, S ⊂ Λ, and r > 0, we define BΛ(λ0, r) to be
the ball in the space Λ of the radius r > 0 centered at λ0, S to be the closure of S in Λ. We
denote by Rm the m-dimensional real vector space with the norm | · | and by Ω – a compact
subset of R2. Denote by Lp(Ω× (−π

2
, π
2
],R2) and Cp(Ω× (−π

2
, π
2
],R2) the Banach spaces

of integrable and continuous, respectively, functions from Ω× (−π
2
, π
2

to R2), which are π-
periodic in the second variable. For all T > 0, we denote ΞT = [0, T ]× Ω× (−π

2
, π
2
] and

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON BI-LAMINAR NEURAL FIELD MODELS 179

defineBCp(ΞT ,R2) to be the Banach space of bounded continuous functions ϑ : ΞT → R2 π-
periodic in the third variable with the norm ||ϑ||BCp(ΞT ,R2) = max

x∈ΞT

|ϑ(x)| and L∞,p(ΞT ,R2×2)

– to be the Banach space of 2× 2 matrix functions from ΞT to R with essentially bounded
components π-periodic with respect to the third variable.

In this section we consider the solvability of a Cauchy problem for the bi-laminar neural
field equation (1.2) in the case of continuous activation functions (i.e. gd = fd, gs = fs) and
justify the possibility to approximate the bi-laminar model (1.2) with a two-layered neuronal
network. We start out with the system

∂tud(t, x) = −τdud(t, x) +
∫
Ω

ωd(x, y)fd(ud(t, y))dy

+νd

π
2∫

−π
2

fs(us(t, x, ψ))dψ,

∂tus(t, x, φ) = −τsus(t, x, φ) +
∫
Ω

π
2∫

−π
2

ωs(x, φ, y, ψ)fs(us(t, y, ψ))dψdy

+νsfd(ud(t, x))

(2.3)

together with the initial condition

ud(0, x) = ûd(x), us(0, x, φ) = ûs(x, φ) (2.4)

where ûd : Ω → R, ûs : Ω× (−π
2
, π
2
] → R are continuous and lim

φ→−π
2

ûs(x, φ) = ûs(x,
π
2
).

Assume that
(Aω) The neuronal connectivity functions ωd, ωs : Ω× Ω → Rn are continuous;
(Af ) The neuronal activation functions fd, fs : R2 → [0, 1] are Lipschitz continuous;
Theorem 2.1:
Let assumptions (Aω) and (Af ) be satisfied. Then there exists a unique solution of the
problem (2.3), (2.4), which is a continuous function from [0,∞)× Ω× (−π

2
, π
2
] to R2.

Proof
The system (2.3) can be written as

∂tu(t, x, φ) = −τu(t, x, φ) +
∫
Ω

π
2∫

−π
2

V (x, y, φ, ψ)F (u(t, x, φ))dψdy, (2.5)

where

F (u) =

(
fd(ud)
fs(us)

)
, V (x, y, φ, ψ) =

(
ωd(x, y)/π νsδ(x− y)
νdδ(x− y)/π ωs(x, φ, y, ψ)

)
.

We will consider the problem of solvability of the system (2.3), (2.4) in a more general
setting that reads as follows:

u(t, x, φ) = û(x, ψ) +

t∫
0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)F (u(s, y, ψ))µ1(dψ)µ2(dy)ds (2.6)

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



180 E. BURLAKOV, I. MALKOV

where

u(t, x, φ) =

(
ud(t, x)
us(t, x, φ)

)
, û(x, φ) =

(
ûd(x)
ûs(x, φ)

)
,

W (t, s, x, y, φ, ψ) =

(
exp(−τd(t−s)) 0

0 exp(−τs(t−s))

)
V (x, y, φ, ψ),

and µ1 and µ2 are complete σ-additive measures defined on R and R2, respectively, and finite
on bounded subsets of their ranges of definition.

Introduce the following two conditions:

(AW ) For any T > 0 and any (t, x, φ) ∈ ΞT , the function W (t, ·, x, ·, φ, ·) belongs to
L∞,p(ΞT ,R2×2), the function (t, x, φ) 7→ ∥W (t, ·, x, ·, φ, ·)∥L∞,p(ΞT ,R2×2) is bounded,
and for any measurable set I ⊂ ΞT and any (t0, x0, φ0) ∈ ΞT , it holds true that

lim
(t,x,φ)→(t0,x0,φ0)

∫ ∫ ∫
I∩
(
[0,t]×Ω

) W (t, s, x, y, φ, ψ)µ1(dψ)µ2(dy)ds

=

∫ ∫ ∫
I∩
(
[0,t0]×Ω×(−π

2
,π
2
]
) W (t0, s, x0, y, φ0, ψ)µ

1(dψ)µ2(dy)ds.

(AF ) The function F : R2 → [0, 1]2 is Lipschitz continuous (hereinafter, we denote [0, 1]2 =
[0, 1]× [0, 1]).

Choose T > 0. We define uT ∈ BCp(ΞT ,R2) to be a T -local solution to (2.6) if uT
satisfies the equation (2.6) on the set ΞT . We consider a continuous function u∞ : [0,∞)×
Ω× (−π

2
, π
2
] → R2 to be a global solution to (2.6) if for any T > 0, the restriction of u∞ to

ΞT is a T -local solution to (2.6).
Let us formulate the statement on solvability of the integral equation (2.6) in terms of the

definitions given above.

Lemma 2.1:
Let assumptions (AW ) and (AF ) be satisfied. Then for any T > 0, the equation (2.6) has a
unique T -local solution that is the restriction to ΞT of the unique global solution to (2.6).

Proof of Lemma 2.1
We choose arbitrary T > 0 and take any two functions u1, u2 ∈ BCp(ΞT ,R2). We esti-

mate ∥ITNTu1 − ITNTu2∥BCp(ΞT ,R2) = max
(t,x,φ)∈ΞT

|(ITNTu1)(t, x, φ)− (ITNTu2)(t, x, φ)|,

where for any ξ ∈ BCp(ΞT ,R2), the operator IT is defined by the relation

(IT ξ)(t, x, φ) =

t∫
0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)ξ(s, y, ψ)µ1(dψ)µ2(dy)ds (2.7)

and, due to condition (AW ), has the image in BCp(ΞT ,R2) (see e.g. [14], Chapter 3, § 5.5);
the operator NT defined as (NT ξ)(t, x, φ) = F (ξ(t, x, φ)) acts from the space BCp(ΞT ,R2)
to itself due to (AF ). By the virtue of (AF ), we have

max
(t,x,φ)∈ΞT

|(ITNTu1)(t, x, φ)− (ITNTu2)(t, x, φ)|

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON BI-LAMINAR NEURAL FIELD MODELS 181

= max
(t,x,φ)∈ΞT

∣∣∣∣∣
t∫

0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)F (u1(s, y, ψ))µ
1(dψ)µ2(dy)ds

−
t∫

0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)F (u2(s, y, ψ))µ
1(dψ)µ2(dy)ds

∣∣∣∣∣
≤ max

(t,x,φ)∈ΞT

t∫
0

∫
Ω

π
2∫

−π
2

|W (t, s, x, y, φ, ψ)||F (u1(s, y, ψ))− F (u2(s, y, ψ))|µ1(dψ)µ2(dy)ds

≤ max
(t,x,φ)∈ΞT

t∫
0

∫
Ω

π
2∫

−π
2

|W (t, s, x, y, φ, ψ)|lf |u1(s, y, ψ)− u2(s, y, ψ)|µ1(dψ)µ2(dy)ds

≤ lF max
(t,x,φ)∈ΞT

t∫
0

∫
Ω

π
2∫

−π
2

|W (t, s, x, y, φ, ψ)|µ1(dψ)µ2(dy)ds ∥u1(s, y, ψ)− u2(s, y, ψ)∥

where lf > 0 is the Lipschitz constant of F . By choosing T = T1 > 0 in a way that

lF max
(t,x,φ)∈ΞT1

T1∫
0

∫
Ω

π
2∫

−π
2

|W (t, s, x, y, φ, ψ)|µ1(dψ)µ2(dy)ds < 1

and applying Banach fixed point theorem (see e.g. [15], Chapter 2, § 14) to the problem

u = IT1NT1u (2.8)

in the space BCp(ΞT1 ,R2) we prove the existence of a unique solution uT1 ∈ BCp(ΞT1 ,R2)
of (2.8), which is a unique T1-local solution of the integral equation (2.6).

We introduce a new time-variable t′ = t− T1. Now we consider the problem (2.6) with
t = t′. We will refer to this new problem as (2.6)′ (with the initial problem u′d(−T1, x) =
û′d(x), u

′
s(−T1, x, φ) = û′s(x, φ)). We apply the procedure described above and prove the

existence of a T ′
1-local solution uT ′

1
to (2.6)′ for some T ′

1 > 0. We thus obtain a T2-local

solution uT2 to (2.6) where T2 = T1 + T ′
1, u =

{
uT1 t ∈ [0, T1]
uT ′

1
t ∈ [T1, T2]

. The T2-local solution uT2

is continuous at (T1, x) for any x ∈ Ω, that is, uT2 ∈ On the next step, we choose any T2-local
solution uT2 ∈ to the equation (2.6) We introduce a new time-variable t′′ = T − T2 and repeat
the procedure

We thus obtain a strictly increasing sequence {Ti}, i = 1, 2, . . . and the corresponding
sequence of of local solutions uTi

, i = 1, 2, . . . such that for any i1 < i2, uTi1
is the restriction

of uTi2
∈ BCP (ΞTi2

,R2) to the set ΞTi1
. We find lim

i→∞
Ti = T̂ . Take any t∗ ∈ (0, T̂ ). For

some number i, t ∈ (Ti−1, Ti) and ut∗ therefore is a t∗-local solution. We have constructed
the mapping t∗ 7→ ut∗ . Prove that {Ti}, i = 1, 2, . . . is not bounded. Indeed, assuming the
contrary, we get Ti < T ∗ for some T ∗ <∞ and all i = 1, 2, . . ., so that the norms of the

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



182 E. BURLAKOV, I. MALKOV

corresponding solutions satisfy the relation lim
T→T ∗−0

∥uT∥BCp(ΞT ,R2) = ∞ which contradicts

to (AW ). We thus proved that {Ti}, i = 1, 2, . . . is not bounded and, hence that the sequence
of local solutions constructed in the proof has a unique limit that is a global solution to (2.6).
Thus, Lemma 2.1 is proved.

Note that the validity of conditions (Aω) and (Af ) naturally provides the fulfillment of
(AW ) and (AF ).

Now, applying Lemma 2.1 to (2.6) in the case when µ1(dψ) is the Lebesgue measure on
R µ2(dy) is the Lebesgue measure on R2, we prove the theorem.

The validity of the following statement is implied by Lemma 2.1.

Remark 2.1:
For any T > 0, the restriction of the unique (global) solution obtained in Theorem 2.1 to the
set ΞT is the unique solution to the problem (2.3), (2.4) on the set ΞT .

Consider now a two-layer neural network

∂tv
i
d(t, n)=−τdvid(t, n)+

n∑
j=1

ωij
d(n)fd(v

j
d(t, n))+νd

m∑
l=1

fs(v
il
s (t,m)),

∂tv
ik
s (t, n,m)=−τsviks (t, n,m)+

n∑
j=1

m∑
l=1

ωijkl
s (n,m)fs(v

jl
s (t, n,m))+νsfd(v

i
d(t, n))

(2.9)

having n ”spatial” elements in each of the layers and m ”directions” in the orientation
columns layer, and parameterized by the dimensions n and m of its layers. For any natural n
and m, the values vid(t, n), v

ik
s (t, n,m) correspond to the neuronal activity of the deep layer

and the superficial layer in the network, the constants ωij
d(n), ω

ijkl
s (n,m) define the strengths

of connections inside each of the layers, and the constants νd, νs define the strengths of
connections between the layers.

The following statement establishes correspondence between the bi-laminar neural field
model (2.3) and the two-layer neuronal network (2.9).

Proposition 2.1:
Let assumptions (Aω), (Af ) be satisfied. For each naturalm and n, let {δ1k (m), k = 1, ...,m}
and {δ2i (n), i = 1, ..., n} be finite families of open subsets of R and R2, respectively,
satisfying the conditions

m⋃
k=1

δ1k (m) = (−π
2
,
π

2
],

n⋃
i=1

δ2i (n) = Ω,

lim
m→∞

max
k=1,...,m

mes(δ1k (m)) = 0, lim
n→∞

max
i=1,...,n

mes(δ2i (n)) = 0,

where mes(·) denotes the Lebesgue measure. Let ψk(m) and yi(n) (k = 1, ...,m, i = 1, ..., n)
be arbitrary points in δ1k (m) and δ2i (n), respectively.

Then for each natural n and m and any α(n,m) ∈ Rnm+n, there exists a unique
continuous solution to the system (2.9) considered together with the initial conditions(

vid(n)
)
(0) = αi

d(n),
(
viks (n,m)

)
(0) = αik

s (n,m), (2.10)

which is a function v(n,m) = (vd(n), vs(n,m)) from [0,∞) to Rnm+n.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON BI-LAMINAR NEURAL FIELD MODELS 183

Moreover, the sequence of solutions vik(n,m) = (vid(n), v
ik
s (n,m)) to the initial value

problem (2.9), (2.9), where

ωij
d(n) = mes(δ2i (n))ωd(yi(n), yj(n)),

ωijkl
s (n,m) = mes(δ2i (n))mes(δ1k (m))ωs(yi(n), yj(n), ψk(m), ψl(m)),

αik
d(n) = ûd(0, yi(n)), α

ik
s (n,m) = ûs(0, yi(n), ψk(m))

(2.11)

converges to the solution u(t, x, φ) (t ≥ 0, x ∈ Ω, φ ∈ (−π
2
, π
2
]) of the initial value problem

(2.3), (2.4), as n,m→ ∞, in the following sense:

lim
n→∞
m→∞

max
t∈[0,T ]

(
sup
1≤i≤n
1≤k≤m

(
sup

x∈δ2i (n)
φ∈δ1k (m)

|u(t, x, φ)−
(
vik(n,m)

)
(t)|

))
= 0 (2.12)

for any T > 0.

Proof
By a reasoning similar to the one applied in the proof of Theorem 2.1, we can conclude
that for any natural n and m, the problem (2.9), (2.10) is equivalent to the equation (2.6)
with µ1 = µ1

1/m, µ2 = µ2
1/n, where µ1

1/m and µ2
1/n are the sums of m Dirac point measures

concentrated at the points ψk(m) ∈ (−π
2
, π
2
] and n Dirac point measures at the points

yi(n) ∈ Ω, respectively. We define µ1
0 and µ2

0 to be the Lebesgue measures on R and R2,
respectively. Therefore, solvability of (2.9), (2.10) for each natural n and m follows from
Theorem 2.1. The proposition conditions imply that the measures µ1

(·) and µ2
(·) are weakly

right-continuous at 0 on the sets (−π
2
, π
2
] and Ω, respectively. Indeed, for any continuous

function Υ(x, φ) : (−π
2
, π
2
]× Ω → R2,

∫
Ω

π
2∫

−π
2

Υ(y, ψ)µ1
1/m(dψ)µ

2
1/n(dy)

=
n∑
i=1

m∑
k=1

Υ(yi(n), ψk(m))mes(δ1k (m))mes(δ2i (n))

=

∫
Ω

π
2∫

−π
2

Υ(y, ψ)µ1
0(dψ)µ

2
0(dy)

(2.13)

as n,m→ ∞. Finally, assumptions (Aω), (Af ), together with the relations (2.11) and the
property (2.13) provide the convergence (2.12).

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



184 E. BURLAKOV, I. MALKOV

3. BI-LAMINAR NEURAL FIELD MODEL WITH HEAVISIDE-TYPE ACTIVA-
TION FUNCTIONS

In this section we derive conditions for solvability of (1.2) in the case of discontinuous
Heaviside-type activation functions (i.e., gd = Hd, gs = Hs):

∂tud(t, x) = −τdud(t, x) +
∫
Ω

ωd(x, y)Hd(ud(t, y))dy

+νd

π
2∫

−π
2

Hs(us(t, x, ψ))dψ,

∂tus(t, x, φ) = −τsus(t, x, φ) +
∫
Ω

π
2∫

−π
2

ωs(x, φ, y, ψ)Hs(us(t, y, ψ))dψdy

+νsHd(ud(t, x))

(3.14)

where

(AH) The components of the neuronal activation function H = (Hd, Hs), H : R2 → {0, 1}
are Heaviside-type activation functions:

Hk(u) =

{
0, u ≤ hk,
1, u > hk,

where hk is the threshold of activation, k = d, s.

Following the procedure described in the proof of Theorem 2.1 we rewrite the problem
(3.14), (2.4) as follows:

u(t, x, φ) = û(x, ψ) +

t∫
0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)H(u(s, y, ψ))dψdyds (3.15)

where H =

(
Hd

Hs

)
. Using the ideas of A.F. Filippov (see [16], Chapter 2, § 4), we address

the problem of solvability of the integral equation (3.15) and, hence, the problem (3.14),
(2.4), in the sense of the so-called generalized solutions. We define a generalized solution
to the problem (3.14), (2.4) to be a continuous function from [0,∞)× Ω× (−π

2
, π
2
] to R2,

which satisfies the inclusion

u(t, x, φ) ∈ û(x, ψ) +

t∫
0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)H(u(s, y, ψ))dψdyds (3.16)

where

(AH) The set-valued function H : R2 ⇒ [0, 1]2 is defined as

H = (Hd,Hs), Hk(u) =

{
0, u < hk,

[0, 1], u = hk,
1, u > hk,

k = d, s (hk are the same as in (AH)).

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON BI-LAMINAR NEURAL FIELD MODELS 185

Theorem 3.1:
Let assumptions (Aω) and (AH) be satisfied. Then there exists a generalized solution to the
problem (3.14), (2.4), which is a continuous function from [0,∞)× Ω× (−π

2
, π
2
] to R2,

Proof
We start out with proving the solvability of the inclusion (3.16) in the following sense. For
any T > 0, we define uT ∈ BCp(ΞT ,R2) to be a T -local solution to (3.16) if uT satisfies
the inclusion (3.16) on the set ΞT . We say that a continuous function u∞ : [0,∞)× Ω×
(−π

2
, π
2
] → R2 is a global solution to (3.16) if for any T > 0, the restriction of u∞ to ΞT is a

T -local solution to the inclusion (3.16).
Lemma 3.1:
Let assumptions (AW ) and (AH) be satisfied. Then, for any T > 0, the inclusion (3.16) has
a T -local solution. Any T -local solution can be extended to a global solution to (3.16).
Proof of Lemma 3.1

We choose arbitrary T > 0 and represent the mapping on the right-hand side of (3.16) as
follows:

(ITNTu)(t, x, φ) =

t∫
0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)H(u(s, y, ψ))dψdyds

where IT : BCp(ΞT ,R2) → BCp(ΞT ,R2) is defined by (2.7) and for any ξ ∈ BCp(ΞT ,R2),
(NT ξ)(t, x, φ) = H(ξ(t, x, φ)).

By the virtue of (AW ) we have that IT is a continuous operator. The operator
H is upper-semicontinuous due to (AH). As H is upper-semicontinuous and IT is a
linear continuous operator, for Mξ

r = BC(Ω×(−π
2
,π
2
],R2)(ξ, r), the composition ITNT ξ : Mξ

r →
C(Ω× (−π

2
, π
2
],R2) is a convex valued mapping. Show that the composition ITNT ξ is upper-

semicontinuous and closed. Choose ui and ϑ such that ϑ ∈ INui and

∥ui − u0∥BCp(ΞT ,R2) → 0, ∥ϑi − ϑ0∥BCp(ΞT ,R2) → 0

where u0 ∈ BCp(ΞT ,R2), ϑ0 ∈ BCp(ΞT ,R2) are some limit points and vi → v0 ∈
BCp(ΞT ,R2). Choose also wi ∈ N vi such that ϑi = Iwi.

Consider the sequence {wi} ⊂ L as the sequence of Bochner integrable mappings wi :
[0, T ] → L(Ω× (−π

2
, π
2
]). By the virtue of condition (AH) and convergence vi → v0 we can

apply Kolmogorov-Riesz compactness theorem [17] and consequently Proposition 4.2.1 [18],
and conclude that wi → w0 weakly, w0 ∈ L. Further, using Mazur’s lemma (see [19], Section

5.1, Theorem 2) we have ŵi =
∞∑
j=i

βijwj such that

∥ŵi − w0∥ → 0 (3.17)

where the coefficients βij satisfy the following conditions:

•
∞∑
j=i

βij = 1 for all i = 1, 2, ...;

• one can find a number j0 such that βij = 0 for all j > j0 and for each i = 1, 2, ....

The relation (3.17) implies (see Section 41, Theorem 4 [15]) the existence of a
subsequence of the sequence ŵi converging to w0 almost everywhere.

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186 E. BURLAKOV, I. MALKOV

Due to upper semicontinuity of H, for almost all (t, x, φ) ∈ ΞT and for any ε > 0 there
exists a number i0 = i0(t, x, φ, ε) such that for all i > i0 it holds true that

H(t, x, φ, vi) ⊂ B((H(t, x, φ0))(t, x, φ), ε)

We thus have wi ∈ B((H(t, x, φ0))(t, x, φ), ε). As an ε-neighborhood of a convex set is
convex, we obtain ŵi ∈ B((H(t, x, φ0))(t, x, φ), ε). By the virtue of the closedness of H,
the latter relation implies that w0 ∈ B((H(t, x, φ0))(t, x, φ), ε) so that w0 ∈ Nu0. Putting

ϑ̂i = IT ŵi = IT

∞∑
j=i

βijwj =
∞∑
j=i

βijITwj =
∞∑
j=i

βijϑj

we obtain ∥ϑ̂i − ϑ∥ → 0, which due to the continuity of IT implies that

ϑ = ITw0 ∈ ITNTu0.

Thus, the closedness and, consequently, the upper semicontinuity of the composition ITNT

is proved.
Now we choose some sufficiently large T > 0 and put r = 2max(û). Using (AW ) and

(AH), find the maximal T1 ∈ (0, T ] such that

max
(t,x,φ)∈ΞT1

∣∣∣∣∣
t∫

0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)ξr(s, x, ψ)− û(x, φ)dψdyds

∣∣∣∣∣ ≤ r.

Thus, we have IT1NT1(Mξ
r) ⊂ Mξ

r, we apply Bohnenblust–Karlin theorem and prove the
existence of a fixed point uT1 ∈ ΞT1 that is a T1-local solution to the problem (3.16).

Choose any T1-local solution uT1 to problem (3.16). We introduce a new time-variable
t1 = t− T1. Now, we consider the problem (3.16) with t = t1. We apply the procedure
described above and prove the existence of a T ′

1-local solution u′T ′
1

to (3.16) for some
T ′
1 > 0. We thus obtain a T2-local solution uT2 to (3.16), where T2 = T1 + T ′

1. u ={
uT1 t ∈ [0, T1]
u′T ′

1
t ∈ [T1, T2]

. The T2-local solution uT2 is continuous at (T2, x, φ).

In the next step, we select any T2-local solution uT2 to the problem (3.15), (2.4).
We introduce a new time-variable t2 = t− T2 and repeat the procedure above. We thus
obtain a strictly increasing sequence {Ti}, i = 1, 2, . . . and the corresponding sequence of
of local solutions uTi

, i = 1, 2, . . . such that for any i1 < i2, uTi1
is the restriction of uTi2

∈
BCP (ΞTi2

,R2) to the set ΞTi1
. We find lim

i→∞
Ti = T̂ . Take any t∗ ∈ (0, T̂ ). For some number

i, t ∈ (Ti−1, Ti) and ut∗ therefore is a t∗-local solution. We have constructed the mapping
t∗ 7→ ut∗ . Prove that {Ti}, i = 1, 2, . . . is not bounded. Indeed, assuming the contrary, we
get Ti < T ∗ for some T ∗ <∞ and all i = 1, 2, . . ., so that the norms of the corresponding
solutions satisfy the relation lim

T→T ∗−0
∥uT∥BCp(ΞT ,R2) = ∞ which contradicts to (AW ). We

thus proved that {Ti}, i = 1, 2, . . . is not bounded and, hence that the sequence of local
solutions constructed in the proof has a unique limit that is a global solution to (2.6). Thus,
Lemma 3.1 is proved.

Lemma 3.1 implies that the problem (3.14), (2.4) possesses a generalized solution, which
proves the theorem.

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ON BI-LAMINAR NEURAL FIELD MODELS 187

4. CONTINUOUS DEPENDENCE OF SOLUTIONS TO BI-LAMINAR NEURAL
FIELD MODEL UNDER THE TRANSITION FROM CONTINUOUS TO
HEAVISIDE-TYPE NEURONAL ACTIVATION

We introduce here the following parameterized version of (2.3)

∂tud(t, x) = −τdud(t, x) +
∫
Ω

ωd(x, y)f
i
d(ud(t, y))dy

+νd

π
2∫

−π
2

f i
s(us(t, x, ψ))dψ,

∂tus(t, x, φ) = −τsus(t, x, φ) +
∫
Ω

π
2∫

−π
2

ωs(x, φ, y, ψ)f
i
s(us(t, y, ψ))dψdy

+νsf
i
d(ud(t, x))

(4.18)

with a natural parameter i.
The next statement presents the main result of this section.

Theorem 4.1:
Let assumption (Aω) be fulfilled and for all natural i, the functions f i

d, f
i
s : R → [0, 1] satisfy

assumption (Af ).
Then for any natural i, there exists a unique solution, say ui, to the equations (4.18) with

the initial condition (2.4), which is a continuous function from [0,∞)× Ω× (−π
2
, π
2
] to R2.

Moreover, if for any ε > 0, one can find a number iε such that

|f i
k(u)−Hk(u)| < ε, u ∈ R \BR(hk, ε), i > iε, k = d, s, (4.19)

a continuous function u0 : [0,∞)× Ω× (−π
2
, π
2
] → R2, u0 = (u0

d, u
0
s ), obtained from the

relation
lim
T→∞

lim
i→∞

max
(t,x,φ)∈ΞT

|ui(t, x, φ)− u0(t, x, φ)| = 0

in the case if it satisfies the condition

mes
(
{(t, x) ∈ [0,∞)× Ω, u0

d(t, x) = hd, u
0
s (t, x, φ) = hs}

∪{(t, x, φ) ∈ [0,∞)× Ω× (−π
2
,
π

2
], u0

d(t, x) = hd, u
0
s (t, x, φ) = hs}

)
= 0,

(4.20)

is a generalized solution to the problem (3.14), (2.4).

Proof

Similarly to the previous sections we can rewrite (4.18) as

u(t, x, φ) = û(x, ψ) +

t∫
0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)
(
F(u,

1

i
)
)
(s, y, ψ)dψdyds, (4.21)

where

F(u,
1

i
) =

(
f i
d(ud)
f i
s(us)

)
.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



188 E. BURLAKOV, I. MALKOV

Unique solvability of the problem (4.18), (2.4) for all natural i follows from Theorem 2.1.
Choose arbitrary T > 0. Assumptions (Aω) and (Af ) imply that the set

∞⋃
i=1

t∫
0

∫
Ω

π
2∫

−π
2

W (t, s, x, y, φ, ψ)
(
FT (BCp(ΞT ,R2),

1

i
)
)
(s, y, ψ)dψdyds

and, hence, the set of solutions to the problem (4.18), (2.4) defined on the set ΞT , say
ui
T , (for all natural i) is relatively compact in BCp(ΞT ,R2). Choose any limit point, say

u0
T ∈ BCp(ΞT ,R2), of the set

∞⋃
i=1

ui
T . Denoting

FT (u, 0) =

(
Hd(ud)
Hs(us)

)
we notice that the relations (4.19), (4.20) and the properties of set-valued integral (see
e.g. [20], §1.5.1) provide that for any ε > 0, there exists a number iε such that

ITFT (u
i,
1

i
) ∈ BBCp(ΞT ,R2)(ITF(u0, 0), ε)

for all i > iε. The latter allows to get the following relation:

BBCp(ΞT ,R2)(u
0, ε/2) ∋ ui = ITF(ui,

1

i
) ∈ BBCp(ΞT ,R2)(ITF(u0, 0), ε/2),

which means that u0 is a limit point of ITFT (u
0, 0) = ITNTu

0 (see Section 3). Noticing
that the values of ITFT are closed (see e.g. the proof of Lemma 3.1), we prove that
u0 ∈ BCp(ΞT ,R2) is a generalized solution of the problem (3.14), (2.4) on the set ΞT . Taking
into account the arbitrary choice of T > 0, we finalize the proof.

5. CONCLUSIONS AND OUTLOOK

In this research we established fundamental properties of the mathematical model for the
macro- and meso-scale electrical activity in the primary visual cortex based on the neural field
concept. This opens several perspectives of further applications of the modelling framework
(1.2).

In recent studies of the human brain travelling waves, which are the most well-identified
phenomenon of the brain electrical activity, underlying the brain functioning in both normal
regime and in various pathological states (such as e.g. epilepsy and Parkinson’s disease), the
neural field equation (1.1) was successfully used in the simulations of travelling waves in
the sensorimotor cortex [21]. The model (1.1) was supplemented by an extra component
formalizing a slow negative feedback in the neural media due to the presence of the
inhibitory neurons (see e.g. [22]). A much more intriguing problem from the point of
view of neurophysiology is to investigate and model the interaction of travelling waves in
the visual cortex with the orientation columns under a presentation of spatially oriented
stimuli. In a similar way to (1.1), the models (2.3) and (3.14) can be equipped with the
corresponding negative feedback components for mathematical modelling in the framework
of the aforementioned studies. Moreover, the presentation of spatially oriented stimuli can be
treated as an impulse control problem for the modelling system (1.2), which can be studied
e.g. based on the ideas developed in [23].

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON BI-LAMINAR NEURAL FIELD MODELS 189

The models of the form (1.2) can be used in the framework of the neurofeedback
paradigm, where the target characteristics of the brain activity are converted into human-
interpretable visual, auditory or tactile stimuli [24, 25]. For example, a person undergoing
a neurofeedback session learns to regulate the activity of his own central nervous system
by performing task of maintaining the stimulus in a certain state trying to keep the
angle of a rotating arrow displayed on the monitor screen within certain limits, which
corresponds to maintaining the target characteristic of brain activity in the required range.
This paradigm is used both for the correction of the psycho-emotional state, and for training
to improve the efficiency of cognitive functions, as well as the treatment of a wide range of
neurodegenerative diseases, including epilepsy [26].

Another paradigm involves stimulation and alteration (transcranial magnetic, using direct
or alternating current) of brain activity, depending on the current state of the central nervous
system [27, 28]. It is aimed at suppressing pathological activity or inducing a specific
behavioral response, which is used e.g. to suppress tremor in patients with Parkinsonism
or reduce the probability of an epileptic seizure.

The paradigms described can be related to minimization problems constructed based
on the mathematical framework (3.14), as it is more suitable for computer simulations
compared to the model (2.3). In these minimization problems, the discontinuity in the
nonlinear activation functions implies difficulties in using the standard theory that relies on
the smoothness of the mappings involved. However, we conjecture that due to the presense of
the Heaviside-type activation functions in (3.14), one can construct isotone operators acting in
an appropriate ordered space and apply the results on minimization of functionals in partially
ordered spaces developed in [29] to prove the solvability of the minimization problem.

ACKNOWLEDGEMENTS

The present research is supported by the Russian Science Foundation (grant no. 22-21-
00756).

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Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)


	Introduction
	Bi-laminar neural field model with continuous activation functions
	Bi-laminar neural field model with Heaviside-type activation functions
	Continuous dependence of solutions to bi-laminar neural field model under the transition from continuous to Heaviside-type neuronal activation
	Conclusions and outlook