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Neural Mass model, Jansen and
Rit 1995
Andr´es Sandoval Abarca
July 12, 2016
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Introduccti´on
The Jansen-Rit model is a neural population model of a
local cortical circuit. The single area model contains three
interconnected neural populations: one for the pyramidal
neurons and two for excitatory and inhibitory interneurons
forming feedback loops (single area). A multiple area
models includes interactions between multiples neural
masses.
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Model
First each Neural mass has an average membrane
potential (state variable)
This average membrane potential is the results of
different imputs
In Neural mass model these imputs represents average
pulse density or firing rate, we have that average
membrane potential and average pulse densite are the
two main quantities of this model.
these two quantities are converted via two
transformation, the first is a called Post Synaptic
Potential function (PSP) and the second is a
Potential-to-rate function.
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Model
That means, each imput to a Neural mass is converted
from an average pulse density via PSP function to an
average potential membrane, each of those potential is
multiplied by some constants modeling the average
number of synapses to the population.
Figure: scheme of the transformations
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Model
Based in the model of Jansen and Rit the PSP function
is a second diferential operator, wich is given by:
∂x
∂t
= y
∂y
∂t
= Qqz − 2qy − q2
x
Here z(t) is the imput to the PSP (firing rate) ,x(t) is
the output and the constants depend of the exitatory
or inhibitory case
Jansen used also the next potential-to-rate function:
S(v) =
2e0
(1 + expr(v0−v))
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Model
In this work is we are considering the Neural mass like
three blocks: pyramidal neurons, local excitatory and local
inhibitory neurons. The scheme is showed in the next
figure:
Figure: scheme of the single Neural mass model of Jansen
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We have the following O.D.E



∂2y0(t)
∂t2 = AaS(C2y1(t) − C4y2(t)) − 2a∂y0(t)
∂t
−a2
y0(t)
∂2y1(t)
∂t2 = Aa(p/C2 + S(C1y0)) − 2a∂y1(t)
∂t
−a2
y1(t)
∂2y2(t)
∂t2 = BbS(C3y0)) − 2b∂y2(t)
∂t
− b2
y2(t)
A = 3.25, B = 22, C1 = 135, C2 = 0.8C1,
C3 = C4 = 0.25C1, e0 = 2.5, v0 = 6, r = 0.56 y p entre 120
y 300. y0, y1 and y2 represening the firing rate of pyramidal,
excitatory and inhibitory neurones.
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The Fourier Transform
Lemma 1: Let f a periodic function, that is,
∃T > 0 : f(t + T) = f(t)∀t ∈ R then can be written as
Fourier’s Series:
f(x) =
a0
2
+
∞
n=1
ancos(ωnx) + bnsen(ωnx)
Con ωn = n2π
T
, a0, an and bn ∈ R. Note that ω1 = 2π
T
Definici´on 1: given a function f ∈ L1
(R), We define the
Fourier’s Transform of f as the following aplication:
{f} : ξ → ˆf(ξ) =
R
f(x)e−2πiξx
dx
.
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Lemma 2: Let f a periodic function , then:
arg max
R+
ˆf(ξ) = ω
That’s why we use the Fourier Transform, to get the
frequency of the cylcles.
Figure: F. Transform of a periodic function
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Some problems
not periodic function
F.T.
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Solutions
I used some literature values of parameters to get
periodic oscilations
Sometimes I used Matcont to find Hopf Bifurcations
Figure: Hopf bifurcation of equilibrium point
I moved parameters C, p usually.
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Some frequency Maps
With the correct parameters values i got the following
Frequency maps:
Figure: Frequency maps
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The main idea of all this is to find zones where the
frequency change, for example.
Figure: Moving A
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Figure: Another Frequency map, changing the p value
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We got a decreasing relation to the frequency
Figure: In this case we are moving B
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Coupled system
Figure: Phased signales