Synchronization of two coupled Hindmarsh-Rose Model
with electrical and chemical synapses
K. Usha
P.A. Subha
Department of Physics,
Department of Physics, Farook College,
University of Calicut, India
University of Calicut, India
ushavyasa@gmail.com
pasubha@gmail.com
ABSTRACT:
In this paper we analyze the synchronization of two coupled
chaotic Hindmarsh Rose (HR) neurons with electrical and
chemical synapses. The control inputs are derived such that it
makes the time derivative of Lyapunov function a negative
definite. We have analyzed the behaviour of the coupled
system by applying the controllers. Numerical simulation
verifies that the proposed controllers are effective in
stabilizing the error states at the origin and there by achieving
synchronization.
KEYWORDS:
HR Model, Electrical and chemical synapse, Synchronization.
I INTRODUCTION:
In physical, chemical and biological systems
synchronization of chaos is a fascinating subject and has
attracted research interest due to its potential applications [1,
2]. Synchronization of chaos refers to a process where
chaotic systems adjust a given property of their motion to a
common behaviour due to a coupling or due to a forcing [3-7].
The study of synchronization process for populations of
interacting neurons is basic to the understanding of some key
issues in neuroscience [8]. The collective behaviour of
neurons and their synchronization helps in information
processing in the brain [9]. Recently, many researchers have
focused on the synchronization of coupled neurons, which is
one of the fundamental issues in understanding the neuronal
behaviour [10]. The selective synchronization of neural
activity has been suggested as a mechanism for binding
spatially distributed object into a coherent one [12, 13]. Partial
synchrony in cortical networks is the reason for generating
several brain oscillations, such as the alpha and gamma EEG
rhythms [14 - 17]. The brain activities are mainly dependent
on the ability of neurons to communicate with each other. The
transmission of impulses between neurons takes place at
special intercellular pathways known as gap junctions. Gap
junctions are clusters of aqueous channels that connect the
cytoplasm of adjoining cells and they are responsible for the
development of cell polarity and the left/right
symmetry/asymmetry in animals [18, 19]. They allow the
direct transfer of ions and small molecules between cells
without leakage to the extracellular fluid. Studies have shown
that, embryos with areas of blocked gap junction fail to
develop normally [20]. As the gap junctions play an important
role in the process of information transmission, they become a
research focus in neural system [21]. Neurons can be coupled
to each other by electrical or chemical synapses at the gap
junction. These two modes of synaptic transmission interact
with each other, both during the development and in the adult
brain. For the proper functioning of brain, interneuronal
communication and interaction between these two modes are
required. The distance between synaptic junctions is shorter in
the case of electrical synapse and such electrical impulses
travel faster than chemical impulses. The connection of
neurons by electrical synapse increases at the time of
neuronal injury. The ischemic increase in neuronal gap
junction coupling is regulated by glutamate via group II
metabotropic glutamate receptors (mGluRs) [22,23].
In this paper, we analyze the synchronization of two
coupled chaotic HR neuron with electrical and chemical
synapses. In section 2, the dynamics of HR model with
electrical synapse for weak and strong coupling strength is
described and also the control laws are derived using
Lyapunov function method. Section 3 deals with the
synchronization of HR model with chemical synapse. Section
4 concludes the study.
2, HR NEURONS WITH ELECTRICAL
SYNAPSE:
The phenomenological neuron model proposed by
Hindmarsh and Rose [24] is either a generalization of the
Fitzhugh Nagumo equations or a simplification of the
physiologically realistic model proposed by Hodgkin and
Huxley. The HR model is able to reproduce many of the
dynamical behavior such as quiescence, spiking, bursting,
irregular spiking and irregular bursting [25]. Bursting is a
complex oscillatory rhythm generated by neurons, consisting
of a rapid sequence of spikes followed by a quiescent state.
The HR model is described by three coupled nonlinear
differential equations,
ẋ 1 = y1 + ax12 − x13 − z1 + I
ẏ 1 = 1 − bx12 − y1
ż 1 = r[R(x1 − e) − z1 ],
(1)
where ‘x’ is the membrane potential, ‘y’ is the fast
current (recovery variable), ‘z’ is called slow current
(adaptation variable), a=3.0, b=5.0, R=4.0, e = -1.61, r =
0.006 ( ratio of fast/slow time scale) and ‘I’ is an external
current or the control parameter. For I=3.1 mA, the HR model
shows chaotic bursting. Chaotic oscillations generated by HR
neuron given by eqn. (1) are shown in Fig. 1.
Figure 1: Chaotic bursting of HR model (time verses the
membrane potential).
Interacting bursting neurons may exhibit different
forms of synchrony. The emergence of synchronous rhythms
in coupled neuron is closely related to the properties of the
individual bursting neurons and the type of synaptic coupling
[26]. The coupling function specify the manner in which the
neurons i and j are coupled. If the neurons i and j are
connected via electrical synapse, then the coupling function
takes the form,
𝑔 (𝑥𝑖 (𝑡), 𝑥𝑗 (𝑡)) = 𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡) [27].
Figure 2: The error dynamics of (a) membrane potential.
(b) recovery variable and (c) adaptation variable for weak
coupling (g=0.1) without the control inputs.
HR neuron with electrical synapse is,
ẋ 1 = y1 + ax12 − x13 − z1 + I − g(xi − xj )
ẏ 1 = 1 − bx12 − y1
(2)
ż 1 = r[R(x1 − e) − z1 ]
ẋ 2 = y2 + ax22 − x23 − z2 + I − g(xj − xi )
ẏ 2 = 1 − bx22 − y2
(3)
ż 2 = r[R(x2 − e) − z2 ] .
For two coupled system i, j = 1,2 and ‘g’ represents the
coupling strength.
We take I=3.1mA, to ensure the neurons are in
chaotic region [28]. We have analyzed the system numerically
by changing the value of ’g’ using Fourth order Runge-kutta
method. The error dynamics of the membrane potential,
recovery variable and adaptation variable of the coupled
system before synchronization are shown in Figs. 2(a), 2(b)
and 2(c) respectively. From the figures it is clear that the state
variables of the two systems show irregular behaviour. We
have analyzed the system for synchronization by increasing
the coupling strength of gap junction. For g > 0.6, Figs. 3(a),
3(b) and 3(c) shows that strong coupling can synchronize a
system of two coupled HR neurons with electrical synapse.
Figure 3: The error dynamics of (a) membrane potential
(b) recovery variable and (c) adaptation variable for
strong coupling (g>0.6) without the control inputs.
2.1, Control functions for weak coupling:
We develop control functions to synchronize the system
modeled by eqns. (2) and (3) for weak coupling (i.e., g < 0.6)
using the lyapunov function method. The modified system of
two coupled HR neurons with the controller is expressed as,
ẋ 1 = y1 + ax12 − x13 − z1 + I − g(x1 − x2 )
ẏ 1 = 1 − bx12 − y1
(4)
ż 1 = r[R(x1 − e) − z1 ]
ẋ 2 = y2 + ax22 − x23 − z2 + I − g(x2 − x1 ) + Ux
ẏ 2 = 1 − bx22 − y2 + Uy
(5)
ż 2 = r[R(x2 − e) − z2 ] + Uz ,
where Ux(t), Uy(t) and Uz(t) are the control functions.
The error states are given by,
ex (t) = x2 (t) − x1 (t)
ey (t) = y2 (t) − y1 (t)
ez (t) = z2 (t) − z1 (t) .
Figure 4: The error dynamics of (a) membrane potential.
(6)
(b) recovery variable and (c) adaptation variable for weak
On the synchronization manifold the errors ex(t),
ey(t) and ez(t) tends to 0 as t tends to infinity. Then the error
dynamics takes the form,
eẋ = ey + a(x22 − x12 ) − (x23 − x13 ) − ez − 2gex + Ux
ė y = −b(x22 − x12 ) − ey + Uy
(7)
ė z = rRex − rez + Uz .
We choose the lyapunov function as,
1
V = (e2x + e2y + e2z ) .
2
(8)
Then,
V̇ = ex ey + aex (x22 − x12 ) − ex (x23 − x13 ) − ex ez − 2ge2x +
ex Ux − bey (x22 − x12 ) − e2y + ey Uy + rRex ez − re2z
+ ez Uz .
In order to make the time derivative of lyapunov function
negative, we choose the controllers as:
Ux = −ey − a(x22 − x12 ) + (x23 − x13 ) + ez
Uy = b(x22 − x12 )
(9)
Uz = −rRex − rez .
Substituting the values of Ux, Uy and Uz in the eqn. for 𝑉̇ ,
we get,
V̇ = −2ge2x − e2y − 2re2z .
(10)
Since g is positive, 𝑉̇ is always negative.
We have studied eqns. (4) and (5) by applying
the control laws given in eqn. (9) for ‘g=0.1’. When these
controllers synchronize the system the error states given in
eqn. (6) satisfies the condition,
lim 𝑒𝑥,𝑦,𝑧 → 0 .
t→∞
coupling after applying the control inputs.
3, HR NEURONS WITH CHEMICAL
SYNAPSE:
For a coupling via chemical synapse the coupling function is
given by,
h(xi , xj ) = (xi − V) g Γ(xj ),
where ’g’ is synaptic coupling strength, 𝛤 is called synaptic
modeled by a sigmoid function with a threshold and
Γ(xj ) = 1/(1 + exp(−λ(xj − θ) ).
where the reversal potential V can set the threshold for
excitation or inhibition and Γ is the threshold reached by
every action potential for a neuron. The HR neurons are
identical and the synapses are fast and instantaneous [29, 30].
HR neurons with chemical synapse can be expressed as,
1
ẋ 1 = y1 + ax12 − x13 − z1 + I − (x1 − V) g (
1+exp(−λ(x2 −θ) )
ẏ 1 = 1 − bx12 − y1
)
(12)
ż 1 = r[R(x1 − e) − z1 ]
ẋ 2 = y2 + ax22 − x23 − z2 + I − (x2 − V) g (
1
1+(−λ(x1 −θ))
ẏ 2 = 1 − bx22 − y2 + Uy
) + Ux
(13)
ż 2 = r[R(x2 − e) − z2 ] + Uz .
(11)
The error states are plotted and plots are shown in Figs. 4(a),
4(b) and 4(c). The plot depicts that the two systems
represented by eqns. (2) and (3) evolving from different initial
conditions are synchronized in the presence of controllers.
We have analyzed the dynamics of coupled systems
given by eqns. (12) and (13) without any control inputs. The
error dynamics of membrane potential, recovery variable and
adaptation variable of the coupled system without controllers
are shown in Figs. 5(a), 5(b) and 5(c) respectively. From the
figures it is clear that without the control inputs the variables
of the two systems show irregular behaviour. We have
analyzed the system for synchronization by increasing the
coupling strength. Synchronization occurs only when the
coupling strength reaches very high values (i.e., g > 1.5). In
order to synchronize the system in the range (0 < g < 1.5), the
controllers Ux (t), Uy (t) and Uz (t) are applied in eqn. (13).
In this case the error dynamics takes the form,
eẋ = ey + a(x22 − x12 ) − (x23 − x13 ) − ez
1
− x2 g (
)
1 + (−λ(x1 − θ))
1
+Vg (
)
1 + (−λ(x1 − θ))
1
+ x1 g (
)
1 + (−λ(x2 − θ))
1
− Vg (
) + Ux
1 + (−λ(x2 − θ))
ė y = −b(x22 − x12 ) − ey + Uy
ė z = rRex − rez + Uz .
(14)
We choose the lyapunov function as,
1
V = (e2x + e2y + e2z ) .
(15)
2
Then,
V̇ = ex ey + a ex (x22 − x12 ) − ex (x23 − x13 ) − ex ez −
x2 g ex (
x1 g ex (
b
ey (x22
1
) + V g ex (
1+(−λ(x1 −θ))
1
) − Vg ex (
1+(−λ(x2 −θ))
− x12 ) − e2y + ey Uy
1
1+(−λ(x1 −θ))
1
1+(−λ(x2 −θ))
)+
)+ ex Ux −
+ r R ex ez − r e2z + ez Uz .
In order to make the time derivative of lyapunov function
negative, we choose the controllers as,
Ux = −ey − a (x22 − x12 ) + (x23 − x13 ) + ez
1
+ x 2 g ex (
)
1 + (−λ(x1 − θ))
1
− x1 g ex (
)
1 + (−λ(x2 − θ))
2
2
(16)
Uy = b(x2 − x1 )
Figure 6: The error dynamics of (a) membrane potential.
(b) recovery variable and (c) adaptation variable for
chemical synapse (g=1) after applying the control inputs.
4. RESULTS AND CONCLUSIONS:
Uz = −rRex − rez .
Substituting the values of 𝑈𝑥 , 𝑈𝑦 and 𝑈𝑧 in the eqn. for 𝑉̇ ,
we get,
1
1
V̇ = V g ex ( (
)−(
) − e2y − 2re2z ),
1+(−λ(x1 −θ))
1+(−λ(x2 −θ))
which is found to be a negative definite.
We have studied eqns. (12) and (13) by applying
the control laws given in eqn. (16). The error states are plotted
and plots are shown in Figs. 6(a), 6(b) and 6(c). The plot
shows that the two systems evolving from different initial
conditions are synchronized in the presence of controllers.
In this work we study the synchronization
phenomena which occur in HR neurons coupled with
electrical and chemical synapse by the method of lyapunov
function. In the case of electrical coupling, for small values of
coupling strength the two systems exhibit irregular behaviour.
We have analyzed the system by varying the coupling
strength. Numerical results have shown that strong coupling
can synchronize a system of two coupled HR neurons with
electrical synapse. The control laws for synchronizing the
system for weak coupling is derived using lyapunov function
method. The time derivative of lyapunov function is found to
be a negative definite which ensure the errors 𝑒𝑥,𝑦,𝑧 →
0 𝑎𝑠 𝑡 → ∞. We have also studied the system using a chemical
synapse. In the case of chemical synapse very strong coupling
is needed to synchronize the system in the absence of
controllers unlike electrical synapse. We have synchronized
the system in the range (0< g< 1.5), by applying the
controllers and the control inputs are derived such that V̇ is
negative. Numerical simulation verifies the effectiveness of
the proposed synchronization scheme.
ACKNOWLEDGEMENTS:
UK would like to thank UGC for providing financial
assistance through JRF scheme for doing the research work.
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