Chaos, Solitons & Fractals, Amsterdam, November 2013
Synchronization in time-discrete model
of two electrically coupled spike-bursting
neurons
Vladimir Nekorkin, Institute of Applied Physics,
Nizhny Novgorod (Russia)
E-mail: [email protected]
Spike-bursting oscillations of a neuron
Spike-bursting oscillations have two
distinct phases characterized by two
different time scales, an active phase
comprising a sequence of fast spikes,
and a passive phase with slow variation
of the membrane potential.
xn 1  xn  F ( xn )   H ( xn  d )  y n ,
y n 1  y n   ( xn  J ).
Model of neural dynamics
The model was proposed in
xn 1  xn  F ( xn )   H ( xn  d )  y n ,
y n 1  y n   ( xn  J ).
Courbage M., Nekorkin V.I., Vdovin L.V. Chaotic
oscillations in a map-based model of neural activity. –
Chaos 17, 043109, 2007.
x-variable describes qualitatively the membrane potential dynamics of a nerve cell, y
is responsible for the total effect of all ionic currents (recovery variable). The positive
parameter ε determines a time scale of the recovery variable, the parameters β, d, J
control the shape of a signal generated. On the one hand, the model is based on a
discrete version of the FitzHugh-Nagumo model with a cubic nonlinearity F(x) and
the Heaviside step function H(x) added.
F (x)
 1, x  0
H ( x)  
0, x  0.
On the other hand, for fixed
y, the 1st map of the system
is a Lorenz-type map which
can display chaotic behavior
shown on the right for
y=const.
Different regimes produced by the map
Regular regimes (phase portraits
and waveforms) and
chaotic regimes
Dynamics of the two coupled neurons
 x1  x1  F ( x1 )  y1  H ( x1  d )  c( x2  x1 ),

 y1  y1   ( x1  J1 ),

 x2  x2  F ( x2 )  y2  H ( x2  d )  c( x1  x2 ),
 y2  y2   ( x2  J 2 ).

с ~ coupling strength
Fast-slow dynamics
System of fast motions
 x1  x1  F ( x1 )  y10  H ( x1  d )  c ( x 2  x1 )

0
 y1  y1  const

0
 x 2  x 2  F ( x 2 )  y 2  H ( x 2  d )  c( x1  x 2 )
 y  y 0  const.
 2
2
System of slow motions
 y1  y1   ( x1  J )
y  F(x )
 1
1

 y 2  y 2   ( x2  J )
 y 2  F ( x2 ).
Fast motions of the system
Structure of the attractors.
Bifurcations of the fast subsystem
A typical bifurcation of fixed points in the system is a
saddle-node one. Chaotic attractors are typically
destroyed through boundary crisis that may be
accompanied by the appearance of homoclinic orbits to
unstable fixed points and fractal basin boundaries.
Structure of chaotic attractors. There are
two types, foliated structure typical for
many chaotic attractors, and densely
covered by trajectories
Slow motions
Fast
variables
xi≈xi*,
where xi*, (i=1,2) are the
coordinates of the stable
fixed point of the fast
subsystem.
Fast variables xi (i=1,2)
vary within one of the
chaotic attractors of the
fast subsystem.
Relaxation dynamics
Regular slow motions
Saddle-node
bifurcation of the fixed
points of the fast
subsystem
Fast motions on a
chaotic attractor of the
fast subsystem and
slow varying of yi.
Boundary crisis of a
chaotic attractor of the
fast subsystem
…
Synchronization of spike-bursting
oscillations
The value of σ characterizes
the ratio of the total time of
coincidence of active phase of
oscillations to the total duration
of all the active phases.
Tidep are the depolarization
periods; Δtj denote periods of
time during which neurons are
in the active phase.
Dynamics of the system as a Markov chain
For understanding transitions of the system between different states, we
have applied techniques of the theory of Markov topological chains.
X 01 X 11
X 01 X 11
X 01 X 11
X 00 X 10
X 00 X 10
X 00 X 10
The probability of the event that the 1st passive neuron
and the 2nd active will stay at the same state after one
time step (down) and after two time steps (up).
X 01 X 11
X 01 X 11
X 00 X 10
X 00 X 10
We define four activity states of the coupled neurons: X00 – both are passive, X10 –
the 1st neuron is active, and the 2nd is passive, X01 – vice versa, X11 – both are
active. It is seen that the difference between the probabilities indicated is not
vanishing with increasing coupling although it is decreasing which indicates a loss of
memory under coupling increase.
Dynamics of the system as a Markov chain
X 01 X 11
X 00 X 10
X 00 X 10
X 01 X 11
X 01 X 11
X 01 X 11
X10
X 00 X 10
X 00 X 10
X 00
The probability of the event that the 1st passive neuron
becomes active and the 2nd one is in any state (up) and
the same under the condition that one time step ago
the 2nd neuron was active (down). The influence of the
active state of the 2nd neuron on the transition of the 1st
neuron from passive state to active one becomes
significant beginning from some coupling strength cmax.
X 01 X 11
Considering the transition from the passive phase to
the active one of the 1st neuron, we are interested to
know how could it be influenced by the activity of the
2nd neuron. Under coupling, the system acquires a
memory effect that increases to a maximum value
corresponding to a regime of active phase
synchronization and then decreases for strong
coupling.
Conclusions
• Relaxation dynamics of two electrically
coupled neurons with spike-bursting
chaotic oscillations has been studied.
• A method for quantifying spike-bursting
synchronization was proposed.
• Dynamics of the system in terms of a
Markov topological chain was analyzed.
Acknowledgements
• I thank my coauthors Maurice Courbage
and Oleg Maslennikov.
• Thank you for your attention.
Reference: M. Courbage, O.V. Maslennikov and V.I. Nekorkin,
Synchronization in time-discrete model of two electrically coupled spikebursting neurons, Chaos, Solitons & Fractals 45(5), 645–659 (2012).