IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2013

Synchronization Phenomena of Coupled Rulkov Maps
with STDP for Modeling Epilepsy
Naohiro Shibuya†, Charles Unsworth‡, Yoko Uwate† and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering, The University of Tokushima,
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Email:{shibuya, uwate, nishio}@ee.tokushima-u.ac.jp‡Dept. of Engineering Science, The University of Auckland,
70 Symonds Street, Auckland 1001, New Zealand
Email:c.unsworth@auckland.ac.nz
Abstract—Epilepsy of the neuropsychiatric disorder is provoked from an imbalance in the long-term potentiation (LTP)versus long-term depression (LTD) of the synapses in the hippocampus. The LTP and LTD are replicated by using the Spike
Timing Dependent Plasticity (STDP). Additionally, the spiking
activity of the synapses in the hippocampus can be expressed byusing the Rulkov Maps. In this paper, we consider a simulation
model which is constructed by using Rulkov Maps with STDP.
We explore the effect of 5 connective arrangements on spiking
activity, as basic simulation for constructing the approximate
simulation model of epilepsy. From the result, the tendency offiring has been to improve of synchronization accuracy with theincreasing of connective number.

II. S PIKE T IMING D EPENDENT P LASTICITY
STDP is a temporally asymmetric form of Hebbian learning
induced by tight temporal correlations between the spikes ofpre- and postsynaptic neurons. STDP provokes the LTP of thesynapses, if the presynaptic spike arrival a few milliseconds
before postsynaptic spikes. Whereas, it provokes the LTD of
the same synapse, if presynaptic spike arrival after postsynaptic spikes.

I. I NTRODUCTION
Epilepsy is known as one of neuropsychiatric disorder andinvolved with a large part of the hippocampus. The hippocampus has the lowest seizure threshold in the brain, therefore
it has indicated the beginning of most epilepsy seizures.
Moreover, the seizure-related neuronal electrical activity has
feature of synchrony arising regularly [1],[2]. In the normalstatus, some neurons suppress the abnormal firing. If theneuronal firing property of neurons or potentiation and depression balance are altered slightly, the supranormal excitability
is diffused and lead to seizure. We consider that the eachpotentiation and the depression corresponds approximately toeach long-term potentiation (LTP) and long-term depression(LTD).
In this study, we apply Spike Timing Dependent Plasticity (STDP) to Rulkov Maps for constructing the smallscale simulation models. STDP is a generic model used toreplicate the LTP and the LTD [3],[4], and Rulkov Map
produces two-dimensional spiking-bursting behavior like real
biological neurons [5]-[7]. By incorporating the STDP into
the original Rulkov Map, we can explore the synchronous
behavior like real biological spiking activity. We explore theeffect of 5 connective arrangements on spiking activity, asbasic simulation for constructing the approximate simulation
model of epilepsy. Moreover, we investigate a relationshipbetween synchronous firing and LTP, LTD.

- 61 -Fig. 1.

The STDP function of changing synaptic connection.

The weight change ∆wij depends on the relative timing
between presynaptic spike arrivals and postsynaptic spikes.
The total weight change ∆wij induced by a simulation protocol with pairs of pre- and postsynaptic spikes is shown asfollowing.

∆wij =N∑W (tfj − tfi )

(1)

f =1

where W (x) denotes one of the STDP functions in Fig. 1.
The method to choose for the STDP function W (x) is shownas following.
{A+ exp(−x/τ+ ) (x < 0)
W (x) =−A− exp(x/τ− ) (x > 0)(2)

which has been used in fits to experimental data and models.
The parameters A+ and A− may depend on the current valueof the synaptic weight wij . In this article, the parameter A+and A− are fixed 0.05, and the time constants are on the orderof τ+ = 10ms and τ− = 10ms.
Moreover, the STDP function as in Eq. (2) can be implemented in an on-line update rule using the following equation.
∑dwij= A+ (wij )xj (t)δ(t − tfi )dtf∑−A− (wij )yi (t)δ(t − tfj )(3)

f

IV. C OUPLED RULCOV M APS WITH STDPFirst, we explain about the coupled Rulkov Map. Theequation of the coupled Rulkov Maps are shown as following.
xm,n+1 = f (xm,n , xm,n−1 , ym,n )1+ wij (xm+1,n − 2xm,n + xm−1,n )
2ym,n+1 = ym,n − µ(xm,n + 1) + µσ + µσm,n1+ wij (xm+1,n − 2xm,n + xm−1,n )(6)
2
where parameter wij shows the coupling weight of the connection between the maps. Each parameters of x1 , x2 sets
individually.
The coupling weights wij are updated by STDP function. If
STDP provokes the LTP, the coupling weights are updated topositive direction. Whereas, if it provokes LTD, the couplingweights are updated to negative direction. We consider that theRulkov Maps show in-phase/reverse-phase synchronization bychanging the coupling weight.
A. 2 Coupled Maps

Fig. 2. Top: A presynaptic spike trace xj (t) with postsynaptic spike. Bottom:
A postsynaptic spike trace yi (t) with presynaptic spike.

III. RULKOV M AP
In recent years, a simple model which replicates the dynamics of spiking and spiking-bursting activity of real biologicalneurons has proposed by N. F. Rulkov. The model is a twodimensional map that produces chaotic spiking-bursting neuralbehavior. It is demonstrated that the results of this model are inagreement with the synchronization of chaotic spiking-bursting
behavior experimentally found in real biological neurons. The
expressions of the Rulkov map are shown as following.

xn+1= f (xn , xn−1 , yn )yn+1

= yn − µ(xn + 1) + µσ + µσn

In this section, we observe fundamental synchronization
phenomena of the coupled Rulkov Map. We consider two
coupled Rulkov Maps with STDP function as shown in Fig. 3.
Additionally, we set the parameter α = 10, 15, and explore thespiking activity and coupling weight, if the coupling direction
is changed. The spiking activity of coupled Rulkov Maps andcoupling weights are shown in Fig. 4.

Fig. 3.

2 coupled Rulkov Maps with STDP.

(4)

where x is the fast and y is the slow dynamical variables. αand µ, σ shows the parameters of the maps. In this paper, we
set parameter α with arbitrary value, and µ, σ are fixed to0.002, 0.24 each. Here, these parameters are the control thedynamics parameters and the behavior shows the typical of theneurons. Specifically, the amplitude of waveform is changingby parameter α. The nonlinear function f (xn , xn−1 , yn ) isshown as following:α/(1 − xn + yn ), (xn ≤0)f (xn , xn−1 , yn )= α + yn , (0<xn <α+yn and xn−1 ≤0)−1, (xn ≥α+yn or xn−1 >0)(5)- 62 -(a) α1 = 15, α2 = 10.

(b) α1 = 10, α2 = 15.
Fig. 4. The spiking activity of 2 coupled Rulkov Maps and the couplingweights. (a) In-phase synchronization. (b) Reverse phase synchronization.

From the result of Fig. 4, we can see that the synchronization phenomenon shows inverting phase by changing
the coupled direction. Additionally, we can observe the inphase/reverse-phase synchronization if the coupling weight is
set to positive/negative value. However, the coupling weight
is not symmetric about the x-axis. Therefore, we consider thatthe some neurons are able to interconnect.
B. 4 Coupled Maps with 5 Connective Arrangements
In this section, we consider the 4 coupled Rulkov Maps.
We have defined these connective arrangements to be: linear
unidirectional , linear bidirectional , box unidirectional ,box bidirectional , and all-to-all [8]. Each configuration isshown in Fig. 5. We explore the effect of 5 connective arrangements on neuronal dynamics of spiking activity. Moreover,
we investigate an relationship between synchronous firing andLTP, LTD.

explored the effect of 5 connective arrangements on spiking
activity, as basic simulation for constructing the approximate
simulation model of epilepsy. Moreover, we investigated anrelationship between synchronous firing and LTP, LTD. From
the result, if STDP provokes the LTP, the coupling weights areupdated to positive direction. Whereas, if it provokes LTD, thecoupling weights are updated to negative direction. Overall,
the tendency of firing has been to improve of synchronization
accuracy with the increasing of connective number.
R EFERENCES[1] P. A. Schwartzkroin, “Role of the hippocampus in epilepsy,” Hippocampus, vol. 4, pp. 239-242, 1994.
[2] K. Nakashima, H. Hayashi, O. Shimizu and S. Ishizuka, “Long-termchange in synaptic transmission in CA3 circuits followed by spontaneous
rhythmic activity in rat hippocampal slices,” Neuroscience Research, vol.
40, pp. 325-336, 2001.
[3] A. Morrison, M. Diesmann and W. Gerstner, “Phenomenological modelsof synaptic plasticity based on spike timing,” Biological Cybernetics, vol.
98, pp. 459-478, 2008.
[4] J. P. Pfister, W. Gerstner, “Triplets of spikes in a model of spike timingdependent plasticity,” Journal of Neuroscience, vol. 26, pp. 9673-9682,2006.
[5] N. F. Rulkov, I. Timofeev and M. Bazhenov, “Oscillations in LargeScale Cortical Networks: Map-Based Model,” Journal of Computational
Neuroscience, vol. 17, pp. 203-223, 2004.

Fig. 5. Connective arrangements of 4 coupled Rulkov Maps. (a) Linearunidirectional. (b) Linear bidirectional. (c) Box unidirectional. (d) Box
bidirectional. (e) All-to-all.

The parameter (α1 , α2 , α3 , α4 ) are fixed (15, 10, 20, 10).
Moreover, we define the synchronization probability is shownas follows.
Y(7)X
where X is the number of burst waveform, and Y is thenumber of synchrony burst waveform.
The simulation results of spiking activity (from 0 to 30000)
and the synchronization probability (from 0 to 15000, from15001 to 30000) are shown in Fig. 6. The linear bidirectional model shows different clustering phenomena as thelinear unidirectional by interconnecting the neurons. Thebox bidirectional model also shows different clustering
phenomena as the box unidirectional by interconnecting
the neurons. From these results, we consider that the effects
of interconnecting of neurons are making stable waveform,
and improving the synchronization accuracy. Finally, the allto-allmodel shows high accuracy in-phase/reverse-phasesynchronization with duration. Overall, we can see that thetendency of firing has been to improve of synchronization
accuracy with the increasing of connective number.
I=V. C ONCLUSION
In this paper, we considered a simulation model whichis constructed by using Rulkov Maps with STDP. We have

- 63 -[6] A. L. Shilnikov, N. F. Rulkov, “Origin of chaos in a Two-dimensional
Map Modeling Spiking-Bursting Neural Activity,” Int. J. Bifurcation andChaos, vol. 13, pp. 3325-3340, 2003.
[7] N.F. Rulkov, “Modeling of Spiking-Bursting Neural Behavior using Twodimensional Map,” Physical Review E, vol. 65, 041922, 2002.
[8] D. Cumin, C. P. Unsworth, “Generalising the Kuromoto Model for the
study of Neuronal Synchronisation in the Brain,” Phycsica D: Nonlinear
Phenomena, vol. 226, pp. 181-196, 2007.

Fig. 6.

The spiking activity of 4 coupled Rulkov Maps and the synchronization probability. α1 = 15, α2 = 10, α3 = 20, α4 = 10.

- 64 -