2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Clustering and Synchronous Firing of Coupled Rulkov Maps with STDP
Naohiro Shibuya1 , Charles Unsworth2 , Yoko Uwate1 and Yoshifumi Nishio1
1

The University of Tokushima
2-1 Minami-Josanjima, Tokushima, Japan
Phone/FAX:+81-88-656-7470/+81-88-656-7471
E-mail: {shibuya, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
Epilepsy of the neuropsychiatric disorder is provoked froman imbalance in the long-term potentiation (LTP) versus longterm 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 some easy
simulation models which are constructed by using Rulkov
Maps with STDP. We explore the effect of 2 and 3 connective arrangements on spiking activity, as basic simulation for constructing the approximate simulation model ofepilepsy. From the result, the coupled Rulkov Maps show
in-phase/anti-phase synchronization by relationship of presynaptic and post-synaptic neuron.

2The University of Auckland
70 Symonds Street, Auckland, New ZealandPhone/FAX:+64-09-373-7599/+64-09-373-7468
E-mail: c.unsworth@auckland.ac.nz

2. Spike Timing Dependent Plasticity
STDP is a temporally asymmetric form of Hebbian learning induced by tight temporal correlations between the spikesof pre-synaptic and post-synaptic neurons. STDP provokes
the LTP of the synapses, if the pre-synaptic spike arrival afew milliseconds before post-synaptic spikes. Whereas, it
provokes the LTD of the same synapse, if pre-synaptic spikearrival after post-synaptic spikes.

1. Introduction
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]. In the normal status, some neurons suppress the abnormal firing. If the neuronal 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 each potentiation 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 small-scale
simulation models. STDP is a generic model used to replicate the LTP and the LTD [2], and Rulkov Map produces
two-dimensional spiking-bursting behavior like real biological neurons [3],[4]. By incorporating the STDP into the original Rulkov Map, we can explore the synchronous behavior
like real biological spiking activity. We explore the effectof 2 and 3 connective arrangements on spiking activity, as
basic simulation for constructing the approximate simulation
model of epilepsy.

Figure 1: The STDP function of changing synaptic connection.
The weight change ∆wij depends on the relative timing
between pre-synaptic spike arrivals and post-synaptic spikes.
The total weight change ∆wij induced by a simulation protocol with pairs of pre-synaptic and post-synaptic spikes is
described as following.

∆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.

- 606 -{W (x) =A+ exp(−x/τ+ ) (x < 0)−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 each 0.05 and 0.055, and the time constants
are on the order of τ+ = 10ms and τ− = 10ms.
3. Coupled Rulkov Maps

4.1 CA3 Hippocampus Network Model
The network model of CA3 hippocampus is shown in
Fig. 2. CA3 hippocampus network consist of pyramidal cells(excitability) and intercalated cells (inhibitory). The pyramidal cells are interconnected with 8 pyramidal cells of neighborhood. Moreover, the intercalated cells are interconnectedwith 16 pyramidal cells of neighborhood.

In recent years, a simple model which replicates the dynamics of spiking and spiking-bursting activity of real biological neurons has proposed by N. F. Rulkov. The model is a
two-dimensional map that produces chaotic spiking-burstingneural behavior. It is demonstrated that the results of thismodel are in agreement with the synchronization of chaotic
spiking-bursting behavior experimentally found in real biological neurons. The expressions of the Coupled Rulkov
maps are shown as following.
Figure 2: Network structure of CA3 hippocampus.
xm,n+1 = f (xm,n , xm,n−1 , ym,n )Additionally, the pyramidal cells and the intercalated cells
exhibit spiking/bursting activity as shown in Fig 3. We
set the parameter of Rulkov Maps for replicating the spiking/bursting activity of the pyramidal cells and intercalatedcells. The parameter α is set to 4.7 ˜ 5.5, and σ is set to 0.1
for the pyramidal cells. Moreover, the parameter α is set to4.5, and σ is set to 0.33 for the intercalated cells.

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 )2(3)where x is the fast and y is the slow dynamical variables andthe parameter wij shows the coupling weight of the connection between the map. The coupling weights wij are updated
by STDP function. If STDP provokes the LTP, the couplingweights are updated to positive direction. Whereas, if it provokes LTD, the coupling weights are updated to negative di- Figure 3: (a) Spiking activity of pyramidal cells. (b) Spiking
rection. The nonlinear function f (xn , xn−1 , yn ) is shown as activity of intercalated cells.
following:f (xm,n , xm,n−1 , ym,n )α/(1 − xm,n + u), (xm,n ≤0)= α + u,(0<xm,n <α+u and xm,n−1 ≤0)−1,(xm,n ≥α+u or xm,n−1 >0)(4)

α and µ, σ shows the parameters of the maps. In this paper, we set parameter α and σ with arbitrary value, and µ is
fixed to 0.001. Here, these parameters are the control the dynamics parameters and the behavior shows the typical of theneurons. Specifically, the amplitude of waveform is changing
by parameter α.
4. Network Models and Simulations
In this section, we consider some easy simulation models
to analyze CA3 hippocampus network model.

4.2 2 Coupled Maps
In this section, we observe fundamental synchronization
phenomena of the coupled Rulkov Map. Moreover, we define the synchronization probability to provide quantitative
analysis of synchronization accuracy. The synchronization
probability is defined as follows.
Y(5)X
where X is the number of burst waveform, and Y is the number of synchrony burst waveform. The coupled Rulkov Mapsshows inphase synchronization, if the synchronization probability closes in the value to 1. Whereas, it shows antiphasesynchronization, if the synchronization probability closes inthe value to 0.
We consider 2 coupled Rulkov Maps with STDP as shownin Fig. 4. We explore the spiking activity and synchronization probability, if the parameters α are changed. The spiking- 607 -I=

activity of P-P model is shown in Fig. 5. Moreover, Fig. 6
shows the synchronization probability if the parameter αpreand αpost are changed from 4.7 to 5.5 each. The red areaof cubic graph shows synchronization probability, if it is bigger than 0.5. Whereas, the blue area of cubic graph shows
synchronization probability, if it is smaller than 0.5.

Figure 6: Synchronization probability of P-P model.

Figure 4: Connective arrangements of 2 coupled maps.

Figure 7 shows the synchronization probability of I-P andP-I models if the parameter αpyramidal cell is changed from4.7 to 5.5.

Figure 7: Synchronization probability of I-P and P-I models.
From Fig 7, the coupled Rulkov Maps show a anti-phasesynchronization to all of the value of α. Thus, the intercalated
cells provoke the anti-phase synchronization, or decrease thesynchronization probability.
4.3 3 Coupled Maps
In this section, we consider 3 coupled Rulkov Maps withSTDP as shown in Fig. 8. We offer an explanation of phenomena of 3 coupled maps from the result of 2 coupled maps.
Figures 9-12 show the synchronization probability between
the neuron 1 and the neuron 3.

Figure 5: Spiking activity of P-P model.
Figure 5 shows spiking activities of P-P model between
8000 and 10000[ms]. The upside of Fig. 5 shows anti-phasesynchronization, and downside one shows in-phase synchronization. From Fig. 6, the coupled Rulkov Maps show antiFigure 8: Connective arrangements of 3 coupled maps.
phase synchronization, if the parameter αpre is bigger thanαpost . Whereas, the coupled Rulkov Maps show in-phase
Figure 9 shows synchronization probability of P-P-Psynchronization, if the parameter value of αpre is smaller thanαpost . If αpre is same as αpost , the coupled Rulkov Maps model, if the parameter α1 is fixed 4.7 and α2 , α3 are changed
from 4.7 to 5.5 each. From this result, we can see that theshow perfect synchronism.

- 608 -in-phase place of synchronization probability is increased ascompared with Fig. 6. The neuron 1 and 2 show in-phase synchronization because α1 is fixed 4.7. Thus, the neuron 3 gets
involved in in-phase synchronization of the neuron 1 and 2.

Figure 12: Synchronization probability of P-P-I model.

Figure 9: Synchronization probability of P-P-P model.
Figures 10-12 show the synchronization probability of IP-P, P-I-P and P-P-I models if the parameter α of pyramidal cells changed from 4.7 to 5.5. From these result, thein-phase place of synchronization probability is decreased ascompared with Fig. 6. Moreover, the peak values of synchronization probability are changed by position of intercalated cell. Because synchronization probability represent thedifferent values if the intercalate cell is pre-neuron or postneuron, from the result of Fig. 7.

5. Conclusions
In this paper, we considered the simulation model whichis constructed by using Rulkov Maps with STDP. We have
explored the effect of 2 and 3 connective arrangements on
spiking activity. From the result, the coupled Rulkov Maps
show anti-phase synchronization, if the parameter αpre is bigger than αpost . Whereas, the coupled Rulkov Maps show
in-phase synchronization, if the parameter value of αpre issmaller than αpost . In case of 3 coupled Rukov Maps,
the spiking activity and synchronization probability become
more complex as compared with 2 coupled Rulkov Maps.
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[3] A. L. Shilnikov, N. F. Rulkov, Origin of chaos in a Twodimensional Map Modeling Spiking-Bursting Neural Activity, Int. J. Bifurcation and Chaos, vol. 13, pp. 33253340, 2003.

Figure 10: Synchronization probability of I-P-P model.

[4] N.F. Rulkov: Modeling of Spiking-Bursting Neural Behavior using Two-dimensional Map, Physical Review E,
vol. 65, 041922, 2002.
[5] D. Cumin, C. P. Unsworth: Generalising the Kuromoto
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Figure 11: Synchronization probability of P-I-P model.

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