2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Examining the Eﬀects of Noise Contamination during Epilepsy using Low
Complexity Coupled Rulkov Maps and Spike Timing Dependent Plasticity
Naohiro Shibuya1 †, Charles Unsworth2 ‡, Yoko Uwate3 † and Yoshifumi Nishio4 ††Department 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‡Department of Engineering Science, The University of Auckland
70 Symonds Street, Auckland 1001, New Zealand
Email: c.unsworth@auckland.ac.nz
Abstract—Epilepsy is a neurological disorder provokedfrom an imbalance between long-term potentiation (LTP)
and long-term depression (LTD) of the synapses. This results in an abnormal synchrony of the neurons in the brain,visible in the electroencephalogram (EEG). In certain types
of epilepsy such as tonic-clonic (grand mal) seizures synaptic disturbance occurs. In this work, we explore the eﬀectof how noise contamination in a coupled lattice of neurons alters the synchrony in such circuits. We build a low
complexity model using Rulkov maps to model the spiking
activity and Spike Timing Dependent Plasticity (STDP) tomodel the LTP and LTD. From the simulation results, weobserved that the chaotic noise STDP model has resistancecharacteristic of diverging of membrane potential.

2. Spike Timing Dependent Plasticity
Spike Timing Dependent Plasticity (STDP) [3],[4] is atemporally asymmetric form of Hebbian learning inducedby tight temporal correlations between the spikes of presynaptic and post-synaptic neurons. STDP provokes theLTP of the synapses, if the pre-synaptic spike arrival occurs
a few milliseconds before post-synaptic spikes. Whereas,
STDP provokes the LTD of the same synapse, if presynaptic spike arrival after post-synaptic spikes. (Fig. 1).

1. Introduction
Epilepsy is a neurological disorder eﬀecting 1 in 100
people worldwide [1]. During epilepsy, the excitationand inhibition of neurons becomes unbalanced leading tosupra-normal excitability which leads to seizure. Whenepilepsy occurs the neurons are said to undergo an abnormal synchrony which is prevalent in the electroencephalogram (EEG). There are approximately 40 diﬀerent
types of epilepsy each having their own very diﬀerent EEG
signature. In certain types of epilepsy such as tonic-clonic(grand mal) seizures synaptic disturbance occurs. Moreover, we consider some noises which occur in our daily
lives aﬀect the development of epilepsy. In this work, weexplore the eﬀect of how noise contamination in a coupledlattice of neurons alters the synchrony in such circuits.
In this study, we construct a low complexity, grid latticemodel of neurons using Rulkov maps to produce a twodimensional spike-bursting behavior of similar real biological neurons [5],[6]. In addition, we have defined both excitatory neurons and inhibitory neurons to exist in the lattice which express long-term potentiation (LTP) and longterm depression (LTD) [2] respectively, using Spike Timing Dependent Plasticity (STDP) [3],[4]. Thus, incorporating noise into the original STDP, we can explore the synchronous behavior of biological spiking activity.

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

∆wi j =N∑W(t jf − tif )

(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) isshown as following.

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A+ exp(−x/τ+ ) + z(x) (x < 0)
W(x) = −A− exp(x/τ− ) + z(x) (x > 0)(2)

which is in alignment to experimental models. The parameters A+ and A− depend on the current value of the synaptic
weight wi j , and z(x) is the time-series of noise. Additionally, the parameters A+ and A− are fixed to the value of 0.05
and the time constants are τ+ = τ− 10ms which parameters
are determined by reference to previous study [9].
In this study, we consider the 3 kinds of STDP function; Basic STDP (i.e., z(x) = 0), STDP with random noise(uniform random numbers between −0.004 and 0.004) andSTDP with chaotic noise (Eqs. (3), (4) and Fig. 2). The
chaotic noise which is generated by logistic map is given
as following equation.
ẑ(x + 1) = αẑ(x)(1 − ẑ(x)),(3)

it is known that the map produces intermittent bursts justbefore periodic-windows appear. We apply the intermittency chaos time series after the following normalization
to STDP function W(x).

z(x + 1) =ẑ(x) − z̄
,σz

(4)

where z̄ and σz are the average and the standard division ofẑ(x), respectively. Figure 2 shows the intermittency chaos
near the three-periodic window obtained from Eq. (4) for
α = 3.828.

xn+1yn+1==f (xn , yn ),yn − µ(xn + 1) + µσ,(5)Where, x represents the fast and y is the slow dynamicalvariables. The nonlinear function f (x, y) is described inequation (6) as follows:α/(1 − x) + y, (x ≤ 0)

f (x, y) = 
α + y,(0 < x < α + y) − 1,(x ≥ α + y)(6)The equation of the coupled grid lattice of Rulkov maps
was then derived to be as in equation (7).
1xm,n+1 = f (xm,n , ym,n ) + wi j (xm+1,n − 2xm,n + xm−1,n ),21ym,n+1 = ym,n −µ(xm,n +1)+µσ+ wi j (xm+1,n −2xm,n +xm−1,n ),2(7)Where, parameter m is the number of neurons, n is the number of iterations, and wi j shows the coupling weight of theconnection between the maps. The coupling weights wi j
are then updated by the STDP function. When the STDPprovokes LTP, the coupling weights are updated as positive.
Whereas, if STDP provokes LTD, the coupling weights areupdated in a negative manner (Fig. 1). In the following section, we demonstrate how Rulkov maps demonstrate various synchronization phenomena due to coupling weight.
4. Network Model and Simulations

Figure 2: Time series obtained from logistic map (α =3.828).

3. Coupled Rulkov Maps
In recent years, a simple model which replicates thedynamics of spiking and spiking-bursting activity of real
biological neurons has proposed by N. F. Rulkov [5],[6].
The model is a two-dimensional map that produces chaotic
spiking-bursting neural behavior. It is demonstrated thatthe results of this model are in agreement with the synchronization of chaotic spiking-bursting behavior experimentally found in real biological neurons [8]. The expressionsof the Rulkov map are shown in equations (5-7) below.

In this work, we consider the 5 coupled maps which isconstructed by excitatory neurons and the inhibitory neurons as shown in Fig. 3. The network of excitatory neurons
and the inhibitory neurons exhibit spike/bursting activity.
The excitatory neurons exhibit the bursting activity which
amplitude is around 60mV and 5-10 spikes burst all at oncein a bursting wave, and the inhibitory neurons exhibit thespike activity which spike amplitude is 60-70mV and spikeinterval is 10ms in the biological neurons [10]. Therefore,
the parameter α, σ and µ of the Rulkov Maps were set to50, 0.1 and 0.04-0.06 respectively for the excitatory neurons and the parameter α, σ and µ were set to 40, 0.33 and0.6 respectively for the inhibitory neurons in the network.
In this study, the parameters of coupled Rulkov Maps are
fixed as Tab. 1.

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Figure 3: Connective arrangement of 5 coupled maps.

with random noise STDP model. Each excitatory neuronshows high accuracy in-phase synchronous firing around50000ms, whereas the neuron 5 (inhibitory neuron) does
not get involved in synchronous firing of excitatory neurons (i.e., it exhibits spike activity without being bursting
activity). The membrane potentials do not increase and diverge for the above reason. By the way, the reason why thechaotic noise STDP model shows diﬀerent spiking behavior compared to the random noise STDP model is that thelogistic map represents the value close to extreme.

Table 1: The Parameters of Coupled Rulkov Maps
Neuron #ασµ
1
50
0.1
0.04
250
0.1
0.05
350
0.1
0.06
450
0.1
0.05
540
0.33
0.6

Figure 4-(a) shows the coupling weights (0-50000ms)
and the spiking activities (20000-20500ms, 2500025500ms, 30000-30500ms) on 5 coupled Rulkov mapswith STDP. The color bar and the number on the bottomleft of Fig. 4 describe the number of neurons and neuronal connection on the result of coupling weight, andthe red/blue waveforms show the spiking activity of excitatory/inhibitory neuron each. The reason why the coupling weights are not updated from around 40000ms is thecoupled Rulkov maps are convergence and do not exhibit
the bursting activity. The magnitude relation of couplingweight are fixed relatively early compared to another models. In addition, we consider that the synchronization accuracy between the excitatory neurons has an eﬀect on diverging of membrane potentials from what the coupling weights
between the excitatory neurons increase sharply just before
the membrane potentials diverge. The spiking activity ofneuron 5 (inhibitory neuron) has already exhibited recognizable bursting activity around 20000ms, and it becomesinvolved in high accuracy in-phase synchronization of neuron 1, 2 and 4 around 30000ms. In basic STDP model,
the synchronization accuracy has been increased with time,since the membrane potentials will increase at an accelerating pace and diverge (i.e., lead to seizure) from around40000ms.
Figure 4-(b) shows the coupling weights (0-50000ms)
and the spiking activities (20000-20500ms, 3000030500ms, 40000-40500ms, 49500-50000ms) on 5 coupled
Rulkov maps with random noise STDP. The magnitude relation of coupling weight are hard to converge compared
to another models. Moreover, the excitatory neuron 1, 3
and 4 show high accuracy in-phase synchronous firing, andthe spiking activity of excitatory neuron 2 shows anti-phasesynchronization with another excitatory neurons. The spiking activity of neuron 5 (inhibitory neuron) has exhibitedrecognizable bursting activity and the diverging of membrane potentials around 40000ms. Also in random noise
STDP model, the synchronization accuracy has been increased with time, since the membrane potentials will increase at an accelerating pace and diverge (i.e., lead toseizure) relatively late compared to basic STDP model.
Figure 4-(c) also shows the coupling weights (050000ms) and the spiking activities (20000-20500ms,30000-30500ms, 40000-40500ms, 49500-50000ms) on 5
coupled Rulkov maps with chaotic noise STDP. The magnitude relation of coupling weight are stable compared

5. Conclusions
In this work, we built a low complexity model using
Rulkov maps to model the spiking activity and Spike Timing Dependent Plasticity (STDP) to model the LTP andLTD. Using the aforementioned models, we then proceed
to explore the eﬀect of noise contamination on the spiking activity and the synchronous provoked. From the simulation results, we observed that the membrane potentials
increase at an acceleration pace and diverge (i.e., lead toseizure) , if the inhibitory neuron obtains involved in synchronous firing of excitatory neurons. Additionally, thechaotic noise STDP model has resistance characteristic ofdiverging of membrane potential. Basically, certain kind ofnoise prevents development of epilepsy.
Acknowledgments
The authors would like to thank NOLTA2014 organizing committee members for their fruitful suggestions andcomments.
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Figure 4: Spiking activity and coupling weight of 5 coupled maps with: (a) Basic STDP (b) Random noise STDP and(c) Chaotic noise STDP.

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