Studying a chaotic spiking neural model

International Journal of Artificial Intelligence & Applications (IJAIA), Vol. 4, No. 5, September 2013
DOI : 10.5121/ijaia.2013.4508 107
STUDYING A CHAOTIC SPIKING
NEURAL MODEL
Mohammad Alhawarat1
, Waleed Nazih2
and Mohammad Eldesouki3
1,2
Department of Computer Science, College of Computer Engineering and Sciences,
Salman Bin Abdulaziz University, Kingdom of Saudi Arabia
3
Department of Information Systems, College of Computer Engineering and Sciences,
Salman Bin Abdulaziz University, Kingdom of Saudi Arabia
ABSTRACT
Dynamics of a chaotic spiking neuron model are being studied mathematically and experimentally. The
Nonlinear Dynamic State neuron (NDS) is analysed to further understand the model and improve it.
Chaos has many interesting properties such as sensitivity to initial conditions, space filling, control and
synchronization. As suggested by biologists, these properties may be exploited and play vital role in
carrying out computational tasks in human brain. The NDS model has some limitations; in thus paper the
model is investigated to overcome some of these limitations in order to enhance the model. Therefore, the
model’s parameters are tuned and the resulted dynamics are studied. Also, the discretization method of
the model is considered. Moreover, a mathematical analysis is carried out to reveal the underlying
dynamics of the model after tuning of its parameters. The results of the aforementioned methods revealed
some facts regarding the NDS attractor and suggest the stabilization of a large number of unstable
periodic orbits (UPOs) which might correspond to memories in phase space.
KEYWORDS
Nonlinear dynamics, chaotic spiking neuron, artificial neural networks, chaos, Rössler, Eigen space
1. INTRODUCTION
Biologists suggest that chaos may play an important role in human brain [1-8]. Chaos has many
interesting properties that might be exploited in carrying out information processing tasks. Such
properties are: sensitivity to initial conditions, space filling, control, synchronization and rich
dynamics that can be accessed using different control methods. When Artificial Neural Networks
(ANNs) are equipped with chaos they might provide an access to large number of rich dynamic
behaviours. These can be accessed if appropriate control mechanisms is chosen such as feedback
control [9-11]. If such approaches are applied to chaotic neural models then the model might
stabilize into one of many UPOs that a chaotic attractor encompasses.
In recent years, many chaotic neural models have been devised to study how such rich dynamic
behaviours might be exploited in carrying out information processing tasks. One of these models
is the NDS model [12] which is based on Rössler system [13]. It is a simple chaotic system with
one nonlinear term. Rössler system has been studied in different contexts such as control and
biology to name a few [14,15].
The NDS model was first devised in 2003 [16] and it represents a chaotic spiking neural model.
The authors introduced the model in [16] to represent an infinite state machine using the rich

108
dynamic behaviours that model encompasses. The authors have demonstrated that using input
signals along with a control mechanism could stabilize the model attractor to one of its UPOs.
The control mechanism used was a modified version of Pyragas [11]. The authors have studied
small networks of 2-3 neurons, the results shows that the network has been stabilized to an UPO
according to a periodic length that is relative to the input pattern.
Using only one NDS neuron, its attractor might have access to large number of UPOs that can be
stabilised to. These might be mapped to memories in phase space. In contrast, a Hopfield neural
network can give only 0.15n memory size (where n is the number of neurons in the network).
The NDS model has been studied in [12, 17-20]. In [17] Lorenz attractor has been used instead
of Rössler in an application of human motion detection from a sequence of video frames. In
another work [19] the authors suggested that chaos may equip mammalian brain with a
mechanism that helps in solving hard nonlinear problems. In [20] networks of NDS neurons
have been studied in terms of Spike Time Dependent Plasticity (STDP) which is a property of
cortical neurons. Experiments results suggested that NDS neurons may have the propriety of the
realism of biological neural networks. In [18] the NDS model has been investigated thoroughly.
The author has studied the chaotic behaviour of the model both experimentally and analytically.
The study has shown interesting results.
This paper is organized as follows: in section 2 the original Rössler model is described, in
section 3 the NDS model is introduced, in section 4 the discretization method that has been used
to devise the NDS model is analysed and studied, in section 5 the experimentation setups that
are carried out to tune the parameters of the NDS model are described, section 6 is devoted
to the analysis of the dynamics of the NDS attractor, section 7 includes discussions and finally
section 8 concludes the paper.
2. RÖSSLER CHAOTIC ATTRACTOR
Rössler system [13] is a simple chaotic dynamical system and is represented by the following
equations:
uyx −−=′ (1)
yaxy *−−=′ (2)
)( cxzbz −+=′ (3)
Where a and b are usually fixed and c is called the control parameter. The usual parameter
settings for the Rössler attractor are a = 0.2, b = 0.2, and c = 5.7, and the attractor for
such settings is shown in figure 1.
Fig. 1 The Rössler chaotic attractor with parameters a = 0.2, b = 0.2, and c = 5.7.

109
3. DESCRIBING THE NDS MODEL
The NDS model is first proposed in [12]. The NDS neuron is a discretized model that is based
on Rössler’s attractor [13] as described by equations 1-3 in section 2.
Large number of orbits with varying periodicity might be stabilised by varying different
system’s parameters such as time delays, period length τ and initial conditions.
The NDS model simulates a novel chaotic spiking neuron and is represented by:
))()(()()1( tutybtxtx −−+=+ (4)
))()(()()1( taytxctyty ++=+ (5)



≤+++−++
>
=+
θ
θη
)()()())())()((()(
)(
)1( 0
tutIntFtkutxtuvdtu
tu
tu (6)
∑=
−=
n
j
jj twtF
1
)()( τγ (7)
∑=
=
n
j
j tItIn
1
)()( (8)



≤
>
=+
θ
θ
γ
)(0
)(1
)1(
tu
tu
t (9)
Fig. 3 The chaotic behaviour of a NDS neuron without input (a) the time series of u(t) and γ(t) and (b) the
phase space of x(t) versus u(t)
where x(t), y(t) and u(t) describe the internal dynamics of the model, γ(t) is the model’s binary
output, F(t) is the feedback signal, In(t) represents the external input spike train, and the
parameters of the model are: a = 0.002, v = 0.002, b = 0.03, c = 0.03, d = 0.8, k = −0.057, θ =
−0.01, η0 = −0.7 and τj is the period length of the feedback signal.
The discretization method that is used in constructing the NDS model has been carried out by
scaling the system variables x(t), y(t) and u(t) using different scaling constants: b, c, d which
been tuned by carrying out experiments until major dynamics of the Rössler system are
preserved.

110
Figure 3 depicts the dynamics of a single NDS neuron without input, whereas figure 4 shows the
NDS dynamics when it is stabilised to period-4 orbit due to the feedback control mechanism F
and a time delayed feedback connection.
Fig. 4 The stabilizing of period-4 orbit of a NDS neuron with feedback connection.
4. DISCRETIZATION METHOD
There are different discretization methods that are used to convert a continuous system into
discrete. One of the nonstandard methods is Euler’s Forward differentiation that is used in
formulating simple simulation systems. A time step TS is used in calculating the approximate
value of next step of a continuous system. For example, when Euler’s Forward differentiation
method is applied to equation 1 then it will become:
)()(
)()(
)( 1
kk
kk
k tuty
TS
txtx
tx −−=
−
= +
& (10)
and then solving for )( 1+ktx :
))()(()()( 1 kkkk tutyTStxtx −−+=+ (11)
if )(tx is used instead of )( ktx , then the equation becomes:
))()(()()1( tutyTStxtx −−+=+ (12)
Note that TS is chosen to be parameter b with the value 0.03 when equation 12 is compared to
equation 4.
For simulation purposes TS is usually chosen to be small and it is preferable to be chosen
according to:
max
1.0
λ
≤TS (13)
Where max
λ is the largest absolute eigenvalues for the system under study. For the NDS model
and according to the mathematical analysis results that is achieved in [18], 5.68698max
=λ .
When substituting this value in equation 13:
0176.0≤TS (14)
If this value is compared with b it is obvious that the time step that has been chosen doesn’t
follow the simulation preferable setup.

111
Also, it is clear from equation 5 that TS is chosen to be c = 0.03 and a scaling factor is applied to
y variable using a = 0.002.
For the u variable TS is chosen to be d =0.8. Moreover, authors in [12] have scaled down
constant c to 0.057 and modified its sign. Finally, sign of x and k have also been changed.
The aforementioned modifications have affected the original Rössler system properties in phase
space and in Eigen space as well. According to [18], the two spiral saddle points of fixed points
of original Rössler system have become two spiral repellors in the NDS model. This major
change might be due to the change to both scaling factors and sign of x and k.
The change in the fixed points indicate that the NDS model have weak connection to the original
Rössler system. This conclusion was shown in [18] when the author indicated that UPOs exist
only as the results of both the acting forces of the spiral repellors of the attractor along with the
applied reset mechanism. Without the reset mechanism the spiral repellors will enforce any
nearby trajectory to approach infinity.
When Euler’s forward differentiation method is applied to Rössler system then equations 1- 3
become:
))()(()()1( tutyTStxtx −−+=+ (15)
))()(()()1( taytxTStyty ++=+ (16)
)))()((()()1( ctxtzbTStutu −++=+ (17)
Where TS = 0.0055, a = 0.2, b = 0.2, and c = 5.7.
Fig. 5 The Discrete version of Rössler attractor based on equations 15- 17
An experiment setup has been prepared to make sure the new equations will preserve the original
Rössler system attractor.
The following changes need to be made to equations 15-17 to follow the NDS equations:
1. Set TS to 0.0055.
2. Change b and c in equation 17 to v and k.
4. Change TS in equations 15,16 and 17 to b,c and d and set them to the value of TS.
7. Change the value of a and v and to 0.2.
8. Change the value of k to 5.7
9. Rename variable z to be u.
10. Change the sign of the term (x(t)−k) in equation 17 to become (−x(t)+k)

112
When applying the previous changes, then equations 15- 17 become:
))()(()()1( tutybtxtx −−+=+ (18)
))()(()()1( taytxctyty ++=+ (19)
)))()((()()1( ktxtuvdtutu +−++=+ (20)
Where a = 0.2, b = c = d = 0.0055, and k = 5.7.
To verify the new model an experiment has been carried out to depict the attractor that
represents equation 18-20. This is depicted in figure 6. Note that the original Rössler attractor
has disappeared.
Fig. 6 The attractor of the model that is based on equations 18- 20.
5. TUNING THE PARAMETERS OF THE NDS MODEL
In this section the details of tuning the parameters of the NDS model is shown. To achieve this,
many experiment setups have been prepared. The parameters that considered are a,v,b,c,d and k,
the other parameters, viz., jτηθ ,, 0 have already been studied in [18]. In all experiments one
NDS neuron is used where the feedback control is applied after time step of 1000. After that the
experiment runs for 9000 iterations.
Wide range of settings have been chosen. Values of the variables of the model are recorded and
then are depicted in phase space. Values of the parameters are considered valid if an attractor
exists for each setting. The results of valid ranges for the model parameters are shown in table 1.
Table 1 Parameter value’s ranges
Parameter a,v b,c D K
Value 0.001-0.1 0.01-0.055 0.8-0.9 -(0.055-0.058)
Different parameter settings have been chosen as stated in table 2. The NDS original parameter
setup is used in setup 7 to compare this setup with other settings to highlight areas of
enhancements of the model.

113
Table 2 Parameter settings with different selected values from the ranges appear in table1
Parameter a,v b,c D K
Setup 01 0.001 0.03 0.8 -0.057
Setup 02 0.01 0.03 0.8 -0.057
Setup 03 0.1 0.03 0.8 -0.057
Setup 04 0.002 0.001 0.8 -0.057
Setup 05 0.002 0.02 0.8 -0.057
Setup 06 0.002 0.05 0.8 -0.057
Setup 07 0.002 0.03 0.8 -0.057
Setup 08 0.002 0.03 0.85 -0.057
Setup 09 0.002 0.03 0.9 -0.057
Setup 10 0.002 0.03 0.8 -0.055
Setup 11 0.002 0.03 0.8 -0.056
Setup 12 0.002 0.03 0.8 -0.058
Setup 13 0.01 0.05 0.85 -0.055
Setup 14 0.002 0.015 0.8 -0.058
Setup 15 0.1 0.04 0.8 -0.056
Another experiment setup has been prepared to verify the capacity of the attractor in terms of
UPOs that it might encompasses. The average number of successfully stabilized UPOs is
computed over 1000 run according to the parameter settings that appear in table 2 and then
depicted in figure 7.
Fig. 7 Average number of Stabilized UPOs over 1000 run based on the parameter settings that appear on
table 2.
Figure 7 suggests that slightly better parameter settings exist such as Settings 14 and Settings 05
increased the attractor capacity when compared to the NDS original settings (Setting 07).

114
6. EIGEN SPACE ANALYSIS
In this section the fixed points of the NDS attractor will be calculated for the different parameter
settings that appear in table 2. Also Eigen space will be computed and compared to the NDS and
original Rössler attractors. Before that a gentle introduction to Eigen space terminology will be
prefaced. For more information about nonlinear dynamics and mathematical details used in this
section please refer to [21-28].
6.1 Fixed points and Eigen space Analysis
In a chaotic system there are poles that organize and control the behaviour of the trajectories that
pass nearby them, and are known as fixed points. Around each of these points there is a basin of
attraction or repulsion. Once a trajectory enters one of these basins of attraction its behaviour
will be affected, then it will either be attracted and end up in a periodic or fixed point behaviour
or it will be repelled and move to another basin of attraction. These fixed points restrict the
behaviour of trajectories of a chaotic system in a fractal-bounded area and determine the shape
of the attractor in the phase space.
The types of fixed points of a dynamic system can be determined by finding its characteristic
values. These characteristic values are then used to analyze and understand the different system
behaviours around the fixed points including the stability of these points. The types of fixed
points available depend on the dimensions of the dynamic system under study.
Finding and analyzing fixed points in three-dimensional state space is more complicated than in
one and two-dimensional state space. That is because systems in three-dimensional state space
require:
• Solving cubic equations which is usually hard and sometimes is impractical.
• A larger Jacobian matrix which requires more calculations.
• Calculating the three characteristic values that are used to determine the nature of each
fixed point of the system.
Finding the characteristic values from the Jacobian matrix includes solving quadratic equations.
There are three roots that result from solving cubic equations, which are the characteristic
values. The combination of these characteristic values together determines the type of the system
fixed point(s).
Then the characteristic values could be in one of these categories:
• All real numbers:
o All negative. The fixed point is of type Node: All trajectories head directly to it.
o All positive. The fixed point is of type Repellor: All trajectories move away directly
from it.
o Two negative and one positive. The fixed point is of type Saddle point Index-1:
Trajectories attracted by the saddle point on a surface (two stable manifolds) and
repelled on a curve (one unstable manifold).
o Two positive and one negative. The fixed point is of type Saddle point Index-2:
Trajectories repelled by the saddle point on a surface (two unstable manifolds) and
attracted on a curve (one stable manifold).
The index number here indicates the number of the unstable manifolds.
• Mixed real and imaginary numbers:
o All negative; one real and two complex conjugate pair. The fixed point is of type Spiral
Node.

115
o All positive; one real and two complex conjugate pair. The fixed point is of type Spiral
Repellor.
o Two negative (complex conjugate pair) and one real positive. The fixed point is of type
Spiral Saddle point Index-1: Trajectories spiral around the saddle point while they are
attracted by the two stable manifolds.
o Two positive (complex conjugate pair) and one real negative. The fixed point is of type
Spiral Saddle point Index-2: Trajectories spiral around the saddle point on a surface (two
unstable manifolds) while they are moving away from the saddle point.
6.2 Calculating Fixed points and Eigen space values
According to [18], the two fixed points for the NDS model are of type spiral repellor, where the
original Rössler system fixed points are of type spiral saddle point.
An extensive mathematical calculations need to be carried out to reveal the effect of parameter
tuning that has been discussed in the previous section. Then an experiment setup has been
prepared to calculate the fixed points and the Eigen space for the different parameter settings
according to table 2.
First the fixed points have been calculated for each parameter setup by solving the quadratic
equations of the system. Second, the Jacobian matrices have been belt for each equilibrium
vector, and then the Eigen vectors and Eigen values have been calculated. Finally, the fixed
point type is specified based on the classification that is aforementioned in subsection 6.1. The
results of running the experiment show different fixed points and Eigen values as shown in
tables 4, 5 and 6.
Table 4 Fixed points for each parameter settings that appears in table 2.
Fixed Points
First Fixed Point Second Fixed Point
X y Z x y z
Setup 01 -0.05702 57.01754 -57.01754 0.00002 -0.01754 0.01754
Setup 02 -0.05870 5.87035 -5.87035 0.00170 -0.17035 0.17035
Setup 03 -0.13248 1.32482 -1.32482 0.07548 -0.75482 0.75482
Setup 04 -0.05707 28.53504 -28.53504 0.00007 -0.03504 0.03504
Setup 05 -0.05707 28.53504 -28.53504 0.00007 -0.03504 0.03504
Setup 06 -0.05707 28.53504 -28.53504 0.00007 -0.03504 0.03504
Setup 07 -0.05707 28.53504 -28.53504 0.00007 -0.03504 0.03504
Setup 08 -0.05707 28.53504 -28.53504 0.00007 -0.03504 0.03504
Setup 09 -0.05707 28.53504 -28.53504 0.00007 -0.03504 0.03504
Setup 10 -0.05507 27.53632 -27.53632 0.00007 -0.03632 0.03632
Setup 11 -0.05607 28.03567 -28.03567 0.00007 -0.03567 0.03567
Setup 12 -0.05807 29.03444 -29.03444 0.00007 -0.03444 0.03444
Setup 13 -0.05676 5.67617 -5.67617 0.00176 -0.17617 0.17617
Setup 14 -0.05807 29.03444 -29.03444 0.00007 -0.03444 0.03444
Setup 15 -0.13185 1.31846 -1.31846 0.07585 -0.75846 0.75846

116
Table 5 Eigen values for the first fixed point and the corresponding fixed point type for each parameter
setting that appears in table 2.
Eigen Values For the First Fixed Point Fixed Point Type
Setup 01 1.0000+1.1702i 1.0000-1.1702i 1.0000 Spiral Repellor
Table 6 Eigen values for the second fixed point and the corresponding fixed point type for each parameter
setting that appears in table 2.
Eigen Values For the Second Fixed Point Fixed Point Type
Setup 03 1.0842 1.0091 0.8038 Repellor
Setup 15 1.1010 1.0123 0.7852 Repellor
6.3 Analyzing the results
It is obvious from table 4 that setup 1, 2, 3, 13 and 15 are affected by tuning the parameters.
Some values of the fixed points for the other setups have been slightly changed, but these have
not affected the types of the fixed points as shown in both tables 5 and 6.
The change in the values of the fixed points for setup 1, 2, 3, 13 and 15 have affected the type of
the fixed points for setup 3 and 15 only as shown in table 6. However, the change from spiral
repellor to repellor will not affect the dynamic behaviour of the attractor as both of them will
repel trajectories heading toward them in all directions: x, y and z.

117
7. DISCUSSION
There results attained in this paper suggest that there are weak connections between the NDS
model and original Rössler system. The discretization method that is used in constructing the
NDS model doesn’t follow known methods that are used in simulating continuous systems. The
authors in [12] has used different time steps and nested scaling factors. These and the change
that is made to the sign of both x and k has led to a change in attractor properties such as fixed
points. Tuning the parameters of the NDS model has led to slight changes in the capacity of the
attractor in terms of the number of stabilized UPOs. The mathematical analysis of the dynamics
of the model for different parameter’s settings has shown that the fixed point’s types remained
unchanged.
The two spiral repellors of the NDS model force any nearby trajectory to evolve to infinity, but
when NDS model is equipped with a suitable reset mechanism then trajectories will remain in a
fractal dimension as shown in figure 3. This is one of the main chaos properties. It has already
proven that the NDS model is chaotic according to the Lyapunov exponent estimates obtained in
[29]. In contrast, the two spiral saddle points of the Rössler attractor will keep nearby trajectories
evolving between the forces of attracting sides of both of saddle points.
When comparing the attractors of both NDS and original Rössler as depicted in figure 1 and
figure 3, it is clear that the shape of the attractor is preserved in x-u plot except that the attractor
in the NDS model is flipped due to the sign change in x and k. However, when comparing the
fixed points, eigenvalues and eigenvectors it is obvious that they are different.
The results obtained in this paper also suggest that both the reset mechanism and the feedback
signal are vital for the NDS attractor to exist.
8. CONCLUSION
Chaotic spiking neural models have been studied and investigated in recent years to explore the
possibilities of exploiting dynamics of such models in carrying out information processing tasks.
The NDS model encompasses a large number of UPOs that are stabilised using a feedback
control mechanism. Although the NDS model has weak connections to the original Rössler
system, still much rich dynamic behaviours are inherited from the Rössler system which can be
noticed when comparing figure 1 and figure 3.
The discretization method that is used in devising the NDS model along with the change in sign
in the term u(t)(x(t)−k) has affected the shape and properties of the NDS attractor when
compared with Rössler system.
Different experiments have been carried out to tune parameters of the NDS model. The valid
ranges of the model have not changed the properties of the NDS model both in phase space and
Eigen space. The mathematical analysis of the attractor for different parameter settings revealed
that the underlying dynamics of the attractor remained the same.
Although there are weak connections between the NDS and the Rössler models, the NDS
attractor encompasses large number of UPOs and wide range of dynamic behaviours that may be
exploited to carry out information processing tasks.

118
ACKNOWLEDGEMENTS
I would like to thank the deanship of scientific research at Salman Bin Abdulaziz University in
KSA for supporting this work under grant number 40/H/1432.
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Authors
Mohammad Alhawarat has got his Ph.D. in chaotic neural networks from School of
Technology of Oxford Brookes University, United Kingdom, in 2007. He has worked as an
assistant professor in Petra University in Jordan for 1 year, he then joined the College of
Computer Engineering and Sciences of Salman Bin Abdulaziz University since 2008 as an
assistant professor of computer science. His research interests are chaotic neural networks,
data mining and machine learning including natural language processing.
Waleed Nazih has got his M.Sc. degree of Information Technology from Faculty of Computers &
Information, Cairo University, 2007. He has worked as lecturer in the College of Computer Engineering
and Sciences of Salman Bin Abdulaziz University since 2008. His research interests includes machine
learning including speech recognition and natural language processing.
Mohammad Eldesouki has got his M.Sc. in Information Systems from the Faculty of Computers
and Information, Helwan University, 2002. He has worked as lecturer at the department of
information systems, College of Computer Engineering and Sciences of Salman Bin Abdulaziz
University since 2008. His research interests are databases and data mining.

Studying a chaotic spiking neural model

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