International Journal of Computer Applications (0975 8887)
Volume 99 - No. 17, August 2014

arXiv:1501.02192v1 [] 9 Jan 2015

Investigation of a chaotic spiking neuron model

M. Alhawarat

T. Olde Scheper,

N. T. Crook

Department of Computer Science,
Salman Bin Abdulaziz University,
Al Kharj, 11942, Kingdom of Saudi Arabia
m.alhawarat@sau.edu.sa

Dept. of Computing and Comm. Tech.,
Oxford Brookes University,
Wheatley, OXFORD OX33 1HX, UK
tvolde-scheper@brookes.ac.uk

Head of Computing and Comm. Tech.,
Oxford Brookes University,
Wheatley, Oxford OX33 1HX, UK
ncrook@brookes.ac.uk

ABSTRACT
Chaos provides many interesting properties that can be used
to achieve computational tasks. Such properties are sensitivity
to initial conditions, space filling, control and synchronization.
Chaotic neural models have been devised to exploit such properties.
In this paper, a chaotic spiking neuron model is investigated
experimentally. This investigation is performed to understand the
dynamic behaviours of the model.
The aim of this research is to investigate the dynamics
of the nonlinear dynamic state neuron (NDS) experimentally.
The experimental approach has revealed some quantitative
and qualitative properties of the NDS model such as the
control mechanism, the reset mechanism, and the way the
model may exhibit dynamic behaviours in phase space. It
is shown experimentally in this paper that both the reset
mechanism and the self-feed back control mechanism are
important for the NDS model to work and to stabilise to
one of the large number of available unstable periodic orbits
(UPOs) that are embedded in its attractor. The experimental
investigation suggests that the internal dynamics of the NDS
neuron provide a rich set of dynamic behaviours that can be
controlled and stabilised. These wide range of dynamic behaviours
may be exploited to carry out information processing tasks.

General Terms:
Chaotic Neural Model, Chaos

Keywords:
Rössler Attractor, Chaotic Spiking Neural Model, Nonlinear
Dynamics Investigation

1. INTRODUCTION
It has been suggested by biologists that chaos may be one of
the main ingredients in carrying out information processing tasks
in human brain[?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?]. Plugging chaos
into Artificial Neural Networks (ANN) may enrich its state space
with a large number of dynamic behaviors when compared to
hopfield networks. These behaviors can be utilized through chaos
control methods[?, ?, ?]. Results of applying such control methods
to chaotic attractors showed stabilization into one of its UPOs
[?, ?, ?, ?, ?].

Chaotic neural network models have been devised to exploit the
rich dynamic behaviours that chaos provides. The main idea is
to apply appropriate control methods to map a specific input to
one of the internal dynamic behaviours that the chaotic attractor
encompasses. One way to do so is to use spikes as input signals.
One system that have implemented the aforementioned idea is the
NDS model [?]. The model has been proposed initially in [?]
where the authors defined a chaotic neuron that is based on a
discrete version of Rössler system [?]. The idea is to provide a
large number of memories using only one NDS neuron. A modified
version of Pyragas [?] has been used to control the dynamics of
the NDS model. The authors in [?] have shown that it is possible
to retrieve previously stored periodic patterns using the modified
control mechanism.
The authors in [?] have used Lorenz attractor instead of Rössler.
They have used the idea of transient computation machine to detect
human motion from sequences of video frames. In another research
paper, it has been proposed that chaos may equip mammalian
brain with the equivalent of kernel trick for solving hard nonlinear
problems[?] using the NDS neuron model. Spike Time Dependent
Plasticity (STDP) has been studied in networks of NDS neurons
[?]. The results of experiments suggest that the NDS neuron may
own the realism of biological neural networks.
Even though the NDS neuron model has many interesting
properties, it has some drawbacks including: the way the control
and reset mechanisms work are unclear, the analysis of the
dynamics of the model is missing, analysis of the stability of
networks of such neurons is also missing, and learning algorithms
need to be developed to exploit the large number of UPOs that
the attractor of the NDS encompasses. In this paper the control
and the reset mechanisms are investigated experimentally. This
investigation is important to understand how the control mechanism
works, and will show how important the reset mechanism is for the
model to work.
The paper is organized as follows: section 2 describes the NDS
model, section 3 is devoted to investigate reconstructing the
internal dynamics in NDS model, where section 4 investigates
the NDS control mechanism and section 5 investigates the NDS
reset mechanism. In section 6 the results of the experiments are
discussed and finally section 7 concludes the paper.

2.

DESCRIBING THE NDS MODEL

Crook et al. have proposed a chaotic spiking neuron model [?],
viz., the NDS neuron. The NDS neuron is a conceptual discretized

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International Journal of Computer Applications (0975 8887)
Volume 99 - No. 17, August 2014
model that is based on a modified version of Rössler’s chaotic
system [?].
D(t) = F (t) + I(t)

F (t) =

25

n
X

(7)

wj γ(t − τj )

(8)

j=1

20
15
z

I(t) =

n
X

Ij (t)

(9)

1 u(t) > θ
0 u(t) ≤ θ

(10)

j=1

10
5

γ(t + 1) =
0
10
15

0

10
5

−10
y

0
−20

−5
−10

x

Fig. 1. Rössler chaotic attractor with parameters a = 0.2, b = 0.2, and
c = 5.7.

Rössler system [?] is a simple dynamical system that exhibits chaos
and has only one nonlinear term in its equations. Rössler built
the system in 1976, and it describes chemical fluctuation and is
represented by the following differential equations:
x′ = −y − z

(1)

y ′ = x + ay

(2)



where x(t), y(t) and u(t) describe the internal state of the
neuron, γ(t) is the neuron’s binary output, F (t) represent the
feedback signals, In(t) is the external input binary spike train,
and the constants and parameters of the model are: a = v = 0.002,
b = c = 0.03, d = 0.8, k = −0.057, θ = −0.01, η0 = −1 and τj
is the periodP
length of the feedback signals.
n
The term
w γ(t − τj ) in equation 8 represents the
j=1 j
self-feedback mechanism which is the summation of spikes fired
from the NDS neuron at time t = t − τj each multiplied by the
corresponding connection weight wj .
By varying the system parameters such as period length τ ,
connection time delays and initial conditions, large number of
distinct orbits with varying periodicity may be stabilised. The
dynamics of a single NDS neuron without input is shown in
Figure 2.

1

0

0.5

u(t+1) =



x(t + 1) = x(t) + b(−y(t) − u(t))

(4)

y(t + 1) = y(t) + c(x(t) + ay(t))

(5)

η0
u(t) > θ
u(t) + d(v − u(t)x(t) + ku(t)) + D(t) u(t) ≤ θ

(6)

0
−1

−0.5

u(t)

Where a and b are usually fixed and c is varied. The familiar
parameter settings for Rössler attractor are a = 0.2, b = 0.2, and
c = 5.7, and the corresponding attractor is shown in figure 1.
The NDS model is a discrete version of Rössler system. The
main reason for discretization is that spikes should occur in
discrete time. The discretization has been carried out by scaling
the system variables x(t), y(t) and u(t) using different scaling
constants: b, c and d. The values of these constants have been tuned
experimentally until the dynamics of Rössler system are preserved.
If the values of these constants are larger, then a system trajectory
will miss many dynamic evolutions while moving from one discrete
iteration to the next.
The NDS model simulates a novel chaotic spiking neuron and is
represented by:

−0.5
γ(t)

(3)

u(t)

z ′ = b + z(x − c)

−1

−1.5

−1.5
−2
−2
−2.5
9500

9600

9700

9800

9900 10000

−2.5
−0.4

t

−0.2

0
x(t)

0.2

(a) Time series

(b) phase plot: u(t) vs x(t)

0.4

Fig. 2. 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)

The NDS model was built as a spiking neural model, therefore
Rössler system equations have been modified to cope with the
reset mechanism after spikes are emitted. The modification that
was made to Rössler system was the change of the sign of
the term u(t)x(t) in equation 3 from positive to negative (u
here is equivalent to z), the term then becomes −u(t)x(t). This
modification will flip the fin of Rössler attractor in the x − u plane
horizontally. This effect can be noticed once figures 2 and 1 are
compared.
Before the control and reset mechanisms are investigated; the
method of reconstructing the internal dynamics is explained first.

2

3. RECONSTRUCTING INTERNAL DYNAMICS IN
NDS
One of the main properties of the NDS model is the ability
of reconstructing the original periodic internal dynamics of the
neuron. Experimental results have shown this to be possible by
stimulating an NDS neuron with a previously recorded firing
pattern of a stabilised orbit [?]; this forces the chaotic dynamics
of the neuron into the same orbit. Even though the mechanism
of the stabilization, namely periodic forcing [?], is completely
different from the feedback control, the orbit is the same in
both cases. An example is shown in Figure 3 where (a) and (c)
are the original dynamics, and (b) and (d) are the successfully
reconstructed dynamics which can be seen to be identical.
A

B

u(t)

(a)

γ(t)

1

0

−2
3800

3900
t

4000

−1

(b)

−2
9800

0.5

0.5

0

0

−0.5
−1
−1.5
−0.4

(c)

0

u(t)

−1

u(t)

u(t)

γ(t)

1

9900
t

10000

0
x(t)

0.2

−0.5

−1.5
−0.4
(d)

2000
1800
1600
1400
1200
1000
800
600
400
200
0

0

200

400

600
period length τ

800

1000

1200

Fig. 4. The number of successfully stabilised UPOs for different
periodicity τ lengths of a NDS neuron with time-delayed self-feedback
connection.

4.1 Comparing the variables of the NDS model

−1

−0.2

Number of successfully stabilised orbits out of 2000 runs

International Journal of Computer Applications (0975 8887)
Volume 99 - No. 17, August 2014

−0.2

0
x(t)

0.2

Fig. 3. The reconstruction of period-3 orbit with time delay 100. (a) and
(c) show the behaviour of the feedback controlled neuron. (b) and (d) show
the behaviour of the reconstructed period with forcing inputs.

Figure 3 shows just one example of the stabilised orbits, however
there are very large number of stabilised orbits for different period
lengths τ . To demonstrate this, an experimentation has been set up
to calculate the reliability of the NDS model in stabilizing orbits. A
single NDS neuron with a time-delayed self-feedback connection is
used with different period lengths τ ∈ [50, 1000]. For each τ 2000
different initial conditions have been used, and the system then run
and the time series of the system variables are recorded. Figure 4
shows how reliable is the stabilization of orbits for different periods
of length τ using forcing. Also the stabilization reliability was
calculated and it is 99.9922%.
The control mechanism is a very important aspect of the NDS
model and will be investigated in some detail in the following
section.

4. NDS CONTROL INVESTIGATION
In the following set of experiments the control mechanism that is
used in the NDS model will be investigated in some detail. The
purpose of these experimentations is to understand how the internal
dynamics respond to either the self-feedback signals or external
input signals. There will be two experimental setups:
—Comparing the variables of the NDS model.
—Replacing the feedback signal with a fixed value.

This set of experiments investigates the control mechanism in the
case of applying time-delayed self-feedback signals in terms of the
internal dynamics of the NDS. The idea is to compare two sets of
points ρ(t) and ρ(t − τ ) where ρ ∈ [x, y, u].
The results will be considered for each variable separately and
then also for all three state variables x, y and u combined in 3D
phase space. In all of the experimentations the self-feedback control
mechanism is applied at time step t = 1001. This may help in
eliminating any transient behaviour.
In this experiment setup one NDS neuron is used with a self
feedback connection with time delay τ = 100 and connection
weight of 0.3 which is initially turned off. The other parameters
values are the same as described in equations 4, 5 and 6. The
internal dynamics evolves until time step t = 1000 when the
self-feedback connection is turned on and the internal dynamics
continue evolving until time step t = 5000. The time series of all
the variables and output spikes are recorded.
Figure 5 depicts the absolute difference between the time series
of the variables x(t), y(t) and u(t) along x(t − τ ), y(t − τ )
and u(t − τ ). Note from the figure that once the self-feedback
connection is turned on the values of x, y and u variables at
time step t tend to synchronize with the values at time step
t − τ and a corresponding spike pattern emerges. Note also how
the absolute difference represents a nonlinear decreasing pattern
(which will be used later as a ’difference pattern’) which reflects
the dynamics encapsulated within the variables. For variable x
the dynamics is the product of the nonlinear term −ui (t)xi (t) of
equation 6.However, for variable y the absolute difference is not as
nonlinear as the one shown with the x variable and is more likely to
be linear pattern. This is because the variable y appears in a linear
form in the NDS system equations. It is important here to mention
that the behaviour that appears in figure 5 is not an average and is
the result of one run that starts from a random initial condition.
Finally, variable u has more nonlinearity in the difference pattern
when compared with x difference pattern. This difference in
nonlinearity is due to the nonlinear term that is found in the u
variable equation which is the only variable being updated based

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International Journal of Computer Applications (0975 8887)
Volume 99 - No. 17, August 2014

1.4
abs(x(t)−x(t−τ))
abs(y(t)−y(t−τ))
abs(u(t)−u(t−τ))

1.2

0.15

0.15

0.1

0.1

0.05

0.05

0.8

0.6

y(t)−y(t−tau)

y(t)−y(t−tau)

NDS Model variables

1

0

0

−0.05

−0.05

−0.1

−0.1

0.4

0.2

0

0

500

1000

1500

2000

2500

(a)

Time t

(b)

−0.2 −0.1 0 0.1 0.2
x(t)−x(t−tau)

Fig. 7. Comparing internal variable values of x and y at time step t with
the same variables values at time step t − τ

on a nonlinear term. Even though the nonlinear term affects the x
variable value, as the u variable is included in the equation of the
x variable, still the dynamics are not as strong as appears in the u
variable values as in the difference pattern mentioned previously.
One observation in this experimentation setup is the high sensitivity
of the model. On average, the spike output is stabilised by around
t = 1385. However, the underlying dynamics often do not stabilize
until much later (average of t = 5000). One example is shown in
figure 6. This demonstrates the dynamical behaviour of the system
and may be due to the nonlinearity expressed by the u variable.

in circles until it reaches 0. Note that moving from one circle to the
other looks like a transient which is mainly the effect of a neuron
firing a spike. These are obviously seen in the figure where the
difference trajectory is not smooth.

2.5
abs(x(t)−x(t−tau))
abs(y(t)−y(t−tau))
abs(u(t)−u(t−tau))

2
1.5
1
0.5

1

1

0.9

0.9

0.8

0.8

3D Euclidean distnace of x,y and u

−15

3D Euclidean distnace of x,y and u

Fig. 5. Comparing internal variable values of x, y and u at time step t with
the same variables values at time step t − τ

x 10

NDS Model variables

−0.2 −0.1 0 0.1 0.2
x(t)−x(t−tau)

0.7
0.6
0.5
0.4
0.3
0.2
0.1

0

0
(a)

−0.5
−1

Full Control Occurs
at time t = 5214

4700

4800

4900
5000
Time t

0.6
0.5
0.4
0.3
0.2
0.1

500 10001500200025003000
t

0
(b)

500 10001500200025003000
t

Fig. 8. The Euclidean distance in three dimensions of x, y and u at time
step t with the same variables values at time step t − τ

−1.5
4600

0.7

5100

5200

5300

Fig. 6. Full dynamic control while comparing internal variable values of
x, y and u at time step t with the same variables values at time step t − τ

To see how the x and y variables behave together in phase space,
the absolute difference between these variables time series at
current time and the previous period of time is depicted in figure 7.
Figure 7 shows how the difference of the variables values together
between two successive periods of times of length τ spirals around

In figure 8 the Euclidean distance in three dimensions, between
the values of x, y and u variables at time step t and those at time
step t − τ , is calculated and depicted in order to investigate the
way the system internal dynamics synchronize to itself. The figure
shows that the difference pattern is an average of the dynamic
behaviour of the three variables together, therefore the difference
pattern seems to have less dynamical behaviour compared to those
shown in figure 6.
These experimental results which are shown in figures 7- 8 help in
understanding how the control mechanism might be working and
might also address the importance of the self-feedback signal.

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International Journal of Computer Applications (0975 8887)
Volume 99 - No. 17, August 2014

1

0.5

0

u(t)

The control mechanism seems to be a self - synchronization
process. This appears from Figures 7- 8 where the time series of
one period of time τ of the system variables is coming closer
to the stabilised orbit each τ period of time. The reason behind
this self-synchronization is the time-delayed self-feedback signal
that encourages the u variable value to cross the threshold, even
though it is not guaranteed to happen every τ period of time and
depends on the value of the u variable just before receiving the
signal. If the value of the u variable crosses the threshold θ then
the reset occurs and the u variable is assigned the value of η0 .
Because the time-delayed self-feedback will incur a signal every τ
period of time, then the value of the u variable will have a periodic
value when reset occurs at specific times. These specific times will
organize the time-series of period τ and the value of the u variable
at these time steps will affect the values of both x and y variables,
and hence the internal system dynamics are driven to synchronize
with itself and stabilize to one of the attractor UPOs.
The results also show the strength of the dynamics of individual
variables and suggest that: y variable has a weak dynamics, x
variable has an intermediate dynamics, and u variable has the
strongest dynamics amongst the three. At the same time the average
of these dynamics is shown in figure 8 where the dynamics of the
difference pattern has less rapidly changing dynamics than that of
the u variable and more dynamics than the y variable has.

−0.5

−1

−1.5

−2
−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x(t)

Fig. 9. x(t) vs. u(t) stabilised orbit using fixed value of 1 being assigned
to u variable rather than the self-feedback connection value.

1

4.2 Replacing the feedback signal with a fixed value

0
−1
−2
−3
u(t)

The aim of this experimental setup is to investigate the importance
of timing of time-delayed self-feedback connection compared to
the connection weight in the process of controlling the NDS
internal dynamics. One method to investigate this is by replacing
the self-feedback signal with a fixed value to be assigned to the
u variable periodically. Then the same experimental set up that
has been used previously will be used with one change: replace
the time-delayed self-feedback connection with a fixed value to be
assigned to the u variable in a specific time every τ period of time.
In one example of this experimental setup the fixed value is chosen
to be 1, which will be assigned to the u variable at a fixed
time of t = 3 every τ period of time as an input. Initially, the
internal dynamics evolves from a randomly chosen initial point
until t = 1000 without input. After that, the fixed value is added to
the variable u values at the times like t = 1003, 1103, 1203, ...etc
until t = 30000. Then the time series of all the variables and output
spikes are recorded.
The results of this experimentation have shown that UPOs can be
stabilised, and one of these UPOs is depicted in figure 9. In one
run, the time series for x, y and u variables have been recorded
and have been depicted for variable u as shown in figure 10. The
figure shows that it takes the internal dynamics a very long time
compared to the time it takes an UPO to be stabilised when using
time-delayed self-feedback connection.
The spike train for the stabilised orbit shown in Figure 9
is (3,84,103,194, 203,286,303,384,403,494) and is of period
length 5τ . Note that the periodicity of the stabilised orbit has
increased dramatically compared to UPOs with periodicity τ
when self-feedback connection is used. This happened because the
external input will not drive the internal dynamics to synchronize
with itself every τ time steps. This increase in periodicity occurs
despite the fact that the fixed value signal is fed every τ time
steps. The spike train mentioned above is fed to an NDS neuron
that is equipped with a time-delayed self-feedback connection
when t ∈ ([1000, 1500] and an UPO was stabilised accordingly
which has a periodicity of τ and with spike train of (3, 38, 62, 82)

−4
−5
−6
−7
−8
−9

0

0.5

1

1.5
t

2

2.5

3
4

x 10

Fig. 10. u(t) time series using fixed value of 1 being added to u variable
rather than the self-feedback connection value.

for one chosen initial condition. The difference in periodicity of
the two stabilised UPOs is due to the time-delayed self-feedback
connection. In the first case without the self-feedback connection
the internal dynamics are not restricted to the periodicity τ , because
the output spikes of the system are not periodic in the sense that
they are not fed back to the model, therefore the output spikes every
τ period of time are different from those in the previous τ period of
time, and consequently it took a longer time to stabilize to a UPO.
On the other hand, when the model is equipped with a self-feedback
connection, the output spikes are fed back to the model by means
of the connection and hence encourages the internal dynamics to
repeat every τ of time.
Even though UPOs can still be stabilised when no connection is
present, it takes a very long time to stabilize a UPO, and those
stabilised UPOs cannot be reconstructed by feeding the spike train
to an NDS neuron with a self-feedback connection. Moreover,
in the case of feeding the spike train (3, 84, 103, 194, 203, 286,

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International Journal of Computer Applications (0975 8887)
Volume 99 - No. 17, August 2014

0.5

0

u(t)

303, 384, 403, 494) to an NDS model without a self - feedback
connection, there was no stabilised orbit even when the input spike
train is being fed to the neuron model during all the running time.
Based on the previous results, a UPO of an NDS model can be
defined as the orbit that has output spikes occurring within a
periodicity of τ , which can be reconstructed by feeding these spikes
to an NDS model with or without a self-feedback connection with
time delay τ .
The previous discussions substantiate that the self-feedback signal
is an important ingredient for the NDS model to be stabilised to one
of the large number of UPOs that the attractor encompasses. Also
the self-feedback control is powerful in stabilizing orbits in terms
of time; because the firing times are being adapted to a specific
pattern, which means the spike train will converge to one of the
available UPOs.

−0.5

−1

−1.5

0

1000

2000

3000

4000

5000

t

5. RESET MECHANISM INVESTIGATION
In this section the reset mechanism of the NDS neuron model is
investigated to understand its role in stabilization. Investigation
includes studying the NDS stabilization process in the case of
replacing the reset fixed value by a relative reset value. Also,
finding the range for the reset value will help in utilizing the
NDS attractor. In the coming two subsections the aforementioned
investigations is explained.

5.1 Replace the reset fixed value by a relative reset
value
The reset mechanism is a crucial element of the NDS model
because without it the internal dynamics will evolve to infinity.
However to investigate the role of the reset mechanism in
stabilizing UPOs then the model will be studied with a relative reset
value rather than an absolute value.
The same setup used previously is used here with the exception of
replacing the fixed reset value η0 with a relative reset mechanism.
The idea is to add the reset value η0 to the value of the u variable
instead of assigning η0 to the u variable. This mechanism will allow
the u variable to have different positions when reset occurs until a
UPO is stabilised. Then the reset positions will be a periodic pattern
and each position is assigned with an output spike.
The results of one experimental run are shown in figure 11. Note
that it takes on average 350 time steps for the spike output to be
stabilised. This is a little lower than the average when using fixed
reset value, and the reason behind this is that the values of the u
variable after reset are a little greater than those when using fixed
reset value, therefore the time the value of the u variable needs
to reach threshold is shorter, and hence the firing rate is greater.
However, the average time for the internal dynamics to be fully
stabilised when using fixed reset is 4035 which is lower than when
compared to when using relative reset mechanism (4085). This
might be the result of the different positions of the values of the
u variable for every reset before stabilization occurs, which affects
the speed in which the u variable drives the variables x and y.
The previous discussions suggest that the reset mechanism is
working because the value of the u variable is being reset to
a specific area in the phase space. However, to investigate the
previous conclusion; the next experimentation setup will show the
range of values that can be used as a reset value for the NDS model
with a fixed reset mechanism.

Fig. 11. u(t) time series with relative reset value is used rather than fixed
reset value.

5.2 Finding the range for the reset value
The same experimental setup used previously is used here except
that the reset value η0 will be adjusted to different values. These
values are chosen to be in the range η0 ∈ [−2, 0.1] for threshold
θ = −0.01, this range represents values above threshold, around
threshold, and below threshold. Therefore results of experiments
are divided into three regions as the following paragraphs
demonstrate.
The results for runs performed with the reset value being assigned
to a value above threshold show that the system internal dynamics
is reduced to two dimensions composed of x and y variable. The
behaviour of the trajectory is a spiral repellor around point (0, 0) in
the x − y plane. The results of one the runs are shown in figure 12
where the reset value was chosen to be 0.1. Note that the trajectory
goes in a transient first until it reaches the threshold value, and then
it spirals around the origin. The reason for this behaviour is that
the u variable value is always above threshold and is not changing,
therefore the only changing variables are x and y which represent
a simple oscillating system.
The results for runs performed with the reset value being assigned
to a value around and less than the threshold in the range of
η0 ∈ [θ − 0.035, θ] show that the system internal dynamics is still
linear but it appears in three dimensions. First the trajectory goes
into a transient, and then spirals around the origin in the x − y plane
with the u variable alternating between two values. The results of
one run is shown in figure 13 where the reset value chosen to be
−0.02. The reason for this behaviour is that the u variable value is
always very near to the threshold and it takes only a few time steps
until the u value reaches threshold value, therefore the variables x
and y are dominating the system behaviour and again represent a
simple oscillating system.
Finally, the third case is when the reset value is defined to be
in the range η0 ∈ (−2, θ − 0.035]. Many experiment runs have
been carried out with the reset value changed from η0 = −0.05 to
η0 = −1.2. The results for these values show that the NDS attractor
is reliable with percentage of around 97%. Reliable here means
stabilizing UPOs quickly and more important keeping the system
working and preventing it from approaching infinity. So a reliability
of 97% means that out of 100 runs there were 97 successfully

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International Journal of Computer Applications (0975 8887)
Volume 99 - No. 17, August 2014

2
0
0.5
−2

0

−4

−1

u(t)

u(t)

−0.5

−6

−1.5
−8
−2
10

−10
5

10
5

0
−5
−10

y(t)

−12

0

−5
−10

x(t)

−14

0

5000

10000

15000

t

Fig. 12. NDS attractor for reset value η0 = 0.1
Fig. 14. u variable time series for reset value η0 = −1.59 and initial
conditions of (-0.1556,0.4469,-0.3596)

6.

u(t)

1
0
−1
1.5
1

2
0.5
1

0
−0.5

0
−1

y(t)

−1

−1.5
−2

−2

x(t)

Fig. 13. NDS attractor for reset value η0 = −0.02

stabilised orbits, and in 3 times the system trajectories approach
infinity. This happens because if the reset value is a deep negative
number, then chances for the trajectory to leave the attractor are
increased because if the x variable exceeds a negative limit value
then it will approach infinity.
However, when the reset value changed to −1.4 the reliability
percentage was reduced. The reset value continued to be changed
until −2 where the reliability percentage becomes around 5%. Also
it is important here to mention that the time it takes for a UPO
to be stabilised increases when decreasing the reset value. In one
experiment run the reset value was chosen to be η0 = −1.59 and
the trajectory stabilised to one UPO after around 12500 time steps
as shown in Figure 14 where the time series of the u variable values
are depicted.
To summarise, optimal control is reliably achieved for values of
η0 in the range [−0.05, −1.2]. This allows accessibility to rich
dynamics with high reliability.

DISCUSSION

The NDS model has been developed for the purpose of exploiting
the rich dynamics that its attract provides in carrying out
information processing tasks. This exploitation can be maximized
if the way these dynamic behaviours that are exhibited in phase
space is known. This has been carried out by experimental
investigations of the internal dynamics of the NDS model. The
results of these investigations partly explain how UPOs are
stabilised in phase space and suggest optimal parameter settings
to enlarge the capacity and ensure stability in NDS neurons.
The results of all the previous experiments suggest that both the
reset mechanism and the feedback signal are the major ingredients
for the NDS model to work and to be stabilised to one of its
available UPOs.
Note from figure 9 that even though the system is fed with single
signal each τ time steps, the system has still been stabilised to
one of its UPOs. This single signal represent a special case of
the time-delayed self-feedback mechanism with the exception of
ignoring all the output spikes and only repeating one signal. This
means that the feedback control is very important element in the
NDS model.
Also note from figure 14 that the reset mechanism has led the
internal dynamics to be stabilised after 125τ time steps even though
the reset value has increased to −1.59 which is almost double the
original value defined for the NDS model (−1). The reset values
range that the NDS model performance is high has been found to
be in [−0.05, −1.2]. The reset mechanism is crucial for the NDS
model to work because without the reset mechanism the internal
dynamics will approach infinity.
The question that might be raised here is why both of them are
important for the NDS model to be stabilised? The other question
is how the control mechanism leads the internal dynamics to
synchronize with itself? The experiments that have been conducted
so far provide some insight into these issues. However, an analysis
of the NDS model from a mathematical and dynamical systems
perspective may suggest answers to these questions.

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International Journal of Computer Applications (0975 8887)
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7. CONCLUSION
One chaotic model, viz., the NDS model has been investigated in
this paper. The capabilities of the NDS neuron are considerable in
terms of UPOs. These UPOs can be stabilised using a feedback
control mechanism. The NDS model is a modified version of
Rössler chaotic system. The rich dynamics of Rössler system is
inherited by the NDS model. This is shown by the large number
of UPOs that can be stabilised, and also is obvious from the NDS
model attractor as shown in figure 2.
First, the NDS model is described and its dynamics has
been depicted and explained. This explanation gives an initial
understanding for the behaviour of the NDS model and suggests
how the model works.
Different experimental setups have been prepared to understand
the dynamic behaviour of the NDS model in terms of control
and reset mechanisms. The results of these experimentations have
revealed that both the reset mechanism and the self-feed back
control mechanism are important for the NDS model to work and
to stabilise to one of the large number of UPOs that are embedded
in the attractor. This wide range of dynamic behaviours may be
exploited to carry out information processing tasks.
However these conclusions are based only on experimental results,
therefore a mathematical analysis of the NDS model is needed to
shed some light on these properties of the model, which will also
help in understanding the different dynamical behaviours that the
NDS model exhibits in phase space. This is being prepared and
will be published soon.

8. ACKNOWLEDGEMENTS
This work is a modified and updated part of a thesis submitted to the
school of technology at Oxford Brookes University[?]. This work
was supported by the University of Petra, Amman, Jordan. I’m very
grateful to this support and would like to thank all the staff who
were involved in helping and encouraging me.

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