TU/e Mechanical Engineering

Bachelor Final Project

Output layer synchronization

of Hindmarsh-Rose neurons

in layered networks

M.P.A. Spoelstra

DC 2010.052

Coaches: Ir. E. Steur

Prof. dr. H. Nijmeijer

Eindhoven University of Technology

Department of Mechanical Engineering

Dynamics and Control Group

Eindhoven, September, 2010

2

Abstract

The subject of this study is the synchronization of specific nodes in an interconnected

network of nodes. The system that will represent a node in these networks is the

Hindmarsh-Rose model for neuronal activity. The neuronal networks that are treated are

of a specific class, namely layered networks. In such a network there are three different

types of layers to distinguish. There is an input layer, an output layer and an arbitrary

number of so-called hidden layers in between. The objective is to only synchronize the

nodes in the output layer. Output layer synchronization is for example useful for solving

the binding problem, which occurs in the visual cortex of the brain.

There exist sufficient conditions for (partial) synchronization of networks

consisting of Hindmarsh-Rose systems. Nodes are connected to each other using

diffusive coupling. The eigenvalues and eigenvectors of the coupling matrix determine

the coupling strengths needed for synchronization of every layer as well as for full

synchronization of a layered network.

The theoretical synchronization thresholds are used to find out which layered

networks show output layer synchronization. Layered networks with symmetric coupling

strengths, and layered networks with asymmetric coupling strengths between nodes are

evaluated. A network may contain multiple different coupling strengths. First networks

with uniform symmetric coupling strengths are studied. Then layered networks

containing two different coupling strengths are examined in symmetric and asymmetric

coupled networks. Several network configurations are evaluated with simulations.

Finally an experimental setup is used to demonstrate the theoretical findings. Up

to eighteen electronic equivalent Hindmarsh-Rose neurons can be connected to each

other with this setup. The fact that these electronic neurons are not identical affects and

restricts synchronization. The presence of noise effects also influences synchronization.

3

Contents

1. Introduction 4

1.1. Neuronal oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2. Layered networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3. Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4. Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. The Hindmarsh-Rose model and

synchronization of coupled H-R neurons 10

2.1. The Hindmarsh-Rose model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2. Diffusive coupling and synchronization of multiple H-R neurons . . . . . . . 12

2.2.1. Sufficient conditions for synchronization . . . . . . . . . . . . . . . . . . . . . 13

2.2.2. Partial synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. Output layer synchronization of layered networks

with uniform symmetric coupling strengths 17

3.1. All-to-all intra hidden layer connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2. All-to-non intra hidden layer connectivity . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4. Output layer synchronization of layered networks

with non-uniform coupling strengths 29

4.1. Symmetric coupling and variable inter-layer coupling strength . . . . . . . . . 29

4.2. Asymmetric coupling for networks consisting of two layers . . . . . . . . . . . 31

4.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5. Experimental synchronization 36

5.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2. Practical synchronization and synchronization robustness . . . . . . . . . . . . . 39

5.3. Practical limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.4. Experimental synchronization of multiple neurons . . . . . . . . . . . . . . . . . . . 44

5.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6. Conclusions and recommendations 50

6.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.2. Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Bibliography 52

A. Characteristic equation of layered networks

with uniform symmetric coupling strengths 54

B. Simulation results 57

B.1. Simulations accompanying Section 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

B.2. Simulations accompanying Section 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B.3. Simulations accompanying Section 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4

Chapter 1

Introduction

In this chapter the main subjects of this study will be introduced. This research is about

the synchronization of neuronal oscillators. Neurons are connected to each other to form

a network. The objective is to choose the coupling between the nodes in such a way that

some but not all nodes will show synchronous behavior. An introduction will be given

about neuronal oscillators, a certain class of networks called layered networks, and

synchronization. The objective and outline of this study will be presented as well.

1.1 Neuronal oscillators

A short overview will be given here about the relevant mechanisms of a biological

neuron to be able to describe some electrical behavior of such a cell. Over the years an

abundance of models have been created to describe the electrical behavior of neurons. In

this study the Hindmarsh-Rose model will be used, which will be discussed in Chapter 2.

The brain is made out of many neurons. There are different types of neurons, but a

neuron generally consists of a cell body with dendrites and axon attached to it, see Figure

1.1. Dendrites can be viewed as input channels to the cell body, whereas the axon can be

considered as the output channel. Neurons can have several dendrites, but only one axon.

However it is certainly possible for an axon to have branches through which the neuron

can connect to many other neurons.

Figure 1.1: Schematic representation of a common neuron

The electrical state of a neuron can best be described by the membrane potential m

V of

the cell body. This potential is influenced by several ionic currents flowing in or out of

the neuron. The change in the concentration of the different ions in the cell results in a

change of the membrane potential. When the membrane potential reaches a certain

5

threshold the cell releases an ionic current that flows through the axon out of the cell

body to lower the membrane potential again. The build-up of the membrane potential and

the sudden discharge due to reaching the threshold is called action potential or spike. The

fluctuations of the membrane potential as a function of the time can be classified into

three states:

Resting: The membrane potential is constant and remains at the resting potential of

around -70mV.

Tonic spiking: Here the neuron produces spikes with a steady rate.

Bursting: A burst can be described as a collection of spikes followed by a relative

long period of quiescence. When the number of spikes in the bursts is not

constant, then one can speak of chaotic bursting.

The states, other than the resting state, are depicted in the following figure.

Figure 1.2: The three different states of the membrane potential.

Over the years several neuronal models have been developed that mimic the behavior of

real neurons. These models try to describe the dynamical changes of the membrane

potential. The models can be divided in two categories; biologically plausible models and

mathematical models. Biologically plausible models describe the dynamics of the

membrane potential by using the membrane capacity and the membrane potential

dependent ionic currents. Probably the most well known biologically plausible model is

the Hodgkin-Huxley (H-H) model [Hodgkin and Huxley, 1952].

Mathematical neuronal models try to match the various dynamics that biologically

plausible models describe, but with less computations in the evaluation. In this study one

of these mathematical models will be used; the Hindmarsh-Rose (H-R) model. According

to [Izhikevich, 2004] this is one of the most extensive mathematical models available, in

the sense that all the different neuronal behavior that this model describes is also

described by the H-H spiking model. However the H-R neuron is not biologically

plausible. In other words, the behavior of the neuron is approximated for more efficient

evaluation. The number of floating point operations needed to compute the H-R model is

about ten times less than for the H-H model.

6

1.2 Layered networks

When multiple neurons are connected to each other they form a network. The topology or

structure of the network is determined by the connections. One class of network

topologies is the so-called layered network topology. A layered network is a network

where the neurons can be grouped into layers, on the basis of similar connectivity.

Layered networks are also present in, for instance, the cortex of the brain [Shepherd,

1990].

The cerebral cortex forms the outer part of the brain, and is most enlarged in humans

relative to other mammals. It can be seen as a shell surrounding the rest of the brain, i.e.

this cortex surrounds the subcortical areas. An abundance of data supports the idea that

this part of the brain is the place where much of the neuronal activity regarding cognition

takes place [O’Reilly and Munakata, 2000]. The cortex can be divided into several

different cortical areas that are specialized for different kinds of cognitive data

processing. However, despite this functional specialization, the general structure of the

cortex is consistent across the different cortical areas as shown in Figure 1.3. The dots in

the four columns of this figure represent the neuron density. Cortical neurons in the most

recent part (in the evolution) of the cerebral cortex, the neocortex, are organized into six

distinct layers [Shepherd, 1990]. These layers have been identified on anatomical

grounds and are important for understanding the physiology of the cortex. To simplify

this laminar structure, three functional layers are introduced: the input, hidden, and output

layers. These layers are a clustering of the cortical layers by function, connectivity and in

some cases neuron type [O’Reilly and Munakata, 2000].

Figure 1.3: Laminar structure of the cortex from different areas of a monkey brain

[Shepherd, 1990].

7

The top of a column, in Figure 1.3, is the outer most part of the neocortex. Column A

shows specialization of input layer 4 in the primary visual input area, while in column B

there is an emphasis on the hidden layers 2 and 3 higher up in the visual processing

stream. Column C shows emphasis of output layers 5 and 6 in a motor output area. And

in Column D there is a relatively even blend of layers in a prefrontal area [Shepherd,

1990].

The anatomical connectivity between the different brain areas suggests that information

comes into the input layer after which it is primarily transmitted to the hidden layers and

then on to the output layers [White, 1989]. A more detailed description of the three

functional layers is given below.

Input layer: The input layer (cortical layer 4 in Figure 1.3) usually receives sensory

input from a subcortical area called the thalamus. The thalamus in turn

receives information from the retina and other sense organs. One could say

that the input layer filters the data that is available from the subcortical

areas for processing by the hidden layers.

Output layer: The output layer (cortical layers 5 and 6) sends motor commands to

muscles if this layer is situated in the motor cortex. In other cortical areas

the output can for example contain information regarding object

recognition or linguistic information.

Hidden layer: The hidden layer (cortical layers 1, 2 and 3) locally receives input from

other cortical layers. This layer is called hidden because non-cortical brain

areas are only connected to the input or output layers, therefore this layer

is ‘hidden’ from these areas. The role of the hidden layers is to process (or

transform) the data, which is given by the input layer, and hand the result

over to the output layer.

The three functional layers, which were introduced as a clustering of the six layers of the

neocortex, will be used in the layered networks presented in this study. H-R neurons will

represent the nodes in these networks.

8

1.3 Synchronization

In some cases it would be convenient if the neurons in the output layer of a neuronal

network would synchronize to be able to perform certain actions. In the brain an output

layer of the neocortex can be connected to neurons that drive the muscles. In some cases

there is a need to simultaneous drive particular muscles as one might image. In this

section it is also shown that synchronization plays an important role when it comes to

image processing in the brain.

One of the cortical areas of the cerebral cortex is the primary visual cortex. This is where

the binding problem occurs, and synchronization is thought to play an important role in

how the brain solves this problem [Raffone and van Leeuwen, 2003; Singer, 1999]. For

example, if a human sees two different shapes that have different colors, then some

neuron regarding the shapes will activate and some neurons regarding the colors will

activate. But how can the brain tell which color belongs to which shape. If the neurons

concerning the shape and the corresponding color are firing synchronously then the

problem could be solved. This synchronization, where individual neurons fire their action

potentials at the same time, should occur at the output layer for the following reason. The

input layer of the primary visual cortex receives raw data from the retina through the

thalamus, the hidden layers processes the information to find the shapes and colors in this

case, and the output layer ‘tells’ it to some other brain area. To properly notify the other

area which color belongs to which shape the output layer could synchronize to include

this piece of information.

Another case where output layer synchronization is useful is in a motor area in the

neocortex. There are situations thinkable where simultaneous muscle contractions are

required. Playing a musical instrument is an example that might benefit from

synchronization of the motor cortex.

Probably the most general definition of synchronization is the following one [Pikovsky et

al., 2003]:

Synchronization is the adjustment of rhythms of oscillating objects due to their

weak interaction.

This definition suggests that the system under examination must be able to perform

oscillating motion. The definition does not say anything about whether the oscillation is

periodic or non-periodic, e.g. chaotic.

The definition also suggests that there is an interaction between the systems. The

type of interaction is not specified and thus can be chosen freely. The type of coupling

used in this study will be introduced in Section 2.2. The direction of the connection is not

specified as well, so it could be unidirectional or bidirectional. Unidirectional coupling is

a master-slave connection type: a secondary system is changing its rhythms to become

synchronized with the primary system which is not influenced by the secondary system.

The primary system can however be influenced by some other system in the network that

is connected to it. Obviously a bidirectional connection implies that two systems

9

influence each other directly. When the connection is the same in both directions and the

connection strengths are equal, then one can speak of symmetric coupling.

1.4 Objective

This research will focus on synchronization of neurons in layered networks. As stated in

Section 1.3 it is interesting to look at synchronization of the nodes in the output layer.

This particular synchronization is termed output layer synchronization.

First networks with uniform symmetric coupling strengths will be examined, i.e.

networks where every symmetric coupling in the network has the same connection

strength, to see whether output layer synchronization (OLS) can occur in those layered

networks. In these layered networks, two types of hidden layers will be inspected in

particular: the type where all nodes in a hidden layer are connected to every other node in

that layer, and the type where all nodes in a hidden layer are connected to none of the

other nodes in that hidden layer.

The next step is to use asymmetric coupling. Asymmetric coupled neurons will be

considered in layered networks with only an input layer and an output layer.

The results will be validated experimentally with the use of an experimental setup

that enables the creation of neuronal networks consisting of up to eighteen H-R neurons.

1.5 Outline

This report is organized as follows. In Chapter 2 the H-R model will be specified and the

relevant dynamics of the model will be shown as well. Also the type of coupling and the

conditions for synchronization will be presented in this chapter. After Chapter 2 the

networks with uniform symmetric coupling strengths are tackled in Chapter 3 and the

networks with non-uniform coupling strengths are tackled in Chapter 4. Experimental

demonstrations of the networks presented in Chapter 3 and 4 are given in Chapter 5. And

finally conclusions and recommendations are presented in Chapter 6.

10

Chapter 2

The Hindmarsh-Rose model and

synchronization of coupled Hindmarsh-Rose neurons

In this chapter the Hindmarsh-Rose model for neuronal activity will be presented. The

coupling between neurons will also be introduced and sufficient conditions for

synchronization of coupled Hindmarsh-Rose neurons will be presented as well.

2.1 The Hindmarsh-Rose model

The final version of the 1984 H-R model [Hindmarsh and Rose, 1984] is given by the

following system of coupled differential equations:

( )( )

3 2

1 2

2

1 1

2 0 2

3 ,

1 5 ,

,

y y y z I z

z y z

z r s y y z

= − + + + −

= − −

= − −

�

�

�

where 1r � and s are positive constants and 0y is the y-component of the stable

equilibrium corresponding to resting ( 0I = ). The y state is the output, i.e. the membrane

potential, of the model, 1z and 2z are internal states, and I is the input. In this study a

modified version of the final H-R model will be used that is better suited for realizing the

electronic equivalent of the model which is used in the experiments. Due to a coordinate

transformation concerning the y-state [Steur et al., 2007] and an additional linear

coordinate transformation by [Neefs, 2009], the modified H-R equations are now given

by:

( )( )

3 2

11 12 13 14 1 15 2 16 17

2

1 21 22 23 1

2 31 32 33 2

1,

1,

1,

y c y c y c y c z c z c c IT

z c y c y c zT

z c c y c zT

= − + + + − − +

= − − −

= + −

�

�

�

where T is a time scaling factor, and the non-negative parameters i

c are given by

11 12 13 14 15 16 17

21 22 23

31 32 33

1, 0, 3, 5, 1, 8, 1

1, 2, 1

0.005, 4, 1.1180

c c c c c c c

c c c

c c c

= = = = = = =

= = =

= = =

11

The time scaling factor is set to be 1000T = , and the constant input will in this study

always be 3.3I = . At this input the model produces chaotic bursting. The number of

spikes in a burst is not fixed when the system is in this mode. The H-R model can also

show tonic spiking and bursting behavior, which occur at different constant inputs I.

In Figure 2.1 the model’s behavior is depicted for 3.3I = . The top side shows the

trajectories of the three states as a function of the time, and on the bottom side the

corresponding attractor is displayed.

Figure 2.1: Simulated responses of the H-R model for input 3.3I =

12

2.2 Diffusive coupling and synchronization of multiple Hindmarsh-Rose

neurons

An array of n coupled identical H-R oscillators can be described by the following

equations:

( )( )

3

1, 2,

2

1, 1,

2, 2,

13 5 8

12

10.005 4 1.1180

i i i i i i

i i i i

i i i

y y y z z I uT

z y y zT

z y zT

= − + + − − + +

= − − −

= + −

�

�

�

(2.1)

Here, the states have a subscript 1,..,i n= denoting the number of the neuron in the

network.i

y is the output of neuron i. The additional term i

u at the end of the first

equation represents the linear coupling between the nodes. The coupling that will be used

is diffusive coupling, and it is defined as follows:

( )1,

n

i ij i j

j i j

u y yγ= ≠

= − −∑ , (2.2)

where 0ij ji

γ γ= ≥ . Let ( )1col ,..., nu u=u and ( )1col ,..., ny y=y , then = −u Γy with

1 12 1

2

21 2 2

1, 2

1

1 2

1

n

j n

j

n

j n

j j

n

n n nj

j

γ γ γ

γ γ γ

γ γ γ

=

= ≠

−

=

− −

− − =

− −

∑

∑

∑

Γ

�

�

� � � �

�

(2.3)

The coupling strength ij

γ is the coupling strength on the connection from node j to node

i. The coupling from a node to itself is not considered since a node is always

synchronized with itself. The diffusive coupling is defined as a symmetric coupling, thus

the coupling matrix Γ will be symmetric as well and therefore the eigenvalues and

eigenvectors are real. Furthermore by applying Gershgorin’s theorem it can be shown

that the eigenvalues of the coupling matrix are non-negative. A network must also be

strongly connected resulting in the following eigenvalues of the coupling matrix:

1 20 ...n

λ λ λ= < ≤ ≤

13

2.2.1 Sufficient conditions for synchronization

The diffusive coupling between the neurons is useful when it comes to synchronization.

Synchronization and partial synchronization are defined as follows.

Definition 2.1. ((Partial) Synchronization)

The states of H-R neuron i are 1, 2,, ,i i i i

y z zΤ

= x and the solutions ( ) ( )1 ,..., nt tx x of n

coupled H-R neurons (2.1) with initial condition ( ) ( )1 0 ,..., 0nx x are called (partial)

synchronized if:

( ) ( )lim 0i jt

t t→∞

− =x x for all (or some) , 1,...,i j n= (2.4)

In [Pogromsky and Nijmeijer, 2001] a theoretical framework is presented that can

guarantee full synchronization of diffusively coupled oscillators under certain conditions.

The Hindmarsh-Rose model fits this framework [Neefs, 2009]. The theoretical

framework states that a (strongly connected) network of diffusively coupled identical H-

R systems will fully synchronize, for all initial conditions, if the smallest non-zero

eigenvalue of the coupling matrix Γ is large enough, i.e. 2ˆλ γ≥ . The eigenvalues of Γ

contain the coupling strengths as well as the connectivity. This means that a network with

fixed connections will synchronize if the coupling strengths are large enough. The

threshold of full synchronization can therefore be expressed in terms of coupling

strengths.

To investigate OLS in layered networks consisting of H-R neurons the conditions under

which partial synchronization occurs are also needed. The conditions for full

synchronization are still useful, because when studying OLS the intention is to let the

nodes in the output layer synchronize before the whole network synchronizes. Thus the

OLS threshold should be sufficiently lower than the threshold of full synchronization.

However to specify any synchronization threshold the positive value γ̂ needs to be

known. Simulation results from two diffusely coupled H-R neurons give an almost ideal

value of ˆ 1γ ≈ (see Appendix B).

By using the Wu-Chua conjecture [Wu and Chua, 1996] this threshold can be

used to determine the full synchronization threshold for an arbitrary network. The

conjecture states that the coupling strength needed to reach full synchronization is inverse

proportional to the smallest non-zero eigenvalue of Γ . Consequently the positive value

γ̂ is the same for all networks of coupled H-R systems with 3.3I = . Consider for

example a network consisting of two diffusively coupled H-R neurons with a symmetric

coupling strength k and the following network with a uniform symmetric coupling

strength k:

14

Figure 2.2: All-to-all network consisting of four nodes

The coupling matrices of the two node network and the four node network are

respectively

2

1 1

1 1k

− = −

Γ , and 4

3 1 1 1

1 3 1 1

1 1 3 1

1 1 1 3

k

− − − − − − = − − − − − −

Γ

The smallest non-zero eigenvalues of these two networks are

( ) ( )2 2 2k kλ λ∗= =Γ Γ

( ) ( )4 4 4k kλ λ∗= =Γ Γ

The Wu-Chua conjecture states that fsˆk λ γ∗ = , with fsk the coupling strength needed for

full synchronization, and λ∗ the smallest non-zero eigenvalue of the coupling matrix.

Therefore ( ) ( )2 2 4 4k kλ λ∗ ∗=Γ Γ , with 2k the coupling strength needed to synchronize the

two node network, and 4k the coupling strength needed to synchronize the network in

Figure 2.2. 2 0.5k = as shown in Appendix B, this means that 40.5 2 4k⋅ = ⋅ and the four

node network will therefore synchronize for 4 1 4k k= = .

15

2.2.2 Partial synchronization

When a network of n H-R systems contains some symmetries then these symmetries

should be present in the coupling matrix ΓΓΓΓ . Let n n×∈RΠΠΠΠ be a permutation matrix that

describes a permutation of some elements in the coupling matrix, i.e. the rearrangement

of some nodes, which leaves the network invariant. Given such a permutation matrix the

set

( ){ }3

3 3| kern

n∈ ∈ − ⊗x x I IΠΠΠΠR , (2.5)

defines a linear manifold A of a network of diffusely coupled identical H-R systems that

corresponds with partial synchronization.

When the coupling matrix ΓΓΓΓ commutes with the permutation matrix ΠΠΠΠ then the set (2.5)

contains a globally asymptotically stable subset if the smallest non-zero eigenvalue of ΓΓΓΓ

is larger than the positive constant γ , but only if the corresponding eigenvector is in the

( )range n −I ΠΠΠΠ [Pogromsky et al., 2002]. From simulations and experiments done during

this study it is reasonable to say that ˆγ γ≈ .

As an example consider the following network, with the symmetric coupling strengths

12 13 1kγ γ= = and 23 2kγ =

Figure 2.3: All-to-all network containing three nodes

The coupling matrix of this network with accompanying eigenvalues and eigenvectors

are

1 1 1

1 1 2 2

1 2 1 2

2k k k

k k k k

k k k k

− − = − + − − − +

Γ ,

1

2 1

3 1 2

0,

3 ,

2 ,

k

k k

λ

λ

λ

=

=

= +

1

1

1

1

=

v , 2

2

1

1

− =

v , 3

0

1

1

= −

v

16

A permutation matrix that commutes with the coupling matrix is

1 0 0

0 0 1

0 1 0

=

ΠΠΠΠ

This permutation matrix defines the following linear manifold

{ }9

1 2 3|A = ∈ ≠ =x x x xR

The eigenvector 3v is in the ( )3range −I ΠΠΠΠ . This means that if 3λ γ> then node 2 and

node 3 will synchronize. This synchronization regime will not coincide with full

synchronization if 2ˆλ γ< .

17

Chapter 3

Output layer synchronization of layered networks

with uniform symmetric coupling strengths

This chapter discusses networks with uniform symmetric coupling strengths. This means

that all diffusive couplings in the network have the same coupling strength. Symmetric

coupling is illustrated in Figure 3.1.a with coupling strength k. By choosing uniform and

symmetric coupling strengths it will be shown that the number of network topologies that

show output layer synchronization (OLS) (see Definition 3.1) is very limited. In part this

is because nodes in the hidden layers have connections to other nodes in the same layer,

i.e. intra layer connections; see Section 3.1. When these intra layer connections are

dropped then the number of networks that show OLS will increase slightly. But before

any of this is discussed in more detail, the structure of the connections in a general

layered network will be defined.

As stated in Chapter 1.2, a layered network can contain three different kinds of layers.

The network always consists of an input layer and an output layer. The number of hidden

layers in between can be chosen freely. To emphasize the layered structure of the

networks, every node in a layer is connected to every node in the neighboring layers. For

example the nodes in the first hidden layer of Figure 3.1.b are connected to the nodes in

the input layer and the nodes located in the second hidden layer.

Figure 3.1: a) A connection between two nodes. b) An example network consisting of ten

nodes conform to our definition of the structure of a layered network.

The arrows used for the connections in the network of Figure 3.1.b point in both

directions. Figure 3.1.a shows that every arrow in Figure 3.1.b actually consists of two

connections with equal coupling strength k, i.e. symmetric coupling. In a larger network,

like the one depicted in Figure 3.1.b, the two arrows are combined to make the network

structure more clear.

18

One has to note that in Figure 3.1.b the hidden layers have intra layer connectivity while

the input and output layer do not. To make OLS more interesting the output layer has no

intra layer connectivity. Because when node 9 and node 10 in Figure 3.1.b would be

connected to each other, then the threshold for which they will synchronize would

decrease.

The input layer nodes have no connections to each other as well. This is done

because the input layer can be seen as the provider of information (or signals) to the rest

of the network. And when these input nodes are connected to each other, they will

influence each other. The role of the input layer is to filter and prepare the available

information for processing by the hidden layers, as stated in Chapter 1.2. Also if the input

nodes were connected to each other, the synchronization threshold of this layer would be

lower. A synchronized input layer means that all the input nodes act as one node, and that

is not useful.

Whether the hidden layers have intra layer connectivity or not is not yet specified.

Section 3.1 will discuss the case where every node in a hidden layer is connected to every

other node in the same hidden layer, i.e. all-to-all intra hidden layer connectivity. By

making this choice OLS will only show up in a limited number of network topologies.

Therefore Section 3.2 additionally discusses the case of all-to-non intra hidden layer

connectivity, where every node in a hidden layer has no connections to any of the other

nodes in that hidden layer.

An arbitrary layered network consisting of n H-R neurons must have at least two layers;

the input layer and the output layer. To be able to specify the number of layers in a

network and the number of nodes per layer, the row matrix n is introduced:

[ ]1 2 ln n n=n … , so the total number of nodes 1

l

i

i

n n=

=∑ ,

with i

n the number of nodes in layer i and 2l ≥ the number of layers in the network.

Since the structure of a network is fixed, the matrix n fully characterizes the topology of

a layered network with uniform symmetric coupling.

The definition of output layer synchronization that will be used throughout this report is

the following one.

Definition 3.1. (Output Layer Synchronization)

Consider a layered network consisting of n H-R neurons (2.1) which couple via (2.2). The

neurons are said to output layer synchronize if only the neurons in the output layer

synchronize, i.e.

( ) ( )lim 0i jt

t t→∞

− =x x , i j≠ , only if { }, 1, 2,...,l li j n n n n n∈ − + − +

19

3.1 All-to-all intra hidden layer connectivity

In a network with all-to-all intra hidden layer connectivity, every node in a hidden layer

is connected to every other node in the same hidden layer. It will be shown that some

eigenvalues of the coupling matrix can be determined for an arbitrary layered network. It

seems that every layer synchronizes at a certain threshold, if a layer contains more than

one node. The thresholds are determined by these eigenvalues and their corresponding

eigenvectors. The objective is to only synchronize the output layer nodes. The output

layer should therefore synchronize at a lower threshold than any other layer in the

network.

The coupling matrix Γ of an arbitrary layered network with uniform symmetric coupling

strength k and all-to-all intra hidden layer connectivity can be written as

1 1,2

2,1 2

2, 1

1, 2 1 1,

, 1

0 0

0 0

0 0

l l

l l l l l

l l l

k − −

− − − −

−

− − −= − − −

D J

J D

JΓ

J D J

J D

�

� � �

� �

� �

�

, (3.1)

with ,i jJ a matrix of ones of size i jn n× , 11 2 nn= ⋅D I with

1nI the identity matrix of size

1 1n n× , 1 ll l nn −= ⋅D I of size l ln n× , and

1 1

1

1

1 1

i

i

i

d

d

− − − = − − −

D

�

� � �

� � �

�

,

with 1 1 1i i i i

d n n n− += + + − for 2,..., 1i l= − of size i i

n n× .

20

To find the thresholds of (partial) synchronization regimes the eigenvalues and

eigenvectors need to be determined. This is done by solving i i i

λ=Γv v . In order to do so,

the eigenvalues of k′ =Γ Γ are obtained first by ( )det 0nλ′ − =Γ I . The determinant can

be reduced to the following:

(1) (1)

1 1,2

(1) (1)

2,1 2

(1)

2, 1

(1) (1) (1)

1, 2 1 1,

(1) (1)

, 1

0 0

00 0

0 0

l l

l l l l l

l l l

− −

− − − −

−

−

−

⋅Λ =−

− −

−

D J

J D

J

J D J

J D

�

� � �

� �

� �

�

, (3.2)

with

( ) ( )

( ) ( ) ( )

1 2

1

1 1

2 1

11 1 1

2 1 1 1

2

, if 2

, if 2l i

n n

ln n n

l i i i

i

n nl

ln n n n n

λ λ

λ λ λ

− −

−− − −

− − +

=

− −=

Λ = >− − + + −

∏

where (1)

,

0 0

0 0

1 1

i j

=

J

�

� �

�

�

, (1)

1

2

1 0 0 1

0 0

1 1

0 0 n λ

− = −

−

D� �

� �

�

, (1)

1

1 0 0 1

0 0

1 1

0 0

l

ln λ−

− = −

−

D� �

� �

�

,

(1)

1 0 0 1

0 0

0 0 1 1

1 1

i

id λ

− = − − − −

D� �

�

, for 2,..., 1i l= − .

Ultimately ( )det nλ′ −Γ I can be reduced to ( )det 0n tot tot⋅Λ = Λ =I , with tot

Λ the

characteristic polynomial (see for examples Appendix A). In (3.2)

0Λ = , (3.3)

since ( )det 0⋅ ≠ . The result is that some but not all eigenvalues of ′Γ can be obtained

from (3.3).

21

These eigenvalues are

1 2nλ = , with an algebraic multiplicity of 1 1n − ,

1l lnλ −= , with an algebraic multiplicity of 1

ln − ,

1 1i i i in n nλ − += + + , with an algebraic multiplicity of 1

in − ,

for 2,..., 1i l= − and only if 2l >

When a permutation matrix ΠΠΠΠ commutes with the coupling matrix Γ then the set (2.5) is

globally asymptotically stable if the smallest non-zero eigenvalue of Γ is larger than the

positive value γ , and if the corresponding eigenvector is in the ( )range n −I ΠΠΠΠ .

The eigenvectors are needed to link a partial synchronization regime to an eigenvalue of

the coupling matrix. A permutation matrix defines a partial synchronization regime

according to the set (2.5). It seems that the eigenvalues resulting from 0Λ = define the

synchronization thresholds of the layers in an arbitrary layered network with uniform

symmetric coupling.

The following permutation matrix, for example, defines synchronization of the input

layer nodes:

( )1 1 1 ,

n n ndiag −

E IΠ =Π =Π =Π = with,

1

1

1

0 1

0 1 1

1 0

n

n

−

= =

0 IE

0

� �,

a permutation matrix of size 1 1n n× .

This permutation does commute with Γ since

1 1 11 2 1n n n

n⋅ = ⋅ = ⋅D E E E D

With this permutation matrix the set (2.5) defines the following linear manifold

{ }1 1

3

1 1 2 1: ... ...n

n n nA += ∈ = = = ≠ ≠ ≠x x x x x xR , with 1, 2,i i i i

y z zΤ

= x

22

If the eigenvector 1v corresponding to the eigenvalue 1λ , from (3.3), is in the

( ) 1range n −I ΠΠΠΠ , then the synchronization threshold of the input layer will become

1kλ γ> ⇒ 1

kγ

λ>

The threshold of a layer synchronization regime in general is

i

kγ

λ> , for 1,...,i l= , (3.4)

where i

λ are the eigenvalues obtained from (3.3). And it seems that the eigenvectors of

these eigenvalues always are in the ( ) range n i−I ΠΠΠΠ , with iΠΠΠΠ the permutation matrix that

describes the synchronization of layer i.

Consider for example the layered network [ ]1 2 1 1 2=n , which is illustrated in

Figure 3.2.

Figure 3.2: Layered network as described by [ ]1 2 1 1 2=n

The eigenvalues from (3.3) with their corresponding calculated eigenvectors are

2 4λ = , 2 0 1 2 1 2 0 0 0 0 = − v ,

5 1λ = , 5 0 0 0 0 0 1 2 1 2 = − v

Synchronization of the second and last layer can be described by the following two

permutation matrices respectively

[ ]( ) 2 1 2 4, ,diag I E IΠ =Π =Π =Π = ,

[ ]( ) 5 5 2,diag I EΠ =Π =Π =Π = ,

where 1

1

i

i

− =

0 IE

0.

23

These two permutation matrices commute with the coupling matrix of this network.

The eigenvector 5v is the only eigenvector of the coupling matrix of this network that is

in the ( )7 5range −I ΠΠΠΠ . Therefore the output layer will synchronize for 1k γ> . The first

hidden layer synchronizes at 4k γ> , because 2v is the only eigenvector that is in the

( )7 2range −I ΠΠΠΠ .

By using 1γ = the theoretical thresholds can be verified with simulations. For a

coupling strength of 0.2k = none of the nodes are synchronized as expected, but when

the coupling strength is increased to 0.25k = the first hidden layer synchronizes, as can

be seen in the following figure.

Figure 3.3: The network [ ]1 2 1 1 2=n for a coupling strength of 0.25k =

For a coupling strength of 1k = the output layer is synchronized, see Figure 3.4.

The whole network should synchronize for ˆ 0.359 1 0.359 2.787k γ≥ ≈ = , since

0.359λ = is the smallest non-zero eigenvalue of the coupling matrix of this layered

network with all-to-all intra hidden layer connectivity. At a coupling strength of 2.75k =

the simulation results are as in Figure 3.5.

24

Figure 3.4: The network [ ]1 2 1 1 2=n for a coupling strength of 1k =

Figure 3.5: The network [ ]1 2 1 1 2=n for a coupling strength of 2.75k =

When the coupling strength 1k = the output will synchronize but the first hidden layer is

already synchronized at this point. If we want to let the output layer synchronize first then

one could try to increase the number of nodes in the fourth layer to five nodes, so

25

[ ]1 2 1 5 2=n . This provides a threshold of 5 1 5k γ λ> ≈ for OLS. However by

making this change the synchronization threshold of the fourth layer itself will become

4 1 8k γ λ> ≈ . Obviously this does not solve the issue.

To accomplish that OLS is the first of the layer synchronization (LS) regimes to show up,

the output layer should be preceded by one or more layers all containing only one node,

i.e. [ ]1 1 ln=n � . In this case only one synchronization regime can be identified

with (3.3); OLS.

It is shown in Appendix A that OLS and full synchronization do not coincide if

4l ≥ with a network of the form [ ]1 1 ln=n � . The OLS threshold in these

networks is always 1 1k > , while the threshold for full synchronization is

1 0.5188 1.93k ≥ = or higher, for 4l ≥ (see Figure A.1). To illustrate these findings, the

network [ ]1 1 2=n and the network [ ]1 1 1 2=n are investigated with the use of

simulations. Figure B.8 shows that at a coupling strength of 0.95k = none of the nodes

in the network [ ]1 1 2=n are synchronized with each other, but when the coupling

strength is increased to 1k = , as shown in Figure B.9, all nodes are synchronized. No

partial synchronization is observed between 0.95k = and 1k = . Simulations on the

network with four layers [ ]( )1 1 1 2=n reveals OLS at a coupling strength of 1k = ,

which is shown in Figure B.11. The theoretical threshold for full synchronization of this

network lies at 1.93k ≥ .

The only other network topology, with uniform symmetric coupling strengths, that shows

OLS before any other LS regime is a two layered network [ ]1 2n n=n , where

1 2 2n n> ≥ . Such a network has two partial synchronization regimes; one for the input

layer and one for the output layer. The output layer will synchronize first with a coupling

strength of 11k n> . When the coupling strength is increased to 21k n≥ the input layer

will also synchronize. However this synchronization regime coincides with full

synchronization, because 2nλ = is also the smallest non-zero eigenvalue. The

characteristic polynomial of an arbitrary two layered network can easily be calculated

(unlike networks with more layers) and is given by

( ) ( ) ( )1 21 1

2 1

n n

totn n nλ λ λ λ

− −Λ = − − − , with 1 2n n n= +

The two layer network [ ]3 2=n was examined with simulations. OLS should occur at

1 3k > . Figure B.13 shows the results for a coupling strength of 0.35k = .

26

3.2 All-to-non intra hidden layer connectivity

In this section the connections within the hidden layers are removed. This means that the

coupling strengths in the network are still uniform and symmetric. By making this

adjustment it is now possible to have OLS in networks with hidden layers consisting of

more than one node.

The hidden layers behave now much like the input or output layer when it comes to

synchronization thresholds, as we will see. This is because the diagonal blocks i

D of size

i in n× from (3.1) can now be written as

( )1 1 ii i i nn n− += + ⋅D I , for 2,..., 1i l= −

The rest of the coupling matrix Γ remains unchanged. The reduction of

( )det 0nλ′ − =Γ I therefore yields

( ) ( )

( ) ( ) ( )

1 2

1

1 1

2 1

11 1 1

2 1 1 1

2

, if 20

, if 2l i

n n

ln n n

l i i

i

n nl

ln n n n

λ λ

λ λ λ

− −

−− − −

− − +

=

− −=

= Λ = >− − + −

∏

(3.5)

The result is that the thresholds (3.4) will now be determined by the following

eigenvalues

1 2nλ = , with an algebraic multiplicity of 1 1n −

1l lnλ −= , with an algebraic multiplicity of 1

ln −

1 1i i in nλ − += + , with an algebraic multiplicity of 1

in −

for 2,..., 1i l= − and only if 2l >

In this case the eigenvalues i

λ for 2,..., 1i l= − are smaller. This means that the threshold

of hidden layer synchronization will be higher according to (3.4).

Let’s again look at the network [ ]1 2 1 1 2=n of the preceding section. With all-to-

non intra hidden layer connectivity this network looks as follows:

Figure 3.6: Layered network as described by [ ]1 2 1 1 2=n

27

The eigenvalues from (3.5) with their corresponding eigenvectors are

2 2λ = , 2 0 1 2 1 2 0 0 0 0 = − v ,

5 1λ = , 5 0 0 0 0 0 1 2 1 2 = − v

In this case the first hidden layer will synchronize at the threshold 2k γ> . The output

layer will synchronize at 1k γ> . To achieve OLS, the third hidden layer should have

four nodes. By adding three nodes to the third hidden layer [ ]( )1 2 1 4 2=n , the

threshold of OLS will decrease to 4k γ> . The third hidden layer itself will synchronize

with a coupling strength of 3k γ> . This addition to the network did not affect the

threshold of the first hidden layer, so now the output layer will synchronize first since it

has the lowest threshold. The smallest non-zero eigenvalue of this network is 0.359λ = ,

which means that the synchronization of the output layer will also not coincide with full

synchronization.

The network [ ]1 2 1 4 2=n was also examined with simulations. And as it

turns out the output layer is indeed the first layer to synchronize at 0.25k = (see Figure

B.15). After that the fourth layer is synchronized at 0.35k = and the second layer is

synchronized at 0.5k = , which can be seen in Figure B.16 and Figure B.17 respectively.

It is possible for the output layer to have the largest eigenvalue in some (additional)

network topologies, implicating the smallest threshold for layer synchronization. In

practice this seems to be only possible when

2l

n ≥ ,

1 2l l

n n− ≥ + ,

2 1l

n − = ,

under the assumption that

1 2ln n− > , if 1 2n ≥ and

( )1 1 1l i in n n− − +> + for 2,.., 2i l= − , if 2i

n ≥

This assumption is of course derived from the eigenvalue inequality 1lλ λ> and

l iλ λ>

respectively. The network [ ]1 2 1 4 2=n meets these requirements, but the

network [ ]1 2 2 4 2=n does not, and therefore it will not show OLS first. This is

due to the fact that the third layer now has its own eigenvalue; 3 6λ = . This will set the

synchronization threshold of this layer to 3 1 6k γ λ> ≈ , which is lower than the

threshold for OLS 5 1 4k γ λ> ≈ . Even if the second layer only has one node, then the

threshold of the third layer is still lower than the threshold for OLS, i.e. 3 1 5k γ λ> ≈ . It

28

is easy to see that increasing or decreasing the number of nodes in the fourth layer does

not solve this issue, since the number of nodes in the fourth layer influences both the

synchronization threshold of the output layer and the third layer.

3.3 Summary

In this chapter we discussed layered networks with uniform symmetric coupling strength

and all-to-all intra hidden layer connectivity as well as all-to-non intra hidden layer

connectivity. If a layered network, with uniform coupling strength, has only two layers,

then it is possible to output layer synchronize when the input layer has more nodes than

the output layer. A layered network with hidden layers does not show OLS in most cases,

especially with all-to-all intra hidden layer connectivity. If all the hidden layers consist of

only one node then OLS is possible when the total number of layers exceeds three. If

there are hidden layers with more than one node, then OLS will show up first in some

cases if there is all-to-non intra hidden layer connectivity. The last hidden layer should

have two more nodes than the output layer and the penultimate hidden layer should

contain only one node.

29

Chapter 4

Output layer synchronization of layered networks

with non-uniform coupling strengths

When one wants a layered network to have OLS then the permutation matrix ΠΠΠΠ used in

the set (2.5) is known. This means that the coupling matrix Γ must meet certain

requirements in order to commute with this permutation matrix. The coupling matrix

does not have to be symmetric in order to commute with a permutation matrix though. If

the coupling matrix is not symmetric then some (or all) coupling strengths ij

γ in (2.2) are

asymmetric, i.e. ij ji

γ γ≠ , where ij

γ is the coupling strength from node j to node i, while

jiγ is the coupling strength from node i to node j. It can be shown that the theory for

partial synchronization can also be applied to asymmetric coupled nodes, if the network

is strongly connected [Steur, 2010].

Chapter 4.2 deals with these asymmetric coupling strengths in networks consisting of

only two layers. But first we look at a variation of the networks discussed in the previous

chapter.

4.1 Symmetric coupling and variable inter-layer coupling strength

This section can be seen as an extension to Section 3.2. In that chapter the network

[ ]1 1 2 4 2=n with all-to-non intra hidden layer connectivity did not show OLS,

because the third layer has a lower synchronization threshold than the output layer, i.e.

3 5λ = , while 5 4λ = . This problem can be solved by using two coupling strengths in the

network instead of one. The first possible coupling strength ij

kγ = is still the same as

always. The second possible coupling strength is a linear function of the first one;

ijkγ β= , with ( ]0,1β ∈ . And like in Section 3.2 there is no intra hidden layer

connectivity in this section.

Every node in a layer is symmetrically coupled to every node in a neighboring layer with

either the coupling strength k or the coupling strength kβ . This restriction is made to

limit the possibilities in assigning the coupling strengths to connections in the network.

By using two different coupling strengths, the OLS problem of the network

[ ]1 1 2 4 2=n can be solved. This is demonstrated with the use of the last three

layers of this network, i.e. [ ]2 4 2=n . However, the solution given below can also be

applied to the full network.

30

The connections between the first and second layer of [ ]2 4 2=n are the ones with

the new coupling strength kβ , resulting in the following coupling matrix

1 1,2

2,1 2 2,3

3,2 3

k

β

β

−

= − − −

D J

Γ J D J

J D

with 1 24β=D I , ( )2 42 2β= +D I , 3 24=D I and ,i jJ a matrix of ones of size

i jn n× .

By making the reduction of the ( )det 0nλ′ − =Γ I , as in (3.2), then the following

eigenvalues are obtained:

1 4λ β=

2 2 2λ β= +

3 4λ =

Any value of 0 1β< < seems to lead to OLS. To make the difference between partial

synchronization regimes more noticeable let 0.5β = . This means that the output should

synchronize for 3 1 4k γ λ> ≈ . Figure 4.1 shows that this is indeed correct.

Figure 4.1: The network [ ]2 4 2=n for a coupling strength of 0.25k =

The second layer will synchronize for 2 1 3k γ λ> ≈ (see Figure B.20) and the first layer

for 1 1 2k γ λ> ≈ . The smallest non-zero eigenvalue of ′Γ is 1 2λ λ= = , therefore the

synchronization of the first layer will coincide with full synchronization, as shown in

Figure B.21.

31

4.2 Asymmetric coupling for networks consisting of two layers

Output layer synchronization can be achieved with much less nodes if the couplings are

asymmetric. Therefore only two layers will be used in this case; the input layer and the

output layer.

The diffusive coupling as defined in (2.2) is a symmetric coupling. In this section the

diffusive coupling will be defined as

( )1,

n

i ij i j

j i j

u y yγ= ≠

= − −∑ , with 0 0ij ji

γ γ≤ ≠ ≥ for some or all , 1,...,i j n= (4.1)

Let’s first look at the simplest layered network relevant to OLS; the three node network

[ ]1 2=n . To ensure OLS the permutation matrix should be

1 0 0

0 0 1

0 1 0

=

ΠΠΠΠ

According to (2.5) the linear manifold A corresponding with partial synchronization now

becomes

{ }9

1 2 3:A = ∈ ≠ =x x x xR

However the permutation matrix ΠΠΠΠ must commute with Γ in order to have a globally

asymptotically stable subset of (2.5) when the smallest non-zero eigenvalue of Γ is

larger than the positive γ , if the corresponding eigenvector can be taken from

( )3range −I ΠΠΠΠ . The general coupling matrix for the network [ ]1 2=n is as follows

11 12 13

21 22

31 33

0

0

γ γ γ

γ γ

γ γ

− − = − −

Γ , with 1,

n

ii ij

j i j

γ γ= ≠

= ∑ ,

The coupling strengths between the two output nodes are a priori set to zero, i.e.

23 32 0γ γ= = . If Γ commutes with ΠΠΠΠ then

11 12 13 11 13 12

31 33 21 22

21 22 31 33

0 0

0 0

γ γ γ γ γ γ

γ γ γ γ

γ γ γ γ

− − − − ⋅ = − = − = ⋅ − −

Γ ΓΠ ΠΠ ΠΠ ΠΠ Π

32

Therefore the following coupling strengths need to be equal in order to let the coupling

matrix commute with the permutation matrix.

12 13

21 31

22 33

γ γ

γ γ

γ γ

=

=

=

Now suppose 12 1kγ = and 21 2kγ = ,

then Γ will be:

1 1 1

2 2

2 2

2

0

0

k k k

k k

k k

− − = − −

Γ

The eigenvalues and corresponding eigenvectors of this asymmetric coupling matrix are

1

2 2

3 1 2

0

2

k

k k

λ

λ

λ

=

=

= +

1

1

1

1

=

v , 2

0

1

1

= −

v ,

1 2

3

2

1

1

k k− =

v

It is easy to see that 2v is in the ( )3range −I ΠΠΠΠ , therefore 2 2kλ = is the eigenvalue

belonging to output layer synchronization and OLS should occur when 2 2 1kλ γ= > ≈ .

However if the coupling strength 1 0k ≥ , then OLS will coincide with full

synchronization, as 2 2kλ = is also the smallest non-zero eigenvalue of Γ . OLS is only

possible when the coupling strength 1k is negative, because if 3ˆ0 λ γ≤ < when 2λ γ>

then 3λ is the smallest non-zero eigenvalue of Γ . This means that

2 21

ˆ0

2 2

k kk

γ− −≤ <

21 0

2

kk− ≤ < ,

when 2 1k > and ˆ 1γ ≈ .

33

To avoid the use of negative coupling strengths the three node network is extended with

one additional node in the input layer, i.e. [ ]2 2=n . The permutation matrices that

correspond with input layer synchronization and OLS are respectively

1

0 1 0 0

1 0 0 0

0 0 1 0

0 0 0 1

=

ΠΠΠΠ , 2

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

=

ΠΠΠΠ

The general coupling matrix of the network [ ]2 2=n is

11 13 14

22 23 24

31 32 33

41 42 44

0

0

0

0

γ γ γ

γ γ γ

γ γ γ

γ γ γ

− − − − = − − − −

Γ , with 1,

n

ii ij

j i j

γ γ= ≠

= ∑ ,

and with no intra layer connectivity as always with the input and output layer. This

general coupling matrix must commute with 2ΠΠΠΠ for OLS to be possible.

11 13 14 11 14 13

22 23 24 22 24 23

2 2

41 42 44 31 32 33

31 32 33 41 42 44

0 0

0 0

0 0

0 0

γ γ γ γ γ γ

γ γ γ γ γ γ

γ γ γ γ γ γ

γ γ γ γ γ γ

− − − − − − − − ⋅ = = = ⋅ − − − − − − − −

Γ ΓΠ ΠΠ ΠΠ ΠΠ Π

Thus the following coupling strengths should be equal

13 14

23 24

γ γ

γ γ

=

=,

31 41

32 42

γ γ

γ γ

=

=, and 33 44γ γ=

Let 13 23 1kγ γ= = and 31 32 2kγ γ= = , then the coupling matrix becomes:

1 1 1

1 1 1

2 2 2

2 2 2

2 0

0 2

2 0

0 2

k k k

k k k

k k k

k k k

− − − − = − − − −

Γ

34

Figure 4.2: The layered network [ ]2 2=n

The eigenvalues and eigenvectors of this matrix are

1

2 1

3 2

4 1 2

0

2

2

2 2

k

k

k k

λ

λ

λ

λ

=

=

=

= +

1

1

1

1

1

=

v , 2

1

1

0

0

− =

v , 3

0

0

1

1

= −

v ,

1 2

1 2

41

1

k k

k k

− − =

v

The eigenvector 2v is in the ( )4 1range −I ΠΠΠΠ , while eigenvector 3v is in the

( )4 2range −I ΠΠΠΠ . Therefore OLS occurs when 3 22kλ γ= > . The objective is to have OLS

without the need of negative coupling strengths. This can be achieved by choosing

2ˆ0 λ γ≤ < and 3λ γ> , consequently 2λ will be the smallest non-zero eigenvalue of Γ .

Note that by choosing 2ˆ0 λ γ≤ < and 3λ γ> input layer synchronization will coincide

with full synchronization.

So if 1ˆ0 2 1 2k γ≤ < ≈ then OLS will occur for any 2 1 2k > and full synchronization

will not be possible. Only when 1 1 2k ≥ , in combination with 2 1 2k > , will full

synchronization occur.

If for instance 1 0.25k = then only OLS will occur for 2 0.5k >

Figure 4.3: The network [ ]2 2=n for 1 0.25k = and 2 0.5k =

35

When 1k is increased to 1 0.5k = with 2k still at 2 0.5k = then full synchronization

occurs.

Figure 4.4: The network [ ]2 2=n for 1 0.5k = and 2 0.5k =

4.3 Summary

In this chapter we first studied layered networks with symmetric coupling using two

different coupling strengths in the network. In the previous chapter we saw that, by

removing the connections within the hidden layers, the eigenvalues from (3.3) did not

change very much. These eigenvalues were still mostly dependent on the structure of the

layered network. By using multiple different coupling strengths the eigenvalues are more

dependent on the coupling strengths instead of the network structure. This gives more

control over the thresholds of layer synchronization regimes, if the structure of the

layered network is fixed. The new coupling strength should be assigned to the inter layer

connections, because the number of nodes in neighboring layers used to determine the

layer synchronization thresholds mostly.

Asymmetric diffusive coupling with a number of different coupling strengths is

more efficient in terms of getting OLS with as few nodes as possible. Therefore only

layered networks consisting of two layers were considered. And it turns out that the

smallest network that shows OLS is a network with two nodes in the input and two nodes

in the output layer. Two coupling strengths were used in that network.

36

Chapter 5

Experimental synchronization

The first experimental setup was build by Steur [Steur, 2007] with which up to four

electronic neurons could be connected to each other. The electronic implementation of

the H-R model was based on earlier publications on this subject by [Merlat et al., 2006]

and [Lee et al., 2004]. Later Neefs improved the electronic neuron design with voltage

regulation, input separation and junction reduction. Also a new coupling interface was

developed to be able to couple up to eighteen electronic neurons in an arbitrary network

topology [Neefs, 2009].

First the current experimental setup will be given in Section 5.1. After which

synchronization robustness will be discussed in Section 5.2. The electronic neurons are

not identical, unlike the H-R neurons in simulation. This is due to tolerances in the

electrical components of the experimental systems. Therefore the solutions of the coupled

systems will only reach the synchronization manifold within some bound ε and

therefore a new definition of synchronization will be introduced; practical

synchronization. In Section 5.2 the experimental synchronization of two diffusely

coupled neurons will also be presented. In Section 5.3 the practical limitations that were

encountered will be pointed out. And in Section 5.4 the experimental results of the

networks discussed in Chapter 3 and 4 will be presented.

5.1 Experimental setup

The experimental setup used in this research is depicted in the following figure.

Figure 5.1: The experimental setup

37

Hereafter the different parts of the setup will be explained in more detail.

1. DC voltage source: The DC voltage source supplies the circuits of the electronic

neurons with power. The supplied voltage is kept as constant as possible with a

voltage regulation circuit on every electronic neuron, to ensure that the model

parameters are constant over time [Neefs, 2009]. The voltage source’s range is

[ ]20 V± .

2. Eighteen electronic neurons: Here the eighteen electronic neurons are housed. For

the circuit of the electronic H-R neuron the reader is referred to [Neefs, 2009].

Every electronic neuron can be connected to the coupling interface, see Figure

5.2.

3. Coupling interface: The coupling interface provides the coupling between the

neurons in a network. The output of a neuron that is being used is connected to

one of the inputs of the coupling interface. A schematic representation of the

coupling interface is given in Figure 5.3. The outputs of the neurons ( )iy t go first

through an analog-to-digital converter (ADC). Then the signals are being

processed by the microcontroller (ARM9SAM9260). Software that has been

flashed into the memory of the microcontroller uses the outputs of neurons in the

network to calculate the inputs to the neurons in the network. The diffusive

coupling functions need to be defined in the source code. After the source code

has been compiled it can be flashed into the microcontroller through a serial

connection between the computer and the coupling interface. Finally the

manipulated digital signals are converted back to analog signals. And after some

signal reshaping the signals are used as inputs ( )iu t of the electronic neurons in

the network. The total time it takes for a signal to go through the coupling

interface is about 5

int 8 10 [sec]τ −≈ ⋅ when all eighteen channels are used,

according to [Neefs, 2009].

4. Data acquisition: For the data acquisition two National Instruments devices are

used. Each containing sixteen measurement channels. It is possible to measure

both the inputs and the outputs of the neurons in a network.

5. Computer: The computer is used for modifying coupling functions and for data

acquisition. The software package LabVIEW SignalExpress is used to view and

store the signals acquired by the National Instruments measurement devices.

38

Figure 5.2: Example of a connection between neuron and synchronization interface

Figure 5.3: Schematic representation of the synchronization interface [Neefs, 2009]

39

5.2 Practical synchronization and synchronization robustness

The electronic H-R neurons are not identical because of tolerances in the analog

components on the printed circuit boards. Components like resistors and capacitors used

in the realization are of the shelf, and the manufacturer made these components with a

certain tolerance. These tolerances lead to small differences of the parameters in the H-R

model. Having non-identical neurons, means that the solutions of the coupled systems

can only reach the synchronization manifold within a certain bound. Therefore a slightly

different definition of synchronization is presented here:

Definition 5.1. (Practical (Partial) Synchronization)

The states of H-R neuron i are 1, 2,, ,i i i i

y z zΤ

= x and the solutions ( ) ( )1 ,..., nt tx x of n

coupled H-R neurons (2.1) with initial condition ( ) ( )1 0 ,..., 0nx x are called practically

(partially) synchronized if:

( ) ( )

limsup i jt

t t ε∞→∞

− ≤x x , for all (or some) , 1,...,i j n= , (5.1)

with a sufficiently small constant 0ε >

Note that only the y-state is measured during the experiments (and simulations), as it is

the natural output of the H-R model. When the y-states of two diffusely coupled H-R

neurons are synchronized, then the internal states, 1z and 2z , of the two systems will also

be synchronized [Neefs, 2009].

Practical synchronization implies that if the error ( ) ( )supi j ij

y t y t ε− = is small enough

for large enough time t then two nodes will be called practically synchronized. This error

is called the maximum synchronization error.

The maximum synchronization error should be smaller than ε . This is similar to when

the solutions of the coupled systems need to reach the linear synchronization manifold

within the bound 0ε > . However the bound ε is perpendicular to the synchronization

manifold, while the maximum synchronization error is in line with the axes. In the phase-

plane representation, like in Figure 5.4, the synchronization manifold is obviously a

diagonal in the ideal case (indicated by the dotted line). The bound ε is perpendicular to

the diagonal, therefore 2ε ε= .

In this study the value ε is chosen to be 0.35ε = ( 0.25ε = ). Figure 5.5 shows a

zoomed-in plot of the outputs of two symmetrically coupled electronic H-R neurons with

a coupling strength of 0.7k = . The plot is focused at a burst of three spikes. The

maximum synchronization error with this coupling strength was 12 0.348ε ε= < ,

therefore these two neurons can be called practically synchronized. The synchronization

40

error is the largest during a spike. This is because a spike consists of relative steep slopes;

a small difference in spike-timing can lead to a relative large synchronization error.

Figure 5.4: A visualization of the synchronization manifold for two identical coupled

H-R systems and two non-identical H-R systems [Neefs, 2009].

Figure 5.5: The synchronization error as a function of the time for two synchronized

electronic H-R neurons.

41

The size of a spike is about 2.5 [V], therefore the maximum synchronization error is

0.35 2.5 14%= of the total neuron output range. Setting ε to a lower value is not

desirable, because in that case a higher coupling strength is needed to practically

synchronize the neurons. A higher coupling strength k also implicates a higher γ

according to (3.4).

Several electronic neurons were selected to determine the coupling strength needed for

synchronization of two diffusively coupled neurons. For practical purposes, coupling

strengths were incremented with 0.05k∆ = . The average coupling strength needed to

synchronize two electronic neurons is 0.65k ≈ . The eigenvalues of the coupling matrix

of two diffusively coupled nodes are 1 0λ = and 2 2kλ = , consequently ˆ 1.3γ ≈ .

5.3 Practical limitations

The electronic H-R neurons are not identical due to tolerances in the electrical

components on the circuit boards. Moreover the electronic neurons contain some output

noise. These two factors influence the maximum synchronization error as a function of

the overall coupling strength of a network, as sketched in Figure 5.6. As long as the

synchronization error is more dominant than the output noise, then this error will

decrease for increasing coupling strength. Notice that if there was no output noise, then

the maximum synchronization error would asymptotically descent to zero.

Figure 5.6: Schematic representation of the maximum experimental synchronization

error as a function of the coupling strength

However at some point the output noise starts to dominate. The result is that the coupling

terms ( )iu t starts to become non-zero due to the noise. The maximum synchronization

error will therefore increase if the coupling strength is increased further.

42

Consider for example the network [ ]1 4=n with uniform symmetric coupling strength

k, see Figure 5.7. Experiments show that this network is fully synchronized for 1.2k ≥ .

When the coupling strength is increased to 5k = , the synchronization between the node

in the first layer and the nodes in the second layer starts to disintegrate, as is depicted in

Figure 5.8. This is the point where the noise starts to dominate. If the coupling strength

5k > , then the maximum synchronization error will only increase, with 2.5ij

ε ≈ as

maximum, since 2.5 [V] is approximately the output range of a H-R neuron with 3.3I = .

Figure 5.7: The layered network [ ]1 4=n with uniform symmetric coupling

Figure 5.8: The network [ ]1 4=n for a coupling strength of 5k =

Figure 5.9: The network [ ]1 4=n for a coupling strength of 5.5k =

43

Figure 5.9 shows the case where the coupling strength 5.5k = . When a coupling strength

is in the noise dominant range, then there is a possibility that saturation of the coupling

interface occurs. This happens when the output noise is amplified too much by the

relative high coupling strength. For the network [ ]1 4=n this happens at 5.5k = as can

be seen from Figure 5.10. The range of the coupling interface is limited to [ ]10 V± .

Figure 5.10: Input to the neuron in the first layer of the network [ ]1 4=n

One may have noticed that the nodes in the output layer of the network [ ]1 4=n do not

experience any noise effects. This is because the number of connections to the output

layer nodes is different from the number of connections to the input layer node; the input

node has four connections, while the output nodes only have one connection.

The threshold for full synchronization is in most cases higher for larger layered

networks. It is therefore sometimes not possible to reach full synchronization in some

larger networks, where noise effects start to appear at a lower coupling strength than the

coupling strength needed for full synchronization.

44

5.4 Experimental synchronization of multiple neurons

The most relevant network topologies of Chapters 3 and 4 will be demonstrated

experimentally. Some neurons synchronize at a lower coupling strength than others. This

means that the synchronization thresholds given in this section are not the mean

thresholds of those synchronization regimes. During the experiments a choice needs to be

made about which neurons should be connected to which neurons. Electronic neurons

that synchronize more easily with each other were put inside one layer whenever

possible, to aid (output) layer synchronization.

To be fairly certain that the solutions of the coupled systems are in a stable subset, the

outputs of the neurons are recorded for three seconds, of which only half a second will be

plotted to avoid too much overlapping lines in the figures. Note that an experiment is

initialized when the microcontroller of the coupling interface is reset. The recorded

solutions will therefore not be the solution directly after the initialization. In most cases

not all combinations of solutions ( )iy t will be plotted against each other to save space;

only the most important ones will be shown.

The first layered networks with hidden layers and uniform symmetric coupling strenghts

that showed OLS were the networks [ ]1 1 1 ln=n . In Chapter 3 it was explained that

the networks [ ]1 ln=n and [ ]1 1 ln=n only showed full synchronization. The

eigenvalues of these two networks are in both cases 0λ = , 1λ = and nλ = with the

appropriate algebraic multiplicities. At ˆ 1k γ≥ the network should fully synchronize.

However some unexpected partial synchronization occurs in the network [ ]1 2=n right

before full synchronization occurs that did not occur in simulations.

Figure 5.11: The layered network [ ]1 2=n with uniform symmetric coupling

Figure 5.12: The network [ ]1 2=n for a coupling strength of 1.1k =

45

Figure 5.13: The network [ ]1 2=n for a coupling strength of 1.2k =

The neuron in the first layer synchronizes with the two nodes in the second layer before

the output layer nodes are synchronized with each other (see Figure 5.12). This occurs

when the sequence of spiking is in the following order: neuron 2 spikes, neuron 1 spikes

and then neuron 3 spikes (or in reverse). In such a case, the synchronization error

between the input node and either of the output nodes will be smaller than the

synchronization error between the output nodes themselves. Furthermore it seems that

23 12 13ε ε ε≈ + .

For 1.2k ≥ this network is fully synchronized (see Figure 5.13). The same applies to the

network [ ]1 3=n . Yet when a fourth node is added to the output layer, then only full

synchronization can be observed at 1.2k ≥ with no partial synchronization regime

happening before full synchronization. Similar results can be obtained from the networks

[ ]1 1 ln=n .

The networks [ ]1 1 1 ln=n are the first networks of this kind to show OLS. The

network with 2l

n = and the network with 4l

n = were analyzed experimentally:

Figure 5.14: The layered networks [ ]1 1 1 2=n and [ ]1 1 1 4=n

46

Figure 5.15: The network [ ]1 1 1 2=n for a coupling strength of 1.35k =

Figure 5.16: The network [ ]1 1 1 2=n for a coupling strength of 1.8k =

OLS occurs at a coupling strength of 1.35k = when [ ]1 1 1 2=n , as one can see in

Figure 5.15. At 1.8k = the third layer, i.e. node 3, synchronizes with the output layer and

at 2.2k = the network fully synchronizes. For 1.8 2.2k< < no nodes join the

synchronization regime of the last two layers. The network [ ]1 1 1 3=n again gives

similar results to this network.

The network [ ]1 1 1 4=n is a different story. Here OLS coincides with the

synchronization of the last two layers. So for a coupling strength 1.4k < no neurons are

synchronized with each other, while at 1.4k = suddenly both the third and the fourth

layer are synchronized. Full synchronization of this network occurs at a coupling strength

of 2.5k ≥ .

Figure 5.17: The network [ ]1 1 1 4=n for a coupling strength of 1.4k =

When another hidden layer, containing one node, is added to the network described

above another effect occurs. The network [ ]1 1 1 1 2=n behaves similar to the

47

network [ ]1 1 1 2=n , but the network [ ]1 1 1 1 4=n cannot fully synchronize

due to significant output noise amplification at 4.25k > . The theoretical threshold for

full synchronization of this network is ˆ 0.2774 4.7k γ≥ ≈ .

The thresholds of different practical synchronization regimes of the networks

[ ]1 1 1 1 ln=n as well as similar networks are summarized in the following table.

Some of these networks were discussed earlier in this section.

Network topology Output layer

synchronization

Synchronization of

the last two layers

Full synchronization

[ ]1 2=n 1.2 1.2 1.2

[ ]1 4=n 1.2 1.2 1.2

[ ]1 1 2=n 1.2 1.2 1.2

[ ]1 1 4=n 1.2 1.2 1.35

[ ]1 1 1 2=n 1.35 1.8 2.2

[ ]1 1 1 4=n 1.4 1.4 2.5

[ ]1 1 1 1 2=n 1.4 2.6 3.65

[ ]1 1 1 1 4=n 1.5 1.5 -

Table 5.1: Synchronization thresholds for layered networks with hidden layers consisting

of only one node

Layered networks with three nodes in the output layer are left out of the table, because

the results from these networks are similar to the same networks with only two nodes in

the output layer. Adding a fourth node does however make a noticeable difference in the

thresholds of synchronization regimes.

In Section 4.2 we looked at how to achieve OLS with as few nodes as possible. This

could be done when the coupling strengths were not symmetric. The network with the

least number of nodes that shows OLS is the following network.

Figure 5.18: The layered network [ ]2 2=n with asymmetric coupling

48

The eigenvalues of the coupling matrix of this network are:

1

2 1

3 2

4 1 2

0

2

2

2 2

k

k

k k

λ

λ

λ

λ

=

=

=

= +

In this case the second eigenvalue 2λ determines the synchronization threshold of the

input layer, and the third eigenvalue 3λ the synchronization threshold of the output layer.

When the coupling strength 1k is set 1 0.25k = and 2 0.6k ≥ then there will be OLS. Full

synchronization will not be possible by changing 2 0k ≠ .

Figure 5.19: The network [ ]2 2=n for a coupling strength of 1 0.25k = and 2 0.6k =

Only when 1ˆ 2k γ≥ and 2

ˆ 2k γ≥ will there be full synchronization. This is shown in the

following figure.

Figure 5.20: The network [ ]2 2=n for a coupling strength of 1 0.6k = and 2 0.6k =

49

5.5 Summary

In this chapter it was shown that the theoretical thresholds for OLS are usable in practice,

where one has to deal with non identical neurons. Synchronization can however only be

reached within a certain bound. This has to do with tolerances in the analog components

used to realize the electronic equivalent of the H-R system. The electronic neurons also

produce some noise. Because of this noise, synchronization is not possible for relative

high coupling strengths. The effects of noise, and saturation of the coupling interface, set

a restriction on the maximum coupling strength that can be applied in practice. The limit

is dependent on the network topology. This makes it impossible for some nodes to

synchronize in some larger layered networks, and it is therefore not possible to evaluate

full synchronization in relatively large layered networks.

50

Chapter 6

Conclusions and recommendations

6.1 Conclusions

The objective is to have the output layer synchronize before any other nodes synchronize.

Layer synchronization thresholds are derived for an arbitrary layered network

with uniform symmetric coupling strengths, i.e. the same coupling strength on every

connection. Diffusive coupling is used to create connections. This coupling is in general

defined as a symmetric coupling, meaning that the coupling strength between two nodes

is the same in both directions. A layer synchronization threshold is determined by the

number of nodes in the preceding, and succeeding layers and also by the number of nodes

in the current layer if there is intra layer connectivity.

For networks with uniform symmetric coupling strengths there are not that many

topologies that show OLS. If such a network only has two layers, then it is possible when

the input layer has more nodes than the output layer. A network with hidden layers does

not show OLS in most cases, especially when the nodes in the hidden layers are

connected to all the other nodes in their hidden layer. If all the hidden layers consist of

only one node then OLS is possible when the total number of layers exceeds three.

However if there are hidden layers with more than one node, then OLS will show up first

in some cases if there is all-to-non intra hidden layer connectivity. The last hidden layer

should have two more nodes than the output layer and the penultimate hidden layer

should contain only one node.

The layer synchronization thresholds can still be derived when a layered network

uses two coupling strengths instead of one. The additional coupling strength should

however be a linear function of the original one. Having a second coupling strength, in an

otherwise uniform symmetrically coupled network, gives more control over the layer

synchronization thresholds. The new coupling strength should be assigned to the inter

layer connections, because the number of nodes in neighboring layers determine the layer

synchronization thresholds mostly. Note that dropping the intra hidden layer connectivity

did not result in significant more topologies that show OLS.

Asymmetric diffusive coupling with a number of different coupling strengths is

more efficient in terms of getting OLS with as few nodes as possible. Therefore only

layered networks consisting of two layers are considered. And it turns out that the

smallest network that shows OLS is a network with two nodes in the input and two nodes

in the output layer. Two coupling strengths were used in this network.

The Experiments that were performed give results that are comparable to

simulation results. However, because the electronic H-R neurons are not identical,

synchronization can only be reached within a certain bound. Also the effects of noise and

saturation of the coupling interface set a restriction on the maximum coupling strength

that can be applied. The limit is dependent on the network topology. This makes it

impossible for some nodes to synchronize in some larger layered networks.

51

6.2 Recommendations

A number of recommendations can be made for (possible) future work.

One could look at the effects of time delays on synchronization of the output layer. In

[Neefs, 2009] time delays are already used in complex networks of H-R neurons. The

results from that work could be used as guidance when applying time delays to layered

networks. Time delays become relevant when, for example, the distance between layers

is larger than the distance of connections within layers, and if the connection speed is

finite.

Also the algebraic relation between the network topology and the eigenvalues and

eigenvectors should be derived. The eigenvalues are used to determine thresholds of

partial synchronization regimes. The corresponding eigenvectors essentially determine

which partial synchronization regimes it is. Some eigenvalues could already be

determined algebraically, but the corresponding eigenvectors still need to be derived.

52

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54

Appendix A

Characteristic equation of layered networks with uniform

symmetric coupling strengths

The layered networks that are going to be discussed in this appendix are layered

networks, where only the output layer contains multiple nodes:

[ ]1 ln=n

[ ]1 1 ln=n

[ ]1 1 1 ln=n

where 2l

n ≥ are the number of nodes in the output layer. The characteristic equation is

obtained by reducing the ( )det 0nλ′ − =Γ I . At the beginning of the derivation of the

characteristic equation, the number of output nodes is set to 3l

n = . Later on it will be

reset to l

n .

[ ]1 ln=n

3 1 1 1 3 1 1 1 3 0 0 3

1 1 0 0 0 1 0 1 0 1 0 0

1 0 1 0 0 0 1 1 0 0 1 0

1 0 0 1 1 0 0 1 1 0 0 1

λ λ λ

λ

λ

λ λ λ

− − − − − − − − − −

− − −= Λ = Λ

− − −

− − − − − −

with ( ) ( )2 1

1 1ln

λ λ−

Λ = − = −

( )( )( )

1

111 3

3

l l

l

n na

n

λ λλ

λ λ

− − −−= − + ⋅ =

− −

Therefore the characteristic equation is

( ) ( ) ( ) ( )( )

( ) ( )( ) ( ) ( )

1

1

1 1

3 1 1

1 1 1 0

l

l l

n

tot l l

n n

l

a n n

n n

λ λ λ λ

λ λ λ λ λ λ

−

− −

Λ = Λ − = − − − −

= − − + = − − =

with n the total number of nodes in the network. OLS will occur at 1k γ> in this

network. However full synchronization also occurs at ˆ 1k γ≥ , with γ̂ γ= . The next

network has the same problem.

1

3 0 0 0

0 1 0 0

0 0 1 0

0 0 0 a

λ−

= Λ

55

[ ]1 1 ln=n

1 1 0 0 0 1 1 0 0 0

1 4 1 1 1 1 4 0 0 3

0 1 1 0 0 0 0 1 0 0

0 1 0 1 0 0 0 0 1 0

0 1 0 0 1 0 1 0 0 1

λ λ

λ λ

λ

λ

λ λ

− − − −

− − − − − − − −

= Λ− −

− −

− − − −

with ( ) ( )2 1

1 1ln

λ λ−

Λ = − = −

( )( )( )

1

1 1 114

1 1

ln

aλ λ

λλ λ

+ − − −−= − + =

− −

( )( ) 1

2

1 1

111 3

la n

aa a

λλ

− −−= − + ⋅ =

Therefore the characteristic equation is

( ) ( ) ( ) ( )( )( )

( ) ( )( ) ( ) ( )

1

1 2 1 1 1 1 1 1

1 2 1 0

l

l l

n

tot l l

n n

l

a a n n

n n

λ λ λ λ λ

λ λ λ λ λ λ

−Λ = Λ − = − − + − − − −

= − − + = − − =

OLS and full synchronization still coincide at a threshold of 1 1k > .

The next layered network is the first network of the kind of networks discussed in this

appendix that shows different thresholds for OLS and full synchronization.

[ ]1 1 1 ln=n

1 1 0 0 0 0 1 1 0 0 0 0

1 2 1 0 0 0 1 2 1 0 0 0

0 1 4 1 1 1 0 1 4 0 0 3

0 0 1 1 0 0 0 0 0 1 0 0

0 0 1 0 1 0 0 0 0 0 1 0

0 0 1 0 0 1 0 0 1 0 0 1

λ λ

λ λ

λ λ

λ

λ

λ λ

− − − −

− − − − − −

− − − − − − − −= Λ

− −

− −

− − − −

1

2

1 0 0 0 0

0 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0

a

a

λ−

= Λ

56

with ( ) ( )2 1

1 1ln

λ λ−

Λ = − = −

( )( )( )

1

2 1 112

1 1a

λ λλ

λ λ

− − −−= − + =

− −

( )( ) 1

2

1 1

4 114

aa

a a

λλ

− −−= − + =

( )( ) 2

3

2 2

111 3

la n

aa a

λλ

− −−= − + ⋅ =

Therefore the characteristic equation is

( )

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( )( )

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( )

1 2 3

1

1

1 2 2

1 22 2

1

1 1 4 2 1 1 1 3 2 1 1

1 1 1 2 1 1 1 2 1 1

1 1 1 3 1 1 3 1

1 2 1 3 1 1

l

l

l

l

tot

n

n

l l

n

l l

n

l

a a a

n n

n n

n

λ

λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ λ

λ λ λ λ λ λ

−

−

−

−

Λ = Λ −

= − − − − − − − − − − − −

= − − + − − − − − − − − − −

= − − + − − + − − − − +

= − − + + − + − −

( ) ( ) ( ) ( )( )1 3 2 1 5 3 7 3 0ln

l l ln n nλ λ λ λ λ

−

= − − + + + − + =

The OLS threshold is obviously still 1k γ> , but the full synchronization threshold is

now the smallest root of the polynomial

( ) ( ) ( )3 25 3 7 3l l ln n nλ λ λ− + + + − +

The smallest non-zero eigenvalues are plotted in Figure A.1 for different number of

output nodes 2l

n ≥ . In this figure it is also shown that the threshold of full

synchronization becomes higher when the number of hidden layers, containing one node,

is increased.

Figure A.1: The smallest non-zero eigenvalue as a function of the number of output nodes

1

2

3

1 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0

a

a

a

λ−

= Λ

57

Appendix B

Simulation results

The theoretical findings of Chapter 3 and 4 are verified with the use of Matlab Simulink.

First the value γ̂ is obtained from simulations by inspecting a network of two mutually

coupled nodes. The coupling matrix of this network is as follows

1 1

1 1k

− = −

Γ

The eigenvalues of this matrix are 1 0λ = and 2 2kλ = . There will be full

synchronization if ˆ 2k γ≥ . Simulations reveal that this network is synchronized for a

coupling strength of 0.5k ≥ ; ˆ 1γ ≈ (see Figure B.1). According to the Wu-Chua

conjecture this value holds for any layered network.

Figure B.1: Simulation results of two diffusively mutual coupled H-R neurons

Each simulation takes three seconds of which the last two seconds are recorded. The first

second will be discarded to be fairly certain that no transient behavior is recorded. The

initial conditions are chosen randomly in the range [-1, 1]. The results are inspected and

if the linear manifolds that correspond with synchronization are stable, then half a second

of the solutions will be depicted. Only a small portion of the recorded data is displayed to

avoid too much overlapping lines in the plots. In most cases not all combinations of

solutions ( )iy t will be plotted against each other to save space; only the most important

ones will be plotted.

58

B.1 Simulations accompanying Section 3.1

Figure B.2: The layered network [ ]1 2 1 1 2=n

Figure B.3: The network [ ]1 2 1 1 2=n for a coupling strength of 0.2k =

Figure B.4: The network [ ]1 2 1 1 2=n for a coupling strength of 0.25k =

59

Figure B.5: The network [ ]1 2 1 1 2=n for a coupling strength of 1k =

Figure B.6: The network [ ]1 2 1 1 2=n for a coupling strength of 2.75k =

60

Figure B.7: The layered network [ ]1 1 2=n

Figure B.8: The network [ ]1 1 2=n for a coupling strength of 0.95k =

Figure B.9: The network [ ]1 1 2=n for a coupling strength of 1k =

Figure B.10: The layered network [ ]1 1 1 2=n

61

Figure B.11: The network [ ]1 1 1 2=n for a coupling strength of 1k =

Figure B.12: The layered network [ ]3 2=n

Figure B.13: The network [ ]3 2=n for a coupling strength of 0.35k =

62

B.2 Simulations accompanying Section 3.2

Figure B.14: The layered network [ ]1 2 1 4 2=n

Figure B.15: The network [ ]1 2 1 4 2=n for a coupling strength of 0.25k =

63

Figure B.16: The network [ ]1 2 1 4 2=n for a coupling strength of 0.35k =

Figure B.17: The network [ ]1 2 1 4 2=n for a coupling strength of 0.5k =

64

B.3 Simulations accompanying Section 4.1

Figure B.18: The layered network [ ]2 4 2=n

Figure B.19: The network [ ]2 4 2=n for a coupling strength of 0.25k =

65

Figure B.20: The network [ ]2 4 2=n for a coupling strength of 0.35k =

Figure B.21: The network [ ]2 4 2=n for a coupling strength of 0.5k =