Y OKOHAMA C ITY U NIVERSITY
D OCTORAL T HESISNoise influence on spike activation
in a Hindmarsh-Rose small-world
neural network

Author:Supervisor:

Sun Z HE

Prof. Ruggero
M ICHELETTO

Yokohama City University
in theMicheletto Sensory Information Laboratory
Department of Materials System Scienceii
acknowledgements I wish thank all those people that helped and supported
me to learn knowledge and explore the unknown: my parents, my supervisor, and all of my friends.
Since I was a primary school student, I wanted to become a good scientist.
I believe without my supervisor Professor Ruggero Micheletto, I will neverachieve this dream.
I also would like to thank Professor Andrzej Cichocki and Professor Keiichi Kitajo in RIKEN Brain Science Institute, who have given me a lot of advice for my research.
I would like also to thank all of the help from the professors and researches in Yokohama City University and RIKEN Brain Science Institute.
I really appreciate Rotary-Yoneyama Memorial foundation, Donghua ChinaJapan Education and Cultural Exchange foundation, Nitori Scholarship foundation. I am so grateful they funded my Master course and Ph.D. course.

1

Chapter 1

Introduction
1.1

Reasons to study the brain

Without any doubt the brain is the most fabulous machine in the universe.
Animals have brains and we humans have brains. We actually are brains.
All our activities and bodily functions are directed by the brain under controlof the brain.
In the past human were thinking that science was a discipline of research
separated from the science of the mind. We human, were convinced that notliving things, like materialistic matter like physics, engineering, chemistry or
biology, were fields that could be studied using science. However, there wasthe conviction that brain was a separate field of study.
Studying the brain was out of question. Brain was considered to be amysterious entity, in which mind, consciousness and soul reside. At the
beginning of the century, many scientists in the life science field started toinvestigate on the brain using simple tools, first doing post-mortem examinations in animals and humans, and later with modernization studying thebrain more systematically.
It was soon discovered that brain shows a lot of electrical activity and thesole active element in the brain is the neuron, a specialized cell. This cell,that I will describe later, is a simple element that essentially does only one
simple thing: it gives a quick electric impulse, called spike, or action potential.
I will describe this functions later in details. However, our brain is simply acomplex connection of a great number of neurons.
Currently science have no idea how all our creativity, our consciousness
and our ability to think and change the world around us, are realized simply
by connecting many neurons together. We can imagine that one day we will
be able to understand how the brain works, when that day will arrive, wewill be able to construct artificial brains. This will give an enormous impulse
to human technology and human society as a whole.

2

Chapter 1. Introduction

Brain is not a magic box, something that we do not have the scientificmeans to study. Brain is a machine, a computational machine. It is made byconnected neurons, and for small animals with small brains, we even know
almost perfectly the connections configuration. Brain give small animals perfect control on their bodies and functions. A bee for example, can solve very
difficult problems, like flying from one place to the other, arriving on kmaway flower filed, land exactly to a flower, take the pollen, lift off and findthe way fly back home.
A bee is even able to communicate to other bees the position of the field,
using complex body language communication. All of this, plus many otherthings, are realized by the very small brain of the insect, that have less than
one million neurons. Take in account that a neuron can emit its spikes at maximum rate of 200 Hz, whereas our best computers cannot yet land a plane
automatically. We need a brain to do that.
Even if brains are so slow and so simple, they give animals full controlof their bodies, they allow the solutions of very complex problems and therealization of very structured societies (consider the societies of Ants, thathave a brain with only 250.000 neurons). Brains also have huge memory,consider for example the enormous amount of things that we adult humanknow, but also small ants and small animals, are able to memorize a lot ofthings, with minimal brains sizes.
What are currently the facts about the brain ? We will show later that
we know exactly how one neuron works. We have currently mathematicalmodels that can emulate almost perfectly the electrical functioning of a single
neuron.
Even if brains are simply many neurons connected in an intricate way, we
have no theory about how the brain works. If we could understand that, it
will be for sure the most important discovery in human history.
One of the most important motivation to study the brain, is not only thepossible technological applications that we can achieve with artificial-brain
science, but by the fact that up to few years ago brains were basically studied
solely by life-science researchers. These scientists studied the brain from thefunctionality point of view, trying to understand what are the inner functionsa brain, but leaving aside the more complex theoretical and mathematical
reproductions of these functions. For this goal, the goal to achieve understanding and reproduce artificially brain functions, scientists like physicists,
mathematicians and engineers are more qualified, because they are more accustom to construct and simulate physical systems and machines.

1.1. Reasons to study the brain

3

In other words, the brain is a computational machine, so we need also
physicists, mathematicians and engineers to understand it and reveal its secrets. G. Neves (Nature Reviews Neuroscience, 9, 65-75, 2008) states in apaper on Synaptic plasticity that we do not know yet how networks of neurons work. In particular in his study they concentrate on the how networks
encode and represent memory. Even the most simple memory element, thebinary minimal entity yes or no or 0/1 case, is not clear how it is represented
in our brain. For example we may have decide if to go to the cinema tomorrow or not. Once this decision has been taken, the information yes I will go,
or no I will not will be stored somewhere in our brain. Even this very small
information bit is not known how it is stored by the brain.
Buonomaro [11] in Nature Reviews in 2009 have shown that we do nothave yet a computational framework for the brain. This theoretical background remains elusive and we have very few insight on how to proceed.
Some authors like Karl Friston, in Nature Reviews Neuroscience, vol 11,
2010, put forward the problem of a unified brain theory. We need a paradigm
that can help us to understand and decode the brain functioning and unfortunately we do not have that yet.
The study of brain principles is at the beginning of its history. Any small
incremental contribution to the mathematical and theoretical investigationon the brain could be a discovery of paramount importance. There is a great
potential for fundamental discovery in the study of the finding of how thebrain works and how we could cure it. Life-science brain research is also
concerned on why certain functions are working in a certain way and how
this is inserted in the big picture of an organism.
However, physicists, mathematicians and engineering have a different
perspective. The final goal of these researchers is to reproduce the functionsof the brain on machines. The intermediate goal is to establish a theoreticalframework, that is currently almost completely missing. We want to understand what is the mathematics behind brain working, and what are the algorithms running in the brain. Brain for us is a computational machine, so weneed computational theory understanding to solve the mystery of the brain
functioning. The ultimate goal of this is to implement brain algorithms on
machinery, networks and computers in order, not only to understand better
how we work, but also to realize new intelligent devices, much better, fasterand reliable than us.
Currently brain research, from this latter point of view, is beginning to be

4

Chapter 1. Introduction

found on journals like Physical Review E, Physica A, Physical Review Letters, Journal of Physics A to cite a few prominent Physics journals. Other
physic related journals that publish on brain and cognitive research are Institute of Physics Chaos, PLoS one and of course more generalist an highly
cited magazines like Science or Nature

1.2

The neuron and brain

Through human sense (vision, auditory, touch sense, taste, smell, equilibrium) brain gets external information and process the information. The functioning of the brain is very complex, the basic unit is neuron. The humanbrain is made up of billions of neurons. Each neuron has a cell body, an axon,
synapses and many dendrites.
Neurons have a difference in voltage between the inside of the cell and its
exterior environment. Each neuron maintains a potential difference across its
membrane inside is about –70 mV relative to outside. This flux of ions creates
the a difference in the concentration of certain ions, and therefore in voltage,
between its outside and inside. For one neuron system, the most importantions are potassium (K + ), sodium (N a+ ), chlorine (Cl− ), and calcium (Ca2+ ),and each ion channel accepts only one of these ion species.
As in figure 1.1, N a+ and Cl− concentrations are higher outside; K + concentration is higher inside When the neuron is stimulated by chemical or
physical elements, such as receiving a neurotransmitter (chemical synapse)
or electrical synapse (gap junction), some of N a+ channels will open, allowing positively charged N a+ ions to go inside the neuron. This N a+ invasion
depolarizes the neuron membrane potential as in the figure 1.3. If this depolarization continues and the potential surpass a specific threshold value,
an action potential (spike) will be generated. Much more N a+ ion channels
suddenly open, allowing the neuron to flood with positively charged Na+
ions, so the membrane potential will generate a peak. We call the electricalpulse during this period action potential or spike. Action potential is the basisof information transfer and processing between neurons in the brain.

1.3

Neuron Model

In 1952, in order to develop an mathematical model to describe how a neuron’s membrane potential evolves and the variety of action potentials (spikes),scientist A.L. Hodgkin and A.F. Huxley observed the action potentials in the1.3. Neuron Model

5

+

_

Na Cl

K+
Outside0 mV-70 mV

Inside_++K Na ClF IGURE 1.1: This movement of ions will change the membranepotential.
Input

SynapseDendrite

OutputSpike
Cell Body

Axon

Information
Processing
Functional model of a biological neuron

F IGURE 1.2: Function model of a neuron

squid’s giant axon and developed the so called Hodgkin-Huxley equation.
The equation is described as following:
I = Cdv/dt + ḡN a m3 h(V − VN a ) + ḡk n4 (V − Vk ) + ḡL (V − VL ),dm/dt = (Sm (V ) − m)/Tm (V ),dh/dt = (Sh (V ) − h)/Th (V ),(1.1)

dn/dt = (Sn (V ) − n)/Tn (V ).
C = 1µF/cm2 ,ḡN a = 120.0mS/cm2 ,ḡL = 0.3mS/cm2 ,VN a = 55.0mV,

VL = −49.387mVḡk = 36.0mS/cm2
VK = 55.0mV,

6Chapter 1. Introduction

Sodium Channels
openMore Sodium
Channels open

Sodium Channels
Close

Potassium Channels
open

Potassium Channels
Close+30 mV
0 mV
Action Potential (Spike)-70 mVTime (ms)F IGURE 1.3: In this diagram of an action potential, you can
see the membrane potential spike that peaks around +30mv (asyou can see, this neuron’s resting potential is a little higher thanthe usual −70mV ). This kind of electrical pulse is called action
potential (spike)In this equation:
V is membrane potential;
m is N a+ activation (0 < m < 1);
n is the K + activation (0 < n < 1);h is the N a+ inactivation (0 < h < 1);
t is time (ms);
I is current (µA/cm3 ).
As in equation 1.1, Hodgkin-Huxley model has four variables with four
equations. gN a and gK are the electrical conductance of the N a+ and K + ,parameters n, m, h are the gating unit variables to describe the probabilitiesof opening ion gate. Hodgkin-Huxley model succeed for the explanation ofaction potential generation in a neuron.
In 1982, depend on the action potentials of snail neurons, Hindmarsh andRose constructed the Hindarmarsh-Rose neuron model. Now, this model isused to simulate the dynamics of membrane potential observed in one neuron. The system is characterized by three independent variables that represent the membrane potential and the two ion channel currents. The single
neuron model is described by the following differential equation:
32
 ẋ = y − ax + bx − z + Iẏ = c − dx2 − yż = r[s(x − χ) − z]

(1.2)

1.3. Neuron Model7

In this equation, x is time dependent and represents the membrane potential, whereas y and z are often called the spiking and bursting variables, respectively[18]. Those variables are all expressed in arbitrary units and do
not have a direct translation in biologically realistic parameters[27]. And in
the following figure, it shows the simulation of membrane potential phenomenon in one burst spiking neuron.

F IGURE 1.4: Simulation of Hindmarsh-Rose model and it
shows burst spiking activity of one neuron.

Another very efficient model for neuron dynamics simulating is the model
developed by Izhikevich[30]:dv= h1 v 2 + h2 v + h3 − u + I ext + I syn ,dt
du= a(bv − u).
dt
After the neuron emits a spike, action potential will reset:(ifv ≥ 30mV,

then

v←c
u ← u + d.

(1.3)
(1.4)

(1.5)

This is a reduced, bi-dimensional canonical model that reproduces mostprominent behaviour of mammal cortex neurons[13, 61]. The variable u is
the membrane recovery variable that represents the activation of K+ channels or inactivation of Na+ or both the phenomena. Parameter a, b, c, d areused for regulating the types of neurons. In fact, a represents the time scaleof variable u to recover from a firing, and b determines its sensitivity to thefluctuations of variable v. Moreover, c is the after-spike reset value of themembrane potential v and, in the same fashion, u + d is the value that the8Chapter 1. Introduction

recovery variable u resets immediately after-spike. In the equation, I ext denotes a generalized external stimulus and I syn is the synaptic current from aconnected pre-synaptic neuron.
In chapter 1, I studied a two coupled-neurons system, as an element andbuilding block of larger networks. Differently from literature studies, we
simulated two isolated neurons coupled by finely controlled connection strengthparameter w to elucidate its discontinuous effect on the system dynamics(Stich and Verlade used a dynamically changing w whereas Wei et al. tested
w in few discrete points). Three general classes of neurons were simulated:bursting, non-bursting with frequency adaptation and frequency acceleration properties: a chattering (CH, bursting) neuron, a regular spiking (RS,frequency adaptation) neuron and a fast spiking (FS, frequency acceleration)neuron. To account for the representation of a brain system, noise and heterogeneity are incorporated in the equations in a realistic fashion.
In chapter 2, we emulated a triggering event in the brain through a single
active neuron that we call initiator xo and, in controlled noise conditions, westudy the flow of spike activity along the network to evaluate the role of noise
amplitude on the signal propagation. We elucidate in a quantitative manner
the role of noise and we found for the first time a linear relationship between
the amplitude expressed in decibel and the delay interval between initiator
xo first spike and the network activation.

9

Chapter 2

Non-linear and threshold effects ofsynaptic connectivity on thecorrelation parameter of a systemof two coupled neurons.
2.1

Coupled Neurons and synaptic connectivity

The analysis of human brain’s neural network behavior is very important
for the development of next generation intelligent devices[38, 80]. However,practical applications on how to realize pattern recognition or natural language processing is still a quite difficult problem[49, 12, 42, 25, 74] where theunderstanding of actual brain mechanism may be groundbreaking. Key torealize progress is the quantitative understanding of the fundamental mechanism related to learning and memory function in our brain, that is to shedlight on the so called synaptic plasticity, the variation of connection strength
between neurons[44]. In this process, temporal relation between spike trains
can alter the connection strength between neurons and it is the crucial mechanism for learning and memory[68, 69, 3].
Even if the behaviour of a single neuron is known and its representation
with analytical functions possible, the study of two or more connected neurons is difficult because of the extreme non-linearity of neuronal models andthe lack of proper tools. The study of complex networks of neurons is baffling
despite great efforts because we do not have an established theory on how
the Brain encodes, transmits and decodes information. Inspired by the workof Wei, Perez and Cerdeira[78], and the more recent and prominent studyof Stich and Velarde[70], the main motivation of this study is to simplify asmuch as possible the problem and study a two coupled-neurons system, asChapter 2. Non-linear and threshold effects of synaptic connectivity on the10
correlation parameter of a system of two coupled neurons.
an element and building block of larger networks. Differently from literature studies, we simulated two isolated neurons coupled by finely controlledconnection strength parameter w to elucidate its discontinuous effect on thesystem dynamics (Stich and Verlade used a dynamically changing w whereas
Wei et al. tested w in few discrete points). Three general classes of neurons
were simulated: bursting, non-bursting with frequency adaptation and frequency acceleration properties: a chattering (CH, bursting) neuron, a regularspiking (RS, frequency adaptation) neuron and a fast spiking (FS, frequency
acceleration) neuron. To account for the representation of a brain system,
noise and heterogeneity are incorporated in the equations in a realistic fashion.

2.2

Model

A very efficient and widely applicable simulation method is the Izhikevichmodel[35]:dv= h1 v 2 + h2 v + h3 − u + I ext + I syn ,
(2.1)
dt
du
= a(bv − u).
(2.2)
dt
After the neuron emits a spike, action potential will reset:(ifv ≥ 30mV,

then

v←c
u ← u + d.

(2.3)

This is a reduced, bi-dimensional canonical model that reproduces mostprominent behaviour of mammal cortex neurons[13, 61]. In this equation,
h1 = 0.04, h2 = 5 and h3 = 140 are constants scaled by Izhikevich in orderto have the time t in milliseconds and the voltage v in millivolts. The variable u is the membrane recovery variable that represents the activation of K+
channels or inactivation of Na+ or both the phenomena. Parameter a, b, c, d
are used for regulating the types of neurons. In fact, a represents the timescale of variable u to recover from a firing, and b determines its sensitivity to
the fluctuations of variable v. Moreover, c is the after-spike reset value of themembrane potential v and, in the same fashion, u + d is the value that therecovery variable u resets immediately after-spike[35]. In the equation, I ext
denotes a generalized external stimulus and I syn is the synaptic current froma connected pre-synaptic neuron.

2.2. Model11

In this framework, the analytical expression of two neuron connectedwith a strength w is given by the solution of this system of differential equations:(v̇m = h1 vm 2 + h2 vm + h3 − um + Im ext(2.4)
v̇n = h1 vn 2 + h2 vn + h3 − un + In ext + w ∗ (vm − vn ),where we indicated with the index m the pre-synaptic and n the post-synaptic
neurons (the signal transfers from pre-synaptic neuron to post-synaptic neuron). We would like to have completely analytical expression relating thepost-synaptic membrane potential to the pre-synaptic one in function of theconnection strength w. However, the study of this is difficult because of thenon-derivable nature of the threshold in equation (2.3), so numerical simulation is adopted. Figures 2.1, 2.2, 2.3 show the spike train of three classes ofneurons simulated with the model described in details below. The so-called
chattering (CH) neurons exist in superficial layers of the cortex and generate
synchronous oscillations in the visual cortex[24]. They are excitatory neurons
and generate repetitive bursts and short-duration spikes. The regular spiking(RS) neurons are typically spiny pyramidal cells which widely exist in thebrain[41]. These are excitatory neurons and characterized by spike-frequency
adaptation, that is the slow increase in inter-spike interval at constant stimulation.
The third class used is the so-called fast spiking (FS) that have similar highfiring rates, but show the opposite phenomena of spike-frequency acceleration. Some cortical inhibitory inter-neurons belong to this type and are important for the generation of gamma waves in the brain. In general FS neurons and RS neurons exhibit no bursts and inter-spike interval of RS neurons
are broader than CH and FS neurons[55, 23, 33] (for other details see Izhikevich 2007[30]).
We used two configurations to couple the two neurons. In the first configuration, to induce spiking the pre-synaptic neuron m is excited by a constant current I ext [9]. We want to study the bare effect of stimulation, so postsynaptic neuron is not stimulated, but reacts solely to the pre-synaptic signals(configuration A). In another test (configuration B), both of the two neurons
are connected to an external stimulus. These two configurations exist in realistic neural system. A correlation parameter ρ (introduced later in equation(2.9)) was used to estimate the correlation between the spike train of the two
neurons in function of connection strength w.

Chapter 2. Non-linear and threshold effects of synaptic connectivity on the12correlation parameter of a system of two coupled neurons.

CH
RS
FS

a
0.02
0.02
0.1

b
c d
0.2 -50 2
0.2 -65 8
0.2 -65 2

TABLE 2.1: Values of parameters a, b, c, d. These four parameters are used for determining the types of neurons.

The already mentioned figures 2.1, 2.2, 2.3 show the resulting action potential variable v of three types of neurons CH, RS, FS over time t. Simulations have been performed with the model described in equations (2.1), (2.2)
and (2.3) with Iext = 8mA where values of the parameters a, b, c, d are defined as in Table 2.1. No other input signals, internal or external noises are
introduced. T

F IGURE 2.1: Spike train of a CH neuron, the external stimulus
I ext was fixed at 8mA. The value of the membrane potential iscalculated every 0.01 ms for 50000 data points.

.
Neuron m and neuron n are coupled by a single electrical synapse thatmediates the two action potentials vm (t) and vn (t) (see figure 2.5). The connection strength w ranges from 0.0 to 1.0. To avoid artifacts due to the representation of identical neurons and the systematic summation of numericalerrors, we introduced heterogeneity and a small level of noise, elements thatare present in actual biological systems. Heterogeneity was introduced bydefining CH and RS neurons with a and b parameter fixed accordingly totable (2.1) and variable parameters cm and dm . Those are centered in table2(2.1)’s c and d and spread randomly accordingly to the formulas cm = c+15rm
2
and dm = 8 − 6rm, rm is a uniform-distributed random variable on the interval [0,1] (Izhikevich, 2003[35]). In the case of FS neurons, we defined
am = a + 0.08rm and bm = b − 0.05rm and c and d are constant accordingly2.2. Model13

F IGURE 2.2: Spike profile for a RS neuron, the external stimulus
I ext is 8mA, time step 0.01 ms and there are 50000 points in thegraph

F IGURE 2.3: A plot of the action potential of the FS neuron described in the text. The external stimulus I ext is fixed at 8mA.
Time interval is 0.01 ms for a total of 50000 data points.

again to table (2.1). Concerning the external noise term, we consider that in areal biological system, adjacent spiking neurons will generate a noise that has
the statistical distribution of the sum of each of those neurons. The central
limit theorem states that the sum of a great number of arbitrarily distributed
event tends to a Gaussian distribution, so the noise in our simulations is distributed in the same way (Bennet, 2004[8]).
Firstly, we generated a Gaussian distributed random variable X ∼ N (µ, σ 2 ).
Here X is generated internally by a standard numerical library (python numpy),µ is the mean value around which the Gaussian is centered and σ is the corresponding standard deviation.
The differential equations are integrated with Euler method and the index
i represents the ith integration loop. For each simulation conditions a trial
running is done to evaluate the average amplitude of the action potential A,

Chapter 2. Non-linear and threshold effects of synaptic connectivity on the14
correlation parameter of a system of two coupled neurons.
this value is then used as a constant parameter in the actual tests,
A=

Σi vi
.
N

(2.5)

Accordingly to literature[16, 86], a reasonable value for noise level in realA
brains is about 20dB, so SNR(signal-to-noise ratio) was defined as 10 log10 ( Anoise)2 .
The SNR was kept at 20 dB, so Anoise is
A2noise =

A2
SN R

10 10
εi = Anoise Xi ,

,

(2.6)

where εi is the noise variable calculated at each integration loop. In our experiment, we used δt to estimate one step of the time interval. Equation (2.1)
becomes:vi+1 = vi + (h1 vi2 + h2 vi + h3 − ui + Iiext + Iisyn + i )δt.

(2.7)

To avoid artifacts due to unrealistic instantaneous transfer of information, afixed delay τ between two neurons interaction was introduced[39, 20, 28].
The time delay in mammalian neocortex is estimated by several biologicalexperiments[72, 71], here the constant delay was fixed to τ = 0.01 ms[32].
We also fixed the single integration step (index i) δt as 0.01 ms in our Euler
integration method. This will suffice to avoid artifacts and represent existentneuronal delay in a certain amount.
The equations governing the system of two neurons are described as thefollowing:
(

2
ext
˙
v(t)
(t) + εm (t)m = h1 vm (t) + h2 vm (t) + h3 − um (t) + I(2.8)
˙ = h1 vn (t)2 + h2 vn (t) + h3 − un (t) + I ext (t) + w ∗ (vm (t − τ ) − vn (t)) + εn (t).
v(t)
nThe term w ∗ (vm (t − τ ) − vn (t)) represents the synaptic current for the postsynaptic neuron n with the information coming from the pre-synaptic neuron
m retarded by a delay τ . The coefficient w is called here connection strengthand ranges from zero (system decoupled) and one (a perfectly coupled system)[31]. The expressions relative to the recovery variable u and the threshold behavior are calculated accordingly to above equations (2.2) and (2.3) .

2.2. Model

15

Synchronization and phase locking of the system was estimated by the correlation coefficients ρ with the following equations:
Cov(Sm , Sn ).
ρ= pV ar(Sm )V ar(Sn )(2.9)

In this equation, Cov is the covariance between the vectors Sm and Sn and
V ar is the variance relative to the same vectors. To suppress sub-threshold
activity from the correlation calculation, S is an one-dimension vector which
stores over-thresholds intervals counts over a time-based slider window.

F IGURE 2.4: An exemplification of the method used to remove
baseline values from correlation: a sliding window of duration
W T is shifted along time. For every time step in which the action potential is over a threshold (−30mV ), the counter for thatwindow is increased of one unit. An array S is filled with integers that counts the time intervals in which the membranepotential was over the threshold, this happens for each time
windows and then the window is shifted to the next adjacentposition. The red box indicates the window range and the number 5 is just an example that coincide with the number of spikes
visible. Note that, depending of the time step duration, actual
count may differ from the spike count. In our study the window width is fixed to 3 msec and time step to 0.1msec.

If ∆ is the duration of the window (in figure 2.4, ∆ = W T ), T is the lengthof a experiment[17], the spike train of membrane potential v is converted to
the following binary data train M P :(
M P (t) =

1 if
0 if

v(t) ≥ −30mVv(t) < −30mV.

(2.10)The threshold was fixed as −30 mV to eliminate any sub-threshold activity in the correlation calculation. Any value over −50mV or less than 20mV ,for example, could have been a good choice too; this particular thresholdvalue was chosen in order to have an accurate representation of the spike

Chapter 2. Non-linear and threshold effects of synaptic connectivity on the16
correlation parameter of a system of two coupled neurons.

(a)

(b)

F IGURE 2.5: The two configurations used in our simulations. In
panel (a) the pre-synaptic neuron m receives the external stimulus of Im = 8 mA. In panel (b) both the neurons receive an
identical stimulus of I = 8 mA. The synaptic current from neuron m is mediated by the connection strength parameter w.

event with our type of neurons. See figures 2.1, 2.2 and 2.3 for the exact spike
profiles of them, see also figure 2.4 for a sketch of the method. According to
the above, we determine the counts from time t0 to t + ∆. Then the time windows is shifted along the time axis and the counts are stored into the vector
S as below:

S(t) = Σt+∆
M P (t),tt ∈ {∆, 2∆, 3∆..., T − ∆}.

(2.11)In this formula, ∆ is the length of time windows and T is the length of simulation time, ∆ is fixed as 3 ms in all our tests.

2.3

Results and discussion

We used two configurations to couple the two neurons. As shown in figure
2.5(a) and figure 2.5(b), in Configuration A neuron m has an external stimulusthat forces it to spike regularly, neuron n spike trains are due solely to thestimulation received from m. In Configuration B instead, both of two neurons
receive external stimuli. This configuration is used to test if effects observedin A are due to neuron m drive, or instead due to other implicit characteristicsof neuron n.
First of all, we observed how the action potential changed when the connection strength w between the neurons is varied in Configuration A. In figure 2.6, we simulated two CH neurons and varied the connection strength
w accordingly to configuration A. At first, low values of w do not induce
spiking activity on n until values of about w = 0.15. Beyond this n generates
spikes and the correlation of the system increases until the two neurons reach2.3. Results and discussion

17very high synchronization. At values higher than w = 0.45, neuron m is giving the pace to its coupled n neuron that reaches locking, with correlation ρof about 0.8. Even if this is the most simple network realizable, we can saythat m acts as a sort of pacemaker of the system.
Interestingly, lag-synchronization is noticeable in the third plot of figure
2.6, where the two spikes trains appear to be locked, but with a shift in time(lag) that keeps the overall correlation coefficient low (ρ = 0.156). This fact
is important[62, 10], however here we do not want to investigate on lag synchronization for reason of uniformity and clarity in respect to the correlationand connection strength dependence.

F IGURE 2.6: The figure shows spike trains of two coupled CH
neurons. Only the pre-synaptic neuron is connected to an external stimulus of I = 8 mA. When w is increased, neuron n starts
to generate spikes. The two neurons reveal very high synchronization when w reaches a maximum value.

From the dynamical system point of view, this means that for small w
the pulses that neuron n receives from m, act as perturbation, too small in
amplitude to alter the state variables of the neuron that remains around inits equilibrium point at −65mV . This is represented in the phase diagram of
figure 2.7(a) where vm potential is seen to range from −60mV to +20mV flat
against an almost constant vn that drift around few mV interval. When w
grows over a certain threshold, these perturbations raise in amplitude until
the equilibrium is broken and spiking begins. Moreover, the phase diagramof figure 2.7(d) shows a bifurcated shape for ρ = 0.74 indicating that thephases are locked at two different states in the two neurons (neuron n spikes

Chapter 2. Non-linear and threshold effects of synaptic connectivity on the18
correlation parameter of a system of two coupled neurons.
when m is about −50mV , whereas m spike seems to cover a wider span of npotentials, ranging from −70 mV to -40 mV ).
(a)

(b)

(c)

(d)

F IGURE 2.7: Strong non-linearity in w observed in the two neurons phase diagram. In this group of figures, the vertical axis is
the action potential of the post-synaptic neuron n and the horizontal axis that of the pre-synaptic m, both the neurons are oftype CH. In these simulations only neuron m is connected to theexternal stimulus to induce regular beating. In (a) the connection parameter w is very low ( w = 0.01), in (b) ten times bigger(w = 0.1). Nevertheless in these two plots, the two neurons are
not synchronized, their correlation coefficient ρ is negligible. In
panel (c) and (d) the activity of the pre-synaptic neuron induce
the other to spike in levels of synchronization proportional to w.
In (c) w = 0.6 and ρ = 0.66, in (d) the connection is maximum
at w = 1.0 and the correlation reaches ρ = 0.74.

Most interestingly, the behaviours observed indicate a strongly non-linear
dependence of correlation against connection strength. In other words there2.3. Results and discussion

19

is not a linear or smooth transition to the two neurons system synchronization, the profile of ρ against w shown in figure 2.8 reveals a net threshold behaviour that is reproduced at different values for all types of neurons tested.

F IGURE 2.8: An example of dependence between correlation ρand connection strength w between neuron m and n. A clear
threshold and plateau behaviour is noticeable in most cases(this plot represent the interaction of two Chattering neurons(CH-CH system), the post-synaptic one has no external input).
The dots are the correlation coefficients data, a sigmoid function is fitted along them to emphasize the non-linear, step-likebehaviour.

Our simulations show that the electrical voltage coupling factor w dependence is non-linear and discontinuous, as shown for example shown later infigure 2.15 (a) where several neurons types exhibit abrupt rise in correlation
with increased coupling value w. Nevertheless, data can be fitted with a sigmoid with three degree of freedom represented by three parameters α, β and
w0 in the function:
Σ(w) =

α.
1 + exp−β(w−w0 )(2.12)

where w0 represents the center of the sigmoid, in other words it represents the
position of the threshold, whereas α and β represent the height and sharpnessof the curve respectively. In table 2.2 we summarize the fitting parameter for
all cases discussed.
The inter-spike interval (ISI) is key to several models of information coding in the nervous system[88] and can be used in a method to reconstruct thesystem dynamics[64]. The non-linear dependence of w on synchronizationis reflected on a similar strong non-linearity in the inter-spike dynamics. We
record the inter-spike interval mean µISI and standard deviation σISI for thesystem for the two different configurations. Using vector P T to record thespiking time of the post-synaptic neuron, fixing the spiking time index as j,Chapter 2. Non-linear and threshold effects of synaptic connectivity on the20
correlation parameter of a system of two coupled neurons.

CH-CH
RS-RS
FS-FS
CH-FS
CH-RS
FS-RS
FS-CH
RS-CH
RS-FS

αA
0.67
0.65
0.75
0.70
0.58
0.25
0.21
0.79
0.77

w0A
0.25
0.68
0.86
0.30
0.24
0.85
0.85
0.65
0.63

αB
0.90
0.97
0.95
0.87
0.64
0.29
0.77
0.82
1.03

w0B
-0.01
-0.07
0.037
0.26
0.21
0.62
0.57
0.16
0.44

TABLE 2.2: The fitting parameters α and w0 for the sigmoid
interpolation of the correlation function ρ versus connection
strength w for various neuron types combinations (see CH-CH
example in figure 2.8). Interpolation has been done with formula (2.12), α represents the height of the sigmoid plateau, andw0 the center of the curve. The apex A or B of the parameters
represent the two connection configurations in figure 2.5.

then the inter-spike element ISI j can be represented as follows:
ISI j ∈ {P T (j + 1) − P T (j) | j ∈ (1, 2...J − 1)},(2.13)

here J is the total number of spikes. According to the array of inter-spikeintervals ISI, we calculated the mean µISI and standard deviation σISI asfollows:
Σj1 ISIj,js
Σj1 (ISIj2 − µ2ISI )σISI =.
jµISI =(2.14)We show in figure 2.9 and 2.10 the average µISI and the standard deviation
σISI of the post-synaptic inter-spike interval. All possible neurons types andconfigurations were examined. In the neurons of configuration A (Fig 5(a)),
inter-spike activity is absent for low values of w (for clarity in the plot ISI
value is set to zero), then begins at around w = 0.1 for most cases. The results
also show higher values about w = 0.4 or w = 0.5 in the cases of RS-RS or
FS-FS respectively. After the initial spiking, in all cases the average interspike interval decreases and converges to the 2-4 msec range. This meansthat the spiking rates and the spiking intervals of post-synaptic neuron trend
to a stable plateau. Interspike intervals become almost independent to theconnection strength between the two neurons.

2.3. Results and discussion

21

F IGURE 2.9: The y-axis show the average inter-spike intervalµIs of the post-synaptic neuron n, the connection between thetwo neurons is configured as in figure 2.5(a), no external excitation for post-synaptic neuron. And each configuration shownis the average of 10 experiments, inter-spike variability is in theorder of 20%. The figure show that RS neuron and FS neuron need more connection strength to spike compared to othertypes of neurons. The inter-spike interval was measured in ms.
For low values of w, the post-synaptic neuron do not spike (interspike interval is infinite). For reason of clarity, we set the ISI
value as zero in these cases.

F IGURE 2.10: The y-axis shows the standard deviation σIs of
Is. Each configuration shown is the average of 10 experiments.
The inter-spike interval was measured in ms. When w is in therange of 0 0.1, the post-synaptic neuron do not spike and the
ISI value is set to zero.

We now connect an external input to the post-synaptic neuron n to understand the aspect of the influence of external signals on the phase-locking
process. In Figure 2.11 we show the time dependent plot of the two neurons
superimposed as in figure 2.6. Now that both neurons are spiking, phaselocking is reached sooner than before, the correlation dependence with w

Chapter 2. Non-linear and threshold effects of synaptic connectivity on the22
correlation parameter of a system of two coupled neurons.
grows faster and the onset differentiation between neuron types combinations is minimal (figure 2.15 (b)). In this case, the phase space of the system,
shown in figure 2.12, suggests a very quick synchronization between the two
neurons. Also, the system appears to be extremely sensitive to the connection parameter w, since even at w = 0.01 already a certain level of correlation
is observable in the phase space, inset (a) of figure 2.12.

F IGURE 2.11: Two CH-Ch neurons are connected to an identicalexternal current and reach high level synchronization level for
w = 1.0.

This precise phase locking has been for example observed in couple ofneurons by Stich and Verlade[70] using a similar coupling topology. The
main difference of our tests with that study is that synaptic conductance was
freely changing dynamically accordingly to predetermined rules, whereas
in this research we control it systematically to elucidate its influence in thesystem in detail .
The figures 2.13 and 2.14 show the average inter-spike interval µISI andthe corresponding standard deviation σISI . In the figures, both neuron n andneuron m had a stimulus of 8 mA. In this conditions the inter-spike intervalof most of pairs do not change appreciably. However, an exception is the FSRS system, where the properties of frequency acceleration and adaptation ofthese neurons come in to play [30, 48, 22].
Neurons belonging to the same class(CH, RS, FS) exhibit random differences in the four Izhikevich model parameters a, b, c and d to emulate natural heterogeneity and to avoid possible accumulation of systematic artifacts
or errors due to repeated computation on identical systems. For the same

2.3. Results and discussion

23

(a)

(b)

(c)

(d)

F IGURE 2.12: Phase space of the dynamics of the two neurons
connected as in configuration B if m and n are both CH neurons. Vertical axis shows the action potential of neuron n andthe horizontal axis the one of neuron n. In (a) the connection
strength w is 0.01 increasing w to 0.1 in (b) correlation goes to0.59 and then to 0.96 and 0.98 for w = 0.6 and w = 1.0 in (c) and(d) respectively. Compared with tests of figure 2.7, the synchronization reached higher levels and without discontinuity in w.
This plot is made by averaging ten different tests.

reasons, simulations have been run ten times and their values were averaged, each dot in the plots of figure 2.15 (a) and (b) represents 10 different
experiments run with different random seeds. In these plots, the connection
strength w was varied from zero to 1.0 in steps of 0.05, except for the initial range 0 to 0.1 that is divided in more detailed 12 steps of about 0.008 forbetter representing initial threshold crossing transition.
As demonstrated in previous studies[26, 54], low and high frequencies
inputs will influence the synchronization. Our results for the configuration

Chapter 2. Non-linear and threshold effects of synaptic connectivity on the24
correlation parameter of a system of two coupled neurons.

F IGURE 2.13: The y-axis shows the average of inter-spike interval µISI of the post-synaptic neuron n. Each data shown is theaverage of 10 experiments. The inter-spike interval was measured in ms.

F IGURE 2.14: The y-axis shows the standard deviation of theinter-spike interval of the system σISI . Each point shown represents the average of 10 experiments run with different random
seeds. The inter-spike interval was measured in msec.

A (see figure 2.15 (a)), show that if the input spike train has high frequency
components (when the pre-synaptic neuron is of type FS), the system synchronize and lock with difficulties compared with the other cases with presynaptic CH or RS. Noticeably, all curves related to a pre-synaptic FS neuron are grouped at higher connection strength w with threshold of aboutw = 0.5 instead of lower w for the other cases. Also notice that pre-synaptic
RS neurons are found in adjacent position for intermediate w and finally presynaptic CH neurons instead appear to be the best pace maker locking sooner
at lower connection strength of about w = 0.1. This is suggesting that the fast
firing of multiple stimuli in the burst has better chances to find the appropriate time window to make the post-synaptic neuron internal state to resonate
into a synchronized spike[2].

2.3. Results and discussion

A

25

B

F IGURE 2.15: Configuration A (figure (a)): The behaviour ofthe correlation coefficient ρ in function of correlation strength
w. In this plot only the pre-synaptic neuron m has an external
continuous stimulus. Symbols like CH-RS represent the type ofneuron acting on which type of neuron with the pre-synaptic
one on the left and the post-synaptic on the right. Every dot
in the plot represents the average of ten different test simulations. Each of these simulations is done over 100000 points andrun with small random fluctuations on the neuron parameters
to avoid artifacts. Configuration B (figure (b)): A plot of correlation coefficient against connection strength taken in similar
condition as figure 2.15 (a) and figure 2.13. In this case, both of
the neurons have an external continuous stimulus. These twoplot show clear differentiation between type of neurons. Notably CH types appear to be the best to induce synchrony at
low w, whereas FS-RS in all cases show lower level of correlation. Also, FS-RS configuration appear to behave uniquely also
in the plots of figures 2.13 and 2.14, where ISI changed significantly with the variation of connection strength.

In configuration B, as shown in figure 2.15 (b), both of pre-synaptic andpost-synaptic neurons are stimulated by a constant input I = 8 mA. Here,differently from above, if the pre-synaptic and post-synaptic neurons are of
the same kind, they synchronize rapidly at very low values of w. Similarly
to configuration A, pre-synaptic FS class neurons have difficulties to lock thesystem and this results again in a higher threshold value. The threshold valueof w for pre-synaptic CH class neurons and RS class neurons is about 0.2 but
in the case of pre-synaptic FS class neurons, the threshold value moved to
0.5.
The FS-CH case is a noticeable because after an initial growth and plateau
tendency at about w = 0.4 correlation drops again until a second rise at
w = 0.5 and a similar behavior is observed also for the RS-FS couple. Thissuggests that the FS prominent feature of spike frequency adaptation, theChapter 2. Non-linear and threshold effects of synaptic connectivity on the26
correlation parameter of a system of two coupled neurons.
tendency of neurons to relax its inter-spike time so decreasing its instantaneous spiking frequency, hinder conspicuously the phase lock process in astrongly non linear fashion compared to other configurations [15, 30].

27

Chapter 3

Noise influence on spike activation
in a Hindmarsh-Rose small-world
neural network
3.1

Introduction

The control and flow of information in Brains is fundamental for the understanding of high-level processes like attention, visual perception and othercomplex functions[79, 63]. Despite enormous system complexity, it has beenfound that information can be sourced by single neurons even in major sensory pathways[58]. These distinct units can trigger chain of events that contribute to complex perception episodes or high level motor responses. Thisis known to happen for example with features detectors neurons[43] in lower
animals, and similar phenomena are observed in mammal’s neurons of higher
hierarchical position in the visual system, those that respond to complex patterns and activate elaborate responses[50].
Despite the great volume of experimental facts that points toward noise
as a contributing factor in signal transmission in brains[1, 29, 46] or generally
in non-linear networks [46], we do not have yet a universally accepted theoretical framework to quantitatively evaluate the effects of noise to neurons
operations and to the higher level network functions.
Here we want to contribute to this problem with a study in which we emulate a triggering event in the brain through a single active neuron that wecall initiator xo and, in controlled noise conditions, we study the flow of spike
activity along the network to evaluate the role of noise amplitude on the signal propagation. In the first part of this study, we elucidate in a quantitative
manner the role of noise and we found for the first time a linear relationshipbetween the amplitude expressed in decibel and the delay interval between
initiator xo first spike and the network activation. Moreover, studying the28

Chapter 3. Noise influence on spike activation in a Hindmarsh-Rosesmall-world neural network

dependence with network size, we found that as the network grows in dimensions, the spiking activity starts earlier until the role of the initiator iscompletely suppressed beyond about N = 100 neurons. In the second part of
the research, we characterize the network especially focusing on the spectral
characteristics, going in details on its frequency dependence with noise andsize. It is found that high levels of noise introduce chaos in the network andthat, again, network size seems to replace the function of external noise forsimulations with constant noise level and variable network sizes.

3.2

Models and methods

We used a Newman−Watts small world network to simulate the neural network. A realistic neural network is neither regular nor completely random[37]and this network has both deterministic and random properties[52]. Simulation of relatively small artificial networks can be a good metaphor for real
brains as, for example, in 2003 Izhikevich demonstrated that a network ofsimulated spiking neurons exhibited collective waves and frequencies [34] ina range similar to the human brain.
To simulate the neurons we used the Hindmarsh−Rose model[7, 66] changing the intensity of noise and the network size to elucidate the role of spiking
frequencies in function of various parameters. Even though precise arrangement of neural connections in real brains is not known, Small-World networkmodels are widely used to simulate known statistical properties of Brain’s
neural connections, reproducing high clustering coefficient and low shortest
path[53, 6, 85, 51]. The network random connection probability of the SmallWorld structure has been kept to p = 0.4 as a reasonable value with enoughrandom connections to induce fast diffusion of spikes, but still small enoughto represent real biological systems. This network structure and p values are
considered reasonable on numerous literature studies, for example Basset[6]where regions of Human brain are found to have clustering coefficient as low
as 0.14 equivalent to p > 0.4, or also Yan H.Z[87] and Ozer[56] for values of
0.1 < p < 0.6 in small-world neural networks. To measure the global behavior of the network we monitored the integral of the total membrane potential. This is considered to be analogous to the electroencephalogram (EEG)
in a real brain. By means of Fourier analysis, we found that this collective
signal has distincts frequency peaks which are compatible with values observed in biological systems. This fact is not trivial since neuronal activities
of a complex network are very difficult to predict analytically.

3.2. Models and methods

29

The Hindmarsh−Rose[27] system is characterized by three independent
variables that represent the membrane potential and the two ion channel currents. A single neuron model is described by the following differential equation:
32
 ẋ = y − ax + bx − z + Iẏ = c − dx2 − yż = r[s(x − χ) − z].

(3.1)

F IGURE 3.1: Simulation of a Hindmarsh−Rose model for a single neuron not affected by noise or connections. This shows the
dynamical behaviour of the membrane potential. The variable
x(t) represents the difference of voltage between extracellularand intracellular potentials. Spiking Variable y(t) instead describes the rate of change of the fast ion channels (sodium ion,
potassium ion, etc.) and z(t) has similar meaning for the slow
ion channels (calcium).

In this equation, x is time dependent and represents the membrane potential, whereas y and z are often called the spiking and bursting variables,
respectively[18]. Those variables are all expressed in arbitrary units and do
not have a direct translation in biologically realistic parameters[27].
In figure 3.1, it is shown a plot of the three variable phase cycle that induces the spike. The choice of the eight parameters in the model results in thekind of neuron simulated. Here we fixed them as a = 3, b = 3, c = 1, d = 5,
s = 4, r = 0.00, χ = −1.6 determined in order to produce neuronal bursting
dynamics comparable to realistic neurons of "chattering" type. The initiator
neuron xo introduced above has a continuous input stimulus of I = 3 (a.u.)and starts at t = 0 with the initial values x(0)=0.3, y(0)=0.3, z(0)=3. Other
authors[65] use similar values of input currents, ranging from about I = 2

30

Chapter 3. Noise influence on spike activation in a Hindmarsh-Rosesmall-world neural network

F IGURE 3.2: A sketch of the network structure. Each neuron isrepresented by a circle and simulated by the Hindmarsh−Rosemodel. The lines represent neural connections to neighbors
neurons. The random connections that cross the circle arethose due to the p parameter of the Small-World structure. The
grayed neuron indicated by the symbol xo a special initiator thathas a constant external current input Io and it is used to stimulate the whole network and to study the diffusion of spikes inthe system in presence of noise. In our simulations the connections are changed randomly at every run, however the SmallWorld structural parameters are maintained constant. The crossconnections parameter p is kept at p = 0.4 (for image clarity inthis sketch p is lower). In some experiments the number of neurons is also varied.

to I = 10. Other neurons initial current values are set to zero. This neuron is called initiator because of its role of single source of information in the
network. In other terms, the initiator xo is the only neuron that provides stimulated spikes and those are thought as the single source of information in the
network. That is, in absence of external noise, if other neurons are spiking
it is because directly or indirectly they receive the information to do so fromxo . Our study investigates throughout the role of noise in this context. The
entire network is connected as a Small-World system, without exception of xothat receives input from the neighboring neurons accordingly to the network
structure, see Figure 3.2 for a sketch of the configuration used.
The integration of the differential equation (3.1) is done with an Euler
method, a time consuming but straightforward integration method. Timestep is set to ∆t = 0.01 ms [84] and has been verified to not introduce instabilities (slight increase of ∆t does not provoke appreciable variation in the
network response)
Figure 3.3 shows the Fourier transformation of the membrane signal of a

3.2. Models and methods

31

single neuron inserted in a p = 0.4 network. The network itself is intentionally chosen to be smaller (N = 10) and with the minimal noise level (signal
to noise ratio dB = 40) in order to be an example of frequency signature of
the neuron response. The neuron position is chosen at random far from theactive xo . Very interestingly, we can observe the emergence of peaks at 10
Hz, these frequencies in the averages membrane potential can be though as
an analogous of α waves in real brains. This shows biological plausibility insimulations as already established for example by Izhikevich. [35, 57].
50

Neuron Index

40

30

20

10

0
0

100

200

300

400

500

600

time(ms)

F IGURE 3.3: The Fourier transformation of the membrane signal of a single neuron inserted in a p = 0.4 network of N = 10
and with signal to noise ratio dB = 40. This is an example offrequency signature of a single neuronal response. The inset
shows the actual spike plot where the Fourier transformation
was calculated from, the horizontal axis represents 1000 msec
of time. The bar at x = 0.8 indicates the threshold used to separate the baseline activity from the action potential. The neuron
is chosen at random and it is not connected to the initiator neuron xo .

To generate the network, firstly, we create a regular network with a ringover n vertices. Each vertex stands for a neuron. Then, every vertex in thering is connected with its nearest neighbors at both sides. In all our tests,
k = 1, this means that each neuron is connected with two neighbors, one on
its left and the other on its right. We say that the network is of dimension
k = 1 (if k = 2 each neuron has four neighbors, two on the left and twoon the right). Starting from this regular network structure, we create random
shortcuts by adding connections pointing to other neurons chosen at randomwith probability p (p = 0.4 in our case). Each vertex vi (i ∈ {1, 2, 3...N }) can
be connected with any other with probability p. For vertices vi and vj , a
uniform distribution random number in the range zero to one is generated

32

Chapter 3. Noise influence on spike activation in a Hindmarsh-Rosesmall-world neural networkand compared with p. If the number is lower than p the connection betweenvi and vj is created, otherwise not. A similar test is done to the adjacentneuron until all neurons are examined. Neurons have no connections withthemselves and two neurons can be coupled by a unique segment, a single
neuron can receive more than one connections from different neurons[77].
A good representation of biologically plausible noise is controversial, however since the central limit theorem guarantees that a good number or arbitrary distributes sources converge to a Gaussian signal, we opted for thistype of noise[5, 81, 14]. The intensity is modulated using different values ofsignal-to-noise ratio (SNR) defined in decibel as dB = 10 log10 ( AAns )2 . Where
As is the amplitude of the signal and An that of the noise. We use this definition because is standard and logarithmic, however we have to empathizethat bigger values of dB mean a less noisy network.
The random variable is generated with this formula :−(X−µ)2
1
φ(X, µ, σ) = √ e( 2σ2 ) .
σ 2π

(3.2)

Here X is a flat pseudo-random value generated internally by the numericallibrary numpy[36, 75], µ is the mean value around which the Gaussian is centered and σ is the corresponding standard deviation. In our experiment, µwas fixed as 0 and σ as 1[82][19]. We use1
Σ i xi ,
N
r
S
η=
,
dB/10
10
ε = ηφ(X, µ, σ).

S=

(3.3)

N is the total number of neurons, xi the membrane potential of neuroni, η the noise amplitude and ε the final Gaussian noise random variable. It
results to be distributed along a Gaussian curve, with intensity proportionalto η that contains the decibel parameter that is varied in the tests[40]. The
intensity of noise is calculated at each time loop, so several neurons receive
the same noise intensity for the same time step. This is an approximation ofa realistic network where noise do not have spatial specificity.
The differential equation that describes the time evolution of a single neuron xi in the system is:3.3. Results

33


PN
3
2

 ẋi = yi − axi + bxi − zi + εi + gi j=1 aij (xi − xj ) + Ioẏi = c − dx2i − yiżi = r[s(xi − χ) − zi ].

(3.4)

This formula is equivalent to eq. 3.1 but extended for a system of N neurons,variables and parameter have the same meaning explained previously. The
index i represent each of the N neurons in the network. The parameter εiis the Gaussian noise of neuron i, updated at each time loop. To computethe input current that each neuron receives, we calculate gi ΣN
j=1 aij (xi − xj ).
The sum is mediated by an adjacency matrix aij that stores the connection
between the vertices. In it, aij is 1 if there is a connection between the neuron
i and j, otherwise it is 0. xi and xj are the membrane potential of neuron iand j. gi is the coupling strength[76, 4] that is normalized to the number ofconnections of neuron i. In this way:1
gi = PN

j=1 aij.

(3.5)

In this formula, N is the number of neurons in the network. If we do not
impose this normalization, the network will have an unbalanced influence of
signals coming from the neighbors neurons depending on connections number[83, 69].
As stated above, in our simulation a neuron xo receives a specific continuous current stimulus Io that is set to a value of 3 (a.u.) in all our tests. So, inthe first equation of system (3.4) if xi ≡ xo , Io = 3, in all other cases Io = 0.

3.3

Results

We verified that without the contribution of the special initiator neuron xo ,the network exhibits no spiking activity and membrane signal settle to a baseline value.
On the other hand, if neuron xo receives a constant input Io it becomes
active and drives its connected neighbors to spike, and those will induce
spiking on their neighbors. The information will propagate until the wholenetwork will be spiking in a random but stable manner.
In the following tests, corresponding to figures from 3.4 to 3.7, a networkof N =48 neurons was used with connections as in table 3.1, i and j are
the index of neurons. Value 1 in each grid indicates neuron i is wired with

34

Chapter 3. Noise influence on spike activation in a Hindmarsh-Rosesmall-world neural network

neuron j (aij = 1). Empty grid means there is no connection between neuron
i and neuron j (aij = 0).
Neuron i, from 0 to 48 (totally 48 neurons)
0

0
1

10

5
1
1

1
1

1

5

1

1

1

1

1
1

1

10

1
1

1
1

1
1

1
1

1
1

1
1

1

1
1
1

1
1

1

1
1

1
1
1

1
1

1

20

1
1

1
1

1
1

1
1
1

1
1

1
1

1

1

25
30

1

1

1

1
1

1
1

1

1

1
1
1

1
1

1
1

1
1

1
1

1

1

1

35

1
1

1

1

1

1

1

1
1

1

1

1
1

1
1

1
1

1

40

1
1

45

1
1
1

1

1

1
1

1

1

1

1
1
1

1

1

1

1

1

45

40

35
1

1

1

15

30

1

1

Neuron j, from 0 to 48 (totally 48 neurons)

25

20

15
1

1
1
1
1

1
1

1
1

1
1

1
1

1
1

1

1
1

1

TABLE 3.1: An example of network connections. Values of aij
represent the connection strength within the 48 neurons smallworld network. Connection strength can be only of value 1(connected) or zero (not-connected). Zero values are not shownfor clarity).

Noiseless response of such a system is shown in the spike map of figure(3.4). This map is generated using an arbitrary threshold to separate basal
random signal from the spikes. After few tests to evaluate the membranepotential range, the threshold was chosen as x=0.8, see inset of figure (3.3).
When the membrane potential exceeds this value, we assume that the neuron
is firing[59] and plot the mark that is shown in figure (3.4) and the following
spike maps. To evaluate the behavior of the network as a whole, we integrate
the membrane potential for the entire population of neurons. We use thisequation to calculate the parameter E(t):PNi=1 xi (t)

.
(3.6)
N
In figure 3.5 we show the output of the network measured as the average
membrane potential calculated with eq. (3.6) for a network of 48 neuros,
E(t) =

35

Neuron Index

3.3. Results

Time (ms)F IGURE 3.4: The spike map of a noiseless small-world networkof 48 Hindemarsh−Rose neurons, a sketch of it is in figure 3.2.
The neuron indicated as number 0 is a special one that is connected to an external signal of I = 3 (a.u.) constant input.
It functions as an initiator and after a delay of time, provokes
the spiking of the whole network. The same experiment reproduced with identical parameters but with no external input(I = 0), produce no spiking everywhere in the network (blank
result not shown).

p = 0.4 and k = 1. Clearly as the noise increases, the initial transitional
behavior get shorter and shorter. This suggests two things, noise influences
the average baseline of neuronal response and it affects also the spike rate of
the neuronal system.
Figure 3.6 shows the raster plot for a neuron network with 48 neurons,
k = 1 and p = 0.4 as before, for different noise levels. Again, the initiator
neuron xo is spiking since the beginning, whereas the rest of the network isquiescent and responds with a variable delay that depends on noise level.
The signal to noise ratio influences strongly the initiation of secondary
spiking along the network. Calling Ss the time interval by which secondary
spikes are activated, we can interpolate our results with a liner model
Ss = δε + α.
where δ = 14 msec per decibel, ε is the noise level in dB and α is an offset, thelinear regression coefficient is better than 0.9. The influence of the initiator x0
is negligible and the linear dependence is clear. See figure (3.7) for graphicalrepresentation of this with a test with several level of noises. This formulacannot have a general meaning because it is bind to this specific network,
however it shows a quantitative relation to modulate transfer of information

Chapter 3. Noise influence on spike activation in a Hindmarsh-Rosesmall-world neural network

Membrane Potential

36

Time

Neuron Index

Neuron Index

F IGURE 3.5: The behaviour of the neuron population membrane potential in function of noise level. Noise seems to help
the diffusion of spikes along the network. Horizontal axis spansover 1000 ms of simulation calculated in 100,000 steps of 0.01
msec each. The vertical axis is the voltage in arbitrary units.
Each simulation is taken on a network of N = 48 neurons andaveraged on 8 different experiments done with same parameters but different random generator seeds. As the signal tonoise ratio grows (less external noise) the initial transition time
gets longer indicating a worse diffusion of information acrossthe network.

Time (ms)Time (ms)F IGURE 3.6: The spike maps of a network with N = 48 neurons
in different noise conditions. Signal to noise ratio in each plotis (a) dB = 35, (b) dB = 31, (c) dB = 26 and (d) dB = 18. The
signal to noise ratio influences strongly the initiation of regularspiking along the network. On the vertical axis we have theneuron number, neuron indexed as zero is the special initiator
neuron xo . Notice that a better signal to noise ratio means lessexternal noise. Intermediate dB values were calculated but notshown.

with noise intensity in this case.
A network with identical parameters as above was tested in similar way

3.3. Results

37

Io =3
Io =0
Linear fit (Io =3)

F IGURE 3.7: The comparative plot of the delay by which thespiking activity starts in a network with and without an active
initiator xo . Round dots represent the delay with an active (I =3) initiator and the square an inactive one. The influence of
xo is negligible and linear dependence is clear. The red line
represents a linear interpolation of r > 0.9 and inclination δ =14 msec per dB.

but changing the size of the network instead of noise (noise level was kept
constant at 20 dB). As shown in figure 3.8 for small network sizes the initiator neuron xo makes the spread of spikes quicker, but it does not seem toinfluence strongly the transitional delay period after the network grows morethan about N = 100 neurons.

o
o
o
o

Number of neuron s

F IGURE 3.8: The comparative plot between networks with andwithout an active spike initiator xo , in the same fashion as infigure 3.7, but when the network size is changed. For small
network sizes of less than about N = 100 neurons the initiator makes a difference and induce quicker propagation of spike
activity. The noise level is kept to 20 dB in all the tests.

38

Chapter 3. Noise influence on spike activation in a Hindmarsh-Rosesmall-world neural network

To analyze the influence of noise to the collective frequencies of the network, we studied the Fourier spectrum of the variable E in (3.6). The Fourier
spectrum peaks, if present, are very important for recognizing rhythms andregularities in the entire network that mimic the biological phenomena ofslow and fast waves in real brains.

F IGURE 3.9: A plot of the fast Fourier transformation of themembrane potential averaged over the whole network. The
curves represent the frequency spectrum of it averaged over thewhole network. Each line represents a simulation with different level of noises. The horizontal axis is in Hz and the verticalin arbitrary units. The latter has a relative meaning since plots
are shifted vertically for clarity. These show how the network
characteristic frequencies are influenced by different noise levels. The simulations are produced with a total of N = 48 neurons. The small world network is characterized by a connection
probability p = 0.4 and dimension k = 1 and a spiking initiator
neuron xo with input I = 3 (a.u.). The plot considers only thefrequencies at regime. That is, the simulation is for 1000 msec,
the first 300 msec of data are not considered, in order to avoid
low frequencies due to the initial baseline variations. Each plotis the average of 8 identical simulations executed with different
random number seeds. The influence of an active (I = 3) or
passive (I = 0) neuron x_o is null since we obtain equivalent
results in both cases.

The Fourier transformation presents evident peaks at strong signal tonoise ratio (top curve, dB = 35), whereas the more prominent peaks at α andβ seem to decay with increasing noise[35], see figure 3.9. The Fourier transform data are calculated over a simulation period of 1000 msec, cutting off
the first 300 msec to avoid including slow frequencies due to the initial transitional phase discussed above. Interestingly the prominent β peak seems

3.3. Results

39

to shift to the right with noise intensity, a very weak effect that is, however,difficult to prove with mathematical means. We studied the total network
power average among 5 different frequency bands, θ, α, β, γ1 and γ2 . We calculate the Fourier spectrum of a 1000 millisecond simulation as above, thenwe sum up the power of those five frequency bands with this formula
1 7
Σ Ψ(f )df,
Nθ f =4
1 14
Σ Ψ(f )df,
α=Nα f =7
1 30
β=Σ
Ψ(f )df,
Nβ f =14
1 40
Σ
Ψ(f )df,
γ1 =Nγ1 f =30
1 70
γ2 =
Σ
Ψ(f )df.
Nγ2 f =50
θ=

(3.7)

Here Ψ represents the vector that contains the module of the fast Fouriertransformation array. The elements corresponding to frequencies within theband are added up (θ from 4 to 7 Hz, α 7 to 14 Hz, β 14 to 30 Hz, γ1 30 to
40 Hz and γ2 50 to 70 Hz). Each band has different number of elements, so
the value is normalized to the various Nθ , Nα etc., depending on how many
elements each band has. Then we plot the result against the noise, figure 3.10.
In this plot we see that all prominent peaks decrease with noise intensity. This happens because low level of noise produces the transitional phenomena shown in figure 3.6 and 3.5, during which few neurons are spiking, then the network goes to a regime with lower and more distributed frequency peaks. To verify that these patterns are indeed due to the initial transitional phenomena and not by an intrinsic character of the network, we cut
off the first 300 msec of simulation and we instead observe a more intricate
behaviour of the bands power, shown in figure 3.11.
Since each symbol represents a different noise level, the fact that the plots
intersect each other indicates that noise produce a non-linear effect frequency
bands amplitudes in this case. We investigate the network changing the network size as a paradigm of biological internal noise. The analysis is conducted with other important neuron properties and characteristics kept constant to allow comparison and discussion. Small world dimension k = 1,
probability p = 0.4 and every simulation is repeated 8 times with different
random seeds, then averaged as in the previous tests.
As shown in figure 3.12, each spectrum shows distinct peaks in the α andChapter 3. Noise influence on spike activation in a Hindmarsh-Rosesmall-world neural network

Power of Frequency Bands (mV2)40

F IGURE 3.10: The power of five frequency bands for a network of N = 48 neurons. The level of noise is represented byeach symbol according to this scheme: square, dB = 5, plus,
dB = 9, diamond, dB = 13, triangle, dB = 18, triangle (bigger),dB = 22, circle, dB = 26, circle (bigger), dB = 31, plus (bigger), dB = 35. The peaks magnitude seems to keep the same
relative order with noise, this effect is caused by the predominance of the initial transition to regime over the noise influence.
The inset shows the average and standard deviation of E(t) infunction of noise for the same network. The average potential
drops regularly because of the initial transitional period thatdrives potentials to more negative values. Noise reduces thisinitial transition, so the curve appears less negative at low dBvalues. Refer also to figure 3.5. For reason of clarity not all thedecibel labels are indicated.

θ range. Interestingly, this behavior is maintained also for bigger networks.
At present conditions, our system is limiting the simulations to these network sizes, but it would be interesting to increase even more the numberof neurons to see if the diminished lower frequencies peaks phenomenon
observed with external noise increase would be repeated with networks ofmuch higher sizes.

41Power of Frequency Bands (mV2)

3.3. Results

Spectral Density (mV2/Hz)

F IGURE 3.11: The power of five frequency bands for a networkof N = 48 neurons at different noise levels as in figure 3.10. Noticeably, the first 300 msec of simulation where cut off to emphasize the network frequencies at regime. The various plots crosseach other, with no systematic trends. This is indicating that thenoise is influencing the spectrum relative intensities and seems
to shift peaks position (see figure 3.9). For reason of clarity not
all the dB labels are shown. The initiator xo has negligible influence on these plots, see figure 3.7.

Frequency(Hz)F IGURE 3.12: The Fourier transformation of the membrane signal in networks of different sizes and constant 20 dB noise level.
Each line is shifted vertically for clarity, in the same fashion as
figure (3.9). From top to bottom, the number of neurons is decreasing (top 171, bottom 10 with steps of 23 neurons). The
simulation is run for 1000 msec, first 300 msec of data are notconsidered to avoid low frequencies due to the initial baselinevariations. Frequencies peaks emerge in all conditions, but lessprominent for networks of bigger sizes. Each plot is the average of 8 identical simulations executed with different random
number seeds.

43

Chapter 4

Discussion and Conclusion
In this research work I succeeded to clarify two important things, the role ofsynaptic connectivity in a minimal two-neuron system and the way information propagate in a model neural network.
I chapter 1, we have focused on the synchronization of a pair of neurons andstudied the spiking correlation using a coefficient ρ. We have shown for the
first time that correlation is highly non-linear in w with the presence of distinct threshold points for all neurons classes studied. Synchronization in simplified system of two neurons has been analyzed with real isolated coupledneurons by Elson [21] and with analog electronic neurons by Pinto[60]. Innovative numerical simulations has been performed by Wei[78] and Stich[70],however no systematic study on the neurons synchronization against coupling coefficient was elucidated in previous studies. We found that when thethreshold point is crossed two neurons reach a state of synchronous spiking
that is characterized by a bifurcated and asymmetric phase diagram (figure
2.7(d)). This behaviour is similar to what has been found in the work of
Pinto et. al that studied an electric analog neurons model[60]. Non linear
behaviour is evident in the inter-spike interval plots for all class of neurons
tested and clear differentiation between neuron types has been found and reported in figures 2.9 and 2.10. Interspike activity begins at transition points
that make the threshold crossing phenomena more evident. In addition to all
previous literature knowledge, we have shown that neuron class radicallychanges the response of the system to connection strength. Inter-spike interval is kept almost invariable for two neurons stimulated by a constant input
for any connection strength, however evident inter-spike drift is observedfor FS-RS pairs (figure 2.13). The dependence on neuron types is even moreevident in the inter-spike distribution width (represented by the standard deviation in plot of figure 2.14). Prominent threshold behavior is observed andshown in figure 2.15 (a) systematically for each neuron type pair combination. Striking differences in response and threshold values are observed and44

Chapter 4. Discussion and Conclusion

reported, moreover, if the second neuron has his own external constant stimulus, again impressive differences in response, threshold and inclination ofcorrelation curve are reported (figure 2.15 (b)). These quantitative observation and original behavioral features are the elemental foundation that may
pave the way for the development of new brain theoretical models of smallor larger networks.
The fact that discontinuity is found in the connection strength vs correlation plot is not surprising, since neural connections are known to hold astrong threshold dependent relation to input. In fact, simpler artificial neural networks are generally represented with sigmoidal-like curves in input.
However, we found essentially two novel information in our study. The firstis that this threshold is variable, and the variability depends strongly with thetype of neurons that we are using. As visible in our data (figure ??) chattering neurons are driving the post-synaptic neurons to correlation at an earlier
value of connection strength than any other neuron. It is very important to
notice that the threshold at which correlation between the two neuron rises,
is virtually the same for the three type of post-synaptic neurons. The configuration CH-RS, CH-FS and CH-CH shows the same threshold value of about
w0 = 0.2, also shape of the curves is very similar for the three configurations
and a plateau of ρ ≈ 70% is reached for all of them. This striking similarity
for three different neurons is striking and suggests that the level of correlation depends strongly on the pre-synaptic neuron type. Also, our results
indicates that the brief bursting activity of chattering neurons is able to stimulate the dynamics of post-synaptics systems with the same strength even if
the neural characteristics differs.
The fact that we have a real measurable threshold value (w0 ) is useful
to dimension, test and realize actual artificial system knowing in advanceconnection strength level of importance in a dynamical system. For example,
in the case that the pre-synaptic neuron is of CH type, our work give neuralsystems designers the a-priory knowledge that connection strength less than
20% are not influential to the dynamics, as well as 50% or higher ws are notimportant since plateau is already reached. In the case of chattering neuronalsystem, the range of 0.2 < w < 0.5 all what counts.
Same discussion can be deduced for the other neuronal configurations
we studied FS-CH, FS-RS, FS-FS and RS-CH, RS-CH and RS-RS. Very interestingly, when the pre-synaptic neuron is of FS type, whatever is the postsynaptic, we notice a threshold at about w = 40% and a sort of secondary
resonance peak at almost exactly w = 50%. This peak is reproduced only inChapter 4. Discussion and Conclusion45

the case of pre-synaptic of FS type, but not in the case of RS type, even if thethreshold value corresponds. This again support the general conclusion thatwhen a pre-synaptic neuron is driving a not-stimulated post-synaptic one,the pre-synaptic characteristics win on the pos-synaptic neuron. This fact
is very important for the design of neuronal systems. In the right panel of
figure 2.15 we show the situation in which the two connected neurons havetheir own independent stimulation. Also in this case we have have foundvery specific behavior that have meaning in a general neuronal system. The
legend of the figure shows, the three systems where the neurons are identical (CH-CH, RS-RS and FS-FS) show immediate sensitivity to the connection
strength w, without a visible threshold-like behaviour. Moreover, they lockquickly to high values of ρ > 80% reaching a plateau at w < 0.2. Noticeably
the plateau is reached earlier than the initial threshold in the other connection
model.
In chapter 2, We analyzed the collective behaviour of a H-R small world
network of dimension k = 1 and random connection probability p = 0.4.
The influence of noise is studied, especially concentrating on the effects it
has on the propagation of information along the network by studying themembrane average frequencies and spike activity diffusion due to an active
neuron that initiate the spiking. To isolate other effects the network is in
ideal conditions, identical neurons, no plasticity and no delay time are considered. The network have a single active neuron xo connected to an external
stimulus of Io = 3 mA. This neuron is spiking constantly and in absence ofnoise it induces the rest of the network to spike after an initial transitional
time. Most interestingly, it seems that noise favors the initiation of secondary
spikes (the spikes that follow those of xo ), and this spike activity becomes
spontaneous and independent from xo . We found a linear relationship between secondary spikes initiation time and signal to noise ratio in decibels,
with an inclination constant of about δ = 14 millisecond per decibels. Thisphenomenon is instead found to be much weaker or not existent if we increase network size. However, for networks of sizes of N = 100 neurons or
less, the initiator xo still makes a difference, favoring the quicker spread of
spike activity on the network. This suggests that locally active neurons playa role in the spread of information in confined domains outside which theeffect decays with distance. For the sake of simplicity and to avoid complexity this result has a limited meaning bind to the specific Hindmarsh−Rose
network studied here, but we have intention to verify if this function of noise

46

Chapter 4. Discussion and Conclusion

is a general phenomenon in spiking networks or remains limited the conditions and network connection structure used. The importance of noise inneural networks is known, and what we observed reminds phenomena of
stochastic resonance in biological systems[1, 67, 73, 45, 47]. The influenceof noise on frequency have been investigated by Fourier analysis. We haveshown that once the transitional delay is past, noise displaces main frequencies peaks, provoking a slow transition to chaos. Studying the most prominent frequency bands (θ, α, β and γ) we confirmed that noise quenches thosebiologically significant peaks in the network. Still, at regime, a less keen transition to chaos is observed, frequency peaks do reduce and tend to disappear,
but in a slower and more complex trend that seems to conserve some activity
in the theta and alpha band (cfr. figure 3.9 and figure 3.11). In real brains,
the noise is generated internally, so we scaled up the network and used the
network size as a paradigm of the external noise. We found two trends, a
reduction in peaks amplitude and better sharpness and definition of Fourier
peaks, but not an evident degeneration to chaos observed increasing external
noise. This trend must be presumably confirmed with much higher network
sizes, and we have intention to set up the appropriate computational tools to
do that in the near future.
This two studies was related with two points. The first one we used the gap
junction as the coupling way. The results showed the dynamics both in coupled neuron system and neural network with gap junction connections. The
second one is in the two studies, we both analyzed coupled bursting neurons.
Both gap junction and bursting neurons have important roles for information
processing within brain. Recently, since the development of supercomputer,
we can simulate huge networks, but we still don’t have enough theories andbasic methods to analysis the dynamics in complex networks.
Our study gives a basic theory and dynamics analysis method for simple
neural systems. In the future, we would like to contribute to two different
projects.
1. Improving the coupling neuron system what we introduced in Chapter
2 with Spike-timing dependent plasticity rules. We will focus on the influence of the time delay and the initial connection strength on the phases of
spike trains. 2. Instead of small world networks, we will use grid networks
and cube networks to construct our neural network system. In this system
we will adjust the frequency of external stimulus, and we will observe moredetails about how the noise influence the information process in our brain.
In the past years, more and more people realized the importance of brain

Chapter 4. Discussion and Conclusion47

research. Many countries started brain research projects. Even today, many
artificial intelligence progress and deep learning applications show brain-likeability in many areas, but they are still not realistic intelligent systems. On
the other hand, the study of brain’s diseases and other medical related topics
make a lot of social pressure. In brain research, computational neuron sciencehas a very important role. This is because the process of neural information
has a complex spatial-temporal structure. Using mathematical methods andsimulations, we can focus on the significance of the activity of basic neuralsystems. For example we can study individual neurons or coupled neurons.
Once those functions will be elucidated, we can analyze the average activity
in neural networks with huge size or very complex structures with a better
understanding of their functions and better control.
In conclusion using anatomy, EEG or other technologies, researchers havegathered huge of data when the brain perform particular tasks. Computational research like we did in our thesis, help the scientific community tocreate better brain models and using them to emulate particular behavior or
activity that are observed in biological systems. Our study is contributing to
the understanding on how real brains work and then figure out improved
theories of our brain.

49

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