CHAPTER 1. Introduction
taken up and analysed. The authors dedicated themselves to the presentation
and study of the most important and widely spread analytical models as well
as they discussed examples in which the theoretical approach conferred a more
detailed view of the operating mechanism of the nervous system.
We also intend to extend the analysis to a wider number of computational
models and to examine, from a statistical point of view, the spiking activity of a
population of spontaneously reverberating neurons, a probabilistic phenomenon
that occurs in correspondence of an external stimulus and it is manifested as a
peak in the function that describes the membrane potential.
1.3 Work Done
After a preliminary documentation and in-depth examination phase regard-
ing the neurophysiological issues we dealt with, we moved on with the analysis
of statistical methods that describe and explain the creation mechanism of ac-
tion potentials. We examined the spike trains statistics illustrating the use of
different methods of data processing techniques that allow quantitative descrip-
tion of neuron firing configurations. In particular we focused on state-of-the-art
spike timing reliability methods applied to the analysis of the properties of in-
formation processing.
After completing the study regarding statistical methods, we moved on with
the discussion of biological neuron models through the analysis of the physical-
mathematical paradigms that describe their electrical properties. We chose
to carry on with the description starting from the simplest models, known
as Single−compartment models, proceeding with the more complex, known as
Multi−compartment models. This choice is totally arbitrary but allows an easier
understanding of their operating mechanism and the role that they play within
computational neuroscience.
Single neurons can be interconnected to form complex neural networks. The
computational potential of such connections was therefore analysed using both
mathematical analysis and the simulations via software. To fully understand the
behaviour of a neural network, we have also decided to extend the study to the
techniques of information storage within the network. Through the knowledge of
learning phenomena and tapping into data recorded by micro−electrode arrays
(MEA), it was possible to obtain a more detailed description of the behaviour
of neurons. We then went into more depth with the study of spontaneously
reverberating neural networks since these are believed to play an important role
in neural information processing.
After we finished treating physical and mathematical models that describe
2
1.3. Work Done
the behaviour of neural networks, we directed our attention to the populations of
neurons, as these are often the relevant unit for many brain controlled functions.
Lastly we conducted a campaign of simulations using Brian[2] software, a
clock-driven neural network simulator written in Python.
Starting with the simplest neural model, leading up to modelling entire pop-
ulations of spontaneously reverberating neurons, it was possible to represent
from a computational point of view the operating mechanism of the actual
laboratory-grown neural networks.
3
CHAPTER 1. Introduction
4
Chapter 2
Neurophysiologic
Background
Although neurons may differ considerably in shape, size, and many of their
molecular components, their basic ”design principles” are similar: sharing the
same basic morphology and biophysical mechanisms enables neurons to initiate
and propagate action potentials and facilitates communicating them to other
neurons through synaptic connections.
This chapter provides an overview of the basic neuronal biophysics’s building
blocks.
Due to the fact that current research in Neuroscience is working specifically
toward the understanding of neuron populations dynamics, this chapter will also
provide a description of the Micro-Electrode Array (MEA) technology.
2.1 The Nervous System
The nervous system is a network of specialised cells that process and com-
municate information about an animal’s environment and itself[3]. The system
is made of neurons and glia cells, non-neuronal cells that provide support, nu-
trition and maintain homeostasis. The human nervous system can be divided
into two systems: the Central Nervous System (CNS), the largest part of the
nervous system that includes the brain and spinal cord, and the Peripheral Ner-
vous System (PNS) which is composed of sensory neurons and the neurons that
connect them to the nerve cord, spinal cord and brain.
Neurons generate and conduct impulses between and within the two systems.
In response to stimuli, sensory neurons generate and propagate signals to the
5
CHAPTER 2. Neurophysiologic Background
CNS that then processes and conducts signals back to the muscles and glands[4].
2.1.1 Micro-anatomy
The nervous system is, on a small scale, made up mainly of neurons that are
specialised in receiving information from other cells, generating voltage tran-
sients in response to these inputs, and sending them to other neurons. There
is a wide number of different types of neurons: sensory neurons (responding
to stimuli effecting sensory organs and sending signals to the spinal cord and
brain), motor neurons (receiving signals from the brain and spinal cord, causing
muscle contractions and affecting glands), and interneurons (connecting neurons
to other neurons within the brain and spinal cord).
The typical morphology of neurons is made of three main structural compo-
nents:
• The cell body or soma, where the regular cell ”machinery” resides (i.e.
the nucleus, mitochondria, endoplasmic reticule, etc.),
• the dendrites which branch out of the soma like cable structures that
receive inputs from other neurons,
• the axon, a single cable leaving the soma which branches out to connect
to other neurons’ dendrites, insulated by a myelin sheath.
Myelin is composed of Schwann cells wrapped multiple times around the
axonal segment. This forms a thick fatty layer that prevents ions from entering
or escaping the axon. This insulation both prevents significant signal decay as
well as ensuring faster signal speed. This insulation, however, has the restriction
that no channel can be present on the surface of the axon. There are, therefore,
regularly spaced patches of membrane that have no insulation. These nodes of
Ranvier can be considered to be ”mini axons” as their purpose is to boost the
signal in order to prevent significant signal decay.
At the furthest end, the axon loses its insulation and begins to branch into
several axon terminals that end up in forming the second class of synapses, axon
terminal buttons. These buttons have voltage activated calcium channels which
come into play when signalling other neurons.
Glial cells are non-neuronal cells that provide support and nutrition, main-
tain homeostasis, form myelin, and participate in signal transmission in the
nervous system[5]. Glial cells are known as the ”glue” of the nervous system.
A more functional way to classify the human nervous system is the role that
the different neural pathways play: the somatic nervous system is responsible
6
2.2. The Action Potential
for coordinating voluntary body movements; the autonomic nervous system is
responsible for coordinating involuntary functions, such as breathing and diges-
tion.
In turn, these classifications of the nervous system can be further classified
according to the direction in which they conduct nerve impulses:
• Afferent system by sensory neurons, which carries impulses from a somatic
receptor to the CNS,
• Efferent system by motor neurons, which carries impulses from the CNS
to effectors,
• Relay system by interneurons, which transmit impulses between the sen-
sory and motor neurons (both in the CNS and PNS).
The junction between two neurons is called a synapse[6].There is a very
narrow gap (about 20nm in width) between the neurons called the synaptic cleft.
This is where an action potential, the ”message” being carried by the neurons,
is transmitted from one neuron to the next. This is achieved by relaying the
message across the synaptic cleft using neurotransmitters, which diffuse across
the gap. The neurotransmitters then bind to receptor sites on the neighbouring
(post synaptic) neuron, which in turn produces its own electrical/nerve impulse.
This impulse is sent to the next synapse, and so on.
2.2 The Action Potential
In neurophysiology, the action potential is a self-regenerating wave of elec-
trochemical activity that allows nerve cells to carry a signal over a distance. It
is the primary electrical signal generated by nerve cells, and arises from changes
in the permeability of the nerve cell’s axonal membranes to specific ions.
2.2.1 Biophysical and Cellular Context
Electrical signals within biological organisms are generally driven by ions.
The most important cations for the action potential are sodium (Na+) and
potassium (K+): both of these are monovalent cations that carry a single pos-
itive charge[7].
Ions cross the cell membrane under two influences: diffusion and electric
fields. The hydrophobic cell membrane prevents charged molecules from easily
diffusing through it, allowing a potential difference to exist across the mem-
brane. Each neuron is encased in a cell membrane that is nearly impermeable
7
CHAPTER 2. Neurophysiologic Background
to ions. To transfer ions into, and out of the neuron, the membrane provides
two structures:
• Ion pumps use the cell’s energy to continuously move ions in and out. They
create concentration differences, between the inside and outside of the
neuron, by transporting ions against their concentration gradients from
regions of low concentration to regions of high concentration.
• The ion channels use this concentration difference to transport ions down
their concentration gradients from regions of higher concentration to re-
gions of lower one. Ion channels only open and close in response to signals
coming from their environment. The transport of ions through the ion
channels changes the voltage of the cell membrane and these changes are
what brings about an action potential.
The cell membrane acts as a barrier that prevents the internal solution,
the intracellular fluid, from mixing with the external solution, the extracellular
fluid[7]. These two solutions have different ions concentrations. Furthermore,
this difference in concentrations leads to a difference in charge of the solutions
creating a situation whereby one solution is more positive than the other. There-
fore, positive ions will tend to gravitate towards the negative solution. Likewise,
negative ions will tend to gravitate towards the positive solution.
To quantify this property the outside solution is set as the zero voltage; then
the difference between the inside voltage and the zero voltage is determined:
if the outside voltage is 100 mV, and the inside voltage is 30 mV, then the
difference is 70 mV. This is what is commonly referred to as the membrane
potential. Ion channels are integral membrane proteins through which ions can
cross the membrane. The pore is usually so small that ions must pass through
it alone and single-filed. Channels are either fully open or fully closed: the
action potential is a manifestation of different ion channels opening and closing
at different times. In general, closed states correspond either to a contraction of
the pore, making it impassable to the ion, or to a separate part of the protein
stoppering the pore.
Ion channels can be classified basing on how they respond to the environ-
ment: voltage−sensitive ion channels open and close in response to the voltage
across the membrane, ligand−gated ion channels open and close in response to
the binding of a ligand molecule, such as a neurotransmitter, and other ion
channels open and close with mechanical forces or in response to other stimuli,
such as light, temperature or pressure.
The action potential ionic currents flow in response to concentration differ-
ences of the ions across the cell membrane. These differences are established by
8
2.2. The Action Potential
ion pumps, which are integral membrane proteins that carry out active trans-
port. Such ion pumps take in ions from one side of the membrane, decreasing
its concentration there, and release them on the other side, increasing its con-
centration there. The most relevant ion pump to the action potential is the
sodium-potassium pump, which transports three sodium ions out of the cell
and two potassium ions in. Consequently, the concentration of potassium ions
inside the neuron is roughly 20 times larger than the outside concentration,
whereas the sodium concentration outside is roughly 9 times larger than in-
side. Similarly, other ions have different concentrations on the inside and on the
outside of the neuron, such as calcium, chloride and magnesium[8][9].
Ion pumps influence the action potential only by establishing the relative
ratio of intracellular and extracellular ion concentrations. As described above,
the point at which the electric field completely counteracts the force due to
diffusion is called the equilibrium potential. The electric field, in turn, is based
on the charge of the ion in question. Furthermore, in the case of diffusion, it is
based on the difference in concentration between the two solutions. With these
two parameters, the equilibrium potential for any ion can be calculated with the
Nernst equation, which has a physiological application when used to calculate
the potential of an ion of charge z across a membrane[5]:
E = RT
nF
ln(
outside ion concentration
inside ion concentration
), (2.1)
• n = The charge valence of the ion,
• T = The temperature (K),
• R = 8.314 JKmol the molar gas constant,
• F = 96485 Cmol the Faraday constant.
Clearly, even if two different ions have the same charge, they can still have
drastically different equilibrium potentials, provided their outside and/or inside
concentrations differ. However, there is an equilibrium membrane potential Em
at which the net flow of all ions across the membrane is zero: this potential is
calculated by the Goldman equation[10].
A Cartesian coordinate system is used to describe the system, with the z
direction being perpendicular to the membrane. Assuming that the system is
symmetrical in the x and y directions (around and along the axon, respectively),
only the z direction need to be considered. Thus, the voltage Em is the integral
of the z component of the electric field across the membrane. According to
Goldman model, only two factors influence the motion of ions across a permeable
membrane: the average electric field and the difference in ionic concentration
9
CHAPTER 2. Neurophysiologic Background
from one side of the membrane to the other. The electric field is assumed to
be constant across the membrane, so that it can be set equal to Em/L, where
L is the thickness of the membrane. So the Goldman equation results in the
following if we consider a membrane separating two Kx Na1-Clx-solutions:
Em =
RT
F
ln(
PK [K+]out + PNa[Na+]out + PCl[Cl−]in
PK [K+]in + PNa[Na+]in + PCl[Cl−]out ), (2.2)
• Em = The membrane potential (V),
• Pion = the permeability for that ion (m/s),
• (ion)out = the extra cellular concentration of that ion (M/m3),
• (ion)in = the intracellular concentration of that ion (M/m3),
• R = The ideal gas constant (J/KM),
• T = The temperature (K),
• F = Faraday’s constant (C/M).
It is ”Nernst-like” but has a term for each permeant ion. The Nernst equation
can be considered a special case of the Goldman equation for one ion only. The
first term, before the parenthesis, can be reduced to 61.5 log for calculations
at human body temperature (37 C). Note that the ionic charge determines
the sign of the membrane potential contribution. Goldman equation is useful
to electrophysiologists as it allows them to calculate the predicted membrane
potential for any set of specified permeabilities.
2.2.2 Propagation of Action Potential
Action potentials are pulse-like waves of voltage that travel along several
types of cell membranes[11]. A typical action potential is initiated at the axon
hillock when the membrane potential is depolarised sufficiently. As the mem-
brane potential is increased, both the sodium and potassium ion channels begin
to open up: this increases both the inward sodium current and the balancing
outward potassium current. For small voltage increases, the potassium current
overwhelms the sodium current and the voltage returns to its normal resting
value, typically −70mV. However, if the voltage increases beyond a critical
threshold, typically 15mV higher than the resting value, the sodium current
dominates. This results in a runaway condition whereby the positive feedback
from the sodium current activates even more sodium channels. Thus, the cell
”fires”, producing an action potential.
10
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