We have been discussing we have been trying
to understand brain shape.
So, this lecture has 3 segments, we already
discussed the first segment on brain size
intelligence.
We look at some simple metrics that relate
the brain size intelligence, we started with
brains weight and that did not work very well
we looked at brain weight and body weight
ratio, that also had its issues then we looked
at this new interesting quantity called encephalization,
quotient that seemed to work well, but still
it was not too convincing because it does
not give much insight into what aspects of
brain structure it correlates with intelligence.
So, now let us look at the go deeper into
this kind of investigation into properties
of brain shape, structural properties and
arrive at a principle called save wire principle.
So, we will argue that and also show evidence
that brain tries to minimize the total wire
length alright.
And we see that that kind of an attempt to
minimize wire length is also very commonly
seen in engineering domains.
So, let us go deeper into this save wire principle.
So, before we get into that let us look at
some data related to number of neurons and
number of wires and you know various brains
etc, if you look at the number of cells, so
in various brains.
So, in humans it is about 100 billion in adult
human brain in octopus it is about 300 million,
in aplysia it is a very small water creature
it is about 18000 to 20000 in a small worm
called C Elegans or nematode they are exactly
2 neurons So, the number of neurons shows
a lot of variation.
So, shows many orders of magnitude variation.
Now, if you look at the wiring, right the
brains are full of wire and wire is a very
important aspect of brains, because the connections
evolution right as we have seen in the first
lecture.
As recognized that the wire is an important
part of brain structure.
So, if we look at human brain the total length
of myelinated fibers is equal is about 150
thousand kilometers to 180 thousand kilometers.
And the average number of connections per
neuron is about 1000 to 10000.
And the number of synapses on a very large
cell, cell which has a large, large number
of connections is about 1.5 to 2 lakh or 200
thousand connections.
And number of synapses in the cortex in total
entire cortex, he is about 0.15 quadrillion.
Now, number of fibers in human optic nerve
is 1.2 million of number of fibers.
In cat optic nerve is 119000.
And number of fibers in albino rat optic nerve
is about 74000.
So, numbers that you are looking at are very
large when it comes to the wiring let us look
at these numbers slightly differently.
For that we need to define, what is a gray
matter, and what is white matter in common
parlance we use the term gray matter as if
it you know denotes the brains of a person
or intelligence of a person, but basically
gray matter refers to a part of the brain.
So, if you look at this in this slide it shows
a slice of a brain and, this slice shows kind
of a brownish outline and creamy white coloured
interior.
So, this brownish outline is actually what
is called gray matter because that is where
you have a lot of cell body.
So, that is the cortex and the creamy white
interior is the wiring, now this is white
because these wires have a kind of coating
called myelin sheath and myelin is white in
color.
So, the gray matter consists of cell bodies
dendrites and unmyelinated axons.
Whereas white matter consists of this myelinated
axons.
Now, it turns out that the amount of white
matter grows as a brain size grows.
So, for example, in a hedgehog the volume
of white matter right is only 13 percent whereas,
in human your cortex at the volume of white
matter is 42 percent so; that means, as the
brain size grows the volume of the white matter
keeps growing that is a wire part keeps growing.
Right and in this graph it shows the white
matter to gray matter relationship right for
a large number of species.
So, in the x axis you see gray matter volume
and the y axis you see white matter volume,
now these numbers are represented in log scale.
So, that is why it looks linear.
But that does not mean they are proportional
actually there is a power law here and the
power is 1.23 so; that means, the white matter
volume is proportional to the gray matter
volume to the power of 1.23 so; that means,
the white matter volume increases faster than
linear, right with respect to gray matter
volume.
But that is not very surprising because if
you think about it, if you think of brain
as a network right of nodes and then the nodes
are connected by edges you will look at a
small graph which has only 2 nodes, as they
can be connected only in 1 way giving you
1 edge.
If it takes graph of 3 nodes you can have
totally 3 edges.
If you take a graph of 4 nodes you can have
totally 6 edges and so on
So, if you take a graph with 10 nodes.
You can have totally n into n minus 1 by 2
connection.
So, that is about order n squared connections
So, basically if we take a graph in which
every node is connected to every other node
the number of connections increases n squared
where n is the number of nodes.
So, in case of real brains every neuron is
not connected to every other neuron, but it
is right.
So, the power is not two, but and it is not
1 either it is something between 2 that is
it is 1 point 2 three.
So, therefore, you can understand why the
wiring volume increases right rapidly as the
brain size grows.
Now, let us go deeper into the connectivity
patterns in the brain.
So, broadly there are two kinds of connected
to patterns there are feedforward connection
where a region A simply sends projections
to region B, and then there are feedback connections
where region A connection sends projections
to region B and also receives connections
back from region B.
So, let us look at some examples of feed forward
connections.
So, there are point to point connections where
1 region sends projections to other region
B and these present projections are kind of
parallel, what does that mean let us look
at this example.
So, what you see on the slide is a wiring
very approximate wiring pattern of the wires
that take you from the retina, which is inside
the eyes right.
Where the image this received and captured
and surrenders the back of the brain where
you have the primary visual cortex.
So, the green line that you see in this image,
which takes you from the left eye to the left
brain all right, goes through a point on the
retina back to a point in the brain, it is
only the red line versus 1 point on the retina
under the left eye and goes to the back of
the brain in the right brain.
So, thing is if you.
So, if you flash a dot of light on the retina
right that little spot of light activates
a very specific small local set of neurons
in your primary visual cortex.
So, if you move your that dot of light around
on the retina that that will activate a series
of very specific local set of neurons, right
in the primary visual cortex.
So, that is what we mean by point to point
connections a point on in 1 region of the
brain right, will activate a specific point
in another region.
So, then there are converging connections
a where large brain area sends projections
to a very tiny brain area right.
So, that is what is shown in the schematic
at the right bottom part of this slide, and
one example of that is cortex sending projections
to a part of the brain called ventral striatum.
So, cortex is a large area and this item is
much smaller it is deep inside the brain.
So, these projections are examples of convergent
connections.
So, then there are divergent connection which
are like opposite of convergent connections,
where a small brain areas and projections
to a large target area.
So, an example of that is the ventral tegmental
area VTA, setting projections to prefrontal
cortex or PFC you sees that red arrow climbing
from VTA all the way up to a box named PFC.
Then if you go to feedback connections that
again two categories, there are what are called
reciprocal reentrant connections.
So, these connections simply go from 1 area
A to another area B and go back from area
B to area A.
So, it is a simple projection to a destination
and written have written fibers back to the
source.
It is a good example of that kind of connectivity
is the connections from cortex to thalamus.
So, thalamus sends free powered projections
to cortex and cortex sends feedback to the
thalamus then there are more complicated types
of you know feedback connections.
Where region A sends projections to region
B, B sends projections to C, C sends to D,
and D sends projections back to the starting
point which is the A.
An example of these kinds of connections is
cortex sending projections to a region called
newest stratum, which is in the middle to
the right part of the image of the slide.
And the newest stratum projects to the Globus
pallidus external, or GPE and GPE projects
to Globus pallidus internal or GPI, and then
GPI then projects to the thalamus right and
thalamus products back to the cortex.
So, that is a very long loop going over many
stages.
Then, so the all the connections we have seen,
so far are like slightly long range connections
right.
Whereas, there are also very intricate connectivity
pattern at the local level for example, if
we look at the cortex, which is a like I said
before 2 to 5 millimeter thick sheet of neurons
and within the cortex there are 6 layers and.
So, as the named as L 1, L 2 and L 6 up to
l 6 and there are neurons in each of these
layers and these neurons send projections
to neurons in the, neurons in a given layer
send projections neurons in the other layers
above it or layers below it, and also to neighbouring
neurons in the same rate.
So, these kinds of circuits are called local
circuits, so you can see that brain has a
lot of interesting connectivity patterns,
but the question is why is there this kind
of intricate connectivity pattern.
What is the underlying logic of brains connectivity?
What happens if the brain were just 1 big
mass of neural go.
So, it turns out that people have done calculations.
And found that if the cortical neurons were
not arranged, in spatially distinct areas
and merged into 1 big huge mass right, the
human cortex would be 10 times larger to maintain
the same degree of connectedness.
This is something that you experience even
when you pack your bags, when you go on a
travel, but you have a bunch of things that
you need to pack in a box.
If you just put them in some kind of a random
order right they start spilling out of the
box.
So, you need to pack them very judiciously
optimally.
So, that everything and going to the box and
apparently the even courses that which teach
you how to packs you know packed boxes.
So, that you can pack a lot of stuff in the
same volume, so brain seems we are doing something
like that and evolution seems you are pushing
brain to pack the contents of the brain more
optimally more efficiently within the volume
of the cranium.
So, to understand the logic of this kind of
packing we need to ask first of all.
What is a brain?
What is the purpose of the brain?
Why do we have a brain?
Right to understand that let me take a small
example.
So, let us say you are walking and you know
accidentally you step on a pin, right it hurts
and then you withdraw your leg let us say
it is left foot from the pin.
Now when you do that if you just withdrew
your left foot, and if you did not do anything
else right then you would not have a fall
because.
So, until then your body weight is was being
supported by both the legs, and if you suddenly
do your left foot right then your right foot
may not be ready to take all the weight and
you are going to have them fall.
So, in order to for this withdrawal of the
leg to work right you need to make sure that
the left leg can immediately take all the
weight of the body.
And for that to happen the left leg muscles
have to tighten, the antigravity muscles of
the right leg will have to tighten.
So, that the leg now can take the entire bodyweight.
And that is not enough you need to also shift
the body weight to your right side.
So, that all the weight goes through the legs
through the right leg and sometimes you may
even have to stretch your arm.
So, that you balance yourself and you would
not lose balance you might even have to tilt
your head.
So, if you think about it a very tiny microscopic
stimulus like the pin right.
Is producing a whole body responds, right
integrating your entire body that is possible
only because of your brain, the brain is doing
all this coordination.
So, as you are stepping on the pin seems to
produce a coordinated rapid whole body response.
Let us take another example.
So, it is not just humans animals, but even
plants seem to be capable of producing such
you know large scale coordinated responses.
So, take the example of a tomato plant in
this picture you can see a worm eating a certain
leaf on the tomato plant.
So, when that happens the plant launches a
defense mechanism.
So, the leaf where which is being wounded
releases a chemical which is harmful to the
worm that is eating it.
And this release chemical not only is at least
locally within the that leaf, which is wounded
the substance spreads all the way to the or
the entire body of the plant.
So, that at a distant location later on if
another worm starts attacking the plant, right
the newly released toxin will be harmful to
the bug and the prevented from wounding the
plant further.
So, even a plant is able to produce a coordinated
rapid either a time scale of the plant I suppose,
but a coordinated extensive whole body response.
Let me take 1 more, example something that
is more probably related to the experience
of lot of students here.
I am sure a lot of you live in hostels and
let me just consider a situation which is
not uncommon in hostels.
Right imagine one fine night and you are all
sitting along with your friends in the hostel
wing and having a good time doing something
that maybe you're not supposed to do at that
time of the night.
Then then suddenly the warden walks into the
hostel, right and somebody in the ground floor
sees the warden coming in and then sends a
text right to some of his friends right.
Who lives in the same hostel, and the guys
who receive this message then further send
this messages to few more friends and quickly
the news that the hostel warden has just entered
has come for inspection all right spreads
all over the hostel.
So, in this example you can see the hostel
has produced a coordinated rapid whole hostile
response ok.
Now, that is what you need for a brain to
do for the body right, for the body to survive
right in a very hostile environment.
It needs to have some kind of a communication
network, which is spreaded all over the body
and which can help enable communication rapidly
to take place from one part of the body to
another.
So, organisms need you know systems to produce
coordinated rapid whole body responses right
to the shocks of the world.
Now, how does a body produce you know produce
this kind of responses, what how does a nerve
system give the body these kinds of rapid
responses.
So, there is two ways of doing it right.
So, if you want faster response basically
time is distance by velocity.
So, there are two ways to reduce time, one
ways to increase velocity.
So, that will reduce the time.
The other way is reduced distance, what does
that mean and how do you achieve it let us
look at that.
Let's look at the first solution increasing
the velocity of neural conduction.
Now, it turns out that if we want to increase
the velocity of neural conduction if the biophysics
of nerve fibers tells you that, if you increase
the fibre diameter that will increase conduction
velocity.
So, there are different kinds of nerve fibers
there are these thick myelinated nerve fibers
called type A, and then thin myelinated nerve
fibers are type B fibers, and thin unmyelinated
nerve fibers nerve fibers or type C fibers.
So, type A fibers have a diameter of you know
6 to 12 microns and velocity is about 35 to
75 meters per second, and type B thin or thinner,
and the 3 microns diameter and velocity is
3 to 15 meters per second and the type C or
even thinner and even slower.
So, basically as you increase our diameter
you got higher conduction velocities.
But there is a problem with this approach
to achieving faster communication, because
if you use thicker fibers to achieve a higher
conduction velocity and that will make the
brains larger.
And that will require more wire to connect
these right various points in these larger
brains.
So, that will create no longer transmission
delay and to compensate for that you use even
thicker fiber and then you get even larger
brains and so on, so forth.
So, you have a kind of a catch situation,
where this approach does not seem to be very
efficient way of reducing time delays.
The other way of reducing time delay is to
reduce wire length, what does that mean I
look at this simple example shown in the lower
part of the slide.
So, you have 3 target neurons, target neuron
1, target neuron 2, and 3, and then there
is a test neuron which is a green neuron.
Now you are free to put wherever you want
this test neuron, but you would like to place
it.
So, that it is the total wire length connecting
the test neuron to the their target neurons
is minimum.
So, if you do that the total transmission
delay for connecting all these 3 test neurons
to the other 3 neurons, will be minimized
now something like this you can relate to
real world experience also.
Consider the problem of this happy family
that that lives in a city and.
So, there is a mom there is a dad and then
there is a kid now they want to decide where
to pick and choose their home, so that they
all can go to their respective workplaces
right.
So, the mother works at workplaces at A, and
kids school is at B, and father's workplaces
at C. Now P is where they want to live now
where will you put the P in this, but the
place the point P in this triangle.
But; obviously, will place it.
So, that the total distance AP plus BP plus,
you know CP is minimum, because assuming that
the travel costs are the same which are you
travel right this is what you would like to
do.
Now this problem has been solved in geometry
a long ago and this is the scalar point is
called a Fermat point of a triangle right,
and there are ways of constructing the location
of the point P these kinds of.
Problems where you try to minimize the total
wire length are often encountered by electrical
engineers; when they design their you know
logical circuits on chips.
So, let us look at the figure a in this slide
and where you are you are looking at a logic
circuit, right you see all these logic gates
and you know they are numbered from 1 to 8.
So, b is actually just a logical placement
of the gates of figure a, in various boxes
c shows 1 particular configuration right,
by which these logic gates are dissipated
in the boxes and if you look at the total
wire length in c right it adds up to 10.
So, similarly even in d, the total wire length
adds up adds up to 10, but instead of organizing
the gates in 2 columns, if in organize them
as this has 1 big long row right, as shown
in figure e, the total wire length that you
get happens to be 12.
So, if you want to minimize the wire length
right c or d are preferred over the configuration
e.
So, these ideas are relevant to even understanding
brain structure you know how wiring is organized
or is being is organization brain structure,
because it turns out that brain shows a tendency
towards minimizing total wire length this
is this called save wire principle of brains
organization.
Now, there is a lot of evidence to support
this kind this principle and I will just look
at 3 examples, taken from this paper by the
study by cherniak, a paper is published in
1994.
So, in the first of the examples we show we
take a small worm called C Elegans, and show
that the total wire length is minimized in
a nerve system of this worm.
In the second example we show that the adjacency
rule which has is consistent with the save
wire principle I is respected in animal brains.
And the third principle, we argue about the
placement of brain position we ask the question
why is the brain positions in post position
in the head and what is its implication to
the question of save wire principle.
So, in this animation you see the C Elegans,
nerve it is a small worm its common name is
nematode it looks kind of creepy, but actually
it is a very small worm its length is only
1 millimeter it is only 80 microns thick.
So, it is, so small that you need a microscope
to look at it in a culture or in a dish.
So, it has a very small nervous system it
has exactly 3 and 2 neurons, and the connectivity
pattern of all these neurons has completely
worked out.
So, it is an excellent organism for studying
nerve systems and connectivity patterns in
neuroscience.
Here is a simple cartoon picture of various
ganglia that are located inside this in nerve
system.
So, their various names given to the ganglia
I won’t get into that.
And this table tells you which ganglia is
connected to which other ganglia and how many
fibers connect a pair of ganglia.
They are totally 1 plus 10 or 11 ganglia,
there's a head ganglion and then all the other
ganglia.
So, now if you think of the body of the worm
has 1 dimension because it has it is very
it is a very thin worm right.
So, it is body is very thin compared to its
length.
So, the ganglia are placed along the length
of the body of this this worm right.
So, now you can so imagine that given the
connectivity pattern of all this ganglia,
you can move around the ganglia anywhere inside
the body right and calculate the wire length
of the internal system, so the connectivity
is fixed.
But the placement is it can be varied and
for every configuration of the ganglia you
can look at the total wire length.
So, assuming that the ganglia positions are
fixed only the relative ordering is can be
varied right.
So, these 11 ganglia can be permuted, along
these 11 possible positions, in 11 factorial
ways, that suppose a million permutations
it turns out that.
The actual configuration of the C Elegans,
right has a minimal wire length.
So, which is a pretty interesting it is almost
as if an engineer has designed nerve system
of C Elegans.
Then let us look at the adjacency rule right
to explain this rule or to define this rule
we need to understand the organization of
the cortex.
So, cortex in human brain is about 1600 to
2500 millimeter square in a total area.
It is about 50 cytoarchitectonic areas, these
are called Brodman areas.
So, cytoarchitectonic sounds kind of pretty
fancy, but basically it is like a cellular
geographic region.
So, these are certain regions and thus in
the surface of the cortex which can be seen
in this picture.
So, this classification of brain regions is
based on the analysis of the cell types or
neuron types formed in a given in different
parts of the brain surface all right.
So, they are numbered from 1 to 50.
They are easily connected; now thing is if
you assume if you want to prove that the connectivity
of the cortex of the cortical regions satisfies
some kind of a bad minimization principle,
like it makes sense to make sure that the
current nearby regions are connected.
That is connected regions better be contiguous
then you can minimize wires.
So, let us look at the data from cats visual
cortex.
So, and on the rows you can see the connected
pairs number of, so they are totally eighteen
regions which are considered in this study
and in the rows you see there whether the
connected in the pairs of regions are connected
or not connected, in the columns you see whether
they are contiguous or not contiguous.
So, there are totally 70 contiguous pairs
which are also connected.
All right and 108, non-contiguous pairs which
are connected, and there are no contiguous
pairs which are non-connected not connected
and then there are number of cases where the
pairs are not contiguous and not connected
also very big 128.
So, basically it shows that if cells are contiguous
they are completely they are always connected.
So, that is consistent with the save wire
principle.
Now, let us look at the third question the
question of brains position.
Now, what is the question?
What is the problem with brains position;
well 1 interesting question you can ask is
why is the brain inside the head alright why
cannot it be somewhere else inside the body
for example.
Inside the chest, because if you argue that
brain is better a better place inside the
head because it is better by the skull which
is hard you know this bony casing of the brain,
but that argument is not quite valid because
the head is vulnerable because of the neck
and neck is vulnerable.
So, you are probably better off placing the
brain inside the chest right, but why is the
brains at the head to us answer this question
we need to first ask.
What is the head what is what does a head
mean what is special about the head right.
So, we know that.
Both vertebrate and most invertebrate bodies
have a distinct longitudinal axis that that
kind of like a tube right and the head is
usually the place where right.
So, it is a frontal most part of this kind
of a tube like body.
Right, let us look at a simple schematic,
so on the top of this slide you see kind of
a cartoon like animal, who's just a tube like
body and then you can see the longitudinal
axis right and the body is parallel to the
red to the horizontal.
So, now, once you have a body which is like
a tube, right and this body is moving through
the world and because what is moving then
it may moves through the world.
as a body moves through the world right suppose
it was the body it keeps making measurements
on the surrounding world.
Right wants to see what is the head of it
right, it will smell what is a head or what
is around it will try to process the sounds
as I coming around it now imagine a body of
an organism which can only afford.
So, many eyes are, so many photoreceptors
are now.
So, many years or has, so, many the olfactory
receptors inside the nose and, so on.
Now if it has freedom to put them wherever
it wants around its body right, where will
it put them it is logical to put all these
sensors right in the front of the body, because
the front of the body is a part which comes
into contact with the world has this tube
like body moves through the world?
So, you can see that most there is a high
density of sensors like eyes and ears and
nostrils and whiskers, right all these things
and I you see packed very close tightly or
in the front of the animals bodies, which
is the face or the head.
Right if that is the case then you have brain
somewhere inside now let us look at the question
of where should the brain be placed inside
a body let.
Say we are looking at 2 situations, in the
top figure you can see the brain plays slightly
closer to the front, or to the head right
in the bottom figure you see brain plays far
away from the head deeper inside the body.
If the brains are trying to minimize wire
length right and if you look at, if you find
out that this lot of wire go running from
the brain to the head.
Then you are better off putting the placing
the brain closer to the head because that
way you can minimize the wire length, but
if you find that the brains have lot of wire
going not to the head, but remaining parts
of the body then you are better off placing
the brain somewhere in the middle of the body
because that way you can save wire length
right.
So, which is the correct thing, so what is
really happening in real brains let us look
at that.
So, if we look at human nerve system right
if you would look at human wiring patterns.
So, this is a peripheral system, a peripheral
system is there is a consists of a bunch of
nerve fibers called the wires, these come
out of the central nervous system which is,
so basically the brain spinal cord and.
So, there are 2 sets of wires the cranial
nerves which are 12 in number and which basically
run from the brain proper to the 2 parts,
2 points on the head.
Then the spinal nerves which run from the
spinal cord to the rest of the body there
are 31 in number, there are 31 pairs of them
right and going to the both sides of the body.
Now, what we need to see here is, what is
the ratio of anterior to posterior connections
right.
So, the in this figure the green arrow is
anterior connections and the orange arrow
is the posterior connections right.
If the anterior anterior connections are much
higher than posterior connections the brain
has to be in the head and whereas, if it is
other way around brain will be somewhere far
from there.
So, if you look at the data from human’s
right in this table.
So, you see the cranial nerves and they all
add up to about 12 million fibers 4.5 million
fibers these form your anterior connections.
And the fibers of the spinal nerves right
the spinal the dorsal ventral nerves that
all add up to about 2.4 million fibers.
So, the anterior posterior ratio is about
5.25.
So, so when you have a high anterior to posterior
fiber ratio how your head your brain has to
be inside the head and; obviously, this what
happens in case of humans, but if we take
a worm like the C Elegans, and do a similar
calculation.
You will find that the anterior posterior
fiber ratio is only 1.5, I guess therefore,
it does not its forehead is further down it
actually does not have like a brain puppet
it just has kind of a diffused nervous system
unlike in humans, where we have a compact
brain in the spinal cord right and then you
have all these fibers going out of the brain
spinal cord, but in case of C Elegans, it
has a more diffuse nerve system consists of
consisting of a bunch of ganglia distributed
all over the body.
Finally I want to present this interesting
study right is there any connection between
intelligence and brains connectivity.
So, it turns out that I know there is an interesting
connection.
So, the study by Li et. al., they have looked
at the connection between the wiring patterns
of the brain and IQ or intelligence quotient
of individuals.
In this study they have taken seventy eight
subjects their assess their IQ, and the group
them in 2 categories.
People with general intelligence and people
with high intelligence or high IQ, so then
they have scanned the brains of these individuals
using special new imaging technique called.
A diffusion tensor imaging or DTI, and they
looked at the connectivity patterns among
87 cortical areas and the measured a certain
quantity called mean characteristic path length.
Which basically tells you what is the mean
path length of any given pair of cortical
regions, among these 87 cortical regions,
it turns out that mean characteristic path
length is inversely correlated with intelligence;
that means, part of smarter people have more
efficient packing of brain areas inside this
brain volume.
So, in conclusion we set off with the question
of trying to link brain structure with intelligence
and efficiency and things like that.
So, we have kind of concluded that there are
certain structural correlates, anatomical
correlates to intelligence right and we are
basically observed the required to produce
coordinated rapid whole body responses.
And a key mechanism by which they achieve
this seems to be by minimizing wire length
and.
Minimizing wire length seems or also correlates
with intelligence.
So, in the next part of this talk will apply
these principles to account for the brain's
evolution right.
Now as we have this kind of a nerve system
in which there is a brain and a spinal cord
and the peripheral nerve system, right in
mammals and you know higher animals, but in
very primitive creatures you do not have a
brain and spinal cord.
So, there is a, we have covered a long evolutionary
trajectory from a small a creature like say
hydra right to humans now what is the logic
of this evolution.
So, so we will try to apply the same minimum
wire principles to account for this kind of
an evolution.
Thank you.
