You know I've.

Yes there you lay it all.

Every right.
Kind of place that.

Out of Here.

Boy.

Right.

Yeah.

And.

Network.

Well I mean this is.

Thank you Paul.

I'd like to thank Paul for
a very kind introduction.

Although the twenty year many things and

all this makes it sound
like I need a walker.

So I'm not so sure anymore.

So actually given all those things that
Paul mentioned I may tell you about

something completely different.

Now in a physics department as we all
are in you might be wondering why

are you talking about neurons and I hope
by at least slide number three I convince

you that there's an interesting problem in
statistical physics buried in the system.

What I'd like to tie.

Well you about is actually not as
the abstract says which is a little bit of

a lie thinking and consciousness and
all those deep ideas but instead

actually something underlying all of
those I believe essentially what I'd like

to do is tell you about if you like some
of the sub routines that our brains run.

And hopefully the computer will go.

OK.

There we go.

And.

We It went so well.

It jumped twice.

OK we may have all kinds of amusing
computer related events today it seems but

I'd like to do is tell
you two related stories.

What I'd like to do.

First of all is tell you about
a central pattern generator.

It turns out that we have many
pieces inside our central nervous

system that are essentially
Autonomy's hardwired programs.

And these Autonomy's circuits
produce out of simple neurons

complex dynamical properties and it's an
interesting physical question Anderson and

how they work.

I'm going to tell you about one in
particular which is called the pre-bought

singer complex that is actually
responsible for the inspiration

rhythm in breathing so all you're all
sitting here looking at me I'm not so

exciting that you're breathing hard.

And when you're not breathing hard.

In fact your breathing is
controlled by Inspir atory circuit.

That actually activates your diaphragm and
a bunch of other things to make you inhale

and you essentially exhale because of the
Alaska city of your chest at this point.

In fact the pre-bought singer
complex is the evolutionarily fairly

new advance in the thing turns
out it came about with mammals.

Unlike many academics mammals in general
are high energy sort of creatures and

because of that they need a lot of oxygen.

So in fact the evolution of the mammal
involved the evolution of the diaphragm

and the evolution of the pre-bought singer
complex which apparently drives it.

So what I'll tell you about itself tell
you about this collection of about

a thousand neurons fictive
leave all identity.

They're not all exactly
identical like atoms are but

they're very close in their dynamics and
none of them are clocks.

But somehow they conspire to produce
this metronomic clock like signal which

controls our breathing.

What I'd like to do after that is tell
you about something which is in fact

related it turns out that when
we're not thinking too hard

in particular in anesthetized animals.

The interesting part of our brains the
part that actually involves things like

figuring out how the pre-bought
singer complex works.

The neocortex.

Actually produces its own
sort of rhythmic signal.

And I think we can actually
show that simple models

of interacting excited Torrie neurons
naturally produce the sort of rhythm.

And they do so
because of a certain type of non-linearity

built into the response
function of a single neuron.

So I'd like to try to explain these this
picture of simple neurons interacting and

I'm networks generating complex electrical
patterns as the theme of the talk and

I'd like to give you at least two
vignettes of how this goes on.

So here's the basic story.

I'll start with one and here I promise
you there be a slide mentioning.

Why is there some
interesting physics here.

Well.

The key thing to me it's interesting
about the problem is that this to me is

a classic example of emergent behavior
in a fairly simple physical system.

Each one of these pre-bought singer
neurons is not a clock at all.

It's not a bad clock it's
just simply not a clock.

But somehow when you couple them
together with sufficient numbers

they spontaneously collectively oscillate
in this metronome Aker clock like way.

Whereas if you start removing neurons
from the circuit you actually will see

dynamical phase transitions
between clock like behavior and

quiescent behavior or
clock like behavior and chaotic behavior

pending where you are in the phase
diagram that will explain to you.

Where's the new idea here and what the.

Thing which I'm really trying to
pitch to you is the following.

I guess a great deal of work
has been done in trying

to make simple low dimensional
models of neurons.

Of course neurons are highly
complicated devices with many many

independently opening
channels that let ions in or

lead ions out and these interact with
each other in very complicated ways and

if I were talking to people at a drug
company which has in fact never

happened but if I did they would want to
know how to manipulate those channels and

how you make them open and close how
you get more neurotransmitters in or

out things like this but if you don't do
that or you want to take a step back.

You can actually do very well
understanding the dynamics of a single

neuron treating them essentially
as nonlinear dynamical systems and

this of course is bringing coal to
Newcastle coming to Georgia Tech and

talking about non-linear dynamics.

But what I think is the new feature is
that people have spent a great deal of

time particularly Eugene as of
age trying to build nonlinear

dynamical models of different classes
of neurons and asking in what

sort of low What sort of low dimensional
representation of a neuron will reproduce

these sort of characteristic behaviors in
fact here's his pre-bought singer version.

Well what I think is the new feature
is there's a great deal of information

stored not just in the nonlinear
dynamics of the individual neurons.

But stored in the network
in which they are.

So what I'd like to show you is that
top logical properties of the network

of which these guys are connected actually
controls the collective non-linear

dynamics of the system so.

By the way the picture you're probably
wondering why this picture was here.

I set this up to this is actually
from minute one of the Italian.

Now aside from being an artist.

They issued manifestos.

In fact according to Wikipedia
which is that I source of all

facts outside of physics.

The Italian.

Futurists issued five hundred something
manifestos in one year by my count

that's better than one a day so I figure
I can have one manifesto a lifetime.

My man.

If I can join the Futurists for
a minute is the say that

there's interesting physics in looking
not just that nonlinear systems but

nonlinear systems coupled on top
of logically complex networks and

I think features of the network topology
are really what should be focused on

because they haven't been so far so
there I now have a manifesto.

So let me actually show you a picture
of the pre-bought singer complex.

I work with a gentleman named
Jack Feldman at U.C.L.A.

and actually with Robert Burns one
the physics department on this problem and

I don't know which end of the rat has
the brain to be honest with you but

Jack does and if you remove the slice
of the brain stem of a rat.

What you can do is you can actually take
out the pre-bought singer complex and

have it work in a dish in isolation with
no connection to lungs or breast of rat.

So what's nice about this is it tells
you what Tori properties of the system

are truly in dodginess I don't need the
rest of the rat to make this thing work.

I don't need a brain.

I don't need anything else.

What's also nice is it just so
happens that the collective output of this

pre-bought secure complex goes out on a
cranial nerves which is in the same slice.

So you actually have the system pre-wired
for the experimentalists you can actually

hook up your lead to one of
these little bits coming out and

see the integrated activity of the system
whereas you can also stick in Needles in

different spots and
look at the act and if you're clever.

Let's see if this works you first of all
have to find exactly where the pre-bought

singer complex is you can actually
change its chemistry by well.

I don't know how to do these things myself
but Jack knows how to do animations

what he's showing you is that
he can actually apply drugs.

The pre-bought singer complex and
watch how it changes things.

I'll come back to that point
a little bit later in the talk.

So at this state.

I think the movie is not
going to run it will not run.

So it will decide to live with that.

So at this stage what I'd like to do
is give you a very simple mathematical

description for how I want to think about
this obviously very complicated system.

I'm going to take two elements
that people have looked

at before and lump them together.

The first element is what people
call the leaky integrate and

fire model of a neuron.

Basically what this is is it takes
the complex structure of a neuron and

reduces it to one module.

This one module is essentially
an integrator receives but

it's a leaky one so
it has a R.C. time here.

Which is typically on the scale
of milliseconds in fact

that tells you the size of the spikes.

The width of these spikes I should say.

So basically if I try to
charge this thing up.

It will just drain charge back out to
the ground on a timescale of Tao when

it receives a signal from one of the one
of the neurons it's connected to it will

receive some sort of voltage increment at
the cell and now here's the key feature.

I guess I should go through all the terms
first you'll notice the sum is over all

neurons other than yourself for this is
the voltage of the neuron this I'm going

to call the adjacency matrix
of the connectivity matrix.

It's a matrix of ones and zeros.

If M I J is one.

It means that neuron J.

actually sends signals to neuron I and
it had zero it does and.

And this is the firing rate number of
spikes per unit time of the neuron.

Now the key feature in the what reason
that neurons can actually perform

a non-trivial computation is that they
have a nonlinear element in them.

And that fundamental nonlinear element
that everyone worries about is the fact

that the firing rate as a function
of potential measured relative to.

And outside the cell is strongly
the pendant on that potential so

neurons are in the business of maintaining
their potential at something like minus

sixty or seventy millivolts
relative to the outside world and

they do this by pumping things like
calcium and sodium and potassium.

However when you raise their potential
to about minus forty or fifty millivolts

all of a sudden they will start firing
action potentials they will actually.

Very briefly push their potential to about
plus one hundred millivolts and it will

take a down and there's a short refractory
period in which you can't do anything.

It sort of resetting itself and

then it does it again if you
take a pre-bought singer neuron.

Out of the singer complex and
you put it in a dish by itself.

You can change its potential by hand and
ask what its firing rate is.

And in fact you get that sort
of sigmoid all curve like so.

So in fact here you see data
from one lab where they in

fact inject current into
the cell to raise its potential.

And what you see is its resting
potential is about minus fifty nine.

I guess millivolts if I raise it to what
turns out to be about minus forty five or

minus fifty it produces an action
potential it discharges itself a little

bit produce another action potential and
Essentially this would go on for

a long time.

In fact you can see the spacing between
these pulses is quick it's on the scale

of ten milliseconds or so and in fact if
I were to keep stuffing current into this

thing it could keep firing for
a very very long time.

So in a way it's a very
simple computational device.

It's simply asking It's itself is
might be to has my potential been

raised to this threshold point and if so
I'm going to make lots of signals and

Wonderfalls below that I'm going to
make signals typical firing rates

are something like sixty or seventy
hertz for the pre-bought Singur system.

Maybe a few hundred hertz of the fastest.

For some other neurons and for the.

Boxing or
complex even when it's at the resting but

tension of about minus sixty millivolts
it has a firing rate of a Hertz or two.

So that's really what it does for
a living.

How is it that you can take a network of
these things and make a clock out of them.

So you might imagine what you have is
you have the room filled with a bunch of

amplifiers with a speaker and you could
easily imagine a type of exponential

growth of sound in the room due to
feedback feeding back feeding back and

so on one neuron excites another
which excites two more and

you get the sort of runaway effect.

So you could imagine these guys could
all mutually excite each other and

make everything be active.

But it's not easy to see how
you then shut the system off.

Wait is fine.

Period of time turn it back on again.

Shut it off turn it on again and so on.

How do I make a clock out
of such a simple model and

the short answer is you can't
with just this picture.

If we want to understand how it's
possible to make just excited

Torrie neurons coupled together into
something that acts like this metronome.

We have to do something more.

And the something more means we have
to think a little bit more carefully

about how neurons actually
communicate with each other.

So.

Here if I decide to blow up that
point model of a neuron and

actually look inside and
see what's going on.

I realize that neurons actually
have many compartments.

First of all the neuron
has the one point I

was talking about which people call the
soma because the apparently biologists all

learn Greek at some point in their lives.

Neurons have a big antenna array
where they receive signals which

they call a dendrite or dendritic Arbor.

Again the same classical
education pays off.

I guess.

And they have a C. they have one.

Typically ramified output signal
which is called an X. on.

The X. on hill it is actually
where the action potential is

produced which travels
down the axon as this

electro chemical signal until it
reaches what's called the synaptic.

Junction in life as I said
this picture you can download

yourself from within pedia.

And at that point the electro chemical
signal when reaching the sin that junction

causes the release of neurotransmitters
into this nano scale synaptic cleft

which actually opens membrane channels
in the other neuron the what's called

the posts and after neuron allowing the
potential of the dendrite of that other

neuron to change which then
propagates back to the so

much where the integration
if you like happens.

So that's really what we need to know the
key feature here though is the currency of

this exchange is potential but neurons
don't directly control their potential.

They control their conductivity.

The only way that the neuron
actually knows that he gets a signal

is by locally changing the conduct
to video of the dendritic membrane.

That actually means that the dendrites are
inherently nonlinear signal processors.

If I have a signal entering the dendrite
here and propagating down to this way.

If in fact another signal
enters if you like downstream.

Then in fact the local conduct of any of
the section of the dendrite is enhanced

and some of my signal here is now
going to be shunted to ground.

So if you think about it.

I actually have an antenna.

But the way that the antenna is receiving
a signal is by locally manipulating it's

conduct this is been known of course for
a very long time and people said well

that's of course unfortunate because
it makes the problem much harder and

in fact people in the past
that sort of non-linearity and

dendrites the cell tries to avoid
In fact if you look very carefully.

They say at a variety of neurons you'll
find actually on the nano scale there

are lots of little buttons or
what they call spines and

the synapse is actually not on the main
but actually on the little button

that's separated from it and
the idea is that way.

You're not actually changing the conduct
to Vittie of something that is in series

with your transmission cable.

Now it turns out pre-bought singer
neurons don't have the spines.

So in fact they don't
work to avoid this and

my claim is they use this dendritic
non-linearity to produce a signal.

So if I want to include
this idea which Christophe.

Calls dendritic shunting.

What I'm going to have to do is I'm
going to need to have an extra variable

in my extra variable I can call
an adaptation parameter or

of a bold like Mr Feldman I
will call it calcium.

And the reason is as Jack Feldman
has done the following experiment.

He is injected a key leading agent for
calcium into the soma.

So essentially what he's done is he
sucked up spare calcium which is in there

and he said.

Does that change the ability of
the neuron to receive signals and

the answer is No not at first.

Nothing happens.

But if you wait for the calcium to
diffuse out into the dendrites all

of them in the he later to diffuse
out into the dendrites and

suck up calcium in fact what you find
is that the dendrite becomes much less

conductive In other words it
transmits signals better.

So he claims that the rate the key
ingredient here is calcium but

the theory it can remain
exhausted on this point

what I need is a second scalar
variable to make this work.

Really.
So if you look at it this way I call this

the feldman Bill Negra model because all I
really did was take what they told me and

turned it into a set of linear
differential non-linear first

order differential equations and
what they said is the following.

They said.

The response of the somatic potential
to an action potential coming in now.

Depends on the state of this calcium
variable that lives in the dendrites.

Every time I receive
a signal from another cell.

I increment my calcium level a little bit
and the cell actively pumps calcium out.

So there's some sort of
exponential decay of it anyway.

And what we know is that the timescale for

pushing calcium out is long compared
to the time scale or the R C time.

Of the cell in fact this we think is
on the time scale of half a second

where this is on again on
the time scale of milliseconds.

So what you have is you
have a very slow build up

of calcium because you receive
signals in the den droids and

as the calcium level builds up
the increment in voltage you get for

an action potential starts going
down once you hit some threshold and

then calcium sensitive channels
open which makes the thing leaking.

So if you like you can say that the cells
are exchanging currency of information

with your action potentials.

But the value of a unit of
currency of an action potential

starts going getting small once you
hit a certain critical calcium level.

So now we have a system that actually
has a chance to turn off again.

So you can again go back to
the picture of having lots of

these amplifiers with speakers and.

Microphones in the room and you can
say Well first of all I can actually

build up a lot of activity
because one all the cells are in

this regime here are all the gayness
turned out on all my amplifiers.

So I start producing a lot
of noise in the room.

However at that point every
single microphone question.

Threshold.

Yes so I can rescale my c to put Sistar
wherever I want at this point so

I don't have all I really need is
I need this to be fairly sharp.

You can do a little bit of extra
essentially numerical checks to see

how sharp it needs to be and
it's not actually that sensitive.

The thing is that.

Essentially we know these
things are sigmoid old because

most of these channels are essentially
two state systems they either are open or

they're closed and you essentially reach
a type of Fermi function type thing for

a population of them where most of them
are open when you're in this state and

most of them are closed in this state.

So and there's some sort of
threshold type behavior for them.

And that is important.

You definitely need this
to be non-linear but

it that the degree of the slope
of the curve here and so

on is actually pretty irrelevant for
what I'm going to tell you.

OK please please do ask questions
along the way there's nothing worse

than being bored of the talk because you
didn't like some point along the way.

So.

So the key feature what I'd like to do is
I'd like to take the plain old integrated

fire model but now say the value of what
I'm integrating is going down when I

receive lots of signals and actually
the key point is going to be it matters

whether you get tired or the game gets
turned down because you receive signals or

whether the game is turned
down because you send signals.

I'm going to try to convince you.

That's an important feature.

So this is the basic model.

And my claim is that it's actually
physiologically motivated and

as much as most neurons work hard to
try to avoid this sort of then Drake

non-linearity but our friends in the
pre-bought singer complex don't do that.

So let's do a what I call a simple
mean field analysis let me say that

all of that all the neurons are coupled to
all the other neurons and now I have two

non linear equations and I can numerically
integrate them to my heart's content.

And of course you'll
find metronomic behavior.

You'll see that in fact the neurons excite
each other they start the potential goes

up to where they're spiking a lot.

However when they're spiking a lot their
calcium levels in the dendrites go

up once it crosses the threshold.

They can't effectively excite
each other very well anymore.

Now the potential of the neurons
collapses on an R.C. time

constant falls very quickly to this sub
threshold level they don't spike anymore.

The calcium slowly decays
falls below the threshold.

Now they can start exciting each other
again and the process repeats over and

over and over.

So if I reduce the problem to
a system where I throw away all

the details of the connectivity
of the network.

It's very easy to get this to
work in fact if you do this.

It's very easy to show there's
no difference between whether

the calcium level depends on how many
signals I receive or how many signals I

send There's no way to tell the difference
in this sort of picture because what

you've done is you've lumped the whole
circuit into just one effect of neurons.

If you do that.

Of course you lose some of the fun.

If you want to think about this in
a little more detail in sort of

simple according to stroke outs.

You can actually ask yourself what kind
of sticks points there's this non-linear

system have and what you find actually is
that this mean field picture predicts that

if I had just as my free parameters
the number of neurons I have and

if you like the excitability the Basler
excitability of a single neuron in other

words how much of a voltage increment do I
get when the calcium levels set to zero.

Which is another way of saying that you in
the real world you don't actually change

the size of the action potential but
you change either the threshold point for

spiking or you change the basal
potential of the neurons.

But I doesn't really matter in
my model which way I do it.

I'm going to fix the basal potential and
the threshold and

change the size of the the currency
of information transfer.

So another way to say it is of
delta v zero is a big number.

It means a very few spikes Juran one
to excite neuron two whereas delta V.

not is a small number.

I need many many spikes from neuron
one to neuron two to excite it.

So fine.

If I'm in the regime with the sufficiently
high number of neurons and

what I'll call moderate delta v.

I actually find a way to
the this metronomic limit cycle.

So I have a.

Have a civics limit cycle
around this fixed point here.

These by the way or the milk lines of
the potential curve in the calcium or

adaptation parameter curve.

However.

If I go to small enough numbers of neurons
or low enough levels of excitability

this little piece here plunges through and
produces a stable in an unstable fixpoint.

And in fact this point what I'll find is
that there's only one stable fixed point

here which corresponds to a low level
of potential and low level of calcium

at this point I have effectively
a quiescent system in the mean field where

things are spiking very rarely but not
sufficiently to get anyone else excited.

So much like many of the students in my
classes they don't really do a lot and

they don't get terribly excited.

However occasionally you know
like I'll trip or something and

everyone will get excited and
everyone will start talking and

that is this high activity fixpoint So
in fact if I have a sufficient number of

neurons and A actually medium
number of neurons with very

high levels of excitability I find
only one stable fixed point now.

And now I don't have any sort of limit
cycle behavior I actually just have

all the neurons on all the time so

the options you have in this model
are essentially an on off switch.

That's being metronomic leaflet back and

forth leaves the physiologically
desirable state.

This corresponds to not breathing at all.

And that's all you know less desirable.

And this actually corresponds to all
the neurons firing all the time which is

actually seen in experiments too.

And there are again it's not
physiologically desirable of course.

So the question of now is this is a very
simple mean field picture is it actually.

Well here's the fact the phase diagram.

This is the limit of stability for
the S. O. phase if

you're into the stroke got C. in language
of these non of nonlinear dynamics and

I suspect many of you are in fact you're
probably better at the mean this actually

is a type of second order
transition where the period of

the stable oscillations diverges
smoothly as you approach the slide.

And assuming you could adjust
this integer and smoothly and

in fact it's a Sneck if
you like those words.

This on the other hand is a sub critical
hot bifurcation it's more like a first

order phase transition where
you go from either the stable

the oscillating to the high activity face.

And again my axes are numbers of
neurons and this excitability or

maximal excitability or
maximal size of a voltage increment.

Let's look at the stable E. oscillating
phase a little bit more carefully.

If we numerically integrate
these equations for

say a few tens of neurons
using this model.

What you find actually is that in
the stable the oscillating phase.

You can easily identify two
populations of neurons.

There are the neurons that Mr Feldman
likes to call emergent pacemakers.

And these are the ones that have
a a voltage trace that looks like this it

shoots up very quickly at
the beginning of a pulse and

then tires itself out with
other ones talking to it and

decays away and then the rest of
the neurons essentially is to amplify that

signal by coming up a little bit later and
then drifting back down in fact you

can see this sort of shoulder in each
one of the pulses here is due to

the fact that there's a subpopulation
of these guys added on to this.

The reason this is interesting.

Well the reason it's interesting.

Biologically is that there's been
a question in this field about whether you

actually can produce this circuit
with every neuron being identical.

The reason this is an open question.

Isn't that the real world of
the pre-bought singer complex.

We know that there is a sub population of
intrinsically oscillating neurons that

Feldman and others who study this turn
off biochemically which is where they add

the drugs to shut those down.

So there have been a number
of models where X.

something of a controversy in
the business about whether.

You need to have these intrinsic
pacemakers or whether in.

Some since they're a backup system.

We know what they might be doing the
strong version of the intrinsic pacemaker

model or hypothesis would be to say
the system cannot physiologically

oscillate without some sort
of input periodic signal

coming from these pacemakers
the alternate view which is Feldman is

that the intrinsic pacemakers are in
some sense a back up system and

it is not necessary for the physiological
oscillations the problem with that

it is least previously as people
said well if that's the case.

How is it that we always see
certain guys starting the process.

I claim the reason you always see
certain guys starting the process or

at least that's what they believe is
because not because these neurons

are special biochemically but because
these neurons are special because politike

they're special because they're
connected to the right people.

So to try to understand that.

Well clearly in our model since all
the neurons are purely identical.

It must be the case but can we actually
do a better job of explaining that.

And the answer is you can actually do
a fairly good job in the following

way you can do the following company.

You can go through in a numerical
example of say sixty neurons and

look at all of them and you can color in
the ones that are the Pacemakers they're

the ones that shoot up first as opposed So
for instance this blue one is a very good

pacemaker where this orange one is and and
you'll see of course this is a question.

Yeah.

So OK so in my model.

None of the neurons are different.

If you're a pacemaker in my view.

It's because of features of your
connectivity every neuron if you were to

cut it out and put it on the table would
be identical to every other neuron and

why because it's my model
I made it that way.

Yeah.

Now in the real world we know that there
is a subpopulation of pacemaker neurons.

And depending on who you talk
to between twenty percent and

eighty percent of them are pacemakers and
again depending on who you talk to in

the neonatal animal they're more active
than they are in the adult animal.

But again those things
are being debated but

you can turn them off in the experiment.

Yes.

Yes so what.

So the fact I guess I should
have told you that I'm sorry.

So how did I make my network.

It's a great question.

So what I do is I take and neurons.

I connect every neuron to every other
neuron with a fixed probability.

This is in fact a simple way of producing
was called in air to shred a graph.

So in fact you end up with
these exponential distributions

of numbers of connections so
it's not like Laszlo Barabbas E.

with his power law distribution is not
like the World Wide Web in that sense.

So in the World Wide Web guys that
are really well connected are sort

of extra really they're sort of too many
guys that are really well connected.

Whereas here you have this
exponential tail of guys that

are really well connected.

And it's actually very
hard in the pre-bought

Singur complex to know precisely
who's connected to whom.

But what you can do is you can
stick a needle in one neuron.

And then actually make it fire and then
stick another needle in the another and

look for the voltage signal but

it's a little bit like fishing if you
don't see the neurons you stick one in you

find a neuron then you fish
to find which other ones.

So from that essentially what you get is
statistical information you go if I hit

twelve neurons.

What how many of them typically
did I find the signal and

the answer is that roughly each neuron is
connected on average it seems to about

a sixth of the other neurons in the system
and the connection scheme seems to be

independent of distance there is no sense
of there's no real metric in the space.

The whole system is a millimeter
type object or last and

the probability that neuron
A is connected to neuron B.

seems to be as far as they know
independent of where they are in space.

So.

That's how you build the network
now if you go in there and

you ask yourself you can quantify
what you mean by pacemaker and S.

in the following way you
can say who fired first

who fired second every time a burst
started and we'll call that the ranking.

So if you're here you're a very
good pace maker you fired first.

If you're here.

You got on the train at the very end
of us when I was leaving the station.

Now what you can do is you can calculate
how popular a particular neuron is or

how well connected it is this is of course
a very old experimental subject which

goes by the name of high school but
you can actually do this.

Mathematically a little
bit more carefully.

Which you can do is you can
assign a eigenvector central of

the ranking to every neuron in
the following way you can take in the list

of positive integers with positive
numbers as put one on every neuron.

And so consistently to find those
numbers in the following way.

Your popularity is going to be equal
to the sum of the popularity as

of everyone who is connected to you.

And this because this actually if you
think about it looks a little bit like

calculating eigenvectors of your or
your connectivity matrix.

If you do it carefully your
central of the ranking is a set is

the eigenvector associated with
the smallest eigenvalue of that matrix.

Now if in fact we had
a purely linear equation.

This would have to control exactly who
fires first the non-linearity seems to

mess this up in a way that I
don't understand very well so

what we plotted here was
the central of the ranking.

So if you have a low central of the
ranking it means you're the most popular.

So we set it up this way so
that the curve.

If in fact everything was perfect.

This would have a all the points would
lie along the slope here of one.

And what you see is the central of the
ranking does a very good job of predicting

the laggards the guys who are the guys
that do take part here seem to be more

spread out than I can fully understand
from this and it has to be because that.

If I want to say that the pure
the eigenvector central of the controls

everything then in fact I have to
have a linear system of equations and

that would mean I had to linearize this
firing rate which is clearly nonphysical.

So I suspect the reason this fails here.

There's still a strong correlation with
the fact that it fails here in detail

must have something to do
with non-linear interactions.

I don't understand right now.

I should mention in passing
that if you are Google

you use eigenvector central of
the two week pages as well.

So.

Now you could say.

So you've been simulating
this network Alex.

How about you tell the phase
diagram actually look like for

a real more real numerical
simulation as opposed to the.

Your field theory and the answer is
roughly speaking you get what you want if

you have small numbers of neurons and
what not to much excitability there quiet.

What else would they be.

If in fact you have a sufficient number
of neurons and they're fairly excitable.

You get stable the oscillating.

However the phase boundary between
the stable the oscillating regime and

what the mean field theory predicts to be

the high activity phase does not look
like the mean field prediction at all.

In fact if you quantitatively compare this
line calculated numerically to our mean

field prediction for the boundary between
the stable eos elating and quiescent

phases you actually do pretty well
the blue line you can see is horrible.

This is you know at least
in a way this is the C.

plus at best and the interesting thing
is that as in you usually the errors

are the most interesting part of
it look at it it isn't that this

curve is was drawn jerkily there
are actually steps in this.

And in fact I'm going to show you that we
can predict where these steps occur by

looking at purely topple logical features
of the network which have been law.

Lost when you do the mean field theory.

So what's the feature that
we're looking at here.

What are these magic values of N.

at which the phase boundary seems to get
hung up and of course I'll come back

to this magic numbers and really give
you the answer I can't keep a secret or

tell a joke properly as I
always give away the punchline.

The answer is the cake course
which I will try to explain now.

So we're all familiar with percolation
theory at some level of I take

a bread board and I start putting
in resistors at random at some

critical density I will actually find a
conducting pathway of resistors connecting

across my system even in the thermodynamic
limit of really big breadboards and

lots and lots of resistors.

Mathematicians in
the ninety's it appears and

perhaps late eighty's cells I should say
generalized this idea to saying what it

really means if a percolating core of
the system is that you are connected

to at least two other guys two other nodes
who themselves are connected to at least

two other nodes who themselves are
connected to two other nodes ad nauseum.

So if you take the cursive definition
you cannot just talk about percolation

with a two core you could talk about three
cores or seventeen cores you can say let

me isolate the piece of a network that
is four which the nodes are connected

to say seven other nodes who themselves
are connected to seven other nodes.

So the way I like to think about it is
those are the hubs of the airline system

if you like.

So if you think of airports as nodes and

flights as the connections
to the edges of my graph.

Then you can say L.A.X. and land.

And Chicago and a few others
are part of some sort of global

thirty core Why because you can fly from
Atlanta to thirty other airports from

which you could fly to thirty other
airports but if you're in Santa Barbara.

That's not true.

You can fly to L.A.X.
and you can fly to San Francisco and

they're probably part of the thirty
Corps but Santa Barbara isn't as.

It's not connected to thirty other
airports who themselves are connected to

thirty other airports.

So if you want to actually look for
a particular core in a network.

You can follow a simple algorithm and

I think actually explaining it helps
you understand the idea of a cake or.

The algorithm goes ascent.

As follows.

My initial network circle all the guys
that are connected to less than three.

So will look for a three core.

So here we go.

This guy's connected to one no good.

This guy is connected to two no good.

This guy is connected to no good.

This guy's connected the four.

But we'll see what happens to him in
just a minute now remove all the guys

that were not connected to three.

Now you have the system here.

Now go through it and do it again all of
a sudden this guy fell below the threshold

because the two of his friends that he
needed to be connected to for vanished.

So now this guy vanishes and this process
could continue multiple steps but

it wouldn't fit on a slide.

So in fact this is the remaining
three core of this original network.

So in some sense the hub system of this
network I conveniently drew on the top.

Whereas these guys are somehow more
peripheral to the network than

the rest of them.

What's interesting about that in
the thermodynamic limit where I don't draw

little things like this but
I imagine letting the system become

infinite you can ask yourself
at what concentration of bonds.

Do I lose a particular cake or and

in fact it's a phase transition but
it's sort of a funny one.

If we define the order parameter of this
just as you do in percolation to say

what's the probability that
a node chosen at random is

part of the percolating cluster.

Well we know when there you can't span
the system you will actually have zero

probability of any node being part of it
but what you'll find is at the transition

the probability grows smoothly from
zero in a continuous phase transitions.

However if you go to a three core not the
two core which you find in fact is that

the probability that you are part
of the spanning three core is.

Actually goes has a discontinuous jump
to zero at the appropriate density

in fact what's unusual about this system
and something which I don't fully

understand I'm not sure anyone
else does either at this point is

that there seems to be universal scaling
behavior here but a finite jump zero.

So it has the sort of mixed character and

it's something which I think has
been seen through simulations but

is not been well understood in sense
an interesting stat my problem.

Frankly.

But not quite what I
want to talk about today.

What matters for me right now.

Is that there's a discontinuous jump in
the probability of being part of a cake or

and that jump actually
gets to be quite large.

When K. is large.

So in fact if the stars and

show it here but if I look at
the network and I ask where it is

the what does the probability of being
part of the seventeen core look like.

Well essentially the probabilities
are almost one and

when I have such a seventeen core is
the network so multiple He connected.

Everyone's connected to at least
seventeen other guys who are connected

to seventeen other guys that
probability of picking a guy random and

it not being part of that
is just a few percent.

Now I start killing off nodes you could
imagine I start fogging in airports around

the world and at some point I find out
that the entire seventeen core collapses

there are no more seventeen core
nodes left but what I'll find is that

the probability just before the collapse
was something like eighty five percent.

So I go from almost everybody being part
of the seventeen core to nobody being part

of the seventeen core and all that does is
it requires one more node to be removed.

So in fact you can kind
of see that even here.

You kind of fun to play with
if you kill one more guy here

you'll find out that
the three core vanishes.

I think.

I'm not sure if it does on that one but

here I'll show you
an example where it does.

Here's a little test case
let's look at the five core.

So since it doesn't matter where
the network it where the nodes

are I put them on a circle.

By the way this is a.

A network graphing package which comes
out of Bloomington and it's very nicely

set up you give it a connectivity matrix
it will draw these pictures for you.

So here we go here's a little test case
I forty three nodes and the red ones

are part of a five core and you can see
most of this is red on the boundary.

There are a few guys that are part of
the four core which are peripheral now

they're in blue and there's actually four
guys that are part of the three core.

Now I kill off one of these at random and
what I find is that there's more for

core but there's still a five core.

Now I kill off one more guy and
all of a sudden all the red ones or

everything is part of the four core
with a couple of three core ones.

Now I kill off one more nothing
particular happens why it just so

happens that the network I had
had a magic number of forty two

which is of course makes for
a good story given hitchhikers and such.

Now if you want to get rid of the four
core you might have to track down

to thirty six and then all of a sudden all
the blue ones will vanish and you have

nothing but green ones and perhaps some to
Cora left over and something like that.

So what's going on here is that

because of the first order nature
of the transition you get this.

Even for a finite size system you get
this discontinuous collapse of the entire

system going from over
half to none in one step.

So if you go through and take our networks
and you take one network of our pre-bought

singer model and you start killing off
neurons which you'll find is that you'll

cross over from a stable E. oscillating
phase like this to this high activity

phase where you have all kinds of complex
dynamics were different neurons are firing

at different times and as you cross
this line you can map out this boundary.

Now you do it again a different
level of activity and

you'll do it you'll get the same thing or
you'll get a slightly different thing.

And now you can go in and
purely top logically say

let me take this sort of analysis
figure out where the cores vanish and

I'll put those lines on
the graph separately so.

The dashed lines here this dash line might
be the disappearance of the twelve Corps.

This might be the eleven Corps This might
be the ten Corps I don't have the numbers

in front of me currently do this one
is the nine Corps Apparently so.

In fact what you see is that
there's a very good correspondence

between where the horizontal lines are and
where you lose the course.

Now if you redo the network with
a different connectivity for

a small sample you will find that
the magic number will shift a bit.

But then when you go calculate
the phase diagram for

that one you'll find that these
boundaries also shift a bit and

they keep lining up so
this is an example of a general effect.

Now.

What's going on.

Why is it that the cake or is matter.

Well let's think about the problem
physically a little bit.

You have all these active
neurons firing and how is

it that you can synchronize to turning off
of all of them what you need is you need

the calcium levels in the dendrites of
all of them to reach this threshold.

So that they can't receive
signals efficiently.

They're voltage relaxes and they sort of
reset and everything starts over again

the way you end up in this chaotic
high activity state is that some of

them reset and others keep going and then
others reset in the other ones keep going.

It's a little bit like
if you have a riot and

two people sit down because they're tired.

Well they'll get sort of
reenergized they'll keep rioting

it doesn't turn off the riot.

So if you want this to work.

What you need is you need to have it so
that when guys turn off they turn off in

such a synchronized way that anyone
left over can't really talk to anyone.

So the key feature is if you
pick two neurons at random

that are not part of your
take your highest K.-K.

core the probability that they can
actually communicate with each

other without passing through the cake or
is vanishingly small.

So it's a little bit like
saying if you want to fly from.

Champagne Urbana to Paris.

You're almost certain to fly through
Chicago or New York or Atlanta or

one of the other.

Hi K.K. core networks.

So if I were to shut down
all the big airports.

You pretty much can't go anywhere anymore.

So what you need is you don't need
everyone does synchronous turn off.

Turn off in a synchronized fashion but
you need the entire cake or

to turn off in a synchronized fashion so
that increase the level of excitability.

I need to be able to drive the calcium
levels up in my highest cake or

neurons sufficiently.

So that those guys.

Is there the one connection so
they receive the most signals so

they'll be the ones that get tired first.

So if in fact for a given level of
excitability I have a sufficiently high K.

take or left then I can shut down
communication in the network and then

the system can reset and those guys on
the fringes of you like can fire away for

a little while but then they'll get
tired because no one is well they'll

Excite is no one's
talking to them anymore.

No one's exciting them.

So if you shut down the cake or

it appears you essentially shut down
the communication network of the system.

OK I see that I am running out of time.

Let me give you a very brief vignette
of a slightly of the same model

wreaths retold different
with almost the same model.

If we switch from our brain stem to
the cerebral cortex the part where you

actually do the thinking.

You've in doubt that the majority
of the neurons are these

excited Torrie neurons with a complex X.
on going off in a dendritic tree.

This is a pyramidal neuron and there are a
few intern or ons which are inhibitory So

now we have neurons that actively D.
excite the other neurons so

we have a network of guys with
ferromagnetic couplings an anti for

a magnetic couplings.

What's interesting about the system so
schematically you can say I have excited

Torrie neurons that excite themselves and
excite inhibitory neurons even have a Tory

neurons that inhibit the excited Torrie
neurons and actually inhibit themselves.

So I can build a slightly
different schematic picture.

What's amusing about this is that
if you have an anesthetized animal.

So presumably it's not
thinking about very much.

And you actually look at
the potential in its neocortex

you'll find that it actually has
a periodic metronomic like behavior.

This is data from my colleague at U.C.L.A.
Maida and

what you see here is
that he's stuck eight.

He's actually essentially tapped
the phone line of one of these neurons in

the cerebral cortex and what you see is
its potential rides up rides down rides

up rides down and the reason it's doing so
is actually there's

enough firing from other neurons in
the system that the whole of the tension

the whole ground level is floating up and
down a bit but the actual neuron

is firing just rarely it fires one spike
or nine per what's called an upstate.

So this is a down state oscillations
you can actually measure these

things outside your brain with an E.G.

so that you actually get this
sort of brain wave pattern.

So the question is this come out naturally
from a very simple model and why do I want

to include dendritic shunting the reason I
want to include dendritic shunting is that

I want to shut down the network when a guy
is firing only once or no time at all.

So it can't be that this
neuron decides to D.

excite because it's fired too often
because it may not have fired at all.

However these pyramidal neurons have
on average tens of thousands of

synaptic junctions on them so they're
sampling a lot of incoming signals.

Even though they fire very rarely.

So if in fact the system somehow
deciding it's time to dx site.

It can't be because of
the signals it sends it has

to be because of
the signals it's receiving.

So you can do exactly the same sort of
analysis of a mean field picture and

what you'll find is that
within the mean field you can

use the same sort of
the dreaded chanting ma.

To explain the oscillations.

In a sort of musing way which you'll
find actually that you now have

three differential equations one for

the potential of the excited Torrie
neurons one for the inhibitory and one for

this adaptation parameter which we
think is a which we think of as calcium

if I fix the calcium at a sub threshold
level then in fact what I have is

I have a fixed point corresponding to
the down state and I have a fixed point

corresponding to the up state separated
by a saddle point in between them.

Now if I start increasing the level
of calcium what happens is the.

Which guess.

So if I give I increase
the level of calcium

this unstable fixpoint will ride
up the curve of the line here.

And actually annihilate this
the upstate fixpoint So

the system has to fall
into the down stage.

At that point the calcium levels decay and
this is actually a new saddle node split.

Happens and I produce a fix point.

Well I didn't drop for you.

So I guess I won't tell you but

essentially we can think about it is that
the up down state oscillations within

this model occur because
this unstable fixed point.

Alternately shuttles back and
forth to annihilate one or

the other fixed points and
it does so in a periodic fashion.

So in fact here it is.

I don't want to wear out my welcome.

So I want to just hit the main point here.

If you look at this result.

You'll find that it's actually not so
good.

The reason it's not so good is that
you see this ringing in the upstate.

There's no evidence of this sort of high
frequency ringing going on in the data.

So you might say well maybe that's the end
of it and why is there is sort of ringing.

Well if you look at orbits near

the substate fixpoint they spiral into
the fixed point in that spiraling.

Setting this frequency for you.

And the reason is that this
fixpoint is very unusual compared

Well it's certainly different
from the pre-bought singer one

in the pre-bought singer when you can
think of it as that as a pencil that

flops from one side to the other side
has a stable point lying This way

you throw it over here it's stable here
you throw it back over here and so on.

Here you actually have something
kind of amusing happening.

What I didn't tell you is that
the potential at which you are stuck here

is actually corresponds to
the high susceptibility point

the point where the firing rate is
changing most rapidly with potential.

So the system goes to a quiet state
in the down state throws itself

up into a state of maximum
sensitivity to external signals and

falls back into the ground state again.

So it's as if the pencil balances
itself in the state at the top for

a while then falls back.

The reason it does so is that you
actually have excited Torrie neurons that

are exciting inhibitory neurons which are
then exciting the excited Torrie neurons

and the system is tuned in such
a way that it actually reaches with

some pullin Skee calls it
chaotic balanced state.

So with the dendritic chanting We can
go from this some pullin ski style

balanced state here of high susceptibility
back into a quiescent state and

then back up into this
funny balance state.

And the reason is that
the dendritic shunting weakens

the effect of the excited Torrie neuron so
then it falls back off.

So it's like you balance
a pencil on its tip but

because the you've been tilt
gravity just a little bit.

It always collapses the right
direction for you if you like.

So let me skip this turns
out that there's chaos and

all these things which I don't have
a lot of time to tell you about.

But let's do the following
it turns out that.

The up down state oscillations in

your cerebral cortex are remarkably
synchronized over the whole thing.

So and in fact we know that
excited Tori neurons these

pyramidal neurons have a very long at.

Exxon's that connects millimeters away.

So in fact there is the possibility
of long range coupling.

So let's take the simplest possible model.

Let me imagine I take one
downstate oscillator unit and

I weakly couple it to another one.

Now let me tune the strength
of this coupling and

ask Is it possible to tune the strength
of the coupling in such a way

that the Global up down
States are synchronized.

But all the sort of oscillations

the ringing in the up states
are decent criticized.

So to look for that we can compute
the correlation function of

the potential in one guy in
the potential of the other guy and

what you'll find is that
if I turn the coupling off.

Well they're not correlated at
all that makes perfect sense.

I have up down States in my independently.

Now if I tune the coupling
there's actually a window of

about an order of magnitude which if
I had more time I could explain why.

Where in fact you get this
partial synchronization.

And if you look at it.

What happens is all
the states in all of these

up down state modules turn
on together they have non

synchronized dynamics in the up state and
then they all collapse together.

So it's actually a way this is
I believe this is essential

because if you actually want your
neocortex to perform computations.

You can't have complete
synchronization of everything.

Otherwise you just have one circuit
element but which you seem to want to

do if you want to be able to go into this
up state do something non-trivial and

then go back into this down state.

In fact if you don't.

If you mess chemically with the system so
it doesn't go back into the down state

you end up with things that at least
electrically look like epileptic seizures.

So we can talk about that
at the end if you want but

the fact I'm very close to the end.

Let me just review the main point
the only thing I've added to these models

that other people have talked about.

It is what I would call multi compartment
nature of neurons with dendritic shunting

So rather than saying that
a neuron is some sort of nonlinear

processor of signals
that it integrates up.

I'm saying.

Actually you have to take into
account the dendritic tree and

note that signals coming in
interact with each other.

Non-linearly through this
adaptation parameter calcium.

Now that actually leads to a much
more sophisticated question what is

stead of looking at spiking rate models.

What if I actually followed action
potentials the spikes themselves.

Is it possible that the topology
of the dendritic tree actually

in hard wires in a particular
signal processing algorithm.

So if you go back and look at what
people have done in neurophysiology for

one hundred years but what you see is that
they're very they're vastly different

sort of connectivity strategies here
you see that you have a whole bunch of

long lines all connected at one base level

tree here you have something which is this
highly ramified sort of like beta lattice.

So is it possible that this
sort of structure in codes

a different sort of integration of
signals in this sort of structure.

So if fact if I have this sort
of structure a sit ups down at

the bottom has because of shunting has

the power over all the information
that occurs upstream from it.

Now here you have a much more
democratic integrator because if

every synopses on it.

If every sin apps is on a second
a separate dendritic process.

Then in fact you really do
have a point model integrator.

But.

Simple that neurons are actually using the
structure of the dendritic Arbor to do.

Non-trivial computations and
is the idea of a single neuron

being particularly simple in only
complex networks of neurons thinking.

Maybe oversimplified in itself
in fact maybe the right picture.

Is that neurons are endowed genetically.

With some sort of extra non-linear
signal processing long before

it makes a decision about
whether to fire or not.

So I guess an I should think
my collaborators next but

let me just say that.

I would never have heard about the
pre-bought singer complex of Jack Feldman

hadn't given a physics
colloquium at U.C.L.A..

And up down States likewise with my Maida
Robin and I have been collaborating on

this project and a former graduate student
David Schwab did all the work on the K.

course and he's now working with
Bill Bialek and company at Princeton.

So I'd like to thank you for your
attention and take the question if I may

thank you very much.

Yes.

You know the fact that's a really
interesting question because it depends

what kind of animal you look at
you might say that we're smart so

we should have some clever
connection scheme and

lobsters are dumb because you know we
eat them so they can't be too clever.

So they must not have a very
clever connection scheme and

you have it exactly backwards.

There are many people who study
central pattern generators who look at

things like lobsters and
mollusks of various sorts and

what they find actually is when you
have very few neurons like a lobster.

You actually have a genetically
programmed connection scheme so

that fact the circuit diagram
is apparently prescribed and

each neuron has very
different properties in it.

So you actually have a system of
there is a central pattern generator

that makes lobsters stomachs
grind up the food properly.

So there's a pattern of
the muscles that you pull and

you know you that contract
to grind up lobster chow.

So.
So this is the so this circuit of like

ten neurons or twelve neurons and
each neuron is fundamentally different in

its input output characteristics and
they're all wired the same so

every lobster you fish out of
the tank has the same connection

is far as we know it's not true for
higher mammals or us or anything else.

Our pre-bought singer
complexes as far as we know.

Are.

Well described by saying
they're randomly connected.

Then it's possible we just don't know but
one thing you can think about is

the following if you have lots of synaptic
junctions and you have lots of neurons.

There's actually not enough
genetic material in your body

to encode the entire circuit diagram.

So it's if you have enough complexity.

You can't actually prescribe
who's connected to whom anymore

in a reproducible way.

One question people have asked in the past
is if you believe in dendritic shunting is

it still possible to create a network
that will learn a particular signal.

Pattern recognition program or
something and the answer is it can.

At least there are certain
patterns in fact it can do

something that's kind of clever a summer
student and I worked on this a bit and

I think we understand the basic idea.

But you can actually build small networks
of neurons which people like Minsky and

others call perceptron but
the word goes back further.

These things can essentially
act as logic gates but

they can actually with shunting do
a funny type of time ordered logic.

So if you take in two bits of information
in A and B. you can have an AND

gate if signal A comes before signal B.
but it could be in or

gate of signal B. comes before signal way.

So you can do some sort of clever things
like this and in fact you can build

circuits that will do three bits and
four bits with time ordering as well.

So there are a couple fun things
you can do if you prescribe

the system very carefully.

But what's interesting is that at
least some of these things are learned

a bull using.

In learning rules for instance.

That's maybe more than you wanted but it's
the right answer and the rest of it to.

Questions.

Yeah.

Yes So the receiving the signal
is somewhat stochastic

the firing is apparently
quite quite robust.

So if I take the neuron and
I raise its potential by hand.

It will fire a nice clock.

Or the fire can roughly with some fixed
spiking rate as far as I understand

now the reception of the signals
in some cases is actually so

when you actually have an action potential
going down to the synaptic junction.

There's only some probability that enough
neurotransmitter will go out to actually

cause the signal to reach the post
synoptic neuron and there we don't

we don't take that into account but it's
certainly something you can think about.

I should say that.

There's someone else in the physics
community whose thoughts

in particular about
the pre-bought singer system.

Paul Bresso Hoff and
his collaborators now at Oxford.

Have built a model where still casts this
city is very important to their version

is that you have a quiet fixed point and

noise actually throws you off of that
fixed point and then you go around some

long limit cycle and then fall back
to this fixed point essentially.

So your you have a fix

you have a slight You have an almost
stable limit cycles sensually So

when you point you have to kind of wander
around and then fall back into it.

So the upstate is wondering around.

And that's a perfectly good model for
the pre-bought singer complex and

as much is like our mean field model it
will give you roughly stable oscillations

if you have enough noise in
the system to make that work.

The thing which is important
is whether in fact.

The thing I think is important is
whether in fact as you change the number

of neurons you actually see these plateaus
in the phase but in the phase diagram

Jack Feldman is trying to do that
experiment now so if all is right then.

In fact one of the key six
years will be these cake or

steps which was not included in these it
should not come out of these other models.

I should point out that even if
you change that the neurons get

tired because of their own firing.

Then you do not get the cake or
steps either.

So in the sense of statistical physics.

I believe whether you whether
your dendrite gets tired or

whether your SO MY gets worn out is
actually relevant perturbation and

gives you different collective dynamics.

Yeah.

The these the strength of the couplings.

Now I'm really now out of my depth but my
understanding is that they don't change.

So this far as I know this is not a system
that is present actually there serve it.

There's a neonatal training
period apparently.

But how that works I don't
know anything about.

And if you want the full story.

It turns out there's two copies of
this oscillator one that does X.

Aleisha and one that does inhalation.

And they seem to interact with each
other in that the nice the interaction

changes in the neonatal animal
in the actual breathing animal.

And like right now because you're not
running around you're not actually

using that acceleration circuit but
it matters because opiates

turn off the pre-bought singer circuit but
they don't turn off the excellent one.

So it's actually possible that sort
of rescues you know it's the thing

is the reason why the Rolling Stones
still are around.

Or.

They leave the membrane high finance.

You know mystery and red blood cells.

He's also with us.

One kept us for two more days because I'm
leaving Friday early Friday and although

the schedule is relatively full There
are couple slots open here and there.

And if you would like to see here.

Please send e-mail systematic
a Georgia Tech school physics and

with that very much.

Thank you.