piz Pizza how watch I Okay so continue yesterday about Mech model particular the mech
Shing sh and theel that we stud today is the years ago our understanding of how SP has by
so just a brief I will go de um in a few seconds but quickly what we did yesterday is to note
that respons that Sim set up understand so we have that is this structure here there a pipe
that is measuring the electric potential inside itself so the inside and experiment like this
one typical recording looks something like this so this is the C ined neur and this is the poal
of the the differ so L of to line response so the lower line isession the middle line is increase
in poti inrease and then Andre that is followed by this really R events that uh show how
much large response respons okay and uh so there are three features that we want to so
first is in the ABS of any input the me so we done here the me potential is not so there is a
difference in memory potential from the inside to the outside the cell then the second
aspect that we want to understand is this exponential relation so if the steps are small we
have changes and and the third ASP we want to understand are understand now what mean
is to a me explain so yesterday we introduced a few biophysical properties that are
sufficient to understand all these ingredients so we about the fact that neurons are cell so
they are separated from the surrounded by layer that is blocking ion to FL inside outside of
but then on the of the are two typ of communications so open allow the and found that try
to maintain a concentration of different I Parts forance s we've seen yesterday that if we
combine points one and two we can explain the first so if we combine these elements we
can explain this two part so the equilibrium potential so the value of this me potential in the
of input is explained by the fact that the I pumps that contain certain concentration in the
surround and then if we combine them with the that we have when the are open we can
explain this equum and then we der a simple electric C that captur the Dynamics of the neur
response to input and we sh that this electric C can explain this exponential in particular the
time SP of the response is fix by the capacity of the neuron and by the resistance of well so
these are the main result show yesterday okay so the question that remains to be answer is
what mechanism and okay so in order to answer the question we will do many sub steps so
the first thing to do is to generalize the equivalent that we yesterday in a way that we will
explicit two Ty ofum that I will show you is some history of science so the experiments that
ASX have done to come up with that model in particular this experiment will be used F the
parameter that will determine this then one key result that out experiment is that the
conduces that characterize the solum CH are not Conant as the resistance yesterday but
they are and this to be AAL aspect mechanism and now then comb ingred show you what is
this so then the the part we probably discuss next time is about how we can generalize this
model so it turns out that there are there is aity of NE responses so not all NE responses is
respon to C injection look like this so typically suppos to Curr are B enough but the the
temporal arrangement of SP changes so there are cells that first of five cells that the
beginning of the stimulation and then become silent and all these aspects can cap the type
of model that we discuss the theor that people can just include other type of ss that are
specialized in capture this different okay okay so any questions so we start with the first
part of the discussion so relation of the equivalent sa in the cas which we we want to moreit
M so the approach is so set up there is a NE that we always consider as unique this is
something to mention so yesterday the neur as a single comp so we consider for instance
the me potential of the cell is a quantity that poal you can imagine when you have situation
where you have that is poal at this point the might be different potential the or so we can
study this type of model this type of phenomena using what I called compartment mod is
that for each part of this neuron you will model to this one we we will not do that okay so
we focus on this single comp we assume that the neur is a single ened by and so said
yesterday expect to the outside inside of the ne has an excess of negative and uh in a
protction so ingredients so first a capacitor that represents the abity of to separate toate
Char chares the then there a resistance that models the resistance that I FL inside and
outside of the cell enounter this and then another thing that we included is thisor that
model the we app and then and mathematically one way to write this is so this is where is
and and we use this we use differentation of this equation the one that will be useful is this
one in which we explicitly write down this Infinity that is the potential that reaches the
member that the me potential to reach for long enough time so you infity this Infinity is
given by so this is what we discussed yesterday and then the other thing that we discussed
yesterday is that this point here this is fix by so this is fixed by this thetion the twoes discuss
diffusion that we say this okay so what the first thing that you want to do is to merge these
two modeling result so the is fixed by thises dams is given this so simple way to canel thisit
introduce additional term here this additional term that is so if we do that basically what we
have is that if there are inputs to of inputs the neural will converge to this value that we and
if if the current is from Z then the m potenti to that is so we what we is that now we get
automatically the that the new potential and from the perspective this corresponds toif
there what we do is to add what is called a volage generator which ensures that the
difference between potential potential between these two points is equal to here so we can
uh we can show that this using we can show that this is actually equation here so this is so
we have cacity in so so we to the equation describ the using and show that so we use of
relationship that mention so we have so this is the current that is given by and then to just I
didn't understand what is the circle for in the in the in the this one yeah so it's a c so is it like
a battery uh so a battery a fix amount of so you for our purpose of this this is just a machine
that always outputs a f amount of current I well this one is a machine that ures that the
difference in volage between this point and this point is here MH okay so to this one so we
know that the difference in from here to here is e so the difference invol of between these
two points will be B minus now the C that fls here we can compute it using slope so this
current here can be given and okay so the only point was to show you that this is higher that
corresponds to okay the is just another Factory that generates another difference of yes the
only purpose there is to it's so we know thats of any any simulation the voltage should be
equal to here and so this is what happens here so let's say that you take out this cator here
so what would happen is that equilibrium there will be no of Curr so this difference in
voltage will be zero and so the voltage that here will be equal to the voltage now the next
thing that we need is just I'll just you so it is that so if we want to model explicitly uh the of
specific s on the M the way to do it is this one so what we do is that this was our model so
far so let's say that we want to model explicitly potassium s the way which that is to add a
branch that is in par with two that is this one so you have a button that represents the
equilibrium value of the iron corresponding Totum so the the the the difference that have
equilibrium in and then in you have a resistance that is correspond to the resist to and the
same let's that you you want to add also Sol so what you have is that you have another with
resistance and this is potential of so this is modity and then again if you go to experiment
like the one that we show yesterday you have any questions this so the one not yeah now
I'll I'll I was about to to say that so just um okay our first answer your question so the the
point is in general let's say that we know that on the the neur are different type of s
depending on theom that you're interested in you want to you might want to model
explicitly a subset of the two and so add represents all the other quantities that we don't we
don't calcium and chlide forance so we don't want them explicitly we just include them
inside so we can say that our if you so this model all the aspects of the that we don't
describe expc Does this answer your question and then so for every that we want to include
we we we write down our resistance and potential for so for each model use only the S but
uh are we show of this class that are other aspects like the uh adaptations F response that
can captur if you have another C on top of this that is an adaptation so it's still selected for
potassium but that has different by physical prop okay so then this so in then what you have
is that every model of this we Des today you so the general model of the time here can be
expressed in this form so the term the depends on the capacity that is this here and then
you have a term that depends on the injected Curr and then you have minus the mem Curr
and the mem current is the sum of terms of this time that are for instance the Le term that
we WR out here or the so I write theity but you will have a current analogous term for the
potassium and sodium each one of these branes we have a expression okay so okay so the
this concludes the first uh aspect that I want to discuss so I mention that I in order to ince
the asking model to so the next thing I want to show you is using this framework that we
how we can understand the experiment so experiment was done using the D of the so
basically it so you should imagine these experiments were done more than 50 years ago
now and at that time to measure what was going on inside the so I show you that in order
to measure M potential we need to stick an elect that measure the M potential and uh if
you take a cell in our body that's really hard that's the main so the key one of the key
advancements that was done in this experimental preparation is to do the the experiments
in one type of neuron that is really big that is this neuron that mediates an esap an escape
behavior in the so the idea is that the squeen when it's present do an escape behavior that
basically it propels water outside of the mouth and this allows the squid to move really fast
in so this is an behavior and evolutionary pressures shapes the the neural system that rates
this Behavior to make it as fast as possible and it turns out that the propagation of signals on
Axel is faster if the Axel is larger so because Evolution what out is that this giant in the squid
is really large because it is necessary in order to make a fast and so what they realiz is that in
this this animal there are this and so with the tool time they could measure the M potential
so that's the first point then another key ADV that they had is that they were able to take
the a outside of the body put it in a chemical preparation and show that uh inside this
chemical preparation the a condition was still able to produce so basically have a set which
have the neuron embedded in a medium that you control in which you can measure the M
potential and also observe spes and uh so this is an example of now so this is an example of
spy produced inside the animal and this is an example of spy produced by the same type of
neon when it's taken out of the so still so that's problem so now what we do with this is the
f is this why we inject and uh and M to from an experimental preparation like that what I
have is that into cell measure the PO and then what we need is the so that here and set up
they measure they have set up to measure the activity and then control the C and then
what we do is a preparation called the F CL so basically in this setup you say I want the neur
to have a c voltage let's say you want the ne to have a volage vol so the the device that they
and at every time step injects an amount of car such that the voltage measure in the neuron
laes the the target okay so this is what is called and so now if the voltage of the neon is
constant and equal to the Target one what that means is that this Thea in time of the volage
will be zero so let basically the setup is buil in such a way that the voltage of the cell will be
Conant so if we use that in this equation what you have is that your the Curr that you inject
must be equal to the C so this gives you way to measure given the voltage that you that you
select how much are the currents this make sense any questions for this yes so we say that
given the fact to inject the current the result is that the voltage does not CH and we can this
the fact that it stays constant to determine charge the currents so that to do that there is
addition St to make that now but okay so now so there are few steps that need but for the
important thing is if you use this V SC preparation you have a way to measure the man now
couple of things so if voltage is not is large enough so that you are in the region of voltages
where typically spens this here is and in addition this is typic what discuss just forget and so
basically the current is equal to the sum of these two currents now so what you can do in
addition is I mentioned that in this experimental preparation you control what is the
environment the is so thing that you can do is to put it in an environment where there is no
okay let's say you put it an environment in which there is no s so at that point you're sure
that the must be equal to the potassium because you Environ the and this gives you direct
access to this okay so this is an example of what these things looks like okay that's an
experiment so this is the V Target as a function of time I describ here so you have a step in
this V Target this produces the neur response is actually here that's the of the ne Conant as
aun now the response the total that you measure this it is this curve a here so this is a SP
magnified the the SC is so this is only the scale so that's the sum of these two CS but now we
want to know what is the individual contribution of each one of these two the way to do
that is as I was saying you put it the neur in environment there is no s so the total Curr that
you me at the point if you do the same experiment is just thei and this is the here and now if
you do the difference between these two so the difference between the Cent In which you
have both and the Cent which you have the difference between these two gives you now
the Sol and this is this current here see so that's a way to experiment the yes so when
environment only the is fix because of the then if we keep also constant injected current
changes in fact this is so these currents are actually the injected current okay this is it's the
injected current in a different pration this is the injected current in normal conditions this is
the injected current if you have only potassium and this is the so this would be the injected
current if you only other question this so now this gives access to this total C so the product
of the conductors the and now there is a way to also extract this individual contribution so
what is the conductance and what is the and it is that have so basically the experiment gives
you this and you also control this so you can make a pling this voltage so let's say you do the
experiment you have acept voltage you measure C and then you can repeat the
experimenta volage and then you give you a certain function of that relates the the the C
through the so this intersection Point here is this Ral poal and this the slope of this curve is
the condu and so I just want to point out in the case in which uh we uh for conduct for
instance this emission sh was lar what they realize with this experiment is that in the case of
sodium and potassium the conduct is actually a function function so one output of this
result is that forumi this together you can get a sense of what the the Dynamics of is so B
function let's say you look at a that is this here seen before so this is just magnification what
happens this so This Is Your Potential Dynamics and then if you put in all your results what
you have is that initially you have an increase in sodium conduct which means that before
this point there was no s because the so no pass then at this point you start to have non
which means that you have a FL of s ions that will go from the outside inside of the C so this
will produce your increase in Po and then we right L there is a a current of potassium that is
this current here uh and this will produce fluxes theide so this is the of the type of dams you
get questions this okay so this is just an experimental that the the second point that we
have dismissed so these are the results of the experiment of and the next thing that we do is
to give AIC model of this conduct and particular stud I mention that conduct but we have a
differenty of particular the potassium conductance is called aist conductance and I will show
you in a second what it means and then the S conductance is called I will show you in a
second what itans so what we do next is to study a model of channels that capture the type
of dynamics that this conduct have supposed to change the vol okay let's start with
persistent conduces that are relevant for so we want to try to get a simple model of this
condu and so you should imagine CH as a really complicated object so this is just
schematized as a Caron here but so for the S of the discussion we have today we your I think
of having like a door that can allow or not the become slightly more complicated than this
okay so so this is the first uh step model so it turns out that the modeling of this CH is not
deistic but AAS so at every given time you have a probability and this probability is what will
be depend then turns out so this is the the list of things that I mentioned here is something
that was developed empirically we just found what is a fun form that captures the data and
this is just the end result so what they realize is that in order to express this function you
can think of the channels made by four different subunits that work independently you can
imagine that as having four independent doors and it door can be subit can be open with a
probability so this would represent the the probability that each that is open and then the
probability that the is open is the product of these two of these okay so what you have is
that the the case of so again this is uh when it was first introduced it was just found that this
is a functional form that captures data but then when with the development of molecular
Dynamics and crystallography people realize what this means so there is evidence now that
Chan are made by different compartments that can be f as independent of course like this
but that's the the overall mechanism that we have four independent elements and in order
for I uh to flow you need all these elements to be open okay so this is uh the probability that
our potassium CH is open and now with that what you do is to write the conductance of
thisi Channel you write it as a constant that is the maximum conductance times um the
probability of the so this is so this is the function for this mod to to capture the the
dependency on voltage of now where the voltage dependenc in so now what we do next is
equation for this getting and okay any questions this just wanted to make briefly a point so
here I describe the Cur to the conduct and the as but if the channel is closed or open then
this is an on or off process okay so the channel in each channel the C will either flow or not
so in fact this continuous description is basically what you have imagine that you have a
large number of chanels that are the Sur of the neuron and the description that you provide
with this model is just a description of the colletive Dynamics of so this is represented here
with simple simul so if you only have one s described by this type of equation at every time
you would have that the CH is open for so the the a that predicted this EXP this one but if
you only have one at every time you zero or maximum now if you have multiple channels in
parall then the total produced by this channel will approach this prediction that we have
question this is this obvious yes we volage do we mean like the voltage the outside voltage
the voltage coming into the neuron so okay so this is uh okay yeah two points so that's not
for the P but is for the so that's the POS component and then the voltage I haven't described
yet vol but will be the the the so what we typically do is to assume that to take as a refence
the outside voltage that to zero and then the voltage that we talk about is always the inside
voltage any other questions this so this was just a side note that variable and we see them
because we are on the that there is not only one okay okay so now what we want to do next
is to derite an equation Des the Dynamics of the of G so we said is the probability that if you
look at one of these different units here it will be over so each one of this so each unit can
be in two Stace can open or so what we want to D is called that capture how the probability
that one of the subs is open evolves with Okay so let's say that you are uh so how the prob
of being open time so at step you can either be open and this is something that happens
before and then so this is something that pariz with that we so this is the rate of closing so
the probability that is open becomes CL and this weend to decrease probability of so give a
terus then the other thing that happen is that we have a subun that is closed so and this
open and the rate of from close to open we write it as a function okay so this is what is
called atic equation that describes the evolution that a given subunit is open or close as a
function of time and it will only depend on this rate of opening and closing of the the sub
I've been told that you've seen this in stas processes class do you remember that okay okay
so this is our model for the dams we can rewrite this model in different format that is this
one the what I did is just this form in such a way that I have a that will be this and then of
here and the reason for writing it this way is that it is easier to interpret what's going on so if
we consider so if this Con then theend Dynamics will be allow to the Dynamics we describe
as a function of time and we go from initi condition and exponentially this an and this will
happen over a time scale and so when we see cases where we voltage we change how
andage imagine that so let's say that what CL is because okay so so we're trying to comp the
changing time of the probility of so this is given any given time being open depends on so
we can either let's say that close and then we open and so thise or you can be open and
then become so this will basic the fact that the other questions on this okay so then what
ask did is toally fit this functions to data by doing the experiment that I showed as a
functional voltage and then try the measur and these are the typical ex so you have the any
Infinity is a function that inrees volage this is something that becomes smaller as so the larg
voltages the f and now if together we can look at what is the Dynamics of calcium in this
model if potassium is described by this type of so this is here so again the the potassium
current is given by this expr here so the total conduct given by this g k then the probability
of opening be N4 and then you have all so so let's say that you do an experiment in which
you start from- 60 Vol you increase the voltage of the cell using the B PL that I describ this
will change this beta n and Alpha function so that basically Al will increase you will have a
larger rate of opening of sub unit when bet decrease you will have a smaller rate of going
from open to you expect on aage that more sub will be open so the end dynamics that you
get is this one so it response to the step you have an increase in 10 and then the current is
proportional to the four and so you have this Behavior here and the important thing to
remember is that so as long as the voltage is up you you always at Cur and this is why resist
if you apply a constant steing volage it will produce a constant current okay any questions
this so this is theel that we for this persistent model now we do quickly a similar arum for
the just describe an over effective model it just applies to Sol the difference we find is that
rather than being persistent this condu is means that respon to a constant stepping voltage
the fles of sodium the sodium ions ini increase and then go back so you have current a brief
so the way which this Cur model can be schematized using this here so the IDE is that you
have now two type of subunits you have an activation gate that is anous to this doors that
we have here and is described by this variable n and then you have what is called an
inactivation gate that in this is this here and it's aable from 0 to one that is H the the reason
why these two aspects are described differently is that at low voltage this is typically close
and this is open and as you increase voltage you have this this gate that open so this isation
and then this think this now is more likely to be Clos so they have an opposite Behavior with
I will show it that in order to describe quantitively this what you that the probility of
opening depend on the probability that this is open and that this inactivation so the function
for the parts of the data this functional form here so it's m the k k and H this is the and then
you put everything together you have that the sodium current is given by the maximum
condu okay so now for each one of these two sub so both for M and H you have a dynamics
that is is described by equations equations this one it say that now the function will be by
function specific this have Al there is one thing that you should notice that is this one so I
mention that so again you can reite this expression using a different notation so with and
infinity and so M Infinity behaves exactly as INF so it's as you change voltage it goes from
zero to uh to one so for large voltage this activation gate will be one so the gate will be open
no Vol this G will be Clos well for H Infinity has the opposite so at low voltage H isal to one
which represent this part that the B is the tach from the CH so this will allow the current
well for last voltage is close to Zer so it has the opposite Trend with voltage and then the
have a similar Behavior what we matter is that the N varable the activation is much faster
than the this is what so this is if you com this what you have so let's say step voltage of you
have certain change in the alpha and beta function for M and over what you have is
depicted here so H so the inactivation variable is initially something and decreases with
volage so this is what we then the activation m is initially Zer and then increases the so the
total C is the product of time so the total current will be initially zero then increase and then
decree see have so remember I think it's useful that in order to be open you need to have
this this be theable this Tye and this and what changes between the two is that at low
voltage this is Clos and this is open at high voltage theable is open but the is Clos so and uh
trange but so let's that you go from low to Vol you go from close to close it's just that what
happens is that this sub here open much faster than the time it takes for this sun to close so
when you do a steping voltage you have the this opens and then after a while this get mhm
any other questions please okay so that's the the second the second step of this model so
we have now understanding of the Dynamics of the potassium current and the Sol combine
this to to be the okay so just to summarize what we show so far we said that describe these
neurons we put in parall with the capacitor and the Le term of each branches that
correspond to the potassium current and the sodium current and the critical thing that we
show now is that these two conductances here are not constant but they are one depend
and the dependency that we found is but if you together the this is how they like so for the
time constant you see that the time constant for the activation gate of the sodium sorry of
the sodium soundance is much faster than all the other so the activation of uh sodium is
much faster than the all the other variables then the other thing that is uh worth
remembering is that as a function of voltage M and N will one well H will Z okay so now if
you put everything together this is how you bu the the model summarizing this SL so is the
following so this is uh the series of steps that happen when elction potential is let's say
initially when there is no current being injected in the neuron we said the memory potential
is C value is fixed and the there is no sodium current and there is no potassium current so
this is because m is there is no s car because basically the activation m is Clos and there is no
potassium current because each gate can have a like4 probability of being open but now the
currents depend depend on the four and this becomes small number okay so at equilibrium
without current injected these two currents are Zen now let's say that you do a current
injection that drives the M potential by am of is typically around 50 so this point the moves
along this axis here and what you have if you are aroundus 50 m here foret here for the
time you see that m so the activation of the s c is the first is the short so the sodi is the first
respon and the response we said would be something like this curve here so you would
have a fast approach to the value M infinity and M Infinity if you're aroundus 50 m is around
here so it's so which which means that you will start to have opening of this m sub so the
first thing that you have is opening of this m sub this opening of M sub unit now is possible
because there is no inactivation so H will be small but now it is it is slow to exp so H is still
open so for Sol you have that H is open and M is open so you have flux of sodium ions so
you have influx of positive ions so you have B INE you have this influx but now what
happens is that you have a positive feedback and this is what detes in the SP a really rapid
increase in protction B increase influx in positive I but this influx in positive I will increase
further potential and so you go fromus 50 you tend to go higher along and this will continue
after basically the the point in which you reach the reveral potential of C that is around plus
you have a really massive influx of then what happens next is that finally inactivation G up
and this closes the Sol CH and blocks the flux of sod at that point you have um the
potassium that you remember is the is also slow with respect to that now also starts to get
activated so this you start and basically potassium positive go from the inside the of the cell
to the outside so this will to decrease your mation okay this outflux of potassium what
brings down the voltage in the second part of the so now there is a continuous flow of
potassium up to the point in which um the mem potential becomes PR negative and so at
that point and infinity BEC Z so the Al theum and at that point you're basically around here
what Happ just relation value the different so just what happens this is to the point let's say
you are and you have C this leads to an increase in M potential that allows the influx of
sodium this in terms increase sodi the cell to increase in the potential you have POS and this
is really rapid potential that you doing then when you're here you inactivation of this St
variable so the the sodium Channel become Clos now the only thing that means is
potassium that basically flow outside outside of the cell and so it brings down the m prot so
you go from here to here and then when you're at that point also potassium inactivate and
you have relaxation Dynamics so this is the series of steps that you have in when you so any
questions this while but it's pretty straight forward if you think about it for a couple of times
just have to remember that there is this interaction between two different sh of potential of
the spice is given more by The Sod chel yeah so the sodium is what determines this initial
really the potassium is what produces the fast decrease and then all of them together
produces this relaxation effect other so so that's what I'm show you now is the ask this Ser
of St that solity and uh the model recapitulates the type of exp so what I describe is the
response to single step that incre Andreas potential but simate the respon for Conant
injection what you have is a series of spes an to the one that I show you in the first
experiment at the beginning of the class you see so thetion that is not the only possible
responses that seen in experiment so here should imagine each one of these is a single neon
and for each neon you're doing the same type of experiment so you do a step current
injection and you leave the current on for time and then you take down and you see that
some neur respond to different of behavior classical that you see is adaptation that is
depicted here the response to this in you have initial SP then spikes become dly more
separated in time like here and after why the doesn't respond anymore to the then you can
also have bursting that is this type of here you have but having spikes that are regularly
dispers you have this rapid series of spikes that then follow by an interval time the then
again SP again so you have this throughout this experiment the injection is Conant so it's
three property of the and then other things that you can have are like this thing here where
you have a delayed respon so you the current initially you have increasing me potential but
then after a while start to and these are just three example Behavior but so the thing that I
want to emphasize is that all these can be capture by basically adding currents like the one
so the the idea of this models is to you always have in model these things you always have a
term that is to the memory capacit contion and then what you do is to add ter to inside this
so you are this is the the the this is in order to have and then on top of that you can add
other Curr that correspond to other the and all these channels so this is what is powerful
about the model that we today that all this channel can be described by an equation of this
F so you will have your current will be given by a maximum conductance times a probability
that the channel is open theal poal and this probability Express using a combination of
activation and inactivation Val they can be expressed as a c activation variable to certain
Power Times an inactivation variable to another Power and these things in function voltage
and function cany so for instance in the case of adaptation this is an example ofation in the
model looks like this one so this will be injected current this is the sodium current that we
show the Le current the potassium thisr and then in this NE you see that injection become
gradually more delayed up to the point that again you don't have you can capture that by
adding an additional potassium that has the piz Pizza how watch I Okay so continue
yesterday about Mech model particular the mech Shing sh and theel that we stud today is
the years ago our understanding of how SP has by so just a brief I will go de um in a few
seconds but quickly what we did yesterday is to note that respons that Sim set up
understand so we have that is this structure here there a pipe that is measuring the electric
potential inside itself so the inside and experiment like this one typical recording looks
something like this so this is the C ined neur and this is the poal of the the differ so L of to
line response so the lower line isession the middle line is increase in poti inrease and then
Andre that is followed by this really R events that uh show how much large response
respons okay and uh so there are three features that we want to so first is in the ABS of any
input the me so we done here the me potential is not so there is a difference in memory
potential from the inside to the outside the cell then the second aspect that we want to
understand is this exponential relation so if the steps are small we have changes and and
the third ASP we want to understand are understand now what mean is to a me explain so
yesterday we introduced a few biophysical properties that are sufficient to understand all
these ingredients so we about the fact that neurons are cell so they are separated from the
surrounded by layer that is blocking ion to FL inside outside of but then on the of the are
two typ of communications so open allow the and found that try to maintain a
concentration of different I Parts forance s we've seen yesterday that if we combine points
one and two we can explain the first so if we combine these elements we can explain this
two part so the equilibrium potential so the value of this me potential in the of input is
explained by the fact that the I pumps that contain certain concentration in the surround
and then if we combine them with the that we have when the are open we can explain this
equum and then we der a simple electric C that captur the Dynamics of the neur response
to input and we sh that this electric C can explain this exponential in particular the time SP
of the response is fix by the capacity of the neuron and by the resistance of well so these are
the main result show yesterday okay so the question that remains to be answer is what
mechanism and okay so in order to answer the question we will do many sub steps so the
first thing to do is to generalize the equivalent that we yesterday in a way that we will
explicit two Ty ofum that I will show you is some history of science so the experiments that
ASX have done to come up with that model in particular this experiment will be used F the
parameter that will determine this then one key result that out experiment is that the
conduces that characterize the solum CH are not Conant as the resistance yesterday but
they are and this to be AAL aspect mechanism and now then comb ingred show you what is
this so then the the part we probably discuss next time is about how we can generalize this
model so it turns out that there are there is aity of NE responses so not all NE responses is
respon to C injection look like this so typically suppos to Curr are B enough but the the
temporal arrangement of SP changes so there are cells that first of five cells that the
beginning of the stimulation and then become silent and all these aspects can cap the type
of model that we discuss the theor that people can just include other type of ss that are
specialized in capture this different okay okay so any questions so we start with the first
part of the discussion so relation of the equivalent sa in the cas which we we want to moreit
M so the approach is so set up there is a NE that we always consider as unique this is
something to mention so yesterday the neur as a single comp so we consider for instance
the me potential of the cell is a quantity that poal you can imagine when you have situation
where you have that is poal at this point the might be different potential the or so we can
study this type of model this type of phenomena using what I called compartment mod is
that for each part of this neuron you will model to this one we we will not do that okay so
we focus on this single comp we assume that the neur is a single ened by and so said
yesterday expect to the outside inside of the ne has an excess of negative and uh in a
protction so ingredients so first a capacitor that represents the abity of to separate toate
Char chares the then there a resistance that models the resistance that I FL inside and
outside of the cell enounter this and then another thing that we included is thisor that
model the we app and then and mathematically one way to write this is so this is where is
and and we use this we use differentation of this equation the one that will be useful is this
one in which we explicitly write down this Infinity that is the potential that reaches the
member that the me potential to reach for long enough time so you infity this Infinity is
given by so this is what we discussed yesterday and then the other thing that we discussed
yesterday is that this point here this is fix by so this is fixed by this thetion the twoes discuss
diffusion that we say this okay so what the first thing that you want to do is to merge these
two modeling result so the is fixed by thises dams is given this so simple way to canel thisit
introduce additional term here this additional term that is so if we do that basically what we
have is that if there are inputs to of inputs the neural will converge to this value that we and
if if the current is from Z then the m potenti to that is so we what we is that now we get
automatically the that the new potential and from the perspective this corresponds toif
there what we do is to add what is called a volage generator which ensures that the
difference between potential potential between these two points is equal to here so we can
uh we can show that this using we can show that this is actually equation here so this is so
we have cacity in so so we to the equation describ the using and show that so we use of
relationship that mention so we have so this is the current that is given by and then to just I
didn't understand what is the circle for in the in the in the this one yeah so it's a c so is it like
a battery uh so a battery a fix amount of so you for our purpose of this this is just a machine
that always outputs a f amount of current I well this one is a machine that ures that the
difference in volage between this point and this point is here MH okay so to this one so we
know that the difference in from here to here is e so the difference invol of between these
two points will be B minus now the C that fls here we can compute it using slope so this
current here can be given and okay so the only point was to show you that this is higher that
corresponds to okay the is just another Factory that generates another difference of yes the
only purpose there is to it's so we know thats of any any simulation the voltage should be
equal to here and so this is what happens here so let's say that you take out this cator here
so what would happen is that equilibrium there will be no of Curr so this difference in
voltage will be zero and so the voltage that here will be equal to the voltage now the next
thing that we need is just I'll just you so it is that so if we want to model explicitly uh the of
specific s on the M the way to do it is this one so what we do is that this was our model so
far so let's say that we want to model explicitly potassium s the way which that is to add a
branch that is in par with two that is this one so you have a button that represents the
equilibrium value of the iron corresponding Totum so the the the the difference that have
equilibrium in and then in you have a resistance that is correspond to the resist to and the
same let's that you you want to add also Sol so what you have is that you have another with
resistance and this is potential of so this is modity and then again if you go to experiment
like the one that we show yesterday you have any questions this so the one not yeah now
I'll I'll I was about to to say that so just um okay our first answer your question so the the
point is in general let's say that we know that on the the neur are different type of s
depending on theom that you're interested in you want to you might want to model
explicitly a subset of the two and so add represents all the other quantities that we don't we
don't calcium and chlide forance so we don't want them explicitly we just include them
inside so we can say that our if you so this model all the aspects of the that we don't
describe expc Does this answer your question and then so for every that we want to include
we we we write down our resistance and potential for so for each model use only the S but
uh are we show of this class that are other aspects like the uh adaptations F response that
can captur if you have another C on top of this that is an adaptation so it's still selected for
potassium but that has different by physical prop okay so then this so in then what you have
is that every model of this we Des today you so the general model of the time here can be
expressed in this form so the term the depends on the capacity that is this here and then
you have a term that depends on the injected Curr and then you have minus the mem Curr
and the mem current is the sum of terms of this time that are for instance the Le term that
we WR out here or the so I write theity but you will have a current analogous term for the
potassium and sodium each one of these branes we have a expression okay so okay so the
this concludes the first uh aspect that I want to discuss so I mention that I in order to ince
the asking model to so the next thing I want to show you is using this framework that we
how we can understand the experiment so experiment was done using the D of the so
basically it so you should imagine these experiments were done more than 50 years ago
now and at that time to measure what was going on inside the so I show you that in order
to measure M potential we need to stick an elect that measure the M potential and uh if
you take a cell in our body that's really hard that's the main so the key one of the key
advancements that was done in this experimental preparation is to do the the experiments
in one type of neuron that is really big that is this neuron that mediates an esap an escape
behavior in the so the idea is that the squeen when it's present do an escape behavior that
basically it propels water outside of the mouth and this allows the squid to move really fast
in so this is an behavior and evolutionary pressures shapes the the neural system that rates
this Behavior to make it as fast as possible and it turns out that the propagation of signals on
Axel is faster if the Axel is larger so because Evolution what out is that this giant in the squid
is really large because it is necessary in order to make a fast and so what they realiz is that in
this this animal there are this and so with the tool time they could measure the M potential
so that's the first point then another key ADV that they had is that they were able to take
the a outside of the body put it in a chemical preparation and show that uh inside this
chemical preparation the a condition was still able to produce so basically have a set which
have the neuron embedded in a medium that you control in which you can measure the M
potential and also observe spes and uh so this is an example of now so this is an example of
spy produced inside the animal and this is an example of spy produced by the same type of
neon when it's taken out of the so still so that's problem so now what we do with this is the
f is this why we inject and uh and M to from an experimental preparation like that what I
have is that into cell measure the PO and then what we need is the so that here and set up
they measure they have set up to measure the activity and then control the C and then
what we do is a preparation called the F CL so basically in this setup you say I want the neur
to have a c voltage let's say you want the ne to have a volage vol so the the device that they
and at every time step injects an amount of car such that the voltage measure in the neuron
laes the the target okay so this is what is called and so now if the voltage of the neon is
constant and equal to the Target one what that means is that this Thea in time of the volage
will be zero so let basically the setup is buil in such a way that the voltage of the cell will be
Conant so if we use that in this equation what you have is that your the Curr that you inject
must be equal to the C so this gives you way to measure given the voltage that you that you
select how much are the currents this make sense any questions for this yes so we say that
given the fact to inject the current the result is that the voltage does not CH and we can this
the fact that it stays constant to determine charge the currents so that to do that there is
addition St to make that now but okay so now so there are few steps that need but for the
important thing is if you use this V SC preparation you have a way to measure the man now
couple of things so if voltage is not is large enough so that you are in the region of voltages
where typically spens this here is and in addition this is typic what discuss just forget and so
basically the current is equal to the sum of these two currents now so what you can do in
addition is I mentioned that in this experimental preparation you control what is the
environment the is so thing that you can do is to put it in an environment where there is no
okay let's say you put it an environment in which there is no s so at that point you're sure
that the must be equal to the potassium because you Environ the and this gives you direct
access to this okay so this is an example of what these things looks like okay that's an
experiment so this is the V Target as a function of time I describ here so you have a step in
this V Target this produces the neur response is actually here that's the of the ne Conant as
aun now the response the total that you measure this it is this curve a here so this is a SP
magnified the the SC is so this is only the scale so that's the sum of these two CS but now we
want to know what is the individual contribution of each one of these two the way to do
that is as I was saying you put it the neur in environment there is no s so the total Curr that
you me at the point if you do the same experiment is just thei and this is the here and now if
you do the difference between these two so the difference between the Cent In which you
have both and the Cent which you have the difference between these two gives you now
the Sol and this is this current here see so that's a way to experiment the yes so when
environment only the is fix because of the then if we keep also constant injected current
changes in fact this is so these currents are actually the injected current okay this is it's the
injected current in a different pration this is the injected current in normal conditions this is
the injected current if you have only potassium and this is the so this would be the injected
current if you only other question this so now this gives access to this total C so the product
of the conductors the and now there is a way to also extract this individual contribution so
what is the conductance and what is the and it is that have so basically the experiment gives
you this and you also control this so you can make a pling this voltage so let's say you do the
experiment you have acept voltage you measure C and then you can repeat the
experimenta volage and then you give you a certain function of that relates the the the C
through the so this intersection Point here is this Ral poal and this the slope of this curve is
the condu and so I just want to point out in the case in which uh we uh for conduct for
instance this emission sh was lar what they realize with this experiment is that in the case of
sodium and potassium the conduct is actually a function function so one output of this
result is that forumi this together you can get a sense of what the the Dynamics of is so B
function let's say you look at a that is this here seen before so this is just magnification what
happens this so This Is Your Potential Dynamics and then if you put in all your results what
you have is that initially you have an increase in sodium conduct which means that before
this point there was no s because the so no pass then at this point you start to have non
which means that you have a FL of s ions that will go from the outside inside of the C so this
will produce your increase in Po and then we right L there is a a current of potassium that is
this current here uh and this will produce fluxes theide so this is the of the type of dams you
get questions this okay so this is just an experimental that the the second point that we
have dismissed so these are the results of the experiment of and the next thing that we do is
to give AIC model of this conduct and particular stud I mention that conduct but we have a
differenty of particular the potassium conductance is called aist conductance and I will show
you in a second what it means and then the S conductance is called I will show you in a
second what itans so what we do next is to study a model of channels that capture the type
of dynamics that this conduct have supposed to change the vol okay let's start with
persistent conduces that are relevant for so we want to try to get a simple model of this
condu and so you should imagine CH as a really complicated object so this is just
schematized as a Caron here but so for the S of the discussion we have today we your I think
of having like a door that can allow or not the become slightly more complicated than this
okay so so this is the first uh step model so it turns out that the modeling of this CH is not
deistic but AAS so at every given time you have a probability and this probability is what will
be depend then turns out so this is the the list of things that I mentioned here is something
that was developed empirically we just found what is a fun form that captures the data and
this is just the end result so what they realize is that in order to express this function you
can think of the channels made by four different subunits that work independently you can
imagine that as having four independent doors and it door can be subit can be open with a
probability so this would represent the the probability that each that is open and then the
probability that the is open is the product of these two of these okay so what you have is
that the the case of so again this is uh when it was first introduced it was just found that this
is a functional form that captures data but then when with the development of molecular
Dynamics and crystallography people realize what this means so there is evidence now that
Chan are made by different compartments that can be f as independent of course like this
but that's the the overall mechanism that we have four independent elements and in order
for I uh to flow you need all these elements to be open okay so this is uh the probability that
our potassium CH is open and now with that what you do is to write the conductance of
thisi Channel you write it as a constant that is the maximum conductance times um the
probability of the so this is so this is the function for this mod to to capture the the
dependency on voltage of now where the voltage dependenc in so now what we do next is
equation for this getting and okay any questions this just wanted to make briefly a point so
here I describe the Cur to the conduct and the as but if the channel is closed or open then
this is an on or off process okay so the channel in each channel the C will either flow or not
so in fact this continuous description is basically what you have imagine that you have a
large number of chanels that are the Sur of the neuron and the description that you provide
with this model is just a description of the colletive Dynamics of so this is represented here
with simple simul so if you only have one s described by this type of equation at every time
you would have that the CH is open for so the the a that predicted this EXP this one but if
you only have one at every time you zero or maximum now if you have multiple channels in
parall then the total produced by this channel will approach this prediction that we have
question this is this obvious yes we volage do we mean like the voltage the outside voltage
the voltage coming into the neuron so okay so this is uh okay yeah two points so that's not
for the P but is for the so that's the POS component and then the voltage I haven't described
yet vol but will be the the the so what we typically do is to assume that to take as a refence
the outside voltage that to zero and then the voltage that we talk about is always the inside
voltage any other questions this so this was just a side note that variable and we see them
because we are on the that there is not only one okay okay so now what we want to do next
is to derite an equation Des the Dynamics of the of G so we said is the probability that if you
look at one of these different units here it will be over so each one of this so each unit can
be in two Stace can open or so what we want to D is called that capture how the probability
that one of the subs is open evolves with Okay so let's say that you are uh so how the prob
of being open time so at step you can either be open and this is something that happens
before and then so this is something that pariz with that we so this is the rate of closing so
the probability that is open becomes CL and this weend to decrease probability of so give a
terus then the other thing that happen is that we have a subun that is closed so and this
open and the rate of from close to open we write it as a function okay so this is what is
called atic equation that describes the evolution that a given subunit is open or close as a
function of time and it will only depend on this rate of opening and closing of the the sub
I've been told that you've seen this in stas processes class do you remember that okay okay
so this is our model for the dams we can rewrite this model in different format that is this
one the what I did is just this form in such a way that I have a that will be this and then of
here and the reason for writing it this way is that it is easier to interpret what's going on so if
we consider so if this Con then theend Dynamics will be allow to the Dynamics we describe
as a function of time and we go from initi condition and exponentially this an and this will
happen over a time scale and so when we see cases where we voltage we change how
andage imagine that so let's say that what CL is because okay so so we're trying to comp the
changing time of the probility of so this is given any given time being open depends on so
we can either let's say that close and then we open and so thise or you can be open and
then become so this will basic the fact that the other questions on this okay so then what
ask did is toally fit this functions to data by doing the experiment that I showed as a
functional voltage and then try the measur and these are the typical ex so you have the any
Infinity is a function that inrees volage this is something that becomes smaller as so the larg
voltages the f and now if together we can look at what is the Dynamics of calcium in this
model if potassium is described by this type of so this is here so again the the potassium
current is given by this expr here so the total conduct given by this g k then the probability
of opening be N4 and then you have all so so let's say that you do an experiment in which
you start from- 60 Vol you increase the voltage of the cell using the B PL that I describ this
will change this beta n and Alpha function so that basically Al will increase you will have a
larger rate of opening of sub unit when bet decrease you will have a smaller rate of going
from open to you expect on aage that more sub will be open so the end dynamics that you
get is this one so it response to the step you have an increase in 10 and then the current is
proportional to the four and so you have this Behavior here and the important thing to
remember is that so as long as the voltage is up you you always at Cur and this is why resist
if you apply a constant steing volage it will produce a constant current okay any questions
this so this is theel that we for this persistent model now we do quickly a similar arum for
the just describe an over effective model it just applies to Sol the difference we find is that
rather than being persistent this condu is means that respon to a constant stepping voltage
the fles of sodium the sodium ions ini increase and then go back so you have current a brief
so the way which this Cur model can be schematized using this here so the IDE is that you
have now two type of subunits you have an activation gate that is anous to this doors that
we have here and is described by this variable n and then you have what is called an
inactivation gate that in this is this here and it's aable from 0 to one that is H the the reason
why these two aspects are described differently is that at low voltage this is typically close
and this is open and as you increase voltage you have this this gate that open so this isation
and then this think this now is more likely to be Clos so they have an opposite Behavior with
I will show it that in order to describe quantitively this what you that the probility of
opening depend on the probability that this is open and that this inactivation so the function
for the parts of the data this functional form here so it's m the k k and H this is the and then
you put everything together you have that the sodium current is given by the maximum
condu okay so now for each one of these two sub so both for M and H you have a dynamics
that is is described by equations equations this one it say that now the function will be by
function specific this have Al there is one thing that you should notice that is this one so I
mention that so again you can reite this expression using a different notation so with and
infinity and so M Infinity behaves exactly as INF so it's as you change voltage it goes from
zero to uh to one so for large voltage this activation gate will be one so the gate will be open
no Vol this G will be Clos well for H Infinity has the opposite so at low voltage H isal to one
which represent this part that the B is the tach from the CH so this will allow the current
well for last voltage is close to Zer so it has the opposite Trend with voltage and then the
have a similar Behavior what we matter is that the N varable the activation is much faster
than the this is what so this is if you com this what you have so let's say step voltage of you
have certain change in the alpha and beta function for M and over what you have is
depicted here so H so the inactivation variable is initially something and decreases with
volage so this is what we then the activation m is initially Zer and then increases the so the
total C is the product of time so the total current will be initially zero then increase and then
decree see have so remember I think it's useful that in order to be open you need to have
this this be theable this Tye and this and what changes between the two is that at low
voltage this is Clos and this is open at high voltage theable is open but the is Clos so and uh
trange but so let's that you go from low to Vol you go from close to close it's just that what
happens is that this sub here open much faster than the time it takes for this sun to close so
when you do a steping voltage you have the this opens and then after a while this get mhm
any other questions please okay so that's the the second the second step of this model so
we have now understanding of the Dynamics of the potassium current and the Sol combine
this to to be the okay so just to summarize what we show so far we said that describe these
neurons we put in parall with the capacitor and the Le term of each branches that
correspond to the potassium current and the sodium current and the critical thing that we
show now is that these two conductances here are not constant but they are one depend
and the dependency that we found is but if you together the this is how they like so for the
time constant you see that the time constant for the activation gate of the sodium sorry of
the sodium soundance is much faster than all the other so the activation of uh sodium is
much faster than the all the other variables then the other thing that is uh worth
remembering is that as a function of voltage M and N will one well H will Z okay so now if
you put everything together this is how you bu the the model summarizing this SL so is the
following so this is uh the series of steps that happen when elction potential is let's say
initially when there is no current being injected in the neuron we said the memory potential
is C value is fixed and the there is no sodium current and there is no potassium current so
this is because m is there is no s car because basically the activation m is Clos and there is no
potassium current because each gate can have a like4 probability of being open but now the
currents depend depend on the four and this becomes small number okay so at equilibrium
without current injected these two currents are Zen now let's say that you do a current
injection that drives the M potential by am of is typically around 50 so this point the moves
along this axis here and what you have if you are aroundus 50 m here foret here for the
time you see that m so the activation of the s c is the first is the short so the sodi is the first
respon and the response we said would be something like this curve here so you would
have a fast approach to the value M infinity and M Infinity if you're aroundus 50 m is around
here so it's so which which means that you will start to have opening of this m sub so the
first thing that you have is opening of this m sub this opening of M sub unit now is possible
because there is no inactivation so H will be small but now it is it is slow to exp so H is still
open so for Sol you have that H is open and M is open so you have flux of sodium ions so
you have influx of positive ions so you have B INE you have this influx but now what
happens is that you have a positive feedback and this is what detes in the SP a really rapid
increase in protction B increase influx in positive I but this influx in positive I will increase
further potential and so you go fromus 50 you tend to go higher along and this will continue
after basically the the point in which you reach the reveral potential of C that is around plus
you have a really massive influx of then what happens next is that finally inactivation G up
and this closes the Sol CH and blocks the flux of sod at that point you have um the
potassium that you remember is the is also slow with respect to that now also starts to get
activated so this you start and basically potassium positive go from the inside the of the cell
to the outside so this will to decrease your mation okay this outflux of potassium what
brings down the voltage in the second part of the so now there is a continuous flow of
potassium up to the point in which um the mem potential becomes PR negative and so at
that point and infinity BEC Z so the Al theum and at that point you're basically around here
what Happ just relation value the different so just what happens this is to the point let's say
you are and you have C this leads to an increase in M potential that allows the influx of
sodium this in terms increase sodi the cell to increase in the potential you have POS and this
is really rapid potential that you doing then when you're here you inactivation of this St
variable so the the sodium Channel become Clos now the only thing that means is
potassium that basically flow outside outside of the cell and so it brings down the m prot so
you go from here to here and then when you're at that point also potassium inactivate and
you have relaxation Dynamics so this is the series of steps that you have in when you so any
questions this while but it's pretty straight forward if you think about it for a couple of times
just have to remember that there is this interaction between two different sh of potential of
the spice is given more by The Sod chel yeah so the sodium is what determines this initial
really the potassium is what produces the fast decrease and then all of them together
produces this relaxation effect other so so that's what I'm show you now is the ask this Ser
of St that solity and uh the model recapitulates the type of exp so what I describe is the
response to single step that incre Andreas potential but simate the respon for Conant
injection what you have is a series of spes an to the one that I show you in the first
experiment at the beginning of the class you see so thetion that is not the only possible
responses that seen in experiment so here should imagine each one of these is a single neon
and for each neon you're doing the same type of experiment so you do a step current
injection and you leave the current on for time and then you take down and you see that
some neur respond to different of behavior classical that you see is adaptation that is
depicted here the response to this in you have initial SP then spikes become dly more
separated in time like here and after why the doesn't respond anymore to the then you can
also have bursting that is this type of here you have but having spikes that are regularly
dispers you have this rapid series of spikes that then follow by an interval time the then
again SP again so you have this throughout this experiment the injection is Conant so it's
three property of the and then other things that you can have are like this thing here where
you have a delayed respon so you the current initially you have increasing me potential but
then after a while start to and these are just three example Behavior but so the thing that I
want to emphasize is that all these can be capture by basically adding currents like the one
so the the idea of this models is to you always have in model these things you always have a
term that is to the memory capacit contion and then what you do is to add ter to inside this
so you are this is the the the this is in order to have and then on top of that you can add
other Curr that correspond to other the and all these channels so this is what is powerful
about the model that we today that all this channel can be described by an equation of this
F so you will have your current will be given by a maximum conductance times a probability
that the channel is open theal poal and this probability Express using a combination of
activation and inactivation Val they can be expressed as a c activation variable to certain
Power Times an inactivation variable to another Power and these things in function voltage
and function cany so for instance in the case of adaptation this is an example ofation in the
model looks like this one so this will be injected current this is the sodium current that we
show the Le current the potassium thisr and then in this NE you see that injection become
gradually more delayed up to the point that again you don't have you can capture that by
adding an additional potassium that has the