This course provides an introduction to basic computational methods for understanding what nervous systems do and for determining how they function. We will explore the computational principles governing various aspects of vision, sensory-motor control, learning, and memory. Specific topics that will be covered include representation of information by spiking neurons, processing of information in neural networks, and algorithms for adaptation and learning. We will make use of Matlab demonstrations and exercises to gain a deeper understanding of concepts and methods introduced in the course. The course is primarily aimed at third- or fourth-year undergraduates and beginning graduate students, as well as professionals and distance learners interested in learning how the brain processes information.
5.4 A Forest of Dendrites
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Do curso por University of Washington
Neurociência Computacional
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Computing in Carbon (Adrienne Fairhall)
This module takes you into the world of biophysics of neurons, where you will meet one of the most famous mathematical models in neuroscience, the Hodgkin-Huxley model of action potential (spike) generation. We will also delve into other models of neurons and learn how to model a neuron's structure, including those intricate branches called dendrites.
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Rajesh P. N. RaoProfessor
Computer Science & Engineering
Adrienne FairhallAssociate Professor
Physiology and Biophysics
In this final part of the lecture, we're going to continue to go beyond
Hodgkin-Huxley, but this time, in this direction.
We're going to be going back to biophysical reality and addressing the
issue of geometry. How do the complexities of neuronal shape
and structure affect our computation? In the first lecture we invested heavily
in understanding the spike generation process in a patch of membre here at the
easy end. But its a little embarrassing to zoom out
and look at real neurons which have a truly extraordinary range of beauty and
complexity in the geometry of the dendritic abras.
So for moving toward building biophysically realistic models of neuro
processing. It would be good to know how these
structures can contribute to the processing of information.
So, here's what we learnt to model, a compact cell, and here's what real
neurons really look like. Here's even quite a simple version.
So, as with the complexities of ion channel dynamics, what is the appropriate
level of description of a single neuron that's necessary to understand brain
operation? Because we don't yet know the answer to
this, and there probably is not one answer to this, it's important to pursue
models with many different approaches. So, here I'll be introducing you to the
techniques that one can use to handle dendrites, and some ideas about what they
may contribute to computation. So let's start by looking to what extent
dendrites feel what's going on in the soma.
So here, this is an impulse point that's being put in at the soma.
Let's see what that input looks like when it reaches the dendrite.
You can see that it's both delayed, it's reduced in amplitude, and it's broader
Similarly if we put an input here, add in the dendrite, and we look at what happens
at the soma in response to that input, you also see that it's much reduced in
size and it's broadened out. Furthermore, how thin the dendrite is
affects how big a voltage change you could make with a given amount of current
input. The thinner the dendrite the larger the
voltage change but generally the further away the the more that input gets
filtered and attenuated. This tells us the inputs that come along
different parts of the dendrite can have very different effects and very different
influence on firing at the soma. As you can image this can have a
tremendous impact on the information that is integrated and representated by the
receiving neurons The theoretical basis for understanding voltage propogation in
dendrites and axons is cable theory, which was developed by Kelvin in quite a
different context. The voltage, V, is now a function of both
space and time, which means that we're now dealing with partial, rather than
ordinary differential equations. So here's the setup.
We now think about a tube of membrane with sides have the same properties as
our membrane patch. They have both capacitance and
resistance. So we see little elements that look a lot
like our, like our previous patch model, but now they distributed down a cable.
There's additionally the resistance of the cable interior.
Current can flow along the cable as well as through it.
So generally we're not going to worry about the external medium here.
We'll just take it to be infinitely conducting with a resistance of zero.
So the cable equation for a passive membrane, we're not going to deal with
ion channels for now, is derived by considering the changes in current as a
function of space. The current down the cable will be driven
by steps in voltage as a function of x. So, if we have a voltage difference
between two points in the membrane, that's going to drive a current down the
membrane. Of course, current is also passing out of
the membrane. That's the im that we modeled previously.
Now when one puts those 2 things togteher dealing with the way current flows out of
the membrane and the way that it flows down the membrane 1 obtains an equation
that is actually 2nd order in space so it has a 2nd derivative with respect to
space. So this half of the equation you 'll
recognize that we've seen before of the passive membrane now we have an
additional term that, that includes a spacial derivative.
So, some of you will recognize that this equation is not unlike the equation that
describes diffusion or heat propagation, but it has this additional term, so this
part looks like diffusiion, has this additional term that's linear in v.
You might remember that when we rewrote the RC second equation for the passive
membrane to find the time constant of the membrane that gave us sort of the
fundamental time scale of its dynamics. So there's something very similar in this
case too we can rewrite the equation in this form where we bring together all the
dimensional quantities. This will ask us to read off the natural
time scale so going with this time derivative there's a a time.
Constant which is our tow M and now when we look at the, the spacial derivative
this has units of 1 over space squared and there's a space constant out the
front lambda that carries the typical spacial scale from the coefficient of the
space derivative. That's given by the square root of the
ratio of the membrane resistance divided by the internal resistance.
Why is that. So what stops a signal from propagating
is leakage through the cable. The higher the membrane resistance, the
more current stays inside the cable and the further the signal can propagate.
Not surprising the ether space constant also depends on the inverse of the
internal resistance. The less internal resistance, the further
the signal can go and so this combination, and so this combination of
resistances gives us a typical scale over which the current is able to propagate.
So now let's imagine that we're a synapse on a dendrite, trying to get a message
down to the soma, so our message can help trigger a spike and then get handed on to
the next guy. Let's see how far down the dendrite our
message can travel. So here we are.
We're trying to inject a signal into our dendrite at position x equals zero.
Even if we pipe in a constant amount of current, we just keep shouting that
signal, the voltage change that it causes in that cable will only propagate about
as far as the length constant. Now, you see here, it decays
exponentially over a scale given by the space constant lamda.
Now let's put in a brief pause and see how that behaves at T equals 0 we put in
a spike of input at X equals 0. Here's what happens to it, the input gets
broader, spreads out spatially and also decays in time.
This is a lot like diffusion. If you spray a pulse of perfume,
somewhere in a room, and were able to watch what happened to it, it would do
something similar. It would spread out with the same spatial
profile. But for the perfume, there's always the
same amount of perfume. If you were to add up all the molecules
of perfume in the smeared out blob, it would be the same as you started with.
For the voltage, that's not the case. The total voltage signal is decaying away
in time, because of that first order part in the differential equation.
That is, the total voltage is decaying exponentially, just as it did in the RC
circuit. So what are we going to see if we sit
some distance away, and observe the change in voltage?
That's what's shown in this figure. So these are different curves that plot
the voltage that's observed at different locations.
X equals 0, x equals .5, x equals 1. These are all in units of the space
constant of the membrane. At different times, and so sitting at x
equal 0, we see an exponential decay. As we sit a little further away we're
going to see that. That, that pulse of voltage change first
rise, then seal the decay away again. So you can see how rapidly the signal
decays as a function of space. So at about, at one space constant you
still see a reasonably large Deflection in voltage that's caused by that pulse.
But at two space constance away, the signal, the size of the, of the pulse
that we see there is down to five percent of the original.
You can also track the peak of this potential pulse and ask how rapidly does
this signal propagate down the cable? It turns out that that gives us a
velocity of propagation of the signal which goes like, see if you can guess,
so, what would you expect to be the propagation rate down a cable?
Let's call it c. It turns out that it's equal to two times
the space constant over the time constant.
So, what's the typical spacial scale? It's just lamda.
Typical timescale is tow and the propigation velcoity turns out to be the
ratio of those two of those two quantities.
So for some of you who like to see general solutions here's how the voltage
responds to a pulse of input at time T equals zero and position X equals zero
looks at position X and time T. We can see that this solution is made up
of two parts. So here's the diffusive spread for this
Gaussian profile. That's very similar to the way things
spread diffusively. And mutliplying that, there's this
pre-factor that has an exponential decay. So, knowing the solution, one could take
some arbitrary pattern of inputs, decompose it into pulses, as we, as we
put in here, at different central locations.
Say, T prime and X prime, and add together a weighted sum of this solution
form. We've centered up those differnt
locations and times. So now we know how to find solutions for
a very long pass of cable with a fixed radius.
In fact doesn't get us very far in dealing with the real neuron, because of
two things, the inter current branching of dendrites and the fact that many
dendrites are not passive but are active. That is, they have ion channels in them.
So it quickly becomes very difficult to solve anything analytically.
The path forward is by dividing the dendritic arbor into what's called
compartments. So here's an example, one can approximate
the dendritic arbor as a coupled set of compartments.
So these, we're going to break this dendrite into little sub regions In which
the radius and the ion channel density is taken to be constant.
Each compartment will then have an equation that only depends on the time
derivative of the voltage, and not on x. The spatial dependence is incorporated by
coupling each compartment together. So, so [INAUDIBLE] Rall device is a
helpful way to approximate complex dendritic trees of.
Let's consider a branch that divides into 2 daughters.
It turns out that if the diameters of these two branches have this particular
relationship to the diameter of their mother so if the if the diameter of 1 of
them raises to 3 halves plus the diameter of the other also raised to the 3 halves
is equal to the diameter of the mother raised to the 3 halves.
What that means is that these two branches are impedance matched to this
branch, and one can simply accumulate them all together into one long branch of
the same, of the same diameter as the mother.
So one needs to thus compute the effective electrotonic length of these
two additional branch elements. And extend the original cable by that
much. So you can see that one can continue to
do this iteratively. If the same property holds here, lennox
can extend that branch out to an effective branch of this, of D2 diameter.
And then one can agormate those two together.
Until one eventually has a single, a single cable coming out of the soma.
So it turns out that this condition on the diameters it approximately satisfied
by real dendrites and even when it's not exact the resulting approximation is
often quiet accurate. So the role models are very useful for
passive membranes But it doesn't address the issue of ion channels which make the
problem nonlinear. Furthermore ion channels densities often
very along dendrites which can lead to a lot of interesting effects that one might
like to explore. so here's the full approach, given the
geometry and the ion channel density of the dendritic tree One can divide it
where the properties are approximately constant.
One can then write down equations for the membrane potential in each compartment
individually. So let's say compartment 1 we represent
here in terms of a, a similar second model that we saw for the passive
membrane we're going to give that the voltage V1 and now right down an equation
for V1. We can do the same for compartment two
and compartment three, here. These equations will be similar to the
passive membrane equations we looked at for Hodgkin-Huxley, but with the
individual ion conductances, membrane resistance and capacitance set
appropriately for each piece of the cable.
Furthermore, there'll be also two tons that couple the compartment with it's
neighbors. The current input from the neighboring
compartments. Which depends on the voltage difference
between the two compartments and a fixed coupling conductance.
So here for example, let me write down an equation for for v two.
We're going to need to include a current that's coming from, from compartment one
that's going to go like g one, two multiply it by V1 minus v2.
And similarly, there'll be a current that comes into compartment one from
compartment two that's going to have a different coupling conductance and the
opposite voltage difference. These fixed turns, these coupling
conductances, depend on the area of the connection, and whether or not the
compartments straddle a branching point. So if you notice, that in general, these
coupling conductances are not necessarily symmetric, which is why there are 2
values at each of these connections. So there are many models like these that
have been built from microscopic reconstructions of single neurons.
And a great many have been made publicly available on the ModelDB site maintained
at Yale. So if you're in the mood to go explore a
dendritic forest, there's plenty out there, so don't forget your adventure
hat. So what do den-, so what do dendrites add
to neuronal computation? There are many proposals for ways in
which the filtering and active properties of dendrites can work, to shift the way
in which incoming information is Clearly, where an input arrives on the tree can
influence the effective strength of the input because of the passer properties.
Interestingly, it's been found that in the hippocampus, neuronal dendrites have
solved this problem. So that when inputs arrive at the suma/g,
they have a very similar shape no matter where they come in.
This amazing property is known as synaptic scaling.
Filtering through the dendritic tree on the way to the sonar also determines
whether a sequence of successive inputs is integrated to build up to potentially
drive the spike or not. Where two different inputs enter the
dendritic tree can also make a huge difference in how they interact with each
other. For instance, if two inputs come in on
separate branches. They contribute independently.
While if they are on the same branch, they can sum either sublinearly or super
linearly which leads to amplification. Another very important property is that
thanks to their ion channels, dendrites can generate spikes, generally calcium
spikes. [INAUDIBLE] This leads to the possibility
that one could use coincidence of inputs, driving spikes in the dendrites.
Along with back propagating spikes from the soma back to the dendrites to drive
synaptic plasticity. This is a topic you'll be hearing much
more about in the next lectures. So, let's close out by looking at two
ideas for how dendrites might perform a computational role.
The experimental evidence supporting these mechanisms is somewhat mixed, but
the fundamental ideas stand as interesting conceptual models.
First, here's a wonderful example where the propagation of signals through cables
is thought to help out in carrying out a computation.
Nuclei in the auditory brain stem are responsible for sound localization, the
ability that we all have to locate where a sound is coming from.
The que that's thought to be used is the timing difference in the arrival of a
sound at our two years. The sound arise at the two ears at
slightly different times, and the signal those travels through the left and right
auditory pathways at slightly different times.
Imagine that these two signals are piped into the population of neurons The
thresholds are set such that they can only fire, when coincidence signals from
two different inputs arrive at the same time.
Each neuron receives the two inputs with a delay caused by traveling different
distances along the dendrites. The neuron that fires the most is the one
for which the relative timing delays, due to the timing difference between the two
ears Is compensated for by the dendritic delay.
This mechanism turns a tiny difference into a place code where by the label of
the neuron that fires, indicates the timing delay, which can then be
translated into the spatial location of the sound.
Here's a final example Neurons in the retina show direction cell activity.
That is, they respond to individual stimulus moves in one direction, and is
suppressed when it moves in the other. So how might such direction cell activity
begin constructed at the single neuron level?
Imagine that inputs from different spatial locations are coming in to a
dendrite at locations along the dendrite, arranged as in space.
As the dendrite receives a sequence of activations, you can see that if they
receive that input first, at the far end of the dendrite, and it begins to travel
toward the soma, then another input comes in, closer to the soma.
Then the influence of these two inputs can sum and build up as more and more
inputs arrive. So that the net input crosses some
threshold. On the other hand if the nearby location
is stimulated first, then the next one and then the next one and then the first
inputs will die away by the time the later ones arrived.
And you can see how, how that, how that voltage signal at the soma might behave.
So this idea was first proposed by Rall. While it may not fully explain direction
selectivity and retinal ganglion cells, the general idea does predict that the
firing probability of the neuron should be sensitive to the sequence of inputs
along the dendrite. Let's say these inputs occur in some
sequence. One could then scramble the order, and
see whether the firing of the neuron can distinguish those inputs.
Michael Hausser's lab carried out this experiment by simulating sequences of
synaptic inputs into single neurons and found that indeed different input
sequences were discriminable. Okay so here's where we wrap up my
section of the course. I hope you've enjoyed this brief
introduction into the electrical basis of neural signaling in the brain.
Roshesh is going to take it from here, to guide you through the ways in which these
basic cellular components get wired up together, to produce the amazing variety
amount of behaviors that I know the systems are capable of, through
experience and through learning. Have fun.
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