Download Slide 1 - Gatsby Computational Neuroscience Unit

TNI: Computational Neuroscience
Instructors: Peter Latham
Maneesh Sahani
Peter Dayan
TA:
Website:
Phillipp Hehrmann, [email protected]
http://www.gatsby.ucl.ac.uk/~hehrmann/TN1/
(slides will be on website)
Lectures:
Review:
Tuesday/Friday, 11:00-1:00.
Friday, 1:00-3:00.
Homework: Assigned Friday, due Friday (1 week later).
first homework: 2 weeks later (no class Oct. 12).
What is computational neuroscience?
Our goal: figure out how the brain works.
There are about 10 billion cubes of
this size in your brain!
10 microns
How do we go about making sense of this mess?
David Marr (1945-1980) proposed three levels of analysis:
1. the problem (computational level)
2. the strategy (algorithmic level)
3. how it’s actually done by networks of neurons
(implementational level)
Example #1: vision.
the problem (Marr):
2-D image on retina →
3-D reconstruction of a visual scene.
Example #1: vision.
the problem (modern version):
2-D image on retina →
reconstruction of latent variables.
house
sun
tree
bad artist
Example #1: vision.
the problem (modern version):
2-D image on retina →
reconstruction of latent variables.
the algorithm:
graphical models.
x1
r1
x2
r2
^
x1
x3
r3
^
x2
latent variables
r4
^
x3
peripheral spikes
estimate of latent variables
Example #1: vision.
the problem (modern version):
2-D image on retina →
reconstruction of latent variables.
the algorithm:
graphical models.
implementation in networks of neurons:
no clue.
Example #2: memory.
the problem:
recall events, typically based on partial information.
Example #2: memory.
the problem:
recall events, typically based on partial information.
associative or content-addressable memory.
the algorithm:
dynamical systems with fixed points.
r3
r2
r1
activity space
Example #2: memory.
the problem:
recall events, typically based on partial information.
associative or content-addressable memory.
the algorithm:
dynamical systems with fixed points.
neural implementation:
Hopfield networks.
xi = sign( ∑j Jij xj)
Comment #1:
the problem:
the algorithm:
neural implementation:
Comment #1:
the problem:
the algorithm:
neural implementation:
easier
harder
harder
often ignored!!!
Comment #1:
the problem:
the algorithm:
neural implementation:
easier
harder
harder
My favorite example: CPGs (central pattern generators)
rate
rate
Comment #2:
the problem:
the algorithm:
neural implementation:
easier
harder
harder
You need to know a lot of math!!!!!
x1
r1
x2
r2
^
x1
r3
^
x2
r3
x3
r4
^
x3
r2
r1
activity space
Comment #3:
the problem:
the algorithm:
neural implementation:
easier
harder
harder
This is a good goal, but it’s hard to do in practice.
We shouldn’t be afraid to just mess around with
experimental observations and equations.
A classic example: Hodgkin and Huxley.
dendrites
soma
axon
voltage
+40 mV
1 ms
-50 mV
time
100 ms
A classic example: Hodgkin and Huxley.
C dV/dt = –gL(V-VL) – gNam3h(V-VNa) – …
dm/dt = …
…
the problem:
the algorithm:
neural implementation:
easier
harder
harder
A lot of what we do as computational neuroscientists
is turn experimental observations into equations.
The goal here is to understand how networks or single
neurons work.
We should always keep in mind that:
a) this is less than ideal,
b) we’re really after the big picture: how the brain works.
Basic facts about the brain
Your brain
Your cortex unfolded
neocortex (cognition)
6 layers
~30 cm
~0.5 cm
subcortical structures
(emotions, reward,
homeostasis, much much
more)
Your cortex unfolded
1 cubic millimeter,
~3*10-5 oz
1 mm3 of cortex:
50,000 neurons
10000 connections/neuron
(=> 500 million connections)
4 km of axons
1 mm3 of cortex:
1 mm2 of a CPU:
50,000 neurons
10000 connections/neuron
(=> 500 million connections)
4 km of axons
1 million transistors
2 connections/transistor
(=> 2 million connections)
.002 km of wire
1 mm3 of cortex:
1 mm2 of a CPU:
50,000 neurons
10000 connections/neuron
(=> 500 million connections)
4 km of axons
1 million transistors
2 connections/transistor
(=> 2 million connections)
.002 km of wire
whole brain (2 kg):
whole CPU:
1011 neurons
1015 connections
8 million km of axons
109 transistors
2*109 connections
2 km of wire
1 mm3 of cortex:
1 mm2 of a CPU:
50,000 neurons
10000 connections/neuron
(=> 500 million connections)
4 km of axons
1 million transistors
2 connections/transistor
(=> 2 million connections)
.002 km of wire
whole brain (2 kg):
whole CPU:
1011 neurons
1015 connections
8 million km of axons
109 transistors
2*109 connections
2 km of wire
dendrites (input)
soma (spike generation)
axon (output)
voltage
+40 mV
1 ms
-50 mV
time
100 ms
synapse
current flow
synapse
current flow
voltage
+40 mV
-50 mV
time
100 ms
neuron i
neuron j
V on neuron i
neuron j emits a spike:
EPSP
t
10 ms
neuron i
neuron j
V on neuron i
neuron j emits a spike:
IPSP
t
10 ms
neuron i
neuron j
V on neuron i
neuron j emits a spike:
IPSP
t
10 ms
amplitude = wij
neuron i
neuron j
V on neuron i
neuron j emits a spike:
IPSP
changes with
learning
t
10 ms
amplitude = wij
synapse
current flow
A bigger picture view of the brain
x
r
latent variables
peripheral spikes
sensory processing
^
r
action
selection
emotions
cognition
memory
^
r'
“direct” code for
latent variables
brain
“direct” code for
motor actions
motor processing
r'
peripheral spikes
x'
motor actions
r
r
r
r
you are the
cutest stick
figure ever!
r
x
action
selection
emotions
cognition
memory
r
^
r
^r'
r'
brain
x'
Questions:
1. How does the brain re-represent latent variables?
2. How does it manipulate re-represented variables?
3. How does it learn to do both?
Ask at three levels:
1. What are the properties of task x?
2. What are the algorithms?
3. How are they implemented in neural circuits?
Knowing the algorithms is a critical, but often neglected, step!!!
We know the algorithms that the vestibular system uses.
We know (sort of) how it’s implemented at the neural level.
We know the algorithm for echolocation.
We know (mainly) how it’s implemented at the neural level.
We know the algorithm for computing x+y.
We know (mainly) how it might be implemented in the brain.
Knowing the algorithms is a critical, but often neglected, step!!!
We don’t know the algorithms for anything else.
We don’t know how anything else is implemented at
the neural level.
This is not a coincidence!!!!!!!!
What we know about the brain
1. Anatomy. We know a lot about what is where. But be
careful about labels: neurons in motor cortex sometimes
respond to color.
Connectivity. We know (more or less) which area
is connected to which. We don’t know the wiring diagram
at the microscopic level.
wij
2. Single neurons. We know very well how point neurons work
(think Hodgkin Huxley).
Dendrites. Lots of potential for incredibly complex
processing. My guess: they make neurons bigger and
reduce wiring length.
3. The neural code. We’re pretty sure that information is
carried in action potentials. We’re not sure what aspects
of action potentials carry the information. The two main
candidates:
precise timing
firing rate
Once you get away from periphery, it’s mainly firing rate.
4. Recurrent networks of spiking neurons. This is a field that
is advancing rapidly! There were two absolutely seminal
papers about a decade ago:
van Vreeswijk and Sompolinsky (Science, 1996)
van Vreeswijk and Sompolinsky (Neural Comp., 1998)
We now understand very well randomly connected networks
(harder than you might think), and (I believe) we are on
the verge of:
i) understanding networks that have interesting
computational properties.
ii) computing the correlational structure in those
networks.
5. Learning. We know a lot of facts (LTP, LTD, STDP), but it’s
not clear which, if any, are relevant.
Theorists are starting to develop unsupervised learning
algorithms, mainly ones that maximize mutual information.
These are promising, but the link to the brain has not been
fully established.
5. Learning. We know a lot of facts (LTP, LTD, STDP), but it’s
not clear which, if any, are relevant.
Theorists are starting to develop unsupervised learning
algorithms, mainly ones that maximize mutual information.
These are promising, but the link to the brain has not been
fully established.
A word about learning (remember these numbers!!!):
You have about 1015 synapses.
If it takes 1 bit of information to set a synapse,
you need 1015 bits to set all of them.
30 years ≈ 109 seconds.
To set 1/10 of your synapses in 30 years,
you must absorb 100,000 bits/second.
Learning in the brain is almost completely unsupervised!!!
6. Where we know algorithms we know the neural
implementation (sort of). Vestibular system, sound
localization, echolocation, x+y.
1. What we know: my score (1=low, 10=high).
a.
b.
c.
d.
e.
Anatomy.
Single neurons.
The neural code.
Recurrent networks of spiking neurons.
Learning.
7
7
5
4
2
Questions: all answers are “we don’t know”.
1. How does the brain re-represent latent variables?
2. How does it manipulate re-represented variables?
3. How does it learn to do both?
0.001
0.002
0.001
Outline:
1.
2.
3.
4.
Basics: single neurons/axons/dendrites/synapses.
Language of neurons: neural coding.
What we know about networks (very little).
Learning at network and behavioral level.
Latham
Sahani
Latham
Dayan
Outline for this part of the course (biophysics):
1.
2.
3.
4.
5.
What makes a neuron spike.
How current propagates in dendrites.
How current propagates in axons.
How synapses work.
Lots and lots of math!!!