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TNI: Computational Neuroscience
Instructors: Peter LathamManeesh SahaniPeter Dayan
TA: Mandana Ahmadi, mandana@gatsby.ucl.ac.ukWebsite: http://www.gatsby.ucl.ac.uk/~mandana/TNI/TNI.htm
(slides will be on website)
Lectures: Tuesday/Friday, 11:00-1:00.Review: Friday, 1:00-3:00.
Homework: Assigned Friday, due Friday (1 week later).first homework: assigned Oct. 3, due Oct. 10.
What is computational neuroscience?
Our goal: figure out how the brain works.
10 microns
There are about 10 billion cubes ofthis size in your brain!
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: memory.
the problem:recall events, typically based on partial information.
Example #1: memory.
the problem:recall events, typically based on partial information.associative or content-addressable memory.
an algorithm:dynamical systems with fixed points.
r3
r2
r1 activity space
Example #1: memory.
the problem:recall events, typically based on partial information.associative or content-addressable memory.
an algorithm:dynamical systems with fixed points.
neural implementation:Hopfield networks.
xi = sign(∑j Jij xj)
Example #2: vision.
the problem (Marr):2-D image on retina → 3-D reconstruction of a visual scene.
Example #2: vision.
the problem (modern version):2-D image on retina → recover the latent variables.
housesuntreebad artist
Example #2: vision.
the problem (modern version):2-D image on retina → reconstruction of latent variables.
an algorithm:graphical models.
x1 x2 x3
r1 r2 r3 r4
latent variables
low level representation
Example #2: vision.
the problem (modern version):2-D image on retina → reconstruction of latent variables.
an algorithm:graphical models.
x1 x2 x3
r1 r2 r3 r4
latent variables
low level representation
inference
Example #2: vision.
the problem (modern version):2-D image on retina → reconstruction of latent variables.
an algorithm:graphical models.
implementation in networks of neurons:no clue.
Comment #1:
the problem:the algorithm:neural implementation:
Comment #1:
the problem: easierthe algorithm: harderneural implementation: harder
often ignored!!!
Comment #1:
the problem: easierthe algorithm: harderneural implementation: harder
A common approach:
Experimental observation → model
Usually very underconstrained!!!!
Comment #1:
the problem: easierthe algorithm: harderneural implementation: harder
Example i: CPGs (central pattern generators)
rate
rate
Too easy!!!
Comment #1:
the problem: easierthe algorithm: harderneural implementation: harder
Example ii: single cell modeling
C dV/dt = -gL(V – VL) – n4(V – VNa) …
dn/dt = …
…
lots and lots of parameters … which ones should you use?
Comment #1:
the problem: easierthe algorithm: harderneural implementation: harder
Example iii: network modeling
lots and lots of parameters × thousands
Comment #2:
the problem: easierthe algorithm: harderneural implementation: harder
You need to know a lot of math!!!!! r3
r2
r1 activity space
x1 x2 x3
r1 r2 r3 r4
Comment #3:
the problem: easierthe algorithm: harderneural implementation: harder
This is a good goal, but it’s hard to do in practice.
Our actual bread and butter: 1. Explaining observations (mathematically) 2. Using sophisticated analysis to design simple experiments that test hypotheses.
volt
age
100 mstime
-50 mV
+40 mV
dendrites
soma
axon
1 ms
A classic example: Hodgkin and Huxley.
A classic example: Hodgkin and Huxley.
C dV/dt = –gL(V-VL) – gNam3h(V-VNa) – … dm/dt = … …
Comment #4:
the problem: easierthe algorithm: harderneural implementation: harder
some algorithms are easy to implement on a computerbut hard in a brain, and vice-versa.
we should be looking for the vice-versa ones.
it can be hard to tell which is which.
these are linked!!!
Basic facts about the brain
Your brain
Your cortex unfolded
~30 cm
~0.5 cm
neocortex (cognition)
subcortical structures(emotions, reward,homeostasis, much muchmore)
6 layers
Your cortex unfolded
1 cubic millimeter,~3*10-5 oz
1 mm3 of cortex:
50,000 neurons10000 connections/neuron(=> 500 million connections)4 km of axons
1 mm3 of cortex:
50,000 neurons10000 connections/neuron(=> 500 million connections)4 km of axons
1 mm2 of a CPU:
1 million transistors2 connections/transistor(=> 2 million connections).002 km of wire
1 mm3 of cortex:
50,000 neurons10000 connections/neuron(=> 500 million connections)4 km of axons
whole brain (2 kg):
1011 neurons1015 connections8 million km of axons
1 mm2 of a CPU:
1 million transistors2 connections/transistor(=> 2 million connections).002 km of wire
whole CPU:
109 transistors2*109 connections2 km of wire
1 mm3 of cortex:
50,000 neurons10000 connections/neuron(=> 500 million connections)4 km of axons
whole brain (2 kg):
1011 neurons1015 connections8 million km of axons
1 mm2 of a CPU:
1 million transistors2 connections/transistor(=> 2 million connections).002 km of wire
whole CPU:
109 transistors2*109 connections2 km of wire
volt
age
100 mstime
-50 mV
+40 mV
dendrites (input)
soma (spike generation)
axon (output)
1 ms
current flow
synapse
current flow
synapse
volt
age
100 mstime
-50 mV
+40 mV
neuron jneuron i
neuron j emits a spike:
V o
n n
euro
n i
t
10 ms
EPSP
neuron jneuron i
neuron j emits a spike:
V o
n n
euro
n i
t
10 ms
IPSP
neuron jneuron i
neuron j emits a spike:
V o
n n
euro
n i
t
10 ms
IPSP
amplitude = wij
neuron jneuron i
neuron j emits a spike:
V o
n n
euro
n i
t
10 ms
IPSP
amplitude = wij
changes withlearning
current flow
synapse
A bigger picture view of the brain
xr
sensory processing
motor processing
x'
r'
cognitionmemory
action selection
peripheral spikes
latent variables
motor actions
peripheral spikes
brain
r̂ “direct” code forlatent variables
r'̂ “direct” code formotor actions
r
r
r
r
r
you are thecutest stickfigure ever!
r
you are thecutest stickfigure ever!
xr
sensory processing
motor processing
x'
r'
cognitionmemory
action selection
peripheral spikes
latent variables
motor actions
peripheral spikes
brain
r̂ “direct” code forlatent variables
r'̂ “direct” code formotor actions
xr
sensory processing
motor processing
x'
r'
cognitionmemory
action selection
peripheral spikes
latent variables
motor actions
peripheral spikes
brain
r̂ “direct” code forlatent variables
r'̂ “direct” code formotor actions
In some sense, action selection is the most importantproblem:
if we don’t choose the right actions, we don’t reproduce, and all the neural coding and computation in the world isn’t going to help us.
Do I call him and risk rejection and humiliation,or do I play it safe, and stay home on Saturdaynight and eat oreos?
Do I call her and risk rejection and humiliation,or do I play it safe, and stay home on Saturdaynight and eat oreos?
xr
sensory processing
motor processing
x'
r'
cognitionmemory
action selection
peripheral spikes
latent variables
motor actions
peripheral spikes
brain
r̂ “direct” code forlatent variables
r'̂ “direct” code formotor actions
Problems:1. How does the brain extract latent variables?2. How does it manipulate latent variables?3. How does it learn to do both?
Ask at two levels:1. What are the algorithms?2. How are they implemented in neural hardware?
What do we know about the brain?
Highly
biased
wij
a. Anatomy. We know a lot about what is where. But becareful about labels: neurons in motor cortex sometimes
respond to color.
Connectivity. We know (more or less) which areais connected to which. We don’t know the wiring diagramat the microscopic level.
b. Single neurons. We know very well how point neurons work(think Hodgkin Huxley).
Dendrites. Lots of potential for incredibly complexprocessing.
My guess: they make neurons bigger and reduce wiring length (see the work of Mitya Chklovskii).
How much I would bet: 20 p.
c. The neural code.
My guess: once you get away from periphery, it’s mainly firing rate: an inhomogeneous Poisson process with a refractory period is a good model of spike trains.
How much I would bet: £100.
The role of correlations. Still unknown.
My guess: don’t have one.
d. Recurrent networks of spiking neurons. This is a field thatis advancing rapidly! There were two absolutely seminalpapers 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 onthe verge of:
i) understanding networks that have interesting computational properties.ii) computing the correlational structure in those networks.
e. Learning. We know a lot of facts (LTP, LTD, STDP).
• it’s not clear which, if any, are relevant. • the relationship between learning rules and computation is essentially unknown.
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.
e. Learning. We know a lot of facts (LTP, LTD, STDP).
• it’s not clear which, if any, are relevant. • the relationship between learning rules and computation is essentially unknown.
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.
What is unsupervised learning?
Learning structure from data without any help from anybody.
Example: most visual scenes are very unlikely to occur.
1000 × 1000 pixels => million dimensional space.
space of possible pictures is much smaller, and forms a very complicated manifold:
pixel 1
pix
el 2
possible visualscenes
What is unsupervised learning?
Learning structure from data without any help from anybody.
Example: most visual scenes are very unlikely to occur.
1000 × 1000 pixels => million dimensional space.
space of possible pictures is much smaller, and forms a very complicated manifold:
pixel 1
pix
el 2
visual scenes
What is unsupervised learning?
Learning structure from data without any help from anybody.
Example: most visual scenes are very unlikely to occur.
1000 × 1000 pixels => million dimensional space.
space of possible pictures is much smaller, and forms a very complicated manifold:
pixel 1
pix
el 2
visual scenes
What is unsupervised learning?
Learning from spikes:
neurons 1
neu
ron
2
What is unsupervised learning?
Learning from spikes:
neurons 1
neu
ron
2dog
cat
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!!!
f. Where we know algorithms we know the neuralimplementation (sort of):
vestibular system, sound localization, echolocation, addition
This is not a coincidence!!!!
Remember David Marr:
1. the problem (computational level) 2. the strategy (algorithmic level) 3. how it’s actually done by networks of neurons (implementational level)
What we know: my score (1-10).
a. Anatomy. 5b. Single neurons. 6c. The neural code. 6d. Recurrent networks of spiking neurons. 3e. Learning. 2
The hard problems:1. How does the brain extract latent variables? 1.0012. How does it manipulate latent variables? 1.0023. How does it learn to do both? 1.001
Outline:
1. Basics: single neurons/axons/dendrites/synapses. Latham2. Language of neurons: neural coding. Sahani3. Learning at network and behavioral level. Dayan 4. What we know about networks (very little). Latham
Outline for this part of the course (biophysics):
1. What makes a neuron spike.2. How current propagates in dendrites.3. How current propagates in axons.4. How synapses work.5. Lots and lots of math!!!
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