UC Davis, October 18, 2016 MIT Applied Math, Media Lab,...

From brains to machine learningand back again

David RolnickMIT Applied Math, Media Lab, CSAIL

UC Davis, October 18, 2016

Part I: From brains to machine learning

The brain

Deep learning and the visual cortex

Deep learning and the visual cortex

Simple

Simple

Simple

Simple

Simple

Simple

Complex

Complex

Deep learning and the visual cortex● Simple cells are sensitive to different stimuli at different places.● Complex cells pool the results of simple cells - they respond to different stimuli

at *any* place.

● Convolutional neural net: Alternates between convolutional layers (simple cells) and pooling layers (complex cells)

Concepts as attractor networks

dog

cat

skull

pet

bone

meow

brain

word: “dog”

Hopfield model for attractor networks

Each vertex xi can take values ±1,

updates according to:

xi = sign( ∑ Wij xj )

Hopfield networks

Hopfield networks

Memory 1

Hopfield networks

Memory 1 Memory 2

Simulating Markov chain dynamics

p = 0.5

p = 1

p = 0.5

p = 1

Joint work with Haim Sompolinsky, Ishita Dasgupta, and Jeremy Bernstein

Pattern 1

Pattern 2

Pattern 3

● Attractors can be deterministic sequences of patterns● What about non-deterministic sequences?

Motivation

Weather

Rain Sprinkler

Wet roof Wet grass


Motivation


Model outline

Memory attractors Noise attractors

Mixed representationdelay delay


s1

s3s2

p = ⅓ s1

s2

n1

p = ⅔

Stochastic transitions

Deterministictransitions

s1

s3

n2

s1

s3

n3


Memoryattractors

Noiseattractors

Mixed representation

0

0

0

1

1

1

Time

The network in action

Model demo

P = 0.5

P = 1

P = 0.5

P = 1

Markov chain

Memoryattractors

0

1

Time


Applications of Hopfield networks

Joint work with Christopher Hillar, Felix Effenberger, and Sarah Marzen

Hopfield networks on n vertices can store...

● Cover (1965): …at most O(n) randomly selected patterns.● Hillar & Tran (2014): ...at least O(exp(√n)) nonrandom, nontrivial patterns.● Theorem (Effenberger, Hillar, Marzen, R.):

...at least O(exp(n1 - o(1))) nonrandom, nontrivial patterns.

Image processing in the brain

Hermann grid illusion Checker shadow illusion

Image-processing with Hopfield networks

Joint work with Christopher Hillar, Felix Effenberger, and Sarah Marzen

Joint work with Christopher Hillar, Felix Effenberger, and Sarah MarzenData from Tasovanis group at the German Center for Neurodegenerative Diseases (DZNE), Bonn

Denoising images

Continuous neuron dynamics

Joint work with Carina Curto, figures from Morrison et al.

● Threshold linear networks:

● Oscillating behavior observed by Morrison et al. (unproven), limit cycles as attractors.

● Theorem (Curto & R.): Oscillating behavior is indeed a stable state for certain simple network architectures.

Part II: From machine learning to brains

Connectomics

● Input: microscope images of slices of brain tissue.

● Slices aligned and stacked.● Boundaries of neurons are

predicted with deep learning.● Neurons are filled in.● Output: 3D segmentation.

Connectomics with context

● Neurons look like this:

● Standard methods use only local context● Leads to mistakes like this:

Joint work with Nir Shavit, Yaron Meirovitch, and the MIT Computational Connectomics Group

Learning neuron morphologies

Problem: How to learn a distribution on embedded graphs?

One approach: Throw out outliers

● Learn common types of errors

Another approach: Similarity scores

● Compare, cluster graphs based on similarity● Use a library of known graphs to evaluate plausibility of candidate graphs

Fixing merged neurons

Joint work with Nir Shavit, Yaron Meirovitch, and the MIT Computational Connectomics Group

● Neuron as graph embedded in R3

● Smooth embedding, trim short branches● Measure instantaneous direction and radius● Check for coherent splits into subgraphs

Shape context● Original algorithm (Belongie & Malik, 2000)● Pick random sample points on each neuron● Compute Euclidean distance and shortest-path distance between sample points

0.01 0.01 0.05 0.02 0

0.02 0 0.04 0.03 0

0.02 0.06 0.2 0.1 0.01

0.01 0.1 0.15 0 0

0.01 0.04 0.13 0.08 0.01

Example 2D histogram for one sample point

Frac of other points at Euclidean dist.

Frac of other points at shortest-path dist.

400-800

200-400

100-200

50-100

0-50

0-50 50-100 100-200 200-400 400-800

Joint work with Viren Jain and Google Research

Shape context

● Minimum cost perfect matching (linear assignment) in complete bipartite graph● Pair sample points between two neurons● Matching cost (edge weight) = χ2-distance between histograms● Similarity score = normalized minimum cost● Build library of known neuron morphologies: sets of histograms● Compare candidate morphologies against library


Shape context

Partial neuron A: High similarity to A: Low similarity to A:

● Robust to differences in sample preparation● Not sensitive to angles, small variations in spines, etc.● Highly sensitive to erroneous connections


Score distributions, individual neurons


Shape context


● Each point represents a neuron● t-SNE embedding, colors from k-medians clustering

Coming full circle

● Learning the structures of real neural nets with artificial neural nets● Project neurons into 2D images for deep learning:


Thanks to all these people...● Nir Shavit, Yaron Meirovitch, and the MIT Computational Connectomics group● Ed Boyden and the MIT Synthetic Neurobiology group● Viren Jain and Google Research● Haim Sompolinsky, Ishita Dasgupta, and Jeremy Bernstein● Carina Curto● Christopher Hillar, Felix Effenberger, and Sarah Marzen

● This work was also supported by the Center for Brains, Minds, and Machines and the National Science Foundation (grant no. 1122374).

...and thank you!

UC Davis, October 18, 2016 MIT Applied Math, Media Lab,...

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