Omri Barak

Collaborators:Larry AbbottDavid SussilloMisha Tsodyks

Sloan-SwartzJuly 12, 2011

Messy nice stuff&

Nice messy stuff

Neural representation

• Representation of task parameters by neural population.

• We know that large populations of neurons are involved.

• Yet we look for and are inspired by impressive single neurons.

• Case study: Delayed vibrotactile discrimination (from Ranulfo Romo’s lab)

f1 f2

time (sec)

Romo & Salinas, Nat Neurosci Rev, 2003

f1 f2

f1>f2?Y N

time (sec)

Romo & Salinas, Nat Neurosci Rev, 2003

Romo task

• Encoding of analog variable• Memory of analog variable• Arithmetic operation “f1-f2”

Romo, Brody, Hernandez, Lemus. Nature 1999Machens, Romo, Brody. Science 2005

• Striking tuning properties• Lead to “simple / low dimensional” models• “Typical” neurons are used to define

model populations.

Existing models

Machens et al. 2005

Miller et al. 2006Barak et al. 2010

Miller et al. 2003

Not shown: VergutsDeco

Singh and Eliasmith 2006

But… Are all cells that good?

Barak et al. 2010

Brody et al. 2003

35% of the neurons flip their sign

-6 -4 -2 0 2 4 6 8-3

-2

-1

0

1

2

3

Stimulus tuning

Del

ay tu

ning

Jun et al. 2010

prestim -1.5 0 0.5 1.5 2.5 3.5 40

20

40

Time (sec)

10 Hz22 Hz34 Hz

Echo state network

Jaeger 2001Maass et al 2002Buonomano and Merzenich 1995

Echo state network

N

ii

Outi

FBi

N

jjiji

i

rwz

zwrJgxdtdx

1

1

x

r

-4 -2 0 2 4-1

-0.5

0

0.5

1

-2 0 2 40

0.5

1

1.5

2

x

+ Noise

N = 1000 / 2000K = 100 (sparseness)g = 1.5

Implementing the Romo task

f1 f2

f

r

Sussillo and Abbott 2009Jaeger and Haas 2004

Input(f1,f2)

Output

Input(f1,f2)

Output

Unit activity

It works, but…

• How does it work?– After the training, we have a network that is

almost a black box.• Relation to experimental data.

Hypothesis

• Consider the state of the network in 1000-D as the trial evolves

00.5

1 0

0.5

10

0.5

1f1

f2

time (sec)

Hypothesis

• Focus only at the end of the 2nd stimulus.• For each (f1,f2) pair, there is a point in

1000-D space.

0.5 1 1.5 20.5

1

1.5

2

0.51

1.52 0.5

1

1.5

2

-1

0

1

0.51

1.52 0.5

1

1.5

2-2

0

2

Hypothesis

• Focus only at the end of the 2nd stimulus.• For each (f1,f2) pair, there is a point in

1000-D space.• So there is a 2D manifold in the 1000-D

space.• Can the dynamics (after learning) draw a

line through this manifold?

0.5 1 1.5 20.5

1

1.5

2

0.51

1.52 0.5

1

1.5

2

-1

0

1

0.51

1.52 0.5

1

1.5

2-2

0

2

Dynamics or just fancy readout?

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

Yes - YesYes - NoNo - NoFixed - YesFixed - No D

istance in state space

• The two responses are different in network activity, not just through the particular readout we chose.

Saddle point

Searching for a saddle in 1000D

N

ii

Outi

FBi

N

jjiji

i

rwz

zwrJgxdtdx

1

1

ii

FBi

N

jjijii

xfxF

zwrJgxxf

2

1

)()(

)(Vector function:

Scalar function:

Searching for a saddle in 1000D

1

Num

ber o

f uns

tabl

e ei

genv

alue

s

Distance along trajectory

Num

ber o

f un

stab

le e

igen

valu

es

Norm of fixed point

Saddle point

Saddle point

Slightly more realistic

• Positive firing rates• Avoid fixed point between trials. • Introduce reset signal.• Chaotic activity in delay period

= 0

It works

Nice persistent neurons

0 20 40 60 800

0.5

1

1.5

0 20 40 60 800

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Act

ivity

a1-a2 plane

Romo and Salinas 2003

-2 0 2-2

0

2

-2 0 2-2

0

2

-2 0 2-2

0

2

-2 0 2-2

0

2

f1 tuning

f 2 tu

ning

Problems / predictions

• Reset signal• Generalization

Reset

There is a reset (Barak et al 2010, Churchland et al)

There is no reset, and performance shows it (Buonomano et al 2007)

Time (sec)

Cor

rela

tion

Correlation between trials with different frequencies

Generalization

• Interpolation vs. Extrapolation

f1

f 2

Generalization

• Interpolation vs. Extrapolation

f1

f 2

Generalization

• Interpolation vs. Extrapolation

f1

f 2

Extrapolation

Delosh et al 1997

Conclusions

• Response properties of individual neurons can be misleading.

• An echo state network can solve decision making tasks.

• Dynamical systems analysis can reveal function of echo state networks.

• Need to find a middle ground.