Coding and computing with balanced spiking networks Sophie Deneve - - PowerPoint PPT Presentation

Coding and computing with balanced spiking networks

Sophie Deneve Ecole Normale Supérieure, Paris

Poisson Variability in Cortex

Trial 1 Trial 2 Trial 3 Trial 4

Mean Spike Count Variance of Spike Count

Cortical spike trains are highly variable

From Churchland et al, Nature neuroscience 2010

Cortical spike trains are variable

From Churchland et al, Nature neuroscience 2010

Balance between excitation and inhibition

Excitatory, Poisson Inhibitory, Poisson Integrate and fire neuron:

exc inh

V V I I     

exc

I

inh

I

Balance between excitation and inhibition

Excitatory, Poisson Inhibitory, Poisson Integrate and fire neuron:

exc inh

V V I I     

Output is more regular than input

exc

I

inh

I

Balance between excitation and inhibition

Excitatory, Poisson Inhibitory, Poisson Integrate and fire neuron:

exc inh

V V I I     

Output is more regular than input How does Poisson-like variability survives?

exc

I

inh

I

Balanced excitation/inhibition

Excitatory Inhibitory Random walk

Shadlen and Newsome, 1996

exc inh

V V I I     

exc

I

inh

I

Balanced excitation/inhibition

Excitatory Inhibitory Random walk

Shadlen and Newsome, 1996

exc inh

V V I I     

exc

I

inh

I

Variability is conserved when mean excitation =mean inhibition

E/I balance: stimulus driven response

Wehr and Zador, 2003

Okun and Lampl, 2008

E/I balance: spontaneous activity

Two types of balanced E/I

E I E

Feed-forward inhibition: Recurrent inhibition:

E I E Not random: Highly structured…

Balanced neural networks generate their own variability

E I

EE

J

EI

J

IE

J

II

J

EE E IE I EI E II I

J v J v J v J v    

Weak, sparse random connections

e.g C. van Vreeswijk and H. Sompolinsky, Science (1996);

ext

I

Brunel, 2001

Balanced neural networks generate their own variability

E I

EE

J

EI

J

IE

J

II

J

EE E IE I EI E II I

J v J v J v J v    

e.g C. van Vreeswijk and H. Sompolinsky, Science (1996);

Asynchronous irregular regime Low firing rate

ext

I

Balanced neural networks generate their own variability

E I

EE

J

EI

J

IE

J

II

J

EE E IE I EI E II I

J v J v J v J v    

e.g C. van Vreeswijk and H. Sompolinsky, Science (1996);

Asynchronous irregular regime Low firing rate

ext

I

Shuffle one spike by 0.1ms

Balanced neural networks generate their own variability

E I

EE

J

EI

J

IE

J

II

J

EE E IE I EI E II I

J v J v J v J v    

e.g C. van Vreeswijk and H. Sompolinsky, Science (1996);

Asynchronous irregular regime Low firing rate Shuffle one spike by 0.1ms

ext

I

Reshuffle all later spikes

Balanced neural networks generate their own variability

E I

EE

J

EI

J

IE

J

II

J

EE E IE I EI E II I

J v J v J v J v    

e.g C. van Vreeswijk and H. Sompolinsky, Science (1996);

Asynchronous irregular regime Low firing rate Chaotic attractor: Shuffle one spike by 0.1ms

ext

I

Reshuffle all later spikes

Population coding

• Asynchronous, irregular spike trains.
• Population coding.
• E/I balance.

Decoding = summing from large populations Code = Mean firing rates Spikes = random samples “Requiem for a spike”

Continuous variable: Population coding

Georgopoulos 1982

Population Codes

Tuning Curves Average pattern of activity

• 100

100 20 40 60 80 100 Direction (deg) Activity

• 100

100 20 40 60 80 100 Preferred Direction (deg) Activity

s?

   

i i i

s f x x f x   

Noisy population Codes

Pattern of activity (s)

• 100

100 20 40 60 80 100 Preferred Direction (deg) Activity

s?

     

 

exp | !

i

s i i i i

f x f x p s x s  

Poisson noise: Independent:

   

| |

i i

p s x p s x 

Population decoding

x

𝑦=argmin

𝑦

𝑦 − 𝑦 2 𝑠𝑗 = 𝑔

𝑗 𝑦 +Noise

Population vector

Optimal when tuning curves are cosyne, noise gaussian, uniformely distributed over all orientation…

i i i

x r x 

Decoding is easy… but usually suboptimal.

Optimal: Maximum likelihood

Pattern of activity (s)

• 100

100 20 40 60 80 100 Preferred Direction (deg) Activity

s?

x

 

| p s x

Likelihood

ˆ x

Maximum likelihood estimates

 

 

ˆ |

argmax

x

x p s x 

Decoding always optimal … but usually hard.

Optimal Population decoding

Decoding = summing from large neural populations

ˆ

j j j

x r  



x

Efficient population coding

Decoding = summing from large neural populations

ˆ

j j j

x r  



Efficient coding:

   

 

2

ˆ argmin

r

r x x C r   

x

Christian Machens Wieland Brendel Ralph Bourdoukan Pietro Vertechi

In collaboration with

Single neuron Decoding= Post-synaptic integration

input signal x(t) Time

Single neuron Decoding= Post-synaptic integration

input signal x(t) Time

•  

 

ˆ x t r t  

   

*exp r t

• t

  

Single neuron Decoding= Post-synaptic integration

input signal x(t) Time

Where do we place the spikes?

•  

 

ˆ x t r t  

Single neuron

Input signal x(t) Time Minimize: Decoding error

   

 

2

ˆ

t

E x t x t  

Single neuron

Input signal x(t) Time Minimize:

spike no spike t t

E E 

Greedy spike rule:

   

 

2

ˆ

t

E x t x t  

Single neuron

Input signal x(t) Time Minimize:

spike no spike t t

E E 

Greedy spike rule:

   

 

   

 

2 2

ˆ ˆ x t x t x t x t     

   

 

2

ˆ

t

E x t x t  

Single neuron

Input signal x(t) Time Minimize:

spike no spike t t

E E 

Greedy spike rule:

   

 

   

 

2 2

ˆ ˆ x t x t x t x t     

   

 

2

ˆ 2 x t x t     

   

 

2

ˆ

t

E x t x t  

Single neuron

Input signal x(t) Time Minimize:

spike no spike t t

E E 

Greedy spike rule:

   

 

2

ˆ 2 x t x t    

   

 

2

ˆ

t

E x t x t  

Single neuron

Input signal x(t) Time Minimize:

spike no spike t t

E E 

Greedy spike rule:

     

 

2

ˆ 2 V t x t x t     

Membrane potential Threshold Decoding error

   

 

2

ˆ

t

E x t x t  

Single neuron

Input signal x(t) Time Minimize:

spike no spike

E E 

Greedily

ˆ 2 V x x    

Membrane potential Threshold Decoding error

 

2

ˆ E x x  

spike if

x ˆ x

ˆ V x x  

Single neuron

Input signal x(t) Time

2

2 

 

2

V V x x

• 

     

Input Reset Leak Threshold:

 

ˆ V x x   

ˆ x

• x

x    

Leaky Integrate and fire

Neural population

 

1 2

, , ,

J

x x x x 

ˆi

ij j j

x r  



Minimize:

     

2 2

ˆ

t

E x t x t r t    

Decoding error Quadratic cost

Neural population

 

1 2

, , ,

J

x x x x 

Minimize:

spike j no spike j t t

E E 

ˆi

ij j j

x r  



     

2 2

ˆ

t

E x t x t r t    

Neural population

 

1 2

, , ,

J

x x x x 

spike j no spike j t t

E E 

 

2

ˆ 2

T

V x x r         

Minimize:

ˆi

ij j j

x r  



     

2 2

ˆ

t

E x t x t r t    

Neural population

 

1 2

, , ,

J

x x x x 

spike j no spike j t t

E E 

 

2

ˆ 2

T

V x x r         

Decoding error Cost Threshold

Minimize:

ˆi

ij j j

x r  



     

2 2

ˆ

t

E x t x t r t    

Neural population

Time

 

T j j ij i i kj k i i k

V V x x

• 

          

 

Input

Reset +Recurrent

T



T

 

 

1 2

, , ,

J

x x x x 

ˆi

ij j j

x r  



Neural population

Input signal x(t) Time

x

T



ˆ x r  

T

 

Input signal x(t)

Homogeneous network

1

ˆ x

1 

1

x

ˆ x x

ˆ

j j

x r 

2

V

1

V

3

V

Neural variability = Degeneracy

1

ˆ x

1 

1

x

ˆ x x

ˆ

j j

x r 

Equivalent to

“Chaotic”

• 100

100

ˆ , x x

200 600 1000

Time (ms)

1400 1600 2000 2400

c Shift 1 spike by 1ms

x c 

Membrane potentials and spikes

1

ˆ x

1 

1

x 

ˆ x x

ˆ

j j

x r 

2

V

1

V

3

V

Correlated Uncorrelated

Yu and Ferster, 2009

Time (ms)

1

• 1 0

1000 2

Shuffle

Spikes are NOT random samples

1 2

Time (ms)

1000

ˆ , x x ˆ , x x

Input signal x(t)

… But fired exactly at the right moment

1 ˆ x x N 

Why are balanced networks so precise?

x ˆ x

1 N

Time

2 

2  

V 

1 N

Time

x ˆ x

1 N

Balanced network Random spikes

1 ˆ x x N 

1 ˆ x x N 

Why are balanced networks so precise?

x ˆ x

1 N

Time

2 

2  

V 

1 N

Time

x ˆ x

1 N

Balanced network Random spikes

1 ˆ x x N 

Network maintains tight E/I

ˆ

j j j

V x x   

100 ms

1 2

Time (s) Currents

jx

 ˆ

jx



Learning the connections

Cancel the feedforward currents Point in the direction of input currents.

j k

xi

 

2

x r C r   

Find  minimizing

Over- compensation Under- compensation Balanced

∆⌦ −✏

⇤

⌦

∆⌦

✏

¯ ˆ x ˆ x F ˆ − − ˆ x ˆ x Γ − ˆ

⌦

− ∆⌦ −✏

⇤

⌦

∆⌦

✏

¯ ˆ x ˆ x F ˆ − − ˆ x ˆ x Γ − ˆ

⌦

−

Untrained FF. C. Trained FF. C.

D

V

x

^

x

^

x x

Feedforward connections should point in the direction of the input current

• +

∆⌦ −✏

⇤

⌦

∆⌦

✏

¯ ˆ x ˆ x ˆ − − ˆ x ˆ x Γ − ˆ

⌦

− ∆⌦ −✏

⇤

⌦

∆⌦

✏

¯ ˆ x ˆ x ˆ − − ˆ x ˆ x Γ − ˆ

⌦

−

x

x, x

^

x = Dr

^

x = Dr

^

x x F Wr = -Fx =-FDr

^

A C

F

Lateral connections should cancel the feed- forward inputs

Spike-time dependent learning rules

j

∆ij ~ 𝑝

𝑘

𝑦𝑗 − i𝑘

Hebbian STDP Anti-Hebbian STDP

k

xi

kj

 Neuron k spikes

j

V 2

j

V 

 

2

kj k j kj

• V

      

Learning the connections

j

∆ij ~ 𝑝

𝑘

𝑦𝑗 − i𝑘

Hebbian STDP Anti-Hebbian STDP

k

xi

time

pre post

t t 

j

r

k

V

pre post

t t  

Potentiation Depression

• 



2

kj k j kj

• V

      

We can learn networks that respect Dale’s Law E I

WEI WII WIE WEE F

A B

Feed-forward weights Estimate Membrane potential and spikes

Before learning

x2 x1

After learning

Signal

x2 x1

Tuning curves

Quadratic optimization :

• F. Rate (Hz)

xi

20

• 0.1

0.1

• F. Rate (Hz)

10

• 10

360

Direction of x

• 20

ˆ

j j j

x r  



x

   

 

*

2 * *

arg min

r

r x r C r    

Tuning curves

Quadratic optimization :

ˆ

j j j

x r  



x

   

 

*

2 * *

arg min

r

r x r C r



   

Amplitude tuning: Direction tuning:

1 2

x r r   Pocking the network in various ways

1

r

2

r

1 x  2 x  3 x 

1 2

x r r   Pocking the network in various ways

1

r

2

r

Change stimulus:

1 x  2 x  3 x 

1 2

x r r  

1

r

2

r

Inactivate:

Pocking the network in various ways

1

r

2

r

Change stimulus:

1 x  2 x  3 x 

1 2

x r r  

1

r

2

r

Inactivate:

Pocking the network in various ways

1

r

2

r

Change stimulus:

1 x  2 x  3 x 

Instant compensation for neural loss

200 400 600 800 1000 1200

Time (ms)

• 100

100

• 100

100

Time (ms)

400 600 800 1000 1200 200

c

ˆ , x x

Inactivations (e.g. halorhodopsin)

Robust to perturbation, connection noise, background noise, synaptic failure…

Exemple: Neurones integrateurs de position des yeux

Activité Position des yeux ~ 30 neurone de chaque côté

Activité Activité Position des yeux Position des yeux

Activité Activité Position des yeux Position des yeux

Activité Activité Position des yeux Position des yeux

Activité Activité Position des yeux Position des yeux

Direction tuning

x

Direction

Direction tuning

x

Visual orientation tuning

Crook and Eysel, 1992

Population coding

x

𝑦=argmin

𝑦

𝑦 − 𝑦 2 𝑠

𝑗 = 𝑔 𝑗 𝑦 +Noise

ˆ

j j j

x r  



x

   

 

*

2 * *

arg min

r

r x r C r    

Population coding with changing context

x

𝑠

𝑗 = 𝑔 𝑗 𝑦, 𝑏 +Noise

x

   

 

*

2 * *

arg min

r

r x r C r    

Change the output Output unchanged

Pertubation: Adaptation , neural death, lesions…

Conclusions

• Efficient population coding implies:

1. Asynchronous irregular spike trains. 2. Balanced, correlated E/I. 3. Diverse, context dependent tuning curves. 4. Instant changes in those tuning curves to compensate for perturbations.

• This coding can be learnt using a simple spike-time dependent plasticity

rule.

• Balanced spiking networks orders of magnitude more precise (and robust)

than equivalent rate models.

• There is a constant decoder, and thus, an invariant representation at the

population level, but neural responses are themselves highly non- invariant.