Week 10 part 1 : Neuronal Populations 10.1 Cortical ...lcn€¦ · Week 10 – Neuronal Populations Wulfram Gerstner EPFL, Lausanne, Switzerland 10.1 Cortical Populations - columns

Biological Modeling

of Neural Networks:

Week 10 – Neuronal Populations

Wulfram Gerstner

EPFL, Lausanne, Switzerland

10.1 Cortical Populations

- columns and receptive fields

10.2 Connectivity - cortical connectivity

- model connectivity schemes

10.3 Mean-field argument - asynchronous state

10.4 Random Networks - Balanced state

Week 10 – part 1 : Neuronal Populations

Biological Modeling of Neural Networks – Review from week 1

motor

cortex

frontal

cortex

to motor

output

10 000 neurons

3 km wire 1mm

population of neurons

with similar properties

stim

neuron 1

neuron 2

Neuron K

Brain


population activity - rate defined by population average

t

t

tN

tttntA

);()(population

activity ‘natural readout’




Brain

Week 10-part 1: Population activity



Brain

Week 10-part 1: Population activity

population activity

A(t)



Brain

Week 10-part 1: Scales of neuronal processes

Biological Modeling

of Neural Networks:


Wulfram Gerstner







10.4 Random networks

Week 10 – part 1b :Cortical Populations – columns and receptive fields

visual

cortex

electrode

Week 10-part 1b: Receptive fields

Week 10-part 1b: Receptive fields

visual

cortex

electrode

visual

cortex

Neighboring cells

in visual cortex

have similar preferred

center of receptive field

Week 10-part 1b: Receptive fields and retinotopic map

Receptive fields:

Retina, LGN

Receptive fields:

visual cortex V1

Orientation

selective

Week 10-part 1b: Orientation tuning of receptive fields

Receptive fields:

visual cortex V1

Orientation selective


Receptive fields:

visual cortex V1

Orientation selective 20

rate

Stimulus orientation


visual

cortex

Receptive fields:

visual cortex V1

Orientation selective

1st pop

2nd pop

2nd pop

Neighboring neurons

have similar properties

Week 10-part 1b: Orientation columns/orientation maps

visual

cortex

Neighboring cells in visual cortex

Have similar preferred orientation:

cortical orientation map

Week 10-part 1b: Orientation map

population of neighboring neurons: different orientations

Visual cortex

Week 10-part 1b: Orientation columns/orientation maps

Image: Gerstner et al.

Neuronal Dynamics (2014)

I(t)

)(tAn

Week 10-part 1b: Interaction between populations / columns

20

rate

Stimulus orientation

Cell 1

Week 10-part 1b: Do populations / columns really exist?

0

rate

Cell 1 Cell 5

2

Oriented stimulus

Course coding

Many cells

(from different columns)

respond to a single

stimulus with different rate

Week 10-part 1b: Do populations / columns really exist?

Quiz 1, now

The receptive field of a visual neuron refers to

[ ] The localized region of space to which it is sensitive

[ ] The orientation of a light bar to which it is sensitive

[ ] The set of all stimulus features to which it is sensitive

The receptive field of a auditory neuron refers to


[ ] The range of frequencies to which it is sensitive

The receptive field of a somatosensory neuron refers to


[ ] The region of body surface to which it is sensitive

Biological Modeling

of Neural Networks:


Wulfram Gerstner







10.4 Balanced state

Week 10 – part 2 : Connectivity

I(t)

)(tAn

Week 10-part 2: Connectivity schemes (models)

1 population =

What?

population = group of neurons

with

- similar neuronal properties

- similar input

- similar receptive field

- similar connectivity

make this more precise

Week 10-part 2: model population

Week 10-part 2: local cortical connectivity across layers

Here:

Excitatory neurons

(mouse somatosensory cortex)

Lefort et al. NEURON, 2009

1 population =

all neurons of given type

in one layer of same column

(e.g. excitatory in layer 3)

Week 10-part 2: Connectivity schemes (models)

full connectivity random: prob p fixed



random: number K

of inputs fixed

Week 10-part 2: Random Connectivity: fixed p

random: probability p=0.1, fixed




Can we mathematically predict

the population activity?

given

- connection probability p

- properties of individual neurons

- large population

asynchronous

activity

Biological Modeling

of Neural Networks:


Wulfram Gerstner







10.4 Balanced state

Week 10 – part 2 : Connectivity

Population - 50 000 neurons

- 20 percent inhibitory

- randomly connected

Random firing in a populations of neurons

100 200 time [ms]

Neuron # 32374

50

u [mV]

100

0

10

A [Hz]

Neu

ron

#

32340

32440

100 200 time [ms] 50

-low rate

-high rate input

Week 10-part 3: asynchronous firing



Blackboard:

-Definition of A(t)

- filtered A(t)

- <A(t)>

Asynchronous state

<A(t)> = A0= constant



Brain

Week 10-part 3: counter-example: A(t) not constant

Systematic oscillation

not ‘asynchronous’

Populations of spiking neurons

I(t)

?

population activity? t

t

tN

tttntA

);()(population

activity

Homogeneous

network: -each neuron receives input

from k neurons in network

-each neuron receives the same

(mean) external input

Week 10-part 3: mean-field arguments

Blackboard:

Input to neuron i

Full connectivity


Fully connected network

0ijw w

Synaptic coupling

fully

connected

N >> 1 )()()( tItItI netext

blackboard

( ) ( )net f

ij j

j f

I t w t t

All spikes, all neurons

( )f

jt t


N

Jwij

0

All spikes, all neurons

)(tI ext

0( ) ( ) ( )iI t J s A t s ds

fully

connected

All neurons receive the same total input current

(‘mean field’)

Index i disappears

( ) ( )net f ext

ij j

j f

I t w t t I


1s

1

0

ˆ ˆ|Is P t s t ds )( 0hg

0 0 0 0[ ]extI J q A I

frequency (single neuron)

blackboard

Homogeneous network

All neurons are identical,

Single neuron rate = population rate A(t)= A0= const

Single

neuron

0 0( )g I A

Week 10-part 3: stationary state/asynchronous activity

Stationary solution

0I

0A

fully

connected

N >> 1

Homogeneous network, stationary,


Single neuron rate = population rate

0 0( )g I A

0( )g I

0( )g I

0A

0 0 0 0[ ]extI J q A I

Exercise 1, now

Exercise 1: homogeneous stationary solution )( 0hg

0h

0A

fully

connected

N >> 1 Homogeneous network



0A

Next lecture:

11h15

Single Population

- population activity, definition

- full connectivity

- stationary state/asynchronous state

What is this function g?


0 0( )g I A

Examples: - leaky integrate-and-fire with diffusive noise

- Spike Response Model with escape noise

- Hodgkin-Huxley model (see week 2)

Week 10-part 3: mean-field, leaky integrate-and-fire

0( )g I

0 0 0 0

0 0 0 0[ ] /

ext

ext

I J q A I

I I J q A

different noise levels

function g can be calculated

iui

j

Spike reception: EPSP

Spike emission: AP

j f

ijw

Spike emission

Last spike of i All spikes, all neurons

ˆ( )t t

( ) exp[ ]

f

jf

j

t tt t

( )f

jt t

ˆ( )t t

ˆ( )it t ( )f

jt t( )iu t

Firing intensity

( | ) ( )t u f u

Review: Spike Response Model with Escape Noise

Response to current pulse

Spike emission: AP

potential

ˆ( )i iu t t )(th

input potential

Last spike of i potential

( )I t s ds

external input

0

( )( ) ( ( )) exp( )

h tt f h t

instantaneous rate

Blackboard

-Renewal model

-Interval distrib.

Review: Spike Response Model with Escape Noise

Last spike of i

ˆ( )it t

ˆ( )it t

ˆ( )i iu t t

ˆ( )it t

( )s

( )s

1s

1

0

ˆ ˆ|Is P t s t ds0( )g I

0 0 0 0 0[ ]exth RI R J q A I


Homogeneous network


Single neuron rate = population rate A(t)= A0= const

Single

neuron

0 0( )g I A

Week 10-part 3: Example - Asynchronous state in SRM0

1s

fully connected

0hinput potential

1

0

ˆ ˆ|Is P t s t ds0( )g h

0( )g h

0h

0 0 0 0 0[ ]exth RI R J q A I

0A

A(t)=const


typical mean field

(Curie Weiss) Homogeneous network


ˆ( )u t t ˆ( )t t

( )s ds q

Week 10-part 3: Example - Asynchronous state in SRM0

0 0 0

0

1[ ]extA h RI

J qR

Biological Modeling

of Neural Networks:


Wulfram Gerstner







10.4 Random networks

- balanced state

Week 10 – part 4 : Random networks

random connectivity

Week 10-part 4: mean-field arguments – random connectivity

random: prob p fixed random: number K

of inputs fixed full connectivity

I&F with diffusive noise

(stochastic spike arrival)

EPSC IPSC

For any arbitrary neuron in the population

Blackboard:

excit. – inhib.

'

' '

, ', '

( ) ( )f f

i i ik e k ik i k

k f k f

dI I w q t t w q t t

dt

excitatory input spikes

Week 10-part 4: mean-field arguments – random connectivity

Week 10-Exercise 2.1/2.2 now: Poisson/random network

2.

3.

random: prob p fixed

Next lecture:

11h40


random: probability p=0.1, fixed


Neuronal Dynamics (2014) fluctations of A decrease

fluctations of I decrease

Network N=10 000 Network N=5 000

ik

Jw

pN

Week 10-part 4: Random connectivity – fixed number of inputs

random: number of inputs K=500, fixed


Neuronal Dynamics (2014) fluctations of A decrease

fluctations of I remain

Network N=10 000 Network N=5 000 ik

Jw

K

Week 10-part 4: Connectivity schemes – fixed p, but balanced

'

' '

, ', '

( ( ) ( ))f f ext

i i ik e k ik i k

k f k f

du u R w q t t w q t t I

dt

exc inh

make network bigger, but

-keep mean input close to zero

-keep variance of input

eik

e

Jw

pN' 'eJ random

iik

i

Jw

pN' 'iJ random

e e i ip N J p N J

eN iN

Week 10-part 4: Connectivity schemes - balanced



fluctations of A decrease

fluctations of I decrease, but ‘smooth’

Week 10-part 4: leaky integrate-and-fire, balanced random network



Network with balanced excitation-inhibition - 10 000 neurons

- 20 percent inhibitory

- randomly connected

Week 10-part 4: leaky integrate-and-fire, balanced random network








10.4 Random Networks - Balanced state

Week 10 – Introduction to Neuronal Populations

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Week 10 part 1 : Neuronal Populations 10.1 Cortical ...lcn€¦ · Week 10 – Neuronal Populations Wulfram Gerstner EPFL, Lausanne, Switzerland 10.1 Cortical Populations - columns