Connected Populations: oscillations, competition and spatial continuum (field equations)

Lecture 12Course: Neural Networks and Biological Modeling

Wulfram Gerstner, EPFL

Book: W. Gerstner and W. Kistler, Spiking Neuron Models Chapter 9.1

Population of neurons

h(t)

I(t) ?

0t

A(t) ))('),(()( tItIgtA

))()(())(()( dsstIsgthgtA A(t)

A(t) ))(()( tIgtA

potential

A(t) ))(()()( thgtAtAdt

d

t

2 2( ; )( )

t tn t tA t

N t

populationactivity

Blackboard:Pop. activity

Signal transmission in populations of neurons

Connections4000 external4000 within excitatory1000 within inhibitory

Population- 50 000 neurons- 20 percent inhibitory- randomly connected

-low rate-high rate

input

Population- 50 000 neurons- 20 percent inhibitory- randomly connected

Signal transmission in populations of neurons

100 200time [ms]

Neuron # 32374

50

u [mV]

100

0

10

A [Hz]

Neu

ron

#

32340

32440

100 200time [ms]50

-low rate-high rate

input

Theory of transients

low noise

I(t)h(t)

noise-free

(escape noise/fast noise) noise model A

low noise

fast transient

noise model A

I(t)h(t)

(escape noise/fast noise)

high noise

slow transient

))(()( tIgtA

But transient oscillations

))(()( thgtA

High-noise activity equation

noise model A

I(t)h(t)

(escape noise/fast noise)

high noise

slow transient

))(()( thgtA

))(()( thgtA

)()()( tIRththdtd

Population activity

Membrane potential caused by input

blackboard

In the limit of high noise,

Full connectivity

Part I: Single Population - Population Activity, definition - high noise - full connectivity

Fully connected network

N

Jwij

0 f

jtt j f

ijnet wtI )(

All spikes, all neurons

fjtt

Synaptic coupling

fullyconnected N >> 1

)()()( tIRththdtd

)()()( tItItI netext

blackboard

N

Jwij

0

)(tItt extfj

j fiji wtI )(

All spikes, all neurons

)(tI ext dsstAsJtI i )()( 0 fullyconnected

All neurons receive the same total input current (‘mean field’)

)()()( tIRththdtd

Index i disappears

Stationary solution)( 0hg

0h

0A

fullyconnected N >> 1

Homogeneous network, stationary,All neurons are identical,Single neuron rate = population rate

0 0( )g h A

( ) ( ) ( ) ( )ext netwddt h t h t R I t R I t

0( ) ( ) ( ) ( ) ( )extddt h t h t R I t R J ds s A t s

0( ) 0 ( )ddt h t h t h

Exercises 1, now

Next lecture 10:15

Exercise 1: homogeneous stationary solution

)( 0hg

0h

0A

fullyconnected N >> 1

Homogeneous networkAll neurons are identical,Single neuron rate = population rate

A

Connected Populations:Part I: Single Population - Population Activity, definition - high noise - full connectivity - stationary state - mean rate in stationary state

0 0( )A g hWhat is this function g?

Wulfram Gerstner, EPFL

Stationary State/Asynchronous State

1 s

fully connected coupling J/N tt ˆ ttu ˆ| 0h

input potential

1

0

ˆ ˆ|Is P t s t ds

)( 0hg

)( 0hg

extIAqJRRIh 00000

frequency (single neuron)

qdss

blackboard

Homogeneous networkAll neurons are identical,Single neuron rate = population rate

A(t)= A0= const

Single neuron

High-noise activity equation

noise model A

I(t)h(t)

(escape noise/fast noise)

high noise

slow transient

))(()( thgtA

))(()( thgtA

)()()( tIRththdtd

Population activity

Membrane potential caused by input

Note: total membrane potential

)ˆ()()( ttthtu

Effect of last spike

0 0( )A g hWhat is this function g?

Spike Response Model

iuij

fjtt

Spike reception: EPSP

fjtt

Spike reception: EPSP

itt ˆ

itt ˆ

Spike emission: AP

fjtt itt ˆ tui

j fijw

tui Firing: tti ˆ

Spike emission

Last spike of i All spikes, all neurons

hi(t)

)()()( tIRthth iiidtd

Fully connected network

N

Jwij

0

Spike emission: AP

fjtt ( )netw

ijj f

I t wAll spikes, all neurons

fjtt

itt ˆ

Synaptic coupling

fullyconnected N >> 1

)(ˆ thtt ii tui

)()()( tIRththdtd

( ) ( ) ( )ext netwI t I t I t

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

Stationary State/Asynchronous State

1 s

fully connected coupling J/N

tt ˆ ttu ˆ| 0h

input potential

1

0

ˆ ˆ|Is P t s t ds

)( 0hg

)( 0hg

0h

extIAqJRRIh 00000

0A

A(t)=const

extRIhqRJ

A 000

01

frequency (single neuron)

typical mean field (Curie Weiss)

qdss

Homogeneous networkAll neurons are identical,Single neuron rate = population rate

High-noise activity equation

noise model A

I(t)h(t)

(escape noise/fast noise)

high noise

slow transient

))(()( thgtA

))(()( thgtA

)()()( tIRththdtd

Population activity

Membrane potential caused by input

)()()( tItItI netext

)()()( 0 tAqJtItI ext

))(()()( 0 thgqJtItI ext

))(()()()( thgtIRthth extdtd

1 population = 1 differential equation

Connected Populations:Part I: Single Population - Population Activity, definition - high noise - full connectivity - stationary state - mean rate in stationary state

Part II: Multiple Population - Spatial coupling - Background: cortical populations - Continuum model - Stationary state

Wulfram Gerstner EPFL

Microscopic vs. Macroscopic Coupling

I(t)

)(tAn

Cortical columns:Orientation tuning (and coarse coding)

Detour: Receptive fields (see also lecture 4)

visual cortex

electrode

Detour: Receptive field development

visual cortex

Detour: Receptive field development

Receptive fields: Retina, LGN

-+

-

Detour: Receptive field development

Receptive fields: Retina, LGN

Receptive fields: visual cortex V1

Orientation selective

Detour: Receptive field development

Receptive fields: visual cortex V1

Orientation selective

Detour: orientation selective receptive fields

Receptive fields: visual cortex V1

Orientation selective

2

0

rate

Stimulus orientation

Detour: Receptive fields

visual cortex

Neighboring cells in visual cortexHave similar preferred orientation:

cortical orientation map

Detour: orientation selective receptive fields

Receptive fields: visual cortex V1

Orientation selective

2

0

rate

Stimulus orientation

Cell 1

Detour: orientation selective receptive fields

0

rate

Cell 1 Cell 5

2

Oriented stimulus

Course coding

Many cells respond to a single stimulus with different rate

Continuous Networks

Continuous Networks

)(tAi

Several populations

i k

Continuum

Blackboard

Part I: Single Population - Population Activity, definition - high noise - full connectivity - stationary state - mean rate in stationary state Part II: Multiple Population - Background: cortical populations - spatial coupling - continuum model - stationary state - application: head direction cells

Field equations and continuum Models for coupled populations

)(),( AtA

Continuum: stationary profile

Head direction cells (and line attractor)

Neurophysiology of the Rat Hippocampus

rat brain

CA1

CA3

DG

pyramidal cells

soma

axon

dendrites

synapses

electrodePlace fields

Hippocampal Place Cells

Main property: encoding the animal’s location

place field

Head-direction Cells

Main property: encoding the animal ’s heading

Preferred firing direction

r (i

i

Head-direction Cells

Main property: encoding the animal ’s allocentric heading

Preferred firing direction

r (i

i

0

90

180

270300

Basic phenomenology

0

A(theta)

I: Bump formation

strong lateral connectivity

Possible interpretation of head direction cells: always some cells active indicate current orientation

Field Equations:Wilson and Cowan, 1972

Basic phenomenology

x0

A(x)

II. Edge enhancement

Weaker lateral connectivity

I(x)

Possible interpretation of visual cortex cells: contrast enhancement in - orientation - location

Field Equations:Wilson and Cowan, 1972

)(),( AtA

Continuum: stationary profile

Comparison simulation of neurons and macroscopic field equation

Spiridon&Gerstner

See: Chapter 9, book:Spiking Neuron Models,W. Gerstner and W. Kistler, 2002

Exercises 2,3,4 from 11:15-12:45

Exercise 1: homogeneous stationary solution

Exercise 2: bump solution

Exercise 3: bump formation

)( 0hg

0h

0A

Blob forms where the input is

x

0A

x

0A

Part I: Single Population - Population Activity, definition - high noise - full connectivity - stationary state - mean rate in stationary state Part II: Multiple Population - Background: cortical populations - spatial coupling - continuum model - stationary statePart III: Beware of Pitfalls - oscillations - low noise

Field equations and continuum Models for coupled populations

Book: Spiking Neuron Models.W. Gerstner and W. Kistler

Chapter 8

Oscillations

Stability of Asynchronous State

Search for bifurcation points

linearize

tdtAttPtA I

t

ˆ)ˆ(ˆ|)(

)()( 0 tAAtA )()( 0 thhth dsstAsJth )()()(

h: input potential

A(t)

ttieAtA 1)(

0

fully connected coupling J/N

Stability of Asynchronous State A(t)

delayperiod

)()( sess0 for

stable0

03

02

noise

T

)(s

s

)(s

(Gerstner 2000)

High-noise activity equation

noise model A

I(t)h(t)

(escape noise/fast noise)

high noise

slow transient

))(()( thgtA

))(()( thgtA

)()()( tIRththdtd

Population activity

Membrane potential caused by input

))(()()()( thgtIRthth extdtd

Attention: - valid for high noise only, else transients might be wrong - valid for high noise only, else spontaneous oscillations may arise

Random connectivity

Random Connectivity/Asynchronous State

1 s

randomly connected

1

0

,ˆ|ˆ

dststPI ),( 0 hg

),( 0 hg

0h

0A

A(t)=const

frequency (single neuron)

improved mean field

C inputs per neuron

0

0

CJ

ijw

extIAqJI 0000

002 A

C

J

mean drive

variance

(Amit&Brunel 1997, Brunel 2000)

Analogous for column of 1 exc. + 1 inhib. Pop.

The end

Liquid state/Information buffering

Ultra-short-term information buffering (‘echo’)

I(t))()( f

jj f

j ttatout

How much information in out(t) about signal

2)()()( tItoutaE k

)( tI ?

(Maass et al., 2002, H.Jäger 2004)

640 excitatory n.160 inhibitory n.20ms time constant50 connections/n.

population activity

0)( AtA

bias <I>

N parameters

)()( tItout

ms40

Ultra-short-term information buffering (‘echo’)

I(t)

)()( fj

j f

ttatout

dsstAsa )()(

(Mayor&Gerstner)

)(tA

bias <I>

mean-field readoutmicroscopic readout