Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12...
-
date post
22-Dec-2015 -
Category
Documents
-
view
220 -
download
0
Transcript of Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12...
![Page 1: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/1.jpg)
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
![Page 2: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/2.jpg)
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
![Page 3: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/3.jpg)
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
![Page 4: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/4.jpg)
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
![Page 5: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/5.jpg)
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
![Page 6: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/6.jpg)
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,
![Page 7: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/7.jpg)
Full connectivity
Part I: Single Population - Population Activity, definition - high noise - full connectivity
![Page 8: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/8.jpg)
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
![Page 9: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/9.jpg)
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
![Page 10: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/10.jpg)
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
![Page 11: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/11.jpg)
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
![Page 12: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/12.jpg)
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
![Page 13: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/13.jpg)
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
![Page 14: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/14.jpg)
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?
![Page 15: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/15.jpg)
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
![Page 16: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/16.jpg)
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
![Page 17: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/17.jpg)
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
![Page 18: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/18.jpg)
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
![Page 19: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/19.jpg)
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
![Page 20: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/20.jpg)
Microscopic vs. Macroscopic Coupling
I(t)
)(tAn
![Page 21: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/21.jpg)
Cortical columns:Orientation tuning (and coarse coding)
![Page 22: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/22.jpg)
Detour: Receptive fields (see also lecture 4)
visual cortex
electrode
![Page 23: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/23.jpg)
Detour: Receptive field development
visual cortex
![Page 24: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/24.jpg)
Detour: Receptive field development
Receptive fields: Retina, LGN
-+
-
![Page 25: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/25.jpg)
Detour: Receptive field development
Receptive fields: Retina, LGN
Receptive fields: visual cortex V1
Orientation selective
![Page 26: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/26.jpg)
Detour: Receptive field development
Receptive fields: visual cortex V1
Orientation selective
![Page 27: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/27.jpg)
Detour: orientation selective receptive fields
Receptive fields: visual cortex V1
Orientation selective
2
0
rate
Stimulus orientation
![Page 28: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/28.jpg)
Detour: Receptive fields
visual cortex
Neighboring cells in visual cortexHave similar preferred orientation:
cortical orientation map
![Page 29: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/29.jpg)
Detour: orientation selective receptive fields
Receptive fields: visual cortex V1
Orientation selective
2
0
rate
Stimulus orientation
Cell 1
![Page 30: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/30.jpg)
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
![Page 31: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/31.jpg)
Continuous Networks
Continuous Networks
![Page 32: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/32.jpg)
)(tAi
Several populations
i k
Continuum
Blackboard
![Page 33: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/33.jpg)
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
![Page 34: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/34.jpg)
)(),( AtA
Continuum: stationary profile
![Page 35: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/35.jpg)
Head direction cells (and line attractor)
![Page 36: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/36.jpg)
Neurophysiology of the Rat Hippocampus
rat brain
CA1
CA3
DG
pyramidal cells
soma
axon
dendrites
synapses
electrodePlace fields
![Page 37: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/37.jpg)
Hippocampal Place Cells
Main property: encoding the animal’s location
place field
![Page 38: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/38.jpg)
Head-direction Cells
Main property: encoding the animal ’s heading
Preferred firing direction
r (i
i
![Page 39: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/39.jpg)
Head-direction Cells
Main property: encoding the animal ’s allocentric heading
Preferred firing direction
r (i
i
0
90
180
270300
![Page 40: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/40.jpg)
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
![Page 41: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/41.jpg)
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
![Page 42: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/42.jpg)
)(),( 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
![Page 43: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/43.jpg)
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
![Page 44: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/44.jpg)
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
![Page 45: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/45.jpg)
Book: Spiking Neuron Models.W. Gerstner and W. Kistler
Chapter 8
Oscillations
![Page 46: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/46.jpg)
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
![Page 47: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/47.jpg)
Stability of Asynchronous State A(t)
delayperiod
)()( sess0 for
stable0
03
02
noise
T
)(s
s
)(s
(Gerstner 2000)
![Page 48: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/48.jpg)
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
![Page 49: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/49.jpg)
Random connectivity
![Page 50: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/50.jpg)
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.
![Page 51: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/51.jpg)
The end
![Page 52: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/52.jpg)
Liquid state/Information buffering
![Page 53: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/53.jpg)
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
![Page 54: Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.](https://reader036.fdocuments.net/reader036/viewer/2022062516/56649d7d5503460f94a5fccb/html5/thumbnails/54.jpg)
Ultra-short-term information buffering (‘echo’)
I(t)
)()( fj
j f
ttatout
dsstAsa )()(
(Mayor&Gerstner)
)(tA
bias <I>
mean-field readoutmicroscopic readout