lecture 10: mean field theory with fluctuations and correlations reference: a lerchner et al,...
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Lecture 10: Mean Field theory with fluctuations and correlations
Reference: A Lerchner et al, Response Variability in Balanced Cortical Networks, q-bio.NC/0402022, Neural Computation 18 634-659 (also on course webpage)
Mean field theory for disordered systems
In network with fixed randomness (here: random connections):
j
jiji tSJtI )()(
Mean field theory for disordered systems
In network with fixed randomness (here: random connections):
j
jiji tSJtI )()( j
jiji trJtI )()(=>
Mean field theory for disordered systems
In network with fixed randomness (here: random connections):
Different neurons have different number of inputs
j
jiji tSJtI )()( j
jiji trJtI )()(=>
Mean field theory for disordered systems
In network with fixed randomness (here: random connections):
Different neurons have different number of inputs
Different neurons have different mean input currents
j
jiji tSJtI )()( j
jiji trJtI )()(=>
Mean field theory for disordered systems
In network with fixed randomness (here: random connections):
Different neurons have different number of inputs
Different neurons have different mean input currents
j
jiji tSJtI )()( j
jiji trJtI )()(=>
022 III ii
Mean field theory for disordered systems
In network with fixed randomness (here: random connections):
Different neurons have different number of inputs
Different neurons have different mean input currents
Different rates
j
jiji tSJtI )()( j
jiji trJtI )()(=>
022 III ii
022 rrr ii
Mean field theory for disordered systems
In network with fixed randomness (here: random connections):
Different neurons have different number of inputs
Different neurons have different mean input currents
Different rates
j
jiji tSJtI )()( j
jiji trJtI )()(=>
022 III ii
022 rrr ii
Temporal fluctuations: 0)()()()( iiiiii ItIItItItI
Mean field theory for disordered systems
In network with fixed randomness (here: random connections):
Different neurons have different number of inputs
Different neurons have different mean input currents
Different rates
=> Have to include these fluctuations in the theory
j
jiji tSJtI )()( j
jiji trJtI )()(=>
022 III ii
022 rrr ii
Temporal fluctuations: 0)()()()( iiiiii ItIItItItI
Heuristic treatment
Start from (one population for now)
Heuristic treatment
Start from
)(
)()(
tSrJJ
tSJtI
jjj
ijij
jjiji
(one population for now)
Heuristic treatment
Start from
)(
)()(
tSrJJ
tSJtI
jjj
ijij
jjiji
Time average: jj
ijijj
jiji rJJrJtI )()(
(one population for now)
Heuristic treatment
Start from
)(
)()(
tSrJJ
tSJtI
jjj
ijij
jjiji
Time average: jj
ijijj
jiji rJJrJtI )()(
Average over neurons: j
ijj
jiji rJrJtI )(
(one population for now)
Heuristic treatment
Start from
)(
)()(
tSrJJ
tSJtI
jjj
ijij
jjiji
Time average: jj
ijijj
jiji rJJrJtI )()(
Average over neurons: j
ijj
jiji rJrJtI )(
Neuron-to-neuron fluctuations:
qJ
rJrrJJ
III
jij
jj
ijkjjk
ikij
ii
2
22
22
(one population for now)
Temporal fluctuations)()( tSJtI j
jiji Input current fluctuations:
Temporal fluctuations)()( tSJtI j
jiji Input current fluctuations:
Correlations:
)()(
)()()()()(
2
2
ttCJ
tStSJtItI
jij
jjj
ijii
Temporal fluctuations)()( tSJtI j
jiji Input current fluctuations:
Correlations:
)()(
)()()()()(
2
2
ttCJ
tStSJtItI
jij
jjj
ijii
Have to calculate (“order parameters”) self-consistently)(,, ttCqr
2-population model
Like Amit-Brunel model, but with different scaling of synapses(van Vreeswijk-Sompolinsky)
2-population model
Like Amit-Brunel model, but with different scaling of synapses(van Vreeswijk-Sompolinsky)
Populations 0,1,2 (as before)
0
1
2
2-population model
Like Amit-Brunel model, but with different scaling of synapses(van Vreeswijk-Sompolinsky)
Populations 0,1,2 (as before)
Mean number of connections from population b to a neuron inpopulation a:
bb cNK
0
1
2
2-population model
Like Amit-Brunel model, but with different scaling of synapses(van Vreeswijk-Sompolinsky)
Populations 0,1,2 (as before)
Mean number of connections from population b to a neuron inpopulation a:
bb cNK
b
babij
b
b
b
ababij N
KJ
NK
K
JJ 1 prob,0;prob,
0
1
2
2-population model
Like Amit-Brunel model, but with different scaling of synapses(van Vreeswijk-Sompolinsky)
Populations 0,1,2 (as before)
Mean number of connections from population b to a neuron inpopulation a:
bb cNK
b
babij
b
b
b
ababij N
KJ
NK
K
JJ 1 prob,0;prob,
0
1
2
2,0 bJab
Means and variances of synaptic strengths
Means and variances of synaptic strengths
b
bababij N
KJJ Mean:
Means and variances of synaptic strengths
b
bababij N
KJJ Mean:
Variance: b
ab
b
b
b
bab
b
b
b
ababij
abij
abij N
JNK
NKJ
NK
KJ
JJJ2
2
22222 1)()(
Input current statisticsMean (average of time and neurons)
Input current statisticsMean (average of time and neurons)
bbb
abai rKJtI )(
Input current statisticsMean (average of time and neurons)
bbb
abai rKJtI )(
Neuron-to-neuron fluctuations of the temporal mean:
Input current statisticsMean (average of time and neurons)
bbb
abai rKJtI )(
bbab
b
bbj
jb
abij
ai
ai
ai
qJN
KrJ
III
222
22
1)(
Neuron-to-neuron fluctuations of the temporal mean:
Input current statisticsMean (average of time and neurons)
bbb
abai rKJtI )(
bbab
b
bbj
jb
abij
ai
ai
ai
qJN
KrJ
III
222
22
1)(
Neuron-to-neuron fluctuations of the temporal mean:
Temporal fluctuations:
bbab
b
bb
jb
abij
bj
bj
jb
abij
ai
ai
ttCJNK
ttCJ
tStSJtItI
)(1)()(
)()()()()(
22
2
Equivalent single-neuron problem
Single neuron (excitatory or inhibitory) driven by input current
Equivalent single-neuron problem
Single neuron (excitatory or inhibitory) driven by input current
b
ababb
b
bbbaba txq
NK
rKJtI )(1)(2/1
Equivalent single-neuron problem
Single neuron (excitatory or inhibitory) driven by input current
b
ababb
b
bbbaba txq
NK
rKJtI )(1)(2/1
1)(,0 2 abab xxwith
Equivalent single-neuron problem
Single neuron (excitatory or inhibitory) driven by input current
b
ababb
b
bbbaba txq
NK
rKJtI )(1)(2/1
1)(,0 2 abab xxwith )()()(,0)( ttCttt bababab
Can combine the two kinds of fluctuations:
Consider the total correlation function
Can combine the two kinds of fluctuations:
Consider the total correlation function
)()(
)()()(~
tSrtSr
tStSttC
bj
bj
bj
bj
bj
bj
bj
Can combine the two kinds of fluctuations:
Consider the total correlation function
)()(
)()()(~
tSrtSr
tStSttC
bj
bj
bj
bj
bj
bj
bj
)(
)()(
)(~)(~
ttCq
tSrtSr
ttCttC
bb
bj
bj
bj
bj
bjb
Average over neurons
Can combine the two kinds of fluctuations:
Consider the total correlation function
)()(
)()()(~
tSrtSr
tStSttC
bj
bj
bj
bj
bj
bj
bj
)(
)()(
)(~)(~
ttCq
tSrtSr
ttCttC
bb
bj
bj
bj
bj
bjb
Average over neurons
Total input current fluctuations:
bbab
b
bai
ai
ai
ai ttCJ
NK
ItIItI )(~1)()( 2
Balance conditionTotal average current: 0 bb
bab
ai
rKJV
Balance conditionTotal average current: 0 bb
bab
ai
rKJV
i.e., with aiV
Balance conditionTotal average current: 0 bb
bab
ai
rKJV
i.e., with
000
2
1
rKJrKJ abbb
abaiV
Balance conditionTotal average current: 0 bb
bab
ai
rKJV
i.e., with
000
2
1
rKJrKJ abbb
ab
or, defining 2,1,,ˆ0
baJK
KJ ab
bab
aiV
Balance conditionTotal average current: 0 bb
bab
ai
rKJV
i.e., with
000
2
1
rKJrKJ abbb
ab
or, defining 2,1,,ˆ0
baJK
KJ ab
bab
0020
0010
2
1
2221
1211
/
/
ˆˆ
ˆˆ
KrJ
KrJ
r
r
JJ
JJ
aiV
Balance conditionTotal average current: 0 bb
bab
ai
rKJV
i.e., with
000
2
1
rKJrKJ abbb
ab
or, defining 2,1,,ˆ0
baJK
KJ ab
bab
0020
0010
2
1
2221
1211
/
/
ˆˆ
ˆˆ
KrJ
KrJ
r
r
JJ
JJ
aiV
Solution:
)(
)(
ˆˆ
ˆˆ
020
010
020
0101
2221
1211
2
1
KJ
KJ
rJ
rJ
JJ
JJ
r
r
Balance conditionTotal average current: 0 bb
bab
ai
rKJV
i.e., with
000
2
1
rKJrKJ abbb
ab
or, defining 2,1,,ˆ0
baJK
KJ ab
bab
0020
0010
2
1
2221
1211
/
/
ˆˆ
ˆˆ
KrJ
KrJ
r
r
JJ
JJ
aiV
Solution:
)(
)(
ˆˆ
ˆˆ
020
010
020
0101
2221
1211
2
1
KJ
KJ
rJ
rJ
JJ
JJ
r
r
i.e., can solve for mean rates independent of fluctuations/correlations
Correlations/fluctuationsHave to do it numerically:
Correlations/fluctuationsHave to do it numerically:
Start with rb from balance eqn
Correlations/fluctuationsHave to do it numerically:
Start with rb from balance eqn, initial estimates )()(;2 ttrttCrq bbbb
Correlations/fluctuationsHave to do it numerically:
Start with rb from balance eqn, initial estimates )()(;2 ttrttCrq bbbb
Generate Gaussian noisy input current with mean bbb
ab rKJ
Correlations/fluctuationsHave to do it numerically:
Start with rb from balance eqn, initial estimates )()(;2 ttrttCrq bbbb
Generate Gaussian noisy input current with mean bbb
ab rKJ and correlation function
bbbab
b
bai
ai
ai
ai ttCqJ
NK
ItIItI )]([1)()( 2
Correlations/fluctuationsHave to do it numerically:
Start with rb from balance eqn, initial estimates )()(;2 ttrttCrq bbbb
Generate Gaussian noisy input current with mean bbb
ab rKJ and correlation function
bbbab
b
bai
ai
ai
ai ttCqJ
NK
ItIItI )]([1)()( 2
Simulate many trials with different realizations of noise, collect statistics(measure ra, qa, Ca(t) )
Correlations/fluctuationsHave to do it numerically:
Start with rb from balance eqn, initial estimates )()(;2 ttrttCrq bbbb
Generate Gaussian noisy input current with mean bbb
ab rKJ and correlation function
bbbab
b
bai
ai
ai
ai ttCqJ
NK
ItIItI )]([1)()( 2
Simulate many trials with different realizations of noise, collect statistics(measure ra, qa, Ca(t) )
Use these to generate improved noise samples
Correlations/fluctuationsHave to do it numerically:
Start with rb from balance eqn, initial estimates )()(;2 ttrttCrq bbbb
Generate Gaussian noisy input current with mean bbb
ab rKJ and correlation function
bbbab
b
bai
ai
ai
ai ttCqJ
NK
ItIItI )]([1)()( 2
Simulate many trials with different realizations of noise, collect statistics(measure ra, qa, Ca(t) )
Use these to generate improved noise samples
Simulate again, repeat until input and output order parameters agree
When done, compute firing statistics of single neurons:
b
ababb
b
bbbaba txq
NK
rKJtI )(1)(2/1
For a single neuron, with effective input current
When done, compute firing statistics of single neurons:
b
ababb
b
bbbaba txq
NK
rKJtI )(1)(2/1
For a single neuron, with effective input current
Have to hold xab fixed
Experimental background: firing statistics
Gershon et al,J Neurophysiol 79,1135-1144 (1998)
x
xx
x
x
x
Experimental background: firing statistics
Gershon et al,J Neurophysiol 79,1135-1144 (1998)
Variance > mean
x
xx
x
x
x
Experimental background: firing statistics
Gershon et al,J Neurophysiol 79,1135-1144 (1998)
Variance > mean
x
xx
x
x
x
i,e., Fano factor F > 1
Fano factors and correlation functions
T
dttSN0
)(Spike count:
Fano factors and correlation functions
T
dttSN0
)(
rTdttSNT
0 )(
Spike count:
Mean:
Fano factors and correlation functions
T
dttSN0
)(
rTdttSNT
0 )(
Spike count:
Mean:
Variance:
T
T
TTTT
dCTdCtd
ttCtddttStStddtN
0
0000
2
)()(
),()()(
Fano factors and correlation functions
T
dttSN0
)(
rTdttSNT
0 )(
Spike count:
Mean:
Variance:
T
T
TTTT
dCTdCtd
ttCtddttStStddtN
0
0000
2
)()(
),()()(
=> Fano factorr
dCF
)(
Model calculations
Synaptic matrix:
g
gJ s 21
2J
Model calculations
Synaptic matrix:
g
gJ s 21
2J
Js = 0.375Js = 0.75Js = 1.5
Correlation functions:
g = 1
Interspike interval distributions
Js = 1.5
Js = 0.75
Js = 0.375
Spike count variance vs mean
What controls F?
Membrane potential distributions have width ~ Jab = O(1)
What controls F?
Low post-spike reset voltage: takes time (~ ) to recover from reset
Membrane potential distributions have width ~ Jab = O(1)
What controls F?
Low post-spike reset voltage: takes time (~ ) to recover from reset
=> F < 1
Membrane potential distributions have width ~ Jab = O(1)
What controls F?
Low post-spike reset voltage: takes time (~ ) to recover from reset
=> F < 1
Reset near threshold: initial spread of membrane potential distribution
Membrane potential distributions have width ~ Jab = O(1)
What controls F?
Low post-spike reset voltage: takes time (~ ) to recover from reset
=> F < 1
Reset near threshold: initial spread of membrane potential distribution
=> excess early spikes, F > 1
Membrane potential distributions have width ~ Jab = O(1)