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![Page 1: Scaling-up Cortical Representations in Fluctuation-Driven Systems David W. McLaughlin Courant Institute & Center for Neural Science New York University.](https://reader036.fdocuments.us/reader036/viewer/2022062516/56649d445503460f94a21037/html5/thumbnails/1.jpg)
Scaling-up Cortical Representationsin Fluctuation-Driven Systems
David W. McLaughlin
Courant Institute & Center for Neural Science
New York University
http://www.cims.nyu.edu/faculty/dmac/
Cold Spring Harbor -- July ‘04
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In collaboration with:
David Cai
Louis Tao
Michael Shelley
Aaditya Rangan
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Lateral Connections and Orientation -- Tree ShrewBosking, Zhang, Schofield & Fitzpatrick
J. Neuroscience, 1997
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Coarse-Grained Asymptotic Representations
Needed for “Scale-up”
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Cortical networks have a very noisy dynamics
• Strong temporal fluctuations • On synaptic timescale• Fluctuation driven spiking
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Experiment ObservationExperiment ObservationFluctuations in Orientation Tuning (Cat data from Ferster’s Lab)Fluctuations in Orientation Tuning (Cat data from Ferster’s Lab)
Ref:Anderson, Lampl, Gillespie, FersterScience, 1968-72 (2000)
threshold (-65 mV)
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Fluctuation-driven spiking
Solid: average ( over 72 cycles)
Dashed: 10 temporal trajectories
(very noisy dynamics,on the synaptic time scale)
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• To accurately and efficiently describe these networks requires that fluctuations be retained in a coarse-grained representation.
• “Pdf ” representations –(v,g; x,t), = E,I
will retain fluctuations.• But will not be very efficient numerically• Needed – a reduction of the pdf representations
which retains1. Means &2. Variances
• PT #1: Kinetic Theory provides this representationRef: Cai, Tao, Shelley & McLaughlin, PNAS, pp 7757-7762 (2004)
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First, tile the cortical layer with coarse-grained (CG) patchesFirst, tile the cortical layer with coarse-grained (CG) patches
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Kinetic Theory begins from
PDF representations
(v,g; x,t), = E,I
• Knight & Sirovich; • Tranchina, Nykamp & Haskell;
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• First, replace the 200 neurons in this CG cell by an effective pdf representation
• Then derive from the pdf rep, kinetic thry• For convenience of presentation, I’ll sketch
the derivation a single CG cell, with 200 excitatory Integrate & Fire neurons
• The results extend to interacting CG cells which include inhibition – as well as “simple” & “complex” cells.
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• N excitatory neurons (within one CG cell)
• Random coupling throughout the CG cell;
• AMPA synapses (with time scale )
t vi = -(v – VR) – gi (v-VE)
t gi = - gi + l f (t – tl) +
(Sa/N) l,k (t – tlk)
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• N excitatory neurons (within one CG cell)• All-to-all coupling; • AMPA synapses (with time scale )
t vi = -(v – VR) – gi (v-VE)
t gi = - gi + l f (t – tl) +
(Sa/N) l,k (t – tlk)
(g,v,t) N-1 i=1,N E{[v – vi(t)] [g – gi(t)]},
Expectation “E” over Poisson spike train
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t vi = -(v – VR) – gi (v-VE)
t gi = - gi + l f (t – tl) + (Sa/N) l,k (t – tlk)
Evolution of pdf -- (g,v,t): (i) N>1; (ii) the total input to each neuron is (modulated) Poisson spike trains.
t = -1v {[(v – VR) + g (v-VE)] } + g {(g/) }
+ 0(t) [(v, g-f/, t) - (v,g,t)] + N m(t) [(v, g-Sa/N, t) - (v,g,t)],
0(t) = modulated rate of Poisson spike train from LGN;m(t) = average firing rate of the neurons in the CG cell
= J(v)(v,g; )|(v= 1) dg,
and where J(v)(v,g; ) = -{[(v – VR) + g (v-VE)] }
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Kinetic Theory Begins from Moments(g,v,t)(g)(g,t) = (g,v,t) dv(v)(v,t) = (g,v,t) dg1
(v)(v,t) = g (g,tv) dg
where (g,v,t) = (g,tv) (v)(v,t).
t = -1v {[(v – VR) + g (v-VE)] } + g {(g/) }
+ 0(t) [(v, g-f/, t) - (v,g,t)] + N m(t) [(v, g-Sa/N, t) - (v,g,t)],
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Under the conditions,
N>1; f < 1; 0 f = O(1),
And the Closure: (i) v2(v) = 0;
(ii) 2(v) = g2
where 2(v) = 2(v) – (1
(v))2 ,
g2 = 0(t) f2 /(2) + m(t) (Sa)2 /(2N)
G(t) = 0(t) f + m(t) Sa
One obtains:
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t (v) = -1v [(v – VR) (v) + 1(v)(v-VE) (v)]
t 1(v) = - -1[1
(v) – G(t)]
+ -1{[(v – VR) + 1(v)(v-VE)] v 1
(v)}
+ g2 / ((v)) v [(v-VE) (v)]
Together with a diffusion eq for (g)(g,t):
t (g) = g {[g – G(t)]) (g)} + g2
gg (g)
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Fluctuations in g are Gaussian
t (g) = g {[g – G(t)]) (g)} + g2
gg (g)
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PDF of v
Theory→ ←I&F (solid)
Fokker-Planck→
Theory→
←I&F←Mean-driven limit ( ): Hard thresholding
Fluctuation-Driven DynamicsFluctuation-Driven Dynamics
N=75
N=75σ=5msecS=0.05f=0.01
firin
g ra
te
(Hz)
N
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Mean Driven:
Bistability and HysteresisBistability and Hysteresis Network of Simple, Excitatory only
Fluctuation Driven:
N=16
Relatively Strong Cortical Coupling:
N=16!
N
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Mean Driven:
N=16!
Bistability and HysteresisBistability and Hysteresis Network of Simple, Excitatory only
Relatively Strong Cortical Coupling:
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Computational Efficiency
• For statistical accuracy in these CG patch settings, Kinetic Theory is 103 -- 105 more efficient than I&F;
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Average firing rates
Vs
Spike-time statistics
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0 50 100 150 200 250 300
-60
-40
-20
0
20
Time (ms)
Pot
entia
l (m
V)
0 50 100 150 200 250 300
-60
-40
-20
0
20
Time (ms)
Pot
entia
l (m
V)
With NMDA at all times
No NMDA when VD >= -50
Bursting Model:
19 Spikes
16 Spikes
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• Coarse-grained theories involve local averaging in both space and time.
• Hence, coarse-grained theories average out detailed spike timing information.
• Ok for “rate codes”, but if spike-timing statistics is to be studied, must modify the coarse-grained approach
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PT #2: Embedded point neurons will capture these statistical firing properties[Ref: Cai, Tao & McLaughlin, PNAS (to appear)]
• For “scale-up” – computer efficiency• Yet maintaining statistical firing properties of multiple neurons • Model especially relevant for biologically distinguished sparse,
strong sub-networks – perhaps such as long-range connections
• Point neurons -- embedded in, and fully interacting with, coarse-grained kinetic theory,
• Or, when kinetic theory accurate by itself, embedded as “test neurons”
![Page 28: Scaling-up Cortical Representations in Fluctuation-Driven Systems David W. McLaughlin Courant Institute & Center for Neural Science New York University.](https://reader036.fdocuments.us/reader036/viewer/2022062516/56649d445503460f94a21037/html5/thumbnails/28.jpg)
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![Page 29: Scaling-up Cortical Representations in Fluctuation-Driven Systems David W. McLaughlin Courant Institute & Center for Neural Science New York University.](https://reader036.fdocuments.us/reader036/viewer/2022062516/56649d445503460f94a21037/html5/thumbnails/29.jpg)
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![Page 30: Scaling-up Cortical Representations in Fluctuation-Driven Systems David W. McLaughlin Courant Institute & Center for Neural Science New York University.](https://reader036.fdocuments.us/reader036/viewer/2022062516/56649d445503460f94a21037/html5/thumbnails/30.jpg)
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![Page 31: Scaling-up Cortical Representations in Fluctuation-Driven Systems David W. McLaughlin Courant Institute & Center for Neural Science New York University.](https://reader036.fdocuments.us/reader036/viewer/2022062516/56649d445503460f94a21037/html5/thumbnails/31.jpg)
I&F vs. Embedded Network Spike Rasters
a) I&F Network: 50 “Simple” cells, 50 “Complex” cells. “Simple” cells driven at 10 Hz
b)-d) Embedded I&F Networks: b) 25 “Complex” cells replaced by single kinetic equation;
c) 25 “Simple” cells replaced by single kinetic equation; d) 25 “Simple” and 25 “Complex” cells replaced by kinetic equations. In all panels, cells 1-50 are “Simple” and cells 51-100 are “Complex”. Rasters shown for 5 stimulus periods.
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Raster Plots, Cross-correlation and ISI distributions. (Upper panels) KT of a neuronal patch with strongly coupled embedded neurons; (Lower panels) Full I&F Network. Shown is the sub-network, with neurons 1-6 excitatory; neurons 7-8 inhibitory; EPSP time constant 3 ms; IPSP time constant 10 ms.
Embedded NetworkEmbedded Network
Full I & F NetworkFull I & F Network
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ISI distributions for two simulations: (Left) Test Neuron driven by a CG neuronal patch; (Right) Sample Neuron in the I&F Network.
““Test neuron” within a CG Kinetic TheoryTest neuron” within a CG Kinetic Theory
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Cycle-averaged Firing Rate Curves [Shown: Exc Cmplx Pop in a 4 population model): Full I&F network (solid) , Full I&F + KT (dotted); Full I&F coupled to Full KT but with mean only coupling (dashed).] In both embedded cases (where the I&F units are coupled to KT), half the simple cells are represented by Kinetic Theory
The Importance of Fluctuations
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Reverse Time Correlations• Correlates spikes against driving signal• Triggered by spiking neuron• Frequently used experimental technique to
get a handle on one description of the system• P(,) – probability of a grating of orientation
, at a time before a spike
-- or an estimate of the system’s linear response kernel as a function of (,)
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Reverse Correlation
Left: I&F Network of 128 “Simple” and 128 “Complex” cells at pinwheel center. RTC P() for single Simple cell.Below: Embedded Network of 128 “Simple” cells, with 128 “Complex” cells replaced by single kinetic equation. RTC P() for single Simple cell.
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Computational Efficiency
• For statistical accuracy in these CG patch settings, Kinetic Theory is 103 -- 105 more efficient than I&F;
• The efficiency of the embedded sub-network scales as N2, where N = # of embedded point neurons;
(i.e. 100 20 yields 10,000 400)
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Conclusions
• Kinetic Theory is a numerically efficient, and remarkably accurate, method for “scale-up” – Ref: PNAS, pp 7757-7762 (2004)
• Kinetic Theory introduces no new free parameters into the model, and has a large dynamic range from the rapid firing “mean-driven” regime to a fluctuation driven regime.
• Kinetic Theory does not capture detailed “spike-timing” statistics
• Sub-networks of point neurons can be embedded within kinetic theory to capture spike timing statistics, with a range from test neurons to fully interacting sub-networks.
Ref: PNAS, to appear (2004)
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Conclusions and Directions
• Constructing ideal network models to discern and extract possible principles of neuronal computation and functions
Mathematical methods for analytical understandingSearch for signatures of identified mechanisms
• Mean-driven vs. fluctuation-driven kinetic theoriesNew closure, Fluctuation and correlation effectsExcellent agreement with the full numerical simulations
• Large-scale numerical simulations of structured networks constrained by anatomy and other physiological observations to compare with experiments
Structural understanding vs. data modelingNew numerical methods for scale-up --- Kinetic theory
![Page 42: Scaling-up Cortical Representations in Fluctuation-Driven Systems David W. McLaughlin Courant Institute & Center for Neural Science New York University.](https://reader036.fdocuments.us/reader036/viewer/2022062516/56649d445503460f94a21037/html5/thumbnails/42.jpg)
Three Dynamic Regimes of Cortical Amplification:
1) Weak Cortical Amplification
No Bistability/Hysteresis
2) Near Critical Cortical Amplification
3) Strong Cortical Amplification
Bistability/Hysteresis (2) (1)
(3)
I&F
Excitatory Cells Shown
Possible MechanismPossible Mechanismfor Orientation Tuning of Complex Cellsfor Orientation Tuning of Complex CellsRegime 2 for far-field/well-tuned Complex CellsRegime 1 for near-pinwheel/less-tuned
Summed Effects
(2) (1)
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Summary & Conclusion
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Summary Points for Coarse-Grained Reductions needed for Scale-up
1. Neuronal networks are very noisy, with fluctuation driven effects.
2. Temporal scale-separation emerges from network activity.
3. Local temporal asynchony needed for the asymptotic reduction, and it results from synaptic failure.
4. Cortical maps -- both spatially regular and spatially random -- tile the cortex; asymptotic reductions must handle both.
5. Embedded neuron representations may be needed to capture spike-timing codes and coincidence detection.
6. PDF representations may be needed to capture synchronized fluctuations.
![Page 45: Scaling-up Cortical Representations in Fluctuation-Driven Systems David W. McLaughlin Courant Institute & Center for Neural Science New York University.](https://reader036.fdocuments.us/reader036/viewer/2022062516/56649d445503460f94a21037/html5/thumbnails/45.jpg)
Scale-up & Dynamical Issuesfor Cortical Modeling of V1
• Temporal emergence of visual perception• Role of spatial & temporal feedback -- within and
between cortical layers and regions• Synchrony & asynchrony• Presence (or absence) and role of oscillations• Spike-timing vs firing rate codes• Very noisy, fluctuation driven system• Emergence of an activity dependent, separation of
time scales• But often no (or little) temporal scale separation
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Under Under ASSUMPTIONSASSUMPTIONS: 1): 1) 2) 2) SummedSummed intra-cortical intra-cortical low ratelow rate spike events become Poisson spike events become Poisson::
11 N
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g g g t g t gt g g
g g g t g t gt g g
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v v and v vv v
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Closures:Closures:
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Kinetic Theory for Population DynamicsKinetic Theory for Population Dynamics
Population of interacting neurons:Population of interacting neurons:
1-p: Synaptic Failure rate
ii i i
iii j j
j
r EdV
V G t VdtdG S
G f t t p t tdt N
1, ,
0, 1 .j
with probability pp
with probability p
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Kinetic Equation:Kinetic Equation:
Under Under ASSUMPTIONSASSUMPTIONS: 1): 1) 2) 2) SummedSummed intra-cortical intra-cortical low ratelow rate spike events become Poisson spike events become Poisson::
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1 1 1 1
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t v
v v vv v
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Fluctuation-Driven DynamicsFluctuation-Driven Dynamics
Physical Intuition: Fluctuation-driven/Correlation between g and V
Hierarchy of Conditional Moments
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Closure Assumptions:Closure Assumptions:
Closed Equations Closed Equations —— Reduced Kinetic EquationsReduced Kinetic Equations:
1
21 1 1 11 g
r E
r E E
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t v
v vvv v v v g t v
t v v v
2 0vv
2 2gv
2
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Fluctuation Effects Correlation Effects
Fokker-Planck Equation:Fokker-Planck Equation:
Flux:
Determination of Firing Rate:
For a steady state, m can be determined implicitly
Coarse-Graining in Time:Coarse-Graining in Time:
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