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
In collaboration with:
David Cai
Louis Tao
Michael Shelley
Aaditya Rangan
Lateral Connections and Orientation -- Tree ShrewBosking, Zhang, Schofield & Fitzpatrick
J. Neuroscience, 1997
Coarse-Grained Asymptotic Representations
Needed for “Scale-up”
Cortical networks have a very noisy dynamics
• Strong temporal fluctuations • On synaptic timescale• Fluctuation driven spiking
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)
Fluctuation-driven spiking
Solid: average ( over 72 cycles)
Dashed: 10 temporal trajectories
(very noisy dynamics,on the synaptic time scale)
• 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)
First, tile the cortical layer with coarse-grained (CG) patchesFirst, tile the cortical layer with coarse-grained (CG) patches
Kinetic Theory begins from
PDF representations
(v,g; x,t), = E,I
• Knight & Sirovich; • Tranchina, Nykamp & Haskell;
• 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.
• 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)
• 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
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)] }
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)],
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:
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)
Fluctuations in g are Gaussian
t (g) = g {[g – G(t)]) (g)} + g2
gg (g)
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
Mean Driven:
Bistability and HysteresisBistability and Hysteresis Network of Simple, Excitatory only
Fluctuation Driven:
N=16
Relatively Strong Cortical Coupling:
N=16!
N
Mean Driven:
N=16!
Bistability and HysteresisBistability and Hysteresis Network of Simple, Excitatory only
Relatively Strong Cortical Coupling:
Computational Efficiency
• For statistical accuracy in these CG patch settings, Kinetic Theory is 103 -- 105 more efficient than I&F;
Average firing rates
Vs
Spike-time statistics
0 50 100 150 200 250 300
-60
-40
-20
0
20
Time (ms)
Pot
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0 50 100 150 200 250 300
-60
-40
-20
0
20
Time (ms)
Pot
entia
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With NMDA at all times
No NMDA when VD >= -50
Bursting Model:
19 Spikes
16 Spikes
• 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
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”
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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.
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
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
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
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 (,)
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.
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)
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)
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
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)
Summary & Conclusion
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.
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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Coarse-Graining in Time:Coarse-Graining in Time:
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