scaling-up cortical representations in fluctuation-driven systems

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

+ 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=75σ=5msecS=0.05f=0.01

Mean Driven:

Bistability and HysteresisBistability and Hysteresis Network of Simple, Excitatory only

Fluctuation Driven:

Relatively Strong Cortical Coupling:

Mean Driven:

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

Spike-time statistics

0 50 100 150 200 250 300

Time (ms)

0 50 100 150 200 250 300

Time (ms)

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”

DD ED D ID Di

i i i i i

ED DED D ii E

E i E j jDj PE

DBE j j

ID DID D ii I

I i I k kDk PI

DBI k k

r E IdV

V G t V G t Vdt

dG SG f t t p t t

S p t t

dG SG f t T p t T

S p t T

1,, ; , ,

simpleE I simple complex

complex

' ' ' ' ' '' , ' , .B B B Bj j E E I IPoisson spike trains t T are reconstructed from the rate m N m N

B B B BE I

B B B B B BEBE E I E E EB B

B B B B B BIBI E I I I IB B

v U v vt v

vv U v v v g t v

t v v v

vv U v v v g t v

t v v v

t sBDtB B B B

EB E E E E j jDj PE

t sBDtB B B B

IB I I I I k kDk PI

B BD t stEB B B

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Sg t f t S m t e p t t ds

S St f t m t e p t t ds

B BD t stIB B B

IB I I I k kDk PI I I

S St f t m t e p t t ds

DD ED D ID Di

i i i i i

ED DED D ii E

E i E j jDj PE

DBE j j

ID DID D ii I

I i I k kDk PI

DBI k k

r E IdV

V G t V G t Vdt

dG SG f t t p t t

S p t t

dG SG f t T p t T

S p t T

1,, ; , ,

complex

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)

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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, , , , , ,

E IE I

E E I I

EE E I E I

II E I E I

r E Iv vv g gg g

t v g g

ft v g g t v g g t

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EE E E I E I

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Spm t N v g g t v g g t

Under Under ASSUMPTIONSASSUMPTIONS: 1): 1) 2) 2) SummedSummed intra-cortical intra-cortical low ratelow rate spike events become Poisson spike events become Poisson::

E IE I

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m t m tt f t S t f t S

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2 1 2 12 2,E E E I I Iv v v v v v

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Closures:Closures:

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t v v v

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1,, ; , ,

k kk P

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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

V G t VdtdG S

G f t t p t tdt N

0, 1 .j

with probability pp

with probability p

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::

0( , , ) |

, , , ,

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1, , i ii

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21, , ,g

r Evvv g g v g g g t v g

t v g g

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1, , ,

Sg t f t Sm t t f t m t f N finite

1 1 1 1

vvv v g t v v

v v vv v

Fluctuation-Driven DynamicsFluctuation-Driven Dynamics

Physical Intuition: Fluctuation-driven/Correlation between g and V

Hierarchy of Conditional Moments

g g g t g gt g g

vvv v v

22 2 12 2

v g g v dg

v g g v dg v v v

g t f t Sm t

St f t m t

f N finite

Closure Assumptions:Closure Assumptions:

Closed Equations Closed Equations —— Reduced Kinetic EquationsReduced Kinetic Equations:

21 1 1 11 g

vvv v v

v vvv v v v g t v

t v v v

St f t m t

Fluctuation Effects Correlation Effects

Fokker-Planck Equation:Fokker-Planck Equation:

Determination of Firing Rate:

For a steady state, m can be determined implicitly

Coarse-Graining in Time:Coarse-Graining in Time:

2 2 21 g g g EE vv

v g t v g t vv v v v

2g gr EE

vvv g t v v v

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scaling-up cortical representations in fluctuation-driven systems

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