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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

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Lateral Connections and Orientation -- Tree

Shrew Bosking, Zhang, Schofield Fitzpatrick J.

Neuroscience, 1997

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Coarse-Grained Asymptotic Representations

- Needed for Scale-up

Coarse-Grained Reductions for V1

- Average firing rate models (Cowan Wilson .

Shelley McLaughlin) - m?(x,t), ? E,I

Cortical networks have a very noisy dynamics

- Strong temporal fluctuations
- On synaptic timescale
- Fluctuation driven spiking

Experiment Observation Fluctuations in

Orientation Tuning (Cat data from Fersters Lab)

Ref Anderson, Lampl, Gillespie, Ferster Science,

1968-72 (2000)

threshold (-65 mV)

Fluctuation-driven spiking

(very noisy dynamics, on the synaptic time scale)

Solid average ( over 72

cycles) Dashed 10 temporal trajectories

- 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 retains - Means
- Variances
- PT 1 Kinetic Theory provides this

representation - Ref Cai, Tao, Shelley McLaughlin, PNAS, pp

7757-7762 (2004)

First, 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

Coarse-Grained Reductions for V1

- PDF representations (Knight Sirovich

Tranchina, Nykamp Haskell Cai, Tao, Shelley

McLaughlin) - ??(v,g x,t), ? E,I
- Sub-network of embedded point neurons -- in a

coarse-grained, dynamical background - (Cai,Tao McLaughlin)

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- Well replace the 200 neurons in this CG cell by

an effective pdf representation

- 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, Ill 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.

1 - p Synaptic Failure rate

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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)

- 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 ?i1,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) Ngt1 (ii)

the total input to each neuron is (modulated)

Poisson spike trains. - ?t ? ?-1?v (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)

?

- ?t ? ?-1?v (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),
- Ngtgt1 f ltlt 1 ?0 f O(1)
- ?t ? ?-1?v (v VR) g (v-VE) ?
- ?g g G(t)/?) ? ?g2 /? ?gg ?

- where ?g2 ?0(t) f2 /(2?) m(t) (Sa)2 /(2N?)
- G(t) ?0(t) f m(t) Sa

Kinetic Theory Begins from Moments

- ?(g,v,t)
- ?(g)(g,t) ? ?(g,v,t) dv
- ?(v)(v,t) ? ?(g,v,t) dg
- ?1(v)(v,t) ? g ?(g,t?v) dg
- where ?(g,v,t) ?(g,t?v) ?(v)(v,t).
- ?t ? ?-1?v (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),

- ?t ? ?-1?v (v VR) g (v-VE) ?
- ?g g G(t)/?) ? ?g2 /? ?gg ?

Moments

- ?(g,v,t)
- ?(g)(g,t) ? ?(g,v,t) dv
- ?(v)(v,t) ? ?(g,v,t) dg
- ?1(v)(v,t) ? g ?(g,t?v) dg
- where ?(g,v,t) ?(g,t?v) ?(v)(v,t)
- Integrating ?(g,v,t) eq over v yields
- ? ?t ?(g) ?g g G(t)) ?(g) ?g2 ?gg ?(g)

- Integrating ?(g,v,t) eq over g yields
- ?t ?(v) ?-1?v (v VR) ?(v) ?1(v) (v-VE)

?(v) - Integrating g ?(g,v,t) eq over g yields an

equation for - ?1(v)(v,t) ? g ?(g,t?v) dg,
- where ?(g,v,t) ?(g,t?v) ?(v)(v,t)

- ?t ? ?-1?v (v VR) g (v-VE) ?
- ?g g G(t)/?) ? ?g2 /? ?gg ?

- Under the conditions,
- Ngt1 f lt 1 ?0 f O(1),
- And the Closure (i) ?v?2(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 ?1(v) - ?-1?1(v) G(t)
- ?-1(v VR) ?1(v)(v-VE) ?v ?1(v)
- ?2(v)/ (??(v)) ?v (v-VE) ?(v)
- ?-1(v-VE) ?v?2(v)
- where ?2(v) ?2(v) (?1(v))2 .
- Closure (i) ?v?2(v) 0
- (ii) ?2(v) ?g2
- One obtains

- ?t ?(v) ?-1?v (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)

Fluctuation-Driven Dynamics

PDF of v Theory? ?IF

(solid)

Fokker-Planck? Theory?

?IF ?Mean-driven limit (

) Hard thresholding

N75

firing rate (Hz)

N75 s5msec S0.05 f0.01

Fluctuation-Driven Dynamics

PDF of v Theory? ?IF

(solid) Fokker-Planck?

Theory? ?IF ?Mean-driven limit (

) Hard thresholding

N75

firing rate (Hz)

Experiment

N75 s5msec S0.05 f0.01

- Bistability and Hysteresis
- Network of Simple, Excitatory only

N16!

N16

MeanDriven

FluctuationDriven

Relatively Strong Cortical Coupling

- Bistability and Hysteresis
- Network of Simple, Excitatory only

N16!

MeanDriven

Relatively Strong Cortical Coupling

- But we can go further for AMPA (? ? 0)
- ??t ?1(v) - ?1(v) G(t)
- ??-1(v VR) ?1(v)(v-VE) ?v ?1(v)
- ??g2/ (??(v)) ?v (v-VE) ?(v)
- Recall ?g2 f2/(2?) ?0(t)
- m(t) (Sa)2 /(2N?)
- Thus, as ? ? 0, ??g2 O(1).
- Let ? ? 0 Algebraically solve for ?1(v)
- ?1(v) G(t) ??g2/ (??(v)) ?v (v-VE) ?(v)

- Result A Fokker-Planck eq for ?(v)(v,t)
- ? ?t ?(v) ?v (1 G(t) ??g2/? ) v
- (VR VE (G(t) ??g2/? ))

?(v) - ??g2/? (v- VE)2 ?v

?(v) - ??g2/? -- Fluctuations in g

- Remarks (i) Boundary Conditions
- (ii) Inhibition, spatial coupling of CG

cells, simple complex cells have been

added - (iii) N ? ? yields mean field

representation.

New Pdf Representation

- ?(g,v,t) -- (i) Evolution eq, with jumps

from incoming spikes - (ii) Jumps smoothed to diffusion
- in g by a large N

expansion - ?(g)(g,t) ? ?(g,v,t) dv -- diffuses as a

Gaussian - ?(v)(v,t) ? ?(g,v,t) dg ?1(v)(v,t) ? g

?(g,t?v) dg - Coupled (moment) eqs for ?(v)(v,t) ?1(v)(v,t) ,

which - are not closed but depend upon ?2(v)(v,t)
- Closure -- (i) ?v?2(v) 0 (ii) ?2(v) ?g2

, - where ?2(v) ?2(v) (?1(v))2 .
- ? ? 0 ? eq for ?1(v)(v,t) solved algey in terms

of ?(v)(v,t), resulting in a Fokker-Planck eq for

?(v)(v,t)

- Local temporal asynchony enhanced by synaptic

failure permitting better amplification

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Closures

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Simple Complex Cells Multiple interacting CG

Patches

Recall

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1 - p Synaptic Failure rate

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- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Complex Excitatory Cells
- Mean-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Complex Excitatory Cells
- Mean-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Complex Excitatory Cells
- Mean-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Complex Excitatory Cells
- Mean-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Simple Excitatory Cells
- Mean-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Simple Excitatory Cells
- Mean-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Simple Excitatory Cells
- Mean-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Simple Excitatory Cells
- Mean-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Simple Excitatory Cells
- Mean-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Complex Excitatory Cells
- Fluctuation-Driven

- Incorporation of Inhibitory Cells
- 4 Population Dynamics
- Simple
- Excitatory
- Inhibitory
- Complex
- Excitatory
- Inhibitory
- Simple Excitatory Cells
- Fluctuation-Driven

Three Dynamic Regimes of Cortical

Amplification 1) Weak Cortical

Amplification No Bistability/Hysteresis

2) Near Critical Cortical Amplification

3) Strong Cortical Amplification Bistabili

ty/Hysteresis (2) (1)

(3)

IF Excitatory Complex Cells Shown

(2) (1)

Simple Complex Cells Multiple interacting CG

Patches

FluctuationDriven Tuning Dynamics Near

Critical Amplification vs. Weak Cortical

Amplification Sensitivity to Contrast

Ring Model A less-tuned complex

cell Ring Model far field A well-tuned

complex cell Large V1 Model A complex

cell in the far-field

Ring Model of Orientation Tuning Near

Pinwheel Far Field

A Cell in Large V1 Model

Computational Efficiency

- For statistical accuracy in these CG patch

settings, Kinetic Theory is 103 -- 105 more

efficient than IF

- Average firing rates
- Vs
- Spike-time statistics

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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

PT 2 Embedded point neurons will capture

these statistical firing propertiesRef 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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IF vs. Embedded Network Spike Rasters

a) IF Network 50 Simple cells, 50 Complex

cells. Simple cells driven at 10 Hz b)-d)

Embedded IF 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.

Embedded Network

Full I F Network

Raster Plots, Cross-correlation and ISI

distributions. (Upper panels) KT of a

neuronal patch with strongly coupled embedded

neurons (Lower panels) Full IF 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.

Test neuron within a CG Kinetic Theory

ISI distributions for two simulations (Left)

Test Neuron driven by a CG neuronal patch

(Right) Sample Neuron in the IF Network.

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IF vs. Embedded Network Spike Rasters

a) IF Network 40 Simple cells, 40 Complex

cells. Simple cells driven at 10 Hz b)-d)

Embedded IF Networks b) 20 Complex cells

replaced by log-form rate equation c) 20

Simple cells replaced by log-form rate

equation d) 20 Simple and 20 Complex cells

replaced by log-form rate equations. In all

panels, cells 1-40 are Simple and cells 41-80

are Complex. Rasters shown for 5 stimulus

periods.

a) IF Network 50 Simple cells, 50

Complexcells. Simplecells driven at 10 Hz b)

Embedded IF Networks, with Complex IF neurons

receiving input but not outputing to the network.

25 Complexcells replaced by kinetic equations.

In all panels, cells 1-50 are Simple and cells

51-75 are Complex. Rasters shown for 5 stimulus

periods.

Dynamic Firing Rates (Shown Exc Simple Pop in

a 4 Pop Model) Forcing--time-dependent Poisson

process, sinusoidal driving at 10 Hz. In the

embedded models, excitatory neurons are IF and

inhibitory neurons are replaced by Kinetic

Theory.

The Importance of Fluctuations

Cycle-averaged Firing Rate Curves Shown Exc

Cmplx Pop in a 4 population model) Full IF

network (solid) , Full IF KT (dotted) Full

IF coupled to Full KT but with mean only

coupling (dashed). In both embedded cases

(where the IF units are coupled to KT), half

the simple cells are represented by Kinetic Theory

Steady State Firing Rate Curves (for the

excitatory population) IF, Exc. Inh.

(Magenta), Kinetic Theory, Exc. Inh. (Red),

Embedded Network (Blue), Log-form (Green). In

the Embedded model, Excitatory neurons are IF

and Inhibitory neurons are modelled by Kinetic

Theory.

Steady State Firing Rate Curves for Embedded

Sub-networks (Shown, Exc Cmplx Pop) In the

embedded models, half of the simple cells are

replaced by Kinetic Theory. However, in the IF

KT (mean only) , they are coupled back to the

IF neurons as mean conductance drives.

(From left to right) Rasters, Cross-correlation

and ISI distributions for two simulations (Upper

panels) Kinetic Theory of a neuronal patch

driving test neurons, which are not coupled to

each other (Lower panels) KT of a neuronal patch

driving strongly coupled neurons. In both cases,

the CGed patch is being driven at 1 Hz. Neurons

1-6 are excitatory Neurons 7-8 are inhibitory

the S-matrix is (See, Sei, Sie, Sii) (0.3, 0.4,

0.6, 0.4). EPSP time constant 3 ms IPSP time

constant 10 ms.

(From left to right) Rasters, Cross-correlation

and ISI distributions for two simulations (Upper

panels) Kinetic Theory of a neuronal patch

driving test neurons, which are not coupled to

each other (Lower panels) KT of a neuronal patch

driving strongly coupled neurons. In both cases,

the CGed patch is being driven asynchronously,

and are firing at a fixed rate. IF Neurons 1-6

are excitatory Neurons 7-8 are inhibitory the

S-matrix is (See, Sei, Sie, Sii) (0.4, 0.4,

0.8, 0.4). EPSP time constant 3 ms IPSP time

constant 10 ms.

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 systems linear

response kernel as a function of (?,?)

- Reverse-Time Correlation (RTC)
- System analysis
- Probing network dynamics

Time ?

Reverse Correlation

Left IF 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 IF - 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)

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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

understanding - Search for signatures of identified mechanisms
- Mean-driven vs. fluctuation-driven kinetic

theories - New closure, Fluctuation and correlation effects
- Excellent 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 modeling
- New 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)
- IF
- Excitatory Cells Shown
- Possible Mechanism
- for Orientation Tuning of Complex Cells
- Regime 2 for far-field/well-tuned Complex Cells
- Regime 1 for near-pinwheel/less-tuned

(2) (1)

Summary Conclusion

Summary Points for Coarse-Grained Reductions

needed for Scale-up

- Neuronal networks are very noisy, with

fluctuation driven effects. - Temporal scale-separation emerges from network

activity. - Local temporal asynchony needed for the

asymptotic reduction, and it results from

synaptic failure. - Cortical maps -- both spatially regular and

spatially random -- tile the cortex asymptotic

reductions must handle both. - Embedded neuron representations may be needed to

capture spike-timing codes and coincidence

detection. - 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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Closures

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Kinetic Theory for Population Dynamics

Population of interacting neurons

1-p Synaptic Failure rate

Under ASSUMPTIONS 1) 2) Summed

intra-cortical low rate spike events become

Poisson

Kinetic Equation

Fluctuation-Driven Dynamics Physical Intuition

Fluctuation-driven/Correlation between g and V

Hierarchy of Conditional Moments

Closure Assumptions

Closed Equations Reduced Kinetic Equations

Coarse-Graining in Time

- Fluctuation Effects
- Correlation Effects

Fokker-Planck Equation

Flux Determination of Firing Rate For a

steady state, m can be determined implicitly

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