scale-up of cortical representations in fluctuation driven settings david w. mclaughlin courant...

Scale-up of Cortical Representationsin Fluctuation Driven Settings

David W. McLaughlin

Courant Institute & Center for Neural Science

New York University

http://www.cims.nyu.edu/faculty/dmac/

IAS – May‘03

I. BackgroundOur group has been modeling a local patch (1mm2) of a

single layer of Primary Visual Cortex

Now, we want to “scale-up”

David Cai

David Lorentz

Robert Shapley

Michael Shelley

Louis Tao

Jacob Wielaard

Local Patch1mm2

Lateral Connections and Orientation -- Tree ShrewBosking, Zhang, Schofield & Fitzpatrick

J. Neuroscience, 1997

Why scale-up?

From one “input layer” to several layers

II. Our Large-Scale Model • A detailed, fine scale model of a local patch of input layer

of Primary Visual Cortex;• Realistically constrained by experimental data;• Integrate & Fire, point neuron model (16,000 neurons

per sq mm).

Refs: --- PNAS (July, 2000)--- J Neural Science (July, 2001)--- J Comp Neural Sci (2002)

--- J Comp Neural Sci (2002)--- http://www.cims.nyu.edu/faculty/dmac/

III. Cortical Networks Have Very Noisy Dynamics

• Strong temporal fluctuations

• On synaptic timescale

• Fluctuation driven spiking

Fluctuation-driven spiking

Solid: average ( over 72 cycles)

Dashed: 10 temporal trajectories

(fluctuations on the synaptic time scale)

IV. Coarse-Grained Asymptotic Representations

For “scale-up” in fluctuation driven systems

---- 500

----

A Regular Cortical Map

For the rest of the talk, consider one coarse-grained cell,containing several hundred neurons

• We’ll replace the 200 neurons in this CG cell by an effective pdf representation

• For convenience of presentation, I’ll sketch the derivation of the reduction for 200 excitatory Integrate & Fire neurons

• Later, I’ll show results with inhibition included – as well as “simple” & “complex” cells.

• N excitatory neurons (within one CG cell)

• all toall 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” taken over modulated incoming (from LGN neurons) 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):

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)], where

0(t) = modulated rate of incoming Poisson spike train;m(t) = average firing rate of the neurons in the CG cell

= J(v)(v,g; )|(v= 1) dg

where J(v)(v,g; ) = -{[(v – VR) + g (v-VE)] }

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)],

N>>1; f << 1; 0 f = O(1);

t = -1v {[(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

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)

Integrating (g,v,t) eq over v yields:

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)

Integrating (g,v,t) eq over g yields:

t (v) = -1v [(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,tv) dg,

where (g,v,t) = (g,tv) (v)(v,t)

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) v2(v)

where 2(v) = 2(v) – (1

(v))2 .

Closure: (i) v2(v) = 0;

(ii) 2(v) = g2

Thus, eqs for (v)(v,t) & 1(v)(v,t):

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)

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) + g

2/ ((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

Seek steady state solutions – ODE in v, which will be good for scale-up.

Remarks: (i) Boundary Conditions;

(ii) Inhibition, spatial coupling of CG cells, simple & complex cells have

been added;

(iii) N yields “mean field” representation.

Next, use one CG cell to

(i) Check accuracy of this pdf representation;

(ii) Get insight about mechanisms in fluctuation driven systems.

MeanDriven:

Bistability and HysteresisBistability and Hysteresis Network of Simple and Complex — Excitatory only

FluctuationDriven:

N=16

Relatively Strong Cortical Coupling:

N=16!

MeanDriven:

N=16!

Bistability and HysteresisBistability and Hysteresis Network of Simple and Complex — Excitatory only

Relatively Strong Cortical Coupling:

Three Levels 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

(2) (1)

With inhib, simple & cmplx

New Pdf for fluctuation driven systems -- accurate and efficient

(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,tv) 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) v2(v) = 0; (ii) 2(v) = g2 ,

where 2(v) = 2(v) – (1

(v))2 . 0 eq for 1

(v)(v,t) solved algebraically in terms of (v)(v,t), resulting in a Fokker-Planck eq for (v)(v,t)

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

Fluctuation-driven spiking

Solid: average ( over 72 cycles)

Dashed: 10 temporal trajectories

(very noisy dynamics,on the synaptic time scale)

ComplexSimple

CV (Orientation Selectivity)

# of cells 4C

4B

Expt. Results

Shapley’s Lab(unpublished)

Incorporation of Inhibitory Cells

4PopulationDynamics

• Simple: Excitatory Inhibitory

• Complex: Excitatory Inhibitory

Complex Excitatory CellsMean-Driven


4PopulationDynamics



Simple Excitatory CellsMean-Driven


4PopulationDynamics



Complex Excitatory CellsFluctuation-Driven


4PopulationDynamics



Simple Excitatory CellsFluctuation-Driven

scale-up of cortical representations in fluctuation driven settings david w. mclaughlin courant...

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