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Scale-up of Cortical Representations in Fluctuation Driven SettingsPowerPoint Presentation

Scale-up of Cortical Representations in Fluctuation Driven Settings

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### III. Cortical Networks Have Very Noisy Dynamics

### IV. Coarse-Grained Asymptotic Representations

Complex: Complex Excitatory Cells Mean-Driven

Complex: Complex Excitatory Cells Mean-Driven

Complex: Complex Excitatory Cells Mean-Driven

Complex: Complex Excitatory Cells Mean-Driven

Complex: Simple Excitatory Cells Mean-Driven

Complex: Simple Excitatory Cells Mean-Driven

Complex: Simple Excitatory Cells Mean-Driven

Complex: Simple Excitatory Cells Mean-Driven

Complex: Simple Excitatory Cells Mean-Driven

Complex: Complex Excitatory Cells Fluctuation-Driven

Complex: Simple Excitatory Cells Fluctuation-Driven

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. Background

Our 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

1mm2

Lateral Connections and Orientation -- Tree Shrew

Bosking, Zhang, Schofield & Fitzpatrick

J. Neuroscience, 1997

From one “input layer” to several layers

Why scale-up?

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/

Strong temporal fluctuations

On synaptic timescale

Fluctuation driven spiking

spiking

(fluctuations on the

synaptic time scale)

Solid: average

( over 72 cycles)

Dashed: 10 temporal

trajectories

For “scale-up” in fluctuation driven systems

- 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) effective pdf representation
- 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.

effective pdf representationt 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)] }

effective pdf representationt = -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 effective pdf representation

- (g,v,t)
- (g)(g,t) = (g,v,t) dv
- (v)(v,t) = (g,v,t) dg
- 1(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 effective pdf representation

t (g) =g {[g – G(t)]) (g)} + g2 gg (g)

Integrating (g,v,t) eq over g yields: effective pdf representation

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)

effective pdf representationt 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 effective pdf representation(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 ( effective pdf representation 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 effective pdf representation(v)(v,t):

t (v) = v{[(1 + G(t) + g2/ ) v

– (VR + VE (G(t) + g2/ ))](v)

+ g2/ (v- VE)2v(v)}

g2/ -- Fluctuations in g

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

Remarks: (i) Boundary Conditions; effective pdf representation

(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.

- Bistability and Hysteresis effective pdf representation
- Network of Simple and Complex — Excitatory only

N=16!

N=16

MeanDriven:

FluctuationDriven:

Relatively Strong

Cortical Coupling:

- Bistability and Hysteresis effective pdf representation
- Network of Simple and Complex — Excitatory only

N=16!

MeanDriven:

Relatively Strong

Cortical Coupling:

- Three Levels of Cortical Amplification : effective pdf representation
- 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 Mechanism
- for Orientation Tuning of Complex Cells
- Regime 2 for far-field/well-tuned Complex Cells
- Regime 1 for near-pinwheel/less-tuned

With inhib, simple & cmplx

(2) (1)

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 Issues efficientfor 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 efficient

spiking

(very noisy dynamics,

on the synaptic time scale)

Solid: average

( over 72 cycles)

Dashed: 10 temporal

trajectories

Simple efficient

Complex

Expt. Results

Shapley’s Lab

(unpublished)

4B

# of cells

4C

CV (Orientation Selectivity)

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

- Incorporation of Inhibitory Cells efficient
- 4 Population Dynamics
- Simple:
- Excitatory
- Inhibitory

- Excitatory
- Inhibitory

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