Scale up of cortical representations in fluctuation driven settings
Download
1 / 50

Scale-up of Cortical Representations in Fluctuation Driven Settings - PowerPoint PPT Presentation


  • 82 Views
  • Uploaded on

Scale-up of Cortical Representations in 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 (1mm 2 ) of a

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Scale-up of Cortical Representations in Fluctuation Driven Settings' - camden-gordon


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Scale up of cortical representations in fluctuation driven settings
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
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



Lateral Connections and Orientation -- Tree Shrew

Bosking, Zhang, Schofield & Fitzpatrick

J. Neuroscience, 1997



Ii our large scale model
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

III. Cortical Networks Have Very Noisy Dynamics

Strong temporal fluctuations

On synaptic timescale

Fluctuation driven spiking


Fluctuation-driven

spiking

(fluctuations on the

synaptic time scale)

Solid: average

( over 72 cycles)

Dashed: 10 temporal

trajectories


Iv coarse grained asymptotic representations

IV. Coarse-Grained Asymptotic Representations

For “scale-up” in fluctuation driven systems


A Regular Cortical Map

----

500 

----


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) 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 representation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  = -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)],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  = -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)],

N>>1; f << 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


Moments
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,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)


Fluctuations in g are gaussian
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) = -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)


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) v2(v)

where 2(v) = 2(v) – (1(v))2 .

Closure: (i) v2(v) = 0;

(ii) 2(v) = g2


Thus, eqs for  effective pdf representation(v)(v,t) & 1(v)(v,t):

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)


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)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; 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.


N=16!

N=16

Mean­Driven:

Fluctuation­Driven:

Relatively Strong

Cortical Coupling:


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


Scale up dynamical issues for cortical modeling of v1
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)


  • Complex:

    • Excitatory

    • Inhibitory

  • Complex Excitatory Cells

  • Mean-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Complex Excitatory Cells

  • Mean-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Complex Excitatory Cells

  • Mean-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Complex Excitatory Cells

  • Mean-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Simple Excitatory Cells

  • Mean-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Simple Excitatory Cells

  • Mean-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Simple Excitatory Cells

  • Mean-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Simple Excitatory Cells

  • Mean-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Simple Excitatory Cells

  • Mean-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Complex Excitatory Cells

  • Fluctuation-Driven


  • Complex:

    • Excitatory

    • Inhibitory

  • Simple Excitatory Cells

  • Fluctuation-Driven


  • ad