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Title: Scale-up of Cortical Representations in Fluctuation Driven Settings

1
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 May03

2
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
3
Local Patch 1mm2
4
Lateral Connections and Orientation -- Tree
Shrew Bosking, Zhang, Schofield Fitzpatrick J.
Neuroscience, 1997
5
From one input layer to several layers
Why scale-up?
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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/

9
Equations of the Integrate Fire Model
? E,I
vj? -- membrane potential -- ? Exc,
Inhib -- j 2 dim label of location
on cortical layer -- 16000 neurons
per sq mm (12000 Exc,
4000 Inh) VE VI -- Exc Inh Reversal
Potentials (Constants)

10
Integrate Fire Model
? E,I
Spike Times tjk kth spike time of jth
neuron Defined by vj?(t tjk ) 1, vj?(t
tjk ?) 0
11
Conductances from Spiking Neurons
?
?
?
Forcing Noise Spatial Temporal
Cortico-cortical
Here tkl (Tkl) denote the lth spike time of
kth neuron
12
Elementary Feature Detectors

Individual neurons in V1 respond preferentially
to elementary features of the visual scene
(color, direction of motion, speed of motion,
spatial wave-length).
Two important features
Orientation ? of edges in the visual scene
Spatial phase ?

13

Grating Stimuli Standing Drifting
?
Angle of orientation -- ? Angle of spatial phase
-- ? (relevant for standing gratings)
?
14
Cortical Maps

How does a preferred feature, such as the
orientation preference ?k of the kth cortical
neuron, depend upon the neurons location k
(k1, k2) in the cortical layer?

15
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
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/

16
III. Cortical Networks Have Very Noisy Dynamics

Strong temporal fluctuations
On synaptic timescale
Fluctuation driven spiking

17
Fluctuation-driven spiking
(fluctuations on the synaptic time scale)
Solid average ( over 72
cycles) Dashed 10 temporal trajectories
18
IV. Coarse-Grained Asymptotic Representations

For scale-up in fluctuation driven systems

19
A Regular Cortical Map
---- ? 500 ? ? ----
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Coarse-Grained Reductions for V1

Average firing rate models (Cowan Wilson
Shelley McLaughlin)
m?(x,t), ? E,I
PDF representations (Knight Sirovich
Nykamp Tranchina Cai, McLaughlin, Shelley
Tao)
??(v,g x,t), ? E,I
Sub-network of embedded point neurons -- in a
coarse-grained, dynamical background
(Cai, McLaughlin Tao)

To accurately and efficiently describe these
fluctuations, the scale-up method will require
pdf representations
??(v,g x,t), ? E,I
To benchmark these, we will numerically
simulate IF neurons within one CG cell
As an aside, these single CG cell simulations
as well as their pdf reductions, can give us
insight into neuronal mechanisms

23
For example, consider the difficulties in
constructing networks with both simple (linear)
and complex (nonlinear) cells

Too few complex cells
Cells not selective enough for orientation (not
well enough tuned)
Particularly true for complex cells, and when
looking ahead to other cortical layers

Need a better cortical amplifier
But cortical amplification produces
instabilities, such as synchrony far too rapid
firing rates.

25
To begin to address these issues

We turn to smaller, idealized networks of two
types
One coarse-grained cell holding hundreds of
neurons
Idealized ring-models
First, one coarse-grained cell with only two
classes of cells -- pure simple and pure
complex

26
For the rest of the talk, consider one
coarse-grained cell, containing several hundred
neurons
27

Well replace the 200 neurons in this CG cell by
an effective pdf representation
For convenience of presentation, Ill sketch the
derivation of the reduction for 200 excitatory
Integrate Fire neurons
Later, Ill 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 ?i1,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 ? ?-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) ?

?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

31
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)

32
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) ?-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) - ?-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 ?(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 (? ? 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
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.

39
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

41
Bistability and Hysteresis
Red IF, N 1024 Blue Mean-Field solid -
stable dashed - unstable
Mean-field bistability certainly exists, but
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Bistability and Hysteresis
Network of Simple and Complex Excitatory only

N16!
N16
MeanDriven
FluctuationDriven
Relatively Strong Cortical Coupling
45

Bistability and Hysteresis
Network of Simple and Complex Excitatory only

N16!
MeanDriven
Relatively Strong Cortical Coupling
46
Blue Large N Limit Red Finite N,
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Fluctuation-Driven Dynamics
Probability Density Theory?
?IF Theory?
?IF ?Mean-driven limit (
) Hard thresholding
N64
N64
49

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)
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

With inhib, simple cmplx
(2) (1)
50

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

51
Pre-hysterisis in Models
Center Fluctuation-driven (single Complex
neuron, 1 realization trial-average)
Ring Model
Large-scale Model Cortex
52
FluctuationDriven Tuning Dynamics Near
Critical Amplification vs. Weak Cortical
Amplification Sensitivity to Contrast Ring
Model of Orientation Tuning
Ring Model far field A well-tuned complex
cell Ring Model A less-tuned complex
cell Large V1 Model A complex cell in
the far-field
53
Fluctuations and Correlations
Outer Solution Existence of a Boundary
Layer - induced by correlation
54
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)

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Second type of idealized models -- Ring Models
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A Ring Model of Orientation Tuning
I F network on a Ring, neuron preferred qi
4 Populations Exc./Inh, Simple/Complex
Network coupling strength Aij A(qi-qj) Gaussian

Simple Cells Complex Cells
58
Mean-Driven Bump State
Simple Cells Complex Cells
Fluctuation-Driven Bump State
Simple Cells Complex Cells
59
Six ring models From near pinwheels to far
from pinwheels
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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.

62
Preliminary Results

Synaptic failure (/or sparse connections) lessen
synchrony, allowing better cortical
amplification
Bistability both in mean and in fluctuation
dominated systems
Complex cells related to bistability or to pre
-bistability
In this model, tuned simple cells reside near
pinwheel centers while tuned complex cells
reside far from pinwheel centers.

63
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

64
Fluctuation-driven spiking
(very noisy dynamics, on the synaptic time scale)
Solid average ( over 72
cycles) Dashed 10 temporal trajectories
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Complex
Simple
Expt. Results Shapleys Lab (unpublished)
? 4B
of cells
? 4C
CV (Orientation Selectivity)
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