View
26
Download
0
Category
Tags:
Preview:
DESCRIPTION
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 - PowerPoint PPT Presentation
Citation preview
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
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Complex Excitatory CellsMean-Driven
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Complex Excitatory CellsMean-Driven
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Complex Excitatory CellsMean-Driven
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Simple Excitatory CellsMean-Driven
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Simple Excitatory CellsMean-Driven
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Simple Excitatory CellsMean-Driven
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Simple Excitatory CellsMean-Driven
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Simple Excitatory CellsMean-Driven
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Complex Excitatory CellsFluctuation-Driven
Incorporation of Inhibitory Cells
4PopulationDynamics
• Simple: Excitatory Inhibitory
• Complex: Excitatory Inhibitory
Simple Excitatory CellsFluctuation-Driven
Recommended