Scaling-up Cortical Representations in Fluctuation-Driven Systems

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Scaling-up Cortical Representationsin
Fluctuation-Driven Systems

• David W. McLaughlin
• Courant Institute Center for Neural Science
• New York University
• http//www.cims.nyu.edu/faculty/dmac/
• Cold Spring Harbor -- July 04

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• In collaboration with
• ? ? David Cai
• Louis Tao
• Michael Shelley
• Aaditya Rangan

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Lateral Connections and Orientation -- Tree
Shrew Bosking, Zhang, Schofield Fitzpatrick J.
Neuroscience, 1997
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Coarse-Grained Asymptotic Representations

• Needed for Scale-up

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Coarse-Grained Reductions for V1

• Average firing rate models (Cowan Wilson .
  Shelley McLaughlin)
• m?(x,t), ? E,I

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Cortical networks have a very noisy dynamics

• Strong temporal fluctuations
• On synaptic timescale
• Fluctuation driven spiking

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Experiment Observation Fluctuations in
Orientation Tuning (Cat data from Fersters Lab)
Ref Anderson, Lampl, Gillespie, Ferster Science,
1968-72 (2000)
threshold (-65 mV)
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Fluctuation-driven spiking
(very noisy dynamics, on the synaptic time scale)
Solid average ( over 72
cycles) Dashed 10 temporal trajectories
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• To accurately and efficiently describe these
  networks requires that fluctuations be retained
  in a coarse-grained representation.
• Pdf representations
• ??(v,g x,t), ? E,I
• will retain fluctuations.
• But will not be very efficient numerically
• Needed a reduction of the pdf representations
  which retains
• Means
• Variances
• PT 1 Kinetic Theory provides this
  representation
• Ref Cai, Tao, Shelley McLaughlin, PNAS, pp
  7757-7762 (2004)

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First, tile the cortical layer with
coarse-grained (CG) patches
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Kinetic Theory begins from

• PDF representations
• ??(v,g x,t), ? E,I
• Knight Sirovich
• Tranchina, Nykamp Haskell

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Coarse-Grained Reductions for V1

• PDF representations (Knight Sirovich
  Tranchina, Nykamp Haskell Cai, Tao, Shelley
  McLaughlin)
• ??(v,g x,t), ? E,I
• Sub-network of embedded point neurons -- in a
  coarse-grained, dynamical background
• (Cai,Tao McLaughlin)

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• Well replace the 200 neurons in this CG cell by
  an effective pdf representation

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• First, replace the 200 neurons in this CG cell by
  an effective pdf representation
• Then derive from the pdf rep, kinetic thry
• For convenience of presentation, Ill sketch the
  derivation a single CG cell, with 200 excitatory
  Integrate Fire neurons
• The results extend to interacting CG cells which
  include inhibition as well as simple
  complex cells.

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1 - p Synaptic Failure rate
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• N excitatory neurons (within one CG cell)
• Random coupling throughout the CG cell
• 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)

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• N excitatory neurons (within one CG cell)
• All-to-all 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 over Poisson spike train

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• ? ?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) (i) Ngt1 (ii)
  the total input to each neuron is (modulated)
  Poisson spike trains.
• ?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),
  
• ?0(t) modulated rate of Poisson spike train
  from LGN
• m(t) average firing rate of the neurons in the
  CG cell
• ? J(v)(v,g ?)(v 1) dg,
• and where J(v)(v,g ?) -(v VR) g (v-VE)
  ?

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• ?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

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Kinetic Theory Begins from 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).
• ?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),

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• ?t ? ?-1?v (v VR) g (v-VE) ?
• ?g g G(t)/?) ? ?g2 /? ?gg ?
  

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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)

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• 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)

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• ?t ? ?-1?v (v VR) g (v-VE) ?
• ?g g G(t)/?) ? ?g2 /? ?gg ?
  

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• Under the conditions,
• Ngt1 f lt 1 ?0 f O(1),
• And the Closure (i) ?v?2(v) 0
• (ii) ?2(v) ?g2
• where ?2(v) ?2(v) (?1(v))2 ,
• ?g2 ?0(t) f2 /(2?) m(t) (Sa)2 /(2N?)
• G(t) ?0(t) f m(t) Sa
• One obtains

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• ?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
• One obtains

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• ?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)

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Fluctuations in g are Gaussian

• ? ?t ?(g) ?g g G(t)) ?(g) ?g2 ?gg ?(g)

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Fluctuation-Driven Dynamics
PDF of v Theory? ?IF
(solid)
Fokker-Planck? Theory?
?IF ?Mean-driven limit (
) Hard thresholding
N75
firing rate (Hz)
N75 s5msec S0.05 f0.01
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Fluctuation-Driven Dynamics
PDF of v Theory? ?IF
(solid) Fokker-Planck?
Theory? ?IF ?Mean-driven limit (
) Hard thresholding
N75
firing rate (Hz)
Experiment
N75 s5msec S0.05 f0.01
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• Bistability and Hysteresis
• Network of Simple, Excitatory only

N16!
N16
MeanDriven
FluctuationDriven
Relatively Strong Cortical Coupling
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• Bistability and Hysteresis
• Network of Simple, Excitatory only

N16!
MeanDriven
Relatively Strong Cortical Coupling
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• 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)

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• 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

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• Remarks (i) Boundary Conditions
• (ii) Inhibition, spatial coupling of CG
  cells, simple complex cells have been
  added
• (iii) N ? ? yields mean field
  representation.

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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)

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• Local temporal asynchony enhanced by synaptic
  failure permitting better amplification

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Closures
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Simple Complex Cells Multiple interacting CG
Patches
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Recall
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1 - p Synaptic Failure rate
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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Complex Excitatory Cells
• Mean-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Complex Excitatory Cells
• Mean-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Complex Excitatory Cells
• Mean-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Complex Excitatory Cells
• Mean-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Simple Excitatory Cells
• Mean-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Simple Excitatory Cells
• Mean-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Simple Excitatory Cells
• Mean-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Simple Excitatory Cells
• Mean-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Simple Excitatory Cells
• Mean-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Complex Excitatory Cells
• Fluctuation-Driven

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• Incorporation of Inhibitory Cells
• 4 Population Dynamics
• Simple
• Excitatory
• Inhibitory
• Complex
• Excitatory
• Inhibitory
• Simple Excitatory Cells
• Fluctuation-Driven

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Three Dynamic Regimes of Cortical
Amplification 1) Weak Cortical
Amplification No Bistability/Hysteresis
2) Near Critical Cortical Amplification
3) Strong Cortical Amplification Bistabili
ty/Hysteresis (2) (1)
(3)
IF Excitatory Complex Cells Shown
(2) (1)
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Simple Complex Cells Multiple interacting CG
Patches
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FluctuationDriven Tuning Dynamics Near
Critical Amplification vs. Weak Cortical
Amplification Sensitivity to Contrast
Ring Model A less-tuned complex
cell Ring Model far field A well-tuned
complex cell Large V1 Model A complex
cell in the far-field
Ring Model of Orientation Tuning Near
Pinwheel Far Field
A Cell in Large V1 Model
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Computational Efficiency

• For statistical accuracy in these CG patch
  settings, Kinetic Theory is 103 -- 105 more
  efficient than IF

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• Average firing rates
• Vs
• Spike-time statistics

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• Coarse-grained theories involve local averaging
  in both space and time.
• Hence, coarse-grained theories average out
  detailed spike timing information.
• Ok for rate codes, but if spike-timing
  statistics is to be studied, must modify the
  coarse-grained approach

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PT 2 Embedded point neurons will capture
these statistical firing propertiesRef Cai,
Tao McLaughlin, PNAS (to appear)

• For scale-up computer efficiency
• Yet maintaining statistical firing properties of
  multiple neurons
• Model especially relevant for biologically
  distinguished sparse, strong sub-networks
  perhaps such as long-range connections
• Point neurons -- embedded in, and fully
  interacting with, coarse-grained kinetic theory,
• Or, when kinetic theory accurate by itself,
  embedded as test neurons

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IF vs. Embedded Network Spike Rasters
a) IF Network 50 Simple cells, 50 Complex
cells. Simple cells driven at 10 Hz b)-d)
Embedded IF Networks b) 25 Complex cells
replaced by single kinetic equation c) 25
Simple cells replaced by single kinetic
equation d) 25 Simple and 25 Complex cells
replaced by kinetic equations. In all panels,
cells 1-50 are Simple and cells 51-100 are
Complex. Rasters shown for 5 stimulus periods.
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Embedded Network
Full I F Network
Raster Plots, Cross-correlation and ISI
distributions. (Upper panels) KT of a
neuronal patch with strongly coupled embedded
neurons (Lower panels) Full IF Network.
Shown is the sub-network, with neurons 1-6
excitatory neurons 7-8 inhibitory EPSP time
constant 3 ms IPSP time constant 10 ms.
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Test neuron within a CG Kinetic Theory
ISI distributions for two simulations (Left)
Test Neuron driven by a CG neuronal patch
(Right) Sample Neuron in the IF Network.
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IF vs. Embedded Network Spike Rasters
a) IF Network 40 Simple cells, 40 Complex
cells. Simple cells driven at 10 Hz b)-d)
Embedded IF Networks b) 20 Complex cells
replaced by log-form rate equation c) 20
Simple cells replaced by log-form rate
equation d) 20 Simple and 20 Complex cells
replaced by log-form rate equations. In all
panels, cells 1-40 are Simple and cells 41-80
are Complex. Rasters shown for 5 stimulus
periods.
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a) IF Network 50 Simple cells, 50
Complexcells. Simplecells driven at 10 Hz b)
Embedded IF Networks, with Complex IF neurons
receiving input but not outputing to the network.
25 Complexcells replaced by kinetic equations.
In all panels, cells 1-50 are Simple and cells
51-75 are Complex. Rasters shown for 5 stimulus
periods.
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Dynamic Firing Rates (Shown Exc Simple Pop in
a 4 Pop Model) Forcing--time-dependent Poisson
process, sinusoidal driving at 10 Hz. In the
embedded models, excitatory neurons are IF and
inhibitory neurons are replaced by Kinetic
Theory.
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The Importance of Fluctuations
Cycle-averaged Firing Rate Curves Shown Exc
Cmplx Pop in a 4 population model) Full IF
network (solid) , Full IF KT (dotted) Full
IF coupled to Full KT but with mean only
coupling (dashed). In both embedded cases
(where the IF units are coupled to KT), half
the simple cells are represented by Kinetic Theory
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Steady State Firing Rate Curves (for the
excitatory population) IF, Exc. Inh.
(Magenta), Kinetic Theory, Exc. Inh. (Red),
Embedded Network (Blue), Log-form (Green). In
the Embedded model, Excitatory neurons are IF
and Inhibitory neurons are modelled by Kinetic
Theory.
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Steady State Firing Rate Curves for Embedded
Sub-networks (Shown, Exc Cmplx Pop) In the
embedded models, half of the simple cells are
replaced by Kinetic Theory. However, in the IF
KT (mean only) , they are coupled back to the
IF neurons as mean conductance drives.
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(From left to right) Rasters, Cross-correlation
and ISI distributions for two simulations (Upper
panels) Kinetic Theory of a neuronal patch
driving test neurons, which are not coupled to
each other (Lower panels) KT of a neuronal patch
driving strongly coupled neurons. In both cases,
the CGed patch is being driven at 1 Hz. Neurons
1-6 are excitatory Neurons 7-8 are inhibitory
the S-matrix is (See, Sei, Sie, Sii) (0.3, 0.4,
0.6, 0.4). EPSP time constant 3 ms IPSP time
constant 10 ms.
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(From left to right) Rasters, Cross-correlation
and ISI distributions for two simulations (Upper
panels) Kinetic Theory of a neuronal patch
driving test neurons, which are not coupled to
each other (Lower panels) KT of a neuronal patch
driving strongly coupled neurons. In both cases,
the CGed patch is being driven asynchronously,
and are firing at a fixed rate. IF Neurons 1-6
are excitatory Neurons 7-8 are inhibitory the
S-matrix is (See, Sei, Sie, Sii) (0.4, 0.4,
0.8, 0.4). EPSP time constant 3 ms IPSP time
constant 10 ms.
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Reverse Time Correlations

• Correlates spikes against driving signal
• Triggered by spiking neuron
• Frequently used experimental technique to get a
  handle on one description of the system
• P(?,?) probability of a grating of orientation
  ?, at a time ? before a spike
• -- or an estimate of the systems linear
  response kernel as a function of (?,?)

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• Reverse-Time Correlation (RTC)
• System analysis
• Probing network dynamics

Time ?
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Reverse Correlation
Left IF Network of 128 Simple and 128
Complex cells at pinwheel center. RTC P(???)
for single Simple cell. Below Embedded Network
of 128 Simple cells, with 128 Complex cells
replaced by single kinetic equation. RTC P(???)
for single Simple cell.
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Computational Efficiency

• For statistical accuracy in these CG patch
  settings, Kinetic Theory is 103 -- 105 more
  efficient than IF
• The efficiency of the embedded sub-network scales
  as N2, where N of embedded point neurons
• (i.e. 100 ? 20 yields 10,000 ?400)

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Conclusions

• Kinetic Theory is a numerically efficient, and
  remarkably accurate, method for scale-up Ref
  PNAS, pp 7757-7762 (2004)
• Kinetic Theory introduces no new free parameters
  into the model, and has a large dynamic range
  from the rapid firing mean-driven regime to a
  fluctuation driven regime.
• Kinetic Theory does not capture detailed
  spike-timing statistics
• Sub-networks of point neurons can be embedded
  within kinetic theory to capture spike timing
  statistics, with a range from test neurons to
  fully interacting sub-networks.
• Ref PNAS, to appear (2004)

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Conclusions and Directions

• Constructing ideal network models to discern and
  extract possible principles of neuronal
  computation and functions
• Mathematical methods for analytical
  understanding
• Search for signatures of identified mechanisms
• Mean-driven vs. fluctuation-driven kinetic
  theories
• New closure, Fluctuation and correlation effects
• Excellent agreement with the full numerical
  simulations
• Large-scale numerical simulations of structured
  networks constrained by anatomy and other
  physiological observations to compare with
  experiments
• Structural understanding vs. data modeling
• New numerical methods for scale-up --- Kinetic
  theory

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• Three Dynamic Regimes 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

(2) (1)
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Summary Conclusion
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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.

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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

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Closures
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Kinetic Theory for Population Dynamics
Population of interacting neurons
1-p Synaptic Failure rate
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Under ASSUMPTIONS 1) 2) Summed
intra-cortical low rate spike events become
Poisson
Kinetic Equation
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Fluctuation-Driven Dynamics Physical Intuition
Fluctuation-driven/Correlation between g and V
Hierarchy of Conditional Moments
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Closure Assumptions
Closed Equations Reduced Kinetic Equations
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Coarse-Graining in Time

• Fluctuation Effects
• Correlation Effects

Fokker-Planck Equation
Flux Determination of Firing Rate For a
steady state, m can be determined implicitly
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