[PPT] - Synchrony in Stochastic Pulse-coupled Neuronal Network Models Katie PowerPoint Presentation

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Synchrony in Stochastic Pulse-coupled Neuronal Network Models

Katie Newhall1 Gregor Kovaˇ ciˇ c and Peter Kramer1 Aaditya Rangan and David Cai2

1Rensselaer Polytechnic Institute, Troy, New York 2Courant Institute, New York, New York

January 19, 2009 Stochastic Models in Neuroscience Marseille, France

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Motivation and Goals

◮ Motivation: Neuronal Model Behavior

◮ Study the most basic point neuron models, network models,

and coarse-grained models for use in computations

◮ Develop new analytical techniques ◮ Look for simple, universal network mechanisms

◮ Goals: Understand Oscillatory Dynamics of IF Model

◮ Generalize classical problem of perfectly synchronous

Integrate-and-Fire (IF) network oscillations to stochastic setting

◮ Quantitative analysis of mechanism sustaining synchrony ◮ Lessons for understanding more complicated models and types

f oscillations?

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Outline

◮ Network dynamics: model and simulations ◮ Characterization of mechanism for sustaining stochastically

driven synchronized dynamics

◮ Computation of probability of repeated total firing events ◮ Computation of firing rates based on first passage time

⊲ First passage time calculation ⊲ Approximation of first passage time

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Excitatory Neuronal Network

All-to-all connected, current based, integrate-and-fire (IF) network dvi(t) dt = −gL(vi(t) − VR) + Ii(t), i = 1, . . . , N Ii(t) = f

k

δ(t − tik) + S N

j=i
k

δ(t − ˜ tjk)

!" !# !$ !%

VR ≤ vi(t) < VT

◮ VR: reset potential (=0) ◮ VT : firing threshold (=1)

gL: leakage rate (=1) f and S: coupling strengths (> 0)

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Integrate & Fire Dynamics

◮ vi(t) evolves according to the voltage ODE until ˜

tik, when vi(˜ tik) ≥ VT

◮ At ˜

tik, vi(t) is reset to VR: vi(˜ t+

ik) = VR ◮ S/N is added to the voltage, vj, for j = 1, . . . , N, j = i

Reset:

! !

" #

Membrane Potential:

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time Voltage

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Classification by Mean External Driving Strength

Mean voltage driven to VR + fν/gL without network coupling Superthreshold VR + fν/gL > VT

2 4 6 8 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time voltage

f = 0.01, ν = 120 fν = 1.2 > 1 Subthreshold VR + fν/gL < VT

5 10 15 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time voltage

f = 0.01, ν = 90 fν = 0.9 < 1

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Classification by Fluctuation Size

Approximate Poisson point process with rate ν and amplitude f by drift-diffusion process with drift coef fν and diffusion coef f 2ν/2 Diffusion Approx. Valid

2 4 6 8 10 0.5 1 voltage Poisson Process Driven 2 4 6 8 10 0.5 1 time voltage Drift−Diffusion Process Driven

f = 0.005 ≪ (VT − VR)/gL ν = 2400, fν = 1.2, N = 1 Diffusion Approx. Not Valid

2 4 6 8 10 0.5 1 Poisson Process Driven voltage 2 4 6 8 10 0.5 1 Drift−Diffusion Process Driven time voltage

f = 0.1 =∼ O((VT − VR)/gL) ν = 12, fν = 1.2, N = 1

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Types of Synchronous Firing: Partial Synchrony

All-to-all connected network of N = 100 neurons

2 4 6 8 10 20 40 60 80 100 time number of firing events 2 4 6 8 10 20 40 60 80 100 time neuron number

f = 0.01, fν = 1.2, S = 0.4

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Types of Synchronous Firing: Imperfect Synchrony

All-to-all connected network of N = 100 neurons

2 4 6 8 10 20 40 60 80 100 120 time number of firing events 2 4 6 8 10 20 40 60 80 100 time neuron number

f = 0.1, fν = 1.2, S = 2.0

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Types of Synchronous Firing: Perfect Synchrony

All-to-all connected network of N = 100 neurons Consistent total firing events; “synchronizable network”

2 4 6 8 10 20 40 60 80 100 120 time number of firing events 2 4 6 8 10 20 40 60 80 100 time neuron number

f = 0.001, fν = 1.2, S = 2.0

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Model for Self-Sustaining Synchrony

Fluctuations in input desynchronize the network

◮ Between total firing events N neurons behave independently

Pulse coupling synchronizes the network

◮ First neuron firing causes cascading event, pulling all other

neurons over threshold due to synaptic coupling (S) After each total firing event, all neurons reset, process repeats Which systems are synchronous? What is the mean firing rate of the network?

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(1) What is the probability to see repeated total firing events?

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Model for Synchronous Firing

Voltages of the other N − 1 neurons when the first neuron fires: Cascade-Susceptible

0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 voltage number of neurons

f = 0.008, fν = 1.0 Not Cascade-Susceptible

0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 voltage number of neurons

f = 0.08, fν = 1.0 S = 2, N = 100 (Bin size = S/N)

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For What Parameters is Network Synchronizable?

0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 voltage number of neurons < 15 S/N

Neuron will fire in a cascade if instantaneous coupling from other firing neurons pushes its voltage over threshold Probability all neurons included in cascading event: P(C) Cascade-susceptibility described by event: C =

N−1

k=1

Ck where Ck : VT − V (k) ≤ (N − k) S N C, Ck ∈ [VR, VT ]N and V (1) ≤ V (2) ≤ · · · ≤ V (N−1)

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Computation of Cascade-Susceptibility Probability

◮ Condition upon time of first threshold crossing:

P(C) = ∞ P(C | T (1) = t)p(1)

T (t)dt. ◮ Approximate condition: max neuron at threshold at time t

P(C | T (1) = t) ≈ P(C | V (N)(t) = VT )

◮ Manipulate using elementary probability,

P(C | V (N)(t) = VT ) = 1 − P(Cc | V (N)(t) = VT ) = 1 −

N−1

j=1

P(Aj | V (N)(t) = VT ) where Ak ≡ Cc

k N−1

j=k+1

(Cj) denotes event that cascade fails first at kth neuron

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Computation of P(Aj | V (N)(t) = VT)

A4:

VT ◮ Reduce to elementary combinatorial calculation of how

neurons are distributed in bins (width S/N), each independently distributed with pv|V (N)(x, t) ≡      pv(x, t) VT

VR pv(x′, t) dx′

for x ∈ [VR, VT ],

therwise,

◮ Approximate single-neuron freely evolving voltage pdf pv(x, t)

as Gaussian

◮ Sum probabilities of each configuration consistent with Aj

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Probability that Network is Cascade-Susceptible

Larger fluctuations need larger coupling to maintain synchrony

0.01 0.02 0.03 0.04 0.2 0.4 0.6 0.8 1 S/N P(C)

N=1000 N=500 N=250 N=100 5 10 0.5 1 S P(C)

fν = 1.2, f = 0.001

0.01 0.02 0.03 0.2 0.4 0.6 0.8 1 S/N P(C) f=0.0001 f=0.0005 f=0.001 f=0.002 f=0.004

fν = 1.2, N = 100

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Probability that Network is Cascade-Susceptible

What about for networks where connections are not all-to-all? Synaptic failure: pf

1 2 3 4 5 0.2 0.4 0.6 0.8 1 S P(C) pf=0.00 pf=0.25 pf=0.50 pf=0.75

N = 100, fν = 1.2, f = 0.001 Sparsity: pc

1 2 3 4 5 0.2 0.4 0.6 0.8 1 S P(C) pc=0.00 pc=0.25 pc=0.50 pc=0.75

N = 100, fν = 1.2, N = 100 Adjust P(C) by taking pv|V (N)(x, t) → (1 − pf) (1 − pc) pv|V (N)(x, t)

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Probability that Network is Cascade-Susceptible

What about scale-free networks? Effect of local topology

◮ Distribution of connections:

P(K = k) ∝ k−3

◮ Account for topology in P(A2):

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# !

red only neurons that fire blue neurons result of clustering neuron A connects to B ◮ Consider higher order terms: P(kB) ⇒ P(kB|kA, A → B)

0.2 0.4 0.6 0.8 1.0 0.02 0.04 0.06 0.08 P(C) S Thr.: m=30 Sim.: m=30 Thr.: m=50 Sim.: m=50 Thr.: m=100 Sim.: m=100

N = 4000, fν = 1.2, f = 0.001

(also with M. Shkarayev)

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Probability that Network is Cascade-Susceptible

What about non instantaneous synaptic coupling? P(C) = 0 Synaptic Delay: TD

2 4 6 8 10 200 400 600 800 1000 time neuron number

N = 1000, S = 0.1, TD = 0.002 fν = 1.2, f = 0.001 Synaptic Time Course: H(t) t

τ 2

E e−t/τE

2 4 6 8 10 200 400 600 800 1000 time neuron number

N = 1000, S = 0.1, τE = 0.002 fν = 1.2, f = 0.001,

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(2) What is the mean time between total firing events?

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Synchronous First Exit Problem

◮ N neurons - don’t want to solve N-dimensional mean exit

time problem!

◮ Obtain PDF of one neuron’s exit time, pT (t), and find PDF of

minimum exit time, p(1)

T , of N independent neurons

2 4 6 8 10 20 40 60 80 100 time neuron number

Mean first passage time of N neurons t = ∞ tp(1)

T (t)dt

p(1)

T (t) = NpT(t)(1 − FT (t))N−1

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Synchronous First Exit Problem

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 voltage t=1.6 t=3.3 t=4.9 t=6.6

pv(x, t)

2 4 6 8 10 0.2 0.4 0.6 0.8 1 time

1 − FT (t)

◮ Neuron reaches threshold - removed from system (absorbed) ◮ Probability neuron not fired is probability still in VR → VT

P(T ≥ t) = 1 − FT (t) = VT

VR

pv(x, t)dx

◮ pv(x, t) solves Kolmogorov Forward Equation (KFE)

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Single Neuron Kolmogorov Forward Equation

KFE: ∂ ∂tpv(x, t) = ∂ ∂x [gL(x − VR)pv(x, t)] + ν [pv(x − f, t) − pv(x, t)] Taylor Expand KFE - Diffusion Approximation: ∂ ∂tpv(x, t) = ∂ ∂x [(gL(x − VR)−fν)pv(x, t)] + f 2ν 2 ∂2 ∂x2 pv(x, t) Boundary Conditions:

◮ absorbing at VT : pv(VT , t) = 0 ◮ reflecting at VR: J[pv](VR, t) = 0

Flux: J[pv](x, t) = − [(gL(x − VR) − fν)pv(x, t)] − f2ν

2 ∂ ∂xpv(x, t)

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Solution to Single Neuron KFE

Eigenfunction expansion: pv(x, t) = ps(x)

∞

n=1

AnQn(x)e−λnt Define: z(x) = gL(x−VR)−fν

f√νgL

Eigenfunctions are Confluent Hypergeometric Functions: Qn(x) =    c1M

−λn

2gL , 1 2, z2(x)

+ c2U
−λn

2gL , 1 2, z2(x)

for z(x) < 0

c3M

−λn

2gL , 1 2, z2(x)

+ c4U
−λn

2gL , 1 2, z2(x)

for z(x) ≥ 0

Stationary Solution: ps(x) = Ne−z2(x) Initial Conditions: pv(x, 0) = δ(x − VR) An =

Qn(VR) R VT

VR ps(x)Q2 n(x)dx

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Synchronous Gain Curves

dashed - Gaussian approx. solid - first passage time circles - simulations black - deterministic rate

◮ Good agreement

where diffusion approximation valid

◮ For fixed

(superthreshold) fν, dependence ∼ √f

0.5 1 1.5 0.5 1 1.5 2 fν firing rate

(a)

f=0.05 f=0.01 f=0.002 f=0.0005 0.1 0.2 0.5 1

m − 1/ˆ τ √f N=100, S=5.0

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Synchronous Gain Curves

dashed - Gaussian approx. solid - first passage time circles - simulations

◮ For fixed

(superthreshold) fν, dependence ∼ ln N

0.7 0.8 0.9 1 1.1 1.2 0.2 0.4 0.6 0.8 1 fν firing rate

(b)

N=1000 N=500 N=100 6 8 0.35 0.4 0.45

m − 1/ˆ τ ln(N) f = 0.01, S = 20.0

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