Synchrony in Stochastic Pulse-coupled
Neuronal Network Models
Katie Newhall1
Gregor Kovačič and Peter Kramer1
Aaditya Rangan and David Cai2
1 Rensselaer
Polytechnic Institute, Troy, New York
Institute, New York, New York
2 Courant
January 19, 2009
Stochastic Models in Neuroscience
Marseille, France
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
of oscillations?
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
Excitatory Neuronal Network
All-to-all connected, current based, integrate-and-fire (IF) network
dvi (t)
= −gL (vi (t) − VR ) + Ii (t), i = 1, . . . , N
dt
X
S XX
Ii (t) = f
δ(t − tik ) +
δ(t − t̃jk )
N
k
v1
v3
v2
v4
j6=i
k
VR ≤ vi (t) < VT
◮
VR : reset potential (=0)
◮
VT : firing threshold (=1)
gL : leakage rate (=1)
f and S: coupling strengths (> 0)
Integrate & Fire Dynamics
◮
◮
◮
vi (t) evolves according to the voltage ODE until t̃ik , when
vi (t̃ik ) ≥ VT
At t̃ik , vi (t) is reset to VR : vi (t̃+
ik ) = VR
S/N is added to the voltage, vj , for j = 1, . . . , N, j 6= i
Reset:
Membrane Potential:
1
0.9
0.8
VR
VT
Voltage
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
Time
1.5
2
Classification by Mean External Driving Strength
Mean voltage driven to VR + f ν/gL without network coupling
Subthreshold
VR + f ν/gL < VT
1
1
0.9
0.9
0.8
0.8
0.7
0.7
voltage
voltage
Superthreshold
VR + f ν/gL > VT
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
2
4
6
time
f = 0.01, ν = 120
f ν = 1.2 > 1
8
10
0
0
5
10
time
f = 0.01, ν = 90
f ν = 0.9 < 1
15
20
Classification by Fluctuation Size
Approximate Poisson point process with rate ν and amplitude f by
drift-diffusion process with drift coef f ν and diffusion coef f 2 ν/2
Diffusion Approx. Not Valid
Diffusion Approx. Valid
Poisson Process Driven
Poisson Process Driven
1
voltage
voltage
1
0.5
0
0
2
4
6
8
0.5
0
0
10
Drift−Diffusion Process Driven
6
8
10
1
voltage
voltage
4
Drift−Diffusion Process Driven
1
0.5
0
0
2
2
4
6
8
time
f = 0.005 ≪ (VT − VR )/gL
ν = 2400, f ν = 1.2, N = 1
10
0.5
0
0
2
4
6
8
10
time
f = 0.1 =∼ O((VT − VR )/gL )
ν = 12, f ν = 1.2, N = 1
Types of Synchronous Firing: Partial Synchrony
100
100
80
80
neuron number
number of firing events
All-to-all connected network of N = 100 neurons
60
40
20
0
0
60
40
20
2
4
6
8
10
0
0
2
time
4
6
time
f = 0.01, f ν = 1.2, S = 0.4
8
10
Types of Synchronous Firing: Imperfect Synchrony
All-to-all connected network of N = 100 neurons
120
100
80
80
neuron number
number of firing events
100
60
40
40
20
20
0
0
60
2
4
6
8
10
0
0
2
time
4
6
time
f = 0.1, f ν = 1.2, S = 2.0
8
10
Types of Synchronous Firing: Perfect Synchrony
All-to-all connected network of N = 100 neurons
Consistent total firing events; “synchronizable network”
120
100
80
80
neuron number
number of firing events
100
60
40
40
20
20
0
0
60
2
4
6
8
10
0
0
2
time
4
6
time
f = 0.001, f ν = 1.2, S = 2.0
8
10
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?
(1) What is the probability to see
repeated total firing events?
Model for Synchronous Firing
Voltages of the other N − 1 neurons when the first neuron fires:
Not Cascade-Susceptible
14
14
12
12
number of neurons
number of neurons
Cascade-Susceptible
10
8
6
4
8
6
4
2
2
0
0.5
10
0.6
0.7
0.8
voltage
f = 0.008, f ν = 1.0
0.9
1
0
0.5
0.6
0.7
0.8
voltage
f = 0.08, f ν = 1.0
S = 2, N = 100 (Bin size = S/N )
0.9
1
For What Parameters is Network Synchronizable?
14
Neuron will fire in a cascade if
instantaneous coupling from other
firing neurons pushes its voltage over
threshold
number of neurons
12
10
< 15 S/N
8
6
4
Probability all neurons included in
cascading event: P (C)
2
0
0.5
0.6
0.7
0.8
voltage
0.9
1
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)
Computation of Cascade-Susceptibility Probability
◮
Condition upon time of first threshold crossing:
Z ∞
(1)
P (C | T (1) = t)pT (t)dt.
P (C) =
0
◮
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 (C c | V (N ) (t) = VT )
=1−
where Ak ≡
Ckc
N\
−1
N
−1
X
j=1
P (Aj | V (N ) (t) = VT )
(Cj )
j=k+1
denotes event that cascade fails first at kth neuron
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


 R pv (x, t)
for x ∈ [VR , VT ],
VT
′ , t) dx′
(x
p
pv|V (N) (x, t) ≡
v
VR


0
otherwise,
Approximate single-neuron freely evolving voltage pdf pv (x, t)
as Gaussian
Sum probabilities of each configuration consistent with Aj
Probability that Network is Cascade-Susceptible
Larger fluctuations need larger coupling to maintain synchrony
0.8
0.6
1
0.4
P(C)
P(C)
0.8
1
N=1000
N=500
N=250
N=100
0.2
0
0
0.02
S/N
0.6
0.4
0.5
0
0
0.01
P(C)
1
f=0.0001
f=0.0005
f=0.001
f=0.002
f=0.004
0.2
5
S
0.03
f ν = 1.2, f = 0.001
10
0.04
0
0
0.01
0.02
S/N
f ν = 1.2, N = 100
0.03
Probability that Network is Cascade-Susceptible
What about for networks where connections are not all-to-all?
Synaptic failure: pf
Sparsity: pc
1
0.8
0.8
0.6
0.6
P(C)
P(C)
1
0.4
0.4
pc=0.00
p =0.00
f
p =0.25
pf=0.25
0.2
c
0.2
pf=0.50
pc=0.50
pc=0.75
p =0.75
f
0
0
1
2
3
4
5
0
0
1
N = 100, f ν = 1.2, f = 0.001
2
3
4
5
S
S
N = 100, f ν = 1.2, N = 100
Adjust P (C) by taking pv|V (N) (x, t) →
(1 − pf )
p (N) (x, t)
(1 − pc ) v|V
Probability that Network is Cascade-Susceptible
What about scale-free networks? Effect of local topology
◮
Distribution of connections:
◮ Consider higher order terms:
P (K = k) ∝ k−3
Account for topology in P (A2 ):
1.0
KA=4
0.8
A
KB=3
B
L
0.6
P(C)
◮
P (kB ) ⇒ P (kB |kA , A → B)
0.4
Thr.: m=30
Sim.: m=30
Thr.: m=50
Sim.: m=50
Thr.: m=100
Sim.: m=100
L
0.2
L=2
0
red only neurons that fire
blue neurons result of clustering
neuron A connects to B
0.02
0.04
S
0.06
N = 4000, f ν = 1.2, f = 0.001
(also with M. Shkarayev)
0.08
Probability that Network is Cascade-Susceptible
What about non instantaneous synaptic coupling? P (C) = 0
Synaptic Time Course: H(t) τt2 e−t/τE
Synaptic Delay: TD
E
1000
1000
800
neuron number
neuron number
800
600
400
200
0
0
600
400
200
2
4
6
8
10
time
N = 1000, S = 0.1, hTD i = 0.002
f ν = 1.2, f = 0.001
0
0
2
4
6
8
10
time
N = 1000, S = 0.1, τE = 0.002
f ν = 1.2, f = 0.001,
(2) What is the mean time
between total firing events?
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
(1)
minimum exit time, pT , of N independent neurons
100
neuron number
80
Mean first passage time of N neurons
Z ∞
(1)
tpT (t)dt
hti =
60
0
40
(1)
pT (t) = N pT (t)(1 − FT (t))N −1
20
0
0
2
4
6
time
8
10
Synchronous First Exit Problem
4
pv (x, t)
3
0.8
1 − FT (t)
3.5
1
t=1.6
t=3.3
t=4.9
t=6.6
2.5
0.6
2
0.4
1.5
1
0.2
0.5
0
0
0.2
0.4
0.6
voltage
0.8
1
0
0
2
4
6
8
time
◮
Neuron reaches threshold - removed from system (absorbed)
◮
Probability neuron not fired is probability still in VR → VT
Z VT
P (T ≥ t) = 1 − FT (t) =
pv (x, t)dx
VR
◮
pv (x, t) solves Kolmogorov Forward Equation (KFE)
10
Single Neuron Kolmogorov Forward Equation
KFE:
∂
∂
pv (x, t) =
[gL (x − VR )pv (x, t)] + ν [pv (x − f, t) − pv (x, t)]
∂t
∂x
Taylor Expand KFE - Diffusion Approximation:
∂
f 2ν ∂2
∂
pv (x, t) =
pv (x, t)
[(gL (x − VR )−f ν)pv (x, t)] +
∂t
∂x
2 ∂x2
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 ) − f ν)pv (x, t)] −
f 2ν ∂
2 ∂x pv (x, t)
Solution to Single Neuron KFE
Eigenfunction expansion:
pv (x, t) = ps (x)
∞
X
An Qn (x)e−λn t
n=1
Define: z(x) =
gL (x−VR )−f ν
√
f νgL
Eigenfunctions are Confluent Hypergeometric Functions:

 c1 M −λn , 1 , z 2 (x) + c2 U −λn , 1 , z 2 (x) for z(x) < 0
2gL 2
2gL 2
Qn (x) =
1
−λn 1 2
−λ
2
n
 c3 M
,
,
z
(x)
+
c
U
,
,
z
(x)
for z(x) ≥ 0
4
2gL 2
2gL 2
Stationary Solution: ps (x) = N e−z
2 (x)
Initial Conditions: pv (x, 0) = δ(x − VR )
An =
Qn (VR )
R VT
VR
ps (x)Q2n (x)dx
Synchronous Gain Curves
◮
◮
Good agreement
where diffusion
approximation valid
For fixed
(superthreshold)
√ f ν,
dependence ∼ f
1.5
firing rate
dashed - Gaussian approx.
solid - first passage time
circles - simulations
black - deterministic rate
2
1
m − 1/τ̂
(a)
1
0.5
0
0
√
0.1
f
0.2
0.5
0
0.5
f=0.05
f=0.01
f=0.002
f=0.0005
1
fν
N=100, S=5.0
1.5
Synchronous Gain Curves
1
dashed - Gaussian approx.
solid - first passage time
circles - simulations
◮
For fixed
(superthreshold) f ν,
dependence ∼ ln N
firing rate
0.8
N=1000
N=500
N=100
0.6
m − 1/τ̂
(b)
0.45
0.4
0.4
0.2
0.35
6
0
0.7
0.8
0.9
1
fν
f = 0.01, S = 20.0
ln(N )
1.1
8
1.2
Summary
◮
Synchronous firing over a wide range of parameters
⊲ Balance between fluctuations and coupling strength
◮
Synchronous firing rate in terms of single neuron properties
⊲ Driven by time first neuron crosses threshold
◮
Network properties are calculated accurately through exit time
problems where diffusion approximation is valid
This work was supported by a NSF graduate fellowship
(Newhall & al. (2010) Comm in Math Sci, Vol. 8, No. 2, pp. 541-600)