introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
Modelization of membrane potentials and
information transmission in large systems of
neurons
Reinhard Höpfner
Johannes Gutenberg Universität Mainz
www.mathematik.uni-mainz.de/∼hoepfner
Marseille 2010
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
1
introduction
2
membrane potential as a (jump) diffusion process
3
Poisson spike trains
4
information transmission in large systems of neurons
theorem
proof
interpretation
5
statistical inference, model verification
comments on level 10 in example 2
comments on example 1
6
references
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
introduction
example 1: membrane potential in a pyramidal neuron emitting spikes
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introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
example 2: pyramidal neuron under different experimental conditions
network activity stimulated by increasing concentration of potassium (K)
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introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
spikes are generated when the membran potential Vt in the soma is high enough
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
view the membrane potential between successive spikes
as a stochastic process of (jump) diffusion type :
synapses ,→ dendrites ,→ soma: additivity and exponential decay
one neuron has O(104 ) synapses, ≈ 90% excitatory, ≈ 10% inhibitory
contribution of incoming spikes to the membrane potential :
left: single exciting synapsis; middle: single inhibitory synapsis; right: 2 exciting and 1 inhibitory synapses combined
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
example 3: spike trains recorded in the visual cortex
210 iid experiments ←- identical visual stimulus (Shadlen-Newsome 98)
hence: view the spike train µ emitted by one neuron
as a random point measure on [0, ∞) with stochastic intensity such that
mean value of intensity at time t corresponds to stimulus at time t
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
jump diffusion process modelization
for the membrane potential between successive spikes
many models are time homogeneous, e.g. mean-reverting Ornstein-Uhlenbeck
(Lansky-Lanska 87, Tuckwell 89, Lansky-Sato 99, Lansky-Sacerdote 01, Ditlevsen-Lansky 05, ...)
or Cox-Ingersoll-Ross
(Lansky-Lanska 87, Giorny-Lansky-Nobile-Ricciardi 88, Lansky-Sacerdote-Tomassetti 95, Ditlevsen-Lansky 06, Brodda-Höpfner 06, ...)
stage 1 (time homogeneous and stationary) : CIR type model (Vt )t≥0
for a neuron belonging to an active neuronal network
p
√
dVt = ([KR + f ] − Vt ) τ dt + σ (Vt − K0 )+ τ dWt
with constants σ, τ > 0, reference levels K0 < KR < KE
K0 : lower bound for possible values of the membrane potential
KR : mean value of membrane potential for neuron ’at rest’
KE : excitation threshold
and some quantity measuring the degree of activity of the network
f ≥ 0 : constant representing strength of external stimulus
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
well known: shifting membrane potential V by K0 , process (Vt − K0 )t≥0
`
´
is ergodic with invariant law Γ σ22 (KR −K0 +f ) , σ22 on [0, ∞)
has (stationary) mean KR −K0 +f and variance
not depending on the time constant τ
σ2
(KR −K0 +f
2
)
(Cox-Ingersoll-Ross 85, Ikeda-Watanabe 89, ...)
time homogeneous CIR model gives
convincing fit for the membrane potential data of example 1
(new electronic stabilization device was used by Kilb)
(Jahn 09)
reasonable fit for some of the membrane potential data in example 2
(at least in levels 8,9,10 where neuron is able to generate spikes)
(Höpfner 07)
but in many data sets which seem CIR compatible
evidence for time dependence concerning term f ≥ 0 in the drift
some indication for presence of jumps
open question; PRO: biological reasons; CONTRA:
sophisticated semimartingale tools (Jacod 09, AitSahalia-Jacod 09) do not work as as they should
for sure, neurons can behave differently: OU, other types of diffusions,
no diffusion at all, ..., but: CIR seems suitable for slowly spiking neurons
belonging to an active network
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
more realistic : between successive spikes
use deterministic fct t → f (t) of time (strength of external stimulus)
introduce Poisson jumps, positive and summable : PRM
µ(dt, dy ) on (0, M)×(0, ∞) with intensity τ e
f (t)dt ν(dy )
independent of BM W , for some deterministic function
t →e
f (t) and
R
some σ-finite measure ν(dy ) on (0, M) such that (0,M) y ν(dy ) < ∞
stage 2 : time inhomogeneous model with jumps :
Z
p
√
dVt = ([KR +f (t)]−Vt ) τ dt + y µ(dt, dy ) + σ (Vt −K0 )+ τ dWt
| {z }
←-e
f (t)
pathwise uniqueness, unique strong solution (Yamada-Watanabe 71, Dawson-Li 06, Fu-Li 08, ...)
explicit Laplace transforms for transition probabilities (Kawazu-Watanabe 71, ...)
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
proposition : shifting the membrane potential by K0 , the process
(Vt − K0 )t≥0
has explicit LT
“
”
λ −→ E e −λ(Vt −K0 ) | (Vs − K0 ) = x
for transition probabilities, of form
„
«
Z tn
o
e v ,t (λ) τ dv
exp −xΨs,t (λ) −
[KR −K0 +f (v )] Ψv ,t (λ) + e
f (v ) Ψ
s
Ψv ,t (λ) =
e −τ (t−v ) λ
1+λ
σ2
2
(1−e −τ (t−v ) )
Z
,
e v ,t (λ) =
Ψ
[1−e −y Ψv ,t (λ) ] ν(dy )
LT analogous to results of Kawazu-Watanabe for time-homogeneous case
(Kawazu-Watanabe 71, Höpfner 09)
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
remarks : a) special case where f (·) ≡ f , e
f (·) ≡ e
f are constant:
«
„
„ Z t
«− 22
σ
λ σ 2 [1 − e −τ (t−s) ]
Ψv ,t (λ) τ dv = 1 +
λ −→ exp −
2
s
”
“
2
is LT of a Gamma law Γ σ22 , σ2 [1−e −τ
(t−s) ] , and the law with LT
„ Z
λ −→ exp −
t
n
«
o
e v ,t (λ) τ dv
[KR −K0 +f ] Ψv ,t (λ) + e
fΨ
−∞
(independent of t and τ ) is invariant for the process (Vt − K0 )t≥0
b) special case where f (·), e
f (·) are T -periodic functions:
have a T -periodic semigroup, an invariant probability on the canonical space
C [0, T ] for T -segments in the path of (Vt − K0 )t≥0 , and thus limit theorems
for a large class of functionals of the process (Vt − K0 )t≥0 (Höpfner-Kutoyants 09)
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
spike trains in the single neuron as point process with random intensity
consider a single neuron whose membrane potential is a stochastic process
V = (Vt )t
driven by (W , µ)
definition : a Poisson spike train is a point process µ indep. of (W , µ) s.t.
µ(ds) is Poisson random measure on (0, ∞) with intensity λ 1[KE ,∞) (Vs ) ds
for some λ > 0 and some excitation threshold KE > KR
(mentioned but not used above)
remark : ’excitation threshold’ is not understood in the usual sense of a fixed
threshold for a first hitting time problem, but defined here as critical level
spikes occur at rate λ > 0 on the random set {t > 0 : Vt ≥ KE }
toy model since
neglects return of membrane potential after spike to some ’restart region’
neglects duration and shape of the spikes, neglects ’refractory period’
but captures one evidence from data:
spikes are not first hitting times to some fixed+deterministic threshold
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
central limit theorem for large systems of neurons
consider stoch. indep. neurons i = 1, . . . , N, . . . processing the same input
external stimulus represented by t → f (t) and t → e
f (t)
and generating Poisson spike trains µ1 , . . . , µN , . . . as defined above :
µi emitted by neuron i ←- membrane potential V i = (Vti )t≥0 in neuron i
(Vti )t≥0
←-
(f , e
f ), (W i , µi )
in very rough approximation, think of
E (Vti )
Z
≈
KR + f (t) + e
f (t)
y ν(dy )
consider now large layers of stoch. indep. neurons processing the same input :
’information transmission from layer to layer’ ←- CLT in pooled spike trains
ΞN :=
1 XN
µi
i=1
N
,
N→∞
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
notations : neurons i = 1, 2, . . . working in parallel, counting processes
N
1 X i
µ (ω, [0, t]) , 0 ≤ t ≤ T
ΞN (t, ω) :=
N i=1
fixed time horizon T , compensators (up to factor λ > 0)
N
1 X i
ΦN (t, ω) :=
A (t, ω) , 0 ≤ t ≤ T
N
Z t i=1
1[KE ,∞) (Vsi (ω)) ds , i = 1, 2, . . .
Ai (t, ω) :=
0
deterministic limit independent of i
Z t
Φ(t) := E (A) =
P(Vs ≥ KE ) ds
,
0≤t≤T
0
work with weak convergence in the following Polish path space L
L := L2 ([0, T ], B([0, T ]), λλ) equipped with Borel σ-field B
and view ω → ΞN (·, ω), ω → ΦN (·, ω), . . . as r.v.’s (Ω, A) → (L, B)
(Grinblat 76, Cremers-Kadelka 86)
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
theorem
theorem : we have weak convergence of pooled spike trains
√
N ( ΞN − λΦ ) −→ W
(weakly in L as N → ∞)
where W = (Wt )0≤t≤T is a Gaussian process with covariance kernel
Z t1Z t2
K (t1 , t2 ) = λ Φ(t1 ∧ t2 ) + λ2
C (r1 , r2 ) dr1 dr2
0
C (r1 , r2 )
:=
`
P Vrj ≥KE , j = 1, 2
0
´
−
2
Y
P(Vrj ≥KE )
j=1
remarks R: a) law L(V ) of membrane potential and deterministic limit
r
Φ : r → 0 P(Vs ≥ KE ) ds depend on
input t → f (t), t → e
f (t) common to all neurons i = 1, 2, . . .
b) first contribution to covariance kernel ←- BM time changed by t → Φ(t)
c) kernel C (., .) measures dependency between events {Vrj ≥KE }, j = 1, 2
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
proof
proof in three steps
on the lines of (Brodda-Höpfner 06)
compensators: prove weak convergence in path space L
√
N ( ΦN − Φ ) −→ W(1)
(Cremers-Kadelka 86)
(1)
where W(1) = (Wt )0≤t≤T is real-valued Gaussian with covariance kernel
Z t1Z t2
(t1 , t2 ) −→
C (r1 , r2 ) dr1 dr2
2
0` 0
Y
´
C (r1 , r2 )
=
P Vrj ≥KE , j = 1, 2 −
P(Vrj ≥KE )
j=1
(can not be strengthened to weak convergence in Skorohod path space D)
point processes: prove weak convergence in D (Jacod-Shiryaev 87)
√
N ( ΞN − λ ΦN ) −→ W(2)
where W(2) is Brownian motion time-changed by the deterministic function
t → λ Φ(t) (’classical’ martingale limit theorem)
prove W(2) independent of W(1) (←- Poisson spike trains !!)
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
interpretation
main idea behind ’information transmission’ as considered here:
first layer of N neurons working in parallel receives incoming stimulus in
f (t)
form of t → f (t) and t → e
in neurons belonging to this first layer, for 0 ≤ t ≤ T , stimulus is
(f ,e
f)
reexpressed as mean value t → E (Vt ) of the membrane potential
collecting all Poisson spike trains emitted by N first layer neurons
into one pooled spike train, this last fct is transformed into a histogram
whose shape is close – by theorem, up to OP (N −1/2 ) errors –
(f ,e
f)
to the function t → λΦ(t) = λ P( Vt
≥ KE )
for suitable choice of KE and λ and thanks to Poisson spike trains,
the shapes of these three functions are not too much different
hence: ’message’ produced by the first layer can be used as
incoming stimulus for a second layer of neurons, and so on ....
possibility to transmit even relatively weak (’subthreshold’) signals !!
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
some simulation pictures illustrating this idea:
(Brodda-H. 04)
bell-shaped stimulus t → f (t) (model without jumps, i.e. e
f (·) = 0)
versus histogram of all spike times collected in the pooled spike train
from N neurons working in parallel, N = 125, 200, 350
explicit Laplace transforms for the transition probabilites in membrane potential
(f ,e
f)
process (Vt )0≤t≤T under stimulus t → f (t) , t → e
f (t)
opens possibility to calculate explicitely (at least in principle)
the successive deformations undergone by the original stimulus
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
short remarks on statistical inference and model verification
here only: time homogenous model without jumps ...
let SDE with drift b(·) and diffusion coefficient σ 2 (·)
dXt := b(Xt ) dt + σ(Xt ) dWt ,
t ∈ [T0 , T1 ]
be observed on a grid of discrete time points with step size ∆
Xi∆ , i0 ≤ i ≤ i1 ,
i0 := d
T0
T1
e , i1 := b c
∆
∆
nonparametric statistical model: b(·) and σ 2 (·) unknown C 1 functions
Florens-Zmirou 93, Hoffmann 99+01, ....
estimate b(·) and σ 2 (·) using nonparametric estimators based on
increments in the time discrete data set whose construction imitates
„
«
Xt+s − Xt
b(x) = lim E
| Xt = x
s↓0
s
«
„
X
t+s − Xt 2
√
σ 2 (x) = lim E [
] | Xt = x
s↓0
s
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
we shall use the following estimators for diffusion coefficient and drift :
(H. 07)
at points a ∈ IR , for some kernel K (·) and some bandwidth h > 0
“
”“X
”2
Xi∆ −a
(i+M)∆ −Xi∆
√
K
i=i0
h
∆·M
c2 (∆,M,h) (a) =
σ
Pi1 −M “ Xi∆ −a ”
i=i0 K
h
Pi1 −M “ Xi∆ −a ” “ X(i+M)∆ −Xi∆ ”
i=i0 K
h
∆·M
b
b(∆,M,h) (a) =
Pi1 −M “ Xi∆ −a ”
i=i0 K
h
Pi1 −M
c2 (a)
σ
:=
b
b(a)
:=
based on M-step ∆-increments in the trajectory, for suitable M
our choices: kernel rectangular or triangular, bandwith h = 0.01, step multiple
M = 20; ∆ imposed by structure of data (moderate variation of M and h ??)
have tightness results in terms of an observable random rate involving
i1 −M
X „ Xi∆ − a «
K
’number of visits near a’
h
i=i
0
thus ’estimation is reliable at points a where the number of visits is high’
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
comments on level 10 in example 2
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out of the 10 experiments (varying K concentration) in example 2, we pick the
’level 10’ data (highest K concentration, neuron emitting spikes) and apply the
above estimators to inter-spike-segments in the membrane potential
(H. 07)
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introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
treating ’level 10’ (15 mM of K) as a time homogeneous CIR (and ignoring
that jumps may be present here), we obtain from figures 3C and 3G (points a
where the number of ’visits near a’ is 300 or more) and linear regression
K0 := zero of regression line for diffusion coefficient
[KR + f ] := zero of regression line for the drift
≈ −37.5
τ σ 2 := slope of regression line for diffusion coefficient
τ := |slope of regression line for the drift|
≈ −40
≈2
≈5
so that the membrane potential (away from the spike times)
p
√
dVt = ([KR + f ] − Vt ) τ dt + σ (V − K0 )+ τ dWt
is estimated in the time homogeneous model as
√ p
dVt = 5(−37.5 − Vt ]) dt + 0.4 (V − K0 )+ dWt
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
the invariant law of this time homogeneous diffusion
„
«
2
2
Γ
(K
−K
+f
)
,
= Γ( ≈ 12.5 , ≈ 5 )
0
R
σ2
σ2
shifted to [K0 , ∞)
produces a good fit to the plot of the overall occupation time of ’level 10’
0.4
0.0
0.2
’local time’ (8) for h=0.01
0.6
0.8
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figure 3D
level 10
comparing occupation time
to the density
Gamma( 10.87 , 4.47 )
shifted by −39.75
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introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
comments on example 1
the data of example 1, investigated in the PhD thesis Jahn 09
with identical methods, yields a good fit to the CIR model
two pictures from his thesis, analyzing the time interval 0 ≤ t ≤ 55 [sec] :
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
end with this picture
————–
thanks for you attention !
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
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Dawson-Li 06: Skew convolution semigroups and affine Markov processes. Ann. Probab. 34, 1103–1142 (2006).
FlorensZmirou 93: On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30 (1993).
Giorny-Lansky-Nobile-Ricciardi 88: Diffusion approximation and first-passage-time problem for a model neuron. Biol. Cybern. 58 (1988).
Grinblat 76: A limit theorem for measurable random processes and its applications. Proc. AMS 61 (1976).
Hoffmann 99: Lp estimation of the diffusion coefficient. Bernoulli 5 (1999).
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Höpfner 09: A time inhomogeneous Cox-Ingersoll-Ross diffusion with jumps. Preprint 2009, arXiv.
Höpfner-Kutoyants 09: Estimating discontinuous periodic signals in a time inhomogeneous diffusion. Preprint 2009, submitted.
Ikeda-Watanabe 89: Stochastic differential equations and diffusion processes. North Holland/Kodansha 1989.
introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference
Jacod-Shiryaev 87: Limit theorems for stochastic processes. Springer 1987.
Jacod 09: Statistics and high frequency data. Preprint 2009, to appear.
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