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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 H¨

• pfner

Johannes Gutenberg Universit¨ at 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

50 100 150 −50 −40 −30 −20 −10 [sec] [mV]

17Sept08_023.asc

data: Kilb and Luhmann, Institute of Physiology, Mainz (in: Jahn 09)

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)

[sec] [mV] 10 20 30 40 50 60 −60 −40 −20 20

1

data: Kilb and Luhmann, Institute of Physiology, Mainz (in: H¨

• pfner 07)

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

• ne 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, ...)

• r Cox-Ingersoll-Ross

(Lansky-Lanska 87, Giorny-Lansky-Nobile-Ricciardi 88, Lansky-Sacerdote-Tomassetti 95, Ditlevsen-Lansky 06, Brodda-H¨

• pfner 06, ...)

stage 1 (time homogeneous and stationary) : CIR type model (Vt)t≥0 for a neuron belonging to an active neuronal network dVt = ([KR + f ] − Vt) τdt + σ p (Vt − K0)+ √τdWt 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 `

2 σ2 (KR−K0+f ) , 2 σ2

´

• n [0, ∞)

has (stationary) mean KR−K0+f and variance σ2

2 (KR−K0+f )

not depending on the time constant τ

(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) (H¨

• pfner 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

• pen 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 some σ-finite measure ν(dy) on (0, M) such that R

(0,M) yν(dy) < ∞

stage 2 : time inhomogeneous model with jumps : dVt = ([KR+f (t)]−Vt) τdt + Z y µ(dt, dy) | {z }

← ֓e f (t)

+ σ p (Vt−K0)+ √τdWt 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 exp „ −xΨs,t(λ) − Z t

s

n [KR−K0+f (v)] Ψv,t(λ) + e f (v) e Ψv,t(λ)

• τdv

« Ψv,t(λ) = e−τ(t−v) λ 1 + λ σ2

2 (1−e−τ(t−v))

, e Ψv,t(λ) = Z [1−e−yΨv,t(λ)] ν(dy) LT analogous to results of Kawazu-Watanabe for time-homogeneous case

(Kawazu-Watanabe 71, H¨

• pfner 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: λ − → exp „ − Z t

s

Ψv,t(λ) τdv « = „ 1 + λ σ2[1 − e−τ(t−s)] 2 «− 2

σ2

is LT of a Gamma law Γ “

2 σ2 , 2 σ2[1−e−τ(t−s)]

” , and the law with LT λ − → exp „ − Z t

−∞

n [KR−K0+f ] Ψv,t(λ) + e f e Ψv,t(λ)

• τdv

« (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

(H¨

• pfner-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 = (V i

t )t≥0 in neuron i

(V i

t )t≥0

← ֓ (f , e f ), (W i, µi) in very rough approximation, think of E(V i

t )

≈ KR + f (t) + e f (t) Z yν(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 N XN

i=1 µi

, 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 ΞN(t, ω) := 1 N

N

X

i=1

µi(ω, [0, t]) , 0 ≤ t ≤ T fixed time horizon T , compensators (up to factor λ > 0) ΦN(t, ω) := 1 N

N

X

i=1

Ai(t, ω) , 0 ≤ t ≤ T Ai(t, ω) := Z t 1[KE ,∞)(V i

s (ω)) ds

, i = 1, 2, . . . deterministic limit independent of i Φ(t) := E(A) = Z t P(Vs ≥ KE) ds , 0 ≤ t ≤ T work with weak convergence in the following Polish path space L L := L2([0, T], B([0, T]), λ λ) 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 K(t1, t2) = λ Φ(t1 ∧ t2) + λ2 Z t1 Z t2 C(r1, r2) dr1 dr2 C(r1, r2) := P ` Vrj ≥KE , j = 1, 2 ´ −

2

Y

j=1

P(Vrj ≥KE) remarks : a) law L(V ) of membrane potential and deterministic limit Φ : r → R 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

• n the lines of (Brodda-H¨
• pfner 06)

compensators: prove weak convergence in path space L

(Cremers-Kadelka 86)

√ N ( ΦN − Φ ) − → W(1) where W(1) = (W(1)

t )0≤t≤T is real-valued Gaussian with covariance kernel

(t1, t2) − → Z t1 Z t2 C(r1, r2) dr1 dr2 C(r1, r2) = P ` Vrj ≥KE , j = 1, 2 ´ −

2

Y

j=1

P(Vrj ≥KE) (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 form of t → f (t) and t → e f (t) in neurons belonging to this first layer, for 0 ≤ t ≤ T, stimulus is reexpressed as mean value t → E(V (f ,e

f ) t

) 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 – to the function t → λΦ(t) = λ P( V (f ,e

f ) t

≥ 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 process (V (f ,e

f ) t

)0≤t≤T under stimulus t → f (t) , t → e f (t)

• pens 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 σ2(·) 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 := ⌈T0 ∆ ⌉ , i1 := ⌊T1 ∆ ⌋ nonparametric statistical model: b(·) and σ2(·) unknown C1 functions

Florens-Zmirou 93, Hoffmann 99+01, ....

estimate b(·) and σ2(·) using nonparametric estimators based on increments in the time discrete data set whose construction imitates b(x) = lim

s↓0 E

„ Xt+s − Xt s | Xt = x « σ2(x) = lim

s↓0 E

„ [ Xt+s − Xt √s ]2 | Xt = x «

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 ∈ I R, for some kernel K(·) and some bandwidth h > 0 c σ2(a) := c σ2(∆,M,h)(a) = Pi1−M

i=i0

K “

Xi∆−a h

” “ X(i+M)∆−Xi∆

√ ∆·M

”2 Pi1−M

i=i0

K “

Xi∆−a h

” b b(a) := b b(∆,M,h)(a) = Pi1−M

i=i0

K “

Xi∆−a h

” “ X(i+M)∆−Xi∆

∆·M

” Pi1−M

i=i0

K “

Xi∆−a h

” based on M-step ∆-increments in the trajectory, for suitable M

• ur 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

i=i0

K „Xi∆ − a h « ’number of visits near a’ 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

• ut 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)

[mV] sigma^2(.) estimated −40 −39 −38 −37 −36 −35 5 10 15 20 25 30

3A

[mV] drift b(.) estimated −40 −39 −38 −37 −36 −35 −20 −10 10 20

3E

[mV] ’local time’ (8) for h=0.01 −40 −39 −38 −37 −36 −35 200 400 600 800

• 3D

[mV] sigma^2(.) estimated −40 −39 −38 −37 −36 −35 5 10 15 20 25 30

3B

[mV] drift b(.) estimated −40 −39 −38 −37 −36 −35 −20 −10 10 20

3F

[sec] [mV] 10 20 30 40 50 60 −40 −38 −36 −34 −32 −30 spikes (truncated) ’pulses’

3H

[mV] sigma^2(.) estimated −40 −39 −38 −37 −36 −35 5 10 15 20 25 30

3C

[mV] drift b(.) estimated −40 −39 −38 −37 −36 −35 −20 −10 10 20

3G

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 ≈ −40 [KR + f ] := zero of regression line for the drift ≈ −37.5 τσ2 := slope of regression line for diffusion coefficient ≈ 2 τ := |slope of regression line for the drift| ≈ 5 so that the membrane potential (away from the spike times) dVt = ([KR + f ] − Vt) τdt + σ p (V − K0)+ √τ dWt is estimated in the time homogeneous model as dVt = 5(−37.5 − Vt]) dt + √ 0.4 p (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 Γ „ 2 σ2 (KR−K0+f ) , 2 σ2 « = Γ( ≈ 12.5 , ≈ 5 ) shifted to [K0, ∞) produces a good fit to the plot of the overall occupation time of ’level 10’

[mV] ’local time’ (8) for h=0.01 −40 −38 −36 −34 0.0 0.2 0.4 0.6 0.8

• figure 3D

level 10

comparing occupation time to the density Gamma( 10.87 , 4.47 ) shifted by −40 comparing occupation time to the density Gamma( 10.87 , 4.47 ) shifted by −39.75

even if we have neglected possibility of jumps or slight time inhomogeneities ...

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,

References

AitSahalia-Jacod 09: Testing for jumps in a discretely observed process. Ann. Statist. 37, 184–222 (2009). Brodda-H¨

• pfner 06: ... information processing in large systems of neurons. J. Math. Biol. 52 (2006). Preprint with simulations 2004.

Cox-Ingersoll-Ross 85: A theory of the term structure of interest rates. Econometrica 53 (1985). Cremers-Kadelka 86: On weak convergence of integral functionals of stochastic processes ... Stoch. Proc. Appl. 21 (1986). Ditlevsen-Lansky 05: Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model. Physical review E 71 (2005). Ditlevsen-Lansky 06: Estimation of the input parameters in the Feller neuronal model. Phys. Rev. E 73 (2006). 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). Hoffmann 01: On estimating the diffusion coefficient: parametric versus nonparametric. Ann. IHP PS 37 (2001). H¨

• pfner 07: On a set of data for the membrane potential in a neuron. Math. Biosciences 207, 275–301 (2007).

H¨

• pfner 09: An extension of the Yamada-Watanabe condition for pathwise uniqueness to SDE with jumps. ECP 2009.

H¨

• pfner 09: A time inhomogeneous Cox-Ingersoll-Ross diffusion with jumps. Preprint 2009, arXiv.

H¨

• pfner-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.

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