modelization of membrane potentials and information

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, model verification references

Modelization of membrane potentials andinformation transmission in large systems of

neurons

Reinhard Hopfner

Johannes Gutenberg Universitat Mainzwww.mathematik.uni-mainz.de/∼hoepfner

Marseille 2010


1 introduction

2 membrane potential as a (jump) diffusion process

3 Poisson spike trains

4 information transmission in large systems of neuronstheoremproofinterpretation

5 statistical inference, model verificationcomments on level 10 in example 2comments on example 1

6 references


introduction

example 1: membrane potential in a pyramidal neuron emitting spikes

0 50 100 150

−50

−40

−30

−20

−10

0

[sec]

[mV

]

17Sept08_023.asc

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


example 2: pyramidal neuron under different experimental conditionsnetwork activity stimulated by increasing concentration of potassium (K)

[sec]

[mV

]

0 10 20 30 40 50 60

−6

0−

40

−2

00

20 1

data: Kilb and Luhmann, Institute of Physiology, Mainz (in: Hopfner 07)


spikes are generated when the membran potential Vt in the soma is high enough


view the membrane potential between successive spikesas 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


example 3: spike trains recorded in the visual cortex210 iid experiments ← identical visual stimulus (Shadlen-Newsome 98)

hence: view the spike train µ emitted by one neuronas a random point measure on [0,∞) with stochastic intensity such thatmean value of intensity at time t corresponds to stimulus at time t


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-Hopfner 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 + σp

(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


well known: shifting membrane potential V by K0, process (Vt − K0)t≥0

is ergodic with invariant law Γ`

2σ2 (KR−K0+f ) , 2

σ2

´on [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) (Hopfner 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 neuronsbelonging to an active network


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 ef (t)dt ν(dy)

independent of BM W , for some deterministic function t → ef (t) andsome 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 +

Zy µ(dt, dy)| z

←ef (t)

+ σp

(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, ...)


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) + ef (v) eΨv,t(λ)

oτdv

«Ψv,t(λ) =

e−τ(t−v) λ

1 + λσ2

2(1−e−τ(t−v))

, eΨv,t(λ) =

Z[1−e−yΨv,t (λ)] ν(dy)

LT analogous to results of Kawazu-Watanabe for time-homogeneous case(Kawazu-Watanabe 71, Hopfner 09)


remarks : a) special case where f (·) ≡ f , ef (·) ≡ ef are constant:

λ −→ exp

„−Z t

s

Ψv,t(λ) τdv

«=

„1 +

λσ2[1− e−τ(t−s)]

2

«− 2σ2

is LT of a Gamma law Γ“

2σ2 ,

2

σ2[1−e−τ(t−s)]

”, and the law with LT

λ −→ exp

„−Z t

−∞

n[KR−K0+f ] Ψv,t(λ) + ef eΨv,t(λ)

oτdv

«(independent of t and τ) is invariant for the process (Vt − K0)t≥0

b) special case where f (·), ef (·) are T -periodic functions:have a T -periodic semigroup, an invariant probability on the canonical spaceC [0,T ] for T -segments in the path of (Vt − K0)t≥0, and thus limit theoremsfor a large class of functionals of the process (Vt − K0)t≥0 (Hopfner-Kutoyants 09)


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 fixedthreshold 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 ≥ KEtoy 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


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 → ef (t)

and generating Poisson spike trains µ1, . . . , µN , . . . as defined above :

µi emitted by neuron i ← membrane potential V i = (V it )t≥0 in neuron i

(V it )t≥0 ← (f ,ef ), (W i , µi )

in very rough approximation, think of

E(V it ) ≈ KR + f (t) + ef (t)

Zyν(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=1µi , N →∞


notations : neurons i = 1, 2, . . . working in parallel, counting processes

ΞN(t, ω) :=1

N

NXi=1

µi (ω, [0, t]) , 0 ≤ t ≤ T

fixed time horizon T , compensators (up to factor λ > 0)

ΦN(t, ω) :=1

N

NXi=1

Ai (t, ω) , 0 ≤ t ≤ T

Ai (t, ω) :=

Z t

0

1[KE ,∞)(Vis (ω)) ds , i = 1, 2, . . .

deterministic limit independent of i

Φ(t) := E(A) =

Z t

0

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 ]), 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)


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) = λΦ(t1 ∧ t2) + λ2

Z t1

0

Z t2

0

C(r1, r2) dr1 dr2

C(r1, r2) := P`Vrj≥KE , j = 1, 2

´−

2Yj=1

P(Vrj≥KE )

remarks : a) law L(V ) of membrane potential and deterministic limitΦ : r →

R r

0P(Vs ≥ KE ) ds depend on

input t → f (t), t → ef (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


proof

proof in three steps on the lines of (Brodda-Hopfner 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

0

Z t2

0

C(r1, r2) dr1 dr2

C(r1, r2) = P`Vrj≥KE , j = 1, 2

´−

2Yj=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 − λΦN ) −→ W(2)

where W(2) is Brownian motion time-changed by the deterministic functiont → λΦ(t) (’classical’ martingale limit theorem)

prove W(2) independent of W(1) (← Poisson spike trains !!)


interpretation

main idea behind ’information transmission’ as considered here:

first layer of N neurons working in parallel receives incoming stimulus inform of t → f (t) and t → ef (t)

in neurons belonging to this first layer, for 0 ≤ t ≤ T , stimulus is

reexpressed as mean value t → E(V(f ,ef )t ) of the membrane potential

collecting all Poisson spike trains emitted by N first layer neuronsinto one pooled spike train, this last fct is transformed into a histogramwhose shape is close – by theorem, up to OP(N−1/2) errors –

to the function t → λΦ(t) = λP( V(f ,ef )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 asincoming stimulus for a second layer of neurons, and so on ....

possibility to transmit even relatively weak (’subthreshold’) signals !!


some simulation pictures illustrating this idea: (Brodda-H. 04)

bell-shaped stimulus t → f (t) (model without jumps, i.e. ef (·) = 0)versus histogram of all spike times collected in the pooled spike trainfrom N neurons working in parallel, N = 125, 200, 350

explicit Laplace transforms for the transition probabilites in membrane potential

process (V(f ,ef )t )0≤t≤T under stimulus t → f (t) , t → ef (t)

opens possibility to calculate explicitely (at least in principle)the successive deformations undergone by the original stimulus


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 := dT0

∆e , i1 := bT1

∆c

nonparametric statistical model: b(·) and σ2(·) unknown C1 functionsFlorens-Zmirou 93, Hoffmann 99+01, ....

estimate b(·) and σ2(·) using nonparametric estimators based onincrements in the time discrete data set whose construction imitates

b(x) = lims↓0

E

„Xt+s − Xt

s| Xt = x

«σ2(x) = lim

s↓0E

„[Xt+s − Xt√

s]2 | Xt = x

«


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

cσ2(a) := cσ2(∆,M,h)(a) =

Pi1−Mi=i0

K“

Xi∆−ah

”“X(i+M)∆−Xi∆√

∆·M

”2

Pi1−Mi=i0

K“

Xi∆−ah

”bb(a) := bb(∆,M,h)(a) =

Pi1−Mi=i0

K“

Xi∆−ah

”“X(i+M)∆−Xi∆

∆·M

”Pi1−M

i=i0K“

Xi∆−ah

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

our choices: kernel rectangular or triangular, bandwith h = 0.01, step multipleM = 20; ∆ imposed by structure of data (moderate variation of M and h ??)

have tightness results in terms of an observable random rate involving

i1−MXi=i0

K

„Xi∆ − a

h

«’number of visits near a’

thus ’estimation is reliable at points a where the number of visits is high’


comments on level 10 in example 2

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 theabove estimators to inter-spike-segments in the membrane potential (H. 07)

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treating ’level 10’ (15 mM of K) as a time homogeneous CIR (and ignoringthat jumps may be present here), we obtain from figures 3C and 3G (points awhere 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 + σp

(V − K0)+√τ dWt

is estimated in the time homogeneous model as

dVt = 5(−37.5− Vt ]) dt +√

0.4p

(V − K0)+ dWt


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]

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l tim

e’ (

8) fo

r h=

0.01

−40 −38 −36 −34

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figure 3Dlevel 10

comparing occupation timeto the densityGamma( 10.87 , 4.47 )shifted by −40

comparing occupation timeto the densityGamma( 10.87 , 4.47 )shifted by −39.75

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


comments on example 1

the data of example 1, investigated in the PhD thesis Jahn 09with identical methods, yields a good fit to the CIR modeltwo pictures from his thesis, analyzing the time interval 0 ≤ t ≤ 55 [sec] :


end with this picture ————– thanks for you attention !


References

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

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Tuckwell 89: Stochastic processes in the neurosciences. CBMS-NSF conf. series in applied mathematics, SIAM, 1989

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modelization of membrane potentials and information

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