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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Modelling neuronal membranes with Piecewise Deterministic Processes

Martin Riedler

work with Evelyn Buckwar

Heriot–Watt University, Edinburgh, UK

Stochastic Models in Neuroscience, CIRM Marseille, Jan 19th, 2010

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Outline

1

The Modelling Problem

2

Piecewise Deterministic Processes (PDPs) in finite dimensions

3

Spatial Dynamics – PDPs in infinite dimensions

4

Outlook

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Scale levels and modelling approaches

• I. atomic level
• II. protein level
• III. cellular level

charged ions involved in large numbers;

• ne single channel consists of hun-

behaviour of transients important is their distribution dreds of amino acids, hence

• ver length scales of 1µm

close to the membrane and their flux channel ≫ ion and possibly only to 1m; fitting of a deter- across the membrane; flux rate is a few channels involved ∼ low ministic model to the average 103 − 106 ions/ms per open channel; channel density or small fibres; emergent behaviour;

macroscopic model hybrid stochastic models continuous stochastic models "adding" noise deterministic limit macroscopic continuous modelling continuous noise approximation SDE/SPDE models ODE/PDE models discrete particle system microscopic description discrete modelling discrete modelling continuous approximation

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Motivation for and aim of our work

Motivation several hybrid algorithms proposed that reproduce noisy excitable behaviour for neuron models [Skaugen/Walloe 79, Clay/DeFelice

83, Chow/White 96, etc.]; ”noise = channel noise”

in context of neuronal modelling, . . . . . . hardly any theory or rigorous analysis of hybrid processes and their numerical approximation . . . hardly any analysis of continuous stochastic approximations, their appropriateness and quality Our approach and aims use PDPs which provide accurate mathematical description hybrid model (in the space-clamped setting) well known class of stochastic processes with existing theory extend PDP theory to include spatial dynamics develop tools for theoretically analysing hybrid algorithms

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Building blocks of a hybrid stochastic model Single channel model – continuous-time Markov chain Example: (simplified) Na-channel

am(v) m0h0 ⇋ m1h0 bm(v) ah(v) ⇃ ↾ bh(v) ah(v) ⇃ ↾ bh(v) am(v) m0h1 ⇋ m1h1 bm(v)

→ waiting time distr., e.g.

P[τ > t] = e−(am(v)+ah(v))t

→ post-jump value distr., e.g.

pm1h0 = am(v) am(v) + ah(v), pm0h1 = ah(v) am(v) + ah(v).

→ transmembrane potential constant over time

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Building blocks of a hybrid stochastic model Single channel model – continuous-time Markov chain Example: (simplified) Na-channel

am(v) m0h0 ⇋ m1h0 bm(v) ah(v) ⇃ ↾ bh(v) ah(v) ⇃ ↾ bh(v) am(v) m0h1 ⇋ m1h1 bm(v)

→ waiting time distr., e.g.

P[τ > t] = e−(am(v)+ah(v))t

→ post-jump value distr., e.g.

pm1h0 = am(v) am(v) + ah(v), pm0h1 = ah(v) am(v) + ah(v).

→ transmembrane potential constant over time (Space-clamped) membrane model – ordinary differential equation C ˙ v =

m

• i=1

gi(v) (Ei − v) conductances gi ∝ fraction of open channels

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

combining these building blocks: transmembrane potential is not constant → single channel models are not Markovian anymore the correct waiting distribution for a channel in time-varying potential s → v(s) is, e.g., Example cont. P[τ > t] = e−

R t

0 am(v(s))+ah(v(s))ds

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

combining these building blocks: transmembrane potential is not constant → single channel models are not Markovian anymore the correct waiting distribution for a channel in time-varying potential s → v(s) is, e.g., Example cont. P[τ > t] = e−

R t

0 am(v(s))+ah(v(s))ds

→ using this waiting time distribution [Clay/DeFelice, 83] propose an approximation algorithm for a space-clamped membrane → no analysis of the algorithm; particularly, a strong Markov property of the hybrid process ”(voltage, channel state)” needed for the algorithm to be correct

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

(finite-dimensional) Piecewise Deterministic Process - Definition

Davis, Markov Models and Optimization, 1994

c` adl` ag process (y(t))t≥0, where y(t) = (x(t), r(t)) ∈ Rd × R, with |R| < ∞. continuous, deterministic dynamics – cont. component x(t) a family of ODEs ˙ x = gr(x) with solutions t → φr(t; x0) for all x(0) = x0 ∈ Rd and every r ∈ R discrete, stochastic dynamics – pwc. component r(t) jump rate λ : Rd × R → R+, s. t. for the jump times 0 < τ1 < τ2 < . . .

• f r(t) with τk < T

P[τk+1 − T > t|r(T) = r, x(T) = x] = exp “ − Z t λ ` φr(s; x), r ´ ds ” transition measure Q : Rd × R → P(R) for post-jump values P[r(τk+1) = ¯ r|r(τk+1−) = r, x(T) = x] = Q(φr(τk+1 − T; x), r)({¯ r})

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Properties under certain assumptions ... strong, homogeneous Markov process theoretically, an exact simulation algorithm exist; practically, for each inter-jump interval ˙ x = gr(x) and, simultaneously, to obtain the next jump time − log U = t λ

• φr(s; x), r
• ds,

with U ∼ U(0,1), has to be solved numerically. generator A of the Markov process y(t) and its domain D(A) exactly defined, i.e., for f ∈ D(A) Af(x, r) = fx(x, r)gr(x) + λ(x, r)

• R

[f(x, p) − f(x, r)]Q(x, r)(dp) → starting point for the approximation with continuous processes generalisation to include spatial dynamics [Buckwar, R., in prep.]

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Example: space-clamped, leaky membrane with N channels

am(v) m0h0 ⇋ m1h0 bm(v) ah(v) ⇃ ↾ bh(v) ah(v) ⇃ ↾ bh(v) am(v) m0h1 ⇋ m1h1 bm(v)

˙ v = ¯ gNarm1h1 (ENa − v) − gLv | {z }

=gr(v)

r = (rm0h0, rm1h0, rm0h1, rm1h1), ri = # channels in state i

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Example: space-clamped, leaky membrane with N channels

am(v) m0h0 ⇋ m1h0 bm(v) ah(v) ⇃ ↾ bh(v) ah(v) ⇃ ↾ bh(v) am(v) m0h1 ⇋ m1h1 bm(v)

˙ v = ¯ gNarm1h1 (ENa − v) − gLv | {z }

=gr(v)

r = (rm0h0, rm1h0, rm0h1, rm1h1), ri = # channels in state i

R =

• r ∈ R4 : ri ∈ {0 : 1 : N} and
• i

ri = N

• jump rate

λ(v, r) = r·

• am(v)+ah(v), bm(v)+ah(v), am(v)+bh(v), bm(v)+bh(v)
• transition probabilities, e.g.

P[r → r + (−1, 1, 0, 0)] = rm0h0 am(v) λ(v, r)

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

0.5 1 50 100 150 monophasic stimulus # channels =100 0.5 1 50 100 150 # channels = 500 0.5 1 50 100 150 # channels =1000 0.5 1 50 100 150 preconditioned stimulus 0.5 1 50 100 150 0.5 1 50 100 150

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

infinite-dimensional Piecewise Deterministic Process - Definition

[Buckwar, R., Preprint in prep.]

separable, real Hilbert space E continuously and densely embedded in and a Borel subset of another separable, real Hilbert space H; further H ⊂ E∗; c` adl` ag process (y(t))t≥0, where y(t) = (u(t), r(t)) ∈ E × R, with |R| < ∞. continuous, deterministic dynamics – cont. component u(t) a family of abstract ODEs ˙ u = Lru + Gr(u)

• Lr : E → E∗ linear operators
• Gr : E → E∗ nonlinear mappings

with solutions t → ψr(t; u0) for all u(0) = u0 ∈ E and every r ∈ R discrete, stochastic dynamics – pwc. component r(t) jump rate Λ : E × R → R+, transition measure Q : E × R → P(R)

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Properties under certain assumptions ... strong, homogeneous Markov process if the strong (Fr´ echet-) derivative of f ∈ D(A) w.r.t. the u component exists, then the extended generator of y is given by Af(u, r) = Lru+Gr(u), fu(u, r)+Λr(u)

• R

[f(u, p)−f(u, r)]Qr(dp; u) with ·, · duality pairing of E and fu the unique element of E s.t. df dx(x, r) ◦ y = (y, fx(x, r))E ∀ y ∈ E where df

du(u, r) ∈ E∗ Fr´

echet derivative of f w.r.t. u at (u, r). → in case of higher (spatial) regularity we can use the inner product on H instead of ·, ·

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Example cont. Ω = [a, b] ⊂ R, E = H1

0(Ω), H = L2(Ω) and

C ∂v(t, x) ∂t = d 4R ∂2v(t, x) ∂x2 +

N

• i=1

gi

Na(x; r) (ENa − v(t, x))

v(·, x) = 0, for x = a, b with gi

Na(x; r) = ¯

gNa I{m1h1}(ri) δxi(x) ∈ H−1 (= E∗) ⇒ ∀ v(0) ∈ H1

0(Ω) ∃! (weak) solution in C([0, T], H1 0(Ω)).

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Example cont. Ω = [a, b] ⊂ R, E = H1

0(Ω), H = L2(Ω) and

C ∂v(t, x) ∂t = d 4R ∂2v(t, x) ∂x2 +

N

• i=1

gi

Na(x; r) (ENa − v(t, x))

v(·, x) = 0, for x = a, b with gi

Na(x; r) = ¯

gNa I{m1h1}(ri) δxi(x) ∈ H−1 (= E∗) ⇒ ∀ v(0) ∈ H1

0(Ω) ∃! (weak) solution in C([0, T], H1 0(Ω)).

R = {m0h0, m1h0, m0h1, m1h1}N, Λ(v, r) =

N

X

i

h` am(v(xi)) + ah(v(xi)) ´ Im0h0(ri) + . . . i , P[ri = m0h0 → ri = m1h0] = am(v(xi)) Λ(v, r)

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Outlook

Conclusion Based on the combination MC model of single channels and the

• cont. modelling of the charge flow, PDPs are the appropriate description
• f the resulting stochastic process and provide a reference model.

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Outlook

Conclusion Based on the combination MC model of single channels and the

• cont. modelling of the charge flow, PDPs are the appropriate description
• f the resulting stochastic process and provide a reference model.

Work in progress: derive and analyse numerical methods to simulate PDPs use PDP model as reference model for existing hybrid algorithms, convergence and error analysis use PDP model as the starting point to derive and analyse approximations by continuous stochastic models → SDE/SPDE models Further extensions: allow for SDEs/SPDEs instead of the det. inter-jump behaviour

• ther applications, e.g., calcium dynamics

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The Modelling Problem fin.-dim. PDPs infin.-dim. PDPs Outlook

Thank you for your attention!

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