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;
important is their distribution
close to the membrane and their flux
across the membrane; flux rate is
103 − 106 ions/ms per open channel;
one single channel consists of hundreds of amino acids, hence
channel ion and possibly only
a few channels involved ∼ low
channel density or small fibres;
behaviour of transients
over length scales of 1µm
to 1m; fitting of a deterministic model to the average
emergent behaviour;
continuous
approximation
discrete
modelling
microscopic description
discrete
modelling
hybrid stochastic models
discrete particle
system
macroscopic
continuous
modelling
continuous
noise
approximation
continuous stochastic models
deterministic limit
macroscopic model
ODE/PDE
models
SDE/SPDE
models
"adding" noise
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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) N a-channel
→ waiting time distr., e.g.
P[τ > t] = e−(am (v)+ah (v))t
m 0 h0
am (v)
bm (v)
ah (v) bh (v)
m 0 h1
m1 h0
ah (v) bh (v)
am (v)
bm (v)
→ post-jump value distr., e.g.
am (v)
,
am (v) + ah (v)
ah (v)
=
.
am (v) + ah (v)
p m 1 h0 =
m1 h1
p m 0 h1
→ 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) N a-channel
→ waiting time distr., e.g.
P[τ > t] = e−(am (v)+ah (v))t
m 0 h0
am (v)
bm (v)
ah (v) bh (v)
m 0 h1
→ post-jump value distr., e.g.
m1 h0
ah (v) bh (v)
am (v)
bm (v)
am (v)
,
am (v) + ah (v)
ah (v)
=
.
am (v) + ah (v)
p m 1 h0 =
m1 h1
p m 0 h1
→ transmembrane potential constant over time
(Space-clamped) membrane model – ordinary differential equation
C v̇ =
m
X
gi (v) (Ei − v)
i=1
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 7→ v(s) is, e.g.,
Example cont.
P[τ > t] = e−
Rt
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 7→ v(s) is, e.g.,
Example cont.
P[τ > t] = e−
Rt
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
càdlàg 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
ẋ = gr (x)
with solutions t 7→ 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 < . . .
of r(t) with τk < T
“ Z t `
´ ”
P[τk+1 − T > t|r(T ) = r, x(T ) = x] = exp −
λ φr (s; x), r ds
0
d
transition measure Q : R × R → P(R) for post-jump values
P[r(τk+1 ) = r̄|r(τk+1 −) = r, x(T ) = x] = Q(φr (τk+1 − T ; x), r)({r̄})
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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 ẋ = gr (x) and,
simultaneously, to obtain the next jump time
Z t
− log U =
λ φr (s; x), r ds,
0
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)
Z
Af (x, r) = fx (x, r)gr (x) + λ(x, r) [f (x, p) − f (x, r)]Q(x, r)(dp)
R
→ 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
m 0 h0
am (v)
bm (v)
ah (v) bh (v)
m 0 h1
m1 h0
ah (v) bh (v)
am (v)
bm (v)
m1 h1
v̇ = ḡN a rm1 h1 (EN a − v) − gL v
|
{z
}
=gr (v)
r = (rm0 h0 , rm1 h0 , rm0 h1 , rm1 h1 ),
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
m 0 h0
am (v)
bm (v)
ah (v) bh (v)
m 0 h1
m1 h0
v̇ = ḡN a rm1 h1 (EN a − v) − gL v
|
{z
}
=gr (v)
ah (v) bh (v)
am (v)
bm (v)
m1 h1
r = (rm0 h0 , rm1 h0 , rm0 h1 , rm1 h1 ),
ri = # channels in state i
n
o
X
ri = N
R = r ∈ R4 : ri ∈ {0 : 1 : N } and
i
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)] = rm0 h0
am (v)
λ(v, r)
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The Modelling Problem
fin.-dim. PDPs
monophasic stimulus
# channels =100
150
150
100
100
100
50
50
50
0
0.5
1
0
0
0.5
1
0
150
150
150
100
100
100
50
50
50
0
0
0.5
1
0
0
0.5
Outlook
# channels =1000
# channels = 500
150
0
preconditioned stimulus
infin.-dim. PDPs
1
0
0
0.5
1
0
0.5
1
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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 ∗ ;
càdlàg 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
u̇ = Lr u + Gr (u)
◦ Lr : E → E ∗ linear operators
◦ Gr : E → E ∗ nonlinear mappings
with solutions t 7→ 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 (Fréchet-) derivative of f ∈ D(A) w.r.t. the u
component exists, then the extended generator of y is given by
Z
Af (u, r) = hLr u+Gr (u), fu (u, r)i+Λr (u) [f (u, p)−f (u, r)]Qr (dp; u)
R
with h·, ·i duality pairing of E and fu the unique element of E s.t.
df
(x, r) ◦ y = (y, fx (x, r))E
dx
where
df
du (u, r)
∀y∈E
∈ E ∗ Fréchet 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 h·, ·i
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The Modelling Problem
fin.-dim. PDPs
infin.-dim. PDPs
Outlook
Example cont.
Ω = [a, b] ⊂ R, E = H01 (Ω), H = L2 (Ω) and
C
N
∂v(t, x)
∂t
=
d ∂ 2 v(t, x) X i
+
gN a (x; r) (EN a − v(t, x))
4R ∂x2
i=1
v(·, x)
=
0, for x = a, b
with
i
−1
gN
a (x; r) = ḡN a I{m1 h1 } (ri ) δxi (x) ∈ H
(= E ∗ )
⇒ ∀ v(0) ∈ H01 (Ω) ∃! (weak) solution in C([0, T ], H01 (Ω)).
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The Modelling Problem
fin.-dim. PDPs
infin.-dim. PDPs
Outlook
Example cont.
Ω = [a, b] ⊂ R, E = H01 (Ω), H = L2 (Ω) and
C
N
∂v(t, x)
∂t
=
d ∂ 2 v(t, x) X i
+
gN a (x; r) (EN a − v(t, x))
4R ∂x2
i=1
v(·, x)
=
0, for x = a, b
with
i
−1
gN
a (x; r) = ḡN a I{m1 h1 } (ri ) δxi (x) ∈ H
(= E ∗ )
⇒ ∀ v(0) ∈ H01 (Ω) ∃! (weak) solution in C([0, T ], H01 (Ω)).
R = {m0 h0 , m1 h0 , m0 h1 , m1 h1 }N ,
N h
i
X
`
´
am (v(xi )) + ah (v(xi )) Im0 h0 (ri ) + . . . ,
Λ(v, r) =
i
P[ri = m0 h0 → ri = m1 h0 ] =
am (v(xi ))
Λ(v, r)
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The Modelling Problem
fin.-dim. PDPs
infin.-dim. PDPs
Outlook
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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
of 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
of 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
other 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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