multiscale analysis of hybrid processes and reduction of stochastic neuron models. · multiscale...
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Multiscale analysis of hybrid processes and reduction ofstochastic neuron models.
Gilles Wainribjoint work with:
Khashayar Pakdaman and Michele Thieullen
Institut J.Monod- CNRS,Univ.Paris 6,Paris 7 - Labo. Proba et Modeles Aleatoires Univ.Paris 6,Paris 7,CNRSCREA Ecole polytechnique
January, 2010
Part I : Introduction
Deterministic neuron model
Hodgkin Huxley (HH) model (Hodgkin Huxley - J.Physiol. 1952):
CmdV
dt= I − gL(V − VL)− gNam3h(V − VNa)− gK n4(V − VK )
dm
dt= τm(V )−1 (m∞(V )−m)
dh
dt= τh(V )−1 (h∞(V )− h)
dn
dt= τn(V )−1 (n∞(V )− n)
→ Conductance-based neuron model
Time-scale separation and reduction
Sodium activation dynamic is faster than the other variables : τm → 0
m = m∞(V )
Three-dimensional reduced system:
CmdV
dt= I − gL(V − VL)− gNam∞(V)3h(V − VNa)− gK n4(V − VK )
dh
dt= τh(V ) (h∞(V )− h)
dn
dt= τn(V ) (n∞(V )− n)
Reduction of neuron models : key step in theoretical (singular perturbations) andnumerical analysisRinzel 1985, Kepler et al. 1992, Meunier 1992, Suckley et al.2003, Rubin et al. 2007,...
Time-scale separation and reduction
Sodium activation dynamic is faster than the other variables : τm → 0
m = m∞(V )
Three-dimensional reduced system:
CmdV
dt= I − gL(V − VL)− gNam∞(V)3h(V − VNa)− gK n4(V − VK )
dh
dt= τh(V ) (h∞(V )− h)
dn
dt= τn(V ) (n∞(V )− n)
Reduction of neuron models : key step in theoretical (singular perturbations) andnumerical analysisRinzel 1985, Kepler et al. 1992, Meunier 1992, Suckley et al.2003, Rubin et al. 2007,...
Modelling neurons with stochastic ion channels
Single ion channels stochasticity:
• Macromolecular devices : open and close through voltage-inducedconformational changes
Potassium channel
• Stochasticity due to thermal noise
Channel noise : finite size effects responsible for intrinsic variability noise-inducedphenomena (spontaneous activity, signal detection enhancement,...)
Modelling neurons with stochastic ion channels
Single ion channels stochasticity:
• Macromolecular devices : open and close through voltage-inducedconformational changes
Potassium channel
• Stochasticity due to thermal noise
Channel noise : finite size effects responsible for intrinsic variability noise-inducedphenomena (spontaneous activity, signal detection enhancement,...)
Modelling neurons with stochastic ion channels
Deterministic model X = (V , u)
dV
dt= F (V , u)
du
dt= (1− u)α(V )− uβ(V ) = τu(V )(u∞(V )− u)
Modelling neurons with stochastic ion channels
Stochastic model XN = (VN , uN )
• Single ion channel i ∈ {1, ...,N} with voltage-dependent transition rates :independent jump Markov process ci (t)
• Proportion of open ion channels (empirical measure) ::
uN (t) =1
N
NXi=1
ci (t)
• Between the jumps, voltage dynamics:
dVN
dt= F (VN , uN )
Modelling neurons with stochastic ion channels
Stochastic model XN = (VN , uN )
• Single ion channel i ∈ {1, ...,N} with voltage-dependent transition rates :independent jump Markov process ci (t)
• Proportion of open ion channels (empirical measure) ::
uN (t) =1
N
NXi=1
ci (t)
• Between the jumps, voltage dynamics:
dVN
dt= F (VN , uN )
Modelling neurons with stochastic ion channels
Stochastic model XN = (VN , uN )
• Single ion channel i ∈ {1, ...,N} with voltage-dependent transition rates :independent jump Markov process ci (t)
• Proportion of open ion channels (empirical measure) ::
uN (t) =1
N
NXi=1
ci (t)
• Between the jumps, voltage dynamics:
dVN
dt= F (VN , uN )
Modelling neurons with stochastic ion channels
Stochastic model XN = (VN , uN )
• Single ion channel i ∈ {1, ...,N} with voltage-dependent transition rates :independent jump Markov process ci (t)
• Proportion of open ion channels (empirical measure) ::
uN (t) =1
N
NXi=1
ci (t)
• Between the jumps, voltage dynamics:
dVN
dt= F (VN , uN )
Modelling neurons with stochastic ion channels
• Modelling framework:Neuron ⇐⇒ population of globally coupled independent ion channels
• Mathematical framework:Piecewise-deterministic Markov process at the fluid limit
⇓ ⇓(Davis, 1984) (Kurtz, 1971)
Modelling neurons with stochastic ion channels
• Modelling framework:Neuron ⇐⇒ population of globally coupled independent ion channels
• Mathematical framework:Piecewise-deterministic Markov process at the fluid limit
⇓ ⇓(Davis, 1984) (Kurtz, 1971)
Modelling neurons with stochastic ion channels
• Modelling framework:Neuron ⇐⇒ population of globally coupled independent ion channels
• Mathematical framework:Piecewise-deterministic Markov process at the fluid limit
⇓ ⇓(Davis, 1984) (Kurtz, 1971)
Modelling neurons with stochastic ion channels
• Modelling framework:Neuron ⇐⇒ population of globally coupled independent ion channels
• Mathematical framework:Piecewise-deterministic Markov process at the fluid limit
⇓
⇓
(Davis, 1984)
(Kurtz, 1971)
Modelling neurons with stochastic ion channels
• Modelling framework:Neuron ⇐⇒ population of globally coupled independent ion channels
• Mathematical framework:Piecewise-deterministic Markov process at the fluid limit
⇓ ⇓(Davis, 1984) (Kurtz, 1971)
Limit Theorems : Law of large numbers
Theorem When N →∞, XN converges to X in probability over finite time intervals[0,T ]
For ∆ > 0, define
PN (T ,∆) := P
"sup
t∈[0,T ]|XN (t)− X (t)|2 > ∆
#
Thenlim
N→∞PN (T ,∆) = 0
More precisely, there exists constants B,C > 0 such that:
lim supN→∞
1
Nlog PN (T ,∆) ≤ −
∆e−BT 2
CT
Pakdaman, Thieullen, W. ”Fluid limit theorems for stochastic hybrid systems withapplication to neuron models” (2009) arXiv:1001.2474
Limit Theorems : Law of large numbers
Theorem When N →∞, XN converges to X in probability over finite time intervals[0,T ]For ∆ > 0, define
PN (T ,∆) := P
"sup
t∈[0,T ]|XN (t)− X (t)|2 > ∆
#
Thenlim
N→∞PN (T ,∆) = 0
More precisely, there exists constants B,C > 0 such that:
lim supN→∞
1
Nlog PN (T ,∆) ≤ −
∆e−BT 2
CT
Pakdaman, Thieullen, W. ”Fluid limit theorems for stochastic hybrid systems withapplication to neuron models” (2009) arXiv:1001.2474
Limit Theorems : Law of large numbers
Theorem When N →∞, XN converges to X in probability over finite time intervals[0,T ]For ∆ > 0, define
PN (T ,∆) := P
"sup
t∈[0,T ]|XN (t)− X (t)|2 > ∆
#
Thenlim
N→∞PN (T ,∆) = 0
More precisely, there exists constants B,C > 0 such that:
lim supN→∞
1
Nlog PN (T ,∆) ≤ −
∆e−BT 2
CT
Pakdaman, Thieullen, W. ”Fluid limit theorems for stochastic hybrid systems withapplication to neuron models” (2009) arXiv:1001.2474
Limit Theorems : Law of large numbers
Theorem When N →∞, XN converges to X in probability over finite time intervals[0,T ]For ∆ > 0, define
PN (T ,∆) := P
"sup
t∈[0,T ]|XN (t)− X (t)|2 > ∆
#
Thenlim
N→∞PN (T ,∆) = 0
More precisely, there exists constants B,C > 0 such that:
lim supN→∞
1
Nlog PN (T ,∆) ≤ −
∆e−BT 2
CT
Pakdaman, Thieullen, W. ”Fluid limit theorems for stochastic hybrid systems withapplication to neuron models” (2009) arXiv:1001.2474
Limit Theorems : Central limit
Theorem:Let
RN (t) :=√
N
„XN (t)−
Z t
0F (XN (s))ds
«When N →∞, RN converges in law to a diffusion process
R(t) =
Z t
0Σ(X (s))dWs
Langevin Approximation XN = (VN (t), uN (t)):
dVN (t) = F (VN (t), uN (t))dt
duN (t) = b(VN (t), uN (t))dt +1√
NΣ(VN (t), uN (t))dWs
Further developments : strong approximation (pathwise CLT), Markov vs. Langevin,large deviations
Limit Theorems : Central limit
Theorem:Let
RN (t) :=√
N
„XN (t)−
Z t
0F (XN (s))ds
«When N →∞, RN converges in law to a diffusion process
R(t) =
Z t
0Σ(X (s))dWs
Langevin Approximation XN = (VN (t), uN (t)):
dVN (t) = F (VN (t), uN (t))dt
duN (t) = b(VN (t), uN (t))dt +1√
NΣ(VN (t), uN (t))dWs
Further developments : strong approximation (pathwise CLT), Markov vs. Langevin,large deviations
Limit Theorems : Central limit
Theorem:Let
RN (t) :=√
N
„XN (t)−
Z t
0F (XN (s))ds
«When N →∞, RN converges in law to a diffusion process
R(t) =
Z t
0Σ(X (s))dWs
Langevin Approximation XN = (VN (t), uN (t)):
dVN (t) = F (VN (t), uN (t))dt
duN (t) = b(VN (t), uN (t))dt +1√
NΣ(VN (t), uN (t))dWs
Further developments : strong approximation (pathwise CLT), Markov vs. Langevin,large deviations
Stochastic reduction ?
Part II : Mathematical analysis
Singular perturbations for jump Markov processes
Figure: Multiscale four-state model. Horizontal transitions are fast, whereas vertical transitions areslow.
Singular perturbations for jump Markov processes
Singular perturbations for jump Markov processes
Singular perturbations for jump Markov processes : general setting
Yin, Zhang, ”Continuous-time Markov Chains and Applications : a singularperturbation approach”, 1998
Assumption There exist n subsets of fast transitions.
E = E1 ∪ E2 ∪ ... ∪ En
• if i , j ∈ Ek then αi,j is of order O(ε−1),
• otherwise, if i ∈ Ek and j ∈ El , with k 6= l then αi,j is of order O(1).
Singular perturbations for jump Markov processes : general setting
Constructing a reduced process:
• quasi-stationary distributions (ρki )i∈Ek
within fast subsets Ek , for k ∈ {1, ..., n}.• aggregated process (X ) on the state space E = {1, ..., n} with transition rates:
αk,l =Xi∈Ek
Xj∈El
ρikαi,j for k, l ∈ E
Singular perturbations for jump Markov processes : first-order
Theorem
• all-fast case For all t > 0, the probability Pεi (t) = P [X εt = xi ] converges when
ε→ 0 to the stationary distribution ρi , for all i ∈ E.
• multiscale case As ε→ 0 the process (Xε) is close to the reduced process (X ).More precisely :
1. EhR T
0
“1{Xε(t)=xik} − ρ
ki 1{Xε=k}
”Φ(xik )dt
i2= O(ε), for any function Φ : E → R,
with k ∈ {1, ..., n} and i ∈ Ek .2. The process X ε converges in law to X .
Singular perturbations for jump Markov processes : first-order
Theorem
• all-fast case For all t > 0, the probability Pεi (t) = P [X εt = xi ] converges when
ε→ 0 to the stationary distribution ρi , for all i ∈ E.
• multiscale case As ε→ 0 the process (Xε) is close to the reduced process (X ).More precisely :
1. EhR T
0
“1{Xε(t)=xik} − ρ
ki 1{Xε=k}
”Φ(xik )dt
i2= O(ε), for any function Φ : E → R,
with k ∈ {1, ..., n} and i ∈ Ek .2. The process X ε converges in law to X .
Singular perturbations for jump Markov processes : second-order
Rescaled process
nε(t) =1√ε
Z T
0
`1{X ε(t)=xi} − ρi
´Φ(xi , s)ds
Theorem The rescaled process nε(t) converges in law to the switching diffusionprocess
n(t) =
Z t
0σ(s)dWs
where W is a standard n-dimensional Brownian motion. The diffusion matrixA = σ(s)σ′(s) is given by:
Aij (s) = Φ(xi , s)Φ(xj , s)ˆρi R(i , j) + ρj R(j , i)
˜where
R(i , j) =
Z ∞0
`Pε(i , j , t)− ρj
´dt
Multiscale analysis of stochastic neuron models
Full model : X εN = (V ε
N , uεN ) with
• uεN empirical measure for a population of multiscale jump processes
• V εN = F (V ε
N , uεN )
Requires two extensions :
1. Population of jump processes
2. Piecewise deterministic Markov process
Multiscale analysis of stochastic neuron models
Full model : X εN = (V ε
N , uεN ) with
• uεN empirical measure for a population of multiscale jump processes
• V εN = F (V ε
N , uεN )
Requires two extensions :
1. Population of jump processes
2. Piecewise deterministic Markov process
Stationnary distribution for populations of multiscale jump processes
Stationnary distributions for the empirical measure→ multinomial distributions
Ex: two-state modelρ(N)(k/N) = C k
N uk∞(1− u∞)N−k
Averaging method for PDMP
Ex (all-fast):
V εN (t) =
Z t
0F (V ε
N (s), uεN (s))ds
with uεN fast
→ FN (VN ) :=
ZF (VN , u)ρ
(N)stat (du) (ergodic convergence)
• Theorem (general case) When ε→ 0, the process (V εN , u
εN ) converges in law
towards a coarse-grained hybrid process:
dVN
dt= FN (VN , uN )
and u reduced jump process with averaged transition rates, functions of V .Faggionato, Gabrielli, Ribezzi Crivellari 2009
• Central limit theorem (ongoing work) → diffusion approximation :
dVN dt = FN (V εN , u
εN )dt +
√εσN (V ε
N , uεN )dWt
Averaging method for PDMP
Ex (all-fast):
V εN (t) =
Z t
0F (V ε
N (s), uεN (s))ds
with uεN fast
→ FN (VN ) :=
ZF (VN , u)ρ
(N)stat (du) (ergodic convergence)
• Theorem (general case) When ε→ 0, the process (V εN , u
εN ) converges in law
towards a coarse-grained hybrid process:
dVN
dt= FN (VN , uN )
and u reduced jump process with averaged transition rates, functions of V .Faggionato, Gabrielli, Ribezzi Crivellari 2009
• Central limit theorem (ongoing work) → diffusion approximation :
dVN dt = FN (V εN , u
εN )dt +
√εσN (V ε
N , uεN )dWt
Averaging method for PDMP
Ex (all-fast):
V εN (t) =
Z t
0F (V ε
N (s), uεN (s))ds
with uεN fast
→ FN (VN ) :=
ZF (VN , u)ρ
(N)stat (du) (ergodic convergence)
• Theorem (general case) When ε→ 0, the process (V εN , u
εN ) converges in law
towards a coarse-grained hybrid process:
dVN
dt= FN (VN , uN )
and u reduced jump process with averaged transition rates, functions of V .Faggionato, Gabrielli, Ribezzi Crivellari 2009
• Central limit theorem (ongoing work) → diffusion approximation :
dVN dt = FN (V εN , u
εN )dt +
√εσN (V ε
N , uεN )dWt
Averaging method for PDMP
Ex (all-fast):
V εN (t) =
Z t
0F (V ε
N (s), uεN (s))ds
with uεN fast
→ FN (VN ) :=
ZF (VN , u)ρ
(N)stat (du) (ergodic convergence)
• Theorem (general case) When ε→ 0, the process (V εN , u
εN ) converges in law
towards a coarse-grained hybrid process:
dVN
dt= FN (VN , uN )
and u reduced jump process with averaged transition rates, functions of V .Faggionato, Gabrielli, Ribezzi Crivellari 2009
• Central limit theorem (ongoing work) → diffusion approximation :
dVN dt = FN (V εN , u
εN )dt +
√εσN (V ε
N , uεN )dWt
Part III : Application to Hodgkin-Huxley model
Application Hodgkin-Huxley model : reduced model (two-state)
Averaging ”m3” with respect to the binomial stationnary distribution
ρ(N)m (k/N) = C k
N mk∞(1−m∞)N−k yields:
CmdV
dt= I − gL(V − VL)− gNam∞(V )3h(V − VNa)− gK n4(V − VK )
− gNah(V − VNa)KN(V) (supplementary terms)
with
KN (V ) =3
Nm∞(V )2(1−m∞(V )) +
1
N2m∞(V )(1 + 2m∞(V )2)
Important remark : Noise strength η := 1N
appears as a bifurcation parameter.
Application Hodgkin-Huxley model : bifurcations of the reduced model
Figure: Bifurcation diagram with η as parameter for I = 0 of system (HHNTS ).
Application Hodgkin-Huxley model : bifurcations of the reduced model
Figure: Two-parameter bifurcation diagram of system (HHNTS ) with I and η as parameters.
Application Hodgkin-Huxley model : bifurcations of the reduced model
1. Below the double cycle curve is a region with a unique stable equilibrium point :ISI distribution should be approximately exponential, since a spike corresponds toa threshold crossing.
2. Between the double cycle and the Hopf curves is a bistable region : ISIdistribution should be bimodal, one peak corresponding to the escape from thestable equilibrium, and the other peak to the fluctuations around the limit cycle.
3. Above the Hopf curve is a region with a stable limit cycle and an unstableequilibrium point : ISI distribution should be centered around the period of thelimit cycle.
Application Hodgkin-Huxley model : bifurcations of the reduced model
1. Below the double cycle curve is a region with a unique stable equilibrium point :ISI distribution should be approximately exponential, since a spike corresponds toa threshold crossing.
2. Between the double cycle and the Hopf curves is a bistable region : ISIdistribution should be bimodal, one peak corresponding to the escape from thestable equilibrium, and the other peak to the fluctuations around the limit cycle.
3. Above the Hopf curve is a region with a stable limit cycle and an unstableequilibrium point : ISI distribution should be centered around the period of thelimit cycle.
Application Hodgkin-Huxley model : bifurcations of the reduced model
1. Below the double cycle curve is a region with a unique stable equilibrium point :ISI distribution should be approximately exponential, since a spike corresponds toa threshold crossing.
2. Between the double cycle and the Hopf curves is a bistable region : ISIdistribution should be bimodal, one peak corresponding to the escape from thestable equilibrium, and the other peak to the fluctuations around the limit cycle.
3. Above the Hopf curve is a region with a stable limit cycle and an unstableequilibrium point : ISI distribution should be centered around the period of thelimit cycle.
Application Hodgkin-Huxley model : stochastic simulations
Figure: A. With N = 30 (zone 3), noisy periodic trajectory. B. With N = 70 (zone 2), bimodalityof ISI’s C. With N = 120, ISI statistics are closer to a poissonian behavior.
Application Hodgkin-Huxley model : stochastic simulations
Figure: Interspike Interval (ISI) distributions
Conclusions and perspectives
• Systematic method for reducing a large class of stochastic neuron models
• Based on recent mathematical developments of the averaging method
• Illustration on HH : enables a bifurcation analysis with noise strength asparameter
• Other applications in neuroscience (synaptic models, networks, biochemicalreactions)
• Open mathematical questions (link with stochastic bifurcations, scaling in thedouble limit N →∞, ε→ 0)
Conclusions and perspectives
• Systematic method for reducing a large class of stochastic neuron models
• Based on recent mathematical developments of the averaging method
• Illustration on HH : enables a bifurcation analysis with noise strength asparameter
• Other applications in neuroscience (synaptic models, networks, biochemicalreactions)
• Open mathematical questions (link with stochastic bifurcations, scaling in thedouble limit N →∞, ε→ 0)
Singular perturbations for jump Markov processes : heuristics
Law evolution :dPε
dt=
„Qs (t) +
1
εQf (t)
«Pε
with initial condition Pε(0) = p0. We are looking for an expansion of Pε(t) of theform
Pεr (t) =rX
i=0
εiφi (t) +rX
i=0
εiψi (t
ε)
Singular perturbations for jump Markov processes : heuristics
Identifying power of ε:
Qf (t)φ0(t) = 0
Qf (t)φ1(t) =dφ0(t)
dt− φ0(t)Qs (t)
...
Qf (t)φi (t) =dφi−1(t)
dt− φi−1(t)Qs (t)
Error control:
1. |Pε(t)− Pεr (t)| = O(εr+1) uniformly in t ∈ [0,T ]
2. there exist K , k0 > 0 such that |ψi (t)| < Ke−k0t
Multiscale analysis of stochastic neuron models : summary
Second order approximation for PDMP
Central limit theorem
1√ε
„V ε
t −Z t
0F (V ε
s )ds
«→Z t
0σF (Vs )dWs