Multiscale analysis of hybrid processes and reduction of stochastic neuron models. Gilles Wainrib joint work with: Khashayar Pakdaman and Mich` ele Thieullen Institut J.Monod- CNRS,Univ.Paris 6,Paris 7 - Labo. Proba et Mod` eles Al´ eatoires Univ.Paris 6,Paris 7,CNRS CREA Ecole polytechnique January, 2010
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Multiscale analysis of hybrid processes and reduction of stochastic neuron models. · Multiscale analysis of hybrid processes and reduction of stochastic neuron models. Gilles Wainrib
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Multiscale analysis of hybrid processes and reduction ofstochastic neuron models.
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