Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Modelling Membrane Potentials by Diffusion Leaky Integrate-and-Fire Models Patrick Jahn DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF COPENHAGEN jahn@math.ku.dk CIRM, Marseille - January 18, 2009 Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Contents 1 Diffusion Leaky Integrate-and-Fire Neuronal Models 2 Modelling Membrane Potentials in Motoneurons by time-inhomogeneous Diffusion Models Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Biological Background Figure: The Neuron Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Membrane Potential recorded by W. Kilb, Mainz −20 potential [mV] −30 −40 −50 0 1 2 3 4 5 6 time [s] Figure: This membrane potential was recorded in vitro from a pyramidal neuron belonging to cortical slice preparation of a juvenile C57bl/6 mouse. Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Diffusion Leaky Integrate-and-Fire Neuronal Model S x 0 . . . � �� � � �� � � �� � T 1 T 2 T n Assume the process X between spikes is a Diffusion process given by d X t = 1 τ ( a − X t ) d t + σ ( X t ) d B t Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Diffusion Leaky Integrate-and-Fire Neuronal Model S x 0 . . . � �� � � �� � � �� � T 1 T 2 T n Assume the process X between spikes is a Diffusion process given by d X t = 1 τ ( a − X t ) d t + σ ( X t ) d B t Goal: Estimate β ( · ) := 1 τ ( a − · ), σ ( · ), x 0 , S from discrete observations of X and from the observation of iid ISIs (level crossing times) T i , i = 1 , . . . , n . . Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Diffusion Leaky Integrate-and-Fire Neuronal Model S x 0 . . . � �� � � �� � � �� � T 1 T 2 T n Assume the process X between spikes is a Diffusion process given by d X t = 1 τ ( a − X t ) d t + σ ( X t ) d B t Goal: Estimate β ( · ) := 1 τ ( a − · ), σ ( · ), x 0 , S from discrete observations of X and from the observation of iid ISIs (level crossing times) T i , i = 1 , . . . , n . . Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Problem of Finding Excitation Threshold and Reset Value −46 −47 −48 potential [mV] −49 S ? −50 −51 x 0 ? −52 1.0 1.5 2.0 2.5 3.0 3.5 time [s] Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Problem of Finding Excitation Threshold and Reset Value 17Sept08_023.asc alle spikes uebereinanderlegen / spikezeiten auf 0 transformieren 0 −10 −20 [mV] −30 −40 −50 −0.04 −0.02 0.00 0.02 0.04 17Sept08_023.asc Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Strategy θ 2 θ 1 � �� � � �� � . . . � �� � T 1 T 2 T n 1. Fixing drift and diffusion coefficient with nonparametric methods proposed by R. H¨ opfner (2006). So assume X is given by d X t = 1 τ ( a − X t ) d t + σ ( X t ) d B t Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Strategy θ 2 θ 1 � �� � � �� � . . . � �� � T 1 T 2 T n 1. Fixing drift and diffusion coefficient with nonparametric methods proposed by R. H¨ opfner (2006). So assume X is given by d X t = 1 τ ( a − X t ) d t + σ ( X t ) d B t 2. Estimate θ 1 = x 0 and θ 2 = S from the observation of iid inter spike times � � t ≥ 0 | X ( θ 1 ) T i := inf = θ 2 , i = 1 , . . . , n . t Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Strategy θ 2 θ 1 � �� � � �� � . . . � �� � T 1 T 2 T n 1. Fixing drift and diffusion coefficient with nonparametric methods proposed by R. H¨ opfner (2006). So assume X is given by d X t = 1 τ ( a − X t ) d t + σ ( X t ) d B t 2. Estimate θ 1 = x 0 and θ 2 = S from the observation of iid inter spike times � � t ≥ 0 | X ( θ 1 ) T i := inf = θ 2 , i = 1 , . . . , n . t Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Some Facts. . . In the most famous cases, where X is an Ornstein-Uhlenbeck (OU) or a Cox-Ingersoll-Ross (CIR) process, no explicit expression for the density of the level crossing time T is known. However, we know its Laplace transfrom (LT) as a ratio of special functions. (Jahn, PhD-Thesis 2009): For the OU and CIR cases the estimation problem of θ 1 = x 0 and θ 2 = S was solved by using Millar’s framework (1984) for minimum distance estimators via the comparison of empirical and theoretical LT with respect to a suitable Hilbert space norm . . . . Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models Some Facts. . . In the most famous cases, where X is an Ornstein-Uhlenbeck (OU) or a Cox-Ingersoll-Ross (CIR) process, no explicit expression for the density of the level crossing time T is known. However, we know its Laplace transfrom (LT) as a ratio of special functions. (Jahn, PhD-Thesis 2009): For the OU and CIR cases the estimation problem of θ 1 = x 0 and θ 2 = S was solved by using Millar’s framework (1984) for minimum distance estimators via the comparison of empirical and theoretical LT with respect to a suitable Hilbert space norm . . . . Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models The Minimum Distance Estimator w.r.t. the LT Theorem (J. 2009) Let X be an OU or a CIR process. Define the empirical LT of the level crossing time T of X from θ 1 to θ 2 and the corresponding family of possibly true LTs by n � L n ( α ) := 1 ˆ e − α T i L θ ( α ) := E θ [ e − α T ] , and n i =1 then the MDE w.r.t. the LT θ 1 <θ 2 � ˆ θ ∗ n := arg inf L n − L θ � H is strongly consistent and asymptotically normal . Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models The Pearson Diffusion Case (work with Jesper L. Pedersen. . . ) Pearson Diffusion � ( X t − c ) 2 + d d B t , d X t = 1 τ ( a − X t ) d t + σ X 0 = θ 1 ≥ 0 Jesper computed the Laplace transform of T : E θ [ e − α T ] = g Re ( θ 1 , α ) − K ( α ) · h Re ( θ 1 , α ) g Re ( θ 2 , α ) − K ( α ) · h Re ( θ 2 , α ) where g , h and K are expressions of hypergemetric functions where the parameters of the diffusion X enter. Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models The Pearson Diffusion Case (work with Jesper L. Pedersen. . . ) Pearson Diffusion � ( X t − c ) 2 + d d B t , d X t = 1 τ ( a − X t ) d t + σ X 0 = θ 1 ≥ 0 Jesper computed the Laplace transform of T : E θ [ e − α T ] = g Re ( θ 1 , α ) − K ( α ) · h Re ( θ 1 , α ) g Re ( θ 2 , α ) − K ( α ) · h Re ( θ 2 , α ) where g , h and K are expressions of hypergemetric functions where the parameters of the diffusion X enter. Goal: Solve the identifiability problem and apply the developed Framework to this case... Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
Diffusion Leaky Integrate-and-Fire Neuronal Models Modelling by time-inhomogeneous Diffusion Models The Pearson Diffusion Case (work with Jesper L. Pedersen. . . ) Pearson Diffusion � ( X t − c ) 2 + d d B t , d X t = 1 τ ( a − X t ) d t + σ X 0 = θ 1 ≥ 0 Jesper computed the Laplace transform of T : E θ [ e − α T ] = g Re ( θ 1 , α ) − K ( α ) · h Re ( θ 1 , α ) g Re ( θ 2 , α ) − K ( α ) · h Re ( θ 2 , α ) where g , h and K are expressions of hypergemetric functions where the parameters of the diffusion X enter. Goal: Solve the identifiability problem and apply the developed Framework to this case... Patrick Jahn - jahn@math.ku.dk Leaky Integrate-and-Fire Models
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