Modelization of membrane potentials and information transmission in - PowerPoint PPT Presentation

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, Modelization of membrane potentials and information transmission in large systems of neurons Reinhard H¨ opfner Johannes Gutenberg Universit¨ at Mainz www.mathematik.uni-mainz.de/ ∼ hoepfner Marseille 2010

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, introduction 1 membrane potential as a (jump) diffusion process 2 Poisson spike trains 3 information transmission in large systems of neurons 4 theorem proof interpretation statistical inference, model verification 5 comments on level 10 in example 2 comments on example 1 references 6

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, introduction example 1: membrane potential in a pyramidal neuron emitting spikes 17Sept08_023.asc 0 −10 −20 [mV] −30 −40 −50 0 50 100 150 [sec] data: Kilb and Luhmann, Institute of Physiology, Mainz (in: Jahn 09)

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, example 2: pyramidal neuron under different experimental conditions network activity stimulated by increasing concentration of potassium (K) 20 1 0 [mV] −20 −40 −60 0 10 20 30 40 50 60 [sec] data: Kilb and Luhmann, Institute of Physiology, Mainz (in: H¨ opfner 07)

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, spikes are generated when the membran potential V t in the soma is high enough

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, view the membrane potential between successive spikes as a stochastic process of (jump) diffusion type : synapses ֒ → dendrites ֒ → soma: additivity and exponential decay one neuron has O (10 4 ) synapses, ≈ 90% excitatory, ≈ 10% inhibitory contribution of incoming spikes to the membrane potential : left: single exciting synapsis; middle: single inhibitory synapsis; right: 2 exciting and 1 inhibitory synapses combined

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, example 3: spike trains recorded in the visual cortex 210 iid experiments ← ֓ identical visual stimulus (Shadlen-Newsome 98) hence: view the spike train µ emitted by one neuron as a random point measure on [0 , ∞ ) with stochastic intensity such that mean value of intensity at time t corresponds to stimulus at time t

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, jump diffusion process modelization for the membrane potential between successive spikes many models are time homogeneous, e.g. mean-reverting Ornstein-Uhlenbeck (Lansky-Lanska 87, Tuckwell 89, Lansky-Sato 99, Lansky-Sacerdote 01, Ditlevsen-Lansky 05, ...) or Cox-Ingersoll-Ross (Lansky-Lanska 87, Giorny-Lansky-Nobile-Ricciardi 88, Lansky-Sacerdote-Tomassetti 95, Ditlevsen-Lansky 06, Brodda-H¨ opfner 06, ...) stage 1 (time homogeneous and stationary) : CIR type model ( V t ) t ≥ 0 for a neuron belonging to an active neuronal network p ( V t − K 0 ) + √ τ dW t dV t = ([ K R + f ] − V t ) τ dt + σ with constants σ, τ > 0, reference levels K 0 < K R < K E K 0 : lower bound for possible values of the membrane potential K R : mean value of membrane potential for neuron ’at rest’ K E : excitation threshold and some quantity measuring the degree of activity of the network f ≥ 0 : constant representing strength of external stimulus

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, well known: shifting membrane potential V by K 0 , process ( V t − K 0 ) t ≥ 0 ` ´ 2 2 is ergodic with invariant law Γ σ 2 ( K R − K 0 + f ) , on [0 , ∞ ) σ 2 has (stationary) mean K R − K 0 + f and variance σ 2 2 ( K R − K 0 + f ) not depending on the time constant τ (Cox-Ingersoll-Ross 85, Ikeda-Watanabe 89, ...) time homogeneous CIR model gives convincing fit for the membrane potential data of example 1 (new electronic stabilization device was used by Kilb) (Jahn 09) reasonable fit for some of the membrane potential data in example 2 (at least in levels 8,9,10 where neuron is able to generate spikes) (H¨ opfner 07) but in many data sets which seem CIR compatible evidence for time dependence concerning term f ≥ 0 in the drift some indication for presence of jumps open question; PRO: biological reasons; CONTRA: sophisticated semimartingale tools (Jacod 09, AitSahalia-Jacod 09) do not work as as they should for sure, neurons can behave differently: OU, other types of diffusions, no diffusion at all, ..., but: CIR seems suitable for slowly spiking neurons belonging to an active network

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, more realistic : between successive spikes use deterministic fct t → f ( t ) of time (strength of external stimulus) introduce Poisson jumps, positive and summable : PRM µ ( dt , dy ) on (0 , M ) × (0 , ∞ ) with intensity τ e f ( t ) dt ν ( dy ) independent of BM W , for some deterministic function t → e f ( t ) and R some σ -finite measure ν ( dy ) on (0 , M ) such that (0 , M ) y ν ( dy ) < ∞ stage 2 : time inhomogeneous model with jumps : Z p ( V t − K 0 ) + √ τ dW t dV t = ([ K R + f ( t )] − V t ) τ dt + y µ ( dt , dy ) + σ | {z } ֓ e ← f ( t ) pathwise uniqueness, unique strong solution (Yamada-Watanabe 71, Dawson-Li 06, Fu-Li 08, ...) explicit Laplace transforms for transition probabilities (Kawazu-Watanabe 71, ...)

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, proposition : shifting the membrane potential by K 0 , the process ( V t − K 0 ) t ≥ 0 has explicit LT “ ” e − λ ( V t − K 0 ) | ( V s − K 0 ) = x λ − → E for transition probabilities, of form „ Z t « n o [ K R − K 0 + f ( v )] Ψ v , t ( λ ) + e f ( v ) e exp − x Ψ s , t ( λ ) − Ψ v , t ( λ ) τ dv s Z e − τ ( t − v ) λ e [1 − e − y Ψ v , t ( λ ) ] ν ( dy ) Ψ v , t ( λ ) = Ψ v , t ( λ ) = , 1 + λ σ 2 2 (1 − e − τ ( t − v ) ) LT analogous to results of Kawazu-Watanabe for time-homogeneous case (Kawazu-Watanabe 71, H¨ opfner 09)

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, remarks : a) special case where f ( · ) ≡ f , e f ( · ) ≡ e f are constant: „ « „ « − 2 Z t 1 + λ σ 2 [1 − e − τ ( t − s ) ] σ 2 λ − → exp − Ψ v , t ( λ ) τ dv = 2 s “ ” 2 2 is LT of a Gamma law Γ σ 2 , , and the law with LT σ 2 [1 − e − τ ( t − s ) ] „ « Z t n o [ K R − K 0 + f ] Ψ v , t ( λ ) + e f e λ − → exp − Ψ v , t ( λ ) τ dv −∞ (independent of t and τ ) is invariant for the process ( V t − K 0 ) t ≥ 0 b) special case where f ( · ), e f ( · ) are T -periodic functions: have a T -periodic semigroup, an invariant probability on the canonical space C [0 , T ] for T -segments in the path of ( V t − K 0 ) t ≥ 0 , and thus limit theorems for a large class of functionals of the process ( V t − K 0 ) t ≥ 0 (H¨ opfner-Kutoyants 09)

introduction membrane potential as a (jump) diffusion process Poisson spike trains information transmission in large systems of neurons statistical inference, spike trains in the single neuron as point process with random intensity consider a single neuron whose membrane potential is a stochastic process V = ( V t ) t driven by ( W , µ ) definition : a Poisson spike train is a point process µ indep. of ( W , µ ) s.t. µ ( ds ) is Poisson random measure on (0 , ∞ ) with intensity λ 1 [ K E , ∞ ) ( V s ) ds for some λ > 0 and some excitation threshold K E > K R (mentioned but not used above) remark : ’excitation threshold’ is not understood in the usual sense of a fixed threshold for a first hitting time problem, but defined here as critical level spikes occur at rate λ > 0 on the random set { t > 0 : V t ≥ K E } toy model since neglects return of membrane potential after spike to some ’restart region’ neglects duration and shape of the spikes, neglects ’refractory period’ but captures one evidence from data: spikes are not first hitting times to some fixed+deterministic threshold