Stochastic Nerve Axon Equations Wilhelm Stannat Institut f ur - PowerPoint PPT Presentation

Stochastic Nerve Axon Equations Wilhelm Stannat Institut f¨ ur Mathematik, Fakult¨ at II TU Berlin Bernstein Center for Computational Neuroscience Berlin Linz, December 13, 2016

Stochastic processes in Neuroscience ◮ Modelling impact of noise in neural systems on all scales - microscopic - e.g., in ion channel dynamics - mezoscopic - on observables in single neurons, e.g. membrane potential - macroscopic - e.g., in neural populations ◮ Analysis - mathematical framework for continuum limits (bridging scales) - multiscale analysis, w.r.t. coherent structures - model reduction, w.r.t. observables, mean field theories ◮ Numerical approximation - strong and weak approximation errors - robust estimation

Hodgkin-Huxley Equations (1952) math. description for the generation of Action Potentials (AP) τ∂ t v = λ 2 ∂ 2 xx v − g Na m 3 h ( v − E Na ) − g K n 4 ( v − E K ) − g L ( v − E L ) + I dp dt = α p ( v )(1 − p ) − β p ( v ) p p ∈ { m , n , h } typical shape of v where ◮ v membrane potential, v = v ( t , x ), t ≥ 0, x ∈ [0 , L ] ◮ m , n , h gating variables, 0 ≤ m , n , h ≤ 1 ◮ τ resp. λ specific time resp. space constants ◮ g Na , g K , g L conductances ◮ E Na , E K , E L resting potentials p e − b 2 v + Ap p ( v + Bp ) α p ( v ) = a 1 p ( v + Ap ) , β p ( v ) = b 1 ◮ p 1 − e − a 2

Hodgkin-Huxley Equations (1952) math. description for the generation of Action Potentials (AP) τ∂ t v = λ 2 ∂ 2 xx v − g Na m 3 h ( v − E Na ) − g K n 4 ( v − E K ) − g L ( v − E L ) + I dp dt = α p ( v )(1 − p ) − β p ( v ) p , p ∈ { m , n , h } biophysiological relevant feature: ∃ I − < I + excitable I < I − oscillatory I ∈ ( I − , I + ) inhibitory I > I +

In reality more like this ... [H¨ opfner, Math. Biosciences, 2007] due to fluctuations between open and closed states of ion channels regulating v

Channel noise Illustration: measurements of single Na-channel in the giant axon of squid (considered by Hodgkin-Huxley) [Vandenberg, Bezanilla, Biophys. J., 1991]

Channel noise impact on APs ◮ spontaneous spiking (due to random openening of sufficient numbers of Na-channels) ◮ time jitter - spike time distribution increases with time ◮ APs can split up or annihilate ◮ propagation failure places limits on the axon diameter (around 0 . 1 µ m ), hence also on the wiring density e.g., [White, et al., Trends Neurosci. 2000, Faisal, et al., Current Biology 2005, Faisal, et al., PLOS 2007]

Adding noise to Hodgkin-Huxley equations Adding channel noise yields a stochastic partial differential equation: Current noise τ∂ t v = λ 2 ∂ xx v − g Na m 3 h ( v − E Na ) − g K n 4 ( v − E K ) − g L ( v − E L ) + I + σ∂ t ξ ( t , x ) dp dt = α p ( v )(1 − p ) − β p ( v ) p , p ∈ { m , n , h } (1) σ = 0 . 2 σ = 0 . 35 σ = 0 . 6

Adding noise to Hodgkin-Huxley equations adding channel noise yields a stochastic pde τ∂ t v = λ 2 ∂ xx v − g Na m 3 h ( v − E Na ) − g K n 4 ( v − E K ) − g L ( v − E L ) + I + σ∂ t ξ ( t , x ) dp dt = α p ( v )(1 − p ) − β p ( v ) p , p ∈ { m , n , h } (2) 100 50 u 0 0 . 5 0 1 100 50 u 0 0 . 5 0 1 x features: subthreshold excitability (well-known already in the point neuron) due to spatial extension: spontaneous spiking, backpropagation, annihilation, propagation failure

Illustration - subthreshold excitability (already known from the point neuron case) I = 6 . 0 , σ = 0 . 0 I = 6 . 0 , σ = 0 . 25

Illustration - spontaneous activation, backpropagation I = 6 . 0 , σ = 0 . 25 I = 2 . 0 , σ = 0 . 36

Subunit noise (classical) diffusion approximation for the Markov chain dynamics p ( t ) modelling ion channel dynamics � t ( N p ) p ( t ) = p (0) + α p ( v ( s ))(1 − p ( s )) − β p ( v ( s )) p ( s ) ds + M t 0 with � t �� � 2 � = 1 ( N p ) E M E ( α p ( v ( s ))(1 − p ( s )) + β p ( v ( s )) p ( s )) ds t N p 0 τ∂ t v = λ 2 ∂ xx v + g L ( v − E L ) − g Na ( X )( v − E Na ) − g K ( X )( v − E Na ) + ξ v ∂ t X = α ( v )(1 − X ) − β ( v ) X + σ ( v , X ) ξ X Rem representation in terms of Wiener noise driven sde causes troubles at reflecting boundaries { 0 , 1 }

Conductance noise leads to pde with random coefficients... τ∂ t v = λ 2 ∂ xx v + g L ( v − E L ) − g Na ( X , ξ Na )( v − E Na ) − g K ( X , ξ K )( v − E Na ) + ξ v ∂ t X = α ( v )(1 − X ) − β ( v ) X + σ ( v , X ) ξ X ◮ comparison and validation of different types, except in case studies, largely open ◮ also derivation from first principles

Illustration: Stochastic Hodgkin-Huxley equations 100 100 u 50 50 u 0 0 0 . 5 0 1 0 . 5 0 1 100 100 50 u 50 u 0 0 0 0 . 5 1 0 0 . 5 1 100 100 50 u 50 u 0 0 0 0 . 5 1 0 0 . 5 1 x x realizations of propagation failure (resp. spotaneous activity) in more realistic models, see Sauer, S., J Comput Neurosci 2016

Propagation failure - numerical studies Tuckwell, et al. (2008,2010,2011) - numerical study of P ( Propagation failure ) w.r.t. σ from [Tuckwell, Neural Computation, 2008]

Propagation failure - computational approach detecting propagation failure � L Φ( v ) := v ( x ) − v ∗ dx , v ∗ = resting potential 0 failure occurs w.r.t. given threshold θ if Φ( v ( t )) < θ for some t ∈ [ T 0 , T ] hence interested in computing � � p σ := P σ T 0 ≤ t ≤ T Φ( v ( t )) < θ min

Spontaneous activity - computational approach detecting spontaneous activity using � L Φ( v ) = v ( x ) − v ∗ dx , v ∗ = resting potential 0 w.r.t. given threshold θ if Φ( v ( t )) > θ for some t ∈ [ T 0 , T ] leads to the probability � � s σ := P σ T 0 ≤ t ≤ T Φ( v ( t )) > θ min

Numerical Illustrations 1 θ = 0 . 5 p σ 0 . 5 θ = 0 . 2 θ = 0 p ref σ 0 0 0 . 3 0 . 6 0 . 9 1 . 2 σ 1 θ = 0 . 2 θ = 0 . 4 θ = 0 . 5 s σ 0 . 5 θ = 0 . 6 0 0 . 2 0 . 4 0 . 6 0 σ typical plots for p σ vs. σ (resp. s σ vs. σ )

Model reduction w.r.t. Φ Assuming the AP ˆ v is loc. exp. attracting with rate κ ∗ , hence d ( v − ˆ v ) ≈ − κ ∗ ( v − ˆ v ) dt + σ d ξ ( t ) implies d Φ( v ( t )) ≈ κ ∗ ( c − Φ( v ( t ))) dt + ˜ σ d β ( t ) where � L �� L σ 2 = σ 2 1 � ◮ c = 0 ˆ v ( t , x ) − v ∗ dx indep. of time, ˜ t Var 0 W ( t , x ) dx ◮ ( β ( t )) - 1-dim BM and reduces the problem to computing first passage-time probabilities of 1-dim. OU-processes d ˜ Φ = κ ∗ ( c − ˜ Φ) dt + ˜ σ d β ( t )

Numerical Illustrations 1 θ = 0 . 2 s σ 0 . 5 s σ ˜ θ = 0 . 3 ˜ s σ 0 0 0 . 2 0 . 4 0 . 6 σ 1 θ = 0 . 7 p σ 0 . 5 ˜ p σ θ = 0 . 5 p σ ˜ 0 0 0 . 3 0 . 6 0 . 9 1 . 2 σ comparison of p σ (resp. s σ ) for the full spde with the 1-dim ou typical plots for p σ vs. σ (resp. s σ vs. σ )

A better fit in the case of FHN parameters for ◮ FHN-system taken from Tuckwell, op.cit. ◮ OU-Approximation: κ ∗ = 0 . 2, c = 8 . 6 ◮ θ = 5

Analysis and Numerical Approximation joint with Martin Sauer (TU Berlin) realization of (2) as (density controlled) (stochastic) evolution equation τ∂ t v = λ 2 ∂ 2 xx v − g Na m 3 h ( v − E Na ) − g K n 4 ( v − E K ) − g L ( v − E L ) + I dp dt = α p ( v )(1 − p ) − β p ( v ) p , p ∈ { m , n , h } crucial properties ◮ eq is neither Lipschitz nor one-sided Lipschitz jointly in ( v , m , n , h ) ◮ condition on ion-channel concentrations X = ( m , n , h ), first part is linear w.r.t. v (one-sided Lipschitz will be sufficient for the general theory) ◮ condition on v , second part is a forward Kolmogorov eq, in particular, p 0 ( x ) ∈ [0 , 1] implies p ( t , x ) ∈ [0 , 1]

Abstract setting Mathematical modelling as stochastic evolution equation on the function space H = L 2 (0 , 1) dv ( t ) = ∆ v ( t ) + f ( v ( t ) , X ( t )) dt + BdW ( t ) , dX i ( t ) = f i ( v ( t ) , X i ( t )) dt + B i ( v ( t ) , X i ( t )) dW i ( t ) . Assumption A Local Lipschitz continuity and conditional monotonicity � 1 + | v | r − 1 � | f ( v , x ) | , |∇ f ( v , x ) | ≤ L (1 + ρ ( x )) for some r ∈ [2 , 4] for ρ i s.th. ρ i ( v ) ≤ e α | v | | f i ( v , x i ) | , |∇ f i ( v , x i ) | ≤ L (1 + ρ i ( | v | )) (1 + | x i | ) ∂ v f ( v , x ) ≤ L (1 + ρ ( x )) ∂ x i f i ( v , x i ) ≤ L

Abstract setting, ctd. Mathematical modelling as stochastic evolution equation on the function space H = L 2 (0 , 1) dv ( t ) = ∆ v ( t ) + f ( v ( t ) , X ( t )) dt + BdW ( t ) , dX i ( t ) = f i ( v ( t ) , X i ( t )) dt + B i ( v ( t ) , X i ( t )) dW i ( t ) . Assumption B Recurrence for voltage & invariance of [0 , 1] for gating variables ∂ v f ( v , x ) ≤ − κ K ∀| v | > K , 0 ≤ x ≤ 1 f i ( v , x i ) ≥ 0 ∀ x i ≤ 0 , v ∈ R f i ( v , x i ) ≤ 0 ∀ x i ≥ 1 , v ∈ R