Modelling of Potential Depolarization Signals in the Hippocampus - PowerPoint PPT Presentation

Modelling of Potential Depolarization Signals in the Hippocampus Andrea Angiuli, Jasmijn Baaijens, Saray Busto Ulloa, Mohit Dalwadi, Moussa Mory Diedhiou, Nitya Dixit, Nataša Džaleta, Aleksis Pirinen Project coordinator: Afaf Bouharguane April 16, 2014

Introduction ◮ Hippocampus: neurons ⇒ Transmission of information ◮ Depolarization ⇒ Propagation of stimulus ◮ Data used: membrane potential on cross-section of hippocampus ◮ Find mapping from one image to the next ◮ Speed of propagation: drugs v.s. control group

Monge-Kantorovich Problem ◮ Find optimal transport plan minimizing total cost: � � R d ρ 0 ( x ) d x = R d ρ 1 ( x ) d x (equality of mass) � R d � x − M ( x ) � 2 ρ 0 ( x ) d x C ( M ) = (cost) ◮ Assumption on data: the mapping minimizes displacement of signal intensity ◮ Solve MKP to analyse data

Non-drugged versus drugged mice

Transport theory ◮ Two distributions, ρ 0 and ρ 1 , that satisfy the following relation � � R 2 ρ 0 ( x , y ) d x d y = R 2 ρ 1 ( x , y ) d x d y . ◮ Introduce a variable t ∈ [ 0 , T ] , through which we can continuously map from one distribution to the other. ◮ Leads to a conservation of mass equation. ◮ Ideally, we want to choose a mapping M that minimises the transportation "cost" C which is defined by � R 2 || M ( x , y ) − ( x , y ) || 2 ρ 0 ( x , y ) d x d y . C ( M ) = (1)

Governing equations ◮ Conservation of mass ρ t + ∇ · ( ρ u ) = 0 . ◮ Optimal transport theory states that u = ∇ φ . ◮ We approximate ρ t and ρ to get � ρ 0 + ρ 1 � ∇ · ∇ φ = ρ 0 − ρ 1 . 2

Finite difference scheme ◮ A second order centred scheme is developed to approximate a solution to the general equation ∇ · ( k ( x , y ) ∇ φ ) = f ( x , y ) . ◮ The following grid is used: ◮ Accuracy is verified by comparing against analytic results for constant k .

Data filtering

Fourier filter

Fourier filter

Fourier filter

Wiener filter ◮ Based on a statistical approach. ◮ Signal + Additive Gaussian Noise ◮ Minimize mean-square error

Comparison of filters

Results ◮ We find that signals in the brain propagate 10% faster in control mice compared to drugged mice.

Future work ◮ Use optimal transport theory (algorithm in appendix) to determine optimal mapping between distributions.

Unclean data ◮ Generalize scheme to incorporate non homogeneous boundary conditions.

Thanks for listening!

Algorithm Algorithm 1 Require: ρ 0 , ρ 1 ≥ 0 , α ∈ ( 0 , 1 ) Initial mapping ψ 0 : Solve ρ 0 + ρ 1 ∇ · ( ∇ ψ 0 ) = ρ 0 − ρ 1 , ( x , y ) ∈ Ω = [ a , b ] × [ c , d ] , (2) 2 ψ 0 = 0 on ∂ Ω (3) Correction: ψ n = ψ 0 res = 10 while res ≥ 10 − 3 do � � ρ ( ξ ) = ρ 1 ( ξ + ε ∇ ψ n ) det ˜ ∇ ξ ( ξ + ε ∇ ξ ψ n ) Solve: ρ 0 + ˜ ρ ∇ · ( ∇ φ ) = ρ 0 − ˜ ( x , y ) ∈ Ω = [ a , b ] × [ c , d ] , (4) ρ, 2 φ = 0 on ∂ Ω (5) ψ n + 1 = ψ n + αφ . res = || ρ 0 − ˜ ρ || ∞ end while