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:

• Rd ρ0(x)dx

=

• Rd ρ1(x)dx

(equality of mass) C(M) =

• Rd x − M(x)2ρ0(x)dx

(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, ρ0 and ρ1, that satisfy the following relation

• R2 ρ0(x, y)dxdy =
• R2 ρ1(x, y)dxdy.

◮ 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 C(M) =

• R2 ||M(x, y) − (x, y)||2 ρ0(x, y)dxdy.

(1)

Governing equations

◮ Conservation of mass

ρt + ∇ · (ρu) = 0.

◮ Optimal transport theory states that u = ∇φ. ◮ We approximate ρt and ρ to get

∇ · ρ0 + ρ1 2 ∇φ

• = ρ0 − ρ1.

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: ρ0, ρ1 ≥ 0, α ∈ (0, 1) Initial mapping ψ0: Solve ∇ · ( ρ0 + ρ1 2 ∇ψ0) = ρ0 − ρ1, (x, y) ∈ Ω = [a, b] × [c, d], (2) ψ0 = 0 on ∂Ω (3) Correction: ψn = ψ0 res = 10 while res ≥ 10−3 do ˜ ρ(ξ) = ρ1(ξ + ε∇ψn)det

• ∇ξ(ξ + ε∇ξψn)
• Solve:

∇ · ( ρ0 + ˜ ρ 2 ∇φ) = ρ0 − ˜ ρ, (x, y) ∈ Ω = [a, b] × [c, d], (4) φ = 0 on ∂Ω (5) ψn+1 = ψn + αφ. res = ||ρ0 − ˜ ρ||∞ end while