Amathematicalmodelforelectrolocationin weakly electric ﬁsh · Thus we neglect electromagnetic waves propagation . =) There exists an electric potential u such that E=—u. ... Probability

A mathematical model for electrolocation inweakly electric fish

Thomas Boulier

Laboratoire d’imagerie biomédicale (UPMC-INSERM)

[email protected]

Electro-activity of biological systems, 11/19/2015

Team

Habib Ammari Josselin Garnier

Wenjia Jing Han Wang Hyeonbae Kang

Active Electrolocation: Physical Principle

Isopotentials of self-emitted electric field.

Active Electrolocation: Behavioral Studies

(Von der Emde, et al. Electric fish measure distance in the dark. Nature, 1998).

Identification: distance, size, geometry, conductivity s ,permittivity e .

Mathematical Model

Measured electric field : E

Main question:How is it possible to localize and identify D from such low-voltage

and low-frequency signal?

Applications: Eletrical Impedance Tomography (EIT)

Underwater robotics Medical imaging

Contents

A Mathematical Model for the Forward ProblemEquationsNumerical Simulations

Target IdentificationPolarizability of an ObjectLocalization from Multifrequency SignalShape Identification with Machine Learning

Contents



Mathematical Model

Emitted electric field (without object) : E0


Problem: knowing (E�E0) ·n over ∂⌦, determine D

(localization, shape identification, tracking).

Mathematical Model





Mathematical Model





Partial Differential Equations

Maxwell equations:

8>>>><

>>>>:

— ·E =r

e

— ·B = 0—⇥E = iwB

—⇥B = µ (js + ji + iweE)

Ohm’s law: ji = sE

Approximate Model

�

L

⌦

Quasi-Static Approximation

With a frequency w ⇠ 1kHz one has a wavelengthl :=

�w

peµ

��1 ⇠ 30km � L= 1m (body size).Thus we neglect electromagnetic waves propagation.=) There exists an electric potential u such that E = —u.

Approximate Model

Boundary conditions

Taking water as reference (s0

⇠ 0.01S ·m�1), the body is highlyconductive (sb = 1S ·m�1), and the skin is very thin (d ⇠ 100µm)and very resistive (ss ⇠ 10�4S ·m�1).=) Impedance boundary conditions across the skin.

Approximate Model

Theorem

If d is “small enough” and sb “big enough”, then the electricpotential u is “close” to the solution of the following system:

8>>>>>>>><

>>>>>>>>:

�u = f , x 2 ⌦,

— · (s + iew)—u = 0, x 2 ⌦c,

∂u

∂n

��= 0, x 2 ∂⌦,

[u]�x

∂u

∂n

��+

= 0, x 2 ∂⌦,

where x := ds

0

/ss is called effective thickness (Assad 1997).

(Proof uses Layer Potential techniques, cf. Zribi 2005, Lanza deCristoforis & Rossi 2004.)

Contents



Numerical Simulations

xi

xi+1xi�1

f

0(xi )'f (xi+1

)� f (xi )

xi+1

� xi.

Numerical simulation = equations discretization.Here, Boundary Elements Method.

Numerical Simulations

Example : ellipse-shaped fish and anomaly D = z+dB withconductivity s and permittivity e .

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Without anomaly. With anomaly.(x = 0). (s = 1010, e = 0).

Contents



Electric Dipole

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1. 2. 3.

1. Fundamental solution: �G (x) = d

0

(x) =) G (x) = 1

2p

log |x |.2. Two charges: +d

�1 at z and �d

�1 at z+dp.3. Electric dipole potential: p ·—G (x� z).

Polarization Tensor

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1. 2. 3.

1. Emitted electric field U,2. Object D = z+dB with conductivity k , electric potential u,3. Equivalent electric dipole: u(x)�U(x)' p ·—G (x� z), with

p =�M(k ,D)| {z }—U(z).

Polarization Tensor

(cf. Ammari-Kang, Polarization and Moment Tensors, 2007).

Contents



Space-Frequency MUSIC

Fact: weakly eletric fishes possess 2 types of electroreceptors. Onetype measures the amplitude of the electric field and anothermeasures its phase.

Idea: use this information to locate the target. We will use analgorithm called Space-Frequency MUSIC, developped by Scholz forbreast cancer imaging.

MUSIC stands for MUltiple Signal Classification, and is originally atool used in signal processing to identify several signals with anadditive noise (Schmidt 1986). It has then been applied to locateinhomogeneities in the context of Electrical Impedance Tomography(Ammari, Borcea, Berryman, Brühl, Griesmaier, Hanke, Kang, Kim,Louati, Papanicolaou, Tsogka, Vogelius...).









Space-Frequency MUSIC: Application

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Electric field. Isopotentials of localization function.

Contents



Objects Database

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Movement and Multifrequency measurements

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=) Extraction of equivalent multifrequency electricdipoles {p(wn)}n from measurements at multiple positions (linearsystem).

Database of Multifrequency Electric Dipoles

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Shape identification: performance under electronic noise

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Pro

ba

bility o

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Different ellipse

Conclusion

I Partial differential equations and model reduction (highlyconductive body, highly resistive and thin skin),

I Numerical simulations (boundary elements method),

I Localization from multifrequency measurements,I Shape identification (machine learning algorithms).

Perspectives

Take-home messages:

I The electric field surrounding the fish can be accuratelysimulated with computational methods;

I Multifrequency and movement contain all the informationnedded to recognize preys.

More to come:

I Application to underwater robotics,I Artificial neural networks (deep learning).

References

Habib Ammari, Thomas Boulier, and Josselin Garnier.Modeling active electrolocation in weakly electric fish.SIAM Journal on Imaging Sciences, 6(1):285–321, 2013.

Habib Ammari, Thomas Boulier, Josselin Garnier, Wenjia Jing,Hyeonbae Kang, and Han Wang.Target identification using dictionary matching of generalizedpolarization tensors.Foundations of Computational Mathematics, 14(1):27–62,2014.Habib Ammari, Thomas Boulier, Josselin Garnier, and HanWang.Shape recognition and classification in electro-sensing.Proceedings of the National Academy of Sciences,111(32):11652–11657, 2014.

Documents

Amathematicalmodelforelectrolocationin weakly electric ﬁsh · Thus we neglect electromagnetic waves propagation . =) There exists an electric potential u such that E=—u. ... Probability