case-14Oct2011

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case-14Oct2011And now for somethingcompletely different.

Models and Capacitiesof Molecular Communication

Andrew W. EckfordDepartment of Computer Science and Engineering, York University

Joint work with: L. Cui, York University P. Thomas and R. Snyder, Case Western Reserve University

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How do these things talk?

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Why do this? … Engineering

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Why do this? … Medicine/Biology

Model and calculate capacity of a molecular communication channel

Show that information theory can predict biological parameters

Show that information theory plays an important role in physiology/medicine

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Communication Model

Communications model

Tx Rx

1, 2, 3, ..., |M|M:

m

Tx

m

Noise

m'

m = m'?

Medium

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Communication Model

Fundamental assumptions

1. Molecules propagate via Brownian motion

2. Molecules don’t interact with each other (no reaction, motions are independent)

3. Molecules don’t disappear or change identity

Say it with Molecules

Cell 1 Cell 2

Identity: Sending 0

Release type A

Say it with Molecules

Cell 1 Cell 2

Identity: Sending 1

Release type B

Say it with Molecules

Cell 1 Cell 2

Identity: Receiving

Measure identity of arrivals

Say it with Molecules

Cell 1 Cell 2

Quantity: Sending 0

Release no molecules

Say it with Molecules

Cell 1 Cell 2

Quantity: Sending 1

Release lots of molecules

Say it with Molecules

Cell 1 Cell 2

Quantity: Receiving

Measure number arriving

Say it with Molecules

Cell 1 Cell 2

Timing: Sending 0

Release a molecule now

Say it with Molecules

Cell 1 Cell 2

Timing: Sending 1

WAIT …

Say it with Molecules

Cell 1 Cell 2

Timing: Sending 1

Release at time T>0

Say it with Molecules

Cell 1 Cell 2

Timing: Receiving

Measure arrival time

Ideal System Model

“All models are wrong,but some are useful”

-- George Box

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Ideal System Model

In an ideal system:

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Ideal System Model

In an ideal system:

1) Transmitter and receiver are perfectly synchronized.

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Ideal System Model

In an ideal system:

1) Transmitter and receiver are perfectly synchronized.

2) Transmitter perfectly controls the release times and physical state of transmitted particles.

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Ideal System Model

In an ideal system:

1) Transmitter and receiver are perfectly synchronized.

2) Transmitter perfectly controls the release times and physical state of transmitted particles.

3) Receiver perfectly measures the arrival time and physical state of any particle that crosses the boundary.

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Ideal System Model

In an ideal system:

1) Transmitter and receiver are perfectly synchronized.

2) Transmitter perfectly controls the release times and physical state of transmitted particles.

3) Receiver perfectly measures the arrival time and physical state of any particle that crosses the boundary.

4) Receiver immediately absorbs (i.e., removes from the system) any particle that crosses the boundary.

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Ideal System Model

In an ideal system:

1) Transmitter and receiver are perfectly synchronized.

2) Transmitter perfectly controls the release times and physical state of transmitted particles.

3) Receiver perfectly measures the arrival time and physical state of any particle that crosses the boundary.

4) Receiver immediately absorbs (i.e., removes from the system) any particle that crosses the boundary.

Everyt

hing i

s perf

ect

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Ideal System Model

Tx

Rx

d0

Two-dimensional Brownian motion

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Ideal System Model

Tx

Rx

d0

Two-dimensional Brownian motion

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Ideal System Model

Tx

Rx

d0

Two-dimensional Brownian motion

Uncertainty in propagation is the main source of noise!

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Ideal System Model

Theorem.

I(X;Y) is higher under the ideal system model than under any system model that abandons any of these assumptions.

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Ideal System Model

Theorem.

I(X;Y) is higher under the ideal system model than under any system model that abandons any of these assumptions.

Proof.

1, 2, 3: Obvious property of degraded channels.

4: ... a property of Brownian motion.

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Ideal System Model

Tx

Rx

d0

Two-dimensional Brownian motion

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Ideal System Model

Tx

Rx

d0

One-dimensional Brownian motion

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Ideal System Model

Tx

Rx

d0

Two-dimensional Brownian motion

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Ideal System Model

Two-dimensional Brownian motion

Tx

Rx

d0

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Ideal System Model

Two-dimensional Brownian motion

Tx

Rx

d0

First hitting time is the only property ofBrownian motion that we use.

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Input-output relationship

Additive Noise: y = x + n

n = First arrival time

fN(n) = First arrival time PDF

fN(y-x) = Arrival at time y given release at time x

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Input-output relationship

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Input-output relationship

Bapat-Beg theorem

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Delay Selector Channel

Transmit: 1 0 1 1 0 1 00 1 0

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Delay Selector Channel

Transmit: 1 0 1 1 0 1 00 1 0

Delay: 1

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Transmit: 1 0 1 1 0 1 00 1 0

Delay: 1

Receive: 0 1 0 0 0 0 00 0 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 00 1 0

Delay: 1

Receive: 0 1 0 0 1 0 00 0 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 00 1 0

Delay: 1

Receive: 0 1 0 0 2 0 00 0 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 00 1 0

Delay:1

Receive: 0 1 0 0 2 0 01 0 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 00 1 0

Delay:1

Receive: 0 1 0 0 2 0 01 1 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 00 1 0

Delay:

Receive: 0 1 0 0 2 0 01 1 0

Delay Selector Channel

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I

Receive: 0 1 0 0 2 0 01 1 0

Delay Selector Channel

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I

Receive: 0 1 0 0 2 0 01 1 0

… Transmit = ?

Delay Selector Channel

Delay Selector Channel

Delay Selector Channel

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Delay Selector Channel

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Delay Selector Channel

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Delay Selector Channel

[Cui, Eckford, CWIT 2011]

Delay Selector Channel

Delay Selector Channel

Max over px

Delay Selector Channel

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Delay Selector Channel

Let m = 1, q1 = 0.5:

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Delay Selector Channel

Let m = 1, q1 = 0.5:

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Delay Selector Channel

Let m = 1, q1 = 0.5:

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Ligand-receptor channels

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Ligand-receptor channels

X = concentration (H/L)

Y = binding state (B/U)

Cell

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Ligand-receptor channels

Cell

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Ligand-receptor channels

Cell

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Ligand-receptor channels

Cell

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Ligand-receptor channels

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Ligand-receptor channels

Theorem:

If αL = 0, αH = 1, β = 1, capacity is log2 ϕ = 0.6942.

Proof sketch:

“BB” subsequences are forbidden at the receiver.

Furthermore, the transmitter always knows the binding state.

[Thomas, Snyder, Eckford, ISIT 2012, in prep]

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Ligand-receptor channels

Conjecture:

In general, feedback-capacity-achieving input distribution is iid.

[Thomas, Snyder, Eckford, ISIT 2012, in prep]

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For more information

http://molecularcommunication.ca

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For more information

Acknowledgments

Lu Cui York Univ., Canada

Peter Thomas, Robin Snyder Case Western Reserve Univ., USA

Research funding from NSERC

Contact

Email: aeckford@yorku.caWeb: http://www.andreweckford.com/Twitter: @andreweckford