Models and Capacitiesof Molecular Communication

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

Joint work with: N. Farsad and L. Cui, York University K. V. Srinivas, S. Kadloor, and R. S. Adve, University of TorontoS. Hiyama and Y. Moritani, NTT DoCoMo

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How do tiny devices communicate?

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How do tiny devices communicate?

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How do tiny devices communicate?

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How do tiny devices communicate?

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How do tiny devices communicate?

Most information theorists are concerned with communication that is, in some way, electromagnetic:

- Wireless communication using free-space EM waves- Wireline communication using voltages/currents- Optical communication using photons

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How do tiny devices communicate?

Most information theorists are concerned with communication that is, in some way, electromagnetic:

- Wireless communication using free-space EM waves- Wireline communication using voltages/currents- Optical communication using photons

Are these appropriate strategies for nanoscale devices?

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How do tiny devices communicate?

There exist “nanoscale devices” in nature.

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How do tiny devices communicate?

There exist “nanoscale devices” in nature.

Image source: National Institutes of Health

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How do tiny devices communicate?

Bacteria (and other cells) communicate by exchanging chemical “messages” over a fluid medium.

- Example: Quorum sensing.

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How do tiny devices communicate?

Bacteria (and other cells) communicate by exchanging chemical “messages” over a fluid medium.

- Example: Quorum sensing.

Poorly understood from an information-theoretic perspective.

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Communication Model

Communications model

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Communication Model

Communications model

Tx Rx

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

m

Tx

m

m'

m = m'?

Medium

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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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Say it with Molecules

Transmit: 1 0 1 1 0 1 0 0 1 0

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

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

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

Communications model

Tx Rx

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

m

Tx

m

Noise

m'

m = m'?

Ideal System Model

What is the best you can do?

Ideal System Model

What is the best you can do?

Ideal System Model

What is the best you can do?

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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.

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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

One-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

One-dimensional Brownian motion

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Ideal System Model

One-dimensional Brownian motion

Tx

Rx

d0

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Ideal System Model

One-dimensional Brownian motion

Tx

Rx

d0

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

Ideal System Model

What is the best you can do?

Ideal System Model

What is the best you can do?

Approaches

Two approaches:

• Continuous time, single molecules• Additive Inverse Gaussian Channel

• Discrete time, multiple molecules• Delay Selector Channel

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Additive Inverse Gaussian Channel

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Additive Inverse Gaussian Channel

Tx

Rx

d0

Two-dimensional Brownian motion

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Additive Inverse Gaussian Channel

Tx

Rx

d0

Two-dimensional Brownian motion

Release: t

Arrive: t + n

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Additive Inverse Gaussian Channel

Tx

Rx

d0

Two-dimensional Brownian motion

First passage time is additive noise!

Release: t

Arrive: t + n

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Additive Inverse Gaussian Channel

Brownian motion with drift velocity v:

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Additive Inverse Gaussian Channel

Brownian motion with drift velocity v:

First passage time given by inverse Gaussian (IG) distribution.

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Additive Inverse Gaussian Channel

Brownian motion with drift velocity v:

First passage time given by inverse Gaussian (IG) distribution.

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Additive Inverse Gaussian Channel

Brownian motion with drift velocity v:

First passage time given by inverse Gaussian (IG) distribution.

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Additive Inverse Gaussian Channel

Brownian motion with drift velocity v:

First passage time given by inverse Gaussian (IG) distribution.

IG(λ,μ)

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Additive Inverse Gaussian Channel

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Additive Inverse Gaussian Channel

Let h(λ,μ) = differential entropy of IG.

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Additive Inverse Gaussian Channel

Let h(λ,μ) = differential entropy of IG.

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Additive Inverse Gaussian Channel

Let h(λ,μ) = differential entropy of IG.

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Additive Inverse Gaussian Channel

Additivity property:

Let a ~ IG(λa,μa) and b ~ IG(λb,μb) be IG random variables.

If λa/μa2 = λb/μb

2 = K, then

a + b ~ IG(K(μa + μb)2, μa + μb).

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Additive Inverse Gaussian Channel

Let h(λ,μ) = differential entropy of IG, E[X] ≤ m.

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Additive Inverse Gaussian Channel

Let h(λ,μ) = differential entropy of IG, E[X] ≤ m.

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Additive Inverse Gaussian Channel

Let h(λ,μ) = differential entropy of IG, E[X] ≤ m.

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Additive Inverse Gaussian Channel

Let h(λ,μ) = differential entropy of IG, E[X] ≤ m.

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Additive Inverse Gaussian Channel

Bounds on capacity subject to input constraint E[X] ≤ m:

[Srinivas, Adve, Eckford, sub. to Trans. IT; arXiv]

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Delay Selector Channel

Transmit: 1 0 1 1 0 1 0 0 1 0

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Delay Selector Channel

Transmit: 1 0 1 1 0 1 0 0 1 0

Delay: 1

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Transmit: 1 0 1 1 0 1 0 0 1 0

Delay: 1

Receive: 0 1 0 0 0 0 0 0 0 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 0 0 1 0

Delay: 1

Receive: 0 1 0 0 1 0 0 0 0 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 0 0 1 0

Delay: 1

Receive: 0 1 0 0 2 0 0 0 0 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 0 0 1 0

Delay: 1

Receive: 0 1 0 0 2 0 0 1 0 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 0 0 1 0

Delay: 1

Receive: 0 1 0 0 2 0 0 1 1 0

Delay Selector Channel

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Transmit: 1 0 1 1 0 1 0 0 1 0

Delay:

Receive: 0 1 0 0 2 0 0 1 1 0

Delay Selector Channel

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I

Receive: 0 1 0 0 2 0 0 1 1 0

Delay Selector Channel

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I

Receive: 0 1 0 0 2 0 0 1 1 0

… Transmit = ?

Delay Selector Channel

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Input-output relationship

For n particles,

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Input-output relationship

For n particles,

First hitting time distribution

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Input-output relationship

For n particles,

Sum over all possible arrival permutations

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Input-output relationship

For n particles,

Sum over all possible arrival permutations

Intractable for any practical number of particles.(Bapat-Beg theorem)

The Delay Selector Channel

The 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]

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Delay Selector Channel

The DSC admits zero-error codes.

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Delay Selector Channel

The DSC admits zero-error codes.

E.g., m=1: 1: [1, 0] 0: [0, 0]

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Delay Selector Channel

The DSC admits zero-error codes.

E.g., m=1: 1: [1, 0] 0: [0, 0]

Receive:0 0 1 0 0 1 1 0 0 0 0 1

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Delay Selector Channel

The DSC admits zero-error codes.

E.g., m=1: 1: [1, 0] 0: [0, 0]

Receive:0 0 1 0 0 1 1 0 0 0 0 1

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Delay Selector Channel

The DSC admits zero-error codes.

E.g., m=1: 1: [1, 0] 0: [0, 0]

Receive:0 0 1 0 0 1 1 0 0 0 0 1

0 0 1 0 1 0 1 0 0 0 1 0

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Delay Selector Channel

The DSC admits zero-error codes.

E.g., m=1: 1: [1, 0] 0: [0, 0]

Receive:0 0 1 0 0 1 1 0 0 0 0 1

0 0 1 0 1 0 1 0 0 0 1 0

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Delay Selector Channel

The DSC admits zero-error codes.

E.g., m=1: 1: [1, 0] 0: [0, 0]

Receive:0 0 1 0 0 1 1 0 0 0 0 1

0 0 1 0 1 0 1 0 0 0 1 0

Ideal System Model

What is the best you can do?

Ideal System Model

What is the best you can do?

About 1 bit/s

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What is the vision?

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What is the vision?

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What is the vision?

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For more information

http://molecularcommunication.ca

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For more information

Acknowledgments

Satoshi Hiyama, Yuki Moritani NTT DoCoMo, Japan

Ravi Adve, Sachin Kadloor, Univ. of Toronto, CanadaK. V. Srinivas

Nariman Farsad, Lu Cui York University, Canada

Research funding from NSERC

Contact

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