an information theoretic framework to analyze molecular ... · an information theoretic framework...

An Information TheoreticFramework to AnalyzeMolecular CommunicationSystems Based on StatisticalMechanicsThis article proposes a mathematical framework to define the main functional blocks ofmolecular communication theory, supported by general models from chemical kineticsand statistical mechanics.

By IAN F. AKYILDIZ , Fellow IEEE, MASSIMILIANO PIEROBON , Member IEEE,AND SASITHARAN BALASUBRAMANIAM, Senior Member IEEE

ABSTRACT | Over the past 10 years, molecular communication

(MC) has established itself as a key transformative paradigm in

communication theory. Inspired by chemical communications

in biological systems, the focus of this discipline is on the

modeling, characterization, and engineering of information

transmission through molecule exchange, with immediate

applications in biotechnology, medicine, ecology, and defense,

among others. Despite a plethora of diverse contributions,

which has been published on the subject by the research

community, a general framework to study the performance

of MC systems is currently missing. This paper aims at filling

this gap by providing an analysis of the physical processes

underlying MC, along with their information-theoretic

underpinnings. In particular, a mathematical framework is

Manuscript received December 2, 2018; revised June 10, 2019; acceptedJune 27, 2019. Date of current version July 19, 2019. This work was supported inpart by the U.S. National Science Foundation under Grant CISE CNS-1763969and Grant CCF-1816969 and in part by the Science Foundation Ireland throughthe SFI VistaMilk research centre (16/RC/3835) and CONNECT research centre(13/RC/2077). (Corresponding author: Massimiliano Pierobon.)

I. F. Akyildiz is with the Broadband Wireless Networking Laboratory, School ofElectrical and Computer Engineering, Georgia Institute of Technology, Atlanta,GA 30332 USA (e-mail: [email protected]).

M. Pierobon is with the Molecular and Biochemical TelecommunicationsLaboratory, Department of Computer Science & Engineering, University ofNebraska–Lincoln, Lincoln, NE 68588 USA (e-mail: [email protected]).

S. Balasubramaniam is with the Telecommunication Software and SystemsGroup, Waterford Institute of Technology, X91 P20H Waterford, Ireland, and alsowith the Department of Electronic and Communication Engineering, TampereUniversity of Technology, 33720 Tampere, Finland (e-mail: [email protected]).

Digital Object Identifier 10.1109/JPROC.2019.2927926

proposed to define the main functional blocks in MC, supported

by general models from chemical kinetics and statistical

mechanics. In this framework, the Langevin equation is utilized

as a unifying modeling tool for molecule propagation in MC

systems, and as the core of a methodology to determine the

information capacity. Diverse MC systems are classified on the

basis of the processes underlying molecule propagation, and

their contribution in the Langevin equation. The classifications

and the systems under each category are as follows: random

walk (calcium signaling, neuron communication, and bacterial

quorum sensing), drifted random walk (cardiovascular system,

microfluidic systems, and pheromone communication), and

active transport (molecular motors and bacterial chemotaxis).

For each of these categories, a general information capacity

expression is derived under simplifying assumptions and

subsequently discussed in light of the specific functional

blocks of more complex MC systems. Finally, in light of the

proposed framework, a roadmap is envisioned for the future

of MC as a discipline.

KEYWORDS | Fokker–Planck equation; information capacity;

Langevin equation; molecular communication (MC); nanonet-

works; Poisson noise; statistical mechanics.

I. I N T R O D U C T I O N

The genesis of molecular communication (MC) asa discipline stands in the observation of the unitsof life, i.e., biological cells, where information is

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https://orcid.org/0000-0002-8099-3529https://orcid.org/0000-0003-1074-6925

Akyildiz et al.: An Information Theoretic Framework to Analyze Molecular Communication Systems Based on Statistical Mechanics

generated, stored, and communicated through molecularprocesses [1]. Molecules are the common substrates usedin cells to represent information, and their chemical reac-tions and transport mechanisms are the key processes thatenable their encoding and propagation. MC aims to buildon top of these processes by modeling, characterizing, andengineering communication systems and devices able totap into a previously uncharted territory, the biochemical,to enable applications where classical communication sys-tems show limitations, i.e., inside the human body and/ordirectly interacting with biological cells [2]. Current andfuture MC applications range from the engineering of com-munication systems between microorganisms [19], to thedevelopment and optimization of biomedical devices [12],and the augmentation of the human body functionalitiesthrough pervasive intrabody deployment of intercom-municating nanotechnology- and biotechnology-enableddevices, i.e., the Internet of Bio-Nano Things (IoBNT) [3].

Since the birth of this field, the research community,largely driven by communication and networking engi-neers, as well as computer scientists, has taken differentelements from the aforementioned biochemical commu-nication processes and abstracted them into theoreticalmodels to assemble and characterize MC systems. Thishas led researchers to develop communication channelmodels based on a wide range of processes for propa-gating information via molecules, ranging from passiveBrownian motion diffusion [17], [18], [60], [83], to thetransport, or advection, in fluid currents [10], [12], [76],and to active processes that require a dedicated energysource to move molecules from a transmitter to areceiver [67], [72], [77]. Subsequent contributions haveexplicitly addressed the estimation or expression of thecommunication capacity with ad hoc studies for some ofthese MC channels, such as time-slotted ON–OFF keying(OOK) [20], one-shot [21], time-slotted [22], [23], andcontinuous [24], [25] timing channels, and multiple sym-bol transmission [26] with perfect transmitter–receiversynchronization for passive Brownian motion and betweenbacteria colonies [33], time-slotted transmission in thecardiovascular system [11] and continuous transmissionin passive Brownian motion [15], [32], and microflu-idic systems [74], [75]. Although these contributionshave validity for specific MC scenarios, a generalinformation-theoretic framework that captures the pecu-liarities of an MC channel over classical communicationsystems is currently missing.

This paper aims at filling the aforementioned researchgap by providing a mathematical framework rooted instatistical mechanics to abstract any MC system and deter-mine or estimate the information capacity of their commu-nication channels. As shown in Fig. 1, by stemming fromthe general formulation of the Langevin equation [9] ofa moving nanoscale particle subject to unavoidable ther-mally driven Brownian forces, we build a general mathe-matical abstraction of an MC system and its main elements.Subsequently, we derive a methodology to determine

Fig. 1. Schematic of the framework proposed in this paper, whichstems from the Langevin SDE.

(or estimate, whenever closed-form analytical solutionsare intractable) the MC channel capacity based on thedecomposition of the Langevin equation into two con-tributions, namely, the Fokker–Planck equation [7] and aPoisson process. We classify any MC system on the basis oftheir representation in terms of the Langevin equation asfollows. MC systems based on random walk, such as cal-cium signaling in cell tissues [39], neuron communicationby means of neurotransmitters [49], and bacterial quorumsensing [55], include only the contribution of the Brown-ian stochastic force f. MC systems based on drifted ran-dom walk, such as MC in the cardiovascular system [12],microfluidic systems [74], and pheromone communicationbetween plants [57], include both f and a drift velocityvn(t) as function of the time t for each molecule n, whichis independent of the Brownian motion. MC systems basedon active transport, such as those based on molecularmotors [72] and bacteria chemotaxis [67], include insteada deterministic force Fn(t) added to f. For each of these cat-egories of MC systems, and based on the aforementionedLangevin equation decomposition, we provide a generalinformation capacity expression under simplifying assump-tions and subsequently discuss these results in light ofthe functional blocks of more specific MC system models,including cases where a closed-form capacity expressioncannot be analytically derived.

The rest of this paper is organized as follows.In Section II, we introduce the framework to model andclassify MC systems based on the Langevin equation,and we introduce a general methodology to determinetheir channel capacity. In Sections III–V, we detail generalcapacity expressions and specific functional block mod-els for MC systems based on a random walk, driftedrandom walk, and active transport, respectively. Finally,in Section VI, we conclude this paper and discuss the futureof MC as a discipline.

II. F R A M E W O R K T O A N A L Y Z EM O L E C U L A R C O M M U N I C AT I O NS Y S T E M S A N D T H E I R C A P A C I T Y

MC is defined as the transmission, propagation, andreception of information by utilizing molecules and their

Vol. 107, No. 7, July 2019 | PROCEEDINGS OF THE IEEE 1231

Akyildiz et al.: Information Theoretic Framework to Analyze Molecular Communication Systems Based on Statistical Mechanics

Fig. 2. Fundamental processes and functional blocks of an MCsystem.

propagation as the medium [1]. Molecules are the small-est identifiable units of a substance, a form of matter withspecific homogeneous chemical composition and proper-ties. Consequently, a molecule is the smallest unit that stillretains information on the substance identity and its abilityto take part in chemical reactions. The size of a moleculeranges from that of diatomic hydrogen (0.074 nm), water(0.275 nm), and carbon dioxide (0.232 nm), to the sizeof a biological macromolecule, such as an average protein(≈2 nm) or a deoxyribonucleic acid (DNA) chain (from2 nm). In an MC system, their dimensions and the strongforces of the chemical bonds that underlie their struc-ture, and the information they carry, are manipulated bychemical reactions, where molecule composition and struc-ture are rearranged. Chemical reactions are the primaryprocesses underlying the MC transmission and reception.Since a single chemical reaction involves single or a fewmolecules of one or more (few) substances, an entireMC system has nanoscale precision and can be containedwithin nanoscale dimensions, and for this reason, MC isidentified as a nanocommunication paradigm [2].

To manipulate and propagate information-bearing mole-cules, the components of an MC system should neces-sarily be immersed in or include a substance in a fluidstate. Brownian motion is the random and independentmovement of molecules suspended in a fluid, and it is anunavoidable consequence of the molecule vibrations for atemperature higher than the absolute zero. An MC systemis, therefore, subject to Brownian motion as a fundamentalstochastic process underlying all its components, and theBrownian motion effects are present in every possibleimplementation of an MC system.

A. Mathematical Models of FundamentalProcesses in MC

With the goal of modeling information propagationin MC, the aforementioned fundamental processes inMC, sketched in Fig. 2, have the following analyticalformulation.

1) Molecules of the same substance, which carry infor-mation in MC, are considered indistinguishable andequivalent to spherical particles of radius r andmass m, where r � d, d being the distance betweenthe transmitter and the receiver in an MC system,defined in the following, and s is the particularsubstance. Consequently, from now on, we will indis-tinguishably refer to molecules or particles.

2) Chemical reactions are processes that convert one ormore input molecules (reactants) into one or moreoutput molecules (products). A reaction j may pro-ceed in forward or reverse directions, which are char-acterized by forward (kf,j) and reverse (kr,j) reactionrates, respectively. We assume to have, in general,S chemical substances andM different chemical reac-tions in their elementary form, i.e., each chemicalreaction happens without any intermediate product.They can be expressed as follows:

R1,js1 + · · · +Rn,jsSkf,j−−−⇀↽−−−kr,j

P1,js1 + · · · + Pn,jsS (1)

where Ri,j and Pi,j are the number of molecules ofthe substance si that participate in a single chemicalreaction j expressed in (1) as reactants or products,respectively. This can be mathematically expressedwith the following reaction rate equation:

Vj = kf,j

S�i=1

[si]Ri,j − kr,j

S�i=1

[si]Pi,j (2)

where Vj is the rate of the reaction, i.e., the rate ofvariation in the molecule concentration [sj ] of thesubstance sj in number of molecules per unit space.Following classical chemical kinetics, the evolution ofthe M chemical reactions can be expressed as:

d[si]

dt=

M�j=1

vi.jVj , 1 ≤ i ≤ S (3)

where vi.j = Pi,j − Ri,j expresses the net change inthe concentration [si] of the substance sj due to thejth reaction.

3) Particle motion in a physical system can be analyti-cally formulated according to the Langevin stochasticdifferential equation (SDE) [9], which states that thelocation pn(t) = {pn,i(t)}i of the particle n at time talong any space dimension i (e.g., one of the 3-D axesX,Y,Z shown in Fig. 3) obeys the following equation:

md2 (pn(t) − vn(t)t)

dt2= Fn(t)

− 6πμr d (pn(t) − vn(t)t)dt

+ f(t) (4)



Fig. 3. Proposed framework to determine MC channel capacity.

where m is the molecule mass, ∂2(·)/∂t2 and ∂(·)/∂tare the second and first time derivative operators,respectively, vn(t) is a drift velocity of the fluid wherethe particle n is located, Fn(t) is a force applied tothe particle n independently of its Brownian motion,μ is the viscosity of the fluid, which we assume homo-geneous in the propagation space, r is the radius ofthe particle, and f(t) is a random process that modelsthe Brownian motion force, whose probability densityfunction is Gaussian and has correlation function< fi(t)fj(t

′) > given by

< fi(t)fj(t′) >= 12πμrkBTδi,jδ(t− t′) (5)

where fi(t) is the component of f(t) in the ith dimen-sion, < · > is the average operator, i and j indi-cate any of the space dimensions, kB is Boltzmann’sconstant, T is the absolute temperature of the fluid,considered homogeneous throughout the space, andδi,j is equal to 1 if i = j and zero otherwise, andδ(t− t′) is the Dirac delta function.

B. Functional Blocks of a MolecularCommunication System

An MC system [38], defined as a set of natural orengineered components that work together to receiveinformation from a source, encode this information intoproperties of molecules emitted at a transmitter, propagatethe information-bearing molecules through a channel, andreconstruct this information through a Receiver, includesthe following main functional blocks based on the afore-mentioned fundamental processes.

1) Information encoding is the modulation of the mole-cule properties according to the source informationX(t), either continuous-time signals or symbols atdiscrete time instants t = tk, k ∈ N . These propertiescan be classified into two main categories, namely,intensive and extensive, following the ways physicalsystems can be characterized. Intensive propertiesdo not depend on the quantity of the molecules,such as their chemical composition and structure(e.g., protein folding), concentration, density, pres-sure, or temperature. Extensive properties are insteadproportional to the quantity of molecules, such astheir number, total mass, occupied volume, enthalpy,or entropy. Some intensive properties can be assignedto a single molecule, e.g., temperature or chemicalcomposition and structure, while others are derived

from the ratio between two extensive properties, e.g.,concentration or density. Some of these propertiesare continuous, e.g., concentration (at high moleculenumber) or temperature, while others are discrete,e.g., molecule number, chemical composition, andstructure. The information encoding results in valuesof these properties as function of the source informa-tion X(t). Consequently, the encoding of an intensiveproperty that can be assigned to a single molecule is,here, formalized as

Intl,n(X(t)) = Al(X(t)) (6)

where Al(·) is the encoding function for the lthintensive property, which determines the intensiveproperty for the nth molecule. The encoding of anextensive property can be formalized as

Extl,m(X(t)) = bm(nl(t)), nl(t) = Cl(X(t)) (7)

where bm is a proportionality constant for the mthextensive property, nl(t) is the number of moleculeswith identical intensive property Al(X(t)) at thetransmitter at time t, and Cl(X(t)) is a function ofthe source information X(t). One of the most usedintensive properties in MC, namely, the concentrationof a substance (characterized by molecules withintensive property l corresponding to a specificchemical composition and structure) can be derivedby dividing the number of molecules nl(t) by theiroccupied volume Extl,m(X(t)), where m denotes aspecific occupied volume.

2) Molecule emission is the release of information-bearing molecules to the molecule propagationmedium. In an MC system, this corresponds tomoving the molecules, whose properties composethe encoded signal from inside to outside the spaceoccupied by the transmitter, into the propagationmedium. Realistic molecule emission processesinclude free diffusion, evaporation, dilution,osmosis/dialysis, pressure gradients (e.g., spray),encapsulation, or release from vesicles/reservoirs.The molecule emission results in molecule locationspn(tn) at the boundary ST that separates thetransmitter from the rest of the space, expressed as

pn(tn) ∈ ST ∀tn, n : NT(tn) > 0, n ∈ NT(tn) (8)

whereNT(tn) =�l nl(tn) is the number of molecules

emitted at time tn at the transmitter, and NT(t) is theset containing all the indices of the emitted particlesfrom time 0 to time t

NT(t) =�� t′

0

NT(τ )dτ

��0 < t′ < t�. (9)



3) Molecule propagation is the process whereby theemitted molecules propagate through space from thetransmitter location to the receiver location by meansof a propagation medium. In an MC system, thispropagation is unavoidably affected by the aforemen-tioned Brownian motion, i.e., the Brownian stochasticforce f, but other processes can be in place to furthercontrol the molecule propagation over a completelyrandom walk, represented by the drift velocity vn(t)and the force Fn(t) in (4), both independent of theBrownian motion, where the former results fromcurrents in the fluid, and the latter from other deter-ministic forces acting on each molecule n. Regardlessof the particular underlying process, the moleculepropagation can be expressed as the translation ofthe spatial coordinates from the location pn(tn) at thetransmitter to a location pn(tn+ΔT ) at time tn+ΔT

pn(tn) → pn(tn + ΔT ), ∀n ∈ NT(tn) (10)

where ΔT is an arbitrary propagation time interval.4) Molecule reception is the detection of the molecules

that propagated to the receiver. The most widespreadprocess for realizing this detection is throughchemical reactions between the information-bearingmolecules at the receiver and other molecules,i.e., chemical receptors, which can be placed atthe receiver boundary or within the receiver spaceSR. Upon detection, molecules can separate fromthe chemical receptors and either degrade/bedegraded (absorbing receiver) or resume theirpropagation (nonabsorbing receiver). The set NR(t)of received molecules at the receiver at time t > tn isrepresented as

NR(t) = {n|pn(t) ∈ SR} . (11)

5) Information decoding is the demodulation of theproperties of the received molecules to obtain anestimate of the source information, which maypossibly include noise or errors in the recognition ofsymbols. Upon effective collision of these molecules,if a chemical reaction takes place, a specific moleculewith composition/structure complementary tothe chemical reception is recognized as beingreceived. By considering the result of chemicalreactions at multiple (different) receptors, in themost general formulation where both intensive andextensive properties are utilized to encode the sourceinformation X(t), the reception process output iscomposed of the estimated values �Intl(t) and ̂Extm(t)of the intensive and extensive properties of thereceived molecules. This is expressed as

Xl,n(t) = A−1l (�Intl(t)) (12)

Xl,m(t) = ̂Extm(t)bm

(13)

where Xl,n(t) and Xl,m(t) are the estimated value ofthe source information X(t) from the lth intensiveproperty of the received molecule n and fromthe mth extensive property of molecules with lthintensive property, respectively. A−1l (·) is the inverseof the encoding function Al(·) for the lth intensiveproperty, and bm is the proportionality constantdefined in (7). The received information Y (t) is thenobtained from Xl,n(t) and Xl,m(t). The expressionsin (12) and (13) are intended to be general andinclude any possible information encoding scheme onmolecule properties. For example, information couldbe encoded into the sequence of the nucleotides ofdifferent DNA strands (intensive properties) and ina different number of copies of each different DNAstrand (extensive properties). In the case where thesame source information has been encoded both inintensive and extensive properties of the emittedmolecules, the received information can be obtained,e.g., through averaging, as follows:

Y (t) =1

L

L�l=1

�NR(t)�n=1

Xl,n(t)NR(t)

+M�m=1

Xl,m(t)M

� (14)where we average over the total number L ofintensive properties used for encoding the sourceinformation X(t). For the intensive properties,we also average over the number of receivedmolecules NR(t) at time t, while for the intensiveproperties, we average over the total number ofextensive properties M used for encoding.

C. General Principles of MolecularCommunication Channel Capacity

The capacity C of an MC channel in [bit/sec] is, here,defined as the maximum rate of transmission betweenthe molecule emission process and the reception process,where this maximum is with respect to all possible prob-ability distributions of the emission process [14]. This isexpressed by the general formula from Shannon [6], whichdefines the capacity as the maximum mutual informationI(E;P ) between the transmitted signal (emitted mole-cules) E = {tn, pn(tn)}n, where n is the index of an emit-ted molecule at time tn in the set NT(tn), and the receivedsignal (received molecules) P = {tnR ,pnR(tnR)}nR ,where tnR is the time of reception of one or more mole-cules, and nR is the index of a received molecule in theset NR(t), with respect to the probability density functionfE(e) in all the possible values of the transmitted signal

C = maxfE(e)

{I(E;P )} . (15)



The mutual information I(E;P ) in [bit/sec] is definedas

I(E;P ) = H(E) −H(E|P ) = H(P ) −H(P |E)

= H(E) +H(P ) −H(E,P ) (16)

where H(E) is the entropy per second of the transmittedsignal E [6], H(E|P ) is the entropy per second of thetransmitted signal E, given the received signal P , H(P |E)is the entropy per second of the received signal P , giventhe transmitted signal E, and H(E,P ) is the joint entropyper second of the transmitted signal E and the receivedsignal P .

The relationship between the transmitted signal E andthe propagated signal P in an MC system is expressed,in general, by the aforementioned Langevin SDE (4).According to statistical mechanics, this propagation modelcan be separated into two distinct contributions, as shownin Fig. 3. The Fokker–Planck equation [7], which is a deter-ministic partial differential equation (PDE) to computethe probability density of the particles in the propaga-tion space, and a Poisson point process [15], which is astochastic process that results in the assignment of theparticle locations pn in the space based on the result ofthe Fokker–Planck equation. In the following, we detailhow to exploit these properties of the molecule propaga-tion process to define general principles to determine thechannel capacity in MC systems.

The Fokker–Planck equation describes the evolution ofthe particle propagation in the space in the variable ρ(p, t),which is the probability distribution of the location of aparticle as function of the space coordinates p = {pi} andtime t. The expression of this equation for MC systemsaccounts for the aforementioned molecule emission as asource of particles at the transmitter. This translates intoan additional term, namely, (1/n)δ(|p − pn(tn)|)δ(t − tn),which corresponds to the contribution of one particle attime tn and location pn(tn) to the total number n ofpropagating particles up to time tn. We make the followingassumptions: 1) the diffusing particles have a sphericalshape; 2) the diffusing solute particles are in low con-centration; 3) their dimension is much larger than theparticles of the solvent; and 4) their diffusion is isotropicin the considered space. We express this formulation of theFokker–Planck equation as follows:

∂ρ(p, t)∂t

= D∇2ρ(p, t) −∇v(p, t)ρ(p, t)

+

NT(t)�n=1

1

nδ(|p − pn(tn)|)δ(t− tn) (17)

where n is computed from NT(t) according to (8), v(p, t) =(1/m)

�F(p, t)dt [where F(pn, t) = Fn(t) from (4)], andD

is the particle diffusion coefficient, whose expression is as

follows:

D =KbT

6πμr(18)

where Kb is Boltzmann’s constant, T is the absolute tem-perature of the system, μ and r are the aforementionedviscosity of the fluid and the particle radius, respectively.

The Poisson point process is expressed through the sto-chastic process that randomly assigns the location to eachtransmitted particle according to the particle distributionρ(p, t) at each time instant t. This process is a spatialPoisson point process where the expected value is theparticle distribution ρ(p, t), expressed as follows:

pn(t) ∼ Poiss (ρ(p, t)) ∀n ∈ NT (t) (19)

where NT (t) is given by (9). Although specific MC systemimplementations will incorporate other stochastic sources,as described in the next sections of this paper, whichwill impact the performance of the system through noise,we consider the noise generated by the stochastic processin (19) as inevitably present in any MC system describedby our general information-theoretic framework.

The cascade of the aforementioned Fokker–Planck equa-tion and the Poisson point process, as illustrated in Fig. 3,defines a Markov chain [6] in the variables E, ρ, and Pfollowing the order E → ρ → P . This is justified bythe property that E and P are conditionally independentgiven ρ, which is expressed as follows:

fE,P |ρ(e, p) = fE|ρ(e) fP |ρ(p) (20)

since ρ is a function of E from (6)–(8) and theFokker–Planck equation in (17), and the distribution ofP is a function of ρ from [11]–[14] and [19]. The chainrule applied to the joint entropy of E, ρ, and P states thefollowing [6]:

H(E, ρ,P )=H(E,P |ρ)+H(ρ)=H(E|ρ)+H(P |ρ)+H(ρ)(21)

since ρ is a deterministic function of E through the infor-mation encoding in (7) and (12), molecule emission (8),and molecule propagation (17), then the joint entropyper second of E, ρ, and P is equal to the joint entropyper second of E and P

H(E,ρ, P ) = H(E,P ). (22)

By applying (21) and (22) to the third expression in (16),we obtain that the mutual information I(E;P ) of thetransmitted signal E and the received signal P as the sumof the mutual information of a communication system,



which includes only the Fokker–Planck equation (mutualinformation I(E;ρ) of the transmitted signal and the par-ticle distribution) and conditional entropy H(ρ|P ) of theparticle distribution given the received signal

I(E;P ) = H(E) +H(P ) −H(E|ρ) −H(P |ρ) −H(ρ)= I(E;ρ) + I(P ;ρ) −H(ρ)= H(ρ)−H(ρ|E) +H(ρ) −H(ρ|P )−H(ρ)= H(ρ)−H(ρ|P ) (23)

where we applied the first two definitions of mutualinformation from (16) and we considered the fact thatH(ρ|E) = 0 since ρ is completely determined by E (deter-ministic function of E).

As a consequence of (23), to determine the aforemen-tioned capacity C, it is necessary to analytically express(or estimate) the entropy H(ρ) of the particle distribu-tion and the conditional entropy H(ρ|P ) of the particledistribution given the received signal. The former dependsexclusively on the Fokker–Planck equation (17), while thelatter depends on the Poisson point process in (19). These,in turn, depend on the processes underlying moleculepropagation in the particular MC system being consid-ered. In particular, as illustrated in Fig. 1, the aforemen-tioned propagation processes can be classified on the basisof the randomness on the trajectory of the propagatingmolecules, which impact the particular expression of theFokker–Planck equation (17). In the following, for eachclass of MC systems, we present a general informationcapacity expression under simplifying assumptions anddiscuss the impact of specific functional block implemen-tations in more realistic MC systems.

III. M O L E C U L A R C O M M U N I C AT I O NV I A R A N D O M W A L K

In MC systems based on random walk, the moleculesemitted by the transmitter propagate to the receiver solelyby means of Brownian motion. Consequently, the moleculepropagation can be modeled by the Langevin equationin (4), where the drift velocity vn(t) of the fluid and theBrownian-motion-independent force Fn are set to zero.MC based on random walk occurs naturally in a numberof biological systems, and it is considered the simplest andmost widespread molecule propagation process in nature.In the following, we obtain a closed-form expression tocompute the capacity of the Brownian motion channelthrough the aforementioned methodology by defining thefunctional blocks of a basic abstraction of an MC systemvia random walk. Subsequently, we provide more specificfunctional block models of key diffusion-based implemen-tations found in nature, namely, cell calcium signaling,communication through chemical synapses between neu-rons, and quorum sensing networks of bacteria.

Fig. 4. Basic abstraction of an MC system based on random walk.

A. Brownian Motion Channel Capacity

With reference to Fig. 4, we describe the functionalblocks of a basic abstraction of an MC system based onrandom walk.

1) Information Encoding: One single molecule type ismodulated in its number NT(t) at time t proportionally tothe source information X(t), expressed as

NT(t) = KX(t), t > 0 (24)

which is derived from (6) and (7) by considering L = 1,M = 1, and C1 = K.

2) Molecule Emission: Molecules are released in a con-tinuous fashion by an ideal point-wise transmitter (sizeequal to zero) at location pTx = {px,Tx, py,Tx, pz,Tx} inan 3-D space, as shown in Fig. 2. At the time tn of emissionof the nth molecule, its location pn(tn) corresponds to thelocation of the transmitter pTx, expressed as

pn(tn) = pTx (25)

where n is a function of NT(t) according to (8) and (9).

3) Molecule Propagation: Molecules propagate throughthe Brownian motion in the 3-D space according to (4)where Fn = vn = 0. For this basic abstraction model, andto derive analytical expressions to determine the channelcapacity, we make the assumption to have a 3-D space withinfinite extent in every dimension.

4) Molecule Reception: The receiver detects the particlesthat are present inside a spherical volume VR centered atthe receiver location and with radius RVR � d, where d isthe distance between the transmitter and the receiver. Thischoice makes the results of the Brownian motion channelcapacity accounting for the simplest ideal receiver possible(e.g., chemical ligand-binding reception will be furtherlimited by a nonnegligible time of unbinding [16]), where



the number of received molecules NR(t) is expressed as

NR(t) = {n|pn(t) ∈ VR} . (26)

5) Information Decoding: This is ideally based on thecount of the number of detected molecules expressed as

Y (t) = #NR(t), t > 0 (27)

where # stands for the cardinality (number of elements)of the set NR(t) defined in (26).

6) Capacity: As a consequence of the aforementionedfunctional blocks, the Fokker–Planck equation (17) for thisMC system corresponds to the inhomogeneous Fick’s sec-ond law of diffusion, or diffusion equation [8], expressedas follows:

∂ρ(p, t)∂t

= D∇2ρ(p, t) +NT(t)δ(|p − pTx|), t > 0. (28)

As a consequence of the aforementioned moleculereception and information decoding functional blocks,the probability distribution of the output signal Y (t) can beexpressed from the aforementioned spatial Poisson pointprocess (19) as follows:

Pr{Y (t)|ρ̄(t)}(N) =(ρ̄(t)VR)

N

N !exp−ρ̄(t)VR (29)

where ρ̄(t) is the average particle distribution inside thereceiver spherical volume VR, for which simplicity is con-sidered equal to the value of the particle distribution at thecenter pRx of VR, expressed as ρ(pRx, t), in agreement withthe aforementioned assumption on the receiver radius.

The entropy H(ρ) of the particle distribution ρ can beanalytically expressed by stemming from the solution tothe aforementioned Fick’s second law, which is expressedas follows:

ρ(pRx, t) = hDiff(d, t) ∗NT(t) (30)

where d = |p − pTx|, ∗ is the convolution operation, andhDiff(d, t) is the impulse response of (28), expressed asfollows:

hDiff(d, t) =e−

d24Dt

(4πDt)3/2. (31)

As a consequence of (24), (30), and (31), Fick’s second lawcorresponds to a linear and time-invariant filter appliedto the modulated number NT(t) of emitted molecules.According to the formula to compute entropy loss in linearfilters [14], the entropyH ′(ρ) per degree of freedom of the

particle distribution as expressed in (30) is as follows:

H ′(ρ) = H ′(NT) +1

W

�W

log2 |HDiff(f)|2 df (32)

where H ′(NT) and W are the entropy per degree offreedom and the bandwidth, respectively, of the numberof molecules NT(t), and HDiff(f) is the Fourier transformof the impulse response in (28). The entropy H(ρ) canbe then computed by multiplying the entropy H ′(ρ) perdegree of freedom by twice the aforementioned band-width W . The expression in (32) can be evaluated byconsidering the following.

1) The modulated number of molecules NT(t) can bedefined as a band-limited ensemble of functions [14]within a bandwidth W , with the following expression:

NT(t) =∞�k=0

NT

�k

2W

�sin [π(2Wt− k)]π(2Wt− k) , k ∈ N

(33)

where the bandwidth W is, here, defined as the maxi-mum frequency contained in the time-continuous sig-nal NT(t) (24), which corresponds to the modulatednumber of molecules as function of the time t. Theentropy H ′(NT) per degree of freedom then equalto the entropy of NT(t) sampled at time instantsk/(2W ), which is the first term of the sum in (33).

2) The Fourier transform HDiff(f) of the impulseresponse in (28) has the following analyticalexpression:

HDiff(f) =e−(1−j)

�2πf2D d

4πDd. (34)

Consequently, the entropy H(ρ) of the particle distributionρ can be derived from (32) and (34)

H(ρ)=2WH ′(NT)− 4√πd

3 ln 2√DW

32 −4W log2 4πDd. (35)

The conditional entropy H(ρ|P ) of the particle distribu-tion given the received signal can be computed from (29)as per time sample of a spatial Poisson counting processwith rate parameter equal to ρ. According to [15], thisbecomes

H(ρ|P) ∼= 23

E [NT]RVRWd

+ ln

�Γ

�2

3

E [NT]RVRWd

��+

�1 − 2

3

E [NT]RVRWd

�ψ

�2

3

E [NT]RVRWd

�(36)

where E [NT] is the average value of emitted molecules ina time interval equal to 1/(2W ), W is the bandwidth ofthe transmitted signal X, ψ(·) is the digamma function,



D is the diffusion coefficient, d is the distance betweenthe transmitter and the receiver, and RVR is the radius ofthe spherical receiver volume VR. The conditional entropyH(ρ|P ) is then equal to (36) multiplied by two timesthe bandwidth W of the modulated number of moleculesNT(t).

The capacity CBrown of the Brownian channel is thenobtained by substituting (35) and (36) multiplied by2W into (23), and by maximizing it according to (15)constrained to the average thermodynamic power P̄H,defined in [15] as the energy necessary to emit the aver-age number E

�N̂T�

of particles per time sample 1/(2W ),divided by the duration of a time sample. The latter isexpressed as

P̄H =3

2KbTE [n̂T ] 2W (37)

where Kb is Boltzmann’s constant, T is the absolutetemperature of the system, and W is the bandwidth ofthe modulated number of molecules NT(t). The capacityCBrown results in the following expression [15]:

CBrown

∼= 2W�

1 + log2P̄H

3 WKbT

�− 4

√πd

3 ln 2√DW

32

− 4W log2 4πDd − 2W2P̄HRVR

9 W 2 dKbT

− 2W ln�

Γ

�2P̄HRVR

9 W 2 dKbT

��− 2W

�1 − 2P̄HRVR

9 W 2 dKbT

�ψ

�2P̄HRVR

9W 2dKbT

�(38)

where ψ(·) is the digamma function, D is the diffusioncoefficient, d is the distance between the transmitter andthe receiver, and RVR is the radius of the spherical receivervolume VR.

B. Calcium Signaling

Calcium signaling is at the basis of biological cellsignaling regulation, where it is one form of juxtacrine sig-naling, which is found in numerous biological regulatoryfunctions in both animals as well as plants [39]. Juxtacrinesignaling is a form of close contact cell-to-cell or cell-to-extracellular matrix information exchange. The regulationof the cellular process resulting from the Ca2+ signalingcan range between millisecond (e.g., protein synthesis andcell division) and minutes as well as hours. This form ofsignaling can exist in both excitable as well as nonexcitablecells, where the elevated Ca2+ concentration can resultfrom triggering of the internal pathways that are due tothe ligand-receptor chemical reaction of specific moleculesat the cell’s membrane. The analysis of Ca2+-based MC isgoverned by biophysical models, which have been devel-oped through experimental work [40], [41], [45]. Fig. 5

Fig. 5. Illustration of an MC system based on calcium signaling.

illustrates the block diagram of a calcium-signaling-basedMC system, where communication is established throughthe diffusion of ions. The calcium propagates through thegap junction to allow the ions to flow between the cells.

1) Information Encoding: The encoding process can beachieved through the elevation of the Ca2+ ion concen-tration NT from (7) within the cytoplasm of the cell. Thisis achieved by releasing the Ca2+ from the organelles(stores), which is controlled by the intracellular Ca2+

signaling pathway, as well as the intake from the extracel-lular space. The increased concentration of the Ca2+ ionswithin the cytoplasm is dependent on various chemicalreaction stimuli, which may include extracellular agonistsand intracellular messengers (for nonexcitable cells). Oneof the most basic models was proposed in [45], and it con-sists of three types of Ca2+ concentration: the cytoplasm(Ccyt), the stores within the organelles (Cstr), and the Ca2+

buffer (B). The kinetic model of the type in (3) for thesethree components is represented as follows:

∂Cstr∂t

= −k1(h+ h0)(Cstr − Ccyt) + V3 C2cyt

k24 + C2str

(39)

∂Ccyt∂t

= k1(h+ h0)(Cstr − Ccyt) − V3 C2cyt

k24 + C2str

+ k5(h+ h0)(Cext − Ccyt) − V6 C2cyt

k27 + C2str

− k2CcytB + k2(Btotal −B) (40)∂B

∂t= −k2CcytB + k(Btotal −B) (41)

where k1 is the rate of Ca2+ release and influx for thestore, k2 is the binding constant of Ca2+ for the buffers,V3 is the maximum rate of Ca2+ intake to the store,k4 is the disassociation constant for the store calcium,Btotal is the total calcium buffer concentration, k7 is thedisassociation constant of the plasma membrane calciumpump (0.6 μm), k5 is the rate of calcium influx from theexternal medium (0.000158 sec−1), V6 is the maximumrate of the plasma membrane calcium pump (1.5 μm/sec),h is the fractional activity of the channels in the store andplasma membrane (h0 ≤ h ≤ 1), h0 is the basal fractional



activity of the channels in the store and plasma membrane(0.4), and Cext is the extracellular calcium concentration(1500 μm). Equation (39) reflects the change in the Ca2+

in the stored organelles, and this is dependent on the stor-age release as well as the recovery rate, and the quantity ofCcyt; (40) reflects the relationship between the change inCcyt, concentration in the intracellular storage, buffer B inthe cytoplasm, extracellular matrix, as well as the releaseand recovery rate; (41) is the process for the Ca2+ bindingto the buffer B in the cytoplasm.

Based on (7), the modulated concentration Ccyt of Ca2+

ion will result in NT(t) and is represented as

NT(t) = CcytX(t), t > 0 (42)

where L = 1, M = 1, and C1 = Ccyt.

2) Molecular Emission: Once the Ca2+ concentration iselevated, this results in ion wave generation. A linear chan-nel model can result from the Ca2+ concentration once itreaches steady state [39]. The steady state can be achievedwhen (39)–(41) becomes zero. At the steady-state point,the concentration Cssp is less than the quantity of externalCa2+ ions and is represented as follows:

Cssp = k7

�k5(h+ h0)Cext

V6 − k5(h+ h0)Cext . (43)

At the same time, the transient response of the Ca2+

wave is approximated as follows:

Ccyt(t) = Cinitcyt +

�Cssp − C initcyt

�(1 − e−tβ) (44)

whereC initcyt is the initial cytoplasmic calcium concentration,and β is the elevation rate of cytoplasmic calcium [39].

Based on the general molecular emission of MC in (8),the Ca2+ ion n released at time tn will have a locationpCa2+,n that corresponds to the location of the transmitterpTx at the center of the cell and is expressed as

pCa2+,n(tn) = pTx. (45)

3) Molecular Propagation: The propagation of the waveis established through physical connections between thecells, where the generated waves will travel from thecytoplasm through gap junctions [46]. The gap junctionsare composed of two connexons situated on each sideof the cell, and they are formed by six proteins calledconnexins. The probability of the number of open gapjunctions sn out of Sn is modeled as a binomial distributionand is represented as [39]

Pr(snopens) =

�Snsn

�ζsnn (1 − ζn)Sn−sn . (46)

The opening of the gap junction is dependent on theelevation of the Ca2+ concentration. The period for theCa2+ waves to travel through the gap junction is τgap.The effective gap junctional transitional rate θn for cell n,which depends on the level of Ca2+, is represented as

θn =1

τgap

P(n)Ca

D(n)Ca

(47)

where PCa is the permeability of the gap junction andDCa is the diffusion constant of Ca2+ ions. Based on this,the received Ca2+ level at the end of the gap junction andthe cytoplasm entry of the next cell n is represented as

Cncyt�l(n)

jct , τ(n)

jct

�=snSnθnC

n−1cyt�l(n−1)jct , τ

(n−1)jct

�(48)

where n is the cell receiving the Ca2+ emitted from theprevious neighboring cell n − 1. ljct is the gap junctionposition, and τjct is the time instant Ca2+ travels throughthe gap junction between cells n− 1 and n. Once the Ca2+wave enters the cytoplasm of cell n − 1, it will propagatethrough diffusion. The modified Fick’s law for this diffusionis represented as

Cncyt(x, t) =1�

4πD(n)Ca τ

(n)cyt

e−(x2/4DCaτ

(n)cyt ) (49)

where x is the 1-D distance that is perpendicular to thegap junction entry into the cytoplasm, and τcyt is the delaypropagation of Ca2+ in the cytoplasm. This is representedas the inhomogeneous Fick’s second law of diffusion asin (28), with boundary conditions that are defined bythe cell’s membrane. The boundary conditions of the cellmembrane ensure that the finite space contains the oscil-lations of the Ca2+ ions within the cytoplasm. We alsoassume that interference between these oscillations andother components of the cell is negligible. From (28), ρis the distribution of Cncyt(x, t), p is the location of the ionswithin the cell, D is the diffusion coefficient D(n)Ca , and NTis the number of Ca2+ ions as in (42).

Although the models that have been presented arebetween two cells, the channel can be extended to multiplecells as the Ca2+ propagates through the tissue.

4) Molecular Reception: The Ca2+ that propagates intoeach cell will be sensed, and the concentration changes willinvoke the subsequent wave generation. The regenerationof the Ca2+ ion waves as it passes from one cell to thenext is based on the internal Ca2+-induced Ca2+ releaseprocess, and this will depend on the increase in Ca2+

concentration that has been received. Once the numberNR(t) (11) of received Ca2+ ions binds to the organellesof the receiving cell, this will invoke the intracellular Ca2+

signaling pathway to restart the generation of Ca2+ wavesfor the next cell.



5) Information Decoding: Depending on the approachtaken for encoding the information into the Ca2+ ionconcentration, the decoding process will sample the wavesthat are received in the cytoplasm. In the case of amplitudemodulation, the receiver will sample the peak as well asthe duration of the arrived Ca2+ waves.

6) Capacity: Given the highly stochastic behavior ofcalcium signaling, a closed-form expression for calciumsignaling has not been developed. In [39], the gain anddelay of Ca2+ waves traveling through a 1-D array of cellswere proposed. This was developed from extracting thelinear channel behavior of Ca2+ waves that are generatedfrom the cells, as well as a stochastic gap junction modelin (46). A capacity expression was developed in [43] fora 1-D array of cells, using empirical measurements forthe entropy of the received Ca2+ wave that was used todetermine the mutual information. The same approachwas developed in [44] for a channel model of a multidi-mensional array of cells representing a tissue. The analysisconsidered the biophysical properties of three differenttypes of cells (e.g., smooth muscle cells, epithelial, andastrocytes), and how these impact on the Ca2+ wavepropagation. The analysis also considered the reflectivebehavior of the Ca2+ waves due to the boundary con-ditions of the tissue. In [62], a channel model was alsodeveloped between the two cardiomyocyte cells, basedon the electrochemical signaling that influences the Ca2+

propagation. The significance of this channel model wasthe introduction of an electrochemical model for Ca2+

signals that are generated from excitable cells. The mutualinformation was developed based on the OOK modulationof Ca2+ waves, out of which the capacity was obtained.

An open issue for the future is the development ofa closed-form expression for the capacity model. Thisclosed-form expression must include the impact of interac-tions between organelles (e.g., endoplasmic reticulum andmitochondria) that contribute toward Ca2+ generation.This model could then be utilized to understand the impactof abnormalities between the intracellular to intercellularsignaling, and how disease can result from this.

C. Neuron Communications

One of the most complex regulatory systems within thehuman body is the nervous system. The nervous systemmimics an information highway that interconnects a num-ber of different organs as well as various physiologicalsubsystems to the brain. This information highway controlsand maintains homeostatic equilibrium while ensuringadaptations as an organism faces varying environmentalchanges [47]. This highly complex system communicatesthrough electrical stimulation based on a compound actionpotential (AP). At a single-cell level, the nervous system aswell as the brain are constructed from neurons, which com-municate through electrochemical impulse signals knownas AP spikes. Through the highly complex interconnectionof neurons, the brain is able to process information, create

Fig. 6. Sketch of an MC system based on neuron communications.

actions through the control of muscles, store informationfor both short- and long-term memory, as well as control-ling emotions, sensations, and perception.

1) Information Encoding: The discrete impulse signal,or the AP, is established through both electrical and chemi-cal impulses that occur in parallel. The electrical impulses,i.e., electrical spikes, travel through the neuron, while thechemical impulses propagate on the surface of the cell. Theinformation that is conveyed between the neurons dependson the frequency of the electrical spikes that travel throughthe cell. As the electrical spike propagates down the axon,this will result in the chemical impulse that depolarizes andrepolarizes the chemical balance of the neuron. Fig. 6 illus-trates this process. Before the electrical spike propagatesdown the axon, there is a chemical balance in the quantityof ions both inside and outside the neuron. At rest, there ismore potassium ions (K+) and negative ions (−ve) insidethe axon, while higher sodium ions (Na+) and positiveions (+ve) outside. As the impulse propagates down theaxon, the depolarization process starts, where the K+ ionswill diffuse outwards from the axon, and the Na+ ionswill diffuse in the opposite direction into the axon. After ashort period, the repolarization process starts, and this willresult in the reverse process. This sequence will continueuntil the impulse arrives at the terminal synapse. Since theinformation traveling through the neuron is dependent onthe type of stimulated and train of spikes, the encodingprocess can be established through the variations in theelectrical spike train. The train of spikes carries the infor-mation to be projected onto the subsequent target neuron,and this is considered as neural codes that are transmittedthrough the network. The train of spikes that carry thecodes is not only affected by the types of neurons but alsoby the synaptic connections to other neurons, which can beeither inhibitory or excitatory. This is one of the elementsof how physical changes in the environment due to sensorychanges (e.g., touch and hearing) can be processed in thebrain as unique information.

As the spike travels down the axon, this will leadto the neuron–neuron communication process. Althoughelectrical synapse can also occur for neuron-to-neuron



communication, where AP is transferred through the gapjunctions of the cells, in this paper, we only consider thechemical synapse through the diffusion of neurotransmit-ters (note that the chemical synapse is different from thechemical signaling that occurs only along the axon of theneuron). Once the spike arrives at the synaptic terminal,the vesicles containing the neurotransmitters will bind tothe terminal membrane to release the neurotransmittersinto the synaptic cleft, by following a stochastic process.A model for this process has been proposed in [61]. Themodel considers the vesicles are packed in a pool, andwhen a spike arrives at the synaptic terminal, a singlevesicle will be released. Two types of vesicle release processare discussed, which are evoked and spontaneous. For theevoked release, given that the spike time begins at t0 andthe duration Δts, the release occurs in the time interval[t0, t0 + Δts]. During this period, the probability of evokedrelease forNT vesicles within the pool is 1−exp(−NTαvΔt)for Δt ≤ Δts and 1 − exp(−NTαvΔts) for Δt > Δts(this is for a release rate αs). We do not consider thestochastic production of NT and assume a fixed quantityof production as well as secretion, because this is highlydependent on the type of cells as well as pattern of APsignals. In the case of spontaneous release, the waitingtime before secretion is approximately 8 min, and for thisreason, the probability of release for Nv vesicles withinthe pool during Δt is 1 − exp(NvΔt/480)). Based on amaximum vesicle capacity Nv in the pool, the release inith time slot (time slot is Δt larger than the refractoryperiod) is determined by the probability F (Nv), expressedas follows:

Fi(Nv)=1−�exp(−NvαvΔts)ps+exp

�−NvΔt

480

�(1−ps)

�(50)

where ps is the probability of a spike arrival at the ithtime slot, which is a Poisson process with rate equal to thesource information X(t), with the following expression:

ps = 1 − exp{−X(t)Δt}. (51)

The model also considers one vesicle vacancy replen-ishment G(τD,Δt), where τD is the mean recovery timefor one vacancy after Δt, and is modeled as a Poissonprocess (G(τD,Δt) = 1 − exp − τ−1D Δt)). After Δt,the vesicle recovery is governed by a binomial distribution(B(NMAX −N,G(τD,Δt))) [61].

2) Molecular Emission: When the released vesicle bindsonto the membrane, it will secreteNT(t) neurotransmittersat the same time t. We assume the neurotransmittersrelease to be point-wise, as defined in (8). This is expressedas in (8), where pn = (pn,x, pn,y, pn,z) and ST correspondsto points in the membrane surface of the neuron facing thesynaptic cleft.

3) Molecular Propagation: These neurotransmitters willpropagate through diffusion in the synaptic cleft (the dis-tance of the synaptic cleft is approximately 20 nm). Theregion of diffusion for the neurotransmitters is a confinedspace. This means that at an initial stage, a number of neu-rotransmitters will bind to the postsynaptic neuron, whilea short period later, a different number will arrive to bind.This is modeled in [60] with the following expression:

ρ(NT,p, t) =NT

(√

4πDt)3e

−(px−pn,x)2−(py−pn,y)24Dt� −1�

k=−∞(2 − Pu)(1 − Pu)−k+1e

−(pz−(2k+1)H)24Dt

+∞�k=0

(2 − Pu)(1 − Pu)ke−(pz−(2k+1)H)2

4Dt

�(52)

where ρ is the probability density of neurotransmitters atlocation p = (px, py, pz) in the synaptic cleft, Pu representsthe uptake probability of the neurotransmitters (whenPu = 0, none of the neurotransmitters have reached thepostsynaptic neuron for uptake, while Pu = 1, means allhave reached the target within a specified time), H is thelength of presynaptic cleft along the z-axis, and D is thediffusion coefficient of the neurotransmitters. The locationpn of each neurotransmitter is based on a Poisson pointprocess as in (19).

4) Molecular Reception: As these neurotransmitters dif-fuse and arrive at the postsynaptic neuron, they will bindonto the receptors forming a ligand-receptor complex.According to [61], the binding probability is based on anexpected neurotransmitter flux that will bind to vacantreceptors during the sampling time t. This means that asthe neurotransmitters arrive at the postsynaptic neuron,the number of vacant receptors will also reduce. This willalso reduce the binding probability as the sampling timeincreases. Therefore, at the kth sampling time, the bindingprobability will be represented as

Pb(kΔt) = a(kΔt)[1 − (1 − Pe(kΔt)N(kΔt))] (53)

where the probability Pe of finding the neurotransmittersinside the effective volume Ve is as follows:

Pe(t) =

��Ve

ρ(NT,p, t)dp (54)

where Δt is the duration of the sampling period, and a isthe availability of a receptor [0, 1]. The probability of theneurotransmitters within the volume space of the synapticcleft will be the basis of the prediction of the quantityof neurotransmitters that will bind to the receptors ofthe postsynaptic neuron. This results in a relationshipbetween the binding process and the patterns of spike



trains generated to carry the information between theneurons. a is also adjusted for the next step and expressedas follows:

a((k + 1)Δt) = a(kΔt) − Pb(kΔt). (55)

Therefore, the expected number NR of neurotransmit-ters bindings in each sampling period is the summation ofbinding probabilities of all the receptors and is representedas follows:

NR(kΔt) = M0Pb(kΔt) (56)

where M0 is the number of receptors at the receiver.

5) Information Decoding: When sufficient neurotrans-mitters bind onto the receptors of the postsynaptic neuron,it will invoke an impulse that will again travel along theneuron, which will open a channel that allows the positiveions to flow into the cell, starting another impulse, whichwas described in the information encoding.

6) Capacity: A physical channel model was developedin [49] for two neurons, considering the mutual informa-tion of the AP and the diffusion of neurotransmitters in thesynaptic cleft. In [51], an upper bound information capac-ity model was developed for both bipartite and tripartiteneural connections using results from optical Poisson chan-nels. The Poisson channel model was used to representthe impulse of AP that is generated from the presynapticneurons, and how this varies depending on the feedbackcontrol from the astrocytes. A multiple access model forneural connections was also developed in [48] based on asequence of spike timings from the presynaptic neurons.This is based on multiple presynaptic neurons that formconnections to the postsynaptic neuron and specifically onthe capacity that is impacted from the variations in theAP. The limitations in all these capacity models are thatthere are no closed-form solutions proposed. This is anopen research problem that needs to be investigated for thefuture. The closed-form expression should also considerthe impact of different types of biophysical propertiesof the neurons, and how this impacts the capacity. Thiscan lead to applications of neuron communication modelsthat can be applied to understanding neuronal disease,such as the correlation between impairments of the sig-naling process and changes in the AP signaling sequences.An extension toward neuronal networks will also need tobe investigated, by understanding how the MC varies assignals propagate through heterogeneous neurons (e.g.,pyramidal and fusiform) in the network.

D. Bacterial Quorum Sensing

Besides the communication process through the transferof DNA, bacteria also have another form of natural commu-nication that uses molecules. This communication is based

Fig. 7. Scheme of an MC system based on bacterial quorumsensing.

on the simple secretion of molecules, and this could becooperative between all the bacteria in the vicinity. Thisform of communication is found in both Gram-negativeand Gram-positive bacteria. One bacterial functionalitythat results from this simple communication is knownas quorum sensing, where the bacteria communicatesthrough molecules known as autoinducers and results insynchronized gene expression of the bacterial population(the autoinducers are known as messenger molecules).This communication is ineffective when the bacteriumis on its own; however, as a population, this leads tonumerous powerful functionalities, and hence the name“quorum.” A number of diverse physiological activitiescan emerge from quorum sensing, and examples includebiofilm formation, antibiotic production, and biolumines-cence. Fig. 7 illustrates an MC system that is based on thebacterial quorum sensing communication process.

1) Information Encoding: The encoding process canbe achieved by stimulating the bacterial populationwith an external chemical signal, in order to produceautoinducers. For example, in [56], signaling moleculeN-(3-Oxyhexanoyl)-L-homoserine lactone, or C6-HSL, wasinjected into engineered E. coli bacteria and in response,this resulted in the cells activating a genetic program toproduce green fluorescent protein (GFP). In this simplesetup, OOK was achieved, where the application of C6-HSLproduced a pulse that represents a single bit. Anotherexample is the generation of pulse-amplitude modula-tion (PAM) using a similar excitation approach [55].

Molecule emissions by the bacteria are initiated fromstimuli excitation, either through the influence of exter-nal chemicals applied to the population such as C6-HSLdescribed above or initiated from a bacterium within thequorum, which will result in a chain reaction of other cellswithin the vicinity to produce the molecules collectively.In this paper, we will only focus on the first case, whereexternal stimuli are applied to the bacterial population.The assumption for this case is that external stimuli, whichcorrespond to administrations of a chemical agent, canbe assumed evenly distributed throughout the population,resulting in all bacteria equally producing the autoinducermolecules to be emitted.



The following equations define the collective productionof molecules by the bacterial population [64], where thesource information X(t) is the basis for the production ofautoinducers A(t), expressed through a chemical kineticmodel (3) as follows:

dA(t)

dt= cA+

kAC(t)

KA+C(t)−k0A(t)−k1R(t)A(t)+k2RA(t)

−poutA(t)+pinE(t) (57)dR(t)

dt= cR+

kRC(t)

KR+C(t)−k3A(t)−k1R(t)A(t)+k2RA(t)

(58)dRA(t)

dt= k1R(t)A(t)−k2RA(t)−2k4RA(t)2+2k5C(t)

(59)dc(t)

dt= k4RA(t)

2+k5C(t) (60)

dE(t)

dt= (poutA(t)−pinE(t))−DA(t). (61)

Equation (57) is the production rate of internal autoin-ducer AT (t) within the bacterium, and this is used tofurther produce more molecules; (58) is the productionrate of receptors inside the bacterium, which will bind tothe internal autoinducers to transform the complex intoa receptor-bound autoinducer (RA) monomer; (59) is theproduction rate of RA monomers, and this depends onthe number of autoinducers (A), receptors (R), as wellas the dimers (C); (60) is the production rate of theRA dimers. A dimer is an association of two monomers,and in this case, it is the association of two RA complex.Equation (61) is the production rate of the autoinducersecreted from each bacterium membrane, which will freelydiffuse into the environment and result in a spatiallyhomogeneous concentration.

2) Molecular Emission: A concentration of autoinducersA(t) is released collectively from the bacterial population,as defined in (8), where ST is now the volume of thecolony.

3) Molecular Propagation: The autoinducers concentra-tion A(t) emitted by the transmitter bacteria will diffuseinto the environment. The diffusion process will follow theinhomogenous second Fick’s law similar to (28), this timeexpressed as:

∂ρ(p, t)∂t

= D∇2ρ(p, t) +A(t)�n=1

1

nδ(|p − pn(tn)|)δ(t− tn)

(62)

where ρ(p, t) is the distribution of autoinducers atlocation p and time t.

4) Molecular Reception: The bacteria receives the autoin-ducer molecules, which leads to an internal signal path-way process in response. The probability of the moleculesbinding to a bacterium’s ith receptors is represented as a

chemical kinetic model (3) and expressed as follows [63]:

dpidt

= −κpi + ρ(p, t)|p∈SRiγ(1 − pi) (63)

where ρ(p, t)|p∈SRi is the distribution of autoinducerswithin the receiver ith bacterium volume SR, γ is the inputgain, and κ is the rate at which the molecules that havebound to the receptor will detach from the receptors.

There is also randomness in the binding process of themolecules to the receptors of the bacterium. Accordingto [59], for each bacterium i, the number of activatedreceptors Xi is a Binomial random variable with para-meters (Mo, pi), where Mo is the number of receptorsand p is the probability of binding for the ith bacteriumas defined in (63). Based on this, NR will be the totalnumber of activated receptors, where NR =

�Moi=1Xi.

However, a reporter mechanism is required in responseto receiving the molecules. This could be through theexpression of GFP, where the fluorescence proteins willreflect a green light when illuminated by an ultravioletlight that can be detected through a photodetector or evenimaging technologies on board of a microscope.

5) Information Decoding: Depending on the reportingmechanism taken, the approach for decoding will bethrough sampling. In the event of fluorescence using GFP,sampling can be performed on the intensity pulse that isgenerated. Sampling efficiency for GFP generated pulsewas investigated in [55] based on the peak value, the totalresponse duration, the ramp-up slope, as well as theramp-down slope.

6) Capacity: Although there has not been anyclosed-form expression for the capacity of bacterialquorum sensing, there has been a number of channelmodels developed based on numerical expressions usingmutual information. In [55] and [56], an experimentalMC model was developed between two populationsof bacteria that communicate through the diffusionof autoinducers, which was part of the NationalScience Foundation MoNaCo project [58]. In [56],the capacity was defined based on the mutual informationin communication-by-silence of bacterial quorum sensing.Communication-by-silence was originally proposed fornoisy wireless channels, and this suits the high latencypropagation of molecules for bacterial quorum sensing.The capacity model was based on the delay betweentwo-pulses representing the start and stop bits that aretransmitted, where the counting process between thedelay represents the information. In [59], a mutualinformation expression for capacity was defined formultihop bacterial network using quorum sensing. Theexpression considers the intracellular to intercellularsignaling that produces the diffused molecules betweenseparate bacterial populations. The next evolutionarystep that is required is a closed-form solution for thebacterial quorum sensing. The closed-form model should



Fig. 8. Basic abstraction of an MC system based on drifted randomwalk.

integrate the types of bacteria as well as molecules that areproduced, and the interactions between different bacteriaspecies, which, to date, have not been investigated inthe MC community. Applications can result from thecommunication process of the bacterial quorum sensing,such as the impact of attacks from different species thatcan be detected from variations of the communicationperformance. An example application is in [54], wherea synchronization between nanomachines was proposedusing the quorum sensing.

IV. M O L E C U L A R C O M M U N I C AT I O N V I AD R I F T E D R A N D O M W A L K

In MC systems based on drifted random walk, the mole-cules emitted by the transmitter not only propagate viaBrownian motion but their locations change with a velocityvn(t) independent of the Brownian motion or the viscosityof the fluid. Consequently, this propagation is modeledby the Langevin equation in (4) with Brownian-motion-independent force Fn(t) set to zero. In the following,we revise the Brownian motion capacity expressed inSection III-A through the definition of a basic abstractionof an MC system via based on Brownian motion with drift.Subsequently, we exemplify more realistic functional blockmodels of system based on drifted random walk, whichhave been studied in recent years, namely, MC based onthe cardiovascular system or microfluidic platforms andpheromone communication between plants.

A. Capacity of the Brownian MotionChannel With Drift

The basic abstraction of an MC system via Brownianmotion with drift is shown in Fig. 8. In the follow-ing, we describe the differences in the channel modelwith respect to the Brownian motion channel detailed inSection III-A.

1) Molecule Propagation: Molecules propagate by sum-ming the Brownian motion components with a constantand homogeneous drift velocity v = {vx, vy , vz} in the3-D space according to (4) where Fn(t) = 0, and vn(t)is constant and equal to v for every particle n. As in

Section III-A, we make the assumption to have a 3-D spacewith infinite extent in every dimension.

2) Capacity: As a consequence of the aforementionedmolecule propagation, the Fokker–Planck equation (17)for this MC system corresponds to the inhomoge-neous Smoluchowski equation, or advection–diffusionequation [7], expressed as follows:

∂ρ(p, t)∂t

= D∇2ρ(p, t) − v∇ρ(p, t) +NT(t)δ(|p − pTx|).(64)

Since this MC system utilizes the same functional blocksfor the molecule reception and information decodingas those described in Section III-A, the Poisson pointprocess (19) again becomes a Poisson counting process asexpressed in (29).

The entropy H(ρ) of the particle distribution ρ can beanalytically expressed similar to Section III-A, but this timebased on (64), which is expressed as follows:

ρ(pRx, t) = hAdv(pTx,pRx, t) ∗NT(t) (65)

where hAdv(pTx,pRx, t) is the impulse response of (64),expressed as follows:

hAdv(pTx,pRx, t) =e−

|pRx−pTx−vt|24Dt

(4πDt)3/2. (66)

As for Fick’s second law in Section III-A, also theadvection–diffusion described in (30), (64), and (66) cor-responds to a linear and time-invariant filter applied tothe modulated number NT(t) of emitted molecules. Conse-quently, we can apply the formula in (32) where the termHDiff(f) is substituted with HAdv(f), which is the Fouriertransform of the impulse response in (66). The latter doesnot have analytical solution such as (31), and the followingexpression has to be solved numerically:

HAdv(f) =

�hAdv(pTx,pRx, t))e

−j2πftdt. (67)

Consequently, the entropy H(ρ) of the particle distribu-tion ρ can be expressed as follows:

H(ρ) = 2WH ′(NT) +�W

log2 |HAdv(f)|2 df. (68)

The conditional entropy H(ρ|P ) of the particle distrib-ution given the received signal can be derived similar toSection III-A, and expressed as in (36)

The capacity CDBrown of the Brownian channel with driftis then obtained by substituting (36) and (68) multipliedby 2W into (23), and by maximizing it according to



Fig. 9. MC functional blocks based on the cardiovascular system.

(15) constrained to the average thermodynamic power P̄Hexpressed in (37). The maximum value expression can beobtained similarly as in [15] even without an analyticalexpression for HDiff(f), since the latter does not dependon the probability distribution of the modulated numberof molecules NT(t). The capacity CDBrown results in thefollowing expression [15]:

CDBrown

∼= 2W�

1 + log2P̄H

3 WKbT

�+

�W

log2 |HAdv(f)|2 df − 2W2P̄HRVR

9 W 2 dKbT

− 2W ln�

Γ

�2P̄HRVR

9 W 2 dKbT

��− 2W

�1 − 2P̄HRVR

9 W 2 dKbT

�ψ

�2P̄HRVR

9W 2dKbT

�(69)

where ψ(·) is the digamma function, D is the diffusioncoefficient, d is the distance between the transmitter andthe receiver, and RVR is the radius of the spherical receivervolume VR.

B. Cardiovascular System

The cardiovascular system is a molecule propagationnetwork in the human body composed of the heart,the blood, and the blood vessels, where the heartpumps the blood through the blood vessels, resulting intoa drift of the molecules that are subject to Brownianmotion within the blood. An MC system has been mod-eled around the cardiovascular system as an MC channel,with the final goal of studying the body distribution ofdrug molecules within particulate drug delivery systems(PDDSs) [10]–[12], and this is illustrated in Fig. 9. In suchsystems, drug molecules are injected into a blood ves-sel at a specific location of the cardiovascular system,they propagate through drifted random walk along theblood vessels, while they distribute through bifurcations

to their branches, until reaching the diseased location ofthe body in need of the drug, where the drug moleculesare absorbed by the tissues. Such a study demonstrates adirect application of MC theory to personalized nanomedi-cine, where the final goal is to provide a methodologyto optimize the PDDSs parameters, such as the injectionlocation and time evolution, according to cardiovascularsystem parameters, where many of those are patient-specific. Moreover, these system models will be essentialto design future communication links to realize pervasivenetworks of nanoscale wearable and implantable devices,i.e., the IoBNT [3]. In the following, we detail the specificfunctional blocks.

1) Information Encoding: The source information X(t)is encoded into a proportional amount of information (ordrug in PDDS context) molecules ND(t) present in thesolution to be injected in the cardiovascular system, similarto (24).

2) Molecule Emission: The information molecules areemitted in the blood vessel at a predefined locationof injection pTx (point-wise transmitter) by following asequence of impulses emitted at a specific time inter-val Ts. According to [11], this models the behavior of acomputer-controlled pump infusion syringe and, in gen-eral, expresses a molecule emission according to pulses(where δ(·) might have a different shape than the Dirac’s),e.g., emitted by engineered cells [13] in an IoBNT sce-nario. This is expressed as

pn(tn) = pTx, n∈�0,

� tn0

Q−1�q=0

ND(t)δ(t−qTs)dt�

(70)

where Q is the total number of injection impulses.

3) Molecule Propagation: The emitted molecules prop-agate through the cardiovascular system according toBrownian motion with a drift velocity v(t,p) (4), whichis a function of the time t and the location p. The timedependence is a function of the heart pumping action whileeach molecule propagates, while the location dependenceis a function of the location of the molecule at each timeinstant.

4) Molecule Reception: The emitted molecules propagateuntil reaching the location in the cardiovascular systemwhere the receiver is located (in the PDDSs case, a bloodvessel is in contact with the targeted tissue to be healed).The received signal corresponds to the particles presentwithin a volume VR surrounding the receiver, with a simi-lar expression as in (26).

5) Information Decoding: The received molecules arerecognized and absorbed by the receiver (diseased tissuein PDDSs, where the information they carry corresponds tothe final healing action at the tissue itself). This is modeledas a quantity proportional to the number of received



molecules according to the molecule reception rate pD(t)as function of the time t, expressed as follows:

Y (t) = pD(t)NR(t) (71)

where pD(t) is expressed as [11]

pD(t) = πr2omRmLe

− χaβw(t)kB Tpr0mR

��aγ

+ψ�Fs+

a2r0Rs

�(72)

where r0 is the radius of the section of an informa-tion molecule, mR is the density of chemical receptorsat the surface of the receiver, mL is the density of lig-and/biomarkers at the surface of the molecules that canbind the receptors, kB is Boltzmann’s constant, Tp is theblood absolute temperature, Fs is the blood molecule dragforce, Rs is the rotational moment of force on the moleculedue to the blood flow, and βw(t) is the blood vessel wallsheer stress as function of the time t. Values or expressionsof these parameters are detailed in [11].

6) Capacity: Given the generally accepted assumptionthat the blood flow in the vessels is laminar [12], the inho-mogeneous advection–diffusion equation of this MC sys-tem is simplified into the Navier–Stokes equation. To deter-mine if the flow is laminar, a metric known as the Reynoldsnumber Re is used, where a laminar flow is characterizedby Re < 2300. For blood flow in vessels, Re ≤ 2000. Theadvection–diffusion equation in this scenario relates theinformation molecule distribution ρ(p, t) in every locationof the cardiovascular system to the blood velocity ul(r, t)as a function of the radial coordinate r and the time tin the artery l of the cardiovascular system, expressed asfollows [12]:

∂ρ(p, t)∂t

= −∇. [−D∇ρ(p, t) + ul (r , t)ρ(p, t)]+ ND(t)δ(|p − pTx|). (73)

The solution to (73) is found by applying the harmonictransfer matrix (HTM) theory [36] and transmission linetheory [35] to express the transfer function of each arteryand bifurcation in the cardiovascular system, as explainedin [12]. Under the aforementioned assumption of laminarblood flow, as well as the assumptions that the bloodvelocity is homogeneous along the longitude of an artery,and that it only depends on the time variable t andthe radial coordinate r in the artery, this corresponds tosolving the Navier–Stokes equation [34], which relates theblood velocity vector ul(r, t) to the blood pressure p(t) asfunctions of the time t. This is expressed as follows [12]:

ρB

�∂ul(r, t)

∂t+ ul(r, t) · ∇ul(r, t)

�= −∇p(t)

+μ∇2ul(r, t) + f(74)

where ρB is the blood density, which we assume homo-geneous, μ is the blood viscosity, and f represents thecontribution of blood vessel wall properties [37].

As a consequence of the aforementioned moleculereception and information decoding functional blocks,the probability distribution in the number of receivedmolecules NR is a Poisson counting process as in (29),where in this case, VR is the volume surrounding the targettissue, as mentioned above.

The capacity CCV of this MC system is computed in [11]by stemming from the aforementioned models. The finalexpression is as follows:

CCV = TS

R�r=1

ψm

�Q�q=1

αq,rAqpr

�(75)

where αq,r summarizes the probability to successfullyreceive and decode an information molecule emitted at theqth interval, defined above, and received at the rth inter-val. An is the maximum nontoxic number of informationmolecules at the time qTs, pD is a coefficient depending onthe aforementioned drug reception rate pD(t) that takesinto account a full reception interval, R is the duration ofthe reception, divided into time intervals of duration Ts,Q is the aforementioned total number of injection pulses,and ψm takes into account noise sources at the injection,as described in [11].

C. Microfluidic Systems

Microfluidics is a technology that enables analysis andcharacterization of fluid dynamics at submillimeter-scale.Through the use of microchannels that allow a mixtureof fluid to flow, the technology can allow integrationof both chemical assay as well as molecular biologyoperations [80]. Examples of these operations include theability to detect as well as separate out specific types ofmolecules on a Lab-on-Chip. MC systems have also beenproposed for microfluidic systems [74], [76], [79]. Net-works of microfluidic channels integrated with MC havebeen proposed to allow multiple steps of automated chem-ical analysis [74]. In [78], microfluidic-based MC was pro-posed for Network-on-chip communication, building onintegrated microchannels that were cooling the computerprocessors.

Since the microfluidic system is considered as a simpleversion of a cardiovascular system, and it is known to beutilized for mixing different molecule types, the microflu-idic structure considered has multiple transmitters and asingle receiver, as illustrated in Fig. 10 [74]. The trans-mitters release molecules, which will diffuse through themicrochannel, where the molecules will propagate underthe influence of a flow with a constant drift velocity v = vxaccording to (4) (we only consider a unidirectional flowalong the x-axis). Therefore, multiple transmitters thatdo not have a centralized controller, such as a dropletregister proposed in [78], can result in interference at



Fig. 10. Illustration of MC based on microfluidic systems.

the receiver, impacting the capacity. This specific structurewill be analyzed and discussed in this section, where thecapacity will be derived based on the model proposedin [74].

1) Information Encoding: The encoding is achievedwhen a concentration of molecules is released to repre-sent the source information X(t). In this particular case,an OOK modulation scheme is used, where each of thetransmitter chamber will produce NT(t) molecules, whichis based on (24).

2) Molecule Emission: The production rate of the mole-cules will depend on the frequency f0 of NT(t) productionfrom a point source pTx . This could be a population ofcells that will coordinate to produce the molecules anddiffuse into the environment [74]. For example, in [56],genetically engineered bacteria are placed in the transmit-ter chamber, and will collectively release molecules uponan external stimulus.

3) Molecule Propagation: The hydrodynamic propertiesof the fluid flow within the microchannel are governed bya Reynolds number Re � 100, which results in a velocityof the flow within the microchannel v(t,p) according tothe Navier–Stokes equation.

4) Molecule Reception: The flow will create an advectiondrift that drives the molecules toward the receive chamber.The chamber will be a volume that receives the molecules,and it is assumed that the space will be large enough tocapture the majority of the molecules (26).

5) Information Decoding: The molecules within thechamber will be sampled to determine the informationthat was transmitted. This will require synchronizationbetween the transmitters as well as the receivers.

6) Capacity: There are similarities in the Fokker–Planckequation that was applied to the cardiovascular and themicrofluidic system. According to (73), the informationmolecule distribution ρ(p, t) is dependent on the bloodvelocity ul(r, t). However, in the case of the microfluidicsystem, the location of the molecules is depending onthe velocity along the x-axis ux(a, b, l), where a, b, and lare the microchannel height, width, and length, respec-tively. The Navier–Stokes equation can be solved toward

an analytical solution for the flow velocity ux(a, b, l) of arectangular-shaped channel, which is represented as

ux(a, b, l) =a2

12μlμl�1 − 0.63a

b

�Δp (76)

where μ is the viscosity of the fluid, and Δp is the pressuredrop for the length of the channel.

According to [74], the received signal is represented as

y = αx+ n (77)

where x is the number of transmitted encoded molecules[equal toNT(t) in the general framework], α is the channelgain, and n is the channel noise. Since we are consideringmultiple transmitters, the gain α in (77) is defined as anend-to-end channel gain αij for interfering signals from jtransmitters to receiver i and is expressed as

αij = αijchα

2tx/rx (78)

where the gain of the transmitter and receiver αtx/rx is

αtx/rx = exp

�−4π

2f20u2tx/rx

D0τtx/rx

�sinc

�atx/rxutx/rx

f0

�(79)

where atx/rx is the width of the transmitter and the receiverchamber, utx/rx is the propagation velocity, τtx/rx is thepropagation delay from the chamber to the microchannel,and f0 is the rate of molecules release from the transmitter.The signal gain of the channel αch is represented as

αch = exp

�−4π

2f20u2ch

Dτch

�(80)

where τch is the propagation delay along the microchan-nel, uch is the propagation velocity, and D is the Taylordispersion adjusted diffusion coefficient for the rectangularmicrofluidic channels.

According to [74], at low frequencies of emitted mole-cules from the transmitter chamber, the spectral densityof the received molecule signal is assumed to be flat. Thismeans that the noise can be considered as additive whiteGaussian noise (AWGN), and this is represented as

σ2 =

�2α4tx/rx

Dτchu2ch

+ 4D0τtx/rxu2tx/rx

α2chα2tx/rx

�4π2f20φ

2. (81)

The magnitude of the interference from the differenttransmitters will also need to be considered. The varianceof the interference from transmitter j on the receiver i isrepresented as

ζ2ij =�αijch�2α4tx/rxφ

2j . (82)



Fig. 11. Illustration of MC based on pheromone communication.

There is also an induced noise from the interferingtransmitters within the microfluidic channel. The varianceof the intefering transmitter j on receiver i is representedas

ξ2ij =�1 − (αijch)2

�

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