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Energy model for synaptic channel in neuro-spike communication
Nano Communication Networks 19 (2019) 102–109
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Nano Communication Networks journal homepage: www.elsevier.com/locate/nanocomnet
Energy model for synaptic channel in neuro-spike communication ∗
Zhen Cheng , Yiming Zhang, Huiting Zhao, Fei Lin, Kaikai Chi School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China
article
info
Article history: Received 25 October 2017 Received in revised form 30 December 2018 Accepted 14 January 2019 Available online 18 January 2019 Keywords: Neuro-spike communication Synaptic channel Energy model Channel capacity Energy efficient
a b s t r a c t The neuro-spike communication (NSC) is inspired by electrochemical communication among biological neurons. In this paper, considering that the energy consumption has important effects on the performance of NSC system, an energy model for the synaptic channel in a point-to-point NSC system is first proposed. Then the energy efficiency which is defined as the channel capacity per unit energy consumption for one spike transmission is analyzed. The numerical results demonstrate that the main parameters including the vesicle release probability, the optimal threshold, the number of the neurotransmitters contained in one vesicle and the side length of the region in which the receptors are located have different impacts on total energy consumption and channel capacity. More importantly, the vesicle release probability is the key factor of energy efficiency. The setting of these system parameters can be used to provide guidelines for designing energy efficient NSC systems. © 2019 Published by Elsevier B.V.
1. Introduction Nanonetworks have important applications which are being considered in biomedical, industrial and environmental areas [1– 3]. In particular, molecular communication (MC) [4], where the information is transmitted, propagated and received in a biological environment by the exchange of molecules, is a promising communication technique to realize nanonetworks [5]. One of the most attractive fields in MC is neuro-spike communication (NSC) [6], which is inspired from signal transmission between neurons. In NSC, electrochemical impulses and neurotransmitters are used for information transmission from the transmitter neuron to the receiver neuron [7]. Generally, the NSC system plays a significant role in transporting information in the body nervous system [8]. In recent years, there have been growing interests and research efforts dedicated to NSC. Many researchers focused on the performance of the NSC system on the basis of the NSC channel model [9,10]. Akan et al. [11] evaluated the throughput and delay performance of the mobile ad hoc molecular nanonetworks by modeling the NSC channel. In [12], Maham designed the optimum binary detector and derived the probability of error at the receiver neuron for the NSC system. In [13], the authors considered a point-to-point NSC model and investigated the effects on the error probability of the NSC channel. Lee and Cho [14] studied the capacity of NSC system based on the process of the information transmission including the axon propagation, vesicle release and ∗ Corresponding author. E-mail addresses: [email protected] (Z. Cheng), [email protected] (Y. Zhang), [email protected] (H. Zhao), [email protected] (F. Lin), [email protected] (K. Chi). https://doi.org/10.1016/j.nancom.2019.01.004 1878-7789/© 2019 Published by Elsevier B.V.
neurotransmitter diffusion. Ramezani and Akan [15] provided a realistic NSC model and evaluated the impacts of availability of vesicles on the channel capacity. The energy consumption requirements constitute the limits on the performance of the MC system. In [16], Schreiber et al. investigated how the factors influenced the energy efficiency of signaling mechanisms. In [17], an energy model for the diffusionbased MC system was proposed and two optimization problems were set up. In [18], the authors proposed and modeled an energy efficient MC system with a simultaneous molecular information and energy transfer relay. Guo et al. [19] analyzed the information delivery energy efficiency of bacteria mobile relays. In [20], an energy model for active transport MC was presented to show that the energy consumption was important in the engineering of this MC system. Ramezani et al. [21] and evaluated the effects of metabolic energy constraints on the sum rate of the multipleinput single-output (MISO) NSC system. However, the metabolic energy is only considered to be consumed by opening and closing of numerous ionic channels on the cell membrane. In this paper, an energy model combination with the whole communication process of the NSC system is proposed, which can be used to provide the guidelines for designing energy efficient NSC systems. The main contributions of our paper are summarized as follows: (1) An energy model for the synaptic channel in a point-to-point NSC system is proposed based on the communication process of one spike transmission. (2) On the basis of this energy model, the energy efficiency which is the channel capacity per unit energy consumption for one spike transmission is formulated. (3) The numerical results show that the main parameters, such as the vesicle release probability, the optimal threshold, the number of the neurotransmitters contained in one vesicle and the
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side length of the region in which the receptors are located have different effects on the total energy consumption and channel capacity. In particular, the vesicle release probability is the key factor of energy efficiency. The remainder of this paper is organized as follows. Section 2 describes the NSC system model. In Section 3, an energy model for the synaptic channel in the point-to-point NSC system is presented and the energy efficiency is analyzed. Numerical results are given in Section 4. Finally, this paper is concluded in Section 5. 2. The NSC system model
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transmission, each spike is transmitted to presynaptic terminals through axon, then the neurotransmitters contained in the vesicles are released to the synaptic cleft and propagate in the synaptic transmission, finally the neurotransmitters bind to the receptors on the postsynaptic neuron to generate the spike in the step of spike generation. Thus one spike transmission between the transmitter neuron and the receiver neuron is finished. The energy consumption is mainly spent in the communication steps including the synthesis of the vesicles and neurotransmitters, neurotransmitters diffusion and neurotransmitter–receptor binding. (1) Synthesis of the vesicles and neurotransmitters
In this section, we introduce the point-to-point NSC system model between two neurons. In the NSC system model, the transmission of information from the transmitter neuron to the receiver neuron is mediated by the electrochemical impulses called spikes. The whole communication process of the NSC system involves three steps including the axonal transmission, the synaptic transmission and the spike generation. The structure of the NSC system model is shown in Fig. 1. (1) The axonal transmission. In the NSC system, the information is encoded with a spiking sequence, then it propagates along the axon which can be considered as a communication channel connecting the soma and the presynaptic terminals. At the end of the axon, there are some presynaptic terminals, by which the vesicles containing the neurotransmitters are released to the gaps between the transmitter neuron and the receiver neuron which are called the synaptic cleft. (2) The synaptic transmission. The synaptic signal can be modeled by a binary random process, in which one and zero represent the events of occurrence and no-occurrence of one spike, respectively. If one spike arrives to the presynaptic terminal, the vesicles are released to the synaptic cleft. Otherwise, no vesicles are released. The second step of the NSC is synaptic transmission which is initiated by the release of the vesicles on the presynaptic terminal to the synaptic cleft. After the neurotransmitters are released by the vesicles, they diffuse in the synaptic cleft. The receptors in the postsynaptic terminal are located at the dendrites of the receiver neuron to capture the diffusing neurotransmitter molecules. Note that for successive synaptic transmissions, neurotransmitters are assumed to be removed from synaptic cleft before next vesicle releases for next spike transmission. (3) The spike generation. When the neurotransmitters bind to the receptors on the postsynaptic terminal of the receiver neuron, the third step of NSC which is called the spike generation, is established by the movement of ions. Then, the ionic channels on the surface of the output neuron open and allow the flow of ions to the receiver neuron. The movement of ions provides excitation and causes the membrane potential of the receiver neuron to change rapidly, and it finally leads to the Excitatory Post Synaptic Potential (EPSP) generation. On this basis, an optimal spike detector based on the postsynaptic voltage in a given time period is used to determine the output is no-occurrence or occurrence of one spike. In this way, one spike from the transmitter neuron to the receiver neuron is completed the transmission. 3. Energy model for the synaptic channel in NSC system In this section, we first propose the energy model and obtain the mathematical expression of the total energy consumption for one spike transmission, then the channel capacity and energy efficiency of this NSC system with synaptic channel is analyzed. 3.1. Derivation of the total energy consumption per spike transmission We build the energy model based on the NSC system structure introduced in Section 2. At first, in the process of the axonal
In the NSC system, the vesicles and neurotransmitters are the basic information units. Then the vesicles and neurotransmitters synthesis at the transmitter neuron is the first step in the communication process. In order to transmit one spike, the number of available vesicles for release in presynaptic terminal which is needed for a realistic NSC system is denoted as Nv . The Ns neurotransmitters contained in each vesicle are transmitted with probability pr which is called the vesicle release probability. The energy cost for synthesis of one neurotransmitter which is a single molecule is denoted as Es and for production of a vesicle is Ev . The total energy cost for synthesis of vesicles and neurotransmitters which is defined as Esynthesis for one spike transmission is Esynthesis = N v pr (Ev + Ns Es ).
(1)
The energy cost of synthesizing one phospholipid molecule is 1 unit of Adenosine Triphosphate (ATP), and ATP is a small molecule used in cells as a coenzyme for energy transfer [22], which equals 83 zJ [23]. Here, the zeptojoule (zJ) is equal to one sextillionth (10−21 ) of one joule. In a vesicle, there are 5 phospholipids in 1 nm2 area [23]. Thus, the energy cost of synthesizing a vesicle with a radius of Rv is Ev = 83 × 5(4π R2v )
zJ.
(2)
The major type of the neurotransmitter in central excitatory synapses is known as Glutamate. Once released into the synaptic cleft, some of these glutamate molecules are captured by receptors on the postsynaptic terminal which can generate EPSP. The energy cost for synthesizing an amino acid is 202.88 zJ [24]. Because the Glutamate molecule is one kind of amino acid, the energy cost of synthesizing a neurotransmitter which is equal to the energy cost of synthesizing a Glutamate molecule is Es = 202.88
zJ.
(3)
According to (1)–(3), the total energy cost of synthesizing the vesicles and neurotransmitters Esynthesis is Esynthesis = N v pr (1660π R2v + 202.88Ns ).
(4)
(2) Neurotransmitters diffusion After the neurotransmitters are released from the vesicles, they diffuse in the synaptic cleft. In the process of neurotransmitters diffusion, each neurotransmitter needs energy to reach the receptors and to bind with the receptors. For each neurotransmitter, it travels 8 nm and spends 1 ATP of energy [25]. It is assumed that the total distance between presynaptic terminal and the postsynaptic terminal that needs to be traveled is roughly equal to the width of the synaptic cleft which is defined as d. pr is the vesicle release probability. Hence, the overall energy cost of the neurotransmitters in this diffusion process Ediffusion is
⌈ ⌉ Ediffusion = 83 where
⌈d⌉ 8
d
8
Nv Ns pr
zJ,
is the ceil function of d/8.
(5)
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Fig. 1. The structure of the NSC system model.
(3) neurotransmitter–receptor binding In the neurotransmitter–receptor binding process, after the vesicles release, some of the neurotransmitters which are the Glutamate molecules bind to the receptors in postsynaptic density (PSD) proteins to form a neurotransmitter–receptor complex. Note that after the bonds are formed, the Glutamate molecules are removed in the synaptic cleft. We assume that the receptors are uniformly distributed on the PSD, a square-shaped region on the postsynaptic terminal with side length Lp . The area of this region Sarea is expressed as Sarea = L2p .
(6)
Let the number of the receptors be Nr . The density of neurotransmitters rereleased in the synaptic cleft and the receptors labeled as Ds is Ds =
Nr + Nv Ns pr Sarea
.
1 − e−kt 1 + (Ds Ka )−1
,
Nb = Nr p(t).
(9)
Assume that the energy cost of each neurotransmitter–receptor binding is a constant which is labeled as E0 , thus the whole energy consumption in this binding process is expressed as Ebinding = Nb E0 .
(10)
According to (4), (5) and (10), the total energy consumption for one spike transmission for the NSC system labeled as Etotal is Etotal =Esynthesis + Ediffusion + Ebinding
= Nv pr (1660π Rv + 202.88Ns + 83 2
⌈ ⌉ d
8
Ns ) + Nb E0 . (11)
3.2. Analysis of the channel capacity per unit energy consumption We model the postsynaptic response to the release of one vesicle by a function h(t), which corresponds to the EPSP waveform represented as an Alpha function [27] hpeak t tpeak
exp(1 −
t tpeak
),
(12)
(aq)J −1 (J − 1)!
e−aq ,
(13)
where a and J are the shape and rate of Gamma distribution, respectively. J can be used to modify the variability of q and J = 1 corresponds to an exponential distribution with the highest variability. For the postsynaptic neuron voltage V (t) coexisting with noise n(t), which models the effect of other membrane noise sources [28] should be estimated. We assume that n(t) is additive white Gaussian noise and bandwidth limited. The power spectral density for n(t) over a bandwidth Bn is denoted by Snn (f ) and is given by
{ Snn (f ) =
σ2 2Bn
,
−Bn ≤ f ≤ Bn ,
0,
(14)
otherwise,
where σ 2 is the variance of n(t). We define SNR as a signal-to-noise ratio on the postsynaptic potential as
(8)
where Ka = kf /kr is the equilibrium association constant and k = Ds kf + kr is the overall rate of reaction. kf and kr are the forward and reverse binding rate constants, respectively. Therefore the number of bonds between the neurotransmitter and receptors in time t which is defined as Nb is
h(t) =
P(q) = a
(7)
Assume either the neurotransmitters or receptors excessively outnumber the other one, p(t) is the probability of forming one bond in time t given by [26] p(t) =
where hpeak is the peak EPSP magnitude and tpeak is the corresponding time-to-peak. Subsequently, the synaptic variability is modeled by multiplying the response h(t) by a random amplitude of q which has a probability distribution P(q) modeled by a Gamma distribution [27]
SNR =
1 Snn (f )
∞
∫
h2 (t)dt = 0
2Bn
σ
2
∞
∫
h2 (t)dt .
(15)
0
The hypothesis H0 represents the case that the postsynaptic neuron voltage V (t) is measured by the noise process n(t). The hypothesis H1 is defined as the case that V (t) corresponds to h(t) gated by stochastic vesicle release W and the noise process n(t) [28]. Thus, the detection of binary signal by the optimal detector can be properly expressed by binary hypothesis testing problem as follows:
{
H0 : V (t) = n(t), H1 : V (t) = qWh(t) + n(t),
(16)
where W represents the vesicle release process which is a binary process. Thus, W = 0 and W = 1 represent a failure and success of a vesicle release, respectively. In particular, the value of W cannot be equal to 0 in the hypotheses H1 . According to the description of the NSC system model introduced above, we summarize the communication process of one spike transmission which is shown in Fig. 2. Let X and Y represent the binary variables denoting the spike occurrence and decision, respectively. If the input X is the occurrence of one spike which is X = 1, the vesicles containing neurotransmitters are released with the vesicle release probability to the synaptic cleft. For nooccurrence of one spike which is X = 0, no vesicles are released. Similarly, Y = 1 and Y = 0 express the decisions that one spike is occurred and not occurred, respectively.
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Fig. 2. Block diagram of one spike transmission under the optimal detector.
Let p1 denote the priori probability of one spike occurrence at the presynaptic neuron. (1 − p1 ) is the probability of no-occurrence of one spike. Then, the probability of error for one spike transmission based on PM and PF is PE = p1 PM + (1 − p1 )PF .
On the basis of (26), the conditional probability of the synaptic channel P(Y |X ) can be written as follows:
Fig. 3. The binary channel model of one spike transmission.
( P( Y | X ) = In Fig. 3, the binary channel model under the hypothesis H0 and H1 is shown. PF is the false alarm probability and PM is the miss detection probability which can be defined as follows: PF = Pr(Y = 1|X = 0), PM = Pr(Y = 0|X = 1), 1 − PF = Pr(Y = 0|X = 0),
(17)
{
(18)
In our energy model for the synaptic channel in the NSC system, we assume that the vesicle release is deterministic and no spontaneous release occurs. According to (17)–(18), PF and PM can be written as PF = Pr(r ≥ θ|X = 0) = Pr(r ≥ θ|W = 0), PM = Pr(r < θ|X = 1)
= pr Pr(r < θ|W = 1) + (1 − pr ) Pr(r < θ|W = 0).
(19) (20)
0 Let PF0 = Pr(r ≥ θ|W = 0) and PM = Pr(r < θ|W = 1) denote the corresponding errors when we have ignored the spontaneous 0 vesicle release. PF and PM can be expressed in terms of PF0 and PM as
PF = PF0 ,
(21)
0 PM = pr PM + (1 − pr )(1 − PF0 ).
(22)
According to the stochastic and deterministic vesicle release 0 processes [7], PF0 and PM can be parametrically expressed in terms of the optimal threshold θ as follows. PF0 = 0 PM =
1 2 1 2
[1 − Erf (θ )], ∫ ∞ √ [1 + Erf (θ − q SNR)P(q)dq],
(23) (24)
0
where Erf (x) is the error function which is defined as 2 Erf (x) = √
π
x
∫
2
e−t dt .
PF
PM
(1 − PF )
)
.
(27)
Using (27), the mutual information I(X ; Y ) between X and Y can be derived as follows [29,30]:
= H(p1 PM + (1 − p1 )(1 − PF )) − (p1 H(PM ) + (1 − p1 )H(PF )), (28) where H(z) denotes the binary entropy function as
At the receiver neuron side, the optimal decision rule is to compare the correlation r between V (t) and h(t) to a threshold θ which can be used to minimize the error probability in detecting one spike transmission. Thus, the optimal decision rule can be written as r < θ ⇒ Y = 0.
(1 − PM )
I(X ; Y ) = H(Y ) − H(Y |X )
1 − PM = Pr(Y = 1|X = 1).
r ≥ θ ⇒ Y = 1,
(26)
(25)
0
However, in practice, the binary channel of the NSC system mainly depends on the false alarm and miss detection errors [7].
H(z) = −z log 2(z) − (1 − z) log 2(1 − z).
(29)
The energy efficiency η which is defined as the channel capacity per unit energy consumption for one spike transmission is
η=
I(X ; Y ) Etotal
.
(30)
4. Numerical results In this section, we use MATLAB to obtain the numerical results of analyzing the total energy consumption, channel capacity and energy efficiency for one spike transmission. The aim is to investigate how the different parameters including the vesicle release probability pr , the optimal threshold θ , the time t, the number of neurotransmitters released in one vesicle Ns and the length of the region on the PSD Lp have impacts on the performance of NSC system. The default parameters used to make analysis of the energy efficiency are given in Table 1 as follows: 4.1. Total energy consumption analysis Fig. 4 shows the tendency that the total energy consumption used per spike transmission Etotal is varying with the time t for different values of Nv and Nr in Fig. 4(a) and 4(b), respectively. On one hand, we can see that the Etotal is increasing with t which is needed in the diffusion process on the basis that the number of bonds formed between the neurotransmitters and receptors is increasing. On the other hand, when Nv and Nr are increasing, Etotal is also increasing. This incidence can be explained by the fact that with the increasing value of Nv and Nr , both the numbers of neurotransmitters and receptors are increasing. According to the formulas of computing the total energy consumption in (11), the larger the value of Nv and Nr , the more the total energy consumption used per spike transmission. In Fig. 5, we plot Etotal in relation to Ns with different values of Nv and pr in Fig. 5(a) and 5(b), respectively. It shows a monotonically increasing trend of Etotal as a function of the value of Ns . We can observe that the larger the values of Nv and pr , the more total energy consumption needed for the NSC system. It can be explained
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Description
Value
Rv d Lp Nv Ns Nr E0 kf kr t a J
The radius of a vesicle The width of the synaptic cleft The side length of the square-shaped region on the PSD The number of the vesicles The number of the neurotransmitters in each vesicle The number of the receptors The energy consumption of one bond formed The forward binding rate The reverse binding rate The time of neurotransmitter diffusion The shape of Gamma distribution The rate of Gamma distribution The optimal threshold The signal-to-noise ratio The vesicle release probability The probability of one spike occurrence
0.05 µm 15 µm 100 µm 10 3000 3×105 1000 zJ 0.003 µm2 /s 0.03 s−1 20 s 1 1 1.1 10 dB 0.5 0.5
θ SNR pr p1
Fig. 4. The total energy consumption used per spike transmission Etotal versus the time t with different values of (a) Nv ; (b) Nr .
Fig. 5. The total energy consumption used per spike transmission Etotal versus Ns with different values of (a) Nv ; (b) pr .
as follows: when Nv and pr are increasing, the number of vesicles is increasing. At the same time, the number of neurotransmitters is also increasing. Thus, according to the formulas (4), the total energy cost of synthesizing the vesicles and neurotransmitters is also increasing.
In Fig. 6, the curves show that Etotal is in relation to Lp with different values of Ns , Nr and pr in Fig. 6(a), 6(b) and 6(c), respectively. First, according to the formulas (8), when Lp is increasing, the probability that the bonds are formed is decreasing. Thus, Etotal is decreasing. Second, for the same value of Lp and the other
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Fig. 6. The total energy consumption used per spike transmission Etotal versus Lp with different values of (a) Ns ; (b) Nr ; (c) pr .
fixed parameters, the larger the value of Ns and pr , the larger the number of the neurotransmitters, then the more the energy consumption used in the synthetic and the diffusion process. In addition, the larger the value of Nr which is the number of the receptors, the more the energy consumption used in the binding process. Therefore, the larger the value of Ns , Nr and pr , the more the total energy consumption. 4.2. The channel capacity and energy efficiency analysis In Fig. 7, we plot the channel capacity I(X ; Y ) versus the probability of one spike occurrence p1 for different values of pr and SNR in Fig. 7(a) and 7(b), respectively. I(X ; Y ) is increasing with p1 , then it reaches its peak at some particular value of p1 , respectively. After this peak value, I(X ; Y ) is decreasing. We also can see that for the same value of p1 , the larger the value of pr and SNR, the larger the value of channel capacity I(X ; Y ). Fig. 7(c) and 7(d) show I(X ; Y ) is changing with the optimal threshold θ for different values of pr and SNR. It is observed that the trends of I(X ; Y ) versus p1 and θ are basically consistent. Fig. 8(a) shows η is varying with t based on the different values of pr . η is decreasing with t. For the same value of t, the larger the value of pr , the larger values of Etotal and channel capacity which are shown in Fig. 4 and Fig. 7(a), respectively. Then the value of η is larger. We can see that the effect of pr on channel capacity is greater than that on Etotal . In Fig. 8(b), we plot η versus the side length of the
square-shaped region on the PSD with different values of pr . First, η is increasing with Lp based on the fact that Etotal is decreasing with Lp which is shown in Fig. 6. Second, for the same value of Lp , the larger the value of pr , the larger value of the channel capacity, then the more the energy efficiency. According to the simulation results in Fig. 8, we can see that the parameter pr plays a vital role in the energy efficiency. 5. Conclusions The objective of our paper is to make analysis of the total energy consumption, channel capacity and energy efficiency for one spike transmission for the synaptic channel in NSC system. The energy model for the synaptic channel in the point-to-point NSC system is first proposed. Then the energy efficiency of this energy model is analyzed. The numerical results show that how the different system parameters including the vesicle release probability pr , the time t, the optimal threshold θ , the number of the neurotransmitters contained in one vesicle Ns and the side length of the region on the PSD Lp have different impacts on the total energy consumption and channel capacity for one spike transmission. More importantly, the effects of the vesicle release probability pr on the channel capacity and the total energy consumption per spike transmission are different which shows the vesicle release probability pr is the key factor of energy efficiency. The setting
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Fig. 7. The channel capacity I(X ; Y ) versus p1 with different values of (a) pr ; (b) SNR; I(X ; Y ) versus θ with different values of (c) pr ; (d) SNR.
Fig. 8. The energy efficiency η versus (a) the time t; (b) Lp with different values of pr .
of the parameters for the NSC system plays a central role in its efficient energy design. As the future work, we plan to extend our work and aim to provide analysis of the energy efficiency for the transmission of spike sequence by considering the NSC channel with interference under more complex communication topologies in the NSC system.
Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos. 61472367, 61432015) and Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LY19F020029).
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Zhen Cheng received B.S. degree from Huanggang Normal University, Hubei, China, in 2004, and received M.S. and Ph.D. degrees from Huazhong University of Science and Technology, Hubei, China, in 2007 and 2010 respectively. She is currently an associate professor in the School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, China. Her current research interests include nanonetworks, molecular communication, wireless networks. She has published more than 30 technical papers in international proceedings and journals.
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Nano Communication Networks journal homepage: www.elsevier.com/locate/nanocomnet
Energy model for synaptic channel in neuro-spike communication ∗
Zhen Cheng , Yiming Zhang, Huiting Zhao, Fei Lin, Kaikai Chi School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China
article
info
Article history: Received 25 October 2017 Received in revised form 30 December 2018 Accepted 14 January 2019 Available online 18 January 2019 Keywords: Neuro-spike communication Synaptic channel Energy model Channel capacity Energy efficient
a b s t r a c t The neuro-spike communication (NSC) is inspired by electrochemical communication among biological neurons. In this paper, considering that the energy consumption has important effects on the performance of NSC system, an energy model for the synaptic channel in a point-to-point NSC system is first proposed. Then the energy efficiency which is defined as the channel capacity per unit energy consumption for one spike transmission is analyzed. The numerical results demonstrate that the main parameters including the vesicle release probability, the optimal threshold, the number of the neurotransmitters contained in one vesicle and the side length of the region in which the receptors are located have different impacts on total energy consumption and channel capacity. More importantly, the vesicle release probability is the key factor of energy efficiency. The setting of these system parameters can be used to provide guidelines for designing energy efficient NSC systems. © 2019 Published by Elsevier B.V.
1. Introduction Nanonetworks have important applications which are being considered in biomedical, industrial and environmental areas [1– 3]. In particular, molecular communication (MC) [4], where the information is transmitted, propagated and received in a biological environment by the exchange of molecules, is a promising communication technique to realize nanonetworks [5]. One of the most attractive fields in MC is neuro-spike communication (NSC) [6], which is inspired from signal transmission between neurons. In NSC, electrochemical impulses and neurotransmitters are used for information transmission from the transmitter neuron to the receiver neuron [7]. Generally, the NSC system plays a significant role in transporting information in the body nervous system [8]. In recent years, there have been growing interests and research efforts dedicated to NSC. Many researchers focused on the performance of the NSC system on the basis of the NSC channel model [9,10]. Akan et al. [11] evaluated the throughput and delay performance of the mobile ad hoc molecular nanonetworks by modeling the NSC channel. In [12], Maham designed the optimum binary detector and derived the probability of error at the receiver neuron for the NSC system. In [13], the authors considered a point-to-point NSC model and investigated the effects on the error probability of the NSC channel. Lee and Cho [14] studied the capacity of NSC system based on the process of the information transmission including the axon propagation, vesicle release and ∗ Corresponding author. E-mail addresses: [email protected] (Z. Cheng), [email protected] (Y. Zhang), [email protected] (H. Zhao), [email protected] (F. Lin), [email protected] (K. Chi). https://doi.org/10.1016/j.nancom.2019.01.004 1878-7789/© 2019 Published by Elsevier B.V.
neurotransmitter diffusion. Ramezani and Akan [15] provided a realistic NSC model and evaluated the impacts of availability of vesicles on the channel capacity. The energy consumption requirements constitute the limits on the performance of the MC system. In [16], Schreiber et al. investigated how the factors influenced the energy efficiency of signaling mechanisms. In [17], an energy model for the diffusionbased MC system was proposed and two optimization problems were set up. In [18], the authors proposed and modeled an energy efficient MC system with a simultaneous molecular information and energy transfer relay. Guo et al. [19] analyzed the information delivery energy efficiency of bacteria mobile relays. In [20], an energy model for active transport MC was presented to show that the energy consumption was important in the engineering of this MC system. Ramezani et al. [21] and evaluated the effects of metabolic energy constraints on the sum rate of the multipleinput single-output (MISO) NSC system. However, the metabolic energy is only considered to be consumed by opening and closing of numerous ionic channels on the cell membrane. In this paper, an energy model combination with the whole communication process of the NSC system is proposed, which can be used to provide the guidelines for designing energy efficient NSC systems. The main contributions of our paper are summarized as follows: (1) An energy model for the synaptic channel in a point-to-point NSC system is proposed based on the communication process of one spike transmission. (2) On the basis of this energy model, the energy efficiency which is the channel capacity per unit energy consumption for one spike transmission is formulated. (3) The numerical results show that the main parameters, such as the vesicle release probability, the optimal threshold, the number of the neurotransmitters contained in one vesicle and the
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side length of the region in which the receptors are located have different effects on the total energy consumption and channel capacity. In particular, the vesicle release probability is the key factor of energy efficiency. The remainder of this paper is organized as follows. Section 2 describes the NSC system model. In Section 3, an energy model for the synaptic channel in the point-to-point NSC system is presented and the energy efficiency is analyzed. Numerical results are given in Section 4. Finally, this paper is concluded in Section 5. 2. The NSC system model
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transmission, each spike is transmitted to presynaptic terminals through axon, then the neurotransmitters contained in the vesicles are released to the synaptic cleft and propagate in the synaptic transmission, finally the neurotransmitters bind to the receptors on the postsynaptic neuron to generate the spike in the step of spike generation. Thus one spike transmission between the transmitter neuron and the receiver neuron is finished. The energy consumption is mainly spent in the communication steps including the synthesis of the vesicles and neurotransmitters, neurotransmitters diffusion and neurotransmitter–receptor binding. (1) Synthesis of the vesicles and neurotransmitters
In this section, we introduce the point-to-point NSC system model between two neurons. In the NSC system model, the transmission of information from the transmitter neuron to the receiver neuron is mediated by the electrochemical impulses called spikes. The whole communication process of the NSC system involves three steps including the axonal transmission, the synaptic transmission and the spike generation. The structure of the NSC system model is shown in Fig. 1. (1) The axonal transmission. In the NSC system, the information is encoded with a spiking sequence, then it propagates along the axon which can be considered as a communication channel connecting the soma and the presynaptic terminals. At the end of the axon, there are some presynaptic terminals, by which the vesicles containing the neurotransmitters are released to the gaps between the transmitter neuron and the receiver neuron which are called the synaptic cleft. (2) The synaptic transmission. The synaptic signal can be modeled by a binary random process, in which one and zero represent the events of occurrence and no-occurrence of one spike, respectively. If one spike arrives to the presynaptic terminal, the vesicles are released to the synaptic cleft. Otherwise, no vesicles are released. The second step of the NSC is synaptic transmission which is initiated by the release of the vesicles on the presynaptic terminal to the synaptic cleft. After the neurotransmitters are released by the vesicles, they diffuse in the synaptic cleft. The receptors in the postsynaptic terminal are located at the dendrites of the receiver neuron to capture the diffusing neurotransmitter molecules. Note that for successive synaptic transmissions, neurotransmitters are assumed to be removed from synaptic cleft before next vesicle releases for next spike transmission. (3) The spike generation. When the neurotransmitters bind to the receptors on the postsynaptic terminal of the receiver neuron, the third step of NSC which is called the spike generation, is established by the movement of ions. Then, the ionic channels on the surface of the output neuron open and allow the flow of ions to the receiver neuron. The movement of ions provides excitation and causes the membrane potential of the receiver neuron to change rapidly, and it finally leads to the Excitatory Post Synaptic Potential (EPSP) generation. On this basis, an optimal spike detector based on the postsynaptic voltage in a given time period is used to determine the output is no-occurrence or occurrence of one spike. In this way, one spike from the transmitter neuron to the receiver neuron is completed the transmission. 3. Energy model for the synaptic channel in NSC system In this section, we first propose the energy model and obtain the mathematical expression of the total energy consumption for one spike transmission, then the channel capacity and energy efficiency of this NSC system with synaptic channel is analyzed. 3.1. Derivation of the total energy consumption per spike transmission We build the energy model based on the NSC system structure introduced in Section 2. At first, in the process of the axonal
In the NSC system, the vesicles and neurotransmitters are the basic information units. Then the vesicles and neurotransmitters synthesis at the transmitter neuron is the first step in the communication process. In order to transmit one spike, the number of available vesicles for release in presynaptic terminal which is needed for a realistic NSC system is denoted as Nv . The Ns neurotransmitters contained in each vesicle are transmitted with probability pr which is called the vesicle release probability. The energy cost for synthesis of one neurotransmitter which is a single molecule is denoted as Es and for production of a vesicle is Ev . The total energy cost for synthesis of vesicles and neurotransmitters which is defined as Esynthesis for one spike transmission is Esynthesis = N v pr (Ev + Ns Es ).
(1)
The energy cost of synthesizing one phospholipid molecule is 1 unit of Adenosine Triphosphate (ATP), and ATP is a small molecule used in cells as a coenzyme for energy transfer [22], which equals 83 zJ [23]. Here, the zeptojoule (zJ) is equal to one sextillionth (10−21 ) of one joule. In a vesicle, there are 5 phospholipids in 1 nm2 area [23]. Thus, the energy cost of synthesizing a vesicle with a radius of Rv is Ev = 83 × 5(4π R2v )
zJ.
(2)
The major type of the neurotransmitter in central excitatory synapses is known as Glutamate. Once released into the synaptic cleft, some of these glutamate molecules are captured by receptors on the postsynaptic terminal which can generate EPSP. The energy cost for synthesizing an amino acid is 202.88 zJ [24]. Because the Glutamate molecule is one kind of amino acid, the energy cost of synthesizing a neurotransmitter which is equal to the energy cost of synthesizing a Glutamate molecule is Es = 202.88
zJ.
(3)
According to (1)–(3), the total energy cost of synthesizing the vesicles and neurotransmitters Esynthesis is Esynthesis = N v pr (1660π R2v + 202.88Ns ).
(4)
(2) Neurotransmitters diffusion After the neurotransmitters are released from the vesicles, they diffuse in the synaptic cleft. In the process of neurotransmitters diffusion, each neurotransmitter needs energy to reach the receptors and to bind with the receptors. For each neurotransmitter, it travels 8 nm and spends 1 ATP of energy [25]. It is assumed that the total distance between presynaptic terminal and the postsynaptic terminal that needs to be traveled is roughly equal to the width of the synaptic cleft which is defined as d. pr is the vesicle release probability. Hence, the overall energy cost of the neurotransmitters in this diffusion process Ediffusion is
⌈ ⌉ Ediffusion = 83 where
⌈d⌉ 8
d
8
Nv Ns pr
zJ,
is the ceil function of d/8.
(5)
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Fig. 1. The structure of the NSC system model.
(3) neurotransmitter–receptor binding In the neurotransmitter–receptor binding process, after the vesicles release, some of the neurotransmitters which are the Glutamate molecules bind to the receptors in postsynaptic density (PSD) proteins to form a neurotransmitter–receptor complex. Note that after the bonds are formed, the Glutamate molecules are removed in the synaptic cleft. We assume that the receptors are uniformly distributed on the PSD, a square-shaped region on the postsynaptic terminal with side length Lp . The area of this region Sarea is expressed as Sarea = L2p .
(6)
Let the number of the receptors be Nr . The density of neurotransmitters rereleased in the synaptic cleft and the receptors labeled as Ds is Ds =
Nr + Nv Ns pr Sarea
.
1 − e−kt 1 + (Ds Ka )−1
,
Nb = Nr p(t).
(9)
Assume that the energy cost of each neurotransmitter–receptor binding is a constant which is labeled as E0 , thus the whole energy consumption in this binding process is expressed as Ebinding = Nb E0 .
(10)
According to (4), (5) and (10), the total energy consumption for one spike transmission for the NSC system labeled as Etotal is Etotal =Esynthesis + Ediffusion + Ebinding
= Nv pr (1660π Rv + 202.88Ns + 83 2
⌈ ⌉ d
8
Ns ) + Nb E0 . (11)
3.2. Analysis of the channel capacity per unit energy consumption We model the postsynaptic response to the release of one vesicle by a function h(t), which corresponds to the EPSP waveform represented as an Alpha function [27] hpeak t tpeak
exp(1 −
t tpeak
),
(12)
(aq)J −1 (J − 1)!
e−aq ,
(13)
where a and J are the shape and rate of Gamma distribution, respectively. J can be used to modify the variability of q and J = 1 corresponds to an exponential distribution with the highest variability. For the postsynaptic neuron voltage V (t) coexisting with noise n(t), which models the effect of other membrane noise sources [28] should be estimated. We assume that n(t) is additive white Gaussian noise and bandwidth limited. The power spectral density for n(t) over a bandwidth Bn is denoted by Snn (f ) and is given by
{ Snn (f ) =
σ2 2Bn
,
−Bn ≤ f ≤ Bn ,
0,
(14)
otherwise,
where σ 2 is the variance of n(t). We define SNR as a signal-to-noise ratio on the postsynaptic potential as
(8)
where Ka = kf /kr is the equilibrium association constant and k = Ds kf + kr is the overall rate of reaction. kf and kr are the forward and reverse binding rate constants, respectively. Therefore the number of bonds between the neurotransmitter and receptors in time t which is defined as Nb is
h(t) =
P(q) = a
(7)
Assume either the neurotransmitters or receptors excessively outnumber the other one, p(t) is the probability of forming one bond in time t given by [26] p(t) =
where hpeak is the peak EPSP magnitude and tpeak is the corresponding time-to-peak. Subsequently, the synaptic variability is modeled by multiplying the response h(t) by a random amplitude of q which has a probability distribution P(q) modeled by a Gamma distribution [27]
SNR =
1 Snn (f )
∞
∫
h2 (t)dt = 0
2Bn
σ
2
∞
∫
h2 (t)dt .
(15)
0
The hypothesis H0 represents the case that the postsynaptic neuron voltage V (t) is measured by the noise process n(t). The hypothesis H1 is defined as the case that V (t) corresponds to h(t) gated by stochastic vesicle release W and the noise process n(t) [28]. Thus, the detection of binary signal by the optimal detector can be properly expressed by binary hypothesis testing problem as follows:
{
H0 : V (t) = n(t), H1 : V (t) = qWh(t) + n(t),
(16)
where W represents the vesicle release process which is a binary process. Thus, W = 0 and W = 1 represent a failure and success of a vesicle release, respectively. In particular, the value of W cannot be equal to 0 in the hypotheses H1 . According to the description of the NSC system model introduced above, we summarize the communication process of one spike transmission which is shown in Fig. 2. Let X and Y represent the binary variables denoting the spike occurrence and decision, respectively. If the input X is the occurrence of one spike which is X = 1, the vesicles containing neurotransmitters are released with the vesicle release probability to the synaptic cleft. For nooccurrence of one spike which is X = 0, no vesicles are released. Similarly, Y = 1 and Y = 0 express the decisions that one spike is occurred and not occurred, respectively.
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Fig. 2. Block diagram of one spike transmission under the optimal detector.
Let p1 denote the priori probability of one spike occurrence at the presynaptic neuron. (1 − p1 ) is the probability of no-occurrence of one spike. Then, the probability of error for one spike transmission based on PM and PF is PE = p1 PM + (1 − p1 )PF .
On the basis of (26), the conditional probability of the synaptic channel P(Y |X ) can be written as follows:
Fig. 3. The binary channel model of one spike transmission.
( P( Y | X ) = In Fig. 3, the binary channel model under the hypothesis H0 and H1 is shown. PF is the false alarm probability and PM is the miss detection probability which can be defined as follows: PF = Pr(Y = 1|X = 0), PM = Pr(Y = 0|X = 1), 1 − PF = Pr(Y = 0|X = 0),
(17)
{
(18)
In our energy model for the synaptic channel in the NSC system, we assume that the vesicle release is deterministic and no spontaneous release occurs. According to (17)–(18), PF and PM can be written as PF = Pr(r ≥ θ|X = 0) = Pr(r ≥ θ|W = 0), PM = Pr(r < θ|X = 1)
= pr Pr(r < θ|W = 1) + (1 − pr ) Pr(r < θ|W = 0).
(19) (20)
0 Let PF0 = Pr(r ≥ θ|W = 0) and PM = Pr(r < θ|W = 1) denote the corresponding errors when we have ignored the spontaneous 0 vesicle release. PF and PM can be expressed in terms of PF0 and PM as
PF = PF0 ,
(21)
0 PM = pr PM + (1 − pr )(1 − PF0 ).
(22)
According to the stochastic and deterministic vesicle release 0 processes [7], PF0 and PM can be parametrically expressed in terms of the optimal threshold θ as follows. PF0 = 0 PM =
1 2 1 2
[1 − Erf (θ )], ∫ ∞ √ [1 + Erf (θ − q SNR)P(q)dq],
(23) (24)
0
where Erf (x) is the error function which is defined as 2 Erf (x) = √
π
x
∫
2
e−t dt .
PF
PM
(1 − PF )
)
.
(27)
Using (27), the mutual information I(X ; Y ) between X and Y can be derived as follows [29,30]:
= H(p1 PM + (1 − p1 )(1 − PF )) − (p1 H(PM ) + (1 − p1 )H(PF )), (28) where H(z) denotes the binary entropy function as
At the receiver neuron side, the optimal decision rule is to compare the correlation r between V (t) and h(t) to a threshold θ which can be used to minimize the error probability in detecting one spike transmission. Thus, the optimal decision rule can be written as r < θ ⇒ Y = 0.
(1 − PM )
I(X ; Y ) = H(Y ) − H(Y |X )
1 − PM = Pr(Y = 1|X = 1).
r ≥ θ ⇒ Y = 1,
(26)
(25)
0
However, in practice, the binary channel of the NSC system mainly depends on the false alarm and miss detection errors [7].
H(z) = −z log 2(z) − (1 − z) log 2(1 − z).
(29)
The energy efficiency η which is defined as the channel capacity per unit energy consumption for one spike transmission is
η=
I(X ; Y ) Etotal
.
(30)
4. Numerical results In this section, we use MATLAB to obtain the numerical results of analyzing the total energy consumption, channel capacity and energy efficiency for one spike transmission. The aim is to investigate how the different parameters including the vesicle release probability pr , the optimal threshold θ , the time t, the number of neurotransmitters released in one vesicle Ns and the length of the region on the PSD Lp have impacts on the performance of NSC system. The default parameters used to make analysis of the energy efficiency are given in Table 1 as follows: 4.1. Total energy consumption analysis Fig. 4 shows the tendency that the total energy consumption used per spike transmission Etotal is varying with the time t for different values of Nv and Nr in Fig. 4(a) and 4(b), respectively. On one hand, we can see that the Etotal is increasing with t which is needed in the diffusion process on the basis that the number of bonds formed between the neurotransmitters and receptors is increasing. On the other hand, when Nv and Nr are increasing, Etotal is also increasing. This incidence can be explained by the fact that with the increasing value of Nv and Nr , both the numbers of neurotransmitters and receptors are increasing. According to the formulas of computing the total energy consumption in (11), the larger the value of Nv and Nr , the more the total energy consumption used per spike transmission. In Fig. 5, we plot Etotal in relation to Ns with different values of Nv and pr in Fig. 5(a) and 5(b), respectively. It shows a monotonically increasing trend of Etotal as a function of the value of Ns . We can observe that the larger the values of Nv and pr , the more total energy consumption needed for the NSC system. It can be explained
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Description
Value
Rv d Lp Nv Ns Nr E0 kf kr t a J
The radius of a vesicle The width of the synaptic cleft The side length of the square-shaped region on the PSD The number of the vesicles The number of the neurotransmitters in each vesicle The number of the receptors The energy consumption of one bond formed The forward binding rate The reverse binding rate The time of neurotransmitter diffusion The shape of Gamma distribution The rate of Gamma distribution The optimal threshold The signal-to-noise ratio The vesicle release probability The probability of one spike occurrence
0.05 µm 15 µm 100 µm 10 3000 3×105 1000 zJ 0.003 µm2 /s 0.03 s−1 20 s 1 1 1.1 10 dB 0.5 0.5
θ SNR pr p1
Fig. 4. The total energy consumption used per spike transmission Etotal versus the time t with different values of (a) Nv ; (b) Nr .
Fig. 5. The total energy consumption used per spike transmission Etotal versus Ns with different values of (a) Nv ; (b) pr .
as follows: when Nv and pr are increasing, the number of vesicles is increasing. At the same time, the number of neurotransmitters is also increasing. Thus, according to the formulas (4), the total energy cost of synthesizing the vesicles and neurotransmitters is also increasing.
In Fig. 6, the curves show that Etotal is in relation to Lp with different values of Ns , Nr and pr in Fig. 6(a), 6(b) and 6(c), respectively. First, according to the formulas (8), when Lp is increasing, the probability that the bonds are formed is decreasing. Thus, Etotal is decreasing. Second, for the same value of Lp and the other
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Fig. 6. The total energy consumption used per spike transmission Etotal versus Lp with different values of (a) Ns ; (b) Nr ; (c) pr .
fixed parameters, the larger the value of Ns and pr , the larger the number of the neurotransmitters, then the more the energy consumption used in the synthetic and the diffusion process. In addition, the larger the value of Nr which is the number of the receptors, the more the energy consumption used in the binding process. Therefore, the larger the value of Ns , Nr and pr , the more the total energy consumption. 4.2. The channel capacity and energy efficiency analysis In Fig. 7, we plot the channel capacity I(X ; Y ) versus the probability of one spike occurrence p1 for different values of pr and SNR in Fig. 7(a) and 7(b), respectively. I(X ; Y ) is increasing with p1 , then it reaches its peak at some particular value of p1 , respectively. After this peak value, I(X ; Y ) is decreasing. We also can see that for the same value of p1 , the larger the value of pr and SNR, the larger the value of channel capacity I(X ; Y ). Fig. 7(c) and 7(d) show I(X ; Y ) is changing with the optimal threshold θ for different values of pr and SNR. It is observed that the trends of I(X ; Y ) versus p1 and θ are basically consistent. Fig. 8(a) shows η is varying with t based on the different values of pr . η is decreasing with t. For the same value of t, the larger the value of pr , the larger values of Etotal and channel capacity which are shown in Fig. 4 and Fig. 7(a), respectively. Then the value of η is larger. We can see that the effect of pr on channel capacity is greater than that on Etotal . In Fig. 8(b), we plot η versus the side length of the
square-shaped region on the PSD with different values of pr . First, η is increasing with Lp based on the fact that Etotal is decreasing with Lp which is shown in Fig. 6. Second, for the same value of Lp , the larger the value of pr , the larger value of the channel capacity, then the more the energy efficiency. According to the simulation results in Fig. 8, we can see that the parameter pr plays a vital role in the energy efficiency. 5. Conclusions The objective of our paper is to make analysis of the total energy consumption, channel capacity and energy efficiency for one spike transmission for the synaptic channel in NSC system. The energy model for the synaptic channel in the point-to-point NSC system is first proposed. Then the energy efficiency of this energy model is analyzed. The numerical results show that how the different system parameters including the vesicle release probability pr , the time t, the optimal threshold θ , the number of the neurotransmitters contained in one vesicle Ns and the side length of the region on the PSD Lp have different impacts on the total energy consumption and channel capacity for one spike transmission. More importantly, the effects of the vesicle release probability pr on the channel capacity and the total energy consumption per spike transmission are different which shows the vesicle release probability pr is the key factor of energy efficiency. The setting
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Fig. 7. The channel capacity I(X ; Y ) versus p1 with different values of (a) pr ; (b) SNR; I(X ; Y ) versus θ with different values of (c) pr ; (d) SNR.
Fig. 8. The energy efficiency η versus (a) the time t; (b) Lp with different values of pr .
of the parameters for the NSC system plays a central role in its efficient energy design. As the future work, we plan to extend our work and aim to provide analysis of the energy efficiency for the transmission of spike sequence by considering the NSC channel with interference under more complex communication topologies in the NSC system.
Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos. 61472367, 61432015) and Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LY19F020029).
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Zhen Cheng received B.S. degree from Huanggang Normal University, Hubei, China, in 2004, and received M.S. and Ph.D. degrees from Huazhong University of Science and Technology, Hubei, China, in 2007 and 2010 respectively. She is currently an associate professor in the School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, China. Her current research interests include nanonetworks, molecular communication, wireless networks. She has published more than 30 technical papers in international proceedings and journals.