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A Nonparametric Stochastic Optimizer for TDMA-Based Neuronal Signaling Junichi Suzuki, Member, IEEE, Dũng H. Phan, and Harry Budiman

Abstract—This paper considers neurons as a physical communication medium for intrabody networks of nano/micro-scale machines and formulates a noisy multiobjective optimization problem for a Time Division Multiple Access (TDMA) communication protocol atop the physical layer. The problem is to find the Pareto-optimal TDMA configurations that maximize communication performance (e.g., latency) by multiplexing a given neuronal network to parallelize signal transmissions while maximizing communication robustness (i.e., unlikeliness of signal interference) against noise in neuronal signaling. Using a nonparametric significance test, the proposed stochastic optimizer is designed to statistically determine the superior-inferior relationship between given two solution candidates and seek the optimal trade-offs among communication performance and robustness objectives. Simulation results show that the proposed optimizer efficiently obtains quality TDMA configurations in noisy environments and outperforms existing noiseaware stochastic optimizers. Index Terms—Multiobjective optimization, neuronal signaling, noise-aware evolutionary algorithms, Time Division Multiple Access (TDMA) communication.

I. INTRODUCTION

A

NANOSCALE system consists of one or more functional machines (or nodes) that perform very simple computation, sensing, and/or actuation tasks on nanometer to micrometer scale. An emerging design strategy for nanoscale systems is to network those nodes for operating in larger physical spaces in higher spatial and temporal resolutions. Although individual nodes are limited in computation, sensing, and actuation capabilities, an assembly of nodes can potentially organize into a “large-scale” network that spreads on millimeter to centimeter scale and collaboratively performs tasks that no individual nodes could. Molecular communication is one of a few options to network nano/micro-scale nodes. It leverages molecules as a communication medium between nodes [1], [2]. Due to its advantages such as inherent nano/micro-meter scale, biocompatibility and energy efficiency, a key application domain of molecular communication is intrabody networks where nodes are networked Manuscript received August 31, 2013; revised August 29, 2014; accepted August 29, 2014. Date of current version September 23, 2014. Asterisk indicates corresponding author. *J. Suzuki is with the Department of Computer Science, University of Massachusetts Boston, Boston, MA, 02125 USA (e-mail: [email protected]). D. H. Phan and H. Budimann are with the Department of Computer Science, University of Massachusetts Boston, Boston, MA, 02125, USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNB.2014.2355015

to perform their tasks in the human body for biomedical and prosthetic purposes [2], [3]. Those tasks include in-situ physiological sensing, biomedical anomaly detection, targeted drug release, medical operations with cellular/molecular level precision, neural signal transduction and neuromuscular implant control. This paper focuses on molecular communication that utilizes neurons as a primary component to build intrabody sensor-actuator networks, each of which consists of a set of nano/micro-scale sensor/actuator nodes and a network of neurons that are artificially formed into a particular topology. It allows individual nodes to interface (i.e., activate and deactivate) neurons and communicate to other nodes with electrochemical signals through a chain of neurons. This paper formulates a noisy multiobjective optimization problem for a neuronal signaling protocol, called Neuronal TDMA, and approaches the problem with a noise-aware evolutionary multiobjective optimization algorithm (EMOA). Neuronal TDMA performs single-bit Time Division Multiple Access (TDMA) scheduling for individual nodes to activate their nearby neurons and multiplex/parallelize neuronal signaling in a given neuronal network. The proposed EMOA heuristically seeks the optimal TDMA schedules for nodes by evolving a set of solution candidates (or individuals) toward the optima through generations. Each individual represents a particular TDMA schedule (i.e., when each node activates its nearby neuron to trigger a signal transmission). The proposed EMOA is designed to address the following three issues inherent in neuronal signaling. • Signal interference: When different signals attempt to travel through a neuron simultaneously, they interfere (or collide) with each other [4]. This leads to the loss or corruption of information encoded with the signals. The proposed EMOA is designed to avoid signal interference to ensure that signals reach the destination while multiplexing/parallelizing signal transmissions to improve communication performance such as latency. • Conflicting objectives: This paper considers two types of optimization objectives: communication performance objectives (e.g., signaling yield and latency) and a communication robustness objective. Here, robustness means the unlikeliness of signal interference (i.e., likeliness for signals to successfully reach the destination). In neuronal signaling, objectives often conflict with each other. For example, latency and robustness conflict. Improving latency often means improving throughput in a neuronal network by multiplexing signals more often. This can degrade robustness because a highly-multiplexed

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SUZUKI et al.: A NONPARAMETRIC STOCHASTIC OPTIMIZER FOR TDMA-BASED NEURONAL SIGNALING

network has a higher risk of signal interference. Conversely, improving robustness often means multiplexing signals less often. This can degrade latency because a rarely-multiplexed network tends to be low-throughput. Since there exists no single optimal solution (i.e., TDMA schedule) under conflicting objectives but rather a set of alternative solutions of equivalent quality, the proposed EMOA is designed to seek the optimal trade-offs among objectives. Thus, it searches Pareto-optimal solutions that are equally distributed in the objective space. As a result, it can produce both extreme TDMA schedules (e.g., the one yielding high robustness and high latency) and balanced schedules (e.g., the one yielding intermediate latency and intermediate robustness) at the same time. Given a set of heuristically-approximated Pareto-optimal TDMA schedules, the proposed EMOA allows a communication protocol designer to examine the trade-offs among objectives and make a well-informed decision to choose one of them, as the best TDMA schedule, according to his/her preferences and priorities. For example, a protocol designer can examine how he/she can trade (or sacrifice) latency for robustness and determine a particular TDMA schedule that yields a desirable/comfortable balance of latency and robustness. • Noise: Noise is inherent in neuronal signaling [5]. For example, thermal, current and chemical noise affects the speed (and latency) of neuronal signal transmission [5]–[7]. Given noise, the same individual (i.e., TDMA schedule) can yield different objective values (e.g., latency measures) from time to time. This degenerates individual-to-individual comparison; for example, the proposed EMOA may mistakenly judge that an inferior individual outperforms a superior one. More importantly, noise can cause signal interference unexpectedly. Even if the proposed EMOA determines a feasible individual, which is designed to perform interference-free signaling, the individual may actually become infeasible, invoking signal interference, due to noise. The proposed EMOA implements and performs a statistical noise-aware operator for individual-to-individual comparison. (Individual-to-individual comparison operators are often called dominance operators in the area of evolutionary multiobjective optimization.) The operator, called -dominance operator, is designed with the Mann-Whitney -test ( -test in short), which is a nonparametric (i.e., distribution free) statistical significance test [8]. The -dominance operator takes objective value samples of given two individuals, performs a -test on the two sample sets to examine whether it is statistically confident enough to judge a superior-inferior relationship between the two individuals, and determines which individual is statistically superior/inferior. Using this operator, the proposed EMOA is designed to seek the individuals (TDMA schedules) that are resilient to noise and high-quality with respect to optimization objectives. Simulation results show that the proposed EMOA efficiently obtains quality TDMA schedules with acceptable computational costs and allows Neuronal TDMA to transmit signals by

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balancing the trade-offs among conflicting objectives in noisy environments. It is also verified that multiobjective analysis is critical to configure and operate neuronal signaling. The proposed EMOA outperforms existing noise-aware EMOAs. II. RELATED WORK Compared to other molecular communication approaches (e.g., molecular motors [9], calcium signaling [10] and bacteria communication [11]), neuron-based communication has such advantages as long distance coverage, high speed signaling (up to 90 meters/second) and low attenuation in signaling [12]. Balasubramaniam et al. examined TDMA-based neuronal signaling [13]. Neuronal TDMA extended it with an EMOA that considers communication performance objectives such as signaling yield and latency [14]. This paper extends [14] by considering communication robustness as well as communication performance in noisy environments and handling noise with the -dominance operator. Communication robustness in noisy environments is never studied in [13], [14]. Tezcan et al. address communication robustness in TDMAbased neural signaling by proposing a signal buffering mechanism with neural delay lines, which parallel fiber delay lines in optical network switching [15]. This buffering mechanism is similar to the proposed EMOA in that both aim to avoid signal interference. However, this paper addresses robustness in the context of optimizing TDMA schedules while Tezcan et al. do not intend to optimize TDMA schedules. In the area of evolutionary multiobjective optimization, there exist various existing work to handle noise/uncertainties in objective functions. Most of them take parametric approaches, which assume particular noise distributions in advance; for example, normal distributions [16]–[20], uniform distributions [21] and Poisson distributions [22], [23]. Given a noise distribution, existing noise-aware dominance operators collect objective value samples from each individual [17], [18], [21]–[23], or each cluster of individuals [19], in order to determine superior-inferior relationships among individuals. In contrast, the -dominance operator assumes no noise distributions in advance because it is not fully known what distribution noise follows in each step/component in the entire neuronal signaling process. Voß et al. [24] and Boonma et al. [25] study similar dominance operators to the -dominance operator in that they assume no noise distributions in objective value samples of each individual and statistically examine the impacts of noise on those samples. Voß et al. propose an operator that considers an uncertainty zone around the median of samples on a per-objective basis. A superior-inferior comparison is made between two individuals only when their uncertainty zones do not overlap in all objectives. The uncertainty zone can be the inter-quartile range (IQR) or the bounding box based on the upper and lower quartiles (BBQ) of samples. Unlike the -dominance operator, this operator is susceptible to asymmetric heavy-tailed noise distributions. Boonma et al. propose an operator that classifies objective value samples with a support vector machine, which can be computationally

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Fig. 1. The structure of neurons.

Fig. 2. Intracellular Ca

concentration.

expensive. In contrast, the -dominance operator is designed lightweight; it requires no classification and learning. III. BACKGROUND This section provides preliminaries on neuronal signaling and neuron-based intrabody sensor-actuator networks. A. Structural and Behavioral Properties of Neurons Neurons are a fundamental component of the nervous system, which includes the brain and the spinal cord. They are electrically excitable cells that process and transmit information via electrical and chemical signaling. A neuron consists of cell body (soma), dendrites and axon (Fig. 1). The diameter of a soma varies from 4 to 100 micrometers. Dendrites are thin structures that arise from the soma. The length of a dendrite is up to a few hundred micrometers. An axon is a cellular extension that arises from the soma. It travels through the body in bundles called nerves. Its length can be over one meter in the human nerve that arises from the spinal cord to a toe. Neurons are connected with each other via synapses, each of which is a junction between two neurons. A synapse contains molecular machinery that allows a (presynaptic) neuron to transmit a chemical signal to another (postsynaptic) neuron. Signals are transmitted from the axon of a presynaptic neuron to a dendrite of a postesynaptic neuron. An axon transmits an output signal to a postsynaptic neuron, and a dendrite receives an input signal from a presynaptic neuron. Presynaptic and postsynaptic neurons maintain voltage gradients across their membranes by means of voltage-gated ion channels, which are embedded in the presynaptic membrane to unbalance intracellular and extracellular concentration of ions (e.g., Ca ) [26]. Changes in the cross-membrane ion concentration (i.e., voltage) can alter the function of ion channels. If the concentration changes by a large enough amount (e.g., approx. 80 mV in a giant squid), ion channels start pumping extracellular ions inward. Upon the increase in intracellular ion concentration, the presynaptic neuron releases a chemical called a neurotransmitter (e.g., acetylcholine), which travels through the synapse from the presynaptic postsynaptic neuron. The neurotransmitter electrically excites the postsynaptic neuron, which in turn generates an electrical pulse called an action potential. This signal travels rapidly along the neuron’s axon and activates synaptic connections (i.e., opens ion channels) when it arrives at the axon’s terminals. This way, an action potential triggers cascading neuron-to-neuron communication.

Fig. 3. An example intrabody nanonetwork.

Fig. 2 shows how Ca concentration changes in a neuron. When the concentration peaks, the neuron releases a neurotransmitter(s) and goes into a refractory period , in which the neuron replenishes its internal Ca store. During , it cannot receive and process incoming signals. The refractory period is approximately two milliseconds in a giant squid. B. Neuron-Based Intrabody Sensor-Actuator Networks This paper utilizes neuronal signaling in a three-dimensional network of neurons that are artificially grown and formed into particular topology patterns. This assumption is made upon numerous research efforts to grow neurons on substrates [27] and form topologically-specific neuronal networks [28]–[31]. Fig. 3 illustrates an example intrabody nanonetwork that contains an artificially-grown (tree-structured) neuronal network and several nodes such as sensors and a sink. Sensors utilize neuronal signaling to deliver sensor data to a sink. As a potential application, sensors may periodically monitor certain physiological status and report physiological data or biomedical anomalies to the sink. The sink may work as a transducer that converts incoming electrochemical signals to electrostatic or electromagnetic signals. Electrostatic signals may carry sensor data to an on-body (i.e., epidermal) device(s) through a body-coupled communication scheme [32], [33]. Electromagnetic signals may carry sensor data to an around-body device(s) such as a smartphone and tablet computer. This paper assumes that nodes interact with neuronal networks in a non-invasive manner. This means that it is not required to insert particular materials (e.g., carbon nanotubes) into neurons so that nanomachines can trigger and receive signals. For example, nanomachines may use chemical agents (e.g., acetylcholine and mecamylamine [13]) or light [34]. Other potential applications are neurointerfaces that leverage in-situ sensing and actuation for prosthetic and rehabilitation devices; for example, neuroprosthetic bladder control (Fig. 4). In a normal bladder, a sensory nerve senses that the bladder is full of urine and transmits a sensory signal (i.e., the bladder’s sense of fullness) to the brain. Whenever appropriate, the brain sends out a control signal through the spinal cord to contract the

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Fig. 5. Neuroprosthetic approach for visual prosthesis.

Fig. 4. Neuroprosthetic bladder control.

bladder, relax the sphincter, and trigger urination. However, nervous system disorders (e.g., spinal cord injuries and subsequent paralysis) can disrupt those signals to/from the brain and eliminate the fullness sensation and muscle control. A person with this kind of disorder is forced to empty his/her bladder with a catheter. In-situ sensing and actuation can help correct incontinence. In Fig. 4, a particular portion of a spinal nerve, called dorsal root ganglion, is teased out and interfaced to sensor nodes. The sensors intercept neuronal signals from the nerve and forward them to the sink node. The sink may determine whether the bladder is full, and if it is full, transmits electrostatic or electromagnetic signals to an on/around-body node(s), which in turn notify the bladder’s fullness to the patient. A neurostimulator(s) connected to the nerve issues high-frequency stimulation to prevent the bladder from emptying itself. Whenever ready to urinate, the patient uses an on/around-body node to signal a subepidermal node, which in turn transmits a neuronal electrochemical signal to in-situ a neurostimulator(s). Each neurostimulator delivers low-frequency stimulation or stops stimulation so that the bladder empties. This intervention can be a less invasive alternative to the current state of the art in neuroprosthetic bladder control (e.g., [35]). Another potential neurointerface application is bio-hybrid visual prosthesis. In a normal retina, photoreceptor cells perform phototransduction, which converts light (visible electromagnetic radiation) to electrical signals in retinal ganglion cells. Ganglion cells are located near the inner surface of the retina, and their axons extend to the brain. Those axons form the optic nerve. Ganglion cells receive visual information from photoreceptor cells by means of electrical signals and transmits it to the lateral geniculate nucleus (LGN) of the thalamus, which in turn projects to the primary visual cortex. One of treatment strategies for retinal disorders is to use retinal implants. For example, an implant bypasses photoreceptor cells and directly stimulates the optic nerve with signals from a phototransduction device that is equipped with a video camera [36]. An emerging strategy is to bypasses the optic nerve as well as photoreceptor cells [37]. A neuron-based nanonetwork can replace the optic nerve and connect an implant with LGN as a transplanted visual pathway (Fig. 5). The

Fig. 6. An example neuronal network.

implant contains sensor nodes that receive signals (visual information) from a phototransduction device. Cultured neurons are attached onto those nodes, and their axons are guided to LGN. This bio-hybrid strategy can be useful for patients with diseases in the optic nerve such as glaucoma and diabetic retinopathy. IV. NOISY OPTIMIZATION FOR NEURONAL TDMA Neuronal TDMA performs a single-bit TDMA communication that periodically assigns a time slot to each signal transmitter. (As illustrated in Fig. 3, signal transmitters are referred to as sensors in the rest of this paper.) Sensors activate (or fire) neurons, one after the other, each using its own time slot. This allows multiple sensors to transmit signals to the sink in a shared neuronal network. Each sensor transmits single-bit information (0 or 1) within a single time slot. This single-bit-per-slot design is based on two assumptions: 1) a signal (i.e., action potential) is interpreted with two levels of amplitudes, which represent 0 and 1, and 2) after a signal transmission, a neuron goes into a refractory period ( in Fig. 2). As described in Section I, an important goal of Neuronal TDMA is to avoid signal interference, which occurs when multiple signals fire the same neuron at the same time and leads to corruption of transmitted sensor data at the sink. Signals can easily interfere with each other if sensors fire their neighboring neurons in uncoordinated manners. Neuronal TDMA is intended to eliminate signal interference by scheduling which sensors fire which neurons with respect to time. The proposed EMOA seeks the optimal TDMA schedules for a set of sensors in a given neuronal network. Fig. 6 shows an example neuronal network that has five neurons ( to ) and four nodes (three sensors, and , and a sink). Fig. 7 illustrates an example TDMA schedule for those sensors to fire neurons. The scheduling cycle period lasts 5 time slots . The sensor fires the neuron to initiate signaling in the first time slot . The signal travels through in the next time slot to reach the sink. transmits a signal on

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Fig. 7. An example TDMA schedule.

in . During , two signals travel in the neuronal network in parallel. The duration of each time slot ( in Fig. 7) is designed as the sum of three time periods: 1) synaptic delay, which is the time for neurotransmitters to travel through a synapse from a presynaptic neuron or a sensor and generate an action potential in a postsynaptic neuron, 2) intracellular transmission delay, which is the time for an action potential to travel within a neuron (i.e., from its dendrite terminal to its axon terminal), and 3) a refractory period. The scheduling problem in Neuronal TDMA is defined as an optimization problem where a neuron-based nanonetwork contains sensors, , and neurons, . Each sensor transmits at least one signals to the sink during the scheduling cycle . denotes the signals that is the total number the th sensor transmits to the sink. of signals that transmits during . A. Optimization Constrains and Objectives The proposed EMOA considers four optimization objectives including a communication robustness objective as well as three communication performance objectives: signaling yield, signaling fairness among sensors and signaling latency. Signaling yield is computed as follows. (1)

denotes the arrival time at which the sink receives the last (the th) signal that transmits. determines the scheduling cycle period . In Fig. 7, . Communication robustness indicates the unlikeliness for signals to interfere with each other on shared neurons . Shared neurons are the neurons that sensors share to transmit their signals to the sink. In Fig. 6, and are shared neurons. ( and share . and share .) Higher robustness means lower chances of signal interference. is computed as follows. It is to be maximized.

(4) denotes the arrival time at which a signal (the th signal from the th sensor ) arrives at a shared neuron . if signals from and interfere on . In Fig. 7, the first signal from and the first signal from pass through in and , respectively. Thus, . Note that if ; otherwise, . The proposed EMOA considers three constraints. The first constraint enforces that at most one signal can pass through each neuron in a single time slot. (Otherwise, signal interference occurs.) The violation for this constraint is measured as the number of signal interferences occurred during . The second constraint enforces each sensor transmits at least one signal to the sink during . When this constraint is violated, constraint violation is 1. The third constraint is the upper limit for signaling latency . The violation in signaling latency is measured as follows. (5)

indicates the total number of signals that the sink receives from all sensors during . It is to be maximized. Signaling fairness is computed as follows. It is to be maximized. (2)

denotes the departure time of the th signal that a sensor transmits to the sink. This objective encourages sensors to equally access the shared neuronal network for signaling in order to avoid a situation where a limited number of sensors dominate the network. Higher fairness means that sensors access the network more equally. Signaling latency is computed as follows. It indicates how soon the sink receives all signals from all sensors. It is to be minimized. (3)

A TDMA schedule is said to be feasible if it never violate constraints (i.e., if and ). If it violates any of constraints, it is said to be infeasible. An example TDMA schedule in Fig. 7 is feasible. B. Impacts of Noise on Objective Functions This paper considers noise that introduces temporal variances in synaptic delay and refractory period and, as a result, varies the speed and latency of signal transmissions. This noise impacts all four objective functions [(1) to (4)]. It impacts and by varying and in (2), (3), and (4). They are redefined as follows for signaling in noisy environments. (6) (7) (8) This way, noise-aware objectives are defined with additive noise ( and ), which can be positive and negative.

SUZUKI et al.: A NONPARAMETRIC STOCHASTIC OPTIMIZER FOR TDMA-BASED NEURONAL SIGNALING

Once ranks are assigned to samples, the rank-sum values, and , are computed for and , respectively. ( sums up the rank values of the samples in .) For a large sample size , the sampling distributions of and are known approximately normal [8]. Therefore, the standardized value of a rank-sum is a standard normal deviate whose significance can be tested under the standard normal distribution. The standardis given as follows. ized value of

Fig. 8. An example individual.

As and vary, signal interference can occur even if the one-signal-per-slot constraint (see Section IV-A) is satisfied. In Fig. 7, ’s signal travels on in , and ’s signal travels on in . This signaling schedule satisfies the one-signalper-slot constraint; however, the two signals can unexpectedly interfere if ’s signal arrives at in and/or if ’s signal arrives at in due to noise. This means that noise can degenerate a feasible TDMA schedule to be infeasible. If signal interference occurs unexpectedly, signaling yield decreases accordingly. Therefore, it is redefined as follows for noisy environments. is a non-negative value. (9) V. THE PROPOSED EMOA The proposed EMOA is designed to solve the noisy TDMA scheduling problem described in Section IV. It iteratively evolves the population of solution candidates (TDMA schedules), called individuals, via several operators (e.g., crossover, mutation, and selection operators) toward the Pareto optima. Each individual represents a particular TDMA schedule for sensors. Fig. 8 shows the structure of an example individual, which encodes the TDMA schedule in Fig. 7. In this example, fires its neighboring neuron, , in the first time slot . and fire and in and , respectively. A. The

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-Dominance Operator

In order to seek Pareto optimality in noisy environments, the proposed EMOA leverages the -dominance operator, which is a noise-aware operator to statistically compare given two individuals and determine which individual is superior than the other. The -dominance operator is designed with the MannWhitney -test ( -test in short), which is a non-parametric (i.e., distribution-free) statistical significance test [8]. It takes objective value samples of given two individuals, estimates the impacts of noise on the samples through a -test, examines whether it is statistically confident enough to judge the superior-inferior relationship between the two individuals, and determines which individual is superior/inferior. Given two sets of objective value samples, and , which are collected from two individuals and , the -dominance operator performs a -test on and in each objective. First, and are combined into a set . The samples in are sorted in the ascending order based on their objective values in an objective in question. Then, each sample obtains its rank, which represents the sample’s position in . The rank of one is given to the first sample in (i.e., the one with the minimum objective value). If multiple samples tie, they receive the same rank, which is equal to the mean of their positions in . For example, if the first two samples tie in , they receive the rank . of

(10) and denote the mean and standard deviation of respectively.

, (11) (12)

-test determines that and With the confidence level of are not drawn from the same distribution if or .( is the cumulative distribution function of the standard normal distribution.) This means that and are significantly different with the confidence level of . Therefore, -dominance operator concludes that is superior than with respect to an objective in question if and that is superior than if . Here, the objective is assumed to be minimized. The proposed EMOA draws 30 samples for each individual whenever it uses the -dominance operator. The operator dominates another individual judges that an individual (denoted by ) with the confidence level if any of the following conditions is held: is feasible, and is infeasible. • • Both and are feasible, and — is not superior than in all objectives using a -test with the confidence level of , and — is superior than in at least one objective using a -test with the confidence level of . • Both and are infeasible, and — is not superior than in all constraints using a -test with the confidence level of , and — is superior than in at least one constraint using a -test with the confidence level of . (Being superior means yielding less constraint violation If does not dominate and does not dominate , the two individuals are said to be non-dominated with each other. B. Evolutionary Optimization Process Algorirthm 1 shows the algorithmic structure of the proposed EMOA. At the 0th generation, feasible individuals are ran(Line 2). Infeasible domly generated as the initial population individuals are excluded in . In each generation , two parent individuals ( and ) with binary tourare selected from the current population naments (Lines 6 and 7). A binary tournament randomly takes two individuals from , compares them with the -dominance operator and chooses a dominating one as a parent. Two parents reproduce two offspring through a one-point crossover (Line 8). In this crossover, each offspring inherits the half of its paramparameters) from a parent and the other eters (

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half (

parameters) from the other parent. If and , like an example shown in Fig. 8, each offspring inherits four (1 4) parameters from a parent and eight parameters (2 4) from the other parent. From Line 9 to 13, mutation randomly chooses a parameter (or parameters) in each and flips its/their offspring with a certain mutation rate value(s). Note that crossover and mutation operators can alter a feasible individual to be infeasible although the initial set of individuals are all feasible. Binary tournament, crossover and mutation operators are exto reproduce offspring. The offspring ecuted repeatedly on are combined with the parent population to form (Line 16). inEnvironmental selection follows reproduction. Best individuals in as the next dividuals are selected from . First, the individuals in are generation population ranked based on the -dominance relationships among them. Non-dominated individuals are on the first rank. The th rank consists of the individuals dominated only by the individuals th rank. Ranked individuals are stored in (Line on the contains the individuals at the th rank. 17). Algorithm 2 shows how to rank individuals with the -dominance operator. From Line 1 to 10, the -dominance relationindividuals. Note that -domships are determined among and inance relationships are not transitive. When is not guaranteed. When objective functions contain high-level noise, the -dominance operator may judge that . If a loop exists in -dominance relationships (e.g., and ), the -dominance operator and , and revokes the -dominance relationships among concludes that they are non-dominated with each other (Line 11 to 13). individuals are ranked in , the half of them move to Once on a rank by rank basis, starting with (Lines 20 to 23 in is less Algorithm 1). If the number of individuals in . Otherthan, or equal to, , the individuals in moves to . The subset is selected based wise, a subset of moves to on the crowding distance metric, which measures the distribution of individuals in the objective space [38] (Lines 24 and 25). The metric computes the distance between two closest neighbors of an individual in each objective and sums up the distances associated with all objectives. A higher crowding distance means that an individual in question is more distant from its neighboring individuals in the objective space. In Line 24, the individuals in are sorted in an descending order based on their crowding distance measures. The individuals with higher crowding distance (Line 25). have higher chances to be selected to Algorithm 1 Optimization Process 1: 2: generated individuals do 3: while ; 4: 5: while do 6: 7: 8: 9: for each parameter in each of do then 10: if

11: Mutate a parameter in question. 12: end if 13: end for 14: 15: end while 16: 17: 18: 19: 20: while do 21: 22: 23: end while 24: sortByCrowdingDistance 25: 26: 27: end while

Algorithm 2 Dominance Ranking: sortDominanceRanking Input: Output:

, A set of

individuals

, Ranked and sorted

individuals

1: for each do do 2: for each 3: Use the -dominance operator to determine the -dominance relationship between and . then 4: if 5: 6: else if then 7: 8: end if 9: end for 10: end for do 11: for each 12: clearDominanceRelationLoop 13: end for do 14: for each then 15: if 16: 17: end if 18: end for 19: 20: while do 21: 22: for each do do 23: for each 24: 25: if then 26: 27: end if 28: end for 29: end for 30: 31: 32: end while 33: return

SUZUKI et al.: A NONPARAMETRIC STOCHASTIC OPTIMIZER FOR TDMA-BASED NEURONAL SIGNALING

Fig. 9. A simulated neuronal network of 40 neurons . The bottom white dot is the sink node. Other white dots are sensor nodes. Green, blue, and red structures indicate somas, axons and dendrites, respectively.

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Fig. 10. Hypervolume measures in noisy and noiseless environments.

B. Simulation Results TABLE I SIMULATION CONFIGURATIONS

VI. SIMULATION EVALUATION This section evaluates the proposed EMOA through simulations. A. Simulation Configurations This paper uses simulated neuronal networks that are obtained with a two-step procedure. The first step utilizes NeuGen [39] to generate a network of neurons. The second step forms a tree structure with those neurons based on a randomized -ary tree construction algorithm. This algorithm generates a rooted tree in which each neuron has no more than child neurons. denotes a normal distribution. and are the mean and the standard deviation of the number of child neurons for each neuron. This paper uses randomized 2-ary (i.e., binary) and 3-ary (i.e., trinary) trees. Each tree-structured neuronal network contains 40 neurons and 10 sensors. Fig. 9 shows an example randomized binary tree. The proposed EMOA is configured with a set of parameters shown in Table I. denotes the number of parameters in an individual ( in Fig. 8). Every simulation result is the average of the results from 20 independent simulations. This simulation study considers noise that varies the time for a signal to travel through a neuron from to as follows. (13) denotes a gamma distribution. This noise model is designed based on existing empirical and theoretical studies on time distributions for synapitic delay and refectory period [40]–[42]. This noise varies and in (2) to (4) and in turn determines and in (6) to (9).

Fig. 10 shows the proposed EMOA’s performance in noisy and noiseless neuronal networks. A noisy network is a (binaryor trinary-tree) network in which noise exists as described in (13). Noise never exist in a noiseless network. Fig. 10 illustrates how individuals increase the union of the hypervolumes that they dominate in the objective space as the number of generations grows. The hypervolume metric quantifies the optimality and diversity of individuals [43]. A higher hypervolume means that individuals are closer to the Pareto-optimal front and more diverse in the objective space. As Fig. 10 demonstrates, in a noiseless network, the proposed EMOA rapidly increases its hypervolume measure in the first 20 generations and converges around the 70th generation. In a noisy network, the proposed EMOA increases its hypervolume measure in the first 60 generations and converges around the 100th generation. In both noiseless and noisy networks, all individuals in the population are feasible and non-dominated at the last (200th) generation. This verifies that the proposed EMOA allows individuals to efficiently evolve and improve their quality and diversity in 200 generations. Fig. 10 also illustrates the impacts of noise on the proposed EMOA’s convergence speed and optimization results. Noise makes convergence speed slower by 20 to 30 generations, which account for 10 to 15% of the total number of generations in a single simulation. In a binary- and trinary-tree networks, noise degrades the hypervolume measure in the last generation by 4.3% and 6.3%, respectively. These results show that the -dominance operator allows the proposed EMOA to make the impacts of noise on convergence speed and optimization results fairly small, considering the amount of noise injected with (13). Fig. 11 shows the hypervolume measures with and without the -dominance operator in noisy networks. When the -dominance operator is disabled, the proposed EMOA uses a classical Pareto dominance operator [38] in its algorithmic structure (Algorithm 1). The proposed EMOA’s convergence speed and optimization results degrade significantly without the -dominance operator. In a binary- and trinary networks, the hypervolume measure in the last generation degrades 43% and 55%, respectively, with the -dominance operator disabled. Fig. 11 demonstrates that the -dominance operator works effectively and efficiently in noisy networks.

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TABLE II OBJECTIVE VALUES WITH ALL FOUR OBJECTIVES ENABLED

TABLE III OBJECTIVE VALUES WITH A SINGLE OBJECTIVE ENABLED Fig. 11. Hypervolume measures with and without noise handling.

Fig. 12. Distribution measures in noisy networks.

Note that, in both Figs. 10 and 11, convergence speed and optimization results are worse in a trinary-tree network than in a binary-tree network because TDMA scheduling optimization is harder in a trinary-tree network. (A trinary-tree has more shared neurons than a binary tree; it has higher chances to multiplex the network and encounter signal interference.) Fig. 12 illustrates how the diversity of individuals changes in the objective space through generations. Only noisy networks are used in this experiment. Diversity is evaluated with the distribution metric, which measures how uniformly distributed individuals are in the objective space: (14) denotes the Euclidean distance between a given individual (the th individual in the population) and its closest neighbor in the objective space. denotes the mean of . denotes the number of individuals in the population. The objective space is normalized to compute the distribution metric. Lower distribution means that individuals are more uniformly (or evenly) distributed. As shown in Fig. 12, the proposed EMOA improves the diversity of individuals as the number of generations grows. The diversity measures converge around the 140th generation in both a binary- and trinary networks. This result is consistent with the ones in Figs. 10 and 11. Table II shows the average, standard deviation (SD), maximum, and minimum of the objective values that individuals

Fig. 13. Yield-fairness tradeoff (binary).

yield at the last generation if all four objectives are considered simultaneously. Table III shows the average and standard deviation of the objective values that individuals yield at the last generation if a single objective is considered at a time. The best (maximum or minimum) objective values in Table II are equal to the average objective values in Table III. (Note that standard deviation results are always 0 in Table III.) For example, in Table II, the best (or minimum) latency is 18 in a binary network. This is equal to the average latency in Table III. These results demonstrate that, in every objective, the proposed EMOA successfully produces the best possible performance when all four objectives are considered simultaneously. Figs. 13, 14, and 15 show two-dimensional objective spaces that plot the individuals obtained at the last generation. (Each figure shows non-dominated individuals with respect to two objectives in question.) Those individuals approximate the Pareto front (i.e., the optimal trade-off) between given objectives. Given the distributions of individuals in objective value ranges, signaling protocol designers can examine how they can trade one objective for another and choose a particular individual as the best TDMA schedule according to their preferences, priorities and/or constraints. For example, Fig. 13 demonstrates that signaling fairness improves 33% as signaling yield doubles

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using a mean-based dominance by 42%. In a trinary-tree network, it outperforms an EMOA using a Gaussian dominance by 72%. Through a single-tail -test and a -test, it is verified that the proposed EMOA significantly outperforms four other EMOAs with the confidence level of 95%. VII. CONCLUSION

Fig. 14. Latency-robustness tradeoff (binary).

This paper focuses on TDMA-based neuronal signaling and formulates a noisy multiobjective optimization problem that consists of communication performance objectives (neuronal signaling yield, fairness and latency) and a communication robustness objective. The problem contains environmental noise inherent in neuronal signaling. The proposed EMOA is designed to solve this problem with nonparametric (i.e., distribution-free) statistical significance test. Simulation results show that the proposed EMOA obtains quality TDMA signaling schedules efficiently and effectively in noisy neuronal networks. The proposed EMOA outperforms existing noise-aware EMOAs. REFERENCES

Fig. 15. Yield-robustness tradeoff (binary). TABLE IV COMPARISON AMONG DIFFERENT EMOAS WITH HYPERVOLUME

from 10 to 20. Fig. 15 illustrates that robustness improves 18% as signaling yield doubles from 10 to 20. Table IV depicts the hypervolume measures that the propose EMOA and four other noise-aware EMOAs yield at the last generation. The four EMOAs use the same algorithmic structure as the one used in the proposed EMOA (Algorithm 1); however, they use different dominance operators: • Median-based dominance operator: It takes multiple samples, obtains the median objective value in each objective and uses median objective values to rank individuals. • Mean-based dominance operator: It takes multiple samples, obtains the mean objective value in each objective and uses mean objective values to rank individuals. • Gaussian dominance operator: It assumes Gaussian noise in objective values in advance and performs noise-aware ranking of individuals (Section II) [18]. • IQR-based dominance operator: It performs IQR-based ranking of individuals (Section II) [24]. As shown in Table IV, the proposed EMOA yields the highest hypervolume measures compared to four other EMOAs. In a binary-tree network, the proposed EMOA outperforms an EMOA

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Junichi Suzuki (M’99) received the Ph.D. degree in computer science from Keio University, Japan, in 2001. He joined the Department of Computer Science, University of Massachusetts, Boston, MA, USA, in 2004, where he is currently an Associate Professor. From 2001 to 2004, he was with the School of Information and Computer Science, University of California, Irvine, Ca, USA (UCI) as a Postdoctoral Research Scholar. Before joining UCI, he was with Object Management Group Japan, Inc., as Technical Director. His research interests include autonomous adaptive distributed systems, biologically-inspired computing, body area networks, cyber-physical systems, molecular communication and multiobjective optimization. Dr. Suzuki is a member of the Association for Computing Machinery (ACM). He serves as the Editor-in-Chief for EAI Transactions on Body Area Networks and Wearable Computing and serves on the editorial boards for nine international journals including Springer Journal on Complex Adaptive Systems Modeling and Elsevier Nano Communication Networks Journal. He has chaired or co-chaired 24 international conferences such as IEEE PIMRC 2014, BICT 2014, BodyNets 2012-2014, BIONETICS 2010 and ICSOC 2009. He is an active participant and contributor of the International Standard Organization SC7 (Software and System Engineering) and the Object Management Group’s Super Distributed Objects SIG.

Dũng H. Phan received a B.S. in computer science at National Defense Academy of Japan in 2010. He is working toward the Ph.D. degree in computer science at the University of Massachusetts, Boston, MA, USA. His research interests include multiobjective optimization, body area networks, cloud integrated systems, and vehicle routing problems.

Harry Budiman received his B.S. degree in business administration from University of Oregon, Eugene, OR, USA. He worked for 5 years as an office manager during which his interest in computing grew. He is currently working toward the M.S. degree in computer science at University of Massachusetts, Boston, MA, USA.