Abstract
Synapses plasticity depends on the gliotransmitters’ concentration in the synaptic channel. And, an abnormal concentration of gliotransmitters is linked to neurodegenerative diseases, including Alzheimer’s, Parkinson’s, and Epilepsy. In this paper, a theoretical investigation of the cause of the abnormal concentration of gliotransmitters and how to achieve its control are presented through a Ca2+-signalling-based molecular communications framework. A feed-forward and feedback control technique is used to manipulate IP3 values to stabilise the concentration of Ca2+ inside the astrocytes. The theoretical analysis of the given model aims i) to stabilize the Ca2+ concentration around a particular desired level in order to prevent abnormal gliotransmitters’ concentration (extremely high or low concentration can result in neurodegeneration), ii) to improve the molecular communication performance that utilises Ca2+ signalling, and maintain gliotransmitters’ regulation remotely. It shows that the refractory periods from Ca2+ can be maintained to lower the noise propagation resulting in smaller time-slots for bit transmission, which can also improve the delay and gain performances. The proposed approach can potentially lead to novel nanomedicine solutions for the treatment of neurodegenerative diseases, where a combination of nanotechnology and gene therapy approaches can be used to elicit the regulated Ca2+ signalling in astrocytes, ultimately improving neuronal activity.
I. Introduction
The field of biological and medical science has witnessed, in recent years, the impact from multidisciplinary research efforts that utilise engineering theories, materials and technologies. Examples of this impact include new approaches for nanomedicine [1], [2], smart drug delivery systems [3] and optogenetics [4], which has seen the fields of nanobiotechnology and information technology being brought together. Telecommunication engineers now are investigating biological communication processes that can either be used to understand the biological signalling processes or develop artificial communication systems at the nanoscale. The latter research topic is known as Molecular Communication [5], and its potential applications include sensor and actuator nanonetworks for the human body, as well as new forms of environmental monitoring for smart cities [6].
Inside the human brain, there is a plurality of communication systems with an interplay of different domains such as the electro-chemical synapses, involving multi-scalability in time and space. This phenomenon has attracted massive attention from the recently-formed molecular communication community due to the many communication issues present within neuronal signalling, neuronal networks and cortical circuits. On the other hand, the historically studied bipartite synapse (neuron-to-neuron) is now accepted as the tripartite synapses, which are formed by a three-way communication of a pre-synaptic neuron, an astrocyte and a post-synaptic neuron [7] [8]. The internal Ca2+ concentration can be influenced by post-synaptic voltage and the communication between astrocytes inside a network of cells and affect the gliotransmitter concentration in the synaptic channel [9]. An example of one gliotransmitter that is triggered by increasing the Ca2+ signals in astrocytes [10] is glutamate. Through the Glutamate Dependent NMDA Receptors (GNMDAR), the astrocytes play a significant role in numerous brain processes such as plasticity, learning and memory processes [11]. Therefore stable regulation of Ca2+ signalling in astrocytes and their communication is critical. Communication problems in the tripartite synapses can lead to serious diseases, including Alzheimer’s, epilepsy, schizophrenia, Parkinson’s and depression [12]. The modelling of these communication processes and the application of control theoretic methods for Ca2+ signalling have been previously suggested as a prospect for prevention of neurodegenerative diseases [13].
To this end, a control theoretic model to achieve stable levels of intracellular Ca2+ signalling is proposed for deliberately influencing the gliotransmitter concentration indirectly and remotely inside astrocyte communications nanonetworks. A theoretical framework is developed based on a feed-forward and feedback control methodology, to maintain stable astrocytes’ cytosolic Ca2+ concentration. Since proteins more easily stimulated, IP3 is used as a regulation point where its control leads to accurate stimulation of Ca2+ ions [14]. This framework opened the door for further contributions including:
Regulation of Ca2+ Signalling in Astrocytes - A mathematical framework is developed to calculate the desired Ca2+ levels based on a given desired IP3 value and also the stability of the system (the system in this context is the point of communication between the astrocyte and neuron inter-cellular signalling). Finally, a disturbance analysis is used to investigate the benefits of having a feed-forward component in the control design. The proposed technique achieves the control of Ca2+ ion levels in the cytosol of astrocytes with proven stability.
Molecular Communication System Design based on Synthetic Biology - Both transmitter and receiver nanomachines are also designed to incorporate the envisioned synthetic biology circuitry that mimics the control theory model and produces the same output. A toggle switch gene transcription is used to activate control of Ca2+ serving as a signal equaliser.
Improved Performance of Molecular Communication System - The use of Ca2+ signalling for molecular communication can result in a high quantity of noise propagating through the tissue, resulting in low data rates. However, the control theoretic method proposed in this paper regulates the Ca2+ levels, resulting in a minimum amount of noise, which in turn achieves superior communication performance with increased the data rate, increased gain and decreased delay.
The paper is organized as follows. §II introduces the tripartite synapses and the intracellular Ca2+ signalling model for astrocytes. §III presents the oscillation behaviour of regular Ca2+ signalling process and the problem statement. §IV presents the feed-forward and feedback control technique for astrocytes’ cytosolic Ca2+ concentration regulation followed by a stability analysis. §V presents the designed toggle switch biological synthetic circuit. §VI presents the Ca2+signallingbased molecular communication system that uses the synthetic circuit to implement the control function. §VII presents the numerical results and analysis of the application of the control technique for elimination of Ca2+ signalling oscillations, disturbance, data rate, molecular gain and molecular delay improvements. §VIII presents a discussion about the future envisioned applications. Finally, §IX concludes the paper.
II. Tripartite Synapses
Fig 1 shows an accurate illustration of the tripartite synapses. The concentration of gliotransmitters in the region connecting both the neurons and the astrocyte is crucial for synapses’ plasticity. Researchers have been able to identify the importance of the astrocytes in the tripartite synapses [8], and the communication process is illustrated in Fig 2. In the tripartite synapses, the stimulation of the IP3 production in the astrocytes will initiate the Ca2+ signalling process. This stimulus starts either from the post-synaptic, pre-synaptic and other astrocytes in the form of gliotransmitters or adenosine triphosphate (ATP). The increased IP3 values triggers the release of Ca2+ ions into the cytosol from the endoplasmic reticulum. High quantity of Ca2+ ions then provoke the release of glutamate1 into the synaptic channel. Glutamate is also released from the pre-synaptic neuron invoking an increase of Ca2+ concentration in the astrocytes. These glutamate molecules go back to the pre-synaptic terminal either to inhibit or assist further glutamate release. Therefore, the intracellular Ca2+ signalling in astrocytes of the tripartite synapses dynamically regulates synaptic transmission. It is still debatable also the influence of different types of neurons on the signalling in astrocytes, and the absence of the modulatory activity from pre-synaptic neurons and interneurons [15]. Therefore, we initially neglect these effects for the sake of defining an initial scenario whereby control can be achieved. Based on this, the focus is on the core of this sequential communication process by concentrating on the intracellular Ca2+ signalling process in astrocytes.
The intracellular Ca2+ signalling model in astrocytes consists of state equations for the Ca2+ concentration in the cytosol (C) (Eq. 1), kinetics of IP3 receptors (h) (Eq. 2) as well as the IP3 concentration (I) (Eq. 3). This model is proposed in [16], and a visual illustration of the model is presented in Fig. 3. The choice of this model is twofold i) needs to be simple and accurate enough to create a low complexity control model ii) it is validated with experiments of astrocytes. The main state equations are defined as follows: where α is the constant degradation time of IP3 concentration, i0 is the IP3 concentration in equilibrium, β is the production rate of IP3 ions, E0 is the pre-synaptic potential and ℋ (.) is the Heaviside function. Ca2+-induced Ca2+ release (CICR) is the trigger process of Ca2+ ions from the sarco(endo)plasmic reticulum by existing Ca2+ ions within the cytosol. The quantities H,τ are defined in (8), (9, (10). The function σ1 models the CICR and is defined as: where v is the maximal CICR rate, c0 is the total cell free Ca2+ concentration depending on the cytosol volume, and C1 is the ratio between the cytosol and endoplasmic reticulum volume.
The IP3 and Ca2+ ion binding process that is responsible for providing stable IP3 kinetics is represented as: where d is the IP3 dissociation constant and d3 is the Ca2+ activation-dissociation constant.
The function σ2 denotes the leakage of Ca2+ ions to the cytosol from the sarco(endo)plasmic reticum and is represented as: where v1 is the maximal rate of Ca2+ ions leakage from the endoplasmic reticulum.
The efflux of Ca2+ from the sarco(endo)plasmic reticulum to the endoplasmic reticulum (SERCA) is represented as: where v2 is the maximal rate of SERCA uptake, and k is Ca2+ binding affinity.
The following equations are important for modelling h: where d1 is the Ca2+ inactivation dissociation constant, d2 is the IP3 dissociation constant and a is the IP3 receptors binding rate for Ca2+ inhibition.
III. Problem Statement
Neurodegenerative diseases are related to the quality of the synapses (plasticity) in neuronal communication. The poor concentration of glutamate inside the synaptic channel will lead to poor propagation in the synapses, causing lack of memory, insomnia, depression which are symptoms of most neurodegenerative diseases. Current treatment techniques of neurodegenerative diseases are based on drugs that are not efficient and only helps to eliminate symptoms and not treat them. Based on what has been already presented, achieving Ca2+ control in astrocytes can indirectly control the glutamate release and potentially improve the synaptic plasticity. Control of Ca2+ has been theoretically achieved in our previous work [17]. The primary challenge now is to provide an analysis on the astrocytes Ca2+ concentration of the tripartite synapses and, therefore, create a theoretical framework that supports the indirect control of glutamate.
For this, the problem of controlling levels of Ca2+ in the cytosol is investigated in this paper and its influence on an astrocyte network communication. More specifically, internal Ca2+ signalling is characterised by oscillations invoked by a particular range of IP3. Fig 4 shows the Ca2+ oscillation at IP3 = 0.5 μM and when production rate of IP3 ions (β - Eqn. 3) varies from 0.1-1.5 μM/s and modulates both the amplitude and frequency of the signal.
Fig 5 shows how the IP3 can affect the intracellular Ca2+ signalling based on the postsynaptic voltage influence. An increase of IP3 in the system is desired for regular Ca2+ concentration levels. Since β is responsible for the IP3 increase, it is highly critical for regulation of Ca2+ concentration levels. As soon as IP3 achieves a constant level, the Ca2+ concentration will drop. The post-synaptic voltage will also be influenced, as seen in Fig. 5 (E0 = 50V in the left, E0 = 35V in the center and E0 = 0V in the right). However, since the synapses happen periodically, we assume only activated neurons and also no inter-synaptic interference in the synaptic channel. The IP3 is, in conclusion, a decisive factor for Ca2+ regulation, in which its increase is controlled by β. We further explore it with a mathematical model that enables the intracellular Ca2+ signalling control by regulating IP3 levels.
IV. Feed-forward and Feedback Control of Intracellular Ca2+ Signalling in Astrocytes
Since IP3 activation can only be performed in highly controlled settings [17], its regulation of Ca2+ concentration is more suitable for in-vitro scenarios at the moment. The proposed approach can be potentially realised with the help of nanoparticles or silicon devices inside a cell to inhibit or activate Ca2+ as part of the control model.
Fig. 6 shows a functional block diagram of the desired Ca2+ signalling set point regulation. Before proceeding further, assume for the rest of the paper that ℋ(E0 − 35) = 1. Note that for other values of the Heaviside function (such as , a similar analysis can be applied. By denoting the column vector state x = [C h I]’, rewrite (1), (2), (3) as the following nonlienar controlled state space system where f(x) = [f1(x) f2(x) f3(x)]′ and f1(.) = σ1+σ2+σ3,, and and B = [0 0 1]′. The control variable u represents the state feedback and feedforward based IP3 regulation parameter given by (see also [14]): where βf is the feed-forward control representing the desired IP3 level If, Cf is the associated desired Ca2+ concentration level and Kf is the linear feedback gain. Note that although not visible in the above equation, there is an associated value of h as well, which is denoted by hf. Denote the entire 0 associated state vector as xf = [Cf hf If]′. Then it follows that where of course, uf = βf. Here, the measured output is the Ca2+ concentration level, so that at the desired level, the output is given by .
Remark 1: Note that without loss of generality, one can assume that in the uncontrolled case, i.e., when β = 0, there is an equilibrium point at the origin for the nonlinear dynamical system [14], implying f(0) = 0. This fact will be used in the stability analysis of the system when the control law (12) is applied in a subsequent subsection.
A similar control law to (12) including the feed-forward term was proposed for disturbance rejection in roll-to-roll manufacturing system [14]. In this model, no specific disturbance variables are implemented, which may be present in cases of diseases that may affect the normal regulated Ca2+ concentration levels. However, since the nature of the problem investigated in this paper is elimination of Ca2+ oscillations, it is clear that not only such a control function perfectly fits into our modelling of the nonlinear controlled Ca2+ regulation system, but also allows us the flexibility of including disturbance factors in future modelling extensions.
A. βf and Cf Relationship
To obtain the required regulation factor βf for a desired Cf, a mathematical relationship between βf and Cf is obtained as follows. Suppose there is an equilibrium point (in the controlled case when β > 0) at (hº cº I0), then at h = hº and at C = cº, and for I = I0. Rewriting Eq. 1 and Eq. 2, obtaining: where and
Here Io is obtained from for Eq. 3, (since ℋ(E0 − 35) = 1 as previously assumed), by solving: which yields
As t → ∞, I becomes a constant Io, which is represented as:
Substituting the above value of Io in the equations for Qo,hº and zo, one can solve the nonlinear equation (14) to obtain cº = Cf for a given Io = βf.
B. System Stability
Since anomalies in the dynamics in Ca2+ signalling can lead to diseases and also tissue death, a stabilising control model for such a system is critically required. In the following, a stability analysis is presented that ensures that how the feed-forward and feedback based regulation of the IP3 level can maintain the Ca2+ level very close to its desired value. This requires a linearization of the nonlinear dynamics around the equilibrium using Taylor’s series expansion, assuming one initializes the system very near the origin. Recall from Remark 1 that is assumed without loss of generality that there is an equilibrium at the origin for the uncontrolled model.
Note that by carrying out a Talyor’s expansion of (11) around the origin, and ignoring the higher order nonlinear terms (since the expansion is obtained very close to the origin) one obtains where Ax is the linear part of f(x). Note that A is given by
It follows easily that
One can also show from linearization of (13) around the origin that the following is true:
Defining ζ = x – xf, having (by subtracting (24) from (24)) where K = [Kf 0 0]. It follows then that the matrix A–BK is given by
Finally, the eigenvalues of A–BK can be easily shown to be . Since all of these eigenvalues are negative, the matrix A – BK is Hurwitz, which implies that x(t)-xf(t) → 0 as t → ∞, or C → Cf as t → ∞. This proves that under the feed-forward and feedback control law proposed in this work, the Ca2+ level can be maintained at the desired level after a sufficiently large time interval. Simulation studies will show that the convergence of C to the desired value Cf is considerably fast.
V. Synthetic Circuit for Control Implementation
The implementation of the control function (12) in astrocytes requires a high-level synthetic circuit design. Synthetic biology has been extensively studied in mammalian cells [18], [19], [20]. However, since the objective of the remainder of the paper is for the validation of the proposed technique, in the following one must find an abstraction of the control function regarding the synthetic circuit. Therefore the implementation of the control function can be expected to be partially achieved in astrocytes. The technique consists of an envisioned programmable genetic circuit that triggers the production of IP3, and therefore, controls Ca2+ through a toggle switch. The advantage of this approach is that there is no need for rewiring Ca2+ signalling pathways. Previously, RNAi has also been used to screen the function of different players in the Ca2+ signalling pathway [21], and the Ca2+ pathway rewiring has been targeted by synthetic biologists [22].
The theoretical two-gene bistable toggle switch has been proposed by Gardner et al [23]. The switch consists of two constitutive promoters (coloured black and grey) and two repressor genes (also coloured black and grey). The black repressor protein silences the grey promoter, which drives production of the grey repressor protein. Conversely, the grey repressor protein silences the black promoter, which drives production of the black repressor protein. Thus, if the black repressor protein were produced, the grey repressor protein could not be produced, and vice versa. This is an example of a synthetic bistable toggle switch similar to the theoretical one in Figure 7. The presented approach has the goal to only theoretically simulate an dependent stimuli of IP3 production based on current Ca2+ levels.
The control function (12) requires monitoring of both Ca2+ and IP3 states. This can be achieved by individual cells simultaneously transfected with recombinant, Ca2+-sensitive and IP3-sensitive fluorescent reporters [24], [25]. This is abstracted here with the assumption of ideal monitoring of both Ca2+ and IP3 levels since the design of the synthetic circuitry is not the focus of the paper.
A. Stochastic Solver
VI. CA2+-signalling-based Molecular Communication System Model
In this section, a single mathematical framework merges the Ca2+-signalling, control and synthetic biology models alongside diffusion and 3D cellular tissue modelling. Barros et. al. has extensively studied this model [26], [27], [28].
Spatio-temporal dynamics is captured by a 3D cellular tissue model, correlating phenomena and variables at different scales and analysing how the control technique will perform. The description is composed of two parts: the 3D modelling of the cellular tissues, and the a stochastic model for the scheduling of reactions within individual cells.
A. 3D Modelling of Cellular Tissue Structure
Consider a cellular tissue space (S) composed of I × J × K cells (c), where ci,j,k (i = 1…I; j = 1,…J and k = 1,…K) denotes an arbitrary cell in the tissue. The cells are connected with a maximum of six neighbouring cells. The topological organisation of the astrocytes is considered following regular specifications in [29]. A Stochastic Solver computes the values of each pool over time, selecting and executing scheduled reactions. The pool is negatively or positively affected by a constant α when a particular reaction is run. More details on the astrocytes’ pools and reactions can be found in Section II.
Modelling diffusion in a cellular tissue area captures the temporal-spatial dynamics of intercellular Ca2+ signaling. The model considers Ca2+ concentration difference for temporalspatial characteristic as follows: where n ∈ (i−1,i+1), m ∈ (j−1,j+1), l ∈ (k−1,k+1), D is the diffusion coefficient, v is the volume of the cell, and ZΔ is the difference in Ca2+ concentration between the cells. p(.) is the probability of the gap junction opening. More details of this model can be found in [27].
At each time step, the Gillespie algorithm [30] (a Stochastic Solver) schedules a particular astrocyte ∈ S, its reaction (R) and the time duration of this particular reaction (t), therefore determining the quantity of each pool over time. Each reaction is allocated a reaction constant (ar). Considering that α0 is the summation of all ar in R, the next reaction chosen ru will be: which follows the roulette wheel selection process, by choosing the events based on their probability values. However, u must satisfy the following restriction: in which ρ2 is a uniform random variable with values in the range (0,1).
At each time step (t), a time lapse (τt) is derived based on α0, and is represented as: in which ρ1 is a uniform random variable with values in the range (0,1). This process ends when , where T is the set of t and tθ is the maximum simulation time.
B. System Design
Both transmitter and receiver design are based on [26], [27], [31] plus the following considerations. Both transmitter and receiver have been engineered with the proposed control function and its toggle switch activation in the form of an equaliser. In this way, both devices are capable of ideally responding to a desired level of IP3 and, therefore, maintain stable levels of Ca2+ concentration during the transmission period Tb. The toggle switch is activated with the two following processes: 1) the external cellular stimuli in the transmitter and 2) the upcoming Ca2+ signals in the receiver. This process is depicted in Fig. 8. The system design also has the following assumptions. Assumption 1: Synchronization between the transmitter and receiver is considered in ideal settings [26]. Assumption 2: The effects of the synthetic circuit on the stimulus of IP3 is also considered ideal. The paper only concentrates on the effect of the Ca2+ control in the astrocyte communication. Assumption 3: The used modulation is OnOff Shifting Keying. We do not extend the analysis for other types of modulation.
VII. Performance Analysis
The performance analysis of the proposed regulation of Ca2+ concentration levels for astrocytes is in the following. It is divided into four parts for a proper understanding of the system under controllable conditions. First, how the control system eliminates intracellular Ca2+ oscillations in astrocytes is shown by solving the problem defined in Section III. This is followed by the disturbance analysis, where Gaussian noise is applied to the intracellular Ca2+ signalling for adding a controlled abnormal behaviour to the system and observing system effectiveness while looking at the feed-forward and feedback techniques separately. Finally, analyses of the performance improvement for the Ca2+-signalling-based molecular communication system of astrocyte networks is shown.
1) Elimination of Intracellular Ca2+ Oscillations: As mentioned in Section III, the elimination of the intracellular Ca2+ oscillation is a desired outcome of the control process. For this Eqs. 1, 2 and 3 are solved using the parameter values found in Table I. For the control technique, the β in Eq. 3 is replaced by Eq. 12, in order to integrate the feed-forward and feedback control element to the system. A value of Cf = 0.32 μM is chosen, which is a central value of the system, and leads to the desired value of βf with an appropriate calibration of the Kf.
A total elimination of the Ca2+ oscillation is obtained using the proposed mechanism, and this is illustrated in Fig 10. Eq. 12, which represents the state feedback and feed-forward control, can efficiently adjust β accordingly and maintain Ca2+ concentration levels throughout the period shown across all Ca2+ signal modulation variations. This positive result demonstrates the effectiveness and potential of utilising the control technique to stabilise the excessive Ca2+ concentration that may either lead to neurodegenerative diseases or artificial molecular communication, which will be further explored in the following subsections. Further results with a different set of scenarios are presented in [17].
2) Analysis of System Disturbance: Feed-forward control techniques are used in control theory to stabilise systems under the presence of disturbances. The presented Ca2+ signalling model so far does not include any disturbances or noise components. To analyse the benefit of the feedback control technique with and without feed-forward separately, a noise component is integrated and determine the effectiveness of the approach to stabilising the disturbance. To achieve this, a disturbance component was added to Eq. 1, through addition of Gaussian noise [32]. This additional noise affects the Ca2+ level in the astrocyte cell’s cytosol.
As illustrated in Fig. 11, the feed-forward control can maintain the levels of Ca2+ even with the additional Gaussian noise. This is obtained by the relation of the current Ca2+ concentration levels with the desired level, and therefore, this stability is achieved through the adjustment of β. This demonstrates the excessive control fluctuations in Ca2+ concentration levels from additional noise.
3) Maintaining Stable Ca2+ Concentration: While the previous section presented the case of maintaining stable Ca2+ concentration due to excessive noise, in this section, the impact of overall fluctuations in the cytosolic concentration of the cell is discussed. The ability to maintain the stability of the cytosolic Ca2+ concentration levels in astrocyte cells can lead to not only a healthy state of the cell but also the synaptic transmission quality in the tripartite synapses. In the event of fluctuations (extreme low or high) in the overall concentration, this can lead to some neurodegenerative diseases. For example, low Ca2+ concentration leads to cell death and poor functioning of neurons that cause depression, whereas high Ca2+ concentration is linked with one of the causes of Alzheimer’s disease [33], [34], [35]. For evaluation of both extreme high and low concentrations, a simple model is presented that shows the effectiveness of the control technique in maintaining the stable zones of Ca2+ concentration levels. We do not claim that this technique can directly be applied to the treatment of neurodegenerative diseases. This method is used only in micro-levels of the astrocyte communication and tripartite synapses. Since neurodegeneration affects higher levels of communication in the brain, there needs to be a extended version of this system that achieves the very challenging distributed astrocyte control.
There is a variation of β in Eq. 3 from 0.1 to 0.9 μM that is compared with the resulting Ca2+ concentration levels with the feed-forward and feedback control technique. In the case of when no control is applied, Ca2+ oscillations are expected. Based on the oscillations, the maximum and minimum values are selected of the final concentrations and used them to define three regions: extreme high region, extreme low region, and stable region. The extreme high region consists of any value higher than the maximum Ca2+ concentration levels, while the extreme low region contains any value lower than the minimum Ca2+ concentration levels. The stable region, which represents the safe level in the Ca2+ concentration, is in between the extreme limits.
Regulation of Ca2+ concentration can maintain stability in the concentration within the safe region for all IP3 values as illustrated in Fig. 12. The Cf from Eq. 12 was selected based on the central value between the maximum and minimum values of C. The βf was computed based on Cf, and by adjusting this, the system will change C to match with Cf. There is a concern of long term usage of the proposed technique in regards to cell function homoeostasis. Even though the Ca2+ concentration levels can be maintained for long periods of time (minutes), we do not guarantee that cellular function will not be changed in longer periods. This will require novel protocol solutions that activate and deactivate the control autonomously under certain condition and can be realized through synthetic biology.
4) Data Rate: One of the issues regarding the use of Ca2+ signalling for molecular communication is the artificial stimulation of the ions for communication purpose, as well as the excessive noise that can result in poor data rate performance. Barros et. al. [26], [27], showed that large Tb is desired to achieve reasonable communication capacity but this results in poor data rate for most types of cells that communicate using Ca2+ ions. Another reason for the long bit transmission periods Tb is due to the refractory time of the Ca2+ oscillation. The refractory time is an inherent process found in Ca2+ intracellular signalling, where the concentration of the fluctuating ions are required to stabilise before they can be stimulated again, Fig. 13. The objective now is to integrate the proposed feed-forward and feedback control technique towards the refractory time elimination using an equaliser.
To show the benefits of the proposed control method for molecular communication, a data rate analysis is presented of a single hop Ca2+ molecular communication system using astrocytes. The data rate in such system can be computed using: 1/(Tb∗Nb), where Nb is the number of bits transmitted. Three Tb values were used (5s, 10s, 50s) and compared with the performance for the case when the system integrates the control model as well as without, and varying Nb in the process. As demonstrated in Fig. 14 the elimination of the refractory time provides substantial benefit for improving the data rate for all the Tb values. On average, the refractory time takes up to five seconds to be completed. This time is needed for a full oscillation cycle from the signalling process of the cell. However, such oscillatory process can also be eliminated with the proposed feed-forward and feedback control technique achieving higher data rate values. For example, with the highest values of data rate (Tb = 5s), without control reaches a limit of 1000bps while with control reaches 2000 bps.
5) Molecular Gain: Due to the equal diffusion direction probability (Eq. 27), Ca2+ may not reach the Rx. Naturally, the amplitude of the received signal is negatively affected by longer distances between the Tx and Rx. This phenomenon is analysed using gain, which is calculated using the following formula [27]: where ΓT(f) is the average peak concentration and ΓT0(f) is the initial peak concentration.
Ideal control of the received Ca2+ ions leads to the maintenance of gain performance over distance, Fig. 15. The equaliser provides total recovery in both transmitter and receiver. However, these results might be different for other modulation schemes different than the OOK. On top of that, there must be a distance limit, where signals cannot be reached, and the activation of the control technique cannot be performed.
6) Molecular Delay: To fairly measure the amount of molecules that are transmitted as well as the time it travels over a certain distance of cellular tissue, the molecular delay proposed in [27] is used to capture this phenomenon. In this way, delay is appropriately measured for each bit, since the information is encoded into a molecular concentration of Ca2+ signals. The proposed method is modelled by the following formula: in which, CR is a function that returns the Ca2+ concentration of a Rx at time t, CT returns the Tx Ca2+ concentration at time t, TT is the time slot length of the Tx and TR is the time slot length of the Rx. Here, the end-to-end delay will be TR.
Delay is considerably reduced due to the contribution of the regenerated molecules in the receiver by the feed-forward feedback control technique in the form of the equaliser, Fig. 16. The elimination of the refractory periods is another contributor to the delay performance since the remaining interbit interference is also minimised. The difference of delay compared to gain performance is that delay performance should be similar even with different modulation schemes that keep the signal energy levels the same. This result is significant for the minimization of distance effects over the delay, decreasing around an average of 60% delay values per unit distance (number of cells away).
VIII. Discussion
In this section, two main applications are explored for the proposed Ca2+ control method, including prevention of neurodegenerative diseases and speeding up molecular communication. A significant impact on these topics will result from the utilisation of the proposed method.
A. Prevention of Neurodegenerative Diseases
Approximately 24 million people worldwide have dementia or neurodegenerative diseases with an annual cost of about $226 billion in the U.S. alone [36], [37]. Causes of such remain unknown, and only conventional symptomatic treatments are available for improving the patients’ health. This is achieved with drugs that target the symptoms alone, neither treating the underlying disease or at best delaying its progression. However, as they progress, brain cells die, and connections among cells are lost, causing the disease symptoms to worsen [38].
A major issue with drug delivery techniques to the brain is the numerous protective barriers that encapsulate the central nervous system, and one of this example is the bloodbrain barrier [38]. Overcoming the blood-brain barrier can be achieved through biotechnology, synthetic biology, as well as nanotechnology, and this can lead to efficient and directed therapeutic tools. The control model proposed in this paper can be developed from a combination of nanoparticles and gene therapy that are used to control the Ca2+ signalling, as well as synthetic biology, towards future in-vivo settings [17]. Novel self-assembly nanoparticles can bypass the blood-brain barrier and stimulate astrocytes based on existing Ca2+ and IP3 values. In the case of synthetic biology, programming of cellular signalling pathways can be achieved that can also lead to stable levels of Ca2+ ions directly, preventing fluctuations that may result in diseases. This is a more complex solution compared to the one proposed in this paper. Some works have investigated mechanisms to use synthetic biology to engineer neurons to achieve sensitivity to light at a particular wavelength, providing a new alternative for externally controlled neural stimulation [39].
B. Molecular Communication
Low data rates are a natural characteristic of molecular communication systems due to many factors including stochastic propagation delay as well as excessive noise in the environments [40]. These are due to the natural biological processes that result in poor communication performance. The contribution provided in this paper is far from solving the problem of integrating control theory and synthetic biology with molecular communication scenarios due to the plurality of biological communication channels that are very specific. Therefore, more integration work of engineered processes will be required to counter these natural processes that can affect the performance. This engineering process will usually come through integration of nanotechnology through components and materials to control the biological process or through manipulation of cell using techniques from synthetic biology.
In Section VII-4, the proposed control technique is beneficial to limit the refractory period to increase the overall performance including data rate, delay and gain improvements. However, the proposed control approach is not just limited to that, but also can be extended to perform other communication functionalities that will improve modulation and noise cancellation. Recent work has proposed a modulation technique for a Ca2+-signalling based molecular communication system [41], where digital modulation schemes such as On-Off ShiftKeying (OOK) were used in conjunction with various error control techniques. In [26], noise in Ca2+-signalling molecular communication was studied and quantified showing that its high concentration can negatively affect the communication system performance. The noise will emanate at the transmitter as the Ca2+ waves are stimulated, along the path as the waves are propagated, and at the receiver, as they stimulate Ca2+ ions to receive digital bits. Therefore, based on this scenario, and also on our technique by which the effect of noise can be limited, the control model can be engineered into the cells of the tissue to cancel noise as they propagate along the channel. This can be achieved by restricting the quantity of IP3 for each cell that represents the transmitter, along the propagation path, as well as the receiver.
IX. Conclusions
In recent years, nanotechnology has brought together some different disciplines including synthetic biology and engineering, where the objective is to develop novel health care solutions to detect, prevent and cure diseases. This includes the field of molecular communication, where its aim is to model and construct biological communication systems for inter and intracellular cellular signalling. This new area of research seeks to develop new approaches for detecting and preventing diseases that can emerge from impairments in the communication process, as well as create artificial communication processes that connect a network of nanomachines. This paper investigated one particular type of molecular communication that utilises Ca2+ signalling between astrocyte cells and pre and post-synaptic neurons. This three-way communication process is known as the tripartite synapse. In particular, the focus is on the application of a feed-forward and feedback control technique to maintain the stability of Ca2+ levels as intercellular signalling is conducted between the cells. The application of the control model achieved firstly, a tight regulation of Ca2+ concentration, demonstrably maintaining a stable level in order to minimise any fluctuations, and secondly, an improvement of the overall performance in molecular communication using Ca2+-signalling in astrocyte cells. Previous studies have shown that Ca2+-signalling in cellular tissue can lead to a significant quantity of noise within the environment, impairing the overall system performance. However, applying the control model has resulted in the reduction in the refractory period of the Ca2+-signalling leading to smaller time-slots for bit transmissions, and higher data rates. The control model proposed in this paper can pave the way for novel techniques for disease prevention, as well as mechanisms to improve the performance of molecular communication systems. Furthermore, future work be carried out to investigate improvements of neural activity by enhancing glutamate propagation through control of astrocyte Ca2+ signalling.
Michael Taynnan Barros was born in Campina Grande, Brazil, 1990. He is currently an Irish Research Council Government of Ireland Postdoctoral research fellow associated with the TSSG, WIT. Michael received his PhD in Telecommunication Software at the Waterford Institute of Technology in 2016, M.Sc. degree in Computer Science at the Federal University of Campina Grande in 2012 and B.Tech. degree in Telematics at the Federal Institute of Education, Science and Technology of Paraiba in 2011. He has published over 40 research papers in diverse journals such as IEEE Transactions on Communications, IEEE Transactions on Nanotechnology, and conferences in the area of wireless communications, optical communications, ad-hoc networks, as well as molecular and nanoscale communications. He is also a reviewer for many journals and participated as technical program committee and reviewer for various international conferences. In 2017, he served as the be the Technical Program Co-chair for the 3rd International Workshop on Nanoscale Computing and Communications (NsCC) held in conjunction with NEW2AN conference, the chair of the 5GPPP Network Management, QoS and Security Working Group and the Chair of the 2nd Network Management, QoS and Security for 5G Networks held in conjunction with the EuCNC. Interests in Molecular Communications, Nanonetworks and 5G Technology for Connected Health.
Subhrakanti Dey (M’96 - SM’06) was born in India in 1968. He received the B.Tech. and M.Tech. degrees from the Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur, in 1991 and 1993, respectively, and the Ph.D. degree from the Department of Systems Engineering, Research School of Information Sciences and Engineering, Australian National University, Canberra, in 1996. e is currently a Professor with the Institute of Telecommunications Research (ITR), University of South Australia, Adelaide. Prior to this, he was the Professor of Wireless Sensor Networks with Dept of Engineering Sciences in Uppsala University, Sweden during 2013-2017, and a Professor with the Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Australia, from 2000 until early 2013. From September 1995 to September 1997, and September 1998 to February 2000, he was a Postdoctoral Research Fellow with the Department of Systems Engineering, Australian National University. From September 1997 to September 1998, he was a Postdoctoral Research Associate with the Institute for Systems Research, University of Maryland, College Park. His current research interests include networked control systems, wireless communications and networks, signal processing for sensor networks, and stochastic and adaptive estimation and control. Professor Dey currently serves on the Editorial Board of the IEEE TRANSACTIONS ON SIGNAL PROCESSING and Elsevier SYSTEMS AND CONTROL LETTERS. He was also an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING during 2007-2010 and the IEEE TRANSACIONS ON AUTOMATIC CONTROL during 2004-2007
Footnotes
E-mail: mbarros{at}tssg.org, Subhra.Dey{at}unisa.edu.au
↵1 One type of gliotransmitter, which activates certain signalling processes in the cells.