A Molecular Communication Perspective of Alzheimer's Disease: Impact of Amyloid Beta Oligomers on Glutamate Diffusion in the Synaptic Cleft
Nayereh FallahBagheri Ozgur B. Akan





















     
   







A Molecular Communication Perspective of Alzheimer's Disease: Impact of Amyloid Beta Oligomers on Glutamate Diffusion in the Synaptic Cleft
    Nayereh FallahBagheri, Student Member, IEEE and
      Özgür B. Akan, Fellow, IEEE


  The authors are with the Center for neXt-generation Communications (CXC), Department of Electrical and Electronics Engineering, Koç University, Istanbul 34450, Turkey  (e-mail: {nbagheri23, akan}@ku.edu.tr).
  
  O. B. Akan is also with the Internet of Everything (IoE) Group, Electrical Engineering Division, Department of Engineering, University of Cambridge, Cambridge CB3 0FA, UK (email: oba21@cam.ac.uk).
  
    This work was supported in part by the AXA Research Fund (AXA Chair
    for Internet of Everything at Koç University)
 
    ^1 Université Paris-Saclay, CEA, Service de Thermo-hydraulique

et de Mécanique des Fluides, 91191 Gif-sur-Yvette, France

^2 Laboratoire de Physique de la Matière Condensée, CNRS,
École Polytechnique, 
Institut Polytechnique de Paris, 91120 Palaiseau, France
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§ ABSTRACT 


Molecular communication (MC) within the synaptic cleft is vital for neurotransmitter diffusion, a process critical to cognitive functions. In Alzheimer’s Disease (AD), beta-amyloid oligomers (Aβos) disrupt this communication, leading to synaptic dysfunction. This paper investigates the molecular interactions between glutamate, a key neurotransmitter, and Aβos within the synaptic cleft, aiming to elucidate the underlying mechanisms of this disruption. Through stochastic modeling, we simulate the dynamics of Aβos and their impact on glutamate diffusion. The findings, validated by comparing simulated results with existing experimental data, demonstrate that Aβos serve as physical obstacles, hindering glutamate movement and increasing collision frequency. This impairment of synaptic transmission and long-term potentiation (LTP) by binding to receptors on the postsynaptic membrane is further validated against known molecular interaction behaviors observed in similar neurodegenerative contexts. The study also explores potential therapeutic strategies to mitigate these disruptions. By enhancing our understanding of these molecular interactions, this research contributes to the development of more effective treatments for AD, with the ultimate goal of alleviating synaptic impairments associated with the disease.

 









Alzheimer's Disease, Amyloidβ Oligomers, Glutamate Diffusion, Synaptic Cleft, Stochastic Modeling, Neurodegenerative Disorders, Molecular Communication, Synaptic Transmission




















§ INTRODUCTION

Alzheimer's disease (AD) is a prevalent neurodegenerative disorder affecting over 32 million individuals globally <cit.>. This condition is marked by memory loss, cognitive decline, behavioral changes, and social challenges <cit.>. A hallmark of AD pathology is the presence of beta-amyloid oligomers (Aβos), which trigger various toxic pathways leading to neuronal damage and widespread neural inflammation. These oligomers negatively impact synaptic functions and memory processes, contribute to abnormal tau protein modifications, and activate microglial cells <cit.>.

The dysregulation of glutamatergic neural circuits is believed to play a critical role in the neurobiological foundation of AD <cit.>. AD progression is notably characterized by the degeneration of large glutamatergic pyramidal neurons <cit.>, <cit.>. Aβos disrupt glutamate transporters <cit.>, alter activity-dependent glutamate release <cit.>, and decrease the surface expression of synaptic N-Methyl-D-Aspartate receptors (NMDARs) <cit.>, which are essential for physiological neurotransmission <cit.>.

Glutamate, the brain's primary excitatory neurotransmitter, plays a vital role in cognitive functions such as learning and memory <cit.>. Its concentration in the brain varies between 5 and 15 millimoles per kilogram of brain tissue, depending on the region, making it the most abundant amino acid in the brain <cit.>.

Neurodegenerative and psychiatric disorders, including AD, are often linked to neuronal death caused by glutamate excitotoxicity or an imbalance between excitatory and inhibitory neuronal activities. This imbalance, particularly the abnormal ratio of excitatory/inhibitory inputs, is suggested to be a fundamental factor in neuropsychiatric conditions such as autism <cit.>, obsessive-compulsive disorder <cit.>, and schizophrenia <cit.>.

The dynamic and heterogeneous nature of Aβos complicates therapeutic targeting, as highlighted by <cit.>. Despite extensive research, the transition from non-fibrillar to fibrillar oligomers remains poorly understood, indicating a significant gap in our knowledge of the biochemical mechanisms involved. Additionally, while various neurotoxic mechanisms of Aβos have been proposed, their exact contributions to AD pathology are still unclear, necessitating further investigation.

Current studies emphasize the pivotal role of Aβos in AD pathogenesis, particularly their strong correlation with neuronal loss and synaptic dysfunction, even more so than amyloid plaques. This understanding is essential for exploring how Aβos disrupt glutamate diffusion and synaptic communication. However, clinical detection and quantification of Aβos are challenging due to the lack of standardized, high-sensitivity assays, leading to inconsistent results across studies. This variability may be attributed to the heterogeneity of Aβos and the diverse assay systems used. Moreover, the precise chemical composition and size-dependent properties of Aβos, likely critical to their role in AD, require further clarification <cit.>.

Research indicates that Aβos specifically bind to NMDARs containing GluN2B subunits and metabotropic glutamate receptors, causing synaptic disruptions. However, the exact receptors mediating the neurotoxicity of soluble Aβos are not well-defined, posing challenges for developing targeted therapeutic strategies for AD. Furthermore, it remains unclear whether Aβ protofibrils interact with specific NMDARs and mGluR1 or if Aβ monomers engage with these receptors at all, highlighting another critical area for future research <cit.>.

This study builds on these findings by employing a stochastic differential equation framework to model Aβo-glutamate interactions. By incorporating variable oligomer sizes and simulating their impact on glutamate diffusion within a realistic synaptic environment, we aim to address these limitations and provide new insights into the mechanisms underlying synaptic dysfunction in AD.

The rest of this paper is structured as follows: In Sec. <ref>, the synaptic cleft and MC are introduced. The system is mathematically modeled in Sec. <ref>. In Sec. <ref>, the interactions between Aβos and glutamate, along with their impacts, are discussed. The final results are detailed in Sec. <ref>, and finally future directions and conclusions are outlined in Sec. <ref>.










§ THE SYNAPTIC CLEFT AND MOLECULAR COMMUNICATION



The brain's myriad cells fall into two primary categories: information-processing neurons and the supportive glial cells, as illustrated in Fig. <ref>. Both are abundant and intricately designed, tightly arranged with tiny spaces between them. These intercellular gaps together form the brain's extracellular matrix (ECM), taking up roughly one-fifth of the brain's total volume <cit.>.




Neuronal synapses are critical junctions between neurons, facilitating the transmission of signals. These synapses consist of presynaptic and postsynaptic terminals, with the synaptic cleft forming the extracellular space between these adjacent cell membranes <cit.>, as illustrated in Fig. <ref>.



 §.§ Structural Overview of the Synaptic Cleft


The synaptic cleft is an essential component of neuronal communication, enabling the diffusion of neurotransmitters such as glutamate <cit.>. This region is heterogeneous <cit.>, with dimensions typically ranging from 0.3 to 0.5 μm in width and 1 to 1.5 μm in height <cit.>.



In this study, we model the synaptic cleft as a cylindrical volume, as depicted in Fig. <ref>, within which neurotransmitter diffusion occurs <cit.>.



 §.§ Molecular Communication Dynamics in the Synaptic Cleft


Molecular communication (MC) represents an advanced approach focused on the transmission of information via the exchange of molecules <cit.>. A key paradigm within MC is neuro-spike communication, which pertains to the signaling processes among neurons <cit.>. Within the synaptic cleft, the release of vesicles leads to interactions between glutamate and Aβos, with Aβos serving as diffusion obstacles. 

Research has shown that synaptic signaling is significantly disrupted in AD due to the presence of Aβos; however, the precise molecular mechanisms underlying this disruption remain only partially understood <cit.>. Exploring MC within this synaptic environment could provide valuable insights into potential therapeutic strategies.









Glutamate plays a critical role as a neurotransmitter in neuronal signaling. Its diffusion within the synaptic cleft, as well as the presence of obstacles along its path, directly influences the outcome of signaling—whether beneficial or pathological. Understanding how glutamate moves through the synaptic cleft and identifying any barriers it encounters during diffusion is essential for unraveling the mechanisms underlying various neuronal disorders.

Glutamate is stored within vesicles and released into the synaptic cleft upon neuronal excitation <cit.>. Based on Fick's law of diffusion and the cylindrical model of the synaptic cleft presented in <cit.>, the normal diffusion of glutamate within the synaptic cleft is described as: 


    ∂ U(t, h, w)/∂ t = D Δ U(t, h, w),

where D represents the diffusion coefficient, and Δ denotes the Laplace operator, indicating how the concentration U changes over time and space due to diffusion. The variable h denotes the distance from the cylinder axis to a point within the cleft, with a range of 0 ≤ h ≤ H. Similarly, w represents the distance along the cylinder axis from the presynaptic membrane to a point within the system, with a range of 0 ≤ w ≤ W. The normal diffusion and concentration of glutamate are illustrated in Fig. <ref>. However, obstacles significantly impact the diffusion process, with their characteristics, positioning, and movement playing crucial roles. The subsequent sections will explore these aspects in greater detail.














§ SYSTEM MODEL AND PROBLEM FORMULATION


In this section, we first describe the characteristics of Aβos, followed by an analysis of their stochastic movement, using experimental data summarized in Table <ref>.


 §.§ Beta-Amyloid Oligomers Characteristics


Aβos are widely recognized as a critical factor in the progression of AD, contributing to neuronal damage and cognitive decline <cit.>. In the brains of individuals affected by AD, both fibrillar and prefibrillar forms of amyloid β_1-42 oligomers have been observed <cit.>. Prefibrillar oligomers exhibit greater mobility compared to their fibrillar counterparts. Despite being formed from different peptides, Aβos share similar structural characteristics and display uniform diffusive behavior on the surface of living cells <cit.>.

Aβos interact with various ions and proteins, leading to random forces that induce their stochastic movement and aggregation. Consequently, the presence of Aβos in the synaptic cleft is inherently stochastic. This observation prompts the use of stochastic methods to model their presence effectively. 



 §.§ Stochastic Modeling of amyloid-beta


Stochastic differential equations (SDEs) incorporate both stochastic (random) and deterministic components, as represented in (<ref>), where the functions b and a denote the stochastic and deterministic components, respectively. This distinguishes SDEs from ordinary differential equations (ODEs), which are purely deterministic. SDEs are particularly well-suited for systems influenced by random factors. The standard SDE formulation for modeling such systems is given as follows <cit.>:


    dY = a(Y_t, t)dt + b(Y_t, t)dW_t.


In addition, the SDE framework is extendable. Given the cylindrical symmetry of the synaptic cleft, a two-dimensional approach is necessary to model the stochastic presence of Aβos. Therefore, we propose the following formulation:

    dX =     [ a(X_t, Y_t, t)dt + b_11(X_t, Y_t, t)dW_t . 
        . + b_12(X_t, Y_t, t)dZ_t ],


    dY =     [ a(X_t, Y_t, t)dt + b_21(X_t, Y_t, t)dW_t . 
        . + b_22(X_t, Y_t, t)dZ_t ],

where X_t and Y_t represent the components of a two-dimensional stochastic process, with dW_t and dZ_t as independent Wiener processes. The function a(X_t, Y_t, t) denotes the drift function that influences both the X and Y axes. The coefficients b_11 and b_22 are the diffusion coefficients along the X and Y axes, respectively. The coefficient b_12 represents the diffusion along the X axis influenced by the Wiener process dZ_t, while b_21 accounts for the diffusion along the Y axis influenced by the Wiener process dW_t. Together, these functions and coefficients define the dynamics of the stochastic process.


The first step in modeling the drift and diffusion functions is to monitor the movement of Aβos. In this context, the mean square displacement (MSD) function is an effective tool for tracking the motion of atoms within large and complex structures <cit.>. MSD can be utilized to model the diffusion matrix in a SDE for Aβos diffusion, as it is directly related to the diffusion coefficient 
(D), a critical component of the diffusion matrix.

Analysis of MSD and the initial D have been reported in the literature <cit.>. According to <cit.>, MSD is determined as:

 
    MSD(n.dt) = 1/N-n∑_i=1^N-n[ (X_i+n - X_i)^2 + (Y_i+n - Y_i)^2 ],

where X_i and Y_i denote a particle's coordinates in the i_th frame, dt is time intervals, N is the trajectory's total frame number, and n is time lag. This technique is crucial for analyzing particles' side-to-side movements, offering insights into both immediate (early D) and extended (movement types) behavior patterns. As the result of MSD calculation for both prefibrillar and fibrillar oligomers using (<ref>) in <cit.>, we have prefibrillar oligomers D as  4.0 × 10^-3μ m^2s^-1 which is nearly four times more that fibrillar oligomers D which  is  9.4 × 10^-4μ m ^2s^-1. This finding from the literature allows for the formulation of the diffusion function matrix for prefibrillar oligomers as follows:

    B = [ 4 × 10^-3         0;         0 4 × 10^-3 ] μm^2/s,

it is inferred that D is isotropic, exhibiting uniform behavior along both the X and Y axes. This suggests that the coefficient remains constant over time, showing no variation with temporal changes.

The drift function represents the deterministic part of the particle movement. It captures how the system's behavior is influenced by its current state, including interactions with other molecules, local environmental conditions, or any factors specific to the system being modeled.

We model the drift function of amyloid stochastic presence in the synaptic cleft based on its root-mean-square velocity (v_rms), which is calculated as:



    v_rms = √(2dD/Δ t) μm/s,

in the above v_rms formula, d represents the dimension of motion, D denotes the diffusion coefficient, and Δ t refers to the time step.

Consequently, the SDE describing the movement of amyloid oligomers in the synaptic cleft can be defined as:


    Δ X_t = -√(2×2 × 4× 10^-3/Δ t)Δ t + 4 × 10^-3Δ W_t,


    Δ Y_t = -√(2×2 ×4× 10^-3/Δ t)Δ t + 4 × 10^-3Δ Z_t.
 

It can be concluded that the stochastic movement of Aβos indeed leads to stochastic distributions of these oligomers within the synaptic cleft.











§ INTERACTION BETWEEN GLUTAMATE AND BETA-AMYLOID OLIGOMERS


In this section, first we analyze the dynamics of collisions; next, we explore the frequency of these collisions; and finally, we examine the consequences of these interactions within the channel.

According to the findings presented in <cit.>, a random distribution of obstacles significantly impedes the diffusion of particles more than a normal distribution does, as it hinders the particles' ability to predict their paths effectively. Similarly, the random presence of Aβos within the synaptic cleft causes random collisions with glutamate. From the moment glutamate starts colliding with Aβos, it initiates Brownian motion, a process where particles move unpredictably due to collisions, as discussed in <cit.>. In this study, as illustrated in Fig. <ref>, we divide the synaptic cleft, the channel for neuronal transmission, into three main regions: the first region, where glutamate undergoes normal diffusion; the second region, where collisions occur; and the third region, where particle movements are influenced by the collisions in the second region.




 §.§ Collision Dynamic



Fig. <ref> illustrates the collisions between glutamate and Aβos. It is evident that the collision between Aβos and glutamate adheres to the principle of momentum conservation, as their respective momenta prior to the collision are approximately equal. Under this assumption, the conservation of momentum for the two entities can be expressed as:

    m_A · v_Ax1 + m_B · v_Bx1 = m_A · v_Ax2 + m_B · v_Bx2,

where m_A and m_B represent the masses of two particles. v_Ax1 and v_Bx1 denote the velocities of particles A and B before the collision, respectively, while v_Ax2 and v_Bx2 are their velocities after the collision.
To analyze the nature of collisions and the behavior of particles after impact, it is essential to determine the coefficient of restitution, denoted as e. This coefficient represents the ratio of the relative velocity of the particles after the collision to their relative velocity before the collision, measured along the line of impact <cit.>. The value of e ranges between 0 and 1. When e = 1, the collision is elastic, meaning no kinetic energy is lost. Conversely, when e = 0, the collision is perfectly inelastic, causing the objects to stick together and move as one after the collision, resulting in the maximum possible loss of kinetic energy within the system. Values of e between 0 and 1 represent partially elastic collisions, where some kinetic energy is lost. e is defined as:


    e = |v_B,x,2 - v_A,x,2|/|v_B,x,1 - v_A,x,1|.









 §.§ Collision Frequency
 

The frequency of collisions between two particles undergoing Brownian motion is calculated as follows <cit.>:

    F = ( 2k_B T/3μ) ( 1/r + 1/R) (r + R),

where k_B is the Boltzmann constant, T represents the absolute temperature, μ denotes the fluid viscosity, and r and R are the radii of the particles. (<ref>) is based on the following assumptions:

    
  * The collision efficiency is defined as one. 
    
  * The fluid motion is laminar.
    
  * All the particles are the same size, a condition known as monodispersity.
    
  * Breaking of particles is not considered in this scenario.
    
  * The particles are spherical and will remain the same after collision.
    
  * Only two particles are involved in the collision.

In the analysis of Aβo and glutamate collisions, we apply the previously mentioned assumptions for collision frequency, with the exception of collision efficiency (α), which can range between 0 and 1. As a result, the number of successful collisions differs depending on the value of α.  Consequently, we propose that the collision frequency can be given by:


    F = α( 2k_B T/3μ) ( 1/r + 1/R) (r + R).





 §.§ Effects of Amyloid Beta Oligomers and Glutamate Collisions





  §.§.§ Impact on Glutamate Diffusion


The stochastic presence of Aβos within the synaptic cleft significantly influences glutamate diffusion and uptake. The obstacles posed by both the size and concentration of Aβos impede the mobility of glutamate molecules. The effective diffusion coefficient, D̃_eff, can be modeled by the equation presented in <cit.>:


    D̃_eff(ϕ) ≈(1 - ϕ/ϕ_c)^λϕ_c,

where ϕ represents the excluded volume fraction, indicating the volume occupied by the obstacles. The parameter ϕ_c denotes the percolation threshold, which is the critical volume fraction at which diffusion is severely hindered. The parameter λ relates to the geometrical configuration and shape of the obstacles.

Fig. <ref> visually demonstrates the impact of Aβo presence and distribution on glutamate diffusion over time. Initially, glutamate spreads relatively uniformly; however, as time progresses and Aβos are introduced, the diffusion pattern becomes disrupted, resulting in localized regions of reduced glutamate concentration corresponding to the locations of Aβos. This visualization reinforces the conclusion that Aβos can significantly hinder glutamate diffusion within the synaptic cleft.










  §.§.§ Impact on Beta-Amyloid Oligomer Movement


When a collision occurs between glutamate and Aβo, the resulting force impels the Aβo in the direction of glutamate's initial trajectory, typically towards the postsynaptic membrane. This directional movement is driven by the transfer of momentum during the collision.

The kinematic equation x = vt can be employed to estimate the displacement of Aβos after the collision, where v is the velocity imparted to the oligomer, and t is the time elapsed. This equation provides a simplified model to understand how the Aβo moves toward the postsynaptic site following its interaction with glutamate.




  §.§.§ Impact on NMDAR Dysregulation


NMDARs are essential for synaptic transmission, plasticity, and the functional maintenance of the nervous system. However, their dysfunction is associated with neurotoxicity, seizures, ischemic stroke, and neurodegenerative diseases such as AD <cit.>. Activation of NMDARs leads to an increase in cytosolic free intracellular calcium ([Ca^2+]), a crucial factor for Long Term Potentiation (LTP) <cit.>. 

Furthermore, emerging evidence increasingly suggests that Aβos disrupt glutamate receptor function, leading to disturbances in glutamatergic synaptic transmission, which are closely linked to early cognitive deficits. NMDARs play a central role in the pathophysiology associated with Aβos for several key reasons. First, NMDAR function is directly impacted by Aβos, making it a critical target. Second, NMDARs are essential mediators of the effects of Aβ on synaptic transmission and plasticity. Third, NMDARs may serve as receptors for Aβos, either through direct or indirect interactions. Finally, NMDAR activity could modulate Aβ production itself <cit.>. In vivo studies suggest a regulatory relationship where lower NMDAR activation is correlated with increased Aβ production, while higher activation levels are associated with a reduction <cit.>. Consequently, as depicted in Fig. <ref>, the binding of Aβos to NMDARs impairs the receptors' ability to receive released neurotransmitters, thus negatively impacting LTP.





∙ Kinetic Model for Aβos Binding to NMDARs

 we can propose the Aβos binding rate to NMDARs by applying the principles of mass-action kinetics, the rate of a first-order reaction A → C is given by k[A], and the rate of a second-order reaction A + B → C is k[A][B], where [X] represents the concentration (or the number of molecules, adjusted for volume) of species X <cit.>.

The binding rate constant k is assumed to vary inversely with the size of the amyloid oligomer:

    k = k_base×(50/size),

where k_base is a base rate constant chosen arbitrarily. The factor 50/size accounts for the hypothesis that larger oligomers have slower diffusion rates and greater steric hindrance, making it more difficult for them to move within the cleft and bind to receptors.

The differential equations based on these reaction rates are:

    
  * d[A]/dt = -k[A][R] for Aβos,
    
  * d[R]/dt = -k[A][R] for NMDA receptors,
    
  * d[AR]/dt = k[A][R] for the complex.


These equations collectively describe a bimolecular reaction where Aβos and NMDA receptors bind to form a complex, following second-order kinetics with a rate constant k. The formation of the complex decreases the concentrations of both free Aβos and NMDA receptors, and the stoichiometry of the reaction is 1:1, assuming no other pathways or reactions are involved.







  §.§.§ Impact on Signal-to-Noise Ratio (SNR)



MC allows implantable devices to transmit information using molecules as carriers. However, a major challenge in MC is molecular noise, which increases the likelihood of communication errors. This noise is particularly relevant in pathological conditions like AD, where cellular disruptions lead to an imbalance in molecular processes, making affected cells more reactive and thus noisier <cit.>. The SNR is a critical metric for evaluating the efficiency of MC within the synaptic cleft, especially in the presence of Aβos. In this context, the signal refers to the successful diffusion and binding of glutamate molecules to postsynaptic receptors, while the noise is represented by the collisions with Aβos, which impede this diffusion process. The SNR quantitatively measures the ability to distinguish the signal from the noise, thereby illustrating the impact of Aβos on synaptic transmission.

To calculate the SNR, the signal power is modeled as the MSD of glutamate in an obstacle-free environment, representing optimal neurotransmitter diffusion. In contrast, the noise power is derived from the variance in glutamate diffusion due to the stochastic presence of Aβos, which act as physical barriers within the synaptic cleft. The SNR is then expressed as the ratio of signal power to noise power, providing insights into the extent to which Aβos impair glutamate transmission.

The SNR can thus be formulated as:

    SNR = MSD_g  (without obstacles)/MSD_g  (with obstacles) - MSD_g  (without obstacles),

where MSD_g represents the mean squared displacement of glutamate.




§ RESULTS AND DISCUSSION


In this section, we discuss the results of our numerical simulations analyzing the interactions between glutamate and Aβos within the synaptic cleft. These simulations were conducted using a mathematical model developed in MATLAB, which builds upon the theoretical framework outlined in previous chapters. Our study focuses on evaluating how the size and concentration of Aβos affect several key parameters: collision frequency, glutamate diffusion and its MSD, movement of Aβos towards the postsynaptic membrane, their binding rate with NMDARs, and the SNR of the synaptic channel. Additionally, we examine the implications of these interactions for synaptic dysfunction, particularly in the context of AD. 

To initiate the analysis, we reference the stochastic differential equations (<ref>) and (<ref>) introduced in Sec. <ref>, which describe the random movement of Aβos within the synaptic cleft. Fig. <ref> illustrates the stochastic distribution of Aβos, highlighting three distinct scales of the Wiener process. This figure demonstrates how variations in the Wiener process significantly impact the random motion, leading to diverse positioning of Aβos within the cleft.






 §.§ Collision Frequency

In Sec. <ref>, the collision frequency has been analyzed, with the corresponding simulation results presented in Fig. <ref>. The parameters used for the simulation are detailed in Table <ref>. The results depict the collision frequency between Aβo and glutamate as a function of Aβo size, considering five different collision efficiencies. The data indicate that as the size of Aβo increases, the collision frequency also rises linearly. Moreover, the analysis suggests that higher collision efficiency (α) results in a proportionately higher collision frequency, with α = 1 yielding the highest frequency across all oligomer sizes.

Notably, for α = 0, the collision frequency remains zero, regardless of the Aβo size, representing a boundary condition where no collisions occur. These findings underscore the importance of considering both the size of Aβo and the efficiency of collisions in evaluating their potential impact on synaptic function.









 §.§ Glutamate Diffusion

In Sec. <ref>, we considered (<ref>) to study the effect of Aβos concentration and size on glutamate diffusion. As shown in Fig. <ref>, the graphs provided depict the effective diffusion coefficient of glutamate as a function of the excluded volume fraction (ϕ) occupied by amyloid obstacles, considering different values of the percolation threshold (ϕ_c) and a parameter λ, which relates to the geometrical configuration and shape of the obstacles. For Fig. <ref> with ϕ_c = 0.3, it is observed that for λ = 1, the effective diffusion coefficient decreases rapidly as ϕ increases, with a slight recovery after a certain point, suggesting a complex interaction between glutamate molecules and amyloid obstacles. For λ = 2 and λ = 3, the diffusion coefficient decreases steadily without recovery, indicating that a higher λ leads to a more substantial reduction in diffusion. This suggests that when ϕ_c is low, indicating a lower percolation threshold, the diffusion of glutamate is significantly hindered even at smaller excluded volume fractions. The effect is more pronounced for higher λ, implying that as the complexity of the obstacle geometry increases, the hindrance to glutamate diffusion also increases. If we consider the size of Aβo as an influential parameter on λ, we can conclude that larger Aβo sizes have a more significant negative impact on glutamate diffusion compared to smaller ones.


For Fig. <ref>, with ϕ_c = 0.5, it is observed that for λ = 1, the diffusion coefficient decreases smoothly yet significantly. For λ = 2 and λ = 3, the curves exhibit a more complex behavior, with a noticeable change in slope around ϕ = 0.5, particularly for λ = 2. This indicates that with a higher percolation threshold of ϕ_c = 0.5, the initial diffusion of glutamate is less hindered compared to Fig. <ref>. However, as ϕ increases, the hindrance becomes more pronounced, especially for λ = 2 and λ = 3. This suggests that the geometrical configuration and shape of the obstacles play a critical role in determining diffusion behavior when the obstacles occupy a larger volume.

In Fig. <ref>, with ϕ_c = 0.7, it is observed that for all values of λ, the effective diffusion coefficient decreases steadily as ϕ increases, with no recovery observed as in Fig. <ref>. The decrease is more gradual compared to Figs. <ref> and <ref>, particularly for lower values of λ. With the highest percolation threshold of ϕ_c = 0.7, the diffusion of glutamate is less affected by lower ϕ values, indicating that obstacles must occupy a larger volume fraction before significantly hindering diffusion. The steady decrease across all λ values suggests that at higher ϕ_c, the shape and geometrical configuration of the obstacles have a less pronounced effect on glutamate diffusion compared to lower ϕ_c values.

In conclusion, both the concentration and size (as reflected by ϕ and λ) of amyloid obstacles significantly influence glutamate diffusion in the synaptic cleft. Lower percolation thresholds (ϕ_c) result in more substantial hindrance to diffusion, particularly at smaller excluded volume fractions, whereas higher thresholds require a more significant volume fraction to produce a similar effect. The parameter λ, which reflects the complexity of the obstacles' geometry, plays a crucial role in determining the rate at which the diffusion coefficient decreases with increasing ϕ.  This result aligns with the findings of <cit.>, which indicate that as the number of obstacles increases, the dispersion of particles becomes more gradual.



 §.§ Glutamate MSD

Fig. <ref> illustrates the influence of Aβos on MSD of glutamate molecules within the synaptic cleft, emphasizing the effects of both amyloid size and concentration. In Fig. <ref>, as the concentration of amyloid oligomers increases while their size remains constant, there is a marked reduction in the MSD of glutamate. This finding suggests that higher concentrations of Aβos create more obstacles, thereby impeding glutamate diffusion. Conversely, Fig. <ref> presents the scenario where the concentration of Aβos is constant, but their size varies according to the range of radii listed in Table <ref>. A similar trend is observed: larger amyloid oligomers cause a significant decrease in glutamate MSD, indicating that larger obstacles more effectively obstruct glutamate movement. These findings collectively demonstrate that both the size and concentration of Aβos are critical factors in disrupting glutamate diffusion, with larger and more numerous amyloid oligomers resulting in a more pronounced hindrance.










 §.§ Aβ Movement

In Section <ref>, we explored how the kinematic equations for Aβos can be employed to estimate the distance these particles travel post-collision with glutamate molecules. Considering the stochastic nature of Aβo movement, which follows Brownian motion, we utilize the (v_rms) as a representative value for the relative velocity of both glutamate and Aβos prior to collisions which are listed in Table <ref>. These velocities are detailed in Table <ref>.

Utilizing the equations for momentum (<ref>) and elasticity (<ref>), Fig. <ref> illustrates the results of three consecutive collisions for Aβos within the synaptic cleft. The simulation parameters used are detailed in Table <ref>.

The graph shows a progressive increase in velocity with each collision, particularly evident in the first graph (Velocity of Aβo over Time). The velocity steps up distinctly after each collision, indicating that the kinetic energy imparted during the collisions propels the Aβos forward.

However, it is crucial to observe that the magnitude of these velocity increases—and consequently the displacement—is dependent on the size of the Aβos. The second graph (Displacement of Aβos over Time) shows that smaller oligomers (e.g., 50 kDa) exhibit much greater displacement over time compared to larger ones (e.g., 250 kDa). This is attributed to their lower mass, which translates to less inertia, allowing them to be more easily accelerated by the collisions.

Conversely, larger Aβos, while still experiencing increases in velocity, show markedly smaller displacements. This is due to their greater mass and inertia, which makes them less responsive to the forces exerted during collisions.

These findings emphasize the significant role that both size and mass play in influencing the post-collision movement of Aβos within the synaptic cleft. The analysis suggests that smaller Aβos are more likely to travel further and potentially reach the postsynaptic membrane, potentially impacting synaptic function more than their larger counterparts.








 §.§ Aβ binding

In Section <ref>, we propose that Aβos are likely to bind to NMDARs as a result of their movement after collisions. Fig. <ref> presents four plots that depict the concentrations of Aβos, NMDARs, and the Amyloid-NMDAR complex over time, with varying masses of Aβos. The initial concentrations are assumed to be: Aβos = 100 and NMDARs = 50.

The figure illustrates a clear trend: smaller Aβ oligomers (50 kDa) exhibit a significantly higher propensity to bind to NMDARs compared to larger ones (e.g., 250 kDa). This trend is observed in the increasing concentration of the Amyloid-NMDAR complex as the mass of Aβos decreases. In contrast, larger Aβ oligomers show a relatively lower increase in complex concentration over the same time period.

This behavior can be attributed to the increased mobility of smaller Aβ oligomers, which enhances their chances of encountering and binding to NMDARs. As these smaller Aβos bind to NMDARs, they obstruct further receptor activation, leading to a weakening of LTP. This interference with normal synaptic function can disrupt subsequent signaling processes, potentially culminating in cellular damage and death <cit.>.

These findings are consistent with the observations in <cit.>, which assert that LTP is primarily mediated by low-n oligomers (small oligomers), rather than Aβ monomers or larger aggregates. The higher binding affinity of smaller Aβos to NMDARs suggests that they play a more critical role in the disruption of synaptic signaling, which is a hallmark of neurodegenerative processes.






 §.§ SNR

Through simulations using mathematical modeling for SNR in Sec. <ref>, we assess the SNR under varying conditions, such as different concentrations and sizes of Aβos. 

The impact of Aβos on the synaptic cleft as a molecular communication channel was evaluated by analyzing the SNR, which serves as a critical metric for understanding the level of noise introduced into the communication channel by these obstacles. The SNR provides a quantitative measure of the channel's efficiency in transmitting signals (in this case, molecular signals such as glutamate) in the presence of noise-inducing factors Aβos. Two primary variables were considered in : the concentration (number) of Aβos and their size (radius).
As shown in Fig. <ref>, the SNR decreases significantly with an increasing number of Aβos, indicating a rise in channel noise. At a low concentration of five Aβos, the channel exhibits a high level of noise, as evidenced by the highly negative SNR of approximately -70. This suggests that even a small number of obstacles can severely degrade the quality of molecular communication within the synaptic cleft. As the number of Aβos increases to ten and fifteen, the SNR improves (becomes less negative), reflecting a non-linear relationship between obstacle concentration and channel noise. The saturation effect observed implies that while additional Aβos continue to introduce noise, their relative impact diminishes as the channel becomes increasingly congested. This trend indicates that the channel noise increases sharply at lower concentrations but stabilizes at higher concentrations, possibly due to the overlapping influence of multiple obstacles.

The analysis of the impact of Aβos size on channel noise, as illustrated in Fig. <ref>, reveals that smaller oligomers (with a radius of 7.5 nm) introduce greater noise into the molecular communication channel, resulting in a lower SNR of approximately -7. As the size of the oligomers increases, the SNR improves, indicating a reduction in channel noise. Larger oligomers, although still obstructive, appear to create a more stable and less disruptive communication environment. This finding suggests that larger oligomers may cause more predictable and consistent noise, leading to a less severe impact on the overall communication process compared to smaller oligomers, which cause more random and disruptive noise patterns.

Finally, the study highlights the significant role that both the concentration and size of Aβos play in determining the noise levels within the synaptic cleft as a molecular communication channel. While higher concentrations of Aβos substantially increase channel noise, the impact becomes less pronounced at higher concentrations, likely due to saturation effects. On the other hand, smaller amyloid Aβos generate more severe noise compared to larger ones, underscoring the importance of obstacle size in the modulation of channel noise. These findings provide crucial insights into the challenges of maintaining effective molecular communication in biological systems, particularly in pathological conditions such as AD, where Aβos are prevalent and contribute to impaired communication by introducing significant noise into the synaptic channel.

Overall, the results of our study are align with the experiments in vitro and in vivo analysis of <cit.> which have demonstrated that cell-derived low-n (small oligomers) Aβos can trigger hippocampal synapse loss and may be important effectors of synaptic dysfunction in AD.








§ CONCLUSION AND FUTURE DIRECTIONS



Our study utilized stochastic modeling to investigate the interactions between Aβos and glutamate within the synaptic cleft, with a particular emphasis on how the size and mass of Aβos influence glutamate diffusion, concentration, and displacement. The findings demonstrate that the size of Aβos is a pivotal factor in determining their impact on synaptic function. Smaller oligomers, due to their higher mobility, are more likely to bind with NMDARs, highlighting them as potential focal points for therapeutic interventions. In contrast, larger oligomers, which exhibit reduced mobility, are less likely to bind with postsynaptic membrane receptors but significantly disrupt glutamate dynamics by reducing its velocity following collisions. These insights emphasize the need to consider the diverse physical properties of Aβos in the development of targeted therapies aimed at mitigating synaptic disruptions in AD.

The findings of this study invite further exploration into several promising areas. An in-depth analysis of the collision dynamics between Aβos and glutamate molecules could elucidate the synaptic disruptions observed in AD. Specifically, understanding how the size of Aβos affects their interaction with glutamate may unveil new therapeutic targets aimed at mitigating synaptic toxicity. Furthermore, strategies to shrink or stabilize larger oligomers should be investigated to reduce their obstructive impact on glutamate diffusion and, consequently, their contribution to synaptic dysfunction.

Advancements in imaging technologies are crucial for progressing this research field. Developing high-resolution imaging systems capable of accurately measuring Aβos size in real-time would significantly enhance our ability to monitor the progression of AD and evaluate the effectiveness of therapeutic interventions. Additionally, the creation of a real-time analytical system to dynamically observe interactions within the synaptic cleft could provide invaluable insights into the mechanistic processes disrupting synaptic function.

On the therapeutic front, targeted interventions should focus on manipulating NMDAR interactions with smaller oligomers. Enhancing these interactions could potentially offset the adverse effects caused by larger oligomers, thus preserving synaptic function. This approach warrants the development of pharmacological agents specifically tailored to the characteristics of Aβos, such as their size and mobility. Such specificity could lead to more effective treatments that directly address the pathological features of AD.






















IEEEtran


































[
    < g r a p h i c s >
]Nayereh FallahBagheri(PhD. Student Member, IEEE)
received her BSc in Electrical Engineering from Semnan State University, Semnan, Iran, in 2018. She is currently working as a research assistant at the Next-generation and Wireless Communications Laboratory and pursuing her Ph.D. degree in Electrical and Electronics Engineering at Koc University, under the supervision of Prof. Dr. Özgür Barış Akan. Her
 research interests include molecular communications, Intrabody Nanonetworks and the Internet of Everything.

[
    < g r a p h i c s >
]Ozgur B. Akan (Fellow, IEEE)
received the PhD from the School of Electrical and Computer En
gineering Georgia Institute of Technology Atlanta,
 in 2004. He is currently the Head of Internet of
 Everything (IoE) Group, with the Department of
 Engineering, University of Cambridge, UK and the
 Director of Centre for neXt-generation Communica
tions (CXC), Koc ¸ University, Turkey. His research
 interests include wireless, nano, and molecular com
munications and Internet of Everything.











