Mathematical modeling for the pathogenesis of Alzheimer's disease

Oscillations in Neuronal Activity: A Neuron-Centered Spatiotemporal Model of the Unfolded Protein Response in Prion Diseases

Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the Unfolded Protein Response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P (healthy) and S (toxic) interacting with heterodimer dynamics (S interacts with P to form two S's). The model takes the form of a coupled system of nonlinear reaction-diffusion equations with a delayed, nonlinear flux for P (delay from the UPR). Through the delay, we find parameter regimes that exhibit oscillations in the P- and S-protein levels. We find that oscillations are more pronounced when the S-clearance rate and S-diffusivity are small in comparison to the P-clearance rate and P-diffusivity, respectively. The oscillations become more pronounced as delays in initiating the UPR increase. We also consider quasi-realistic clinical parameters to understand how possible drug therapies can alter the course of a prion disease. We find that decreasing the production of P, decreasing the recruitment rate, increasing the diffusivity of S, increasing the UPR S-threshold, and increasing the S clearance rate appear to be the most powerful modifications to reduce the mean UPR intensity and potentially moderate the disease progression.

Roles of astrocytes and prions in Alzheimer's disease: insights from mathematical modeling

We present a mathematical model that explores the progression of Alzheimer's disease, with a particular focus on the involvement of disease-related proteins and astrocytes. Our model consists of a coupled system of differential equations that delineates the dynamics of amyloid beta plaques, amyloid beta protein, tau protein, and astrocytes. Amyloid beta plaques can be considered fibrils that depend on both the plaque size and time. We change our mathematical model to a temporal system by applying an integration operation with respect to the plaque size. Theoretical analysis including existence, uniqueness, positivity, and boundedness is performed in our model. We extend our mathematical model by adding two populations, namely prion protein and amyloid beta-prion complex. We characterize the system dynamics by locating biologically feasible steady states and their local stability analysis for both models. The characterization of the proposed model can help inform in advancing our understanding of the development of Alzheimer's disease as well as its complicated dynamics. We investigate the global stability analysis around the interior equilibrium point by constructing a suitable Lyapunov function. We validate our theoretical analysis with the aid of extensive numerical illustrations.

Generating PET scan patterns in Alzheimer's by a mathematical model

Alzheimer disease (AD) is the most common form of dementia. The cause of the disease is unknown, and it has no cure. Symptoms include cognitive decline, memory loss, and impairment of daily functioning. The pathological hallmarks of the disease are aggregation of plaques of amyloid-β (Aβ) and neurofibrillary tangles of tau proteins (τ), which can be detected in PET scans of the brain. The disease can remain asymptomatic for decades, while the densities of Aβ and τ continue to grow. Inflammation is considered an early event that drives the disease. In this paper, we develop a mathematical model that can produce simulated patterns of (Aβ,τ) seen in PET scans of AD patients. The model is based on the assumption that early inflammations, R and [Formula: see text], drive the growth of Aβ and τ, respectively. Recently approved drugs can slow the progression of AD in patients, provided treatment begins early, before significant damage to the brain has occurred. In line with current longitudinal studies, we used the model to demonstrate how to assess the efficacy of such drugs when given years before the disease becomes symptomatic.

Analysis of fractal-fractional Alzheimer's disease mathematical model in sense of Caputo derivative

Alzheimer's disease stands as one of the most widespread neurodegenerative conditions associated with aging, giving rise to dementia and posing significant public health challenges. Mathematical models are considered as valuable tools to gain insights into the mechanisms underlying the onset, progression, and potential therapeutic approaches for AD. In this paper, we introduce a mathematical model for AD that employs the fractal fractional operator in the Caputo sense to characterize the temporal dynamics of key cell populations. This model encompasses essential elements, including amyloid-β ($\mathbb{ A_\beta }$), neurons, astroglia and microglia. Using the fractal fractional operator, we have established the existence and uniqueness of solutions for the model under consideration, employing Leray-Schaefer's theorem and the Banach fixed-point methods. Utilizing functional techniques, we have analyzed the proposed model stability under the Ulam-Hyers condition. The suggested model has been numerically simulated by using a fractional Adams-Bashforth approach, which involves a two-step Lagrange polynomial. For numerical simulations, different ranges of fractional order values and fractal dimensions are considered. This new fractal fractional operator in the form of the Caputo derivative was determined to yield better results than an ordinary integer order. Various outcomes are shown graphically by for different fractal dimensions and arbitrary orders.

How Can We Use Mathematical Modeling of Amyloid-β in Alzheimer's Disease Research and Clinical Practices?

The accumulation of amyloid-β (Aβ) plaques in the brain is considered a hallmark of Alzheimer's disease (AD). Mathematical modeling, capable of predicting the motion and accumulation of Aβ, has obtained increasing interest as a potential alternative to aid the diagnosis of AD and predict disease prognosis. These mathematical models have provided insights into the pathogenesis and progression of AD that are difficult to obtain through experimental studies alone. Mathematical modeling can also simulate the effects of therapeutics on brain Aβ levels, thereby holding potential for drug efficacy simulation and the optimization of personalized treatment approaches. In this review, we provide an overview of the mathematical models that have been used to simulate brain levels of Aβ (oligomers, protofibrils, and/or plaques). We classify the models into five categories: the general ordinary differential equation models, the general partial differential equation models, the network models, the linear optimal ordinary differential equation models, and the modified partial differential equation models (i.e., Smoluchowski equation models). The assumptions, advantages and limitations of these models are discussed. Given the popularity of using the Smoluchowski equation models to simulate brain levels of Aβ, our review summarizes the history and major advancements in these models (e.g., their application to predict the onset of AD and their combined use with network models). This review is intended to bring mathematical modeling to the attention of more scientists and clinical researchers working on AD to promote cross-disciplinary research.

Stochastic model of Alzheimer's Disease progression using two-state Markov chains

In 2016, Hao and Friedman developed a deterministic model of Alzheimer's disease progression using a system of partial differential equations. This model describes the general behavior of the disease, however, it does not incorporate the molecular and cellular stochasticity intrinsic to the underlying disease processes. Here we extend the Hao and Friedman model by modeling each event in disease progression as a stochastic Markov process. This model identifies stochasticity in disease progression, as well as changes to the mean dynamics of key agents. We find that the pace of neuron death increases whereas the production of the two key measures of progression, Tau and Amyloid beta proteins, decelerates when stochasticity is incorporated into the model. These results suggest that the non-constant reactions and time-steps have a significant effect on the overall progression of the disease.

Age-dependent accumulation of tau aggregation in Caenorhabditis elegans

Aging is the primary risk factor for Alzheimer's disease (AD) and related disorders (ADRDs). Tau aggregation is a hallmark of AD and other tauopathies. Even in normal aging, tau aggregation is found in brains, but in disease states, significantly more aggregated tau is present in brain regions demonstrating synaptic degeneration and neuronal loss. It is unclear how tau aggregation and aging interact to give rise to the phenotypes observed in disease states. Most AD/ADRD animal models have focused on late stages, after significant tau aggregation has occurred. There are fewer where we can observe the early aggregation events and progression during aging. In an attempt to address this gap, we created C. elegans models expressing a GFP-tagged version of the human tau protein. Here we examined how tau-gfp behaved during aging, comparing wild-type tau (hTau40), a disease-associated mutation (P301S), and an aggregation-prone variant (3PO). We measured age-dependent changes in GFP intensity and correlated those changes to normal aging in the nematode. We found differences in tau stability and accumulation depending on the tau variant expressed. hTau40GFP and P301SGFP were localized to axons and cell bodies, while 3POGFP was more concentrated within cell bodies. Expression of 3POGFP resulted in decreased lifespan and variations in locomotor rate, consistent with a pathological effect. Finally, we found that the human tau interacted genetically with the C. elegans ortholog of human tau, ptl-1, where the loss of ptl-1 significantly accelerated the time to death in animals expressing 3PO.

Modeling the dynamics of innate immune response to Parkinson disease with therapeutic approach

This paper aims to mathematically model the dynamics of Parkinson's disease with therapeutic strategies. The constructed model consists of five state variables: healthy neurons, infected neurons, extracellular $\alpha$-syn, active microglia, and resting microglia. The qualitative analysis of the model produced an unstable free equilibrium point and a stable endemic equilibrium point. Moreover, these results are validated by numerical experiments with different initial values. Two therapeutic interventions, reduction of extracellular $\alpha$-syn and reduction of inflammation induced by activated microglia in the central nervous system, are investigated. It is observed that the latter has no apparent effect in delaying the deterioration of neurons. However, treatment to reduce extracellular $\alpha$-syn preserves neurons and delays the onset of Parkinson's disease, whether alone or in combination with another treatment.

A continuous-time mathematical model and discrete approximations for the aggregation of β-Amyloid

Alzheimer's disease is a degenerative disorder characterized by the loss of synapses and neurons from the brain, as well as the accumulation of amyloid-based neuritic plaques. While it remains a matter of contention whether β-amyloid causes the neurodegeneration, β-amyloid aggregation is associated with the disease progression. Therefore, gaining a clearer understanding of this aggregation may help to better understand the disease. We develop a continuous-time model for β-amyloid aggregation using concepts from chemical kinetics and population dynamics. We show the model conserves mass and establish conditions for the existence and stability of equilibria. We also develop two discrete-time approximations to the model that are dynamically consistent. We show numerically that the continuous-time model produces sigmoidal growth, while the discrete-time approximations may exhibit oscillatory dynamics. Finally, sensitivity analysis reveals that aggregate concentration is most sensitive to parameters involved in monomer production and nucleation, suggesting the need for good estimates of such parameters.

Neurodegenerative Disorders of the Eye and of the Brain: A Perspective on Their Fluid-Dynamical Connections and the Potential of Mechanism-Driven Modeling

Neurodegenerative disorders (NDD) such as Alzheimer's and Parkinson's diseases are significant causes of morbidity and mortality worldwide. The pathophysiology of NDD is still debated, and there is an urgent need to understand the mechanisms behind the onset and progression of these heterogenous diseases. The eye represents a unique window to the brain that can be easily assessed via non-invasive ocular imaging. As such, ocular measurements have been recently considered as potential sources of biomarkers for the early detection and management of NDD. However, the current use of ocular biomarkers in the clinical management of NDD patients is particularly challenging. Specifically, many ocular biomarkers are influenced by local and systemic factors that exhibit significant variation among individuals. In addition, there is a lack of methodology available for interpreting the outcomes of ocular examinations in NDD. Recently, mathematical modeling has emerged as an important tool capable of shedding light on the pathophysiology of multifactorial diseases and enhancing analysis and interpretation of clinical results. In this article, we review and discuss the clinical evidence of the relationship between NDD in the brain and in the eye and explore the potential use of mathematical modeling to facilitate NDD diagnosis and management based upon ocular biomarkers.

Insulin Resistance at the Crossroad of Alzheimer Disease Pathology: A Review

Insulin plays a major neuroprotective and trophic function for cerebral cell population, thus countering apoptosis, beta-amyloid toxicity, and oxidative stress; favoring neuronal survival; and enhancing memory and learning processes. Insulin resistance and impaired cerebral glucose metabolism are invariantly reported in Alzheimer's disease (AD) and other neurodegenerative processes. AD is a fatal neurodegenerative disorder in which progressive glucose hypometabolism parallels to cognitive impairment. Although AD may appear and progress in virtue of multifactorial nosogenic ingredients, multiple interperpetuative and interconnected vicious circles appear to drive disease pathophysiology. The disease is primarily a metabolic/energetic disorder in which amyloid accumulation may appear as a by-product of more proximal events, especially in the late-onset form. As a bridge between AD and type 2 diabetes, activation of c-Jun N-terminal kinase (JNK) pathway with the ensued serine phosphorylation of the insulin response substrate (IRS)-1/2 may be at the crossroads of insulin resistance and its subsequent dysmetabolic consequences. Central insulin axis bankruptcy translates in neuronal vulnerability and demise. As a link in the chain of pathogenic vicious circles, mitochondrial dysfunction, oxidative stress, and peripheral/central immune-inflammation are increasingly advocated as major pathology drivers. Pharmacological interventions addressed to preserve insulin axis physiology, mitochondrial biogenesis-integral functionality, and mitophagy of diseased organelles may attenuate the adjacent spillover of free radicals that further perpetuate mitochondrial damages and catalyze inflammation. Central and/or peripheral inflammation may account for a local flood of proinflammatory cytokines that along with astrogliosis amplify insulin resistance, mitochondrial dysfunction, and oxidative stress. All these elements are endogenous stressor, pro-senescent factors that contribute to JNK activation. Taken together, these evidences incite to identify novel multi-mechanistic approaches to succeed in ameliorating this pandemic affliction.

Mathematical Model Shows How Sleep May Affect Amyloid-β Fibrillization

Deposition of amyloid-β (Aβ) fibers in the extracellular matrix of the brain is a ubiquitous feature associated with several neurodegenerative disorders, especially Alzheimer's disease (AD). Although many of the biological aspects that contribute to the formation of Aβ plaques are well addressed at the intra- and intercellular levels in short timescales, an understanding of how Aβ fibrillization usually starts to dominate at a longer timescale despite the presence of mechanisms dedicated to Aβ clearance is still lacking. Furthermore, no existing mathematical model integrates the impact of diurnal neural activity as emanated from circadian regulation to predict disease progression due to a disruption in the sleep-wake cycle. In this study, we develop a minimal model of Aβ fibrillization to investigate the onset of AD over a long timescale. Our results suggest that the diseased state is a manifestation of a phase change of the system from soluble Aβ (sAβ) to fibrillar Aβ (fAβ) domination upon surpassing a threshold in the production rate of sAβ. By incorporating the circadian rhythm into our model, we reveal that fAβ accumulation is crucially dependent on the regulation of the sleep-wake cycle, thereby indicating the importance of good sleep hygiene in averting AD onset. We also discuss potential intervention schemes to reduce fAβ accumulation in the brain by modification of the critical sAβ production rate.

On the Conformational Dynamics of β-Amyloid Forming Peptides: A Computational Perspective

Understanding the conformational dynamics of proteins and peptides involved in important functions is still a difficult task in computational structural biology. Because such conformational transitions in β-amyloid (Aβ) forming peptides play a crucial role in many neurological disorders, researchers from different scientific fields have been trying to address issues related to the folding of Aβ forming peptides together. Many theoretical models have been proposed in the recent years for studying Aβ peptides using mathematical, physicochemical, and molecular dynamics simulation, and machine learning approaches. In this article, we have comprehensively reviewed the developmental advances in the theoretical models for Aβ peptide folding and interactions, particularly in the context of neurological disorders. Furthermore, we have extensively reviewed the advances in molecular dynamics simulation as a tool used for studying the conversions between polymorphic amyloid forms and applications of using machine learning approaches in predicting Aβ peptides and aggregation-prone regions in proteins. We have also provided details on the theoretical advances in the study of Aβ peptides, which would enhance our understanding of these peptides at the molecular level and eventually lead to the development of targeted therapies for certain acute neurological disorders such as Alzheimer's disease in the future.

An agent-based model to investigate microbial initiation of Alzheimer's via the olfactory system

BACKGROUND

Alzheimer's disease (AD) is a degenerative brain disease. A novel agent-based modelling framework was developed in NetLogo 3D to provide fundamental insights into the potential mechanisms by which a microbe (eg. Chlamydia pneumoniae) may play a role in late-onset AD. The objective of our initial model is to simulate one possible spatial and temporal pathway of bacterial propagation via the olfactory system, which may then lead to AD symptoms. The model maps the bacteria infecting cells from the nasal cavity and the olfactory epithelium, through the olfactory bulb and into the olfactory cortex and hippocampus regions of the brain.

RESULTS

Based on the set of biological rules, simulated randomized infection by the microbe led to the formation of beta-amyloid (Aβ) plaque and neurofibrillary (NF) tangles as well as caused immune responses. Our initial simulations demonstrated that breathing in C. pneumoniae can result in infection propagation and significant buildup of Aβ plaque and NF tangles in the olfactory cortex and hippocampus. Our model also indicated how mucosal and neural immunity can play a significant role in the pathway considered. Lower immunities, correlated with elderly individuals, had quicker and more Aβ plaque and NF tangle formation counts. In contrast, higher immunities, correlated with younger individuals, demonstrated little to no such formation.

CONCLUSION

The modelling framework provides an organized visual representation of how AD progression may occur via the olfactory system to better understand disease pathogenesis. The model confirms current conclusions in available research but can be easily adjusted to match future evidence and be used by researchers for their own individual purposes. The goal of our initial model is to ultimately guide further hypothesis refinement and experimental testing to better understand the dynamic system interactions present in the etiology and pathogenesis of AD.

Mathematical Models to Shed Light on Amyloid-Beta and Tau Protein Dependent Pathologies in Alzheimer's Disease

The number of patients suffering from dementia due to Alzheimer's disease (AD) is constantly rising worldwide. This has accordingly resulted in huge burdens on the health systems and involved families. Lack of profound understanding of neural networking in normal brain and their interruption in AD makes the treatment of this neurodegenerative multifaceted disease a challenging issue. In recent years, mathematical and computational methods have paved the way towards a better understanding of the brain functional connectivity. Thus, much attention has been paid to this matter from both basic science researchers and clinicians with an interdisciplinary approach to determine what is not functioning properly in AD patients and how this malfunctioning can be addressed. In this review, a number of AD-related articles and well-studied pathophysiologic topics (e.g., amyloid-beta, neurofibrillary tangles, Ca²⁺ dysregulation, and synaptic plasticity alterations) has been literally surveyed from a computational and systems biology point of view. The neural networks were discussed from biological and mathematical point of views and their alterations in recent findings were further highlighted. Application of the graph theoretical analysis in the brain imaging was reviewed, depicting the relations between brain structure and function, without diving into mathematical details. Moreover, differential rate equations were briefly articulated, emphasizing the potential use of these equations in simplifying complex processes in relevance to pathologies of AD. Comprehensive insights were given into the AD progression from neural networks perspective, which may lead us towards potential strategies for early diagnosis and effective treatment of AD.

A mathematical model demonstrating the role of interstitial fluid flow on the clearance and accumulation of amyloid β in the brain

A system of partial differential equations is developed to describe the formation and clearance of amyloid β (Aβ) and the subsequent buildup of Aβ plaques in the brain, which are associated with Alzheimer's disease. The Aβ related proteins are divided into five distinct categories depending on their size. In addition to enzymatic degradation, the clearance via diffusion and the outflow of interstitial fluid (ISF) into the surrounding cerebral spinal fluid (CSF) are considered. Treating the brain tissue as a porous medium, a simplified two-dimensional circular geometry is assumed for the transverse section of the brain leading to a nonlinear, coupled system of PDEs. Asymptotic analysis is carried out for the steady states of the spatially homogeneous system in the vanishingly small limit of Aβ clearance rate. The PDE model is studied numerically for two cases, a spherically symmetric case and a more realistic 2D asymmetric case, allowing for non-uniform boundary conditions. Our investigations demonstrate that ISF advection is a key component in reproducing the clinically observed accumulation of plaques on the outer boundaries. Furthermore, ISF circulation serves to enhance Aβ clearance over diffusion alone and that non-uniformities in ISF drainage into the CSF can lead to local clustering of plaques. Analysis of the model also demonstrates that plaque formation does not directly correspond to the high presence of toxic oligomers.

A Novel Mathematical Model of Glaucoma Pathogenesis

Background

Conventional experimental approaches to understand glaucoma etiology and pathogenesis and, consequently, predict its course of progression have not seen much success due to the involvement of numerous molecular, cellular, and other moieties. An overwhelming number of these moieties at different levels combined with numerous environmental factors further complicate the intricacy. Interaction patterns between these factors are important to understand yet difficult to probe with conservative experimental approaches.

Methods

We performed a system-level analysis with mathematical modeling by developing and analyzing rate equations with respect to the cellular events in glaucoma pathogenesis. Twenty-two events were enlisted from the literature survey and were analyzed in terms of the sensitivity coefficient of retinal ganglion cells. A separate rate equation was developed for cellular stress also. The results were analyzed with respect to time, and the time course of the events with respect to various cellular moieties was analyzed.

Results

Our results suggest that microglia activation is among the earliest events in glaucoma pathogenesis. This modeling method yields a wealth of useful information which may serve as an important guide to better understand glaucoma pathogenesis and design experimental approaches and also identify useful diagnostic/predictive methods and important therapeutic targets.

Conclusion

We here report the first mathematical model for glaucoma pathogenesis which provides important insight into the sensitivity coefficient and glia-mediated pathology of glaucoma.

How to cite this article

Faiq MA, Sidhu T, et al. A Novel Mathematical Model of Glaucoma Pathogenesis. J Curr Glaucoma Pract 2019; 13(1):3–8.

A discrete mathematical model for the aggregation of β-Amyloid

Dementia associated with the Alzheimer's disease is thought to be correlated with the conversion of the β − Amyloid (Aβ) peptides from soluble monomers to aggregated oligomers and insoluble fibrils. We present a discrete-time mathematical model for the aggregation of Aβ monomers into oligomers using concepts from chemical kinetics and population dynamics. Conditions for the stability and instability of the equilibria of the model are established. A formula for the number of monomers that is required for producing oligomers is also given. This may provide compound designers a mechanism to inhibit the Aβ aggregation.

Mathematical Modeling of Protein Misfolding Mechanisms in Neurological Diseases: A Historical Overview

Protein misfolding refers to a process where proteins become structurally abnormal and lose their specific 3-dimensional spatial configuration. The histopathological presence of misfolded protein (MP) aggregates has been associated as the primary evidence of multiple neurological diseases, including Prion diseases, Alzheimer's disease, Parkinson's disease, and Creutzfeldt-Jacob disease. However, the exact mechanisms of MP aggregation and propagation, as well as their impact in the long-term patient's clinical condition are still not well understood. With this aim, a variety of mathematical models has been proposed for a better insight into the kinetic rate laws that govern the microscopic processes of protein aggregation. Complementary, another class of large-scale models rely on modern molecular imaging techniques for describing the phenomenological effects of MP propagation over the whole brain. Unfortunately, those neuroimaging-based studies do not take full advantage of the tremendous capabilities offered by the chemical kinetics modeling approach. Actually, it has been barely acknowledged that the vast majority of large-scale models have foundations on previous mathematical approaches that describe the chemical kinetics of protein replication and propagation. The purpose of the current manuscript is to present a historical review about the development of mathematical models for describing both microscopic processes that occur during the MP aggregation and large-scale events that characterize the progression of neurodegenerative MP-mediated diseases.

The development of a stochastic mathematical model of Alzheimer's disease to help improve the design of clinical trials of potential treatments

Alzheimer's disease (AD) is a neurodegenerative disorder characterised by a slow progressive deterioration of cognitive capacity. Drugs are urgently needed for the treatment of AD and unfortunately almost all clinical trials of AD drug candidates have failed or been discontinued to date. Mathematical, computational and statistical tools can be employed in the construction of clinical trial simulators to assist in the improvement of trial design and enhance the chances of success of potential new therapies. Based on the analysis of a set of clinical data provided by the Alzheimer's Disease Neuroimaging Initiative (ADNI) we developed a simple stochastic mathematical model to simulate the development and progression of Alzheimer's in a longitudinal cohort study. We show how this modelling framework could be used to assess the effect and the chances of success of hypothetical treatments that are administered at different stages and delay disease development. We demonstrate that the detection of the true efficacy of an AD treatment can be very challenging, even if the treatment is highly effective. An important reason behind the inability to detect signals of efficacy in a clinical trial in this therapy area could be the high between- and within-individual variability in the measurement of diagnostic markers and endpoints, which consequently results in the misdiagnosis of an individual's disease state.

Functional dissection of astrocyte-secreted proteins: Implications in brain health and diseases

Astrocytes, which are homeostatic cells of the central nervous system (CNS), display remarkable heterogeneity in their morphology and function. Besides their physical and metabolic support to neurons, astrocytes modulate the blood-brain barrier, regulate CNS synaptogenesis, guide axon pathfinding, maintain brain homeostasis, affect neuronal development and plasticity, and contribute to diverse neuropathologies via secreted proteins. The identification of astrocytic proteome and secretome profiles has provided new insights into the maintenance of neuronal health and survival, the pathogenesis of brain injury, and neurodegeneration. Recent advances in proteomics research have provided an excellent catalog of astrocyte-secreted proteins. This review categorizes astrocyte-secreted proteins and discusses evidence that astrocytes play a crucial role in neuronal activity and brain function. An in-depth understanding of astrocyte-secreted proteins and their pathways is pivotal for the development of novel strategies for restoring brain homeostasis, limiting brain injury/inflammation, counteracting neurodegeneration, and obtaining functional recovery.

Modelling the molecular mechanisms of aging

The aging process is driven at the cellular level by random molecular damage that slowly accumulates with age. Although cells possess mechanisms to repair or remove damage, they are not 100% efficient and their efficiency declines with age. There are many molecular mechanisms involved and exogenous factors such as stress also contribute to the aging process. The complexity of the aging process has stimulated the use of computational modelling in order to increase our understanding of the system, test hypotheses and make testable predictions. As many different mechanisms are involved, a wide range of models have been developed. This paper gives an overview of the types of models that have been developed, the range of tools used, modelling standards and discusses many specific examples of models that have been grouped according to the main mechanisms that they address. We conclude by discussing the opportunities and challenges for future modelling in this field.

The Impact of Mathematical Modeling in Understanding the Mechanisms Underlying Neurodegeneration: Evolving Dimensions and Future Directions

Neurodegenerative diseases are a heterogeneous group of disorders that are characterized by the progressive dysfunction and loss of neurons. Here, we distil and discuss the current state of modeling in the area of neurodegeneration, and objectively compare the gaps between existing clinical knowledge and the mechanistic understanding of the major pathological processes implicated in neurodegenerative disorders. We also discuss new directions in the field of neurodegeneration that hold potential for furthering therapeutic interventions and strategies.

Mathematical model on Alzheimer's disease

Background

Alzheimer disease (AD) is a progressive neurodegenerative disease that destroys memory and cognitive skills. AD is characterized by the presence of two types of neuropathological hallmarks: extracellular plaques consisting of amyloid β-peptides and intracellular neurofibrillary tangles of hyperphosphorylated tau proteins. The disease affects 5 million people in the United States and 44 million world-wide. Currently there is no drug that can cure, stop or even slow the progression of the disease. If no cure is found, by 2050 the number of alzheimer’s patients in the U.S. will reach 15 million and the cost of caring for them will exceed $ 1 trillion annually.

Results

The present paper develops a mathematical model of AD that includes neurons, astrocytes, microglias and peripheral macrophages, as well as amyloid β aggregation and hyperphosphorylated tau proteins. The model is represented by a system of partial differential equations. The model is used to simulate the effect of drugs that either failed in clinical trials, or are currently in clinical trials.

Conclusions

Based on these simulations it is suggested that combined therapy with TNF- α inhibitor and anti amyloid β could yield significant efficacy in slowing the progression of AD.

Computational modeling of cytokine signaling in microglia

Neuroinflammation due to glial activation has been linked to many CNS diseases. We developed a computational model of a microglial cytokine interaction network to study the regulatory mechanisms of microglia-mediated neuroinflammation. We established a literature-based cytokine network, including TNFα, TGFβ, and IL-10, and fitted a mathematical model to published data from LPS-treated microglia. The addition of a previously unreported TGFβ autoregulation loop to our model was required to account for experimental data. Global sensitivity analysis revealed that TGFβ- and IL-10-mediated inhibition of TNFα was critical for regulating network behavior. We assessed the sensitivity of the LPS-induced TNFα response profile to the initial TGFβ and IL-10 levels. The analysis showed two relatively shifted TNFα response profiles within separate domains of initial condition space. Further analysis revealed that TNFα exhibited adaptation to sustained LPS stimulation. We simulated the effects of functionally inhibiting TGFβ and IL-10 on TNFα adaptation. Our analysis showed that TGFβ and IL-10 knockouts (TGFβ KO and IL-10 KO) exert divergent effects on adaptation. TFGβ KO attenuated TNFα adaptation whereas IL-10 KO enhanced TNFα adaptation. We experimentally tested the hypothesis that IL-10 KO enhances TNFα adaptation in murine macrophages and found supporting evidence. These opposing effects could be explained by differential kinetics of negative feedback. Inhibition of IL-10 reduced early negative feedback that results in enhanced TNFα-mediated TGFβ expression. We propose that differential kinetics in parallel negative feedback loops constitute a novel mechanism underlying the complex and non-intuitive pro- versus anti-inflammatory effects of individual cytokine perturbations.

Temporal-logic analysis of microglial phenotypic conversion with exposure to amyloid-β

Alzheimer Disease (AD) remains a leading killer with no adequate treatment. Ongoing research increasingly implicates the brain's immune system as a critical contributor to AD pathogenesis, but the complexity of the immune contribution poses a barrier to understanding. Here I use temporal logic to analyze a computational specification of the immune component of AD. Temporal logic is an extension of logic to propositions expressed in terms of time. It has traditionally been used to analyze computational specifications of complex engineered systems but applications to complex biological systems are now appearing. The inflammatory component of AD involves the responses of microglia to the peptide amyloid-β (Aβ), which is an inflammatory stimulus and a likely causative AD agent. Temporal-logic analysis of the model provides explanations for the puzzling findings that Aβ induces an anti-inflammatory and well as a pro-inflammatory response, and that Aβ is phagocytized by microglia in young but not in old animals. To potentially explain the first puzzle, the model suggests that interferon-γ acts as an "autocrine bridge" over which an Aβ-induced increase in pro-inflammatory cytokines leads to an increase in anti-inflammatory mediators also. To potentially explain the second puzzle, the model identifies a potential instability in signaling via insulin-like growth factor 1 that could explain the failure of old microglia to phagocytize Aβ. The model predicts that augmentation of insulin-like growth factor 1 signaling, and activation of protein kinase C in particular, could move old microglia from a neurotoxic back toward a more neuroprotective and phagocytic phenotype.

A Model for Improving the Learning Curves of Artificial Neural Networks

In this article, the performance of a hybrid artificial neural network (i.e. scale-free and small-world) was analyzed and its learning curve compared to three other topologies: random, scale-free and small-world, as well as to the chemotaxis neural network of the nematode Caenorhabditis Elegans. One hundred equivalent networks (same number of vertices and average degree) for each topology were generated and each was trained for one thousand epochs. After comparing the mean learning curves of each network topology with the C. elegans neural network, we found that the networks that exhibited preferential attachment exhibited the best learning curves.

Unraveling the mechanistic complexity of Alzheimer's disease through systems biology

Alzheimer's disease (AD) is a complex, multifactorial disease that has reached global epidemic proportions. The challenge remains to fully identify its underlying molecular mechanisms that will enable development of accurate diagnostic tools and therapeutics. Conventional experimental approaches that target individual or small sets of genes or proteins may overlook important parts of the regulatory network, which limits the opportunity of identifying multitarget interventions. Our perspective is that a more complete insight into potential treatment options for AD will only be made possible through studying the disease as a system. We propose an integrative systems biology approach that we argue has been largely untapped in AD research. We present key publications to demonstrate the value of this approach and discuss the potential to intensify research efforts in AD through transdisciplinary collaboration. We highlight challenges and opportunities for significant breakthroughs that could be made if a systems biology approach is fully exploited.

Altered intrinsic excitability of hippocampal CA1 pyramidal neurons in aged PDAPP mice

Amyloidopathy involves the accumulation of insoluble amyloid β (Aβ) species in the brain's parenchyma and is a key histopathological hallmark of Alzheimer's disease (AD). Work on transgenic mice that overexpress Aβ suggests that elevated Aβ levels in the brain are associated with aberrant epileptiform activity and increased intrinsic excitability (IE) of CA1 hippocampal neurons. In this study we examined if similar changes could be observed in hippocampal CA1 pyramidal neurons from aged PDAPP mice (20-23 month old, Indiana mutation: V717F on APP gene) compared to their age-matched wild-type littermate controls. Whole-cell current clamp recordings revealed that sub-threshold intrinsic properties, such as input resistance, resting membrane potential and hyperpolarization activated "sag" were unaffected, but capacitance was significantly decreased in the transgenic animals. No differences between genotypes were observed in the overall number of action potentials (AP) elicited by 500 ms supra-threshold current stimuli. PDAPP neurons, however, exhibited higher instantaneous firing frequencies after accommodation in response to high intensity current injections. The AP waveform was narrower and shorter in amplitude in PDAPP mice: these changes, according to our in silico model of a CA1/3 pyramidal neuron, depended on the respective increase and reduction of K(+) and Na(+) voltage-gated channels maximal conductances. Finally, the after-hyperpolarization, seen after the first AP evoked by a +300 pA current injection and after 50 Hz AP bursts, was more pronounced in PDAPP mice. These data show that Aβ-overexpression in aged mice altered the capacitance, the neuronal firing and the AP waveform of CA1 pyramidal neurons. Some of these findings are consistent with previous work on younger PDAPP; they also show important differences that can be potentially ascribed to the interaction between amyloidopathy and ageing. Such a change of IE properties over time underlies that the increased incidence of seizure observed in AD patients might rely on different mechanistic pathways during progression of the disease.

Modeling the Role of the Glymphatic Pathway and Cerebral Blood Vessel Properties in Alzheimer's Disease Pathogenesis

Alzheimer’s disease (AD) is the most common cause of dementia in the elderly, affecting over 10% population over the age of 65 years. Clinically, AD is described by the symptom set of short term memory loss and cognitive decline, changes in mentation and behavior, and eventually long-term memory deficit as the disease progresses. On imaging studies, significant atrophy with subsequent increase in ventricular volume have been observed. Pathology on post-mortem brain specimens demonstrates the classic findings of increased beta amyloid (Aβ) deposition and the presence of neurofibrillary tangles (NFTs) within affected neurons. Neuroinflammation, dysregulation of blood-brain barrier transport and clearance, deposition of Aβ in cerebral blood vessels, vascular risk factors such as atherosclerosis and diabetes, and the presence of the apolipoprotein E4 allele have all been identified as playing possible roles in AD pathogenesis. Recent research has demonstrated the importance of the glymphatic system in the clearance of Aβ from the brain via the perivascular space surrounding cerebral blood vessels. Given the variety of hypotheses that have been proposed for AD pathogenesis, an interconnected, multilayer model offers a unique opportunity to combine these ideas into a single unifying model. Results of this model demonstrate the importance of vessel stiffness and heart rate in maintaining adequate clearance of Aβ from the brain.

The progression towards Alzheimer's disease described as a bistable switch arising from the positive loop between amyloids and Ca(2+)

Alzheimer's disease is a progressive neurodegenerative disorder affecting millions of people. It is characterized by the slow deposition of cerebral amyloid-β peptides in the brain and by dysregulations in neuronal Ca(2+) homeostasis. Numerous experimental studies have revealed the existence of a feed-forward loop wherein amyloids-β disturb neuronal Ca(2+) levels, which in turn affect the production of amyloids. Here, we formalize this positive loop in a minimal, qualitative model and show that it exhibits bistability. Thus, a stable steady state characterized by low levels of Ca(2+) and amyloids, corresponding to a healthy situation, coexists with another 'pathological state' where the levels of both compounds are high. The onset of the disease corresponds to the switch from the lower steady state to the higher one induced by a large-enough perturbation in either the metabolism of amyloids or the homeostasis of intracellular Ca(2+). Numerical simulations of the model reproduce a variety of experimental observations about the disease, as its irreversible character, the threshold-like transition to a severe pathology after the slow accumulation of symptoms, the effect of presenilins, the so-called 'prion-like' autocatalytic behaviour of amyloids and the inherent random character of the apparition of the disease that is well known for the sporadic form. The model thus provides a conceptual framework that could be useful when developing therapeutic protocols to slow down the progression of Alzheimer's disease.

Luteolin reduces primary hippocampal neurons death induced by neuroinflammation

OBJECTIVES

This study examined whether luteolin may exert an anti-inflammatory effect in microglia and may be neuroprotective by regulating microglia activation.

METHODS

We treated BV2 microglia with 1.0 μg/ml lipopolysaccharide (LPS) after incubation with luteolin for 1 hour, the nitric oxide (NO) levels were determined by a Griess reaction, the inducible NO synthase (iNOS), cyclooxygenase-2 (COX-2), tumor necrosis factor-alpha (TNF-alpha), and interleukin 1beta (IL-1beta) mRNA expression were determined by real-time PCR analysis, the iNOS and COX-2 protein induction were determined by Western blot analysis, and the levels of prostaglandin E(2) (PGE(2)), TNF-alpha, and IL-1beta were determined by enzyme-linked immunosorbent assay (ELISA) kits. Rat primary hippocampal neurons were co-cultured with LPS-activated BV2 microglia with 20 μM luteolin for 24 hours, the hippocampal neurons viability was determined by 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay, and the number of apoptotic hippocampal neurons was determined by immunofluorescence detection.

RESULTS

Luteolin significantly inhibited the expression of iNOS and COX-2 in LPS-induced BV2 microglia. Moreover, the compound down-regulated the proinflammatory cytokines (TNF-alpha and IL-1beta) as well as the production of NO and PGE(2) in these cells. When hippocampal neurons were co-cultured with LPS-stimulated BV2 microglia, the administration of 20 μM luteolin increased the neurons viability and reduced the number of apoptotic neurons.

CONCLUSION

These data demonstrate that anti-inflammatory activity of luteolin in microglia contributes to its neuroprotective effect and suggest that it may have a potential therapeutic application in the treatment of neurodegenerative diseases.