Cell ageing: a flourishing field for neurodegenerative diseases

Dora Brites; Dora Brites

doi:10.3934/molsci.2015.3.225

AIMS Molecular Science

2015, Volume 2, Issue 3: 225-258. doi: 10.3934/molsci.2015.3.225

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Review Special Issues

Cell ageing: a flourishing field for neurodegenerative diseases

Dora Brites ^{1,2
,
,}

1.
Neuron Glia Biology in Health and Disease Group, Research Institute for Medicines (iMed.ULisboa), Faculty of Pharmacy, Universidade de Lisboa, Lisbon, Portugal;
2.
Department of Biochemistry and Human Biology, Faculty of Pharmacy, Universidade de Lisboa, Lisbon, Portugal

Received: 19 April 2015 Accepted: 31 May 2015 Published: 03 June 2015

Cellular senescence is viewed as an irreversible cell-cycle arrest mechanism involving a complexity of biological progressive processes and the acquisition of diverse cellular phenotypes. Several cell-intrinsic and extrinsic causes (stresses) may lead to diverse cellular signaling cascades that include oxidative stress, mitochondrial dysfunction, DNA damage, excessive accumulation of misfolded proteins, impaired microRNA processing and inflammation. Here we review recent advances in the causes and consequences of brain cell ageing, including the senescence of endothelial cells at the central nervous system barriers, as well as of neurons and glial cells. We address what makes ageing an important risk factor for neurodegenerative disorders, such as Alzheimer's disease, Parkinson's disease, amyotrophic lateral sclerosis and cerebrovascular disease. In particular, we highlight the importance of defects in mitochondrial dynamics, in the cathepsin activity imbalance, in cell-cell communication, in the accumulation of misfolded and unfolded proteins and in the microRNA profiling as having potential impact on cellular ageing processes. Another important aspect is that the absence of specific senescence biomarkers has hampered the characterization of senescent cells in ageing and age-associated diseases. In accordance, the senescence-associated secretory phenotype (SASP) or secretome was shown to vary in distinct cell types and upon different stressors, and SASP heterogeneity is believed to create subsets of senenescent cells. In addition to secreted proteins, we then place extracellular vesicles (exosomes and ectosomes) as important mediators of intercellular communication with pathophysiological roles in disease spreading, and as emerging targets for therapeutic intervention. We also discuss the application of engineered extracellular vesicles as vehicles for drug delivery. Finally, we summarize current knowledge on methods to rejuvenate senescent cells and we review cell replacement therapeutic strategies. A deeper understanding of the molecular mechanisms underlying the senescence of cells may open novel therapeutic approaches for age-related pathologies and for extending healthy human life span.

Keywords:

Citation: Dora Brites. Cell ageing: a flourishing field for neurodegenerative diseases[J]. AIMS Molecular Science, 2015, 2(3): 225-258. doi: 10.3934/molsci.2015.3.225

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Abstract

1. Introduction

Second order traffic models have been investigated since the early 70's (e.g. [27,33]) and refined with the introduction of the Aw-Rascle-Zhang (ARZ) model [3] and are still subject of current research. Traffic models in general range from individual based microscopic to averaged macroscopic considerations, see [9,15,19,32] and the references therein. The motivation for the consideration of second order traffic models is that on the one hand microscopic models may explode for a large number of cars and, on the other hand, first order models are supposed to be dependent on the traffic density only and hence lack of real-world effects. In particular, macroscopic traffic flow models with delay terms have gained special attention recently, see [30,26]. While second order traffic models without delay terms are well understood on a theoretical and analytical level [3,9,15], models of this type including time delays are less considered so far [26], and if considered, the delay is usually rewritten by a Taylor approximation [30,26]. In contrast, the explicit modeling of reaction times via delays is a well-known approach in microscopic traffic modeling, see e.g. [6], and therefore the intention is that the introduction of an explicit delay can make macroscopic models more reasonable. More precisely, we expect similar effects as in microscopic delayed models such as the developing of stop-and-go waves based on the reaction time.

Starting from car-following type models, our aim is therefore to develop and study second order macroscopic models with an explicit dependence on the delay term. For this purpose, we consider the explicit derivation of delayed second order models from delayed microscopic models in three different ways following [4,22,30]. We will end up with a system of equations for the traffic density $ \rho $ in the sense of mass conservation and a second equation for the velocity $ v $ with an additional delay dependent term on the right-hand side. It turns out that for reasonably small delays all proposed macroscopic delayed models coincide with the classical (or undelayed) ARZ model [3] which reads in Eulerian coordinates for some positive initial state $ \big( \rho_0(x), v_0(x) \big) $:

$\begin{align} \partial_t \rho + \partial_x (\rho v) & = 0, \\ \partial_t (\rho w) + \partial_x (\rho v w) & = 0, \end{align}$ $

(1)

where $ w = v + P(\rho) $ and $ P(\rho) $ is a known pressure function.

For the numerical simulation of the delayed macroscopic models, we use finite differences methods in a straightforward way and combine the method of lines with solvers for delayed differential equations. We analyze numerically the delayed models and compare the solutions to the classical the ARZ model. A comparison to real data and the corresponding fit of parameters (similar to [14]) is also presented.

This work is organized as follows: In section 2 we rigorously derive three second order macroscopic models based on the delayed microscopic model description according to [4,22,30]. The connection to the classical non-delayed ARZ model is also shown. In section 3, we introduce and discuss numerical methods for the solution of microscopic models with delay and second order delayed traffic models using ideas from [5]. In section 4, we present simulation results for comparisons of the different model level of descriptions and particularly stress on phenomena of the delayed macroscopic models.

2. From microscopic to macroscopic models

In this section we derive macroscopic traffic models from delayed microscopic models of the car-following or follow-the-leader-type using three different approaches, cf. [4,22,30]. The resulting models are hyperbolic delay partial differential equations, where the time delay is explicitly kept and appears as an additional term in the equation for the velocity. As we will see, this new type of models gives rise to potential new dynamical behavior.

2.1. Microscopic model

The microscopic model under consideration dates back to the 1950s and belongs to the class of car-following models, see e.g. [6]. In particular, we consider a system of ordinary differential equations for the location $ x_i $ and speed $ v_i $ of the vehicle $ i $ at time $ t \in \mathbb{R}^+ $

$\begin{align} \dot{x}_i(t)& = v_i(t) \\ \dot{v}_i(t)& = C\frac{(v_{i+1}(t-T)-v_i(t-T))}{(x_{i+1}(t-T)-x_i(t-T))^{\gamma+1}}, i = 1, \ldots, N, \end{align}$ $

(2)

with $ T>0 $ the uniform constant reaction time and model constants $ C>0 $ and $ \gamma\geq 0, $ cf. [4]. As initial data, we have to prescribe an initial history function on the time interval $ [-T, 0] $ to get a well-posed solution of the problem starting at $ t = 0. $

Apparently, the acceleration at time $ t $ depends on the reciprocal distance between vehicles and the difference in speed to the leading vehicle. Next, we define $ \Delta X $, $ \Delta X \in \mathbb{R}^+ $, as the length of a vehicle and the coefficient $ C = v_{\text{ref}}\Delta X^{\gamma} $, where $ v_{\text{ref}}>0 $ refers to the reference speed. We assume that the delay $ T $ is always positive and can be therefore interpreted as a reaction time, in contrast to an anticipation $ T<0 $. Then, we rewrite (2) to

$\begin{align} \dot{x}_i(t)& = v_i(t) \\ \dot{v}_i(t)& = v_{\text{ref}}\Delta X^{\gamma}\frac{(v_{i+1}(t-T)-v_i(t-T))}{(x_{i+1}(t-T)-x_i(t-T))^{\gamma+1}}, i = 1, \ldots, N. \end{align}$ $

(3)

The microscopic model equations (3) are the staring point for the macroscopic equations. We focus on three approaches, i.e. reverse spatial discretization [4], coarse graining [22], Taylor method [30], to derive the corresponding macroscopic delayed models. All approaches lead to second order traffic models that which mainly differ in the delay term.

2.2. Reverse spatial discretization

The first approach we are interested in is based on a derivation proposed in [4], where the classical ARZ model is derived from (2). Here, we apply the same technique to the delayed equations (3) by identifying the local density $ \rho_i $ with the free headway, i.e.,

$ \rho_i(t): = \frac{\Delta X}{x_{i+1}(t)-x_i(t)}. $

We further define $ \tau_i(t): = \rho_i^{-1}(t) $ and $ w_i(t): = v_i(t)+P(\tau_i(t)) $, where $ P(\tau) $ can be understood as a function describing the anticipation of the road conditions in front of the drivers:

$ P(\tau): =

$\begin{cases} \frac{v_{\text{ref}}}{\gamma}\tau^{-\gamma} &\text{if}&\gamma > 0 \\ -v_{\text{ref}}\text{ln}(\tau) &\text{if}&\gamma = 0. \end{cases}$ $

Then, the new variables $ \tau_i $ and $ w_i $ allow to equivalently reformulate system (3) as

$\begin{align} \dot{\tau}_i(t)& = \frac{\dot{x}_{i+1}(t)-\dot{x}_i(t)}{\Delta X} = \frac{v_{i+1}(t)-v_i(t)}{\Delta X} \\ \dot{w}_i(t)& = \dot{v}_i(t)+\frac{d}{dt}{P}(\tau_i(t)) = \frac{v_{\text{ref}}}{\Delta X}\frac{(v_{i+1}(t-T)-v_i(t-T))}{\tau_i(t-T)^{\gamma+1}}+P'(\tau_i(t))\dot{\tau}_i (t) \\ & = \frac{v_{\text{ref}}}{\tau_i(t-T)^{\gamma+1}}\frac{(v_{i+1}(t-T)-v_i(t-T))}{\Delta X}-\frac{v_{\text{ref}}}{\tau_i(t)^{\gamma+1}}\frac{v_{i+1}(t)-v_i(t)}{\Delta X} \end{align}$ $

(4)

since $ P'(\tau) = -v_{\text{ref}}\frac{1}{\tau^{\gamma+1}} $ for $ \gamma \geq 0 $. The system (4) can be interpreted as a semi-discretization of the following delay partial differential equation for $ \Delta X\rightarrow0 $:

$\begin{align} \partial_t \tau(x, t)& = \partial_x v(x, t) \\ \partial_t w(x, t)& = v_{\text{ref}}\Big(\partial_x v(x, t-T) \rho(x, t-T)^{\gamma+1}-\partial_x v(x, t)\rho(x, t)^{\gamma+1}\Big), \end{align}$ $

(5)

where $ \tau $ and $ \rho $ are the limiting expressions of $ \tau_i $ and $ \rho_i $. Note that the classical ARZ model (1) in Lagrangian coordinates reads

$\begin{align} \partial_t \tau(x, t) = \partial_x v(x, t), \quad \partial_t w(x, t) = 0. \end{align}$ $

(6)

In fact, a direct computation shows that (5) is equivalent to

$\begin{align} \partial_t \tau(x, t)& = \partial_x v(x, t) \\ \partial_t v(x, t)& = v_{\text{ref}}\partial_x v(x, t-T) \rho(x, t-T)^{\gamma+1} \end{align}$ $

(7)

applying the definitions of $ w $ and $ \tau $ as well as information on $ P'(\tau) $.

Let us take a closer look at the right-hand side of the momentum equation (7). Compared to the classical model, we now get the temporal change in speed dependent on the density and the spatial derivative of speed from an earlier time. For the interpretation of $ w $, which is a Riemann-constant/invariant in the classical ARZ model, we refer to [14], where $ w $ is used as the empty road speed. However, $ w $ is not necessarily constant in the delayed model and changes due to the difference in the product of density and the spatial derivative of speed. This means the empty road speed, i.e. the speed a driver would have on an empty road, is not only dependent on the driver but also on the traffic behavior around leading to a more cautious actions.

So far, we have considered the Lagrangian macroscopic equations (5) and (7), respectively. In a next step, we transform these equations into Eulerian coordinates by introducing the new variables $ (\hat{x}, \hat{t}) $ with

$\begin{equation} \partial_x\hat{x} = \tau, \quad\partial_x\hat{t} = 0,\quad \partial_t\hat{x} = v\quad, \partial_t\hat{t} = 1, \end{equation}$ $

(8)

or, exploiting $ \hat{t} = t $,

$\begin{equation} \partial_t \tau = \partial_{\hat{t}}\tau+(\partial_{\hat{x}}\tau) v,\quad \partial_x v = (\partial_{\hat{x}}v)\tau, \quad\partial _t w = \partial_{\hat{t}}w+(\partial_{\hat{x}}w) v. \end{equation}$ $

(9)

In the following, we write $ t $ instead of $ \hat{t} $ since $ \hat{t} = t $. Then, we obtain from (5) the new equations

$\begin{align} \partial_{t}\rho(\hat{x}, t)+\partial_{\hat{x}}(\rho(\hat{x}, t)v(\hat{x}, t))& = 0 \\ \partial_{t}(\rho(\hat{x}, t)w(\hat{x}, t))+\partial_{\hat{x}}(\rho(\hat{x}, t)v(\hat{x}, t)w(\hat{x}, t)) = v_{\text{ref}}&\big(\partial_{\hat{x}}v(\hat{x}, t-T)\rho(\hat{x}, t-T)^{\gamma}\\ &-\partial_{\hat{x}}v(\hat{x}, t)\rho(\hat{x}, t)^{\gamma}\big) \end{align}$ $

(10)

which are obviously closely related to the undelayed ARZ model.

2.3. Coarse-graining

Another approach to derive a delayed macroscopic equation is the so-called coarse-graining (CG) proposed in [12,22]. Here, we focus on [22] and highlight the most important steps. Although the method is different from reverse spatial discretization (RSD) from subsection 2.2, we note that the resulting equations are quite similar.

We start by defining the density $ \hat{\rho}(x, t) $ and flux field $ \hat{q}(x, t) $ of traffic flow as follows:

$\begin{align} \hat{\rho}(x, t): = \sum\limits_i \delta(x_i(t)-x), \quad \hat{q}(x, t): = \sum\limits_i \dot{x}_i(t)\delta(x_i(t)-x), \end{align}$ $

(11)

where $ x_i(t) $ is the position of the $ i $-th car at time $ t $ and $ \delta $ denotes the $ \delta $-distribution. We now coarse-grain these fields and obtain

$\begin{align} \rho(x, t)&: = \int\Phi(x-x', t-t')\hat{\rho}(x', t')\text{d}x'\text{d}t', \\ q(x, t)&: = \int\Phi(x-x', t-t')\hat{q}(x', t')\text{d}x'\text{d}t' \end{align}$ $

(12)

by applying the coarse graining envelope function $ \Phi(x, t) $ which has a peak at $ (0, 0) $ and is normalized in the sense $ \int\Phi(x, t)\text{d}x\text{d}t = 1 $. According to [22], we can derive the following system of equations from (12) for the density $ \rho $ and flux $ q $:

$\begin{align} &\frac{\partial}{\partial t}\rho(x, t)+\frac{\partial}{\partial x}q(x, t) = 0\\ &\frac{\partial}{\partial t}q(x, t) = \rho(x, t)\langle\ddot{x}_i(t')\rangle_{(x, t)}-\frac{\partial}{\partial x}[\rho(x, t)\langle\dot{x}^2_i(t')\rangle_{(x, t)}] \end{align}$ $

(13)

with the bracketed average of a quantity $ f_i(x, t) $ defined as

$ \langle f_i(x', t')\rangle_{(x, t)} = \frac{1}{\rho(x, t)}\int\Phi(x-x', t-t')\sum\limits_i f_i(x', t')\delta(x_i(t')-x')\text{d}x'\text{d}t'. $

By introducing $ v(x, t) = \langle \dot{x}_i(t')\rangle_{(x, t)} = \frac{q(x, t)}{\rho(x, t)} $ and $ \Theta(x, t) = \langle \dot{x}^2_i(t')\rangle-v^2(x, t) $ we can rewrite the second equation in (13) to

$\begin{align} \rho(x, t)(\frac{\partial}{\partial t}v(x, t)+v(x, t)\frac{\partial}{\partial x}v(x, t)) = \rho(x, t)\langle\ddot{x}_i(t')\rangle_{(x, t)}-\frac{\partial}{\partial x}(\rho(x, t)\Theta(x, t)). \end{align}$ $

(14)

The first equation for the density is obviously the same as in (10) and the derivation so far is independent on microscopic dynamics. However, the second equation for the momentum involves a term $ \langle\ddot{x}_i(t')\rangle_{(x, t)} $ which is dependent on individual drivers' behavior and therefore needs a careful discussion.

Let us first consider the general case of a car-following model of the form

$ \ddot{x}_i(t) = B(\Delta x_i, \Delta \dot{x}_i, \dot{x}_i). $

Then coarse graining leads to the approximated momentum equation

$ \frac{\partial}{\partial t}v(x, t)+v(x, t)\frac{\partial}{\partial x}v(x, t)\approx B(\langle\Delta x_i\rangle, \langle\Delta \dot{x}_i\rangle, v), $

where we further approximate

$\begin{align*} \langle\Delta x_i\rangle&\approx \rho^{-1}(x, t)+\frac{1}{2\rho(x, t)}\frac{\partial}{\partial x}\rho^{-1}(x, t)+\frac{1}{6\rho^2(x, t)}\frac{\partial^2}{\partial x^2}\rho^{-1}(x, t)\\ \langle\Delta \dot{x}_i\rangle&\approx \rho^{-1}(x, t)\frac{\partial}{\partial x}v(x, t)+\frac{1}{2\rho^2(x, t)}\frac{\partial^2}{\partial x^2}v(x, t). \end{align*}$ $

Plugging in and expanding around $ (\rho^{-1}(x, t), 0, v(x, t)) $ yields

$\begin{align*} B(\langle\Delta x_i\rangle, \langle\Delta \dot{x}_i\rangle, v(x, t))\approx& B(\rho^{-1}(x, t), 0, v(x, t))\\ +&B_1(\frac{1}{2\rho(x, t)}\frac{\partial}{\partial x}\rho^{-1}(x, t))+B_2(\rho^{-1}(x, t)\frac{\partial}{\partial x}v(x, t)), \end{align*}$ $

where $ B_i = \partial_{z_i}B(z_1, z_2, z_3)|_{(\rho^{-1}, 0, v)} $. We end up with an equation of type

$\begin{align} \partial_t v(x, t)+v(x, t)\partial_x v(x, t)& = B(\rho^{-1}(x, t), 0, v(x, t))\\ &+\frac{B_1}{2\rho(x, t)}\partial_x \rho^{-1}(x, t)+\frac{B_2}{\rho(x, t)}\partial_x v(x, t). \end{align}$ $

(15)

We remark that it is also possible to include terms of higher order leading to higher order approximations as well as higher order derivatives. Since we are interested in delayed models driven by (3), we can identify $ B $ as follows:

$\begin{align} B(\Delta x_i, \Delta \dot{x}_i, \dot{x}_i) = C\frac{\Delta \dot{x}_{i}(x, t-T)}{\Delta x_{i}(x, t-T)^{\gamma+1}}\\ B_1 = 0, \quad B_2 = C\rho(x, t-T)^{\gamma+1}, \quad B_3 = 0. \end{align}$ $

(16)

The system of equations is then

$\begin{align} \partial_t \tau(x, t) & = \partial_x v(x, t) \\ \partial_tv(x, t)+v(x, t)\partial_xv(x, t)& = v_{\text{ref}}\rho(x, t-T)^{\gamma}\partial_xv(x, t). \end{align}$ $

(17)

In conservative form, we may write the system of equations

$\begin{align} \partial_{t}\rho(x, t)+\partial_{x}(\rho(x, t)v(x, t))& = 0 \\ \partial_{t}(\rho(x, t)w(x, t))+\partial_{x'}(\rho(x, t)v(x, t)w(x, t)) = v_{\text{ref}}&[\partial_{x}v(x, t)\rho(x, t-T)^{\gamma}\\ &-\partial_{x}v(x, t)\rho(x, t)^{\gamma}] \end{align}$ $

(18)

which is (up to the term $ \partial_{x}v(x, t) $) directly comparable to (10).

For completeness, we also state the Lagrangian representation of (18). If we use the inverse transformation from subsection 2.2 and write the equations dependent on $ \tau $ instead of $ \rho $ we end up with

$\begin{align} &\partial_{t}\tau(x, t) = \partial_{x}v(x, t)\\ &\partial_{t}v(x, t) = v_{\text{ref}}\rho(x, t-T)^{\gamma+1}\partial_{x}v(x, t). \end{align}$ $

(19)

2.4. Taylor expansion

The last approach we present is based on a Taylor expansion which was originally introduced in [30]. However, in contrast to the original approach, we derive a second order delayed model instead of a first order model only. Furthermore, we keep the explicit delay while the new model is derived from the microscopic level and do not apply a diffusive approximation.

We start by expanding the delayed variables position $ x_i $ and speed $ v_i $ up to first order. This leads directly to

$ x_i(t-T) = x_i(t)-T\dot{x}_i(t)+\mathcal O(T^2) $

and equivalently

$ v_i(t-T) = v_i(t)-T\dot{v}_i(t)+\mathcal O(T^2). $

By using the microscopic description (3) we can characterize the following delayed system:

$\begin{align} x_i(t-T)& = x_i(t)-T v_i(t)+\mathcal O(T^2) \\ v_i(t-T)& = v_i(t)-T\Big(C\frac{(v_{i+1}(t-T)-v_i(t-T))}{(x_{i+1}(t-T)-x_i(t-T))^{\gamma+1}}\Big)+\mathcal O(T^2), \quad i = 1, \ldots, N. \end{align}$ $

(20)

The definition $ \Delta x_i(t): = x_{i+1}(t)-x_i(t) $ then allows to approximate

$ \Delta x_i(t-T)\approx\Delta x_i(t)-T\Delta v_i(t) $

and

$ \Delta v_i(t-T)\approx \Delta v_i(t)-T \Big(C\frac{\Delta v_{i+1}(t-T)}{(\Delta x_{i+1}(t-T))^{\gamma+1}}-C\frac{\Delta v_i(t-T)}{(\Delta x_i(t-T))^{\gamma+1}}\Big). $

To derive macroscopic equations, we use again $ \rho_i(t): = \frac{\Delta X}{x_{i+1}(t)-x_i(t)} $ and obtain the conservation of mass by introducing $ \tau_i = \rho_i^{-1} $ in the limit $ \Delta X\rightarrow0 $:

$ \partial_t \tau(x, t) = \partial_x v(x, t), $

where $ \tau $ and $ \rho $ denote the respective limit versions of $ \tau_i $ and $ \rho_i $. The second equation for the velocity is derived as follows:

$\begin{align*} &\partial_t v_i(t) = C\frac{\Delta v_i(t-T)}{\Delta x_i(t-T)^{\gamma+1}}\\ &\approx C\frac{1}{(\Delta x_i(t)-T\Delta v_i(t))^{\gamma+1}}\\ &\times\Big(\Delta v_i(t)-T[C\frac{\Delta v_{i+1}(t-T)}{(\Delta x_{i+1}(t-T))^{\gamma+1}}-C\frac{\Delta v_i(t-T)}{(\Delta x_i(t-T))^{\gamma+1}}]\Big)\\ &\approx C\frac{1}{\frac{1}{\rho_i(t)^{\gamma+1}}-(\gamma+1)\frac{1}{\rho_i(t)^{\gamma}}T\Delta v_i(t)}\\ &\times\Big(\Delta v_i(t)-CT\big(\Delta v_{i+1}(t-T)(\rho_{i+1}(t-T))^{\gamma+1}-\Delta v_i(t-T)(\rho_i(t-T))^{\gamma+1}\big)\Big). \end{align*}$ $

For simplicity, we omit the terms $ \mathcal O(T^2) $ in the latter representation. Doing the same scaling procedure and using the same definitions as above, we end up with

$\begin{align} \partial_t \tau(x, t) & = \partial_x v(x, t) \\ \partial_t v(x, t)& = \frac{v_{\text{ref}}\rho(x, t)^{\gamma+1}}{1-(\gamma+1)\rho(x, t)T\partial_x v(x, t)}\\ &\times\partial_x\big(v(x, t)-2v_{\text{ref}}T(\rho(x, t-T)^{\gamma+1}\partial_x v(x, t-T))\big) \end{align}$ $

(21)

in Lagrangian coordinates. However, the interpretation of the right-hand side is now different compared to (7) and (19).

By the use of Taylor expansion, correction terms enter the model. This can be interpreted as a correction towards the fact that drivers do not react immediately but after some delay. These correction terms are, as in the previous models, dependent on the product of density, the spatial derivative of speed and the (current and past) state of the traffic. Note that in the case we ignore the terms depending on $ T $, we recover the classical ARZ model.

3. Properties of delayed macroscopic models

As already seen, the introduction of time delays lead to a right-hand side in the equation for the velocity that includes different states of traffic. This can be interpreted as an anticipation of the change in traffic based on the current and past state. Hence, we are able to model the drivers ability to 'extrapolate' the traffic situation to change his/her driving behavior.

Similar to the microscopic case, the introduction of explicit delays requires the extension of the initial data time span to ensure well-posedness. For all states $ (\rho, v) $ that appear in a delayed form, initial values at $ t = 0 $ are not enough and we need initial histories. These are functions prescribing the states on at least the interval $ [-T, 0] $.

We now aim to discuss some key properties of the delayed macroscopic models. First, we show that for vanishing delays the models coincide with the classical ARZ model. In this context, we also refer to the weak solutions of the ARZ model and investigate the changes caused by the delayed case. Second, we comment on the positivity of solutions. Closely related to this discussion is the question of stability. We inverstigate the system of delayed differential equations (DDEs) resulting from discretizing in space and derive properties to ensure stable solutions.

3.1. Convergence to the classical ARZ model

Lemma 3.1. The macroscopic delayed models (7), (19) and (21) converge to the undelayed ARZ model for $ T\rightarrow0 $.

Proof. The first equation of all proposed models is mass conservation and has no explicit dependence on the delay $ T $. We focus on the second equation in Lagrangian coordinates and need to show that $ \partial_t w(x, t) = 0 $ is satisfied.

Therefore, the velocity equation needs to be expressed in terms of $ w $ for all models (7), (19) and (21). We use that $ w = v+P $ and get the following representation for the models derived by reverse spatial discretization (7) and coarse-graining (19):

$ \partial_t w(x, t) = v_{\text{ref}}\Big(\partial_x v(x, t-T) \rho(x, t-T)^{\gamma+1}-\partial_x v(x, t)\rho(x, t)^{\gamma+1}\Big) $

and

$ \partial_t w(x, t) = v_{\text{ref}}\Big(\partial_x v(x, t) \rho(x, t-T)^{\gamma+1}-\partial_x v(x, t)\rho(x, t)^{\gamma+1}\Big), $

respectively. The velocity equation for the model derived by Taylor expansion (21) reads

$\begin{align*} \partial_t w&(x, t) = \frac{v_{\text{ref}}}{\frac{1}{\rho(x, t)^{\gamma+1}}-(\gamma+1)\frac{1}{\rho(x, t)^{\gamma}}T\partial_x v(x, t)}\\ &\times(\partial_x v(x, t)-2v_{\text{ref}}T\partial_x[\rho(x, t-T)^{\gamma+1}\partial_x v(x, t-T)])-v_{\text{ref}}\partial_x v(x, t)\rho(x, t)^{\gamma+1}. \end{align*}$ $

Then, in the limit $ T\rightarrow0 $, we end up with $ \partial_t w(x, t) = 0 $ for all equations.

3.2. Comparison to weak solutions of the ARZ model

In this section, we discuss the solutions to Riemann problems for the ARZ model (1) and explain how the delayed models fit into this framework. The ARZ model is a system of conservation laws and we have three basic types of weak solutions to the Riemann problem, i.e. contact discontinuities, shocks and rarefaction waves, see [3,15] for an overview. We define $ u = (\rho, \rho w)^T $ and rewrite the fluxes for the ARZ model as

$\begin{equation} f = \begin{pmatrix} f_{\rho} \\ f_{\rho w} \end{pmatrix} = \begin{pmatrix} \rho v \\ \rho wv \end{pmatrix}. \end{equation}$ $

(22)

For further investigations, we consider the ARZ model in the form

$ \partial_t u + \nabla f\partial_x u = 0, $

where $ \lambda_i(u) $ and $ r_i(u) $ are the eigenvalues and eigenvectors of $ \nabla f $, dependent of the state vector $ u $. After defining genuine nonlinear and linear degenerate eigenvalues, we can state the weak solutions. For the ARZ model, the eigenvalues and eigenvectors are

$\begin{align} \lambda_1& = v-\rho P(\rho), r_1 = \begin{pmatrix}1\\w\end{pmatrix}\\ \lambda_2& = v, \quad\quad\quad r_2 = \begin{pmatrix}1\\w+\rho P'(\rho) \end{pmatrix} \end{align}$ $

(23)

with $ \lambda_1 $ being genuine nonlinear and $ \lambda_2 $ being linear degenerate. While lifting this analysis to the delayed models, the first problem occurs since $ \nabla f $ (and its properties) needs to be derived. However, except for $ \gamma = 0 $, a suitable flux function has not been found yet since it would depend on the current and past state, respectively.

Now, let us focus on the weak solutions to Riemann problems with left state $ u_l = (\rho_l, \rho_l w_l)^T $ and right state $ u_r = (\rho_r, \rho_r w_r)^T. $ The contact discontinuity belongs to the linear degenerate eigenvalue and appears when $ \lambda_2(u_l) = \lambda_1(u_r) $. This means $ v_l = v_r $. The solution is then given by the initial date moving with $ v_l = v_r $

$\begin{equation} u(t, x) = u_0(x-v_lt). \end{equation}$ $

(24)

For the delayed models, we do not necessarily get the same solution for $ v_l = v_r $. However, if also the history is such that $ v_l = v_r $ for all $ [-T, 0] $, the solution looks quite similar.

For the shock solution, we consider the Rankine Hugoniot condition

$ f(u_l)-f(u_r) = s(u_l-u_r). $

This gives for every state a curve which can be connected to the latter by a shock. The shock solution belongs to the genuine nonlinear eigenvalue. In the ARZ model, a fixed $ u_l $ can be connected by a shock to states $ u $ with $ w_l = w $. Furthermore, the shock is admissible if $ \rho_l<\rho_r $ and looks like

$\begin{equation} u(t, x) = u_0(x-st). \end{equation}$ $

(25)

It remains unclear how the delayed models fit into this framework due to the missing definition of a flux function and hence the evaluation of the Rankine Hugoniot condition.

Similar to the shock solution, the rarefaction wave also belongs to the genuine nonlinear eigenvalue $ \lambda_j $. The solution is given by

$\begin{equation} u(t, x) = \begin{cases}u_l & x < \lambda_1(u_l) t\\ v(\frac{x}{t}) & \lambda_1(u_l) < x < \lambda_1(u_r)\\u_r & x > \lambda_1(u_r) t\end{cases}, \end{equation}$ $

(26)

where $ v $ is such that $ v'(\zeta) = r_j(v(\zeta)) $ and $ \lambda_j(v(\zeta)) = \zeta $. For the ARZ model, it holds $ v(\zeta) = (\zeta, w_l)^T $. In the case of delayed models, no conclusions can be made since eigenvalues and eigenvectors cannot be determined.

To illustrate the behavior of approximate solutions to the delayed model, we refer to our numerical study in section 4, in particular figures 5, 6 and 7.

Figure 5. Comparison for contact discontinuity: numerical RSD and ARZ solutions vs. analytical ARZ.

	$ \Delta x $	5	1	0.5	0.1
RSD	$ \|\|\cdot\|\|_{2} $-Error	0.074	0.0759	0.0706	0.0335
RSD	$ \|\|\cdot\|\|_{\infty} $-Error	0.038	0.0337	0.027	0.0288
CG	$ \|\|\cdot\|\|_{2} $-Error	0.0727	0.0746	0.0692	0.0308
CG	$ \|\|\cdot\|\|_{\infty} $-Error	0.036	0.0305	0.0236	0.0288
TE	$ \|\|\cdot\|\|_{2} $-Error	0.0651	0.0683	0.0657	0.0486
TE	$ \|\|\cdot\|\|_{\infty} $-Error	0.0223	0.0181	0.0236	0.0288

1.	Dora Brites, Adelaide Fernandes, Neuroinflammation and Depression: Microglia Activation, Extracellular Microvesicles and microRNA Dysregulation, 2015, 9, 1662-5102, 10.3389/fncel.2015.00476
2.	Tao Wang, Yan Yuan, Hui Zou, Jinlong Yang, Shiwen Zhao, Yonggang Ma, Yi Wang, Jianchun Bian, Xuezhong Liu, Jianhong Gu, Zongping Liu, Jiaqiao Zhu, The ER stress regulator Bip mediates cadmium-induced autophagy and neuronal senescence, 2016, 6, 2045-2322, 10.1038/srep38091
3.	Dora Brites, Regulatory function of microRNAs in microglia , 2020, 68, 0894-1491, 1631, 10.1002/glia.23846
4.	Fabíola de Carvalho Chaves de Siqueira Mendes, Luisa Taynah Vasconcelos Barbosa Paixão, Daniel Guerreiro Diniz, Daniel Clive Anthony, Dora Brites, Cristovam Wanderley Picanço Diniz, Marcia Consentino Kronka Sosthenes, Sedentary Life and Reduced Mastication Impair Spatial Learning and Memory and Differentially Affect Dentate Gyrus Astrocyte Subtypes in the Aged Mice, 2021, 15, 1662-453X, 10.3389/fnins.2021.632216
5.	Amanda de Oliveira Ferreira Leite, João Bento Torres Neto, Renata Rodrigues dos Reis, Luciane Lobato Sobral, Aline Cristine Passos de Souza, Nonata Trévia, Roseane Borner de Oliveira, Nara Alves de Almeida Lins, Daniel Guerreiro Diniz, José Antonio Picanço Diniz, Pedro Fernando da Costa Vasconcelos, Daniel Clive Anthony, Dora Brites, Cristovam Wanderley Picanço Diniz, Unwanted Exacerbation of the Immune Response in Neurodegenerative Disease: A Time to Review the Impact, 2021, 15, 1662-5102, 10.3389/fncel.2021.749595
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AIMS Molecular Science

Cell ageing: a flourishing field for neurodegenerative diseases

Related Papers:

Abstract

1. Introduction

2. From microscopic to macroscopic models

2.1. Microscopic model

2.2. Reverse spatial discretization

2.3. Coarse-graining

2.4. Taylor expansion

3. Properties of delayed macroscopic models

3.1. Convergence to the classical ARZ model

3.2. Comparison to weak solutions of the ARZ model

3.3. Positivity of solutions

3.4. Stability of discretized system

4. Numerical results

4.1. Comparison of delayed models

4.1.1. Microscopic versus macroscopic approaches

4.1.2. Classical ARZ model versus macroscopic approaches

4.1.3. Taylor model (TE) versus convection-diffusion flow model [30]

4.2. Applications of delayed macroscopic models

4.2.1. Traffic lights

4.2.2. Data fitting

Conclusion

References

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Catalog

Abstract

1. Introduction

2. From microscopic to macroscopic models

2.1. Microscopic model

2.2. Reverse spatial discretization

2.3. Coarse-graining

2.4. Taylor expansion

3. Properties of delayed macroscopic models

3.1. Convergence to the classical ARZ model

3.2. Comparison to weak solutions of the ARZ model

3.3. Positivity of solutions

3.4. Stability of discretized system

4. Numerical results

4.1. Comparison of delayed models

4.1.1. Microscopic versus macroscopic approaches

4.1.2. Classical ARZ model versus macroscopic approaches

4.1.3. Taylor model (TE) versus convection-diffusion flow model [30]

4.2. Applications of delayed macroscopic models

4.2.1. Traffic lights

4.2.2. Data fitting

Conclusion

References