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...Living cells are complex mixtures of a variety of complex molecules that are constantly undergoing reactions with one another, and such reactions typically have an exponential distribution associated =-=[7]-=-. Furthermore, stochastic fluctuations play a key role in biological processes, both at the molecular level, where low intracellular copy numbers of molecules can fundamentally limit the precision of ...

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...ples and the cell-level properties of the tissue — the macroscopic bidomain equations were not directly connected to the microstructure of the tissue. The advent of more formal homogenisation methods =-=[2]-=- paved the way for a first attempt at a rigorous derivation of the bidomain equations by Neu & Krassowska in 1993 [7]. They used a multiple-scales asymptotic expansion technique to convert the microsc...

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...ng problem for the two counter machine can be reduced to the reachability problem of a particular class of hybrid automata, the reachability problem for hybrid automata itself is not always decidable =-=[1]-=-. However, if H is a T -hybrid automaton and T is a first-order decidable theory, then the reachability through a bounded number of discrete transitions can be characterized with a first-order decidab...

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...lgebra (a subset of stochastic CCS), systems of reactions in the stoichiometric form and ODE systems. Bortolussi and Policriti [6] extend this work by defining a semantics in terms of hybrid automata =-=[11]-=-, defining the requisites according to which a set of processes can be treated as discrete control states, while others have the usual continuous interpretation. Several other methods for associating ...

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...] and they were observed in the thalamus [18], and in the cortex [17]. Many studies suggested that, in the mammalian visual system, neurons signals may be group together through in-phase oscillations =-=[11]-=-. Because of this, the development and analysis of models representing oscillatory phenomena assume a great importance in understanding the neurophysiological activities. A simple continuous model of ...

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...1}. 3 Hybrid semantics and optimal control of infectious processes In this section we define the optimal control law for therapy scheduling in the SIR model by means of Model Predictive Control (MPC) =-=[4]-=-, and we report the optimally controlled solutions with parameters that are typical of the measles [20]. A similar problem has been addressed in [21] for the scheduling of hormone therapy in a model f...

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...hniques 3.1 Noise and Disturbed Automata The density of continuous variables provide an unbounded quantity of memory within a bounded region. As a matter of fact, the undecidability results proved in =-=[13]-=- are based on the possibility of embedding N in (0,1]⊆ R through the function f (n) = 2−n. However, Fränzle in [7] observed that noise disturbes the trajectiories of real hybrid systems, augmenting t...

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...s the differences in bone density by evaluating the dynamics of the bone remodeling process being in a certain state. This kind of data can be elaborated by means of automatic verification procedures =-=[11]-=-. We believe that this kind of quantitative, formal and automated analysis may represent a step ahead in the understanding of the modeling efforts of osteoporosis development, by shifting the attentio...

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...tem, given by the modified bidomain equations (19), and the discrete system, given in [(1),(2),(3),(4)] with the transmembrane current given by (15), using a Beeler-Reuter model for the ionic current =-=[1]-=- in both cases. We use a finite element method [9] with 320 nodes per cell in the discrete case and 80 nodes covering the corresponding area in the continuum case, and use the PETSc library (http://ww...

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...important role in the activities of several brain regions. Some examples of brain locations in which it has been demonstrated the central role of neural oscillations are the cortex [17], the thalamus =-=[18]-=-, and the olfactory information processing [8]. With the aim of understanding neurophysiological activities, we propose the modeling of oscillatory phenomena exploiting hybrid automata. A continuous m...

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...mplified as: φ [X,T]≡ ∃X0(Π1JX0,TK∧Π2JX0,TK∧δ (X0,X)< ε). (8) Similarly, ψ becomes: ψ[X]≡ ∃T(T> 0∧∃X0(Π1JX0,TK∧Π2JX0,TK∧δ (X0,X)< ε)). (9) So we plugged this last formula in Formula 5 and used REDLOG =-=[5]-=- to test it. The formula turns out to be true (the result is computed within few seconds), proving our conjectures. Notice that we used Maxima to compute the exact coordinates of the points on the lim...

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...For these reasons, we study our hybrid automaton exploiting different approximation techniques that introduce noise. In the literature, several approximation techniques have been proposed (see, e.g., =-=[7, 16, 10, 3, 15]-=-). Fränzle in [7] presents a model of noise over hybrid automata. The introduction of noise ensures in many cases the (semi-)decidability of the reachability problem. Another result of (semi-)decidab...

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... variability of the bone density could address the investigation of metabolic syndrome, diabetes and cancer. The logical properties to verify have been formulated in CSL (Continuous Stochastic Logic) =-=[1]-=-, and are: 88 Verification instances for bone pathologies • The cumulative values of bone formation and resorption. f+(t) : R{“boneFormed′′}=?[C ≤ t], f−(t) : R{“boneResorbed′′}=?[C ≤ t], t = 0,10, . ...

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...muli which play an important role in the activities of several brain regions. Some examples of brain locations in which it has been demonstrated the central role of neural oscillations are the cortex =-=[17]-=-, the thalamus [18], and the olfactory information processing [8]. With the aim of understanding neurophysiological activities, we propose the modeling of oscillatory phenomena exploiting hybrid autom...

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...into compartments:Susceptible, Exposed, Infected and Recovered. According to the possible flows among such compartments, different variants arises, like the SIR (Susceptible→Infected→Recovered) model =-=[12]-=-, SIS, SIRS, SEIR and SEIRS. Further compartments can be taken into account like the population of immune infants, usually indicated with M. Similar compartments can be identified not only at the huma...

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...listic automata, and probabilistic timed automata. The PRISM model checker have been successfully employed in a several number of biological case studies, especially for modeling biochemical pathways =-=[12, 6, 2, 16]-=-. However, our framework is not determined by the features of a specific tool, and other similar platforms could have been chosen. 2.2 Definition of the model In our settings, we use Continuous Time M...

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... potential propagation at cell-level, we therefore expect to see the conduction velocity reduce as the wave passes through the gap junction. Indeed, this is supported by experimental data as shown in =-=[11]-=- — the spatial form of the action potential is ‘stepped’, with the reduced conductivity in the gap junction causing a steep jump in membrane potential between one side of the gap junction and the othe...

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...For these reasons, we study our hybrid automaton exploiting different approximation techniques that introduce noise. In the literature, several approximation techniques have been proposed (see, e.g., =-=[7, 16, 10, 3, 15]-=-). Fränzle in [7] presents a model of noise over hybrid automata. The introduction of noise ensures in many cases the (semi-)decidability of the reachability problem. Another result of (semi-)decidab...

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...in regions. Some examples of brain locations in which it has been demonstrated the central role of neural oscillations are the cortex [17], the thalamus [18], and the olfactory information processing =-=[8]-=-. With the aim of understanding neurophysiological activities, we propose the modeling of oscillatory phenomena exploiting hybrid automata. A continuous model of a single oscillator based on an ordina...

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... τ ∗λ ≤ 1, then the point 〈0,0〉 is the unique global attractor of the system, if, otherwise, τ ∗λ > 1, the origin is an unstable equilibrium and all the evolutions converge to a limit cycle attractor =-=[2]-=-. A simulation of f (3,1) is represented in Fig. 4(a). 62 Hybrid Automata and ε-Analysis on a Neural Oscillator Xe Xi Ẋe =−Xeτ Ẋi =−Xiτ +2 v0 Ẋe =−Xeτ −2 Ẋi =−Xiτ v1 Ẋe =−Xeτ Ẋi =−Xiτ −2 v2 Ẋe ...

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...ance, genetic regulatory networks naturally exhibits hybrid dynamics, for which the continuous concentration of proteins is interrupted by the discrete switches dictated by changes in gene expression =-=[3, 10]-=-. Moreover in population biology it happens that some species are present in high concentration, so they can be approximately modelled by continuous variables; conversely, small populations are opport...

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...listic automata, and probabilistic timed automata. The PRISM model checker have been successfully employed in a several number of biological case studies, especially for modeling biochemical pathways =-=[12, 6, 2, 16]-=-. However, our framework is not determined by the features of a specific tool, and other similar platforms could have been chosen. 2.2 Definition of the model In our settings, we use Continuous Time M...

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... are a formal notation initially developed for modelling software systems, but in the last decade it has been extensively used and extended in order to describe biological systems. In a seminal paper =-=[7]-=-, Cardelli showed that a subset of stochastic CCS is powerful enough to encode both systems of reactions in the stoichiometric form and systems of ordinary differential equations. Additionally, a rece...

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...units are finishing [10]. In the normal bone, the number of BMUs, the bone resorption rate, and the bone formation rate are all relatively constant depending on hormones, vascularization and genetics =-=[17]-=-. In cortical BMUs, osteoclasts excavate cylindrical tunnels in the predominant loading direction of the bone. They are followed by osteoblasts, filling the tunnel, creating secondary osteons of renew...

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...which [9, 5]. The formal modelling language employed in this paper is called Disease Chemical Ground Form (DCGF), and it is a variant of CGF for describing complex disease processes. Differently from =-=[17]-=-, where a stochastic process algebra and its continuous semantics are used to describe a classical epidemic model, we attempt to provide a framework for more generic diseases and, although the syntax ...

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...s is presented later in this document (see Section 2.2), and as discussed earlier it is the generally accepted model of cardiac tissue behaviour. A more formal derivation of the model can be found in =-=[5]-=-. Whilst realistic simulations using the bidomain model have been demonstrated, these studies tend to take the bidomain equations as being physiologically correct in all circumstances, and empirical p...

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...the microstructure of the tissue. The advent of more formal homogenisation methods [2] paved the way for a first attempt at a rigorous derivation of the bidomain equations by Neu & Krassowska in 1993 =-=[7]-=-. They used a multiple-scales asymptotic expansion technique to convert the microscopic problem, formulated in terms of the pointwise potentials, to a macroscopic problem formulated in terms of the le...

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...analysis and composition can be made. Trying to linearize the non-linear components of the original continuous model, we first replace in the standard way the sigmoidal behaviours with sign functions =-=[9, 14]-=-, approximating continuos signals with discrete off-on signals. Unfortunately, the behaviour of this model differs from the original one. For this reason, we propose a more sophisticated approximation...

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...For these reasons, we study our hybrid automaton exploiting different approximation techniques that introduce noise. In the literature, several approximation techniques have been proposed (see, e.g., =-=[7, 16, 10, 3, 15]-=-). Fränzle in [7] presents a model of noise over hybrid automata. The introduction of noise ensures in many cases the (semi-)decidability of the reachability problem. Another result of (semi-)decidab...

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...ocess; or in an infection scenario, the cognitive behaviour at the human level that avoids or limits the interactions with other individuals possibly affected by the disease. A work by Bagnoli et al. =-=[1]-=- embeds this aspect of human behaviour into a SIR epidemic model by allowing individuals to perceive the risk of being infected by ill neighbours. Therefore, the use of control-theoretic tools seems p...

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...discrete control states, while others have the usual continuous interpretation. Several other methods for associating hybrid semantics to quantitative process algebras have been proposed, among which =-=[9, 5]-=-. The formal modelling language employed in this paper is called Disease Chemical Ground Form (DCGF), and it is a variant of CGF for describing complex disease processes. Differently from [17], where ...

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...echanosensing micro-fractures and to send signals to the BMU cells. This process is highly dynamic and each BMU has a finite lifetime, so new units are continuously forming as old units are finishing =-=[10]-=-. In the normal bone, the number of BMUs, the bone resorption rate, and the bone formation rate are all relatively constant depending on hormones, vascularization and genetics [17]. In cortical BMUs, ...

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...analysis and composition can be made. Trying to linearize the non-linear components of the original continuous model, we first replace in the standard way the sigmoidal behaviours with sign functions =-=[9, 14]-=-, approximating continuos signals with discrete off-on signals. Unfortunately, the behaviour of this model differs from the original one. For this reason, we propose a more sophisticated approximation...

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...ogical activities, we propose the modeling of oscillatory phenomena exploiting hybrid automata. A continuous model of a single oscillator based on an ordinary differential system has been proposed in =-=[19]-=-. Even if this is a simple model, its analysis and the analysis of its composition in multiple copies is limited due to the non-linearity of the ordinary differential system involved. For this reason,...

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...jor determinant of bone mass and strength [8]. In particular, recent works show that the levels of OPG and RANKL are inversely related to bone density, and may lead to the development of osteoporosis =-=[9]-=-. Our previous work [15] shows how small changes in RANKL for very long periods lead to disease conditions, especially when aging factors are involved. In particular, we have simulated the bone remode...

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...monstrates the effectiveness of a hybrid approach where a process-algebraic specification level is translated into a runnable stochastic agent-based model in the bone remodelling case study, while in =-=[2]-=- the same case study is approached with different semantics according to the biological property to analyse. Consequently, formal languages could represent a wrapping language able to homogeneously de...

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...ow several encodings can be implemented among specifications in the CGF algebra (a subset of stochastic CCS), systems of reactions in the stoichiometric form and ODE systems. Bortolussi and Policriti =-=[6]-=- extend this work by defining a semantics in terms of hybrid automata [11], defining the requisites according to which a set of processes can be treated as discrete control states, while others have t...

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...d of single-scale models, describing the biological process at different characteristic space-time scales, and scale bridging models, which define how the singlescale models are coupled to each other =-=[22]-=-. Higher level languages and formalisms can help in giving a general homogeneous formulation for the different scales in a multiscale biological system. In general, that formal description cannot be d...

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.... In this way, OPG inhibits the production of osteoclasts, thus protecting bone from excessive resorption. The relative concentration of RANKL in bone is a major determinant of bone mass and strength =-=[8]-=-. In particular, recent works show that the levels of OPG and RANKL are inversely related to bone density, and may lead to the development of osteoporosis [9]. Our previous work [15] shows how small c...

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...For these reasons, we study our hybrid automaton exploiting different approximation techniques that introduce noise. In the literature, several approximation techniques have been proposed (see, e.g., =-=[7, 16, 10, 3, 15]-=-). Fränzle in [7] presents a model of noise over hybrid automata. The introduction of noise ensures in many cases the (semi-)decidability of the reachability problem. Another result of (semi-)decidab...

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...For these reasons, we study our hybrid automaton exploiting different approximation techniques that introduce noise. In the literature, several approximation techniques have been proposed (see, e.g., =-=[7, 16, 10, 3, 15]-=-). Fränzle in [7] presents a model of noise over hybrid automata. The introduction of noise ensures in many cases the (semi-)decidability of the reachability problem. Another result of (semi-)decidab...

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...ses where discrete intervention policies has to be modelled. 2.1 The Disease Chemical Ground Form Here we present the Disease Chemical Ground Form (D-CGF), a variant of the Chemical Ground Form (CGF) =-=[8]-=-) targeted to describe complex models of disease in a process-algebraic fashion. The syntax of D-CGF is given by the grammar in Table 1. A D-CGF model is given by a set of species S, an initial popula...

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...deling Liò, Merelli & Paoletti 83 approaches that can handle multiple intracellular and extracellular factors acting on different time and space scales. Previous work has focused on a Shape Calculus =-=[4, 3]-=- approach to bone remodeling [13]. In this paper new models provide a way to integrate both discrete and continuous behavior that are used to represent individual cells (as agents) and concentration o...

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...ired direction would be reverse engineering from patient datasets, which in turn will allow to fine tuning the diagnosis. In addition, we aim at using parametric probabilistic model repair techniques =-=[5]-=- for our model. The problem of model repair is, roughly speaking, “given a probabilistic model M and a formula not satisfied by M, finding a M′ that satisfy the formula and such that the cost of modif...

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...approaches that can handle multiple intracellular and extracellular factors acting on different time and space scales. Previous work has focused on a Shape Calculus [4, 3] approach to bone remodeling =-=[13]-=-. In this paper new models provide a way to integrate both discrete and continuous behavior that are used to represent individual cells (as agents) and concentration or density fields (as environment)...

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...2∧X = 3). Finally, if ϕ[X1, . . . ,Xi, . . . ,Xn] then we may denote the formula ϕJXi/~sK by writing ϕJX1, . . . ,~s, . . . ,XnK. The semantics of a formula is defined in the standard way (see. e.g., =-=[6]-=-). Given a set Γ of sentences and a sentence ϕ , we say that ϕ is a logical consequence of Γ (denoted, Γ |= ϕ) if ϕ is valid in any modelM in which each formula of Γ is valid too (M |= Γ). A theory T ...

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...Xe)+hλ ,α(Xi) , (4) A. Casagrande, T. Dreossi & C. Piazza 63 whose corresponding hybrid automaton is depicted in Fig. 3. A different approximation could be obtained by using the technique proposed in =-=[12]-=-. Xe Xi Xe =−αλ Xe = αλ Xi =−αλ Xi = αλ Ẋe = τ∗λ−ατ∗α Xe− λαXi Ẋi = λαXe+ τ∗λ−α τ∗α Xi v0 Ẋe =−Xeτ − λαXi−1 Ẋi = τ∗λ−ατ∗α Xi−1 v1 Ẋe =−Xeτ Ẋi =−Xiτ −2 v2 Ẋe = τ∗λ−ατ∗α Xe+1 Ẋi = λαXe− Xiτ −1 v...

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...discrete control states, while others have the usual continuous interpretation. Several other methods for associating hybrid semantics to quantitative process algebras have been proposed, among which =-=[9, 5]-=-. The formal modelling language employed in this paper is called Disease Chemical Ground Form (DCGF), and it is a variant of CGF for describing complex disease processes. Differently from [17], where ...

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...cancer. In order to cope with the nonlinear dynamics of the SIR model, the optimal scheduling of multiple therapies is computed by solving an on-line MPC problem in the Multi-Parametric Toolbox (MPT) =-=[13]-=-. P. Liò, E. Merelli & N. Paoletti 163 The optimal control law can be formulated by means of the following constrained finite-time optimal control (CFTOC) problem, where the original continuous-time ...

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... some species are present in high concentration, so they can be approximately modelled by continuous variables; conversely, small populations are opportunely modelled as discrete stochastic variables =-=[14]-=-. Another crucial aspect is related to the self-regulation and control of biological systems: in normal conditions, biological entities and functions are self-regulated and several multiscale control ...

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...t acts as a bacteriostatic agent, and is usually combined with other antibiotics. Based on a recent work of the authors about the modelling of osteomyelitis and the comparison of different treatments =-=[15]-=-, we can define a hybrid version of that model and a treatment strategy combining antibiotic and anti-inflammatory therapies. The model describes how the bone remodelling dynamics of osteoblasts’ (Ob)...

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...ets. 2.2.1 Hybrid dynamical system of D-CGF models Here we describe the semantics of a D-CGF model in terms of a particular class of hybrid dynamical systems, called controlled switched systems (CSS) =-=[16]-=-. While in the general formulation of hybrid dynamical system the external control input and the discrete operation mode are distinct, in a CSS the external controller produces a switching signal (i.e...

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...TMC), ordinary/partial differential equations (ODE/PDE), agent-based models (ABM), or finite state machines (FSM). Such heterogeneous semantics can be wrapped by a common formal language. the authors =-=[18, 19]-=- demonstrates the effectiveness of a hybrid approach where a process-algebraic specification level is translated into a runnable stochastic agent-based model in the bone remodelling case study, while ...

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...TMC), ordinary/partial differential equations (ODE/PDE), agent-based models (ABM), or finite state machines (FSM). Such heterogeneous semantics can be wrapped by a common formal language. the authors =-=[18, 19]-=- demonstrates the effectiveness of a hybrid approach where a process-algebraic specification level is translated into a runnable stochastic agent-based model in the bone remodelling case study, while ...

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...al control law for therapy scheduling in the SIR model by means of Model Predictive Control (MPC) [4], and we report the optimally controlled solutions with parameters that are typical of the measles =-=[20]-=-. A similar problem has been addressed in [21] for the scheduling of hormone therapy in a model for prostate cancer. In order to cope with the nonlinear dynamics of the SIR model, the optimal scheduli...

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...IR model by means of Model Predictive Control (MPC) [4], and we report the optimally controlled solutions with parameters that are typical of the measles [20]. A similar problem has been addressed in =-=[21]-=- for the scheduling of hormone therapy in a model for prostate cancer. In order to cope with the nonlinear dynamics of the SIR model, the optimal scheduling of multiple therapies is computed by solvin...

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...al morphology of the action potential [13]. The model was further refined to study more specific conditions associated with action potential morphology such as early depolarisation [14], spiral waves =-=[3]-=-, and tachycardia [4]. Whilst these models allow efficient and precise analysis of the conditions required for such phenomena, they are not focused on the incorporation of the cell microstructure and ...

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...action potential [13]. The model was further refined to study more specific conditions associated with action potential morphology such as early depolarisation [14], spiral waves [3], and tachycardia =-=[4]-=-. Whilst these models allow efficient and precise analysis of the conditions required for such phenomena, they are not focused on the incorporation of the cell microstructure and the corresponding eff...

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...and σi is the (scalar) conductivity of the intracellular space. The form of the current comes from Ohm’s law which states that current is the product of conductivity and the gradient of the potential =-=[6]-=-. Conservation D.A. Bruce, P. Pathmanathan & J.P. Whiteley 5 6 ff - x σi, φi y σe, φe Ω (a) A representation of cardiac cells in 2D. σi and σe are, respectively, the intracellular and extracellular co...

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...feasible even for very small regions of tissue. It is much more efficient to solve these models using a continuum approximation to the equations, and indeed over the last 20 years many have done this =-=[8]-=-. This continuum model relies on a homogenisation technique to construct a suitable approximation to the discrete model. The homogenisation process uses a multiple-scales method which uses the fact th...

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..., and the discrete system, given in [(1),(2),(3),(4)] with the transmembrane current given by (15), using a Beeler-Reuter model for the ionic current [1] in both cases. We use a finite element method =-=[9]-=- with 320 nodes per cell in the discrete case and 80 nodes covering the corresponding area in the continuum case, and use the PETSc library (http://www.mcs.anl.gov/petsc) to solve the resulting linear...

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...ult being that the mechanism for defibrillation was clarified. It is a version of this derivation that will be presented later in this document. Following this, a recent paper by Richardson & Chapman =-=[10]-=- augmented the above derivation by introducing a co-ordinate transformation that allows for variable tissue structure and ultimately a set of bidomain equations in which the conductivity tensors syste...

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...omputationally desirable to homogenise the microstructure to allow us to pose the problem as a continuum rather than a discrete problem. This led to the proposal of the bidomain model by Tung in 1978 =-=[12]-=- in which the homogenised potentials are solved for via two reaction-diffusion equations using averaged conductivities, where the precise form of averaging arises from the homogenisation technique. A ...

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...c electrophysiology In recent years, models based on hybrid automata have been developed to model networks of excitable cells, with an initial focus on the temporal morphology of the action potential =-=[13]-=-. The model was further refined to study more specific conditions associated with action potential morphology such as early depolarisation [14], spiral waves [3], and tachycardia [4]. Whilst these mod...

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...focus on the temporal morphology of the action potential [13]. The model was further refined to study more specific conditions associated with action potential morphology such as early depolarisation =-=[14]-=-, spiral waves [3], and tachycardia [4]. Whilst these models allow efficient and precise analysis of the conditions required for such phenomena, they are not focused on the incorporation of the cell m...

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...listic automata, and probabilistic timed automata. The PRISM model checker have been successfully employed in a several number of biological case studies, especially for modeling biochemical pathways =-=[12, 6, 2, 16]-=-. However, our framework is not determined by the features of a specific tool, and other similar platforms could have been chosen. 2.2 Definition of the model In our settings, we use Continuous Time M...

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...deling Liò, Merelli & Paoletti 83 approaches that can handle multiple intracellular and extracellular factors acting on different time and space scales. Previous work has focused on a Shape Calculus =-=[4, 3]-=- approach to bone remodeling [13]. In this paper new models provide a way to integrate both discrete and continuous behavior that are used to represent individual cells (as agents) and concentration o...

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..., as a result of mechanisms involving bone remodeling, very often a net bone loss is seen. In order to maintain the mechanical properties of the bones, old bone is continuously replaced by new tissue =-=[14]-=-. However, pathological conditions can alter the equilibrium between bone resorption and bone formation; osteoporosis, and often osteomyelitis and bone cancer, are examples of pathologies with negativ...

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...listic automata, and probabilistic timed automata. The PRISM model checker have been successfully employed in a several number of biological case studies, especially for modeling biochemical pathways =-=[12, 6, 2, 16]-=-. However, our framework is not determined by the features of a specific tool, and other similar platforms could have been chosen. 2.2 Definition of the model In our settings, we use Continuous Time M...