The heights obtained during the countermovement jump and drop jump tests have been measured by numerous studies using different calculation methods and pieces of equipment. However, the differences in calculation methods and equipment used have resulted in discrepancies in jump height being reported.The aim of this systematic review was to examine the available literature pertaining to the different calculation methods to estimate the jump height during the countermovement jump and drop jump.A systematic review of the literature was undertaken using the SPORTDiscus, MEDLINE, CINAHL, and PubMed electronic databases, with all articles required to meet specified criteria based on a quality scoring system.Twenty-one articles met the inclusion criteria, relating various calculation methods and equipment employed when measuring jump height in either of these two tests. The flight time and jump-and-reach methods provide practitioners with jump height data in the shortest time, but their accuracy is affected by factors such as participant conditions or equipment sensitivity. The motion capture systems and the double integration method measure the jump height from the centre of mass height at the initial flat foot standing to the apex of jumping, where the centre of mass displacement generated by the ankle plantarflexion is known. The impulse-momentum and flight time methods could only measure the jump height from the centre of mass height at the instant of take-off to the apex of jumping, thus, providing statistically significantly lower jump height values compared with the former two methods. However, further research is warranted to investigate the reliability of each calculation method when using different equipment settings.Our findings indicate that using the impulse-momentum method via a force platform is the most appropriate way for the jump height from the instant of take-off to the apex of jumping to be measured. Alternatively, the double integration method via a force platform is preferred to quantify the jump height from the initial flat foot standing to the apex of jumping. Jumping is commonly performed during competitive sports, which is an action requiring the coordination of multiple joints and muscles , 2. DuriThere are numerous pieces of equipment available to measure parameters required for JH calculations during both of these jump tasks. For example, force–time data is recorded by force platforms (FP), or the position-time data is recorded by three-dimensional (3D) motion capture systems . SubsequA number of different methods are available to calculate JH , 32, 34.The primary aim of this systematic review was to examine the available literature pertaining to the different calculation methods to estimate JH during the CMJ and DJ tests. More specifically, we sought to critically evaluate the reliability, equipment selections, and the strengths and weaknesses of each method.This systematic review was conducted under the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) statement in 2020 . A revieOriginal and review journal articles were searched from SPORTDiscus, MEDLINE, CINAHL and PubMed electronic databases (publication date from 2000 to 2022). Figure Studies were included if they met the following criteria: (1) implemented at least two or more calculation methods or pieces of equipment in the outcome measure; (2) clearly described equations of each calculation method, equipment information (i.e. type and sampling frequency) and jump actions (i.e. CMJ or DJ); included healthy adult participants (i.e. aged ≥ 18 years old); (3) presented full data (mean and standard deviation) and statistical significance in results; (4) the drafts were written in English and were published in a peer-reviewed journal. For the purpose of this systematic review, the included articles were required to describe methods used to measure JH during CMJ and/or DJ. As such, articles that simply measured JH in their experimental designs were excluded.The quality scoring system used in the present study was adapted and modified from Bishop et al. . Each stA total of 6557 articles were initially returned with an additional four articles included from other sources , 35, 36.Of the 21 articles included in the final analysis  via a single piece of equipment (i.e. the FP) in four studies , 39, 42.The aim of this systematic review was to critically evaluate the available literature relating to different calculation methods to estimate JH during the CMJ and DJ. When collecting data in applied settings, the equipment and the calculation methods employed may have a significant effect on the outcome measure of JH. Given that a variety of equipment is available to collect FT data, the first sub-section will briefly compare the JH-2 values derived from different pieces of equipment, followed by the explanation of why the FT method over- or under-estimated the JH-2 compared to other calculation methods. The subsequent four subsections critically discuss the advantages and disadvantages of the IM method, the double integration method, the jump-and-reach method and the motion capture systems. Thus, the information in this systemic review makes suggestions for how to standardise procedures and use equipment, and which calculation method to use when assessing JH during the CMJ and DJ tests.u equals the initial velocity that is 0 m/s, t is the duration between the take-off and landing instants, where the FT should be half of the t, and a represents the absolute value of gravitational acceleration (− 9.81 m. s−2) [The FT method measures the time intervals between the instant of take-off and landing during vertical jumping JH-2). This time is then used in the following equation of uniform acceleration, as shown in Eq. . This ti m. s−2) . As show m. s−2) , 25.−2) [From the equipment selection perspective, Brooks et al.  used a F−2) ; thus, e−2) .p < 0.05) and ErgoJump Plus  photocell mats using the FT method. In terms of these two devices, the ErgoJump Plus showed a statistically significantly lower JH-2 compared with the FP along with poor to moderate reliability . In contrast, the SportJump System Pro photocell mat showed high reliability  compared to the reference FP . The under-estimation of the optical measurement systems could be because these systems were placed at a small height off the ground , where z et al. , 44. If z et al. . Accordiz et al. . Researcz et al. , 30. In p < 0.001). Reeve and Tyler [p < 0.001). Supported further by Moir [Compared to other calculation methods, the FT method has several limitations, which numerous studies have acknowledged , 34, 36.nd Tyler  suggeste by Moir , JH-2 cap < 0.001) and motion capture systems . In addition, a statistically significantly lower JH-2 was measured by the FT method using the jump mat (38.6 ± 6.5 cm) compared to the JH-1 measured by the double integration method using the FP  in the study by Buckthorpe et al. [p < 0.001) and DJ (p < 0.001). Thus, the rise in height generated by plantarflexion of the ankles prior to the take-off instant largely explains the higher JH-1 values calculated by the double integration method and motion capture systems [p = 0.972). Noting that Leard et al. [p > 0.001 and p > 0.05, respectively). Thus, the FT method provides similar outcomes to the motion capture systems, but only if measuring JH-2 where the take-off height is not considered [The FT method calculates the JH-2 via the time interval from the plantar-flexed take-off to landing on the force–time data, where the take-off height of the jumper is not included in the calculation process. Consequently, this makes the FT method under-estimate JH-2 compared with the double integration method and motion capture systems i.e., JH-1) , 36. Diae et al. . Researc systems , 36, 41. systems  revealedd et al.  did not d et al.  may not d et al.  and Słomd et al.  used the , 36. Dinsidered , 26. Onensidered  recruiteAlthough the accuracy of the FT method is primarily determined by the aforementioned factors, this method is still suitable for various sports testing environments because of its simple operation, fewer data processing and abundant equipment available  . If the J is the net impulse, and tstart and ttake-off are the time at instant of the propulsion phase and take-off, respectively. The vstart (v = 0) and vtake-off are the velocity at tstart and ttake-off, respectively. The FvGRF and Fg are the vGRF and the body mass of the participant, respectively. Finally, the vtake-off is extracted from Eq. vtake-off is subsequently used for the calculation of JH-2 via Eq. g represents the acceleration of gravity (− 9.81 m. s−2).The IM method is based on Newtonian mechanics and related mechanical laws. Specifically, the IM relation is derived from Newton’s law of acceleration, which is also connected to the law of conservation of energy . Accordip < 0.001), along with the accelerometer showing poor reliability (ICC = 0.47) [As previously mentioned, the accelerometer provides reliable but inaccurate JH-2 compared to the FP using the FT method . Not sur = 0.47) . Althoug = 0.47) , 43, 49. = 0.47) .p > 0.05), and both methods presented excellent reliability . To investigate which method is suitable to evaluate the loaded CMJ, Pérez-Castilla et al. [From the calculation method perspective, an early study by Moir et al.  confirmea et al.  recruitea et al. . Resultsa et al. . Althouga et al. , 50. In a et al. , 51. Moia et al.  found tha et al. . Thus, ia et al. , 39, thep < 0.01) and double integration method . Similarly, the JH-2 measured by the IM method (29.8 ± 8.9 cm) was found to be statistically significantly lower than the JH-1 measured by the double integration method  in a study by Chiu and Dæhlin [Nevertheless, like the FT method, the IM method calculates the JH between the COM position at the take-off and the apex of the jump i.e. JH-2), and only accounts for a fraction of the work performed during the jump , 5, 15. , and onld Dæhlin , and reqd Dæhlin . Second,d Dæhlin . Otherwid Dæhlin , 48. Whed Dæhlin , 36, lesd Dæhlin , less thd Dæhlin . Therefod Dæhlin . The 0.3d Dæhlin . This med Dæhlin . Chavda d Dæhlin  in additd Dæhlin , 54.Furthermore, it would be possible to obtain the displacement–time data by twice integrating the force–time data from initial standing still to landing , and thetstart and tlanding are the time at instant of countermovement (or drop movement in DJ) and landing, respectively. The FvGRF and Fg are the vGRF and the body mass of the participant, respectively. The h0 in CMJ is the COM height of jumpers during initial standing still (i.e. h0 = 0 m), and the h0 in DJ is the drop height. It is worth noticing that the DJ measures via the above equation are applicable only when the two-adjacent FP are available [Given that the FT method calculated JH-2 according to the time intervals from take-off to landing , 34, thevailable .p < 0.01). Like the IM method, the double integration also relies on the reading of vGRF from the FP and involves the numerical integration process [p = 0.079 and p > 0.01, respectively). Similarly using the FP with 1000 Hz, JH-1 was not statistically significantly different between the double integration method (0.432 ± 0.15 m) and the motion capture systems  [p < 0.01) was observed [n = 21), and authors in only four studies measured the JH-1 values using the double integration method concurrently with the motion capture systems [From the calculation method perspective, previous studies like Conceição et al. , Wank an process , where t process  and Wank process  revealedthod 0.43 ± 0.15 method 0.4 ± 0.15 mthod 0.43 ± 0.15 method 0.4 ± 0.15 m systems . It woulthod 0.43 ± 0.15 mp = 1.000) when measuring JH-1 via FP. Further to this, no statistically significantly different JH-1 between IM + S method and the motion capture systems  was found by Wade et al. [The double integration method is considered reliable during CMJ measures because this method starts the twice integration prior to the movement initiation of CMJ (i.e. standing with flat feet), where the initial standing height is a constant value, and a ‘truly’ zero acceleration is achieved, which is the requirement for accurate integrations , 32, 36.h0) equals zero as the jumpers have landed, so applying the integration process in reverse makes the calculation of drop height in the forward integration process unnecessary [As proposed by Baca , the douecessary , 36. Notecessary , 58, 59.ecessary , 32, 36,ecessary , and thiecessary , 55. Howecessary . Thus, iecessary , 32, 36,The jump-and-reach method via the vertical jump devices has been proposed to make the JH measurement more convenient for various tests in the field because the method needs less equipment and provides the outcome directly . Practitp < 0.05, p < 0.05 and p < 0.001, respectively). Given that the Vertec device is calibrated using flat feet standing on the floor, the jump-and-reach method (which measures JH-1) involves the positive vertical COM displacement generated by the ankle plantarflexion prior to take-off [p < 0.05). Furthermore, the intersession reliability measures in this study indicated that in female participants, the jump-and-reach method  was less reliable as opposed to the FT method ; in male participants, a higher intersession reliability was found with the jump-and-reach method  rather than the FT method  [p > 0.01). Whitmer et al. [p ≤ 0.01) [When comparing the difference in JH between methods, authors in six studies adopted the jump-and-reach method, and existing results again appeared to be somewhat inconsistent. Bui et al. , Brooks take-off . In conttake-off , 28. In take-off , 60. Thitake-off  required = 0.84) . Of note = 0.84) . Althoug = 0.84) , a lack  = 0.84) , they dir et al.  estimate ≤ 0.01) . However ≤ 0.01) , 41.Despite the appeal of the jump-and-reach method, factors that influence the accuracy of the jump-and-reach method should not be ignored. First, the accuracy depends on the timing of the touch, which is the ability of jumpers to displace the vane or touch the wall at the peak height of jumping. If touching of the device does not appear during the peak height, the measured JH-1 via the jump-and-reach method will be under-estimated , 29. SecIn addition, it is not surprising to see the JH difference between the jump-and-reach method and other calculation methods (e.g. the FT method), as they measured disparate biomechanical constructs, i.e. the reaching height difference versus the FT, which the latter variable is associated with the jumpers’ COM displacement . ConsequThe motion capture systems typically involve high-speed cameras or multiple 3D cameras. The 3D motion capture system acquires the position-time data by tracking the reflective markers placed on the trunk, pelvic and lower extremities , the lefp > 0.01). Similar results have been noted in Conceição et al. [p = 0.079). Although the difference is relatively minor, factors that affect the COM estimation and accuracy of the motion capture systems should be highlighted. For example, researchers in some studies only model parts of the total body for the COM estimation  , 63, whiThe cumulative body of literature indicates that the measured JH is influenced by the calculation methods and equipment employed. For measuring the JH from the COM height at the initial flat feet standing to the apex of jumping (i.e. JH-1), the double integration method via the FP is encouraged for practitioners, as this method measures the most comparable JH-1 values compared to the motion capture systems. Of note as well, when two-adjacent FP are unavailable in the DJ measurement, the double integration method is unable to calculate the initial standing height, and the integration process must be processed reversely. The motion capture systems are not preferred, given that this method requires accurate COM estimations and is primarily determined by the equipment availability. For measuring the JH from the COM height at the instant of take-off to the apex of jumping i.e. JH-2), we recommended that practitioners use the IM method via the FP when estimating JH-2 values in both CMJ and DJ, where the COM displacement before the take-off is ignored. The IM method requires a simpler integration process than the double integration method and shows excellent reliability. The FT method may be of use because of its simple calculative process and an abundant equipment selection enables practitioners to conduct the test when working with large groups of athletes. However, some factors such as the take-off and landing positions reduce the accuracy of the FT method, and practitioners should be aware of this. Similarly, the jump-and-reach method is the most convenient approach to estimate the JH-1 and the maximum jump-and-reach height when testing jumpers in a big squad, despite this method showing lower reliability in some studies. Therefore, if the jump-and-reach method is the only option to quantify JH-1, practitioners are suggested to minimise factors such as the coordination of the jump and arm swing or the timing of touch that affects the accuracy of the jump-and-reach method before conducting the data collection. The findings of this systematic review emphasise the strengths and weaknesses of each calculation method during the calculation of JH in CMJ and DJ (as shown in both Table , we reco


Understanding both the epidemiological and evolutionary dynamics of antimicrobial resistance is a major public health concern. In this paper, we propose a nested model, explicitly linking the within- and between-host scales, in which the level of resistance of the bacterial population is viewed as a continuous quantitative trait. The within-host dynamics is based on integro-differential equations structured by the resistance level, while the between-host scale is additionally structured by the time since infection. This model simultaneously captures the dynamics of the bacteria population, the evolutionary transient dynamics which lead to the emergence of resistance, and the epidemic dynamics of the host population. Moreover, we precisely analyze the model proposed by particularly performing the uniform persistence and global asymptotic results. Finally, we discuss the impact of the treatment rate of the host population in controlling both the epidemic outbreak and the average level of resistance, either if the within-host scale therapy is a success or failure. We also explore how transitions between infected populations (treated and untreated) can impact the average level of resistance, particularly in a scenario where the treatment is successful at the within-host scale. The level of resistance to a given antimicrobial is then a continuous trait by nature referred to as antimicrobial quantitative resistance (qAMR), at least at the population level, and qAMR is key to better understanding the evolutionary dynamics of AMR  resistance are generally based on the dynamical interaction between two parasite strains resulting in a discrete and finite formulation of MICs  is one of the major challenges we face in the modern area  model explicitly linking the within- and between-host evolutionary dynamics. Such a nested structure is particularly important because, over the past few decades, it is clear that ecological and evolutionary dynamics are influenced by processes operating across scales  is the bacterial intrinsic growth rate, and y to mutate towards a level x during the reproduction process. Therefore, the total number of bacteria produced at time x is quantified by x, either by the immune system (k). Thus, t, an individual—with the immune system’s response level An integro-differential equation is used to model the within-host dynamics of the bacterial population. Such a within model formulation is previously proposed in Djidjou-Demasse et al. . Each hoIndividual transmission and loss rates at the between-host scale, t is then given byThe force of infection induced by infected individuals at time The dynamics of newly infected individuals (i.e. hin-host  and betwmentclass2pt{minimmentclass2pt{minimhin-host  are prechin-host . Therefohin-host –1.5).Fi.Fi\documThe rest of this work is organized as follows. In Sect. This section is devoted to the main results of the nested model –1.5). S. S1.5). First of all, for biological feasibility of the nested model –1.5), w, w1.5), k, p are always positive on p is a bounded function on Functions J is bounded and integrable on x.The mutation kernel J decays rather rapidly towards infinity in the sense that The mutation kernel Furthermore, the between-host model’s parameters satisfy the following assumption.Recruitment rate The treatment rates The rates Parameters The transmission rates The dynamical properties of the within-host model  have beeThere exists a unique global solution +1(R) of  with \doThe semi-flow defined by The semi-flow Let Assumption x, within an individual with immune system level i, is calculated asThe basic reproduction number of Model  is stronWhen of Model .When ϑ, Model  has a unof Model  with iniThe bacteria-free equilibrium of Model  is asympWhen Let We recall that the within-host model  is precist model   re re\documi, is given byUsing the next-generation operator approach , t, t\documLet Assumptions n system –1.5) ha ha\documTherefore, the threshold dynamics of Model –1.5) is is1.5) i(i)If f system -1.5) is is\docum(ii)If f system -1.5) is is\documLet Assumptions x can be classically referred to as “sensitive”, “intermediate”, or “resistant” depending on whether Here, we present a series of numerical simulations employing semi-explicit finite difference numerical schemes. We refer to Djidjou-Demasse  for an e Theorem . The pro of type . SpecifiWithin-host parameterization The antimicrobial killing rate function x such that, p is p and k and p can be found in <1 Fig. C. In suc)<1 Fig. A, while )<1 Fig. B. At the)<1 Fig. F–H. Moreize Fig. F–H, withion Fig. E.Fig. 3DIn the second scenario, the treatment is assumed unsuccessful at the within-host level, i.e., the basic reproduction number of treated individuals )>1 Fig. C. In suc)>1 Fig. A,B. Inde)>1 Fig. A, at the∗>0 Fig. C. In con∗>0 Fig. , increaseak Fig. F–H. In fium Fig. E, where ium Fig. E, the avOptimizing the treatment rate in the host population is key to controlling both the epidemic outbreak and the average level of resistance Increasing the treatment rate in the host population contributes to reducing the epidemic size at the between-host scale, although the effect is quite marginal in the treatment failure scenario at the within-host scale  wh whi. We We establish the existence of a positive global solution of the system . We firs, system  becomes5f system  are provLet ation of , and wheFor all The semiflow rated by  is boundThere exists a unique strongly continuous semiflow ution of . That isA is a Hille-Yosida operator. Then standard results apply to provide the existence and uniqueness of a mild solution to (ii)(iii)(iv)By  holds. We have for iii) holds, and then the second claim holds.Now, we show that condition  is nonnegative with its global minimum point By Theorem Step 1: Differentiating Step 2: Note thatStep 3: Finally, combining k, f system  and usinBy using , we obtaby using  and (5.8s, using  and (6.9


Correction to: Journal of Cancer Research and Clinical Oncology 10.1007/s00432-022-04418-yIn Fig. 5 of this article, there were two errors in Fig. 5c, right panel. Firstly, the title of the graph (in the right panel) that had read “Target tumour” should have read “Off-target tumour”. In addition, the graph had presented the raw tumour volume data, and it should have presented the % tumour volume increase data. Asterisks that depict the level of significance had also been missing;The Fig.


However, along the other two directions, i.e., x2 and x3, their properties are constant. The effects of the size of a microstructure (the microstructure effect) on the behaviour of the composites can play a significant role. This effect can be described using the tolerance modelling method. This method allows us to derive model equations with slowly varying coefficients. Some of these terms can depend on the size of the microstructure. These governing equations of the tolerance model make it possible to determine formulas describing not only fundamental lower-order vibrations related to the macrostructure of these composite solids, but also higher-order vibrations related to the microstructure. Here, the application of the tolerance modelling procedure is shown to lead to equations of the tolerance model that can be used for non-periodically laminated solids. Then, these model equations are mainly used to analyse a simple example of vibrations for functionally graded composites with non-periodically laminated microstructure (FGL). Similar problems were investigated in the framework of the homogenised  model ; the resulting equations neglect the microstructure effect.Dynamic problems of elastic non-periodically laminated solids are considered in this paper. It is assumed that these laminates have a functionally graded structure on the macrolevel along the  Therefthe tolerance modelling method (or the tolerance method), to the monographs  can be investigated by applying an alternative approach, named erzbicki , Woźniakerzbicki ,65). ThiVarious mechanical problems of different periodic media were considered in many works using the tolerance method. Fluid-saturated micro-periodic grounds were analysed in . CertainThe tolerance method is also a good tool for modelling and analysing the functionally of structures with tolerance-periodic microheterogeneity. Various aspects of the elastic responses of laminates with functional gradation of properties were shown in the works ,90,91. TThe above-mentioned papers do not cover the full range of problems analysed by the authors using the tolerance method for various media with a periodic or tolerance-periodic structure. They also do not constitute a complete list of such literature.the tolerance model for functionally graded laminates (FGL) and then to show an application of these equations in the analysis of a simple dynamical problem for a special example of functionally graded laminates. The example has been selected in such a way as to obtain simple analytical solutions and to show the extension of the tolerance model in relation to the asymptotic model.The aim of this paper is to present equations of This paper consists of six sections.Introduction includes a short description of a problem under consideration and a review of some papers related to different objects with microstructure.Modelling preliminaries, basic denotations of a classical model of an elastic medium used for laminates are introduced. The known equation of such media is formulated.In Modelling technique includes some basic concepts of the tolerance modelling method, with the reminded definitions of the averaging operator, the tolerance-periodic function, the slowly-varying function, the fluctuation shape function. In the second section, two basic assumptions of the method -the micro-macro decomposition, the tolerance averaging approximation are formulated. The third section has a short description of the tolerance modelling procedure.Model equations, two subsections show the governing equations of the models—the system of equations of the tolerance model of functionally graded laminates (FGL) with tolerance-periodic microstructure, and the asymptotic model of functionally graded laminates (FGL) with tolerance-periodic microstructure.In the An example: vibrations of a special laminated layer consists of three subsections. In the first section, an example under consideration of a laminated functionally graded layer on the undeformable base is described. Some basic assumptions for this example are also introduced. Vibration equations of both the models are shown in the second subsection. Equations of the tolerance model for the considered layer in the first approximation are decoupled on two equations—for the macro- and the micro-vibrations. The third section consists of three points. In the first point, it is assumed that the cell distribution functions and the properties distribution functions are defined. Equations for the macro- and the micro-vibrations are written with the boundary conditions on the upper and the bottom boundaries of the layer, and the new denotations and some additional assumptions in the second point of this subsection. Then these equations with the boundary conditions are rewritten in the non-dimensional form. In the third point, formulas of non-dimensional solutions of the macro- and the micro-vibrations are obtained. Results from these formulas are shown in the form of diagrams.Remarks, general and some detailed remarks of the proposed model for FGL layer and obtained numerical results of the example are formulated.In Ox1x2.For the sake of simplicity, but without losing the generality of the analysis, further considerations are limited to a plane problem of composites in the orthogonal Cartesian coordinate system h1, h1) ×  be the region of the considered laminate, where 2h1 is the dimension of the region along the x1 axis; h2 is the dimension along the x2 axis. Introduce also denotations: x ≡ ; and t for a time coordinate. Then denote x ≡ x1, z ≡ x2. Let ∇ ≡ , ∂ ≡  and 2) denote derivatives of x, x and z, respectively. It is assumed that a distribution of laminas thicknesses is extrapolated by smooth function of cells distribution λ(x), being slowly varying. Let us define a closed subset ∆ of the region Ω as ∆ ≡  × {0} and the region ∆(x), x∈Ω, as ∆(x) ≡ x + ∆, such that Ω = {x∈Ω: ∆(x)⊂Ω}. The subset ∆ can be called the “basic cell” and the region ∆(x)—“the cell with a centre at point x”. Introduce also a set of centres of cells ∆(x) defined as Ω∆ =  × .Let Ω = (−x), ν″(x), satisfying the condition ν′(x) + ν″(x) = 1. Moreover, the non-dimensional function x. It is also assumed that planes xα = const, α = 1, 2, are material symmetry planes, .The laminate under consideration is made of two linear-elastic materials that occupy regions Ω′, Ω″, and where Γ is the system of contact planes between the components. These components are distributed in laminas with varying thicknesses. Every lamina consisted of two sub-laminas made of isotropic homogeneous materials with mass densities ρ’, ρ” and elastic tensors unctions ,89,90,91w = w and a vector of external forces by f, the differential equation of the composite under consideration can be written ast. It is assumed that the vector of displacements w and the stress tensor T = C:∇w satisfy regular conditions, and the condition of continuity for the stresses on contact planes between laminas Γ, i.e., ETFn = 0, where ETF, x ∈ Γ, is a jump of the stress tensor; n is a unit normal vector.Denoting a vector of displacements by The above Equation (1), describing dynamical problems of laminates with functionally graded properties (FGL), has non-periodic (tolerance-periodic), non-continuous functional coefficients. In this form, the equation is not a good tool to consider the special problems of these laminates. Hence, Equation (1) can be replaced by a system of differential equations with slowly varying, continuous functional coefficients, using the tolerance modelling method.SV, tolerance-periodic function TP, and fluctuation shape function FS. They were also shown and applied in different papers, e.g., in  , byDefine f is a slowly varying function in x, but this value is constant for periodic function.The averaged value obtained from (2) for a tolerance-periodic function the tolerance-periodic function, ϕ∈TP; if for every x∈Ω∆, there exists a function ϕx such that the functions ϕ and ϕx are indistinguishable within a certain tolerance, determined by the tolerance parameter δ. The function ϕx can be named a periodic approximation of the function ϕ in the region ∆(x).Function ϕ, integrable and bounded in the region Ω, is called the slowly varying function, ψ∈SV, if, and only if, the following condition is satisfied: the tolerance parameter, δ << 1, related to the considered problems.Function ψ, TP, defined in Ω, having a partly continuous derivative of the order 1.Let us introduce a tolerance-periodic, bounded function φ, φ∈the fluctuation shape function, φ∈FSF, depending on the parameter λ and satisfies the conditions λ∂φ∈O(λ) and <φ>(x) ≈ 0 (for every x∈Ω∆).Function φ is called x. It is assumed in the following form:In the laminates under consideration, this function is a continuous, linear function along the thickness of every sub-lamina, dependent only on the argument x) ≈ 0  ≈ 0 for every x∈ΩΔ, where ρ > 0 is a certain tolerance-periodic function).Because the distribution function of non-homogeneity ν is slowly varying, it can be shown that the averaged value of the function φ is equal to zero in every lamina, <φ> ≅ ψ can be used, according to which quantities of the order O(δ) are negligibly small compared to 1.In the second assumption, called The tolerance modelling procedure was proposed in different forms ,64,65.r(·) within the macro-dynamics, which is defined in the formHere, it is applied similarly to that shown in . The firthe residual orthogonality condition is formulated; it is assumed that the residual field r(·) has to satisfy the following conditions:In the next step, u and the fluctuation amplitude v can be obtained.Then, in applying the conditions (7), combined together with the modelling assumptions, a system of governing equations for the macro-displacement u and v are derived:From the use of the residual orthogonality condition (7) with the tolerance modelling assumptions, the averaged equations of the laminates under consideration for the tolerance model of functionally graded laminates (FGL) with tolerance-periodic microstructure. The above equations have coefficients, being slowly varying functions in x. Moreover, some of the coefficients, underlined here, depend on the parameter λ, determining the thickness of laminas. Hence, the microstructure effect is described by Equation (8) on the overall dynamic behaviour of the considered laminates. Equation (8) is the system of differential equations for the basic unknown factors: the macro-displacement u and the fluctuation amplitude v, which have to be slowly varying functions in x. It can be observed that boundary conditions should be formulated at x = ±h1 only for the macro-displacement u, but on all other edges (at z = const) for every unknown: the macro-displacement u and the fluctuation amplitude v.Equation (8) with the decomposition (5) stands the system of equations of the asymptotic model of functionally graded laminates (FGL) with tolerance-periodic microstructure, omitting the microstructure effect on the overall behaviour of the laminated composites. The above equations have coefficients being slowly-varying functions in x, similarly to Equation (8) for the tolerance model, in the contrast to Equation (1), which have highly oscillating, tolerance-periodic non-continuous, functional coefficients.Results obtained in the framework of the above model derived using the tolerance modelling method can be evaluated by results from the approximate model. Equations of this approximate model neglect the microstructure effect. These equations can be derived using the proper asymptotic modelling procedure . Howeverh2 along the z-axis (z = x2), loaded on the upper boundary by p and rested on the undeformable base are considered. The layer has the length 2h1 along the x-axis (x = x1) and is non-periodically reinforced along this axis. It is made of two components, f = 0.As an example, vibrations of a layer with the thickness u1 and the fluctuation amplitude v1, the following boundary conditions on the boundaries x = ±h1:u1, v1 are equal to zero.Assuming for components of the macro-displacement vector x, i.e.,:Let us assume other components as independent of the argument The tolerance modelUnder assumptions and denotations from - macro-vibrations equation- micro-vibrations equationu describing only macro-vibrations, and one (11) for the fluctuation amplitude v describing only micro-vibrations.The asymptotic modelIt can be observed that for the considered problem of vibrations for FGL layer, Equation (8) can be written as two independent equations—one (10) for the macro-displacement u and has a form identical as Equation (10). Hence, the obtained Equation (12) describes only the macro-vibrations.Under assumptions and denotations from Let us assume the following parameters:- the cell distribution function- for which the properties distribution functions can be written in the form:x, called the macro-loading amplitude, the micro-loading amplitude, respectively; R(x) is an oscillating function.- the loading on the upper boundary:U(z) is the amplitude of the macro-displacement, called the macro-deflection, and V(z) is the amplitude of the fluctuation amplitude, called the fluctuation variable.The basic unknowns in Equations (10) and (11) are assumed asLet us introduce some denotations:the macro-vibrations equationThen let us assume the loadings amplitudes on the upper boundary as- on the upper boundary:- on the bottom boundary:the micro-vibrations equation- on the upper boundary:- on the bottom boundary:Now, the equations of vibrations (23) and (26) with the boundary conditions can be transformed to the non-dimensional form.Introduce non-dimensional parameters:Defining the non-dimensional unknowns asthe non-dimensional macro-vibrations equationthe equations of vibrations (23) and (26) take the following non-dimensional form:the non-dimensional micro-vibrations equationwith the boundary conditions:For the macro-vibrations described by Equation (31) the following solutions are obtained:2 = 0:- the linearly decaying solution for Ω2 > 0:- the oscillating solution for Ω- the resonance for:For the micro-vibrations described by Equation (33), the following solutions are obtained:2 < 0, μ2 = −γ2 = 12χ/η2 − Ω2:- the decaying solution for γ2 = 0:- the linearly decaying solution for γ2 > 0:- the oscillating solution for γ- the resonance for:z/h2.Some results calculated from Formulas (35), (36) and (38)–(40) are presented in From the solutions (35) and (36) and the above figure it can be observed that for the problem of vibrations for the FGL layer under consideration there is one linearly decaying solution of the non-dimensional macro-deflection (for Ω = 0); and are many solutions, which are decaying and oscillating , and are also many solutions, which are decaying and oscillating (for γ2 > 0). Moreover, for higher values of μ, e.g., μ = 5, the solutions become strongly decaying, and for higher values of γ, e.g., γ = 5, the solutions become strongly oscillating. The results shown in Results calculated from the solutions (38)–(40), and presented in The tolerance modelling method applied to the known differential equation of composites replaced this equation with tolerance-periodic (non-periodic), non-continuous coefficients by the system of governing differential equations with continuous, smooth, slowly-varying coefficients.In this paper, vibrations of functionally graded laminated (FGL) composites with tolerance-periodic (non-periodic) microstructures have been considered. The microstructure effect is taken into account in the obtained averaged tolerance model equations on the overall dynamic behaviour of the composites under consideration. Thus, in these problems, this effect can be studied both at the macro and microlevels within the tolerance model.The asymptotic model has been also introduced to evaluate the obtained results within the tolerance model. The governing equations of this approximate model could be derived using the asymptotic modelling procedure. This model, omitting the microstructure effect, allows the investigation of vibration problems only at the macrolevel.u2 = u and the fluctuation amplitude v2 = v), are independent of x1 = x coordinate.In the example, it has been determined that the property distribution function in the FGL layer loaded on the upper boundary, under the condition, and the basic unknowns (the macro-displacement It has been shown that only the tolerance model makes it possible to investigate the effect of the micro-oscillations of the boundary loading, and also the effect of macro-oscillations.The asymptotic model allows for the analysis of only the effects of the macro-oscillations of the loading.Introduced various additional assumptions and restrictions make it possible to obtain the analytical formulas of solutions.It can be observed that the solutions of macro-vibrations can decay linearly, decay and oscillate, or they not exist.Solutions of micro-vibrations can decay exponentially, decay linearly, decay and oscillate, or they not exist.From the example, some specific remarks can be formulated.Other more complicated and interesting problems will be investigated using the tolerance model equations of the considered FGL composites in forthcoming papers.


Games have a long history as benchmarks for progress in artificial intelligence. Approaches using search and learning produced strong performance across many perfect information games, and approaches using game-theoretic reasoning and learning demonstrated strong performance for specific imperfect information poker variants. We introduce Student of Games, a general-purpose algorithm that unifies previous approaches, combining guided search, self-play learning, and game-theoretic reasoning. Student of Games achieves strong empirical performance in large perfect and imperfect information games—an important step toward truly general algorithms for arbitrary environments. We prove that Student of Games is sound, converging to perfect play as available computation and approximation capacity increases. Student of Games reaches strong performance in chess and Go, beats the strongest openly available agent in heads-up no-limit Texas hold’em poker, and defeats the state-of-the-art agent in Scotland Yard, an imperfect information game that illustrates the value of guided search, learning, and game-theoretic reasoning. Student of Games combines search, learning, and game-theoretic reasoning to play chess, go, poker, and Scotland Yard. In the 1950s, Arthur L. Samuel developed a checkers-playing program that used what is now called minimax search  and “rote learning” to improve its evaluation function via self-play , an algorithm that generalizes the class of games in which strong performance can be achieved using self-play learning, search, and game-theoretic reasoning. SoG uses growing-tree counterfactual regret minimization (GT-CFR): an anytime local search that builds subgames nonuniformly, expanding the tree toward the most relevant future states while iteratively refining values and policies. In addition, SoG uses sound self-play: a learning procedure that trains value-and-policy networks using both game outcomes and recursive sub-searches applied to situations that arose in previous searches.SoG achieves strong performance in multiple challenge domains with both perfect and imperfect information—an important step toward truly general algorithms that can learn in arbitrary environments. Applications of traditional search suffer well-known problems in imperfect information games  × Δ[𝒮2(spub)] is a pair of distributions over possible information states for both players representing the beliefs over information states in spub. A basic depiction of the various components of a public belief state is depicted in Imperfect information games introduce additional complexity, as 𝒮1, π2). Denote the expected utility to player to player i as iu and −i as the opponent of player i. A best response to a specific opponent policy πi− is any policy i−: i∣iu = maxi′πiu}. A policy profile π is a Nash equilibrium if and only if π1 is a best response to π2 and π2 is a best response to π1. There are also approximate equilibria: π is an ɛ-Nash equilibrium if and only if i.Suppose players use a policy profile π = , Exploitability(π) = [max1′πu1 + max2′πu2]/2. Also, equilibrium strategies in two-player zero-sum games are interchangeable: If πA and πB are Nash equilibria, then . Here, we focus on the two-player zero-sum setting. To put SoG in context, we begin with a high-level description of several techniques that have been dominant in this setting, and then contrast SoG with existing work.tw, along with a heuristic evaluation function to estimate the value of states beyond the depth limit, h(wt + d), and game-theoretic reasoning to back up values  can be easily beaten by another playing the best response . The Nash equilibrium plays each action with equal probability, which minimizes the benefit of any particular counter-strategy. Similarly, in poker, knowing the opponent’s cards or their strategy could yield substantially higher expected reward, and in Scotland Yard, players have a higher chance of catching the evader if their current location is known. In these examples, players can exploit any knowledge of hidden information to play the counter-strategy resulting in higher reward. Hence, to avoid being exploited, players must act in a way that does not reveal their own private information. We call this general behavior game-theoretic reasoning because it emerges as the result of computing (approximate) minimax-optimal strategies. Game-theoretic reasoning has been paramount to the success of competitive poker AI over the last 20 years.s, ·) ∈ Δ(𝒜) for each player i at each of their information states s in a way that minimizes long-term average regret. As a result, the (appropriately weighted) average policy over all T iterations t, counterfactual values s, a) are computed for each action a ∈ 𝒜(s) and immediate regrets for not playing a, tr = s, a) − ∑a∈𝒜(s)πts, a), are computed and tabulated in a cumulative regret table storing TR = tr. A new policy is computed using regret-matching , if ∑a[tR]+ > 0, or the uniform distribution otherwise.One algorithm for computing approximate optimal strategies is counterfactual regret minimization (CFR)  = [Qt−1 + tr]+ and πt+1 = tQ/∑btQ.CFR+) is one that traverses the public tree of public states, rather than the classical extensive-form game tree of world states (and information states). Quantities required to compute counterfactual values, such as each player’s probabilities of reaching each information state under their policy , are maintained as beliefs. Finally, leaf nodes can be evaluated directly using the ranges, chance probabilities, and utilities } as well as the currently active policy {πCFR[is(h)]} from search: iv normalized by the sum of the opponent’s reach probability at is to resemble state-conditional action values, and the prior policy p obtained from the queries. The second is simply CFR’s average policy at is(h). As soon as the simulation encounters an information state is ∈ spub such that spub is added to ℒ, and visit counts are updated along nodes visited during the trajectory. Similarly to AlphaZero . Optimal policies in perfect information games can be deterministic, and expanding a single action/node is a good way to avoid unneeded computation after unpromising actions. MCTS methods are sound as long as the best action has been added, which is always true in the limit as the tree is completely filled out. In imperfect information games, optimal policies might be stochastic, having nonzero probability over multiple actions. Rather than expanding a single action, SoG thus expands the top spub. In hold’em poker, players generally take turns making actions, and a depth-limited search tree for a re-solving auxiliary game can always be deep enough to contain a state for the opponent’s action. All player actions are fully visible to both players, so the opponent’s maximum counterfactual value in the previous search tree can be used for the next re-solving auxiliary game, no matter what opponent action we are responding to. By working within the single, fixed domain of poker, these properties let DeepStack and Libratus simply retrieve the re-solving summary information from its previous search.The continual re-solving method used by DeepStack rprev. As with DeepStack, this regularization toward the previous opponent policy empirically improves the performance, and we used α = 0.5.Finally, like DeepStack, the gadget for the re-solving auxiliary game is modified by mixing in the opponent’s range from the previous search. As introduced in  be the interior of the tree: all nonleaf, nonterminal public states where GT-CFR generates a policy. Let ℱ(ℒ) be the frontier of ℒ, containing the nonterminal leaves where GT-CFR uses ɛ-noisy estimates of counterfactual values. Let U be the maximum difference in counterfactual value between any two strategies, at any information state, and A be the maximum number of actions at any information state. Then, the regret at iteration T for player i is bounded:RT,full in Theorem 1 is the gap in performance between GT-CFR iterations and the highest-value strategy. Theorem 1 shows that the average policy returned by GT-CFR converges toward a Nash equilibrium at a rate of The regret Theorem 2. Assume we have played a game using continual re-solving, with one initial solve and D re-solving steps. Each solving or re-solving step finds an approximate Nash equilibrium through T iterations of GT-CFR using an ɛ-noisy value function, public states are never removed from the search tree, the maximum interior size ∑spub∈𝒩(ℒT)∣𝒮i(spub)∣ of the tree is always bounded by N, the frontier size of the tree is always bounded by F, the maximum number of actions at any information states is A, and the maximum difference in values between any two strategies is U. The exploitability of the final strategy is then bounded by Theorem 2 is similar to Theorem 1 of (T iterations expanding k children is 𝒪(kT2) public states visited and CVPN network calls. In the special case of perfect information games, the number of network calls can be reduced to 𝒪(T).The computational complexity of a GT-CFR re-solving step with t, in the expansion phase, GT-CFR will traverse a single trajectory through the tree to expand a leaf, and use the CVPN to evaluate the newly expanded children. This requires k network calls and a worst case of ∣tℒ∣ states visited or ⌈logb∣tℒ∣⌉ states visited in a balanced b-ary tree.At every iteration tℒ and uses the CVPN to evaluate every leaf of the tree. Because k child states are added to the tree at each iteration, ∣tℒ∣ ≤ 𝒪(kt), giving the stated bounds.In the regret update phase, GT-CFR visits every state in k = 1, each player range is a single number, and we only need to evaluate a state once because the optimal policy does not depend on the player ranges. If a state is evaluated once with a range of 1 for both players and then stored, any other belief state can by evaluated by scaling the stored result by the opponent’s ranges.In perfect information games, h0 corresponding to the start of the game and produces a sequence of histories . At time t, the agent runs a local search and then selects an action ta, and the next history ht+1 is obtained from the environment by taking action ta at th. Data for training the CVPN are collected via resulting trajectories and the individual searches.SoG generates episodes of data in self-play by running searches at each decision point. Each episode starts at the initial history θ and with searches made at previous public states along the same trajectory . This is a critical requirement for sound search  generated by sound self-play described in the “Data generation via sound self-play” section. Specifically, they are the output policies for all information states within the root public state, computed in the regret update phase of GT-CFR.T iterations of GT-CFR for all the information states within the public state that initiated the search. The specific way that the different value targets are assigned is described by the pseudocode in Supplementary Text and determined by a hyperparameter noted in table S2.Value targets are obtained in two different ways. First, the outcome of the game is used as a [TD(1)] value target for states along the main line of episodes generated by sound self-play. Second, value targets are also obtained by bootstrapping: running an instance of GT-CFR from subgames rooted at input queries. In principle, any solver could be used because any subgame rooted at β has well-defined values. Thus, this step acts much like a policy improvement operator via decomposition described in the “Imperfect information search, decomposition, and re-solving” section. Specifically, the value targets are the final counterfactual values after While the solver is computing targets for a query, it is also generating more queries itself by running GT-CFR. Some of these recursive queries are also added to the buffer for future solving so that the CVPN can produce reasonable answers for all leaves in a search, not just those on the self-play lines. As a result, at any given time, the buffer may include queries generated by search in the main self-play game or by solver-generated queries off the main line. To ensure that the buffer is not dominated by recursive queries, we set the probability of adding a new recursive query to less than 1 .T → ∞ and with very large  memory.One natural question is whether, or under what circumstances, the training process could ensure convergence to the optimal values? The answer is positive: The training process converges to the optimal values, asymptotically, as f(β) that can simply memorize the values and policy for the particular β similar to a tabular value or policy iteration algorithm except with continuous keys. For any subgame rooted at some β with a depth of 1 , the values and policies can be computed and stored for β after T iterations of the solver. This can then be applied inductively: Since CFR is deterministic, for any subgame on the first iteration of GT-CFR, a finite number of queries will be generated. Each of these queries will be solved using GT-CFR. Eventually, the query will be a specific one that is one step from the terminal state whose values can be computed exactly and stored in f(β). As this value was generated in self-play or by a query solver, and CFR is deterministic, it will produce another self-play game with the identical query, except it will load the solved value from f(β), and inductively the values will get propagated from the bottom up. Since CFR is deterministic and T is finite, these ensure that the memory requirement is not infinite despite the continuous-valued keys. Practically, the success of the training process will depend on the representational capacity and training efficacy of the function approximation .Informally, imagine an oracle function For a fully detailed description of the algorithm, including hyperparameter values and specific descriptions of each process described above, see Supplementary Text.


International Journal of Impotence Research 10.1038/s41443-021-00503-x, published online 2 December 2021Correction to: For this article parts of the text have been converted to footnotes. They read as follows:The Brussels Collaboration on Bodily Integrity [10] defines ‘medically necessary’ in the following way: ‘(1) the bodily state poses a serious, time-sensitive threat to the person’s well-being, typically due to a functional impairment in an associated somatic process, and (2) the intervention, as performed without delay, is the least harmful feasible means of changing the bodily state to one that alleviates the threat’.By ‘our’, ‘we’, and ‘us’, I mean all embodied people.


Conventional scenarios assume that K+ ions are tightly bound in the SF, and, thus, they are displaced from their energy well by ion–ion repulsion with an incoming ion. This tight coupling between entering and exiting ions has been called the “knock-on” mechanism. However, this paradigm is contradicted by experimental data measuring the water–ion flux coupling ratio, demonstrating fewer ion occupancies. Here, the results of molecular dynamics simulations of permeation through the KcsA potassium channel revealed an alternative mechanism. In the aligned ions in the SF (an ion queue), the outermost K+ was readily and spontaneously released toward the extracellular space, and the affinity of the relevant ion was ~ 50 mM. Based on this low-affinity regime, a simple queueing mechanism described by loose coupling of entering and exiting ions is proposed.The mechanism underlying ion permeation through potassium channels still remains controversial. KThe online version of this article (10.1007/s12576-019-00706-4) contains supplementary material, which is available to authorized users. The CRw-i metric provides the ion and water permeation pattern in the SF, and the failure of the canonical  mode at low [K+] (< 0.01 M) cannot be overlooked since this may threaten the present “knock-on” and “high-affinity” paradigm.In the knock-on process, the alternating array mode , raising questions as to the fundamental assumption about the permeation process  = 0.15 M). Color traces, denoting individual ions, reveal the progression of ions from the intracellular space (IS) into the nanocavity (NC), followed by progression through the successive sites within the SF before being released into the extracellular space (ES). K+ ions frequently access the entrance of the NC, but only a fraction of them enter the NC. Thus, a water-filled NC is mostly empty of ions and transiently holds up to one K+ ion  , thus warranting further examination of the microscopic details of the permeation process.To examine the concentration-dependent changes in + ion into the SF from the NC  , and two other ions, which previously underwent SF-in and remained in the SF, are denoted as I1 (blue) and I2 (green). The trajectories of the water molecules are drawn in light blue. A snapshot taken at the SF-in moment of I0 (Δt = 0) . Occasionally, zero or two water molecules are intercalated  Fig. c 48]. I. I+ ion I0, I1, and I2 are strongly coupled . The uppermost and lowermost panels represent the distributions at two successive SF-in moments, and the middle panels show the distribution at the “relaxed” state and the next NC-in moment. The number labeling each peak represents the fractional number of the corresponding ion  at the relevant site.The permeation process is quantitatively evaluated by analyzing the time-lapse distributions of ions and water molecules. Figure +] = 0.15 M  and water molecules upstream from I1 are mostly located at S4, where the number of water molecules is 0.86. In single-file permeation, water molecules upstream of I1 are intercalated between I1 and I0 upon the SF-in of I0. The cumulative sum of water-molecule distributions upstream from I1 at S4 and S3 is counted as Nwater = 0.86 + 0.30 = 1.16 at the SF-in moment. This number of water molecules is driven by the ion flux along the channel, leading to alternating (1:1) ion and water distributions within the SF. Thus, the microscopically observed Nwater at the SF-in corresponds to the overall number of water and ion flux ratio during the whole simulation time (CRw–i = 0.99) as well as to the experimentally measured CRw–i values . Thus, the alternating array mode predominates above the [K+] of 0.15 M.At the SF-in moment at [K+] (= 0.05 M) , which is consistent with the experimental CRw–i value. More water molecules were intercalated because I1 shifted outward such that more than half of I1 was located at S2 or S1 (57% = 50% + 7%). Why was I1 shifted further downward at low [K+]? We found that nearly half of I2 had already been released out to the ES (49%) before the SF-in, retaining only one K+ in the SF. In this case, permeation proceeded without passing through the three-ion-occupied knock-on intermediate, posing a question as to the prerequisite role of the transition intermediate for permeation. After the SF-in, two ions (I0 and I1) shifted gradually until the relaxed state, during which half of I1 was spontaneously released . Although this affinity represents an attribute of the outermost ion in the two-ion-occupied state rather than an attribute of a specific site in the SF, this value is consistent with the experimental Kd values of 0.009–0.050 M . From the collection of the trajectories, the elapsed time from the SF-in to the release of either I2 (upper panel) or I1 (lower panel) was plotted as dots , and this de-queueing time is invariable for different concentrations. This immediate alleviation from the transient three-ion-occupied intermediate was expected to be a knock-on scenario, while the time course was first elucidated in this simulation. Another type of de-queueing is spontaneous release (2 → 1: lower right panel), which is significant here for the sparse array mode to proceed. This de-queueing occurs long after the en-queueing (SF-in), with an elapsed time of 16,000 ps at 0.05 M, which is in dramatic contrast with the de-queueing time for the 3 → 2 transition. Hence, such long-overdue time for de-queueing is poorly expressed as the coupling between en-queueing and de-queueing, and the exit process of 2 → 1 is accordingly called the spontaneous release process. During the period of spontaneous release, I0 advances deeper and more water molecules follow, providing higher Nwater or CRw–i values. This mode is frequently seen at the lower concentration of 0.05 M, occurring as high as in half of the cases (51%). Even at high concentrations, the sparse array mode was observed, though infrequently (5% at 1 M).Now, a quantitative evaluation of the MD-simulated trajectories was performed. First, the exit (de-queueing) time was quantified , which are the inverse of the current amplitudes. The en-queueing times for 1 → 2 seemed to be similar to those for 2 → 3 because the arrival time of the ion at the SF was independent of whether the SF occupied either one or two ions.Next, the arrival (en-queueing) time was considered (left panels). The en-queueing occurred upon the SF-in event, and, thus, the en-queueing time could be read with the elapsed time from the less ion-occupied state to the SF-in, which is shown as dots and as the kernel density estimation . The average time for en-queueing of 2 → 3 (upper left panel) gradually prolonged as   the affinity of the outermost ion in the SF is low, and, thus, dynamic equilibrium with respect to the ES governs the occupancy of that ion Fig. b; (ii) t+] Fig. b; (iii) +] Fig. b, c;  because of the geometrical length limit. At low [K+], ion arrival is infrequent and queueing processing in the SF (or de-queueing) is rapid  enough to maintain a short queue. As the frequency of arrival ions increases, a fraction of arriving ions are not allowed to enter the channel because of the limited queue capacity. When the queue elongates up to three ions, the processing rate of the queue is accelerated.Under the framework of the general queueing theory , 54, per+ concentration, whereas the de-queueing rate was constant. In this study, we found that the de-queueing rate of 2 → 1 was high (red line) and the affinity of K+ was deduced as 0.05 M at the intersection of the en-queueing and de-queueing rates. This value is consistent with the experimental values (0.009–0.050 M) [+] below the Kd value, de-queueing (2 → 1) occurs earlier than en-queueing, which occurs long afterward (1 → 2), giving rise to the sparse array mode. At [K+] above the Kd value, en-queueing precedes de-queueing. Once en-queuing (2 → 3) occurs, it immediately de-queues (3 → 2), yielding the predominant alternative array mode. Here, the rate of 3 → 2 deduced from Fig. To characterize the queueing mechanism in comparison to the conventional knock-on mechanism, we expanded Fig. 0.050 M) –52, 55. + binding (< 0.001 M) is implicitly assumed to be much below the physiological extracellular [K+] (0.003–0.005 M), from which the de-queueing rate is estimated as the intersection with the en-queueing rate , using the same potential and channel species as the ones we used, although the driving force was imposed by the concentration gradient. A recent paper by Kopec et al. showed that the CRw–i value through the MthK channel was also 0 at 220–280 mV, and was ∼ 0.1 at ∼ 900 mV  across the membrane, more fractions (> 15%) of permeation should occur via the sparse mode.The + affinity  = 0.15 M was composed of one KcsA channel, 119 dioleoylphosphatidylcholine (DOPC) lipids, 30 K+, 22 Cl−, and 10,520 water molecules. First, the DOPC bilayer was equilibrated with the surrounding KCl solution for 10 ns. MD simulation was performed using the NPT ensemble  with a Berendsen’s barostat [2)]. Simulation of the equilibration using the NVT ensemble (310 K) with a Berendsen’s thermostat [+ and Cl−; the system was then equilibrated.The system at [Kbarostat . The arebarostat . The DOPbarostat . H25, E1barostat . The resbarostat , 61. An ermostat  continue+ ions, and Siu model for the lipids [+–w–K+–w, w-K+–w–K+, K+–w–w–K+, and K+–V–K+–w, where V stands for vacancy) were generated as initial configurations. The initial equilibrium runs, including the ion and water molecule permeation events in the process, were omitted from the analyses.The following empirical potentials were used: the TIP3P model for water, ff94 force field for the channel, Dang model for the Ke lipids , 62–64. e lipids , 66. Ten3) and a constant temperature (310 K) using the Berendsen’s thermostat [MD simulations were performed at a constant volume Supplementary material 2 (MP4 7421 kb)Below is the link to the electronic supplementary material.


This research proposes a deep-learning paradigm, termed functional learning (FL), to physically train a loose neuron array, a group of non-handcrafted, non-differentiable, and loosely connected physical neurons whose connections and gradients are beyond explicit expression. The paradigm targets training non-differentiable hardware, and therefore solves many interdisciplinary challenges at once: the precise modeling and control of high-dimensional systems, the on-site calibration of multimodal hardware imperfectness, and the end-to-end training of non-differentiable and modeless physical neurons through implicit gradient propagation. It offers a methodology to build hardware without handcrafted design, strict fabrication, and precise assembling, thus forging paths for hardware design, chip manufacturing, physical neuron training, and system control. In addition, the functional learning paradigm is numerically and physically verified with an original light field neural network (LFNN). It realizes a programmable incoherent optical neural network, a well-known challenge that delivers light-speed, high-bandwidth, and power-efficient neural network inference via processing parallel visible light signals in the free space. As a promising supplement to existing power- and bandwidth-constrained digital neural networks, light field neural network has various potential applications: brain-inspired optical computation, high-bandwidth power-efficient neural network inference, and light-speed programmable lens/displays/detectors that operate in visible light. Here the authors have realized a programmable incoherent optical neural network that delivers light-speed, high-bandwidth, and power-efficient neural network inference via processing parallel visible light signals in the free space. Its primary process is to train multiple layers of artificial neural networks to find abstractions in domain-specific datasets. The universal approximation theorem states that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact subsets of Rn under mild assumptions pertaining to the activation function3.Deep learning has recently experienced tremendous success in delivering impressive results in various domains, including computer vision, medical analysis, natural language processing, and automatic vehicles7. However, building a programmable ONN using electronically controlled optical components remains an open problem. Note that while electronically controlled optical components are commonly modeled as perfect surfaces, they actually have inconsistently unmeasurable micro-structures and materials, creating unpredictable inter-reflections and interference. In addition to the optics, the electronic circuit and physical environment also introduce unpredictable bias and variance. As such, existing approaches can no longer properly simulate or train programmable ONNs.Modern electronic devices offer strong computing capabilities for the processing of laborious matrix multiplications and memory references, leading to the development of advanced networks with ever-growing amounts of neuron connections. However, the stringent constraints on power and bandwidth hinder the broader applications of deep learning in mobile devices, embedded systems, wireless wearable devices, and robots. The optical neural network (ONN) is a promising alternative mechanism for high-bandwidth, low-latency, and power-efficient neural network inferenceIn this work, we conceive the concept of practically training loose neuron arrays for practical tasks. It is a physical counterpart of the artificial neural network and enables the transformation from handcrafted hardware designs to non-handcrafted designs. One of the most important recent discoveries is that merely connecting many artificial neurons in a non-handcrafted way outperforms sophisticated handcrafted algorithms in a large number of tasks. While connecting artificial neurons is very handy in computers, there is no trivial way to connect and train arbitrary physical neurons because the actual design, material, structure, manufacture, fabrication, and run-time environment are far from ideal models. On the other hand, enabling the connection and training of arbitrary physical neurons releases us from such constraints. It reveals possibilities for hardware design, chip manufacturing, and system control.This concept is instantiated for the realization of incoherent programmable ONNs. We numerically verify the feasibility of various neuron arrays with liquid-crystal (LC) hardware to form programmable ONNs Fig. . The neuWe physically implement a regular-2 array with off-the-shelf optical components, e.g., liquid-crystal display (LCD) panels, polarizers, and a camera, without fine calibration. We call such a programmable incoherent ONN design a light field neural network  is the optics of the LFNN hardware, p is the hardware control parameters, x is the input, and y is the expected output. At a specific inference task, once p is applied to the hardware, the incoherent light signal passing through the device is passively modulated to the target output.Instead, we propose the FL paradigm to control such an uncalibrated optical system. Given the input signals and expected targets as the training set, the parameters of the system are trained as trainable neurons to undertake inference on the input signal:f has no explicit mathematical formulation, we introduce a functional neural network  is an FNN, z denotes the functional space coefficients, p is the primary parameter as trainable neurons, and the two terms are iteratively optimized by p − learning and z − learning through the FL paradigm.Because 2. The function basis block reflects the physical structure of the device in which each input neuron has a potential impact, i.e., light propagation, to each output neural, forming a four-dimensional light field with numerous connections, whose cardinality is equal to input neural size times output neural sizes. The pixels of the LC panel, represented as control parameters, can potentially attenuate arbitrary input-to-output connection and are represented mathematically by multiplication to the input-to-output data flow. In order to depict features in various resolutions, we downsample the inputs and LC control parameters half and merge the results through trainable parameters. More details are in The FNN consists of a physically-inspired function basis block and the following nonlinear combination through five layers of convolutional neural network (CNN) with 3 × 3 × 64 kernel and rectifier linear unit (ReLU)8 to train and test a 1-layer LFNN. The input of the device is a 28 × 28 grayscale image, and the output is the predicted classification probability distribution of the input image, calculated by accumulating the sum of RGB channels at the output plane. Because the viewing angle of our backlight is too narrow to make the contribution of an input neuron cover the entire output plane, we train the probability distribution layout of different classes on the output plane to adapt to the uneven energy distribution. For example, the sum of the intensities in some dispersed areas represents the probability of one class  of every input neuron to the output plane. Then we measure and model the impact of every LC neuron on every input neuron’s PSF as a linear function by switching on and off the corresponding neurons and recording the change on the output plane. In addition, we build look-up tables to calibrate the linearity of input and LC neurons by measuring the system output of the quantized control signal. Training the LFNN with the forward model achieves a 23.50% prediction accuracy. The limitation of this paradigm is that accurately measuring and modeling the tremendous amount of states of the LFNN, i.e., a high-dimensional, non-differentiable, non-linear, and correlated physical system, is beyond reality.In the MNIST test set, the LFNN achieves a 91.02% prediction accuracy, and the FNN predicted accuracy is 91.39%. As a reference, one fully connected digital dense layer running in a computer achieves 92.71% accuracy. In comparison, existing training paradigms have a hard time training a large number of parameters of a high-dimensional system as the LFNN. Using the forward analytical model6 and genetic algorithm11 may work well for a small number of parameters, but are still hard to train a large number of parameters. The finite difference paradigm captures the difference quotient as gradients to optimize the parameters. While facing thousands of parameters, it has to measure the finite difference of every parameter, which requires capturing 6,144 output images from the LFNN for one input image. In contrast, our FNN paradigm captures 1024 output images to implicitly evaluate the gradients for 60,000 images at each epoch. As a result, using finite difference yields an 8.594% prediction accuracy within equal training time. Genetic algorithm faces a similar problem for large-scale optimization. While converging efficiently with a small number of parameters, stochastically exploring a high-dimensional space without gradients is futile. We leverage the scikit-opt genetic algorithm to test training the LFNN. The metric of evaluating an offspring is calculated by randomly drawing 128 images and capturing the outputs. Within equal training time, the genetic algorithm trains the LFNN to achieve a 14.06% prediction accuracy. Table Alternative stochastic training paradigms like finite difference12, followed by the third operation, a simple amplitude filter integrated into the driver of the input plane that clamps the impulse between zero and one. We connected multiple FNNs to train the multi-layer LFNN with the adaption of the single-layer training scheme to multiple layers. More discussion is given in the We also test multi-layer optical neural networks by cyclically feeding the captured outputs onto the input plane. This electronic cycling also conducts energy redistribution and nonlinear activation. We apply three neuron-wise activation operations. We refer to the first operation as X-activation, where the captured output is subtracted from its 180-degree rotated image. X-activation effectively enables consistent value distributions between layers, nonlinearities, and deactivation, i.e., negative operations that are non-trivial in incoherent optical systems. Furthermore, its design uses only simple logic and operators that can be implemented by non-digital hardware in the future to match the activation to speed-of-light processing. The second operation is common neuron-wise batch normalizationWith the 2-layer LFNN, the MNIST classification task shows a 94.35% prediction accuracy, and its FNN counterpart shows 94.52% accuracy. As a reference, a 2-layer digital dense neural network, connected by the ReLU activation function achieves 98.32% accuracy.13, to evaluate a 3-layer LFNN. The FNN predicted accuracy is 46.19%, and the LFNN captured accuracy is 45.62%. As a reference, a 3-layer digital dense neural network achieves 53.52% accuracy. In the neuron array simulation, increasing the LC neuron size can significantly increase the accuracy of ONNs, nearly matching that of the digital dense neural network  to implement the LFNN prototype, the average training time is 4 minutes per epoch. For example, it takes around 400 minutes to train the 100 epochs  has no explicit formulation, we cannot mathematically solve the original problem of Equation  is an implicit function that can be represented by a functional neural network (FNN) as an instance with a learnable functional space. We then approximate the original problem with two sub-problems,gN( ⋅ ) is a FNN , z is the functional space coefficients, and p is the primary parameter, i.e., the hardware parameter in our case. However, both sub-problems are under-determined if we simultaneously optimize the functional space and the parameters that alter the mapping between input and output. Therefore we reformulate the equation as the following, by decoupling the variables to make each sub-problem solvable:p0 and z0 are auxiliary variables. In this formulation, the problem can be solved by alternating minimization, which has shown convergence for non-convex problems30, of each sub-problem. Specifically, at each epoch k, we update p0 ← pk and z0 ← zk+1. The update rules can be then simplified as follows:The optimization of the parameters of a model-free system is challenging. In order to train the hardware parameters of the LFNN, we propose a deep-learning paradigm, functional learning. Specifically, we generalize the learning problem asz-learning and p-learning, respectively.We propose different deep learning schemes to minimize sub-problems of Equation  and EquaFunctional Learning (FL) Paradigm.functionFunctional Learning1: 2:z and p               ⊳ Initialize.3:  Initialize p-optimizer and z-optimizer using gN4:  Initialize p-data ← 5:  k ← 16:  7:whilek <  maximum epoch do8:  u ← p-data. DrawSample(x)9:    v ← LFNN. Capture(u)10:  z-data ←               ⊳ Update z-data.11:   z-optimizer. TrainNetwork  ⊳ Conduct z-learning.12:  p-optimizer. TrainNetwork  ⊳ Conduct p-learning.13:  k ← k + 114:  end while15:  16:end function17: p-variable. The orange lines and the variables of convolutional layers, termed z-variable, are trainable functional space coefficients trained to replace the hidden function f. The p- and z-variables are iteratively updated through p-learning and z-learning. P-learning accepts task-related data sets, e.g., CIFAR10, to optimize the primary parameter. Z-learning dynamically gathers z-data by sending impulses to the LFNN device to capture ground truth (GT) responses. The gradients are computed through backpropagation algorithms, and the variables are updated using the Adam optimizer31 in the PyTorch framework.The FL paradigm departs from existing deep learning paradigms in several ways. The FNN has two sets of trainable variables indicated here in blue and orange Fig. a. The blz-learning and p-learning. Each learning procedure has independent loss functions, optimizers (z-optimizer and p-optimizer), trainable variables (z-variablez and p-variablep), and data set (z-data and p-data). First, we uniformly randomize the variables z and p in . Through the FNN gN, p-optimizer calculates the gradients of p using backpropagation and updates the variables p using the Adam algorithm in the p-learning stage, and z-optimizer is the counterpart for z. At each epoch, z-learning and p-learning are conducted once. The z-data used for z-learning is updated at each epoch by randomly drawing certain input samples from p-data and capturing the corresponding responses of the LFNN device.Algorithm 1 shows the pseudocode of the FL paradigm. Functional learning (FL) uses a weaved learning scheme, which iteratively conducts P-learning. p-learning trains the parameter neurons p, i.e., the hardware parameters for the LFNN device, to optimize the mapping from x to y in a specific function defined by the frozen functional space variables z. The training data set p − data and the loss function depend on the chosen inference task. We use the Adam optimizer with a 0.001 learning rate.Z-learning. z-learning trains the functional space variables z to approximate the unknown function f with gN(z) given a fixed p. Different from classic deep learning, the training data set z-data is updated at each epoch. We give some input signals to the unknown function f, i.e., the LFNN device in our case, and capture its responses. The input signals and output responses constitute the inputs and targets of z-data. Because we use an off-the-shelf low-speed camera to implement our LFNN prototype, capturing the responses of all x of p-data is time-consuming, we only randomly draw a subset with 1024 samples (line 9 of Algorithm 1) every epoch and repeatedly train such a subset ten times while shuffling. This acceleration scheme works well experimentally across the training process. We use the L1 loss and the Adam optimizer with a 0.001 learning rate for z-learning.n-layer LFNN, Equation  Paradigm.functionMulti-Layer Functional Learning1: 2:zi} and {pi} for 1 ≤ i ≤ n ⊳ Initialize.3:  Initialize {p-optimizer and {z-optimizeri} using gN for 1 ≤ i ≤ n4:  Initialize p-data ← 5:  k ← 16:  7:whilek < maximum epoch do8:  forj ← 1 to ndo9:   if j = 1 then10:    uj ← p-data. DrawSample(x)11:     else12:    uj ← vj−113:     end if14:    vj ← LFNN. Capture(uj)15:    z-dataj ←  ⊳ Update z-data.16:    z-optimizerj. TrainNetwork ⊳ Conduct z-learning.17:    p-optimizer. TrainNetwork ⊳ Conduct p-learning.18:      k ← k + 119:     end for20:    end while21:     22:end function23: p-variable, orange represents the functional space variables, z-variable, and black represents fixed connections or neurons that gather inputs with specified operators. z-variable and p-variable are iteratively updated by z-learning and p-learning, respectively.Figure p-variable that represents the hardware parameter of the input plane’s neuron-wise gain, and z-variable that is trained to optimize the functional space. An LC neuron may affect the connection between an input neuron and an output neuron. Thus, we connect every LC neuron to the connections of the light field between the input and output plane  with a 3 × 3 × 64 kernel and the rectifier linear unit (ReLU)2. The function is to non-linearly combine the functional basis block for fitting the non-linear part of the actual hardware. In addition, the output neurons are also weighted by the hardware parameter of the output plane’s neuron-wise gain. The intuition is similar to that of an attention mechanism.The functional basis block consists of various neurons inspired by the light field relationship between the input, LC, and output neurons. An input neuron can make a possible contribution to every output neuron and thus is connected to every output neuron to form a light field Fig. b. The coane Fig. c. The coz-variable.Because the signal passes through some relatively dim and noisy connections, we merge nearby neurons’ connections to build stronger connections with less noise. As shown in Fig. In practice, many connections of LC neurons, especially at the finest resolution, are very weak and therefore can be approximated by connections at coarser resolutions. Removing those connections from the FNN can save memory and accelerate the training. We measure the impacts of connections by alternating the LC neuron parameter between 0 and 1 and capturing the energy change on the output plane. If the energy change of a connection between an input and output neuron pair w.r.t. an LC neuron is below a threshold, we remove that LC neuron’s connection to that pair. For the finest resolution, we experimentally set the threshold to 20% of the maximum energy of valid input and output neuron connections. For the second finest resolution, we experimentally set the threshold to 10%. For the coarsest resolution, we keep all connections.Because there are billions of connections to test and conducting a one-by-one test is time-consuming, we build quad-trees in the neurons of an LC plane and hierarchically perform the test. Figure In summary, this sparsification scheme prunes 99.96% of the LC neuron connections for our LFNN implementation.12. Finally, a simple amplitude filter integrated into the driver of the input plane clamps the impulse between zero and one and also introduces non-linearity.In order to capture multiple-layer LFNN results, we cycle the captured output to the input of the next LFNN layer and conduct certain activation functions with the signal. These activation functions are currently performed on a computer but can be implemented by all-optical or non-digital photoelectric hardware in the future. First, we design a special activation, termed X-activation, to enhance the inference capability of the multi-layer ONN. Specifically, the captured output is subtracted by its 180-degree rotated image. The subsequent X-activation is a common neuron-wise batch normalization methodThis design of X-activation aims for four different targets. First, it enables the deactivation in optical neural networks. The data flow of common artificial neuron networks contains both positive and negative values, but there is no negative light in an incoherent optical system. By introducing X-activation, a neuron can either activate or deactivate an arbitrary subsequent neuron with different LC neuron parameters Fig. c. SecondWe physically prototype a loose neuron array, termed light field neural network (LFNN), to realize the regular-2 array and Bernoulli array by randomly disabling neurons. We use off-the-shelf components, e.g., liquid crystal display (LCD) panels, polarizers, and a machine vision camera, without tedious calibration. As a result, the actual LFNN is not exactly the same as the regular-2 array.μm, the resolution of 1600 × 1200, and the bit depth 12 bit. The focal length and F-number of the camera are 12.5 mm and 1.6, respectively. Because the device is not fine-tuned and the planes have different valid areas, we use the pixels in the center areas of these planes as neurons. Specifically, 15 × 13 pixels in the input plane, liquid crystal 1 (LC1), and liquid crystal 2 (LC2) are jointly controlled as one input neuron and LC neurons. 12 × 15 pixels in the camera’s image plane are read together as one output neuron.The applied LCD panels are expected to show a high transmittance and good linearity between the applied voltage and the polarization rotation angle. Among available LCD panels, we use Chimei Innolux AT070TN83 as the optical layer. The photograph of our prototype is shown in Fig. In our experiment, we use two standard classification datasets. The MNIST dataset contains 60,000 training images and 10,000 test images belonging to 10 classes of handwritten digits. From 0 to 9, each class contains 5,923(980), 6,742, 5,958, 6,131, 5,842(982), 5,421(892), 5,918(958), 6,265, 5,851(974), and 5,949 training(test) samples, respectively. The CIFAR10 dataset consists of 60,000 32 × 32 color images in 10 classes of real-world objects, with 6,000 images per class. There are 5,000 training images and 1,000 test images in each class. The classes include airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck.Supplementary DocumentPeer review file