Drosophila melanogaster wing imaginal disc by Wingless morphogen signalling, where multiple endocytic pathways participate in decoding the morphogen gradient. The geometry of the inference landscape in the high dimensional space of parameters provides a measure for robustness and delineates stiff and sloppy directions. This distributed information processing at the scale of the cell highlights how local cell autonomous control facilitates global tissue scale design.Precise spatial patterning of cell fate during morphogenesis requires accurate inference of cellular position. In making such inferences from morphogen profiles, cells must contend with inherent stochasticity in morphogen production, transport, sensing and signalling. Motivated by the multitude of signalling mechanisms in various developmental contexts, we show how cells may utilise multiple tiers of processing  and parallel branches (multiple receptor types), together with feedback control, to bring about fidelity in morphogenetic decoding of their positions within a developing tissue. By simultaneously deploying specific and nonspecific receptors, cells achieve a more accurate and robust inference. We explore these ideas in the patterning of  Drosophila embryo Precise positioning of cell fates and cell fate boundaries in a developing tissue is of vital importance in ensuring a correct developmental path  and physical (e.g. pH) environments, have been invoked as regulators of biochemical reactions during cellular signalling and development We pose the task of morphogenetic decoding as a problem in local, cell autonomous inference of position from a morphogen input , where e‘reading’ of the morphogen input by receptors on the cell surface,‘processing’ by various cellular mechanisms such as receptor trafficking, secondary messengers, feedback control, and‘inference’ of the cell’s position in the form of a transcriptional readout.reading of the morphogen input is associated with the binding of the morphogen ligand to various receptors with varying degree of specificity, leading to the notion that the information channel describing positional inference must possess multiple branches. Furthermore, the multiple processing steps associated with compartmentalisation of cellular biochemistry and/or signal transduction modules, for example phosphorylation states, provide the motivation for invoking multiple tiers in the channel architecture. At an abstract level, one may think of the branch-tier architecture of the cellular processing as a bipartite Markovian network/graph fast direction (involving multiple branches) consisting of ligand-bound and unbound states along with chemical state changes, and a slower direction (involving multiple tiers) consisting of intracellular transport, fission and fusion, characterised by energy-utilising processes or a flux imbalance. A general developmental context with multiple morphogens may involve several such bipartite Markov networks/graphs with different receptors (or branches) in parallel. Some of these receptors could be shared between different morphogens. We refer to signalling receptors as those which transduce a signal upon binding to their specific morphogen ligand and non-signalling receptors as those that participate in the signalling pathway without directly eliciting a signalling response. At the end of processing, each individual cell may pool information from the various branches for the final inference of position, i.e. a transcriptional readout  estimate in leads to the same qualitative results.chemical rate parameters describing the rates of various reactions in the CRN, receptor parameters describing the receptor concentration profiles, feedback topology in the CRN that is a combination of actuator and rate under regulation, control parameters describing the strength and sensitivity of the feedback/feedforward. With these parameters specified, an input-output relation, calculated as a tier-wise weighted sum of all signalling states, can then be used to infer the cell’s position by In order to calculate the probability of the inferred position given the output external input to the receiving cells, outside the cellular information processing channel. The signal and noise of this external input are captured by the distribution local, cell autonomous morphogenetic decoding that allows the cells to tune their reading dynamically.In the framework described previously, we consider the morphogen ligand as an local, cellular reading to an open-loop control on total (ligand bound plus unbound) surface availability of the signalling We subject the ill form , that isThe range of values for these parameters considered in the numerical analysis are listed in ells see .Note that we are not fixing a receptor profile but taking it from a class of monotonic profiles (including a uniform profile), over which we vary to determine the optimal inference (see Section ‘Performance of the Channel Architectures’ below). Further, in the optimisation scheme (Section ‘Performance of the Channel Architectures’), we allow the receptor concentrations to vary over the space of all monotonically increasing, decreasing or flat profiles, and do not encode the positional information in the receptor profiles. Monotonicity implicitly assumes a spatial correlation in the receptor concentrations across cells – we return to this point in the Discussion.In a single tier channel, all processing is restricted to the cell surface. We represent the bound state of the signalling receptor as The steady-state output In a multi-tiered channel, the receptors go through additional steps of processing before generating an output. We represent the bound state of a receptor in wk, such that The output, realised from all the ligand-bound signalling states, now becomes We consider all rates in the CRN, except the ligand binding and unbinding rates, as potentially under feedback regulation. Any chemical rate r0 as the reference value of the chemical rate in the absence of feedback. The range of values for amplification with With the model in place, we address the Objective discussed previously, by studying the performance of different channel architectures, i.e. number of tiers and branches, and feedback topology. We define a vector back see . While tand implemented by the following algorithm, the details of which are presented in Appendix 3.As discussed previously, cells of a developing tissue face both extrinsic as well as intrinsic sources of noise. We first look at the issue of extrinsic noise in the morphogen input .Introducing a feedback in this one-tier one-branch architecture, either on receptor levels or degradation rate, only partially reduces the inference errors . As seenbranch in the channel architecture opens up several new possibilities of feedback controls, in addition to providing an extra tuning variable. Now, as opposed to the one-tier one-branch case, an inter-branch feedback control  The signalling and non-signalling receptors present opposing optimal profiles – a consequence of the negative inter-branch feedback. (ii) The optimal non-signalling receptor decreases away from the source, indicating that the non-signalling receptor ‘reads’ the ligand input, while the signalling receptor increases away from the source, buffering the noise in the output . A heuriWe next investigate the effects of addition of tiers (compartments) on the inference errors. Our optimisation shows there are two distinct optimised two-tier two-branch architectures, one with inter-branch feedback on the internalisation rate of the non-signalling receptors e errors . Both the errors .robustness to input noise through an increase in signal-to-noise ratio (SNR) It would seem that addition of further tiers, that is more than two, would lead to further improvement in the inference. However, in both these optimised architectures, addition of tiers leads only to a marginal reduction of inference errors  while inSo far, we have only considered noise due to fluctuations in the morphogen profile, that is extrinsic noise. Given that we are considering a distributed channel, intrinsic noise due to low copy numbers of the reacting species in the CRN will have a significant influence on the inference. As discussed in Section ‘Conceptual framework and quantitative models’ and Appendix 3, we solve the stochastic chemical master equations (CMEs) to compute the output distributions and the positional inference. It is here that we find that the addition of tiers contribute significantly to reducing inference errors. A comparison of the one-tier two-branch and two-tier two-branch channel architectures  optimisefeedback . This efitecture , the fluboth decrease away from the source  receptor should decrease (increase) with mean morphogen input. Crucially, in the absence of feedback, performance of the channel diminishes and the optimised receptor profiles e source . FurtherBefore comparing the theoretical results with experiments, we comment on the implications for the cellular control of the signalling Drosophila imaginal disc specific for the various ligands while the non-signalling receptor, being promiscuous, is non-specific. This, as we see below, is the case with the Heparan sulfate proteoglycans (HSPGs) such as Dally and Dally-like protein (Dlp) that participate in the Wingless (Wg) and Decapentaplegic (Dpp) signalling networks This sloppiness in the levels of non-signalling receptor would manifest at a phenotypic level in the context of multiple morphogen inputs as in the case of fidelity landscape around the optimum, and the geometry of the low inference error states. We work with the case of the optimised one-tier two-branch channel  percent changes in inference error rameters , and  on the one-tier two-branch channel architecture with 2 optimum . As show optimum  are more optimum . To explsloppy directions of the Hessian that is along the eigenvectors with small eigenvalues. This suggests that geometry of the low inference error landscape resembles a deep valley, which is shallow along the several sloppy directions and steep along the few stiff directions.where The objective function as defined in where We optimise one-tier one-branch, one-tier two-branch and two-tier two-branch channel architectures for the inference error as defined in  compare , with si compare . HoweverDrosophila melanogasterThe phenomenology of the morphogen reading and processing of Wg in the wing imaginal disc of Wingless (Wg) is secreted by a line of cells (1–3 cells) at the dorso-ventral boundary and forms a concentration gradient across the receiving cells Since an experimental measurement of positional inference error poses difficulties, we measure the cell-to-cell variation in the signalling output for a given position ce error . Larger irection .Let us first discuss the results from the theoretical analysis. The optimised two-tier two-branch channel  shows thIn the experiments, we first established the method by determining the CV of a uniformly distributed signal, CAAX-GFP (expressed using ubiquitin promoter), and observed that the CV of CAAX-GFP is relatively uniform in ng cells . In ordecells , and another, where we disengage one of the tiers by inhibiting the second endocytic pathway using a genetically expressed dominant negative mutant of Garz, a key player in the CG endocytic pathway In principle, the coefficient of variation of the output is affected by all the microscopic stochastic processes that intersect with Wg signalling network in the wing imaginal disc and in the ligand input. Therefore, one has to be careful about interpreting the changes in the coefficient of variation of the output, based on the such perturbation experiments. Notwithstanding, this qualitative agreement between theory and experiment is encouraging.local, cell autonomous morphogenetic decoding that ensures precise inference of position, that is robust to extrinsic and intrinsic noise. We treat the cells as inference channels capable of reading and processing the morphogen input. We describe the architecture of the inference channels in terms of three elements: branches (number of receptor types), tiers (number of compartments) and feedbacks. We ask for properties of the inference channel architectures that allow for precision and robustness in the task of morphogenetic decoding of cellular position.In this paper, we have posed the problem of spatial patterning of cell fates in a developing tissue as a Taking an information theoretic and systems biology approach, we have addressed the issue of accurate and robust morphogenetic decoding of position. For convenience, we summarise our key results in a point-wise manner:multi-branch multi-tier channel architecture with feedback control. This ensures a separation between the reading of the morphogen input and buffering against noise.The main result is that given a noisy morphogen input, cells in a developing tissue can achieve low inference error of their positions by deploying a more elaborate signalling and non-signalling receptors in the channel can significantly improve the performance of cells in their positional decoding.Having a combination of increasing away from the source and the non-signalling receptor decreasing away from the source. This implies that the non-signalling receptor ‘reads’ the morphogen input, while the signalling receptor buffers against noise.For a monotonically decaying morphogen input, the signalling and non-signalling receptors exhibit spatially varying profiles with the signalling receptor inter-branch feedback provides buffering against extrinsic noise.The performance of the multi-tier multi-branch channels is enhanced by having a feedback from the signalling branch to the non-signalling branch. Along with control on the levels of signalling receptor, this cellular compartmentalisation) tempers the effects of intrinsic noise in the channel by stabilising fluctuations of the output at steady-state.Having a multi-tier architecture .Analysis of the geometry of the fidelity landscape reveals that the channel parameters corresponding to feedback, binding rates and profile of the signalling receptor are conjugated state corresponding to a confluence of the signalling and non-signalling branches in a common compartment.The efficacy of inter-branch feedback control is enabled by having a local, cell autonomous control can facilitate the optimisation of a tissue-level task, here morphogenetic decoding of cellular position.Our analysis demonstrates how Drosophila wing imaginal disc. We first show that Wg signalling in the experimental system is equivalent to a two-tier two-branch channel. In the experiments, we use signal-to-noise ratio (SNR) of the output as a proxy for robustness of inference. Perturbation of the architecture, i.e. removal of the non-signalling branch, results in reduction of SNR. In a forthcoming manuscript, we will provide a detailed verification of the predicted opposing receptor profiles.Our theoretical predictions are compared with experimental observations from Wg morphogen system of local geometry of the fidelity landscape around the optimum, and the global geometry of the low inference error states, by perturbing channel parameters and concentration profiles of the receptors.We have explored the sloppy directions and steep along the few stiff directions, the latter being predominantly along the feedback parameters. This dimensional reduction appears to be a recurring feature of such high-dimensional optimisation The local geometry of the fidelity landscape is studied using the Fisher information metric. This shows that steepest variation in the inference error comes from moving along the feedback parameters while perturbations to other channel parameters produces only marginal changes. Further, we explore the global geometry using the spectrum of the Hessian of the inference error. We find that the topography of the low inference error landscape resembles a ravine or a deep valley, which is shallow along the several control - with non-signalling receptor placed under sloppy control. In a context with multiple morphogen ligands setting up the different coordinate axes  or cell fate boundaries need to be specified with any precision. To this end, we have analysed another objective function which partitions the tissue into cell identity segments. The qualitative features of the optimised channel architectures remain unaltered. Depending on the developmental context, one might explore other objective functions. This would be a task for a future investigation.We end our discussion with a list of tasks that we would like to take up in the future. First, the information processing framework established here is very general. Obvious extensions of our models, such as adding more branches, tiers and chemical states, will not lead to qualitatively new features. However, one may alter the objective function – for instance, in the case of short range morphogens like Nodal Next, our optimisation study ignores cellular costs due to compartmentalisation, additional receptors and implementation of feedback controls, and thus possible trade-offs between cellular economy and precision in inference. Nevertheless, the observation that addition of extra tiers beyond two provides only marginal improvements to inference, already suggests a balance between precision and cellular costs.have not encoded the positional information in the receptor profiles.Third, our theoretical result that the optimised surface receptor profiles are either monotonically increasing or decreasing from the morphogen source, suggests that the surface receptor concentrations are spatially correlated across cells. Such correlations could have a mechanochemical basis, either via cell surface tension that could in turn affect internalisation rates external input to the receiving cells, outside the cellular information processing channel. There is no feedback from the output to the receptors and thus no ‘sculpting’ of the morphogen ligand profile. Morphogen ligand profiles  feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.Our editorial process produces two outputs: (i) Decision letter after peer review:eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Naama Barkai as the Senior Editor. The following individual involved in the review of your submission has agreed to reveal their identity: Marcin Zagorski (Reviewer #2).Thank you for submitting your article "Cellular compartmentalisation and receptor promiscuity as a strategy for accurate and robust inference of position during morphogenesis" for consideration by The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.Essential revisions:Both reviewers and reviewing editor agreed that no additional data is needed to support the major conclusions. Therefore a revision focused on improving clarity and presentation will be sufficient. The following two major concerns should be addressed; additional detailed comments and suggestions in the reviews below should be considered for clarity and overall readability.1) One important point to clarify is whether the presented solutions (parameters identified) are representative of a larger class of sub-optimal or optimal solutions. This can be checked by perturbing the solutions. Presumably, the authors have investigated this point already as in the SM there are parts about, robustness, sensitivity and trade-offs, so improving presentation in the main part should solve that. This will also shed more light on the generality of the obtained results.2) The important question to address is whether the choice of Eq. 5, which describes how the cell evaluates its position, influences other results in the manuscript.Reviewer #1 (Recommendations for the authors):– Extrinsic noise coming from the stochastic ligand production at the source is included in the model by letting cells along the direction parallel to the source boundary experience a spatially-fluctuating level of ligand, with statistics given in (1). From what we understand, this noise is quenched in space. Is it clear to what extent this is equivalent to the more realistic annealed disorder originating from Brownian motion and stochastic source production?– In some cases, non-specific receptors  have been shown to increase in expression levels away from the morphogen source. Can the authors comment on this observation in light of their model?– In appendix L, the local inference error for a (optimised) graded receptor expression is compared to that of a uniform receptor expression pattern. We find that the way this result is presented slightly misleading since the expression level in the uniform case is not optimised; the two setups should be compared after optimisation.Reviewer #2 (Recommendations for the authors):– In my opinion, Figure 2 is misleading and is disrupting the flow of the manuscript. Almost the same information is conveyed in Figure 4 and Figure 5. Further Figure 2 suggests some very regular arrangement of the nodes , which is not the case. Presenting a model with tiers and branches, or some different network-like schematic to indicate reaction and flux imbalances could improve presentation.– In lines 85-87, different timescales of signalling processing are mentioned corresponding to branches (fast) and tiers (slow), but this aspect of regulation seems to be not discussed in the later parts of the paper. It might be worth drawing this analogy again when discussing how noise is integrated by different architectures. Possibly, there is the separation of the timescales or signalling integration takes place on the same timescales.– Line 98, Figure 5 is mentioned just after Figure 2 (line 91). Please amend the ordering or rephrase so figures are referenced in the text as they appear in the paper.– Line 114, I would avoid statements "of course any choice consistent with experimental observations would do". For instance, would the model work for stationary wave-like patterns of morphogens that could emerge via Turing mechanism ? Basically, I would rephrase providing a description of acceptable ligand profiles .– The optimization procedure employed is computationally quite-consuming as the intrinsic noise is calculated by solving a chemical master equation. Can the model be solved without directly solving CME? What are the differences? If the differences with and without CME are small this might help to have faster optimization and hence more explicitly explore the space of available signalling architectures that result in optimal or close to the optimal solution.– The intrinsic and extrinsic noises are varied separately as far as I understand. Are there any arguments/heuristics that would indicate the resulting global solutions would be the same if two types of noise would be varied simultaneously?– In appendix C, the screening is described for the model parameters, but it is still slightly obscure what process was carried out.– In table 2, in the middle column, we see that the weight of tier 2 is orders of magnitude greater than that for tier 1. In this case, what role does the first tier really play? And does such a strong weighting make sense biologically?– I don't really understand the need for appendix E, but I might be missing something.– In Appendix G, Figure 8, we see that adding a third tier does not reduce the minimum average inference error very much at all. Were any simulations done for four-tiered systems? What sort of computational cost does adding one extra tier there? Essential revisions:Both reviewers and reviewing editor agreed that no additional data is needed to support the major conclusions. Therefore a revision focused on improving clarity and presentation will be sufficient. The following two major concerns should be addressed; additional detailed comments and suggestions in the reviews below should be considered for clarity and overall readability.1) One important point to clarify is whether the presented solutions (parameters identified) are representative of a larger class of sub-optimal or optimal solutions. This can be checked by perturbing the solutions. Presumably, the authors have investigated this point already as in the SM there are parts about, robustness, sensitivity and trade-offs, so improving presentation in the main part should solve that. This will also shed more light on the generality of the obtained results.fidelity landscape (inference error) around the optimum and the low inference error states. This has involved additional computation and we have presented our detailed results on this as Figure 10 and Figure 11.We thank the reviewers/editor for suggesting this. Indeed the original manuscript has a detailed discussion on sensitivity and robustness to variations in receptor concentrations . We have now included an additional Subsection “Geometry of fidelity landscape” of Results, where we have described the geometry of the n. The eigenspectrum of the FIM shows only a few stiff directions , which have a significant component along the feedback parameters.Briefly, we measure (i) percent change in inference error upon perturbation of the channel parameters, and (ii) the Fisher information metric (FIM) around the optimum. We use the optimised one-tier two-branch channel for this analysis which yields a minimum inference error of We now explicitly connect this detailed perturbative analysis in channel parameters to the discussion on perturbations of receptor profiles in Appendix 13.sloppy directions and steep along the few stiff directions.To address the geometry of low inference error states, we look at the frequency distributions of channel parameters in a small cutoff region around the minimum identified, i.e. This has now been included as a part of Subsection “Geometry of fidelity landscape” in the Results section along with new Figures 10 and 11 in the revised manuscript.2) The important question to address is whether the choice of Eq. 5, which describes how the cell evaluates its position, influences other results in the manuscript.We thank the reviewers for this comment. We presume here that the question refers to Eq. 6 since this describes the choice of objective function and how the cell evaluates its position. On the other hand, Eq. 5 refers to a standard inference of position from an output distribution computed by the cell. We discuss the assumptions behind Eq. 5 in our response to Reviewer #1’s first question.regions or cell fate boundaries, we observe that the general, qualitative results remain the same, i.e. an additional branch helps reduce the inference errors due to extrinsic noise and an additional tier helps with the same when faced with intrinsic noise. Along with these results, we see that the receptor profiles too remain qualitatively the same as before.Indeed the original manuscript contains a discussion on the choice of objective function i.e. inference error . To highlight this important point, we have now included Subsection “ Choice of objective function” of Results on the choice of objective function. With the alternate objective function which specifies positions of certain Reviewer #1 (Recommendations for the authors):– Extrinsic noise coming from the stochastic ligand production at the source is included in the model by letting cells along the direction parallel to the source boundary experience a spatially-fluctuating level of ligand, with statistics given in (1). From what we understand, this noise is quenched in space. Is it clear to what extent this is equivalent to the more realistic annealed disorder originating from Brownian motion and stochastic source production?In our problem setup, each cellular output is calculated from the steady-state distribution of the morphogen concentration in its bound and unbound states within the cell. This implicitly assumes that the relaxation timescale of morphogen concentrations to its steady-state is faster than the timescales associated with individual cellular processing. This is a fair assumption as the timescales associated with diffusive transport (0.1 – 0.01s) and extracellular degradation (seconds to minutes) of the morphogen are faster than cellular processes which are in the order of minutes to tens of minutes.et al) and for free Dpp [10] in Drosophila wing discs.These estimates follow from FCS measurements on the diffusion of free Wg  have been shown to increase in expression levels away from the morphogen source. Can the authors comment on this observation in light of their model?et al. These authors demonstrate that there are two non-signalling receptors, Dally and Dlp, involved in the Wg signalling, one of which Dlp interacts strongly with Wg and facilitates its sculpting. As mentioned in response to Reviewer #1 (comment #4), in the current manuscript, we have treated the morphogen profiles as independent of the receptor dynamics and hence cannot address the issue of morphogen sculpting. We will take up this important issue in a separate study.We thank the reviewer for pointing out this important paper by Han – In appendix L, the local inference error for a (optimised) graded receptor expression is compared to that of a uniform receptor expression pattern. We find that the way this result is presented slightly misleading since the expression level in the uniform case is not optimised; the two setups should be compared after optimisation.We have indeed optimised the chemical rates and feedback parameters in the two-tier two-branch channels with spatially uniform receptor profiles. We have modified the corresponding appendix (Appendix 11 in the revised manuscript) and included Table 3 in the revised manuscript to emphasize this.including flat profiles. Thus we know that any deviation from the optimum will lead to inferior positional inferences. However, the question posed in Appendix 11 of the revised manuscript is the following: “why can’t a flat receptor profile (modified by uncorrelated noise) infer positions accurately from a noisy morphogen gradient?”. If the morphogen gradient was not corrupted by noise, then surely flat receptor profiles would have sufficed. It is because one wants to discriminate between morphogen concentrations in neighbouring cells in a noisy background that the you need a dynamic and spatially varying profile for the receptors. As seen from Appendix 11—figure 1 (of the revised manuscript), the inference from flat receptor profiles reflect the noise in morphogen gradient itself.In arriving at the optimised channel characteristics for a given morphogen profile, we go through all possible monotonic receptor profiles We elaborate on this point in Appendix 11 in the revised manuscript.Reviewer #2 (Recommendations for the authors):– In my opinion, Figure 2 is misleading and is disrupting the flow of the manuscript. Almost the same information is conveyed in Figure 4 and Figure 5. Further Figure 2 suggests some very regular arrangement of the nodes , which is not the case. Presenting a model with tiers and branches, or some different network-like schematic to indicate reaction and flux imbalances could improve presentation.We agree with the reviewer on this point. Thanks to the suggestions, we have now modified the Figure 2 in the revised manuscript. We hope updated figure is more appealing and conveys the point better.– In lines 85-87, different timescales of signalling processing are mentioned corresponding to branches (fast) and tiers (slow), but this aspect of regulation seems to be not discussed in the later parts of the paper. It might be worth drawing this analogy again when discussing how noise is integrated by different architectures. Possibly, there is the separation of the timescales or signalling integration takes place on the same timescales.The timescale separation in signal processing is between the direction of chemical reactions (receptor state transitions) of the same receptor type i.e. within the same branch  and the direction of transport corresponding to tiers . This is reflected in our choice of parameter values  over which all our optimisation is carried out. The unbinding rate, which corresponds to transition in receptor state, is an order of magnitude faster than the internalisation and recycling rates, which correspond to intracellular transport processes.We have added a line to this effect in the first paragraph of the Results section in the revised manuscript.– Line 98, Figure 5 is mentioned just after Figure 2 (line 91). Please amend the ordering or rephrase so figures are referenced in the text as they appear in the paper.Thank you. We have attended to this point in the revised manuscript.– Line 114, I would avoid statements "of course any choice consistent with experimental observations would do". For instance, would the model work for stationary wave-like patterns of morphogens that could emerge via Turing mechanism ? Basically, I would rephrase providing a description of acceptable ligand profiles .Indeed, the morphogen profiles applicable here belong to the family of monotonically behaving functions. We have attended to this point in the revised manuscript.– The optimization procedure employed is computationally quite-consuming as the intrinsic noise is calculated by solving a chemical master equation. Can the model be solved without directly solving CME? What are the differences? If the differences with and without CME are small this might help to have faster optimization and hence more explicitly explore the space of available signalling architectures that result in optimal or close to the optimal solution.Solving the CME is the most accurate approach which is applicable in any parameter regime or any network topology. In some conditions such as low (additive) noise, other approaches e.g. solving an equivalent Langevin equation, may solve for steady-state faster with minor deviations from the CME solution. However, there is no guarantee that the intrinsic noise is small in any given network topology and across the parameter range.– The intrinsic and extrinsic noises are varied separately as far as I understand. Are there any arguments/heuristics that would indicate the resulting global solutions would be the same if two types of noise would be varied simultaneously?robustness of each such optimised channel’s performance to intrinsic noise. This is appropriate considering the timescale of variation in the extracellular morphogen concentrations is taken to be faster than the cellular processing timescales, as discussed in our response to Reviewer #1’s comment #6.Yes, the extrinsic and intrinsic noises have been considered separately in our work. As the reviewer would appreciate, doing an optimisation considering intrinsic and extrinsic noises simultaneously is computationally exorbitant. Instead, for a fixed number of tiers and branches, we first optimise the channel architecture subject to extrinsic noise alone over all possible rates, receptor profiles and feedback topologies. For each of these optimised channels with fixed number of tiers and branches (e.g. as given by Table 2), we calculate the inference errors in face of intrinsic noise. By doing so, we compare different question of what is the optimum channel characteristics subject to both extrinsic and intrinsic noises then we would have had to perform a simultaneous optimisation.This is the scope of our exercise. Were we to address the – In appendix C, the screening is described for the model parameters, but it is still slightly obscure what process was carried out.Thank you for the suggestion. We have now made minor edits to Appendix 3 in the revised manuscript for clarity.– In table 2, in the middle column, we see that the weight of tier 2 is orders of magnitude greater than that for tier 1. In this case, what role does the first tier really play? And does such a strong weighting make sense biologically?As discussed in the original manuscript (Section “Quantitative models for cellular reading and processing” of the revised manuscript), the weights associated with a tier correspond to the residence time of the signal transducing molecules (or secondary messengers) in that tier. We optimise over these weights along with the other channel parameters. A large weight to the second tier in an optimised channel would imply that signal transduction takes place predominantly from the second tier. The first tier, nevertheless, is still important in generating the characteristics of the second tier e.g. through actuating the feedback (or feedforward) control.Drosophila wing disc provides a concrete example. Hemalatha et al., 2016 [4], Seto et al., 2006 [11] and Rives et al., 2006 [12] have shown that endocytosis is important for signalling. Removal of different components of the endocytic network abrogates both short range as well as long range Wg signalling. Intracellular Wg endosomes often appear to be co-localized with the downstream signalling mediator Dishevelled (Dsh). However, a quantitative study comparing the extent of localization of Dsh at the plasma membrane versus endosomal membrane is currently missing. Given the relevance of the endocytic network in Wg signalling and co-localization of downstream adaptor proteins to endosomes containing Wg-DFz2, we believe that these endosomes (corresponding to the second tier) act as signalling hubs to transduce positional information.The Wg signalling in – I don't really understand the need for appendix E, but I might be missing something.minimal channel i.e. with one tier and one branch.In Appendix 5, we are trying to understand why having an appropriate spatial gradient of receptors helps in separating the outputs in nearby cells, thus facilitating accurate positional inference. This is best done in the simplest case of a We have now made minor edits to Appendix 5 in the revised manuscript to reflect the above.– In Appendix G, Figure 8, we see that adding a third tier does not reduce the minimum average inference error very much at all. Were any simulations done for four-tiered systems? What sort of computational cost does adding one extra tier there?The number of network topologies increases as a quadratic function of the number of tiers. Further, the number of parameters to be optimised would also increase linearly with the number of tiers, resulting in a net cubic increase in computation time. Practically, even with parallelisation, the number of computing hours  would increase significantly with additional tiers. From our analysis of two- and three-tier channels, we would predict that further addition of tiers provides a marginal advantage in reduction of inference errors while incurring additional cellular costs.References


Percutaneous vascular interventions are performed for the treatment of haemoptysis and involve embolization of bronchial arteries, pulmonary arteries and pulmonary arteriovenous malformations. There are isolated reports of embolization of pseudoaneurysms forming in the pulmonary vasculature. The migration of components of the coils used in the embolization of vascular pulmonary pathologies is rare.A 46-year-old man presented to the emergency department with cough, haemoptysis, and expectoration of lengths of metal wire. He had an episode of coughing out a wire about a year prior to his admission to our hospital, which he attributed to be present in the can of coke he had consumed at that time and did not report it to the doctors. His past medical history was significant for stab injury to the right chest 17 years ago, for which he underwent right thoracotomy and exploration for bleeding. Injury to the lung parenchyma was noted and repair was performed by suturing the defect. Post operatively the CT scan demonstrated development of pulmonary artery pseudoaneurysm. We report a case of a patient expectorating coils 17 years after embolization of this traumatic pulmonary artery pseudoaneurysm. Radiological imaging demonstrated coils in the perihilar area of the lung parenchyma and in the tracheobronchial lumen. Operative intervention was used to remove the coils.Although percutaneous catheter based vascular interventions have emerged as safe and effective procedures, the long-term complications such as coil migration, recanalization and need for further embolization ought to be considered and patients need to be counselled and followed-up accordingly. To the best of our knowledge, this is the first case of migrated coil post embolization of post-traumatic pulmonary artery pseudoaneurysm. Ultimately, the management of endobronchial coil migration post embolization, be it surgical or bronchoscopic, should be decided on a case-by-case basis, considering the patient’s symptoms and the risk fatal complications. Post-traumatic pulmonary artery pseudoaneurysms are rare, at risk of enlargement and rupture, and associated with high mortality . They caWe report the case of a patient who expectorated a coil 17 yrs following embolization of a post-traumatic pulmonary artery pseudoaneurysm. A 46-year-old man presented to the emergency department with a cough, haemoptysis, and expectoration of lengths of metal wire. 17 years previously he had suffered a significant stab injury to the chest, for which he underwent right thoracotomy and exploration in a different hospital. A suture repair of a parenchymal lung injury was performed. Post-operative computed tomography (CT) scan demonstrated pulmonary artery pseudoaneurysm formation and percutaneous endovascular coils were inserted to prevent rupture and massive haemoptysis. The medical records had been destroyed by the time of this presentation; hence no further information was available.During the current admission, chest x-ray and chest CT demonstrated coils in the right lung parenchyma posterior to the oblique fissure, with a length of coil in the lower lobe bronchus, resulting in adjacent haemorrhagic areas and pneumatoceles Fig. . The patBronchoscopy identified a thin wire emerging from the right lower lobe basal segmental bronchus into the right main bronchus, extending to the proximal trachea and larynx. There was marked inflammation with significant blood-stained purulent secretions emanating from the lower lobe bronchus. The wire could not be pulled out with gentle traction. On entry into the chest, the lung was densely adherent to the chest wall and posterior mediastinum. The adhesions were particularly dense and inseparable from the oesophagus, azygos vein and the whole area was one thick indistinguishable fibrotic mass. A subsequent lower lobe bronchotomy was performed and the wire was traced into the lower lobe. This wire was removed, and lower lobectomy was performed by dissecting out all the relevant structures. The bronchus intermedius was stapled off just distal to uptake of middle lobe bronchus where it was less inflamed. After removal of the lower lobe, it was noted that there was a dense fibrotic mass around the posterior segment of the upper lobe. The upper lobe itself was densely fibrotic and stuck to the chest posteriorly and inseparable from the oesophagus and the azygos vein.After lower lobe removal, the residual fibrotic area was incised to ensure no further wires were left in situ. It was well encapsulated (3*2 cm) and this process revealed necrotic pinkish grey material and a bunch of wires which were subsequently removed. At this point the cavity was closed as there was no evidence for fistulation into the airways or any such tracheobronchial communication. Several coils of metal wires were removed, and the cavity was cleared of all the necrotic debris. The wall of this cavity was inflamed granulation tissue and due to the size, location and adhesions to the mediastinum and oesophagus, the resection was felt not to be possible. In view of this, the cavity was closed using interrupted 2-0 Vicryl suture in two layers ensuring obliteration of the cavity.Four days post-operatively, the patient showed no sign of clinical improvement and this correlated with worsening inflammatory markers and an exaggerated SIRS response. Features of empyema were noted on imaging. The patient was taken back to theatre for rigid bronchoscopy where pus was seen coming out of the right lower lobe bronchial stump soiling the left side. This was vigorously suctioned, and the chest re-opened via the same posterolateral incision (right). The cavity was washed out and it was noted that the previous suture on the capsule of the posterior segment of upper lobe had broken down and resulted in a communication between the capsule and the stump of the lower lobe bronchus. As such, an intercostal muscle bundle was harvested and patched on the opening of the fistulous tract of the lower lobe bronchial stump with multiple 3-0 vicryl. Post patching, there was minimal air leak on dual lung ventilation and at repeat bronchoscopy there was no further purulent discharge from lower lobe stump.Streptococcus anginosus, Bacteroides fragilis and Eikenella corrodens were grown from operative samples. The patient was discharged home with chest drain in situ and a long course of appropriate antibiotics. The chest drain was later removed in the outpatient clinic. The current follow-up period to date is 12 months since discharge.The patient described in this report proceeded to surgery to mitigate against the potential for complications such as haemoptysis, recurrent chest infections and troublesome symptoms of cough. The type of intervention, however, was debateable between bronchoscopic retrieval of the coils or open surgical procedure to resect the lung parenchyma containing the coils. The uncertainty of retrieving all the coils endoscopically, possibility of haemorrhage and perceived difficulty of emergency thoracotomy due to adhesions from the previous surgery led to the decision in favour of a surgical intervention.Some cases of migrated coils fistulating into the bronchus have been removed endobronchially using rigid bronchoscopy . HoweverTo the best of our knowledge, this is the first case of migrated coil following embolization of post-traumatic pulmonary artery pseudoaneurysm. On reflection, placing a muscle bundle or some form of vascularised pedicle over the bronchial stump at the time of the lobectomy, particularly in the setting of such inflammation may have helped to avoid fistulation and propagation of sepsis. Ultimately, the management of endobronchial coil migration post-embolization, be it surgical or bronchoscopic, should be decided on a case-by-case basis, taking into account the patient’s symptoms and the risk of fatal complications.


The fusion of infrared and visible images can improve image quality and eliminate the impact of changes in the agricultural working environment on the information perception of intelligent agricultural systems.In this paper, a distributed fusion architecture for infrared and visible image fusion is proposed, termed RADFNet, based on residual CNN (RDCNN), edge attention, and multiscale channel attention. The RDCNN-based network realizes image fusion through three channels. It employs a distributed fusion framework to make the most of the fusion output of the previous step. Two channels utilize residual modules with multiscale channel attention to extract the features from infrared and visible images, which are used for fusion in the other channel. Afterward, the extracted features and the fusion results from the previous step are fed to the fusion channel, which can reduce the loss in the target information from the infrared image and the texture information from the visible image. To improve the feature learning effect of the module and information quality in the fused image, we design two loss functions, namely, pixel strength with texture loss and structure similarity with texture loss.Extensive experimental results on public datasets demonstrate that our model has superior performance in improving the fusion quality and has achieved comparable results over the state-of-the-art image fusion algorithms in terms of visual effect and quantitative metrics. Infrared images and visible images are important sensing information for intelligent agricultural systems. The key to intelligent agricultural systems is to utilize perceptual data for intelligent analysis and decision-making. The infrared imaging technology with anti-interference solid ability uses the radiation energy released by the target so it can penetrate smoke, fog, rain, snow, etc., in the environment. However, the visible light sensor uses light reflectivity to image with much spectral information and high-resolution characteristics. As the application range in intelligent agricultural equipment gradually broadens and the perceived information environment is usually changeable, a single image imaging technology cannot sufficiently perceive the environmental information. It results in the inability of intelligent agricultural equipment to perceive enough information, which leads to the failure of intelligent agricultural systems to work regularly. Therefore, it is of great significance to study the complementary use of infrared and visible image imaging technology to enhance the information perception ability of intelligent agricultural equipment .High-quality enhanced images can be obtained by fusing infrared and visible images to improve the information perception ability in intelligent agricultural equipment and meet various subsequent visual tasks for intelligent agricultural systems. As a branch of information fusion, image fusion has played an essential role in computer vision since it can generate more informative images for high-level vision tasks such as recognition , detectiAccording to the abstract degree in image information, image fusion is divided into three levels: pixel level, feature level, and decision level . In thisThe traditional methods, in general, cover five approaches: multi-scale transform methods (MST) , sparse In the past several years, deep learning has been widely applied in infrared and visible image fusion to solve the shortcomings in traditional fusion methods. The application of deep learning-based methods for infrared and visible image fusion mainly reflects in convolutional neural network CNN-based network frameworks, such as convolutional sparse representation (CSR) and generative adversarial network (GAN). The CNN-based fusion frameworks for infrared and visible image fusion are divided into two categories: the depth extraction for image features and the construction for fusion networks. In depth feature extraction, VGG-19 , ResNet1Although a variety of networks to improve the performance in image fusion have been proposed by many scholars. The CNN-based network frameworks, such as convolutional sparse representation (CSR), generative adversarial network (GAN), and other many network architectures are applied in infrared and visible image fusion. However, the CNN-based fusion frameworks for infrared and visible image fusion are divided into two categories: the depth extraction for image features and the construction of fusion networks. The extraction for depth features requires a deeper network structure, resulting in weak interpretability, extensive computation, and other problems. The construction of the fusion network is also complex and difficult to control. Although many modelsare superficially similar to RADFNet, they have not abandoned these two categories. To get rid of the dilemma in these two kinds of fusion categories, the RADFNet employs a distributed fusion framework to make the most of the fusion output from the previous step. Two channels utilize residual modules with multiscale channel attention to extract the features from infrared and visible images, which are used for fusion in the other channel. Because it adopts distributed fusion, the fusion network does not entirely rely on the extraction in deep features, and the fusion network is simple to construct, showing strong robustness. The RADFNet solves the limitations from most current fusion networks and shows strong adaptability. The main contributions of our work are summarized as follows:(A) A distributed fusion framework based on residual CNN (RDCNN) for infrared and visible image fusion is proposed in this paper. The distributed fusion framework is distinct from the existing fusion framework in infrared and visible image fusion. It adopts three channels to realize image fusion, wherein two channels are applied to feature extraction and the other channel realizes feature fusion.(B) To obtain coarse-to-fine features and compensate edge information for fused images, the attention mechanism is discussed. In this way, the fused images retain more prominent information and lose less edge information from source images.(C) Two loss functions, including the pixel intensity with texture loss and the structural similarity (SSIM) with texture loss, are designed to train the RADFNet. Through experiments, it is found that networks trained by the two loss functions have their own advantages.(D) Extensive experiments are conducted on public infrared and visible image fusion datasets. Compared with existing state-of-the-art fusion methods, our fusion framework has a promising even better performance in accordance with visual effect and quantitative metrics. In addition, we perform ablation experiments to verify the function in the corresponding module. Last but not least, unregistered source image pairs are fed into the proposed network, emerging the robustness of the proposed framework.Distributed fusion architecture is a classical and typical structure in multi-sensor fusion due to its high speed and reliability . In distIn some tasks, deeper neural networks can extract higher-level features and perform excellently. However, too deep networks may cause the notorious problem of vanishing or exploding gradients and degrade the accuracy. To solve these problems, He et al. proposed a residual network composed of a series of residual blocks . Figurelx and l+1x are the input and output in the l-th layer, and ℱ is the residual function. f is a ReLU , which usually contains 2 or 3 convolutional layers. In many cases, the dimensions of input lx and output l+1x are discrepant, so it is necessary to employ a 1×1 convolution operation to maintain the dimension in input and output consistent, whose schematic diagram is shown in where s a ReLU  functionlx) is the identity skip connection and where h block that adaptively recalibrates the channel feature strength by explicitly modelling the interdependence between channels is introduced in ’ means the kernel size is k×k, the input channel is in and the output channel is out. In the network, BN indicates batch normalization that is utilized to speed up the training and make the training more stable, and ReLU denotes the linear rectification function.The infrared and visible image fusion model RADFNet set out in the present paper is constituted of three channels. The RADFNet structure is exhibited in 1−4 and IR1−4 are the residual networks which extract image features. Because the residual network has the advantages of mitigating gradient disappearance or gradient explosion and protecting the information integrity, the networks we designed can extract meaningful features and ensure the information integrity simultaneously. Besides, ATTi processes the features extracted by residual block VRi or IRi to obtain coarse-to-fine features. EATi acquires the edge information and then compensates edge information for the extracted feature map. The ⓜ refers to the operation for achieving the maximum value in the homologous channel and the homologous position in the feature map. The FUPi generates FUIi by fusing features extracted by the other two branches with the FUIi−1 generated by FUPi−1 when i is not 1. When i is 1, the concatenated infrared and visible image is fed into the FUP1 to generate FUI1. The ⊕ is the concatenation operation in channel-wise, and the 1×1 244 convolution layer in the last fusion layer constructs fusion images.The RADFNet adopts four-layers network structure. The VRi or IRi are used as input in ATTi. Then, the 1×1, 2×2, and 4×4 average pooling operations are performed to generate multiscale features which contain more necessary spatial information. Moreover, the channel attention mechanism is utilized to enhance channel correlation information between features. The multiscale channel attention network is trained to learn the weight i which can be formulated asIn the process of infrared and visible image fusion, image feature extraction is exceptionally significant. However, in practical situations, numerous detailed information loses in the process of feature extraction. Inspired by SENet , the mulG(z) denotes the global average pooling operation. t-th pooling scale in ATTi.  refers to the position in feature map, and H,W means the height and width of the feature map. δ refers to the ReLU function, w1∈ℝk×k and w2∈ℝk×k , σ denotes the sigmoid function. Then the channel-wise multiplication is implemented between where π denotes the instance normalization  means the pixel at position . Moreover, we perform the enhanced operation to obtain the more obvious gradient information:where H={1,…,h–1} and W={1,…,w–1}. The  represent the position at the gradient map. Through the above steps, we get the gradient image G with the abundant enhanced edge information.where i or VRi. The structure diagram of the edge attention module is shown in Vi and EIi layer by layer, which is then used to compensate edge information for the feature maps iV and iI extracted by VRi and IRi respectively. Therefore, compensated feature maps fused to generate the fused images retain more edge information.Subsequently, we feed the gradient images from infrared and visible images into the edge attention module to generate edge attention feature maps with enhanced edge information. Then, the feature maps will be entered into the extraction branch to compensate edge information for the extracted features by IRFor infrared and visible image fusion, it is difficult to provide the ground truth of fused images for networks to train a model. However, the requirement to retain salient target information in the infrared image and the texture information in the visible image is determined. Inspired by this requirement, the loss function we employ is as follows:pixelL constrains the fused image to contain more target information from the image pair facilitating target tracking and the textureL forces the fused images to contain more texture details which can effectively improve the identification of objects in images.where the pixelL is expressed as follows:Specifically, the exact definition of m is the batch size that is the number of training samples used in each iteration. The If means the fused image with the input image pair {irI, visI}, and the max (·) denotes the element-wise maximum selection. Through the maximum selection strategy, the fused images have the prominent target information.where pixelL has very limited constraints on textural details. Therefore, the textureLis introduced to force the fused image to retain more textural information and the textureL is defined as:Moreover, we hope the fused images contain significant target information and simultaneously preserve great textural details from source images. However, the m is the batch size, the fI means the fused image with the input image pair {irI, visI}, and the max (·) denotes the element-wise maximum selection. The ∇ indicates the Sobel gradient operator and the |·| means the absolute operation. The element-wise maximum selection strategy can make the fused images obtain the most significant edge textural information.where the To evaluate the proposed fusion algorithm in many aspects, we conduct extensively qualitative and quantitative experiments on the RoadScene  dataset.a = 10. Adam optimizer  Xeon(R) Gold 6226R CPU.In the training process of the RADFNet model, we use images from the OSU  dataset ptimizer  with β1 The RoadScene dataset contains color visible images, but we employ the input grayscale images to train the proposed network. To get better visuals in the test phase, we adopt the strategy  to proceTo fully evaluate the performance of the RADFNet, we compare the RADFNet with the other six methods on the Roadscene dataset. The Roadscene dataset mainly contains road scenes, including pedestrians and cars, in the daytime and at night. We select two images in the daytime and two in the nighttime for evaluation subjectively so as to exhibit some intuitive fused images on the fusion performance. The fused images of the proposed RADFNet and the other six methods are presented in To avoid human factors and other factors affecting the subjective evaluation. We conduct quantitative assessments with the six approaches and the proposed method. The results of six metrics on the Roadscene dataset, which contains 221 image pairs, are shown in To verify the effectiveness of the edge attention module, we conduct ablation experiments. We employ edge attention and ignore edge attention to create two models, then the same image pair is used as input to test the difference between the two models, and the visual results are presented in In addition, to comprehensively evaluate the impact of edge attention in fused images, we make quantitative evaluations for the four images in FS, which can be defined as follows:For the sake of comprehensively considering the improvement in model training on fused image quality, we design another loss function LtextureL is represented by Equation 17, the value of β is 5, and the SSIML is the structural similarity (SSIM) loss, which can be expressed aswhere SSIM(·) means the structural similarity , edge attention, and multiscale channel attention. The proposed method can most retain the salient target information in the infrared image and the textural details information in the visible image. In addition, we introduce the multiscale channel attention module, which can extract coarse-to-fine features to preserve more information from source images to fused images. We also adopt an edge attention module that can compensate edge information for the fusedimage to make the fused image lose less edge information from source images. The comparative experiments are conducted on the Roadscene dataset, and the results demonstrate that the proposed method has superior performance in improving the fusion qualityand has achieved comparable results over the state-of-the-art image fusion algorithms in terms of visual effect and quantitative metrics. Finally, we send the unregistered image pairs into our network, and the results demonstrate that our method with strong robustness still performs well in fusing images without registration. The RADFNet performs well for infrared and visible image fusion due to the robust feature extraction ability of the network. The distributed fusion framework endows it with strong robustness, but the network parameters are still relatively large, which is not simple enough in the actual project deployment. In the future, it is necessary to improve the parameters of the network and the actual deployment of the model.The original contributions presented in the study are included in the article/supplementary material. Further inquiries can be directed to the corresponding authors.SF is responsible for the writing and theoretical design of the thesis, CW is responsible for the experimental test, and CL and MH are responsible for the verification of the experimental scheme, the analysis of the results and the project funding. All authors contributed to the article and approved the submitted version.


Attention is a particularly important indicator in life, as inattention can lead to many negative consequences. As a non-invasive intervention, real-time neurofeedback training can effectively enhance individuals’ attention adjustment abilities. However, previous studies have neglected to consider differences among individuals. In this study, an individualized neurofeedback training (INT) method based on functional near-infrared spectroscopy (fNIRS) was proposed for attention improvement and compared with non-individualized neurofeedback training (NINT). The neurofeedback channels and thresholds were determined individually for each subject. Then, participants conducted four runs of neurofeedback training. Two attention tests  and attention network test (ANT)) were used to assess the performance of the neurofeedback training. The length of time that the two groups of participants continuously kept their oxygenated hemoglobin concentration above a threshold showed an increasing trend, and the improvement rate of the INT group was higher than that of the NINT group. The reaction times for both groups showed a downward trend, but the INT group declined more significantly. In the fNIRS data, it was observed that the activation degree of the INT group in the middle and dorsolateral prefrontal areas was higher than that of the NINT group. It is preliminarily proved that the proposed INT method can effectively improve the attention level, and its overall performance is better than that of the NINT method. Attention refers to the ability of a person to mentally identify and concentrate on something; it is the tendency to selectively process certain stimuli and ignore others. Many human behaviors require high levels of attention and sometimes rely on sustained attention, such as when listening to a lecture or reading a book . Low attWith the continuous development of neuroimaging technology, neurofeedback through physiological measurements has become a common means of regulating attention-related neural activities . NeurofeAt present, the physiological measurements used in neurofeedback are mainly comprised of electroencephalogram (EEG) and functional magnetic resonance imaging (fMRI) technology. EEG signals feature high time resolution and practicality. There have been numerous EEG-based neurofeedback studies. Some research has suggested that real-time EEG neurofeedback can achieve cortical activation and the voluntary adjustment of attention levels ,6,7. In However, although neurofeedback systems using fMRI for physiological measurements have proven to be effective, the relatively high equipment requirements and restriction of participants’ movement continue to be challenging. In order to reduce cost and enhance mobility, functional near-infrared spectroscopy (fNIRS) has emerged in recent years as a novel physiological measurement method and effective alternative. The technology uses oxygenated hemoglobin (Oxy-Hb) and deoxygenated hemoglobin (Deoxy-Hb), which have different absorption effects in near-infrared light. Thus, fNIRS can detect changes in Oxy-Hb and Deoxy-Hb in brain function areas, indirectly reflecting brain activity . CompareDue to the various advantages of fNIRS, it is a better choice for the field of neurofeedback . Thus, iTo a certain extent, the above studies confirmed the effect and efficiency of fNIRS neurofeedback, but existing neurofeedback research has not taken into account the differences among individuals. Previously, Grandy et al. proposed the individual alpha frequency for EEG , and stuIn summary, in order to improve the effect of fNIRS-based attention neurofeedback training, we propose an INT program. Through pre-experiments, participants’ channels with higher activation level in an attention test task were determined. These channels were used as neurofeedback channels in training. Finally, we calculated the level of blood oxygen concentration that was higher than the threshold during the INT program, as well as the response time change during the attention test task, and compared these with the non-individualized neurofeedback training (NINT) program to evaluate the INT program’s performance.A total of 10 college students  participated in this experiment. All participants were right-handed, had no known respiratory, cardiovascular, psychiatric, or neurological diseases, and had not undergone any type of neurofeedback training. Participants signed an informed consent form before the experiment and received a small amount of remuneration after completion. This research was approved by the Institutional Review Board of the State Key Laboratory of Cognitive Neuroscience and Learning in Beijing Normal University (CNL_A_0010_010), in accordance with the Declaration of Helsinki.All participants were randomly divided into two groups: an INT group using individualized channels and thresholds for neurofeedback and an NINT group using common channels and thresholds. Each of the two groups was comprised of five participants.This experiment used the NIRSport function near-infrared imaging equipment  to detect participants’ brain activity. The equipment uses near-infrared wavelengths of 760 nm and 850 nm. Its working principle is to use the absorption of near-infrared light by different blood components. When the degree is different, the relative concentration changes of Oxy-Hb and Deoxy-Hb are measured. In this experiment, the distance between adjacent light sources and detectors was about 3 cm, and the sampling frequency was 7.8125 Hz.Since the Oxy-Hb signals exhibit superior sensitivity in task-related signal changes and a greater correlation with blood oxygen level-dependent signals in functional magnetic resonance imaging , we usedAttention is not represented anatomically in the brain as a unitary function but through multiple functionally heterogenous regions that jointly integrate to constitute the so-called attention network . The fMRIn this study, a self-developed python-based neurofeedback system was used to train participants’ attention levels. In order to make this system arouse higher levels of interest and avoid boredom caused by long-term training, a cartoon game was adopted. The system’s neurofeedback interface is shown in The participants wore NIRSport electrode caps. The light source and detector were positioned in the forehead area, and the collected Oxy-Hb concentration signal was transmitted to a real-time data preset system through the local area network and the lab’s streaming layer data stream. A real-time data preprocessing module performed filtering preprocessing on the Oxy-Hb concentration data to remove motion artifacts and abnormal signals from the original signal. After the system processed the concentration signal, the neurofeedback information was calculated and fed back to participants in visual form. The participants adjusted themselves according to the visual neurofeedback information received and continued the training task, thereby forming a closed-loop neurofeedback training system.For the INT group, the neurofeedback channel and threshold were individualized. The activation level of each channel of each participant was calculated. For each participant, the four channels with the highest activation level were used as the neurofeedback channels, and the averaged activation level of the four channels was used as the neurofeedback threshold. For the NINT group, the neurofeedback channel and threshold were determinated at the group level. The activation level of each channel was averaged within the group. The four channels with the highest activation level were selected as the neurofeedback channel, and the averaged activation level of these four channels was calculated as the neurofeedback threshold.Before the start of the neurofeedback task, participants were told only to participate in a rocket lift-off game, but not given any strategy for raising the rocket. Individuals had to try all possible methods to raise the rocket. If the strategy adopted did not achieve a positive effect after 20 to 30 s, they could choose to change their strategy.The study was conducted in a separate room, and participants performed their tasks on a computer. The training scene is shown in AX-CPT can be used to assess alertness and response inhibition. In the present research, the main process for this task is shown in ANT was designed to assess alertness, orientation, and conflict attention in a single test session . AlertneThe experiment process is shown in Alert signals were divided into four categories: no alert, central alert, double alert, and spatial alert. The four types of signals appeared randomly with the same level of probability. The target stimulus was five arrows. Participants needed to respond to the direction of the center arrow. The target stimulus was divided into three types: a congruent stimulus with five arrows in the same direction, incongruent stimulus with the central and surrounding arrows pointing in opposite directions, and neutral stimulus with only the arrow in the middle. The direction of the arrow and position (up or down) of the various stimuli were random, with the same level of task parameters and timing for the AX-CPT probability.We extracted fNIRS data from four neurofeedback trainings, separately. Firstly, we filtered these data by a band-pass filtering (0.01–0.1 Hz) to remove the physiological noises, such as heartbeat (1 Hz) and respiration (0.4 Hz). In order to estimate the level of attention, we calculated the percentage of time that the Oxy-Hb concentration of each participant was higher than the threshold, and obtained the proportional change of each participant in the neurofeedback training. Then, we averaged the percentage within the group to compare the proportional change between the two groups.We translated the level of sustained attention into the ability to continuously maintain Oxy-Hb concentrations above the threshold . The lent-test.To assemble statistics for the AX-CPT behavior data for all participants, we calculated the average reaction times and observed changes throughout the six tests. We then calculated the average reaction time and compared the average change trend of the two groups through the slope to verify whether different training modes will lead to different results. In order to test whether there was a significant difference in the reaction time between the two groups after training, we averaged the pre-test and post-test reaction time of the two groups, and then conducted a paired y is the real fNIRS data of the AX-CPT task; x is the design matrix, which is the matrix generated by convolution of the boxcar of the AX-CPT task with the typical hemodynamic function; e is the residual. In the above formula, We used the general linear model (GLM) to perform statistical analyses on the fNIRS data. The formula for the GLM was: Three effect calculation methods were used for ANT. The alert network efficiency calculation was the non-alert signal average response time minus the central alert signal response time. The directional network efficiency calculation was the average of the upper and lower alert signal response times subtracted from the directional alert signal response time. The conflict network efficiency calculation was the average response time of the conflict stimuli minus the average response time of the consistent stimuli . In ordeWe calculated the average reaction time of ANT in each of the two groups, then performed a paired t-test on the pre-test and post-test reaction times to estimate if the reaction time was improved significantly following training. Finally, we compared the trend of the average reaction time between the two groups.The results of the INT group are shown in The results of the INT group are shown in The results of reaction time for the INT group are shown in We used GLM to calculate the t-test results of three effects in the INT group are shown in t-test results of three effects in the NINT group are shown in The In this study, we proposed an INT method that successfully improved participants’ attention levels. We compared its effect to that of a NINT program to verify the feasibility and superiority of the INT method. INT fully takes into account individual differences among participants and calculates INT channels for each, making the training effect more obvious. Data on brain activity and behavior were statistically examined in this study. The results revealed that the INT group outperformed the NINT group.The fraction of oxygenated hemoglobin concentration above the threshold in the INT group was higher than that in the NINT group throughout the neurofeedback process. The reason for this phenomenon could be that in the INT group, each participant’s neurofeedback channels and thresholds are measured separately, which is very appropriate for each participant, making it easier for participants to regulate the activation of relevant channels; in the NINT group, neurofeedback channels and thresholds are obtained by all participants in the group. These channels and thresholds do not correspond to the participants’ personalities, making it difficult for them to adjust.Concerning the length of time that Oxy-Hb concentration continues to be higher than the threshold during training, the average length of time for participants in the NINT group showed an overall upward trend, but there was no continuous increase in the results similar to what was seen for the INT group across all four training sessions. This indicated that for the NINT group, the sustained attention level of participants increased as the runs of neurofeedback increased, but the effect of continuous improvement was not achieved. For the INT group, the average lengths for the four training sessions continually increased, with the longest duration of each participant increasing to varying degrees. The reason for this phenomenon was that participants changed their ability to regulate their brain activity. The sustained attention level of the participants in the INT group increased as the number of trainings increased, and compared with the NINT group, the sustained attention level of the INT group achieved a continuous increase.The behavior results of AX-CPT and ANT were analyzed. In terms of reaction time, both the INT and NINT groups’ average reaction times were on the decline. The decrease of average reaction time indicates that the judgment time of participants to stimuli is shortened, which indicates that the participants’ attention level is improved and the brain can process more stimulus information in a shorter time. The constant decrease in reaction time demonstrates that as the number of neurofeedback runs increases, the participants’ attention level improves.The statistical results of the inter-group comparison revealed that the reaction time of the INT group declined faster than that of the NINT group in the behavioral results of AX-CPT and ANT. This shows that the INT group’s brain processed sensory information faster than the NINT group’s. This demonstrates that the INT strategy we offer is more effective and efficient. The training effect of the INT group is better under the same training times. It is worth noting that the statistical findings of the two tests revealed that, regardless of their beginning performance, participants’ reaction times fell. To put it another way, after neurofeedback training, the participants’ attention level will improve, regardless of their state prior to training.Regarding the brain activation results from the AX-CPT task, the main activation area for the two groups was near the dorsolateral prefrontal area, a result consistent with the area of interest in certain previous studies . The preIn the sixth AX-CPT imaging results, activation of the dorsolateral prefrontal area disappeared for the two groups, but it is worth noting that the activation areas were concentrated in the right anterior area of the prefrontal lobe. Because the sixth test was performed one week after the last training and the second to fifth tests were performed immediately after the neurofeedback training, the activation of the dorsolateral prefrontal area disappeared, possibly because the neurofeedback training lasted only for a short time. Thus, it could not achieve an effect that lasted for a week. In the first test, there was also activation in the dorsolateral prefrontal area. This may have been due to the fact that the participants were more mentally nervous when initially exposed to this paradigm, leading to more obvious activation. In previous studies, the efficacy of rtfMRI neurofeedback was tested in the right lower prefrontal cortex through a randomized controlled experiment. Researchers argued that this area was key, and is often damaged in ADHD patients . This coIn addition, we recorded the strategies adopted by the participants in the neurofeedback training. In the NINT group, the strategies included imagining they were playing a game, imagining that they were driving a fighter plane or fighting a fire, and recalling their excellent performance in basketball. Participants in the INT group used a variety of efficient tactics, including recalling fast-paced songs or movie plots, mental arithmetic, multiplication, and reciting old poems. The strategies adopted by the two groups had certain aspects in common. The things they imagined were focused on the real world. Additionally, most participants said that the more detailed their imagination was, the more obvious was the rocket’s rise. This was because more focused attention is required when imagining details. The two groups adopted the same type of strategy, but the training results of the INT program were better, once again proving the merits of the INT program.In this pilot study, we proposed an individualized fNIRS neurofeedback training system. The experimental results preliminarily prove that the proposed INT method can effectively improve the attention level, and its overall performance is better than that of the NINT method. To solve the limitation of sample number, we will supplement samples to further verify our proposed method. We hope this pilot study could provide a new perspective for attention improvement.


The resources available to fight an epidemic are typically limited, and the time and effort required to control it grow as the start date of the containment effort are delayed. When the population is afflicted in various regions, scheduling a fair and acceptable distribution of limited available resources stored in multiple emergency resource centers to each epidemic area has become a serious problem that requires immediate resolution.This study presents an emergency medical logistics model for rapid response to public health emergencies. The proposed methodology consists of two recursive mechanisms: (1) time-varying forecasting of medical resources and (2) emergency medical resource allocation. Considering the epidemic's features and the heterogeneity of existing medical treatment capabilities in different epidemic areas, we provide the modified susceptible-exposed-infected-recovered (SEIR) model to predict the early stage emergency medical resource demand for epidemics. Then we define emergency indicators for each epidemic area based on this. By maximizing the weighted demand satisfaction rate and minimizing the total vehicle travel distance, we develop a bi-objective optimization model to determine the optimal medical resource allocation plan.Decision-makers should assign appropriate values to parameters at various stages of the emergency process based on the actual situation, to ensure that the results obtained are feasible and effective. It is necessary to set up an appropriate number of supply points in the epidemic emergency medical logistics supply to effectively reduce rescue costs and improve the level of emergency services.Overall, this work provides managerial insights to improve decisions made on medical distribution as per demand forecasting for quick response to public health emergencies. Since the turn of the twenty-first century, there have been numerous outbreaks of large-scale infectious diseases throughout the world. An epidemic occurs when an infectious disease spreads quickly and affects a significant number of individuals. Epidemic infections can spread widely in a short period, usually within 2 weeks or less . The mosEmergency medical logistics in response to public health emergencies is very important, but the research in this area is still inadequate . This woThe use of the mathematical model to analyze the kinetic behavior of diseases dates back to Daniel Bernoulli's  1760 resFor the dispatching and distribution of emergency supplies problem, scholars have established optimization models for allocating emergency materials in the event of a public health emergency, such as an outbreak of infectious diseases –26. SomeThe primary objective of this study is to quantify the actual situation in the epidemic area to distribute supplies, and help decision-maker to make best choices on the allocation of limited emergency medical resources to the appropriate places and quantities to halt the outbreak and mitigate its effects.To be specific, this paper is based on human data collected in each of the affected areas. A particular point in time in the historical data is selected as the basis point. The personnel situation in each epidemic area at the decision time is forecasted based on the information at the basis point using the modified infectious disease model. The number of infected and hospitalized groups is simulated in each area to forecast patients' demand for emergency medical resources. Under the assumption that the demand for emergency medical resources for patients with infectious diseases and inpatients is known, we define emergency indicators for each epidemic area and develop a bi-objective optimization model to determine the optimal medical service allocation plan by maximizing the weighted demand satisfaction rate and minimizing the total vehicle travel distance.In summary, this paper takes sudden infectious public events as the background, emphasizes emergency logistics schedule optimization, and presents an emergency rescue logistics model based on infectious diseases transmission mechanism. First, based on the transmission rules of infectious diseases combined with the local medical treatment capability, the modified susceptible-exposed-infected-recovered (SEIR) model are proposed to predict the material demand of each epidemic area during the entire emergency rescue stage. Second, a mixed-integer programming model for multi-stage and multi-cycle emergency rescue logistics scheduling in multi-epidemic areas is built with fairness  and timeThe local population is relatively steady, regardless of migration, natural birth rate, or death rate.The infected individual requires pharmacological therapy following hospitalization.Individuals who have recovered get permanent immunity.It is assumed that the functional relationship between the demand for emergency medical resources and the number of infected individuals in the epidemic area is known.It is assumed that the demand for supplies in the epidemic area and reserves of relief centers can be forecasted, as well as the number of and geographic location of epidemic areas.This paper does not consider the number of vehicles, transportation mode, transportation capacity limits, or other related factors while implementing quarantine policies during an epidemic.According to the unique characteristics of the spread of infectious diseases, the precise allocation of emergency supplies during an epidemic can be summarized as follows: (1) During the early stages of an epidemic, it is necessary to quantify the extent of the emergency in epidemic areas. The epidemic locations are diverse, and each area has a unique crisis circumstance. (2) Because supplies demand are highly correlated with the number of confirmed infections, it is vital to forecast the demand in the epidemic area based on the number of infected people.Based on the characteristics of the initial spread of infectious disease, hence the SEIHR warehouse models are created. We categorize the affected area's population into six subgroups: susceptible (S), exposed (E), infective (I) who have developed symptoms following the incubation period but have not been hospitalized or isolated, hospitalized infected (H), and recovered (R). N = S + E + I + H + R is used to calculate the total population in each epidemic area. The susceptible population (S) reduce as a result of exposure to exposed, infected, and asymptomatic infected people, and they will enter the incubation period of exposure (moving to class E). As shown in formula 1.The exposed population (E) become infected (moving to class I) with evident symptoms. As shown in formula 2.The infected population (I) enter the inpatient stage (moving to class H) according to the local medical treatment capability. As shown in formula 3.After treatment, the hospitalized population (H) reduces and enter the recovery population (moving to class R). As shown in formula 4.The recovery population is immune to this disease.Mixed-integer linear programming (MILP) is used to develop a deterministic resource allocation model. Based on the descriptions and assumptions of the abovementioned models, the following multi-objective emergency relief supply distribution decision model can be established for each time cycle, to maximize the weighted sum of the demand satisfaction rates and minimize travel distance:i to epidemic area j at the moment ti to epidemic area j at the moment t, it is 1; otherwise, it is 0.Objective 1: To maximize the weighted sum of demand satisfaction rates for each period.Objective 2: To minimize the sum of driving distances per period.Subject toConstraint (9) is the formula for calculating the demand satisfaction rate. When the supply exceeds the demand, the demand satisfaction rate should be equal to one.Constraint (10) establishes a minimal demand satisfaction rate for all epidemic areas, which reflects the fairness principle.According to constraint (11), the quantity of emergency medical resources delivered from the central distribution center shall not exceed its stock.Constraint (12) indicates the relationship between these two variables, that is, only when the quantity of goods The purpose of constraint (13) is to prevent the supply point from receiving more materials than required.K supply point to prevent wasting of transport capacity.Constraint (14) demonstrates that one epidemic area can only be served by Constraints (15) and (16) are constraints on variable values.The key to solving multi-objective –38 problIn this paper, the multi-objective problem is standardized using min-max first, and then the maximization objective function is converted into the minimization objective.An ε-constraint method is a well-known approach used for solving multi-objective problems. This method is to convert a multi-objective problem into a single-objective problem, where only one objective is optimized and the remaining objectives are treated as constraints. The general form of this algorithm is defined by Eq. (18), where X denotes the feasible set of the mathematical model:Weighted sum method is the most frequently used technique to evaluate efficient solutions for a deterministic multi-objective optimization problem. The multi-objective problem is converted into a single-objective problem using the linear weighted grouping method, and assigning a weight to each sub-objective based on its importance. The overall objective function is:According to Formula (16), when the supply of emergency medical resources is sufficient to fulfill demand, the first term on the right end has a constant value of 1. In this case, the model is simplified as the optimization model to minimize the total driving distance. When the supply of medical resources for all points does not exceed the demand, the optimization model is reformulated into a MILP problem.Constraint (9) is transformed into:Suppose that eight epidemic areas  require immediate assistance as a result of the emergence of the epidemic. Four supply points  can provide emergency medical resources for the epidemic areas, with storage of A = 2,400, B = 2,500, C = 1000, and D = 1100, respectively. One supply point can only supply one epidemic area. Because the propagation of the epidemic differs from other situations and has an infectious time, it is difficult to determine the exact start date of the crisis. As a result, a date in the past is chosen as the start time. Different epidemic areas' start time information is shown in The sensitivity analysis of the calculation results is conducted below, and the effect of adjusting the minimum demand satisfaction rates in the model is studied. When we change the number of supply points in an epidemic area, the results are also different.s j change to hospitalized patients of infectious disease model. δ = , to analyze the impact on the number of infected people. Control variable method was adopted, remaining parameters remained unchanged, and SEIHR model was used to simulate the development of infected persons in Region 1, as shown in As indicated in The trend of the number of each population in the infectious disease model over time can be calculated, by using PYTHON programming to enter the parameters in The model is programmed and the computation results are verified using the LINGO software and MATLAB YALMIP toolbox. When fairness is the main objective, the results of the ε-constraint method are shown in in When the fairness and efficiency weights were 0.5, the results of the weighting sum method are shown in Generally speaking, the proposed dispatching and distribution model can satisfy the demand in each epidemic area to the greatest extent, meanwhile ensuring a fair allocation of materials among all the areas. Supplies will be delivered to each area from the nearest distribution center. Decision-makers can choose either objective based on their evaluations. The dual objective mixed-integer programming model, which takes into account both fairness and efficiency of the allocation of medical resources, also provides decision-makers flexibility to choose either objective based on their evaluations. Therefore, this work provides management insights for improving medical delivery decisions based on demand projections to rapidly respond to public health emergencies.This section primarily examines the effect of the difference in the minimum satisfaction rate of all epidemic areas on corresponding decisions, and the results of the ε-constraint method are presented in The black dots in The results reveal that the value of the first objective function increases as the minimum satisfaction rate increases. Thus, the maximization of the weighted sum of the demand satisfaction rates shows a declining trend. This corresponds to the actual situation.When the fairness and efficiency weights are 0.5, the results of the weighting sum method are shown in The black dots (object 1), blue dots (object 2), and yellow dots  represent the values of objective 1, the normalized objective 2, and the weighted value of the total objective at various minimum satisfaction rates, respectively.As shown in The value of the minimum demand satisfaction rate is extremely important, and decision-makers should assign appropriate values to parameters at various stages of the emergency process based on the actual situation, to ensure that the results obtained are feasible and effective.K  = 2, the satisfaction rate of each epidemic area is shown in This section focuses on the influence of the different numbers of supply points on relevant decision-making in all epidemic areas where K = 2, the weighted sum of the demand satisfaction rate is 0.9522, which is greater than when K = 1. When K = 2, the total travel distance is 4740 km, which is larger than the total travel distance of 4,045 km when K = 1. This trend is in line with common sense. Furthermore, when K = 3, the decision result is the same as when K = 2. Thus, even while the overall weighted satisfaction of residents in the epidemic area increases when supplies are delivered from two supply points, it happens that each supply point may deliver a small number of supplies, resulting in a waste of transportation resources. Therefore, for decision-makers, it is necessary to set up an appropriate number of supply points in the epidemic emergency medical logistics supply to effectively reduce rescue costs and improve the level of emergency services.When In comparison to traditional emergency logistics, emergency medical logistics  has threNumerical results show that due to the hidden nature of asymptomatic infection, the number of infected people in infected areas will increase greatly. Decision-makers should assign appropriate values to parameters at various stages of the emergency process based on the actual situation, to ensure that the results obtained are feasible and effective. It is necessary to set up an appropriate number of supply points in the epidemic emergency medical logistics supply to effectively reduce rescue costs and improve the level of emergency services. Overall, this work provides managerial insights to improve decisions made on medical distribution as per demand forecasting for quick response to public health emergencies.Our study is hypothetical, without actual data. This study only evaluates how to predict the demand for medical aid resources and how to allocate resources based on the number of infected and hospitalized people in the epidemic areas. The supply points have also been fixed for the time being; however, these supply points can be adjusted continuously according to the number of supplies, location advantages, and the number of individuals serving in the epidemic area. Furthermore, the impact of population mobility and natural population growth rate on the spread of the epidemic is not considered in this work. Finally, the paper ignores the impact of material production capacity on decision-making.In conclusion, the performance of emergency medical logistics may be improved significantly. Future research can take the material production supply chain, dynamic resource allocation based on locations of supplies, vehicle scheduling, and vehicle routing optimization problem into consideration. These factors help to establish a model that is more in line with the reality in medical resource allocation, thus more scientific and reasonable to solve practical problems.The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.ZW: supervision. GJ: formal analysis, methodology, validation, and writing – original draft. XY, WC, and TY: visualization. QZ: writing – review and editing. BH: funding acquisition. All authors contributed to the article and approved the submitted version.