Paying attention to particular aspects of the world or being more vigilant in general can be interpreted as forms of ‘internal’ action. Such arousal-related choices come with the benefit of increasing the quality and situational appropriateness of information acquisition and processing, but incur potentially expensive energetic and opportunity costs. One implementational route for these choices is widespread ascending neuromodulation, including by acetylcholine (ACh). The key computational question that elective attention poses for sensory processing is when it is worthwhile paying these costs, and this includes consideration of whether sufficient information has yet been collected to justify the higher signal-to-noise ratio afforded by greater attention and, particularly if a change in attentional state is more expensive than its maintenance, when states of heightened attention ought to persist. We offer a partially observable Markov decision-process treatment of optional attention in a detection task, and use it to provide a qualitative model of the results of studies using modern techniques to measure and manipulate ACh in rodents performing a similar task. Paying attention to a stimulus is costly, both in terms of energy and the lost opportunity to pay attention to something else. It is also beneficial, providing more information about its target. Thus, whether and when we pay more or less attention may best be considered as a choice of internal action that responds to this trade-off. Furthermore, measurements and manipulation of the neuromodulator acetylcholine have suggested that it is one of the instruments of attention, providing us with a window onto this choice. Here, we build an abstract model of a task in which an animal must look out for a brief visual stimulus that may or may not occur on each trial. We show that optimal attentional choices in the model depend on many factors, including how likely a signal is to occur across time, the balance between the improvement in information possible by paying greater attention and its increased cost, and whether there are also costs associated with switching between different attentional states. We also show that our model can qualitatively match results from experiments involving acetylcholine. Vigilance is commonly taken to mean the state of being alertly watchful, and is principally equated with sustained attention in the scientific literature [arousal, which was generally conceptualized in more physiological terms as the level of non-specific activation of cerebral cortex [terature , 2. Peopterature –5). Vigiterature , 7, and l cortex –10 (thoul cortex ). Vigilal cortex .However, rather like selective forms of attention , 14, theConcomitantly, EVC conceptions could be enriched by insights from the human and animal paradigms concerned with the neural foundations of vigilance, which, for instance, point to the central involvement of neuromodulatory systems , 20–23. Here, we use an abstract model of the sustained attention task (SAT) of Sarter and colleagues  to elaboIn the SAT, animals face repeated trials in each of which they have to watch out for a potentially extremely brief signal (a flash of light at a particular location) that comes at an uncertain time during an extended epoch. The end of this epoch is indicated by the insertion of two levers into the operant chamber, and the animal indicates whether it believes a signal has been present or absent during the trial by its selection of lever. Correct responses  are rewarded, while incorrect responses  are unrewarded. Withdrawal of the levers, either once the animal has made a response or after the maximum response time of 4s, marks the beginning of the next trial. Uncertainty about whether and when the brief signal will arrive means the animals have to be vigilant across the whole epoch. Many studies –32 have strong or weak attention), with the difference reflected in the SNR of the input, but also with costs for switching into, and maintaining, strong attention (cf. [strong vs. weak) and the ability to control which attentional state is occupied makes for a form of extended signal detection theory problem with explicitly controlled variable signal quality [weak to strong attention. If so, then consistent with the sustained attention styling of the task, it has to decide whether it is worth maintaining the state of elevated attention across successive trials.From a computational perspective, we treat the decision problem faced by an animal in the SAT as a form of partially observed Markov decision process (POMDP) . We suggion (cf. ). Thus,  quality –39. Our weak/strong and top-down/bottom-up information influences in the cortex. Rather, we treat strong attention as a top-down controlled state in which bottom-up information is more reliable , and so exerts a greater influence over hierarchical processing.We interpret the attentional switching as an internal action  mediatedweak to strong attention when a sufficient hint of a signal is detected. We show that even such a simple model of attention can lead to rich patterns of behaviour, both within and across trials, and in a manner that depends in subtle ways on expectations about whether and when a signal may appear, informational quality, and assumed attentional costs. We also report subtle issues concerning the source of misses and false alarms when ACh is manipulated [We first describe the abstraction of the SAT, and then consider the deployment of attention over the course of single trials, showing conditions under which it is adaptive to switch from ipulated , 30. We ipulated . Finallyweakly or strongly—strong attention yields better quality information, but at a cost. We consider the problem of how to optimize this sequence of decisions regarding attentional state, balancing the putative costs of enhanced attention against the benefits of better evidence. With the exception of Section ‘Optogenetic manipulations’, in which we consider how optogenetic manipulations may disturb correct inference, we assume that the actual and believed qualities of the signal are identical. Formally, the decision problem is modelled as a partially observed Markov decision process . Xn ∈ {pre, on, post} is an ‘external’ state component determined by the experimenter, corresponding to the three possible stages of a trial , and is not directly known by the agent; Yn ∈ {weak, strong} is an ‘internal’ variable that reflects the agent’s current attentional state and is always assumed to be known by the agent , but it determines whether the agent begins the next trial in the weak or strong attentional state—a choice of strong here can therefore be unambiguously interpreted as being motivated by future, rather than immediate, benefits.On all time steps 1, 2, …, N of each trial, the agent must indicate whether or not a signal was presented during the trial. Thus, on the final step, the agent in fact selects one of four possible actions since it must choose both an ‘internal’, attentional action, and an ‘external’ action, i.e., aN ∈ {weak, strong} × {nosignal, signal}. In the following, for simplicity, an will always refer to the attentional action unless otherwise indicated.In addition, on the final step p1, and contains no signal  with probability p0 = 1−p1. On all trials, we assume N0 ≥ 1 time steps where it is known that a signal cannot be present , followed by N1 time steps, where a signal may arrive. We write N = N0 + N1 + 1, allowing for the extra time step for the {nosignal, signal} report. A signal, should one occur, is assumed to have equal probability of arriving at any one of the N1 time steps in a trial, so that a signal’s time of arrival τ isp1 that a trial contains a signal, the probability that a signal arrives on step n given it hasn’t arrived sooner  isq per time step  with assumed (correctly or not) probability ime step . It is aN1 time steps means that λn increases over the course of a trial , which is nonzero only for 1, …, N1, can be expressed asN−1, we assume that the attentional state on step n + 1 is completely determined by action an, so that if the choice is weak on step n, then at the start of the next time step we have Yn + 1 = weak, and vice versa.For the sake of completeness, we also note that the assumption that each trial has a fixed length of mean 1/q . The proNote that these ‘external’ (between world states) and ‘internal’ (between agent states) transitions occur independently, so the dynamics can be written in terms of a Kronecker product between the external and internal dynamics, Yn+1|an) .XN−1 and XN is such that any belief in the on state at the end of time step N−1 is transferred to post at the start of N .Some exceptions to the preceding apply. Firstly, the transition between art of N . Secondlal state . One migO1, O2, …, ON−1. As mentioned above, we assume that no observation is provided at N, where the agent must report whether there was a signal or not during the trial. Observations carry information about whether there is currently a signal or not, but just how informative they are is determined by choice of attentional action. In particular, if the agent chooses to attend weakly on the current time step, then we assumestrongly, then we havestrong than in the weak attentional state  reward values. The only positive reward available in the task comes from correctly reporting the presence or absence of a signal.While Rn, and assume that this itself is a sum of two terms,Ri gives the ‘internal’, attentional cost associated with attentional action an in attentional state Yn,Re gives the ‘external’ reward arising from whether it was correctly or incorrectly reported on step N (signified by χn = N) that a signal was present or absent during the trial, which we simply define asno signal or signal action aN depends on whether XN−1 is pre or {on, post}. Note that the cost of attention only depends, in addition to attentional action an, on the subcomponent Yn  of Sn. It can be interpreted as follows: the cost of weak attention (an = weak) is zero, regardless of the previous action, while there is always a cost of strong attention (an = strong); furthermore, it is assumed that choosing to attend strongly given that one is already in the strong attentional state (Yn = strong) is less costly than choosing to attend strongly given one is currently in the weak attentional state (Yn = weak). The intuition behind this latter assumption is that such repeated action can also be conceptualized as maintaining a (non-default) internal state, which may incur less cost than switching between states .We denote the reward function In the current case, the agent’s posterior distribution over states given its observations, or ‘belief state’, coupled with memory of its most recent action, is sufficient to optimize behaviour. In particular, we can formulate the corresponding ‘belief MDP’, and use familiar dynamic programming methods to solve for the optimal policy.bn + 1(x) the agent’s belief that Xn + 1 = x after seeing all the information On1: on time steps 1, …, n, including knowledge of the attentional actions an1: on those steps (and after one further probabilistic transition). , is a deterministic function of On1:, an1: but, for notational convenience, we treat it as a random variable and do not write down this dependency.) That is, Xn + 1 with the known value of Yn + 1 to make a state Using simplified notation similar to , we denoXn + 1 given action an and particular observation On = on can be derived recursively via Bayes’ rule,n, which takes the initial belief state at n, along with the selected action and ensuing observation, and returns the resulting belief state. The transition function over belief states is thenP is the normalizing constant arising in The transition and reward functions for the belief MDP are derived in the usual way. Specifically, the posterior probability over ollowing , we can Using the full state We consider optimal behaviour in two cases: the case of a single trial, and the case of the average reward rate over continuing trials.N. Here, there is no observation to provide additional information and, furthermore, since there are no future trials to be considered, there is nothing to be gained by choosing other than the attentional action an = weak for any state. Then there is the additional choice of whether to report whether a signal was present or absent. The optimal decision here is simply to choose according to whether one’s belief about being in the pre-signal state is greater than 0.5  or not . From these considerations about the optimal policy at N, we can find the corresponding optimal values n < N using the Bellman optimality equationYn at step n to an attentional action an.For the single-trial case, a suitable objective is to determine a policy that maximizes the expected (undiscounted) return over the trial. Such an optimal policy can be found by simple backward induction. First consider the final time step t can affect the starting state in trial t + 1. One suitable objective in this continuing case is to maximize the average rate of reward [g* using value iteration to solve the system of Bellman optimality equationsdifferential value, based on the average-adjusted sum of (undiscounted) future rewards from each state  and unexpected uncertainty  at both relatively slow (for learning) and fast (for inference) timescales. This complements other forms of representation, such as in population codes [When considering optimal (Bayesian decision-theoretic) behaviour, the representation and updating of uncertainty (as well as utilities) are obviously of key importance. It has been previously suggested , 44, 45 on codes , 47. Ouron codes , 32, 48.The earlier ideas about the role of ACh in inference were that it directly reports cases in which prior expectations about the world are vague . In such cases, it is appropriate to weight external information more heavily  and potentially to improve one’s prior from this new information. However, that work involved Bayesian inference rather than Bayesian decision theory, and so did not consider the additional possibility of an attentional action—i.e., the possibility that the SNR of the input could advantageously, though possibly expensively, be changed to increase the net reward rate. We suggest that ACh also plays this additional role in the SAT, such that the release of ACh can be treated as an internal action rather than as a passive reporter of expected uncertainty. In some cases, these roles may be functionally indistinguishable.an = strong. However, when we consider experiments in which ACh is manipulated directly by optogenetics [We generally consider ACh to report genetics , we discIt is important to note that ACh is substantially heterogeneous , 50. In To facilitate understanding of subsequent results, we briefly introduce the main types of plot we use to understand model behaviour. Key measures of performance are also described.bn. Since at any time step n we have 3 possible trial states, Xn ∈ {pre, on, post}, and a belief state must sum to 1, then the belief space at any time can be visualized as a 2-dimensional simplex . The lower panels of Y. Each time step in a trial has a belief space of this form, and so we can visualize changes in value or policy over the course of a trial by appropriate concatenation of such plots  sums to 1 (the upper plot); the evolving attentional state—we simply label the y-axis ‘attention’—is more precisely the probability of selecting strong at each time step, strong at each step is alternatively rendered in terms of shaded squares when systematically varying parameters , along with the ‘marginal’ expected number of steps per trial in which strong is chosen .Another plot type shows the average evolution of the belief state over a trial, along with the average evolution of the attentional state. an = strong (see Section ‘Attention and ACh: Bridging hypothesis’ above), and so is similarly derived directly from Predicted ACh activity under the model  is based on the assumption that ACh reports e et al.  , F is the false alarm rate , and z(⋅) is the quantile function of the standard normal distribution. Note that we can make such measures conditional on the onset and duration of a signal ; that is, while F is the probability of reporting signal when none has been presented, we can separately consider the hit rates H for signals of different duration, disregarding onset , or even examine H for all possible onset-duration combinations . In Fig 8, we report hit and false alarm rates (middle plots) in addition to d′ (upper right plots).In addition to these plots, we report the common sensitivity measure n theory , 52,d′=across trials. Although from a signal detection viewpoint each trial is independent, and so the actual presence of a signal on one trial bears no relationship to its presence on previous or future trials, the agent’s attentional state is allowed to persist across trials. Indeed, as we noted in the Introduction, it might be substantially more expensive to engage strong attention than to maintain it. Then, if attention is already engaged on trial t, it might be optimal to pay the cost of sticking in the strong state for trial t + 1 and thereby avoid the even greater cost of having to re-engage this state, if necessary. In turn, this would mean that the choice to engage attention on trial t can be partly justified by the benefits of strong attention on trial t + 1. Thus, we need to consider an inter-trial timescale as well as an intra-trial one. We start with the case of an isolated trial, and then treat cross-trial influences.We consider both internal and external decision-making over the two key timescales in the task. The most obvious timescale is that associated with each individual trial. Within a trial, the agent has to accumulate evidence about the presence of the signal, decide whether and when to engage or disengage attention, and finally report their external choice . However, we are also concerned with what occurs weak → strong) or disengage (strong → weak) attention within a trial if we only consider optimizing behaviour in a single trial ? We expect multiple model parameters to interact to determine the answer, since the probability that a signal is on and its temporal extent , the relative qualities of information available in different attentional states , and the costs of strong attention  can all be expected to play a role. To gain some insight, we examine a particular regime in detail (these will be our ‘default’ parameters) before more systematically examining the effects of varying parameters.When does it make sense to engage .In our first, ‘default’ regime, we assume that a signal is as likely as not, weak attentional state; the lower plot highlights step n = 7 for clarity. At the decision state (n = 10), state values are higher for more certain beliefs (where one is more likely to be correct), and lowest where there is most uncertainty (where one can expect to perform at chance). Moving backwards in time, the value remains high when convinced there is, or has been, a signal  This region grows over time in accordance with the growing probability of a signal being on. In neither case, however, is there any reason to select strong in the final decision state (n = N = 10)—this is because no information is provided at this state, and so it can have no bearing on performance in the current trial .weak attentional state  or strong (Y1 = strong) attentional state (the difference between which is quite subtle for the current parameters). In all cases, the initial certainty about the pre-signal state decreases, and on average ends up above 0.5 for a non-signal trial, and below 0.5 for a signal trial.For an alternative view, weak state  or not (left) there is actually a signal in the trial, and returns to zero at the decision state (n = 10). While initially the probability of choosing strong is similar for non-signal and signal trials, one can see that this probability reaches a higher level in the non-signal case, and is maintained up to the penultimate time step. By contrast, the probability of choosing strong on a signal trial decreases from time step 6 onwards. This difference comes from the asymmetry already mentioned: once a signal has been detected, there is no reason to pay the cost of remaining in the strong attentional state, since higher quality sensory information cannot change the agent’s mind—a signal can either stay on or switch off. However, if the signal has not yet been detected, it remains theoretically possible (until n = 10) for it to turn on. We also see this effect when starting in the strong state . Regardless of whether the trial is started in the weak or strong attentional state, an earlier onset effectively means that a decision is made earlier that a signal is indeed present, and so attention can be relaxed sooner within the trial. On the other hand, being sure that there has been no signal so far is no guarantee that one will not arise in the future—and so, once in the strong attentional state, it can make sense to maintain this heightened state of vigilance.strong state leads to improved sensitivity , will also be relevant when we consider sequential effects below . Although the overall probabilities of occupying the strong attentional state are much lower , one sees that the engagement of strong attention is more likely on a signal than a non-signal trial. Note, however, the much greater probability of missing a signal, since, again in expectation, In the preceding example, we set rial cf. . Fig 7 sweak’ policy, which enforces constant occupation of the weak attentional state; and the ‘always-strong’ policy, which means the strong attentional state is always occupied.We next measure the effects of varying the default set of parameters systematically, one at a time. In reporting hit/false alarm rates, sensitivities, and average reward rates, we compare the optimal policy to two extreme cases: the ‘always-p1 and q, there are intermediate parameter values which lead to greatest engagement of strong attention; otherwise, as the relative improvement in quality of information from engaging strong decreases , or as the associated costs increase , then there is progressively less engagement of strong attention until a ‘breaking point’ is reached—when it is no longer paying the price of strong attention at all.The overall results  can be sp1 more systematically . On the other hand, if q = 1, then a signal only ever lasts a single time step, and the cost of choosing strong even at a single time step outweighs the benefit in terms of enhanced detection  where it is no longer worthwhile at all , there is again a point (|κ| ≈ 0.15) at which this cost is no longer worth paying , we also observe a decrease in the willingness to occupy this state , then the instantiation of On is indeed generated from μ depends on whether a signal is in fact present (μ = 1) or not (μ = 0), and inference proceeds on this basis  or always-weak (‘supp.’) attentional policies, and the effect on inference is consistent/matched to the policy. As expected, hits increase  under always-strong and decrease under always-weak, while false alarms remain essentially the same (tiny decrease) and increase, respectively , but the qualitative pattern of results holds for a broad range of model parametrizations. strong (‘act.’) or always-weak (‘supp.’) policy  or always-weak (‘supp.’) policies  decrease the false alarm rate.Another type of mismatch is when observations are indeed generated under the optimal attentional policy, but treated as being generated under always-policies : hit ratThe fact that the latter, and not the former, type of mismatch captures the qualitative trend of results is notable, as this allows us to decompose the putative three roles that ACh plays: in order for the (i) increase in the SNR of the input to have an effect, the impact of the bottom-up information concerned has to be increased relative to prior expectations—something that can happen through (ii) boosting the strength of this signal, (iii) the suppression of priors, or both. The mismatch more consistent with the data ; (ii) a strong attentional state is at least sometimes maintained into the next trial following a hit trial (since no increase in ACh was observed for consecutive hits); and (iii) strong → weak shifts do sometimes occur.In terms of our model, attentional dynamics—and, by hypothesis, so too ACh dynamics—that are similar to the reported pattern of ACh activation will arise in the model if the following are satisfied: (i) a weak → strong shift is more likely to occur on a signal trial than a non-signal trial in the model if signals are expected to be rare  occurs. One might draw a distinction between passive and active mechanisms of disengagement: active disengagement would be mandated by the optimal policy (responding to the relative costs and benefits); while passive disengagement would not be so mandated, but rather occur sporadically, somewhat like a ‘lapse’ process, or perhaps even associated with the sort of spontaneous fluctuations in attention apparent in longer-running vigilance tasks [δ in the model was introduced principally to allow for the latter possibility—even if the intention is to begin the next trial in the strong attentional state, one in fact begins in the weak state with probability δ.We already saw that a s trials . What wece tasks , 5. The p1 = 0.5, and is fixed at this level, is not a given. The signal probability needs to be estimated across trials, and may or may not be assumed to be stationary. Since the optimal policy depends on the estimated signal probability, if the latter changes, then so too should the policy .Here, we focus on what is arguably the more interesting case—active disengagement. One possibility, amongst others see , arises strong attentional state across trials, while for lower estimates it is not. Since a signal trial will lead to an increase in strong attentional state if it is occupied; conversely, a non-signal trial will lead to a decrease in strong attentional state.Consider the case where the estimated signal probability weak → strong shift occurs during the trial, then this shift immediately prompts weak → strong shift always indicates a suspicion that a signal may be present, whereas a strong → weak shift does not necessarily indicate a belief that a signal is absent . The second way is in response to feedback  at the end of each trial, which, at least from the experimenter’s point of view, unambiguously indicates whether a signal was or was not in fact present in the trial. Therefore, according to this latter assumption, we deterministically set strong is always maintained across trials when q = 0.2, σw = 1, σs = 0.5, κ = −0.1, ϵ = −0.005, δ = 0.001).In weak → strong shift, and the response was correct (hit); this is an incongruent hit (iH). Since strong attentional state is maintained into Trial 3. This trial is also a signal trial, and the response is correct )—but crucially, since already in the strong state, there is no change in attentional state. Trial 4 is a (incongruent) correct rejection (iCR) trial, and the lack of a signal leads to a reversion of weak → strong shift, and is a hit.The first trial is a miss (M) trial , but the feedback indicates that a signal was in fact present, and so weak → strong shifts , while congruent hits can be associated with no change in attentional state . Indeed, if we run a large number of trials, and treat attentional/ACh dynamics in the model similarly to how Howe et al. treated their ACh data , we find a similar difference between congruent vs. incongruent hits .The point of this example is to show one way in which incongruent hits can be associated with ent hits , right. ent hits , right. weak → strong shift because attention is likely to start weak  and be activated during the course of the current, signal trial; incongruent correct rejections are associated with a strong → weak shift because attention is likely to start strong  and be deactivated at the end of the current, non-signal trial . By comparison, in strong → weak shifts that all occur at the same time . The latter thus leads to a much stronger (and consequently mispredicted) decrease in attention/ACh. In this light, it would certainly be interesting to consider more realistic assumptions about ongoing estimation of Of course, Howe et al.  observedal trial . Our misstrong attention yields better sensory information than weak, but incurs costs. We analysed how optimal allocation of attention  depends on signal characteristics , the relative qualities of information in the different attentional states, and the costs associated with paying strong attention. We also considered what happens when the actual and expected signal properties do or do not match, and possible relationships with the effects on detection performance observed in an optogenetic study.We considered vigilance, or sustained attention, from the point of view of an optimal control problem in which a series of choices of internal, attentional action are made, and where these internal actions may accrue both costs and benefits. In particular, inspired by the sustained attention task  developstrong attention into the next trial, not because of any immediate informational benefit—there is none—but in order to avoid the future cost of a weak → strong shift. We demonstrated that this can lead to trial history effects in when weak → strong shifts occur, and suggested that such sequential effects may help explain when cortical ACh transients have been observed to occur in the SAT.In the multi-trial case, we saw that optimal allocation of attention also takes into account performance in future trials. Here, it can be optimal to pay the costs of maintaining We treated the elective allocation of attention as a characteristic form of internal action . Such inFrom the perspective of a general POMDP , the waystrong and weak, which has the practical benefit of facilitating analysis, but is also consistent with interpretations of behaviour in the SAT in terms of shifts between a small number of task-sets or ‘modes’, such as ‘default’ and ‘detection’ modes [We assumed just two attentional states, n’ modes   and the model ascribed to the agent. While this was somewhat pragmatic in our hands, we suspect such mismatches to be not uncommon. In the particular case of the SAT, subjects ought, if they understand that the feedback at the end of each trial reliably indicates whether a signal was present, to apprehend that the long-run relative frequency of a signal trial is approximately one half—but only then if they also (correctly) believe that this relative frequency is fixed.Particularly when engaging and/or maintaining attention is expensive, it would often be better to have either n theory . Here, wn theory , 37 and n theory , 70. Then theory —most obvn theory , which wigilance , 73. Notigilance , althougOther cases of heterogeneous attentional engagement have been studied when external information signals that the environment has switched into a markedly different state . This hShea-Brown et al.  previousstrong → weak attentional transitions, is an important task for the future.On a rather different timescale are the fluctuations in attention typically observed over the course of minutes to hours in the long-run vigilance tasks often used to interrogate sustained attention in humans , 5, 17. We designed and illustrated our account in the light of the sustained attention task (SAT) of Sarter and colleagues , 30, 33.The observation that stimulating the BF also increases false alarms  is hardeIn contrast to the effects of stimulating the cell bodies of cholinergic neurons in the BF, Gritton et al.  found thNote that the authors of the original study have a different interpretation of the ACh release, at least in PFC. That is, Howe et al.  interprePinto et al.  used optPinto et al.  primarilHowever, we should note that some studies of the activity of ACh neurons do not find the sorts of responses that our analysis would expect. For instance, Hangya et al.   based on interactions between thalamic glutamatergic afferents and heteroreceptor-mediated regulation of cholinergic terminals, rather than reflecting phasic activity of BF ACh neurons [Along with these effects within a single trial is the evidence about the engagement of ACh when considering the sequential structure of trials . Here, p neurons . Howe et neurons  speculatstrong attention being much less than a weak → strong switch, and also being outweighed by the benefit of starting a new trial with strong attention engaged . The second assumption is that attention will sometimes revert to weak, thus requiring another weak → strong shift; in our parameter regime, these latter shifts will happen preferentially on signal trials. We focused on an active mechanism of disengagement  arising from fluctuating estimates in signal probability but, as noted, changes in other parameters could also play a role; utilities, costs, signal-to-noise ratios could all plausibly fluctuate over the course of a trial as an animal’s state of satiety and/or fatigue varies  that the signal has been present. This depends on the cost of maintaining s . The mos  and a detection mode. The costs in switching between the two are then analogous to the switch costs that are prevalent in the task-switching literature [Our account followed Hasselmo and Sarter  in treatterature . We assuOne important predecessor of the current work is by Atkinson , who assOur task and analysis concerns a relatively simple signal—the turning on and off of a light of fixed intensity and location, and of variable duration —and response , and so is likely not ideal for addressing heterogeneity in cholinergic neuromodulation, let alone the substantial additional complexities of cortical control over attentional enhancement  and itsStepping back from the details of this task and model, our simulations exemplify ways in which a neuromodulator might provide central regulation and coordination of the sort of radically distributed processing that otherwise occurs in the context of point-to-point, wired, connections.


Open access funding enabled and organized by Projekt DEAL.The article “Arthroscopic Bankart repair with an individualized capsular shift restores physiological capsular volume in patients with anterior shoulder instability”, written by Helge Eberbach, Martin Jaeger, Lisa Bode, Kaywan Izadpanah, Andreas Hupperich, Peter Ogon, Norbert P. Südkamp and Dirk Maier, was originally published Online First without Open Access. After publication in volume 29, issue 1, page 230–239 the author decided to opt for Open Choice and to make the article an Open Access publication. Therefore, the copyright of the article has been changed to © The Author(s) 2021 and the article is forthwith distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit The original article has been corrected.


In this paper, we aim to explore connections between a Carnapian semantics of theoretical terms and an eliminative structuralist approach in the philosophy of mathematics. Specifically, we will interpret the language of Peano arithmetic by applying the modal semantics of theoretical terms introduced in Andreas (Synthese 174(3):367–383, 2010). We will thereby show that the application to Peano arithmetic yields a formal semantics of universal structuralism, i.e., the view that ordinary mathematical statements in arithmetic express general claims about all admissible interpretations of the Peano axioms. Moreover, we compare this application with the modal structuralism by Hellman , arguing that it provides us with an easier epistemology of statements in arithmetic. Tractatus Logico-Philosophicus   , was verhitehead .Aufbau  appeared. This deviation culminated in a semantics of theoretical terms that is centred on the doctrines of indirect and incomplete interpretation  designates the set of models of A, where A is a set of sentences. To make these ideas about admissible structures precise  closely corresponds to the two cornerstones of a Carnapian semantics of theoretical terms. To recall, (i) theoretical terms are indirectly interpreted by a set of postulates, and (ii) this interpretation may not determine the referents of theoretical terms uniquely. Given this correspondence, how can the modal approach be applied to arithmetic? In other words: how can we explicate truth conditions of arithmetical statements in the modal semantics?To address these questions, let us first fix some terminology.(Peano arithmetic) Let Second-order Peano arithmetic, is axiomatized as follows:(PA1) (PA2) (PA3) (PA4) S’ representing the successor function. Other arithmetical symbols such as the function symbols for addition or multiplication or the numerals ‘2’, ‘54’, etc. can be added by explicit or recursive definitions to and observational terms. As we saw, theoretical terms are those interpreted indirectly through the postulates in question. Observational terms, in turn, have a fixed interpretation in the observable world.Two points of commentary are in order. Notice first that the signature of t is theoretical—relative to a set t is based on some member of t. Now, the crucial question is whether or not we can determine the extension of the numeral ‘0’ and the function symbol ‘S’ without a theory about the natural numbers? To determine the extension of the numeral ‘0’ is to determine the referent of this numeral. What is the referent of the numeral ‘0’? Obviously, we cannot answer this question without having some theory about what the natural numbers are. If we say that the numeral ‘0’ refers to the empty set, we already have some theory about the natural numbers. While van Neumann’s theory and Zermelo’s theory of the natural numbers agree on the interpretation of the numeral ‘0’, these two theories obviously diverge as regards the interpretation of the successor function. If we let the numeral ‘0’ refer to itself, we have yet another theory of the natural numbers. We can certainly apply the numeral ‘0’ at a quotidian level in the sense of determining the cardinality of certain sets, such as the cardinality of the set of elephants in the class room. However, even these simple applications presuppose theories, namely an understanding of natural numbers as cardinal numbers and the comprehension principle of some set theory that allows for Urelements. Note, furthermore, that the semantic dependency of the numeral ‘0’ carries over to the function symbol ‘S’. If we need a theory to determine the referent of ‘0’, we also need a theory to determine the extension of the successor function.How can we apply the theory-observation distinction to the language of Peano arithmetic? Recall that in Sect. In sum, both the numeral ‘0’ and the successor function ‘S’ are theoretical terms in the sense that we cannot determine their referents without having some theory about the natural numbers. To be more precise, these two expressions are theoretical relative to a range of different theories about the natural numbers, each of which yields different interpretations. Hence, It seems as if we can infer Given this, the conjunction of axioms (PA1)-(PA4) can be represented as -admissible structuresD, a distinguished element So, we must wonder whether the set S’ as non-logical and thus theoretical terms.How to think of the truth conditions of arithmetical statements? Given our specification of admissible structures, it should be clear that the modal semantics is also directly applicable to the case of Peano arithmetic. Consider any well-formed sentence in a definitional extension of language R thus presents a total binary relation between any two structures satisfying According to the modal approach sketched above, we can say that an arithmetical sentence Given this general account of arithmetical truth, three comments should be made here. First, the modal semantics outlined allows us to capture the referential indeterminacy of mathematical terms. More generally, this semantics captures two types of indeterminacy. First, the referential indeterminacy of theoretical terms, including individual constants. Second, the semantic indeterminacy of theoretical sentences. A theoretical sentence is semantically indeterminate if and only if it is true in some admissible structure and false in some other structure that is also admissible. In the case of Peano arithmetic, we have referential indeterminacy of non-logical symbols, but not semantic indeterminacy of arithmetical sentences. Semantically indeterminate sentences of Peano arithmetic are ruled out by the fact that Second, the modal approach to arithmetic is in agreement with Carnap’s original discussion of the semantics of theoretical terms in Carnap  and in lFinally, the modal semantics nicely captures the structuralist view of arithmetical knowledge first introduced by Benacerraf.8The modal semantics for mathematical languages outlined above squares well with a non-realist (or eliminative) account of structuralism. It is therefore interesting to see how the approach relates to the most rigorously developed theory of eliminative structuralism, namely Hellman’s modal theory presented in Hellman . In thisAs was mentioned in the introduction, Hellman’s theory was developed against the background of Putnam’s influential article “Mathematics without foundations”  and the possible existence of such objects and of structures in which they occur. Hence, any theorem of number theory, analysis, or any other mathematical discipline can in principle be reformulated, in Putnam’s view, as a modalized conditional statement and thus “essentially a truth of pure modal logic” ; and (2) there are necessary truths of the form “if concrete example of an  a translation of a given mathematical discourse into a formal “structuralist language”, (ii) the specification of a “structuralist theory”, and finally, (iii) a “justification” of the proposed translation scheme . Building on Putnam’s suggestions, Hellman develops a general modal structuralism that is applied to different mathematical theories. Specifically, his account is based on the interpretation of theories of pure mathematics in higher-order modal logic, in which the two modal operators  are treated as primitive. In what follows, we will discuss the three components of Hellman’s modal structuralism for the case of arithmetic, beginning with an exposition of his structuralist translation scheme.The main aim in the first chapter of Hellman’s book is to provide a “modal-structural interpretation” of Peano arithmetic and analysis in a system of modal logic. This is done purely syntactically in terms of a translation scheme that allows one to transform any well-formed arithmetical statement into a purely “structuralist language”, namely a second-order language of modal logic. In Hellman’s view, the translation in question is warranted from a nominalistic point of view since it transforms ordinary arithmetical sentences , a binary relation symbol ‘S’ (for the successor relation), and the numeral ‘0’. Consequently, the axiomatization of second-order Peano arithmetic discussed by Hellman also differs from the above account to the effect that all quantifiers in the axioms are relativized to the domain predicate ‘N’.universally Ramsifying both the axioms of N’, ‘0’, and ‘S’ in these formulas by the universally quantified variables ‘X’, ‘x’, and ‘f’ respectively.Before turning to Hellman’s modal reconstruction of arithmetic in closer detail, it should be noted that his interpretation is based on a slightly different formalization of the Peano axioms than the one presented in the previous section. of the ”metalinguistic” claim thatMoreover, while Hellman’s interpretation of arithmetical statements in modal logic is purely syntactic in character, his underlying motivation is a semantic one. What the modal translation of “mentclasspt{minimaLet us now turn to the second, “categorical” component of the modal reconstruction. As Hellman points out, the modal-structuralist interpretation of number theory is based on the assumption that “ex falso quodlibet, it would follow that the modal translation of any statement expressible in the language of arithmetic would turn out to be true. Thus, the meaningfulness of HYP as a structuralist reconstruction of arithmetical statements presupposes that there possibly exists a model of Roughly put, if it were impossible that locus classicus of [the modal] approach is Putnam’s ‘Mathematics without Foundations’, in which it was suggested that many of the problems plaguing objects-platonism  could be overcome by reinterpreting mathematics, as standardly presented, in a modal language, in which a notion of mathematical or logical possibility is taken as primitive. Since the work of Kripke, we have become familiar with the procedure of providing a set-theoretic semantics for the operators of modal logic. What Putnam proposed was the opposite procedure of working with those operators as primitive, and using them to reconstruct Platonist discourse in such a way that literal quantification over abstract objects could be made to disappear entirely. (ibid. pp. 7-8)The Thus, following Putnam’s original suggestion, the modal operators in Hellman’s structuralist language are also specified purely syntactically, based on an S5 logical system. Hellman’s item (ii) of his list, namely a “structuralist theory” used for the specification of the structural content of arithmetical statements, is thus an S5 second-order modal logic.13Given this brief outline of Hellman’s account, how does it compare to the modal semantics outlined in the previous section? As we saw, both accounts provide a logical analysis of arithmetical knowledge in line with eliminative structuralism. That said, there are several differences between the two approaches. Arguably the central difference concerns the very notion of modal operators. Andreas’ modal approach to theoretical terms is, as we saw, essentially model-theoretic in character. It provides a Kripke frame semantics for theoretical statements based on the concept of admissible structures. Hellman’s “modal-structuralist interpretation” of arithmetic, in contrast, is syntactic in character . His reconstruction consists of a translation manual that transforms arithmetical statements into complex sentences of second-order language with modal operators. It is crucial to emphasize that these operators—or the concepts of “logico-mathematical” possibility and necessity expressed by them—are treated as primitive by Hellman. Thus, unlike in Andreas’ modal semantics, the very idea of describing the logical behaviour of these operators in terms of a set-theoretic semantics is deliberately ruled out by him. The reason for this is Hellman’s mathematical nominalism: providing a Kripke semantics for modal expressions would effectively reduce the “modal description” of mathematics to a set-theoretical and thus to a realist one. Compare Hellman on this point, commenting on Putnam’s modalism in mathematics:The uses of modal logic therefore differ significantly in the two accounts: for Hellman, the modal-logical reconstruction of arithmetic serves primarily the aim of nominalization, that is, of getting rid of any commitment to the actual existence of numbers and number sequences. In the reconstruction of arithmetic based on Andreas’ modal approach to theoretical terms, modal logic is used primarily to represent the referential indeterminacy of the arithmetical vocabulary. The reference to abstract objects such as sets is viewed as permissible for the interpretation of the primitive symbols of Peano arithmetic.This observation leads to a second difference, namely on how precisely the structural content of arithmetic is characterized in the two accounts. Structuralism is essentially viewed as a semantic thesis in our modal approach: as we saw, the structural content expressed by an arithmetical statement can be analyzed in terms of its truth conditions specified relative to a class of theoretically admissible structures. In contrast, in Hellman’s approach, the modal component plays a surprisingly small role in his structuralist reconstruction of arithmetic. Recall that Hellman’s theory is usually characterized as a form of eliminative structuralism, that is, an account that understands talk about mathematical structures as a kind of shorthand for talk about all models of a theory. Thus, eliminative theories of structuralism do not assume the existence of abstract structures in addition to more concrete mathematical systems or models.14What makes Hellman’s modal theory a form of eliminative structuralism in this sense is the universal Ramsification and the “if-thenist” reconstruction of mathematical statements outlined above. More specifically, it is the suggestion to translate any statement of arithmetic Notice, however, that if the structuralist thesis in Hellman’s account is viewed in this way, the two approaches actually turn out to be quite similar. Hellman’s HYP essentially expresses the generalization over all structures admissible for the semantic interpretation of an arithmetical statement. Theoretical admissibility is expressed in the object language here, namely in terms of the Ramsified version of the Peano axioms formulated in the antecedent of HYP.  In our proposed semantics for the arithmetical language, one also generalizes over the admissible structures of an arithmetical statement. However, we have defined the notion of an admissible structure metatheoretically and used it in the formulation of a suitable Kripke semantics.This similarity between the two approaches becomes more evident if one accepts Hellman’s modalized version HYP, but evaluates it semantically by use of Kripke models. We mentioned above that, according to Hellman, a central component of any structuralist interpretation of mathematical theories consist in the “justification” of the suggested translation manual.  In the case of his modal theory, this justification consists in showing the accuracy and adequacy of the translation scheme HYP.  More precisely, it is to show that any statement In the first chapter of his book, Hellman outlines two ways to justify this equivalence for the case of Peano arithmetic: an “external” one which allows one to draw to the resources of set-theory as well as a “justification from within” his nominalist modal framework. We leave aside here the latter approach which centers on the notion of internal categoricity. The external, set-theoretic justification, in turn, is based on the specification of two semantic frameworks—namely a realist notion of ”truth in the standard model” and the modal-structuralist notion of ”truth in any possible model”—that allows one to compare an arithmetical statement and its modal translate . Specifically, Hellman sketches a proof of the following equivalence result in this context: a statement equivalence theorem” :15D is a universal domain of individuals, R a universal accessibility relation. Given this set-up, it can easily be shown that the two accounts are semantically equivalent. Thus, for any statement N’, ‘0’, and ‘S’ in the models of Second, turning to the modal component of his reconstruction, Hellman follows Putnam in that “the Platonist may make reasonable sense of the modality in question by providing a set-theoretic semantics for it  the translation scheme into a structuralist language, (ii) the structuralist theory, and (iii) the justification of the proposed translation scheme.What are the benefits of Hellman’s modal structuralism compared to ours? Arguably, both Hellman’s and our approach work out the components of a structuralist account of mathematics in a coherent manner. The proposals for (ii) and (iii) even exhibit substantial commonalities. However, the nominalism of Hellman’s approach seems to be a key benefit compared to ours. We agree with Hellman, Putnam and Ockham that it is desirable to work with fewer entities if there is a way to do so. Otherwise, we find it difficult to identify clear-cut advantages of Hellman’s structuralism over the ours.What are the benefits of the proposed modal structuralism compared to Hellman’s? First, the translation scheme into a structuralist language is much simpler. Once an arithmetical statement has been translated into the language of second-order Peano arithmetic, we merely need to prefix the translation with a modal box operator. Unlike Hellman’s translation scheme, no second-order variables or additional quantifiers are needed. Hence, our modal structuralism stays closer to the surface grammar of mathematical discourse than Hellman’s. Relatedly, the simplicity of the translation scheme is a virtue in light of standard criteria for assessing theories. If two competing theories capture the same subject matter, we should prefer the simpler one unless the other theory scores better in other dimensions. There is wide-ranging consensus on this being a reasonable criterion for the evaluation of theories in the sciences, mathematics, and philosophy.analytic philosophy, as initiated by Frege, Russell, Carnap, and others. Other types of doing philosophy may be worth pursuing too, but we are not going to do this in this paper. This is not say that Hellman’s account of structuralism parts from the analytic tradition, but to argue that our modal structuralism follows this tradition more strictly.Another benefit we see in our modal structuralism is that modal operators are not left without an explanation. Such operators are rather understood in the standard way. This means that our modal structuralism coheres better with our understanding of the modal operators in a number of different areas, including modal logic as taught in an advanced course to non-classical logic. Moreover, we take it almost for granted that it is a virtue of any account in philosophy not to use highly abstract notions, such as the modalities, without an explanation. This is simply part of the game called external coherence. To finally evaluate the two accounts, let us assume that the following criteria have equal weights: (i) avoiding unnecessary entities, (ii) simplicity, (iii) external coherence, and (iv) the criterion of avoiding primitive notions which are abstract. Then our modal structuralism scores better than Hellman’s. In this sense, we have reason to prefer the former over the latter.In sum, Hellman’s modal structuralism has the benefit of achieving a reduction of entities. No sets are needed in his account. Our modal structuralism has the benefit of a simpler translation scheme into a structuralist language, which comes with the benefit of staying closer to the surface grammar of arithmetical discourse. The latter benefit may or may not be argued to be additional to the former; we do not take a side here. Further, our modal structuralism does not take the modal operators as primitive. Since these operators are understood in the standard way, our account of structuralism coheres better with the understanding of modality in many areas of contemporary philosophy and philosophical logic. Drawing on standard criteria for assessing scientific theories, we can therefore say that our modal structuralism exhibits a higher degree of Andreas’ modal semantics applied to arithmetic yields the following understanding of arithmetical truth: a statement D, there is a closed individual term of Tractatus  with a structuralist account of arithmetic. Specifically, we applied Andreas’ modal semantics for theoretical statements of scientific theories to the semantic evaluation of statements of Peano arithmetic. As was shown, the resulting modal semantics squares well with a particular structuralist account of arithmetic, namely, eliminative structuralism. This was characterized as the view that the structural character of arithmetical knowledge is best expressed in terms of the generalization over all possible interpretations of Peano arithmetic. We saw that the central semantic consequence of this view, namely the indeterminacy of reference of arithmetical terms, is adequately modeled in the modal approach. Given this, we then compared the modal semantics for arithmetic to the most rigorously developed theory of eliminative structuralism, namely Hellman’s modal structuralism. As was shown, there are several interesting points of contact between the semantic view and Hellman’s account. Using standard criteria for assessing theories in science and mathematics, we have argued that our proposed modal structuralism scores better than Hellman’s. However, our proposal fails to meet the nominalist tenets of Hellman’s structuralism.Braithwaite-Ramsey conjecture. This conjecture has recently and independently been formulated by Halvorson (At the end of his seminal “What Numbers Could not Be” , Benaceralvorson , p. 282.