Jean-Philippe Corre

Dorian Obino
email: dorian.obino@pasteur.fr

Pierre Nivoit

Aline Yatim

Taliah Schmitt

Guillaume Duménil
email: guillaume.dumenil@pasteur.fr

Anti-thrombotic treatment enhances antibiotic efficiency in a humanized model of meningococcemia

Keywords: Antibiotic efficiency, Coagulation, Neisseria meningitidis, Platelets, Sepsis

Meningococcal infections remain particularly difficult to treat. Despite antibiotic therapy, the state of the patients often rapidly deteriorates. Early clinical studies suggest that meningococci acquire a form of resistance to antibiotic treatments during infections. Taking advantage of a humanized animal model of infection, we confirm that adherent bacteria become highly resistant to antibiotic treatments as early as 3-6 hours post infection, although fully sensitive in vitro. Within this time frame, meningococci adhere to the endothelium via their type IV pili, proliferate and eventually fill the vessel lumen. Using intravital imaging, we show that rapidly upon infection blood flow is dramatically decreased, thus limiting antibiotic access to infected vessels. Concomitantly, fibrin is deposited inside infected vessels in proximity to bacterial aggregates. Pharmacologically impairing thrombin generation by inhibiting Factor X activity not only improves blood flow in infected vessels, but also enhances the efficacy of the antibiotic treatment. Our results indicate that the combined administration of anticoagulants together with antibiotics might represent a therapeutic approach to treat meningococcal sepsis more efficiently.

Introduction

Despite intensive medical care meningitis and purpura fulminans induced by Neisseria meningitidis remain a major concern worldwide, reflecting our limited understanding of the mechanisms underlying these diseases. Systemic forms of the infection are often the most life-threatening as they account for up to 90% of the mortality associated to invasive meningococcal diseases (IMD) (Brandtzaeg & van Deuren, 2012;[START_REF] Van De Beek | Community-acquired bacterial meningitis in adults[END_REF].

Meningococcal sepsis is characterized by specific clinical manifestations such as the appearance of skin purpuric/petechial lesions and most importantly a particularly fast and severe progression [START_REF] Stephens | Conquering the meningococcus[END_REF][START_REF] Thompson | Clinical recognition of meningococcal disease in children and adolescents[END_REF]. Otherwise healthy patients evolve from an absence of symptoms to being critically ill within 12 to 24 hours. Although made difficult by the relatively unspecific clinical features at disease onset, the early recognition and treatment of meningococcal systemic infections by parenteral administration of antibiotics is key in determining patient outcome [START_REF] Dellinger | Surviving sepsis campaign: international guidelines for management of severe sepsis and septic shock: 2012[END_REF]. Remarkably, despite antibiotic administration, disease often progresses relentlessly towards septic shock making these infections particularly difficult to treat. While it is generally considered that treatments will be inefficient past a certain tipping point, the reasons for these failures remain largely unexplained. A possible explanation would be that bacteria acquire a form of resistance to the treatments at a certain point of infection. Accordingly, biopsies of skin lesions performed on patients suffering from meningococcal septic shock revealed the presence of live bacteria up to 13 hours after the initiation of the antibiotic treatment (de [START_REF] De Greeff | Pre-admission clinical course of meningococcal disease and opportunities for the earlier start of appropriate intervention: a prospective epidemiological study on 752 patients in the Netherlands, 2003-2005[END_REF][START_REF] Van Deuren | Update on meningococcal disease with emphasis on pathogenesis and clinical management[END_REF][START_REF] Van Deuren | Rapid diagnosis of acute meningococcal infections by needle aspiration or biopsy of skin lesions[END_REF]. The mechanisms underlying bacterial antibiotic resistance at play during meningococcal infections remains elusive. Due to the human-species specificity of the bacterium, the lack of reliable animal models of meningococcal infections has precluded any in-depth experimental studies of this process. Today, this limitation has been overcome by the development of a human skin xenograft mouse model that supports the anastomosis of dermal human blood vessels with the murine circulatory system [START_REF] Melican | Adhesion of Neisseria meningitidis to dermal vessels leads to local vascular damage and purpura in a humanized mouse model[END_REF]. This model recapitulates the hallmark features of meningococcal systemic infections, such as the adhesion of bacteria to the human endothelial surface and the occurrence of skin purpuric lesions, hence allowing the exploration of Nm systemic infections in the full complexity of the host tissue in vivo [START_REF] Bonazzi | Intermittent Pili-Mediated Forces Fluidize Neisseria meningitidis Aggregates Promoting Vascular Colonization. Cell Brandtzaeg P, van Deuren M[END_REF][START_REF] Melican | Adhesion of Neisseria meningitidis to dermal vessels leads to local vascular damage and purpura in a humanized mouse model[END_REF]. Combining the use of such a model and intravital imaging, we show that bacteria become resistant to antibiotic treatments as early as 3-6h post infection and that this effect is due to a fast decrease in blood flow induced by the deposition of fibrin inside infected vessels.

Results

Meningococci adhering along the vascular wall rapidly acquire antibiotic resistance

We first sought to evaluate whether the humanized model of meningococcal infection could provide new information on the clinical observations pertaining to antibiotic resistance during infection. In this model, intravenously injected Neisseria meningitidis circulate in the blood stream and a fraction of bacteria specifically colonizes dermal human vessels in the grafted skin, forming tight bacterial aggregates that progressively fill the infected human vessels (Fig 1A-B) [START_REF] Manriquez | Colonization of dermal arterioles by Neisseria meningitidis provides a safe haven from neutrophils[END_REF][START_REF] Melican | Adhesion of Neisseria meningitidis to dermal vessels leads to local vascular damage and purpura in a humanized mouse model[END_REF] 2C), indicating that blood flow reduction was not due to a global collapse of vessels. We recently showed that meningococci can infect dermal arterioles and venules [START_REF] Manriquez | Colonization of dermal arterioles by Neisseria meningitidis provides a safe haven from neutrophils[END_REF] and the same dramatic decrease in blood flow was observed in infected venules and arterioles (Fig 2D), suggesting that the effect observed is independent of the vessel type.

Taken together, our results indicate that by preventing the perfusion of infected vessels starting at 3 hours post infection, vascular colonization by meningococci limits the bioavailability of antibiotics, thus restraining their ability to eliminate intravascular adherent bacteria. We then explored whether this blood flow reduction was due to the bacteria themselves or secondary to intravascular coagulation or thrombosis.

Platelets are not involved in the acquisition of antibiotic resistance

Meningococcal systemic infections are associated with disturbed coagulation ranging from vessel occlusion and thrombus formation to disseminated intravascular coagulation (DIC), as evidenced by the post-mortem analysis of skin biopsies [START_REF] Harrison | Analysis of pathogen-host cell interactions in purpura fulminans: expression of capsule, type IV pili, and PorA by Neisseria meningitidis in vivo[END_REF][START_REF] Pathan | Pathophysiology of meningococcal meningitis and septicaemia[END_REF][START_REF] Sotto | Pathogenesis of cutaneous lesions in acute meningococcemia in humans: light, immunofluorescent, and electron microscopic studies of skin biopsy specimens[END_REF]. Similar observations have been recently made using the humanized mouse model of meningococcal infections [START_REF] Manriquez | Colonization of dermal arterioles by Neisseria meningitidis provides a safe haven from neutrophils[END_REF]. We therefore tested whether infection-induced microvascular thrombosis might restrict the proper access of antibiotics to adherent meningococci.

We first assessed the role of platelets in the process. Visualization of platelets by intravital imaging showed limited platelet aggregation/accumulation, if any, at sites of bacterial adhesion (Fig EV2A). Platelets could be found in spaces where no bacteria were found, and small volumes of blood expected. Interestingly, no evidence of bacteria-platelet interactions could be found, in contrast to what has been observed with other pathogenic bacteria [START_REF] Fitzgerald | The interaction of bacterial pathogens with platelets[END_REF][START_REF] Sun | The interaction between pathogens and the host coagulation system[END_REF]. We observed circulating platelets moving past bacterial aggregates without evidence of adhesive contacts (Movie EV2). We then evaluated the impact of platelet depletion on the efficiency of Cefotaxime treatment (Fig EV2B). To this end, grafted mice were administered an antibody targeting the platelet receptor GPIba/CD42b 6h prior to the infection, as previously described [START_REF] Bergmeier | Structural and functional characterization of the mouse von Willebrand factor receptor GPIb-IX with novel monoclonal antibodies[END_REF][START_REF] Nieswandt | Identification of critical antigen-specific mechanisms in the development of immune thrombocytopenic purpura in mice[END_REF]. This treatment led to the efficient depletion of platelets (Fig EV2C).

Antibiotic treatment was then administrated 16h post infection and bacterial counts assessed 3 hours later. As expected, low numbers of blood-circulating bacteria were found upon antibiotic treatment when compared to the infected but non-treated control grafted animals

Coagulation inhibition restores antibiotic efficiency

Clinical and experimental evidence point to the presence of fibrin clots in the vicinity of infected vessels [START_REF] Faust | Dysfunction of endothelial protein C activation in severe meningococcal sepsis[END_REF][START_REF] Guarner | Pathogenesis and diagnosis of human meningococcal disease using immunohistochemical and PCR assays[END_REF][START_REF] Melican | Adhesion of Neisseria meningitidis to dermal vessels leads to local vascular damage and purpura in a humanized mouse model[END_REF]. Using intravital imaging, we therefore monitored the deposition of fibrin inside infected vessels during vascular colonization by Neisseria meningitidis. In the first 3 hours following infection, no evidence of fibrin deposition could be found. In contrast, starting 3 hours post infection we observed the progressive accumulation of fibrin close to sites of bacterial adhesion ( [START_REF] Osterud | Increased tissue thromboplastin activity in monocytes of patients with meningococcal infection: related to an unfavourable prognosis[END_REF], suggesting a role for TF in triggering disseminated intravascular coagulation (DIC).

Accordingly, TF expression on the surface of monocytes (CD115 + Gr-1 + ) from infected mice increased with kinetics similar to microvascular fibrin deposition (Fig EV4A). In contrast, endothelial cells did not show any increase in TF expression, even though E-Selectin expression strongly increased, as shown previously (Manriquez et al., 2021) (Fig EV4B).

We next sought to determine whether infection-induced fibrin deposition negatively impacts the efficacy of antibiotic therapy by inhibiting the thrombin-dependent cleavage of fibrinogen into fibrin strands. Fondaparinux prevents the formation of fibrin clots by inhibiting Factor Xa, and thus the conversion of prothrombin into active thrombin [START_REF] Bauer | Fondaparinux, a synthetic pentasaccharide: the first in a new class of antithrombotic agents -the selective factor Xa inhibitors[END_REF]. To ensure the constant delivery of the anticoagulant treatment to grafted mice, Fondaparinux was infused subcutaneously using osmotic minipumps. Grafted mice were then infected, and the Accordingly, the perfusion of infected vessels was partially restored upon inhibition of coagulation by . Taken together these results strongly suggest that meningococcal-induced fibrin deposition reduces microvascular patency and thereby the tissue distribution of antibiotics, thus preventing their efficacy.

Discussion

Our results show that within a few hours of infection, meningococcal vascular colonization leads to a dramatic reduction in the perfusion of infected vessels. While bacterial colonization alone is sufficient to partially occlude vessels and reduce blood flow, ensuing thrombus formation leads to the complete cessation of blood flow in affected vessels. This, in turn, has important implications for disease progression. Complete absence of perfusion will result in severe ischemia and irreversible tissue injury, driving the rapid progression of the disease towards multiorgan failure. Another consequence of reduced microvascular perfusion, detailed in this study, is its impact on the access of antibiotics to target tissues. This is observed in the animal model described here but also in human cases where bacteria have been demonstrated to resist intensive antibiotic treatment [START_REF] Van Deuren | Rapid diagnosis of acute meningococcal infections by needle aspiration or biopsy of skin lesions[END_REF].

The molecular process driving thrombin generation during meningococcal infection has been extensively studied clinically and in vitro but remains to be fully established [START_REF] Lecuyer | Pathogenesis of meningococcal purpura fulminans[END_REF]. Our results using a humanized model of Neisseria meningitidis systemic infection show extensive fibrin deposition in close proximity to bacterial aggregates early during infection and provide new insight into meningococci-induced thrombus formation. Our results suggest that the process is largely independent of platelets, which do not stably interact with meningococci, as opposed to many other bacterial infections [START_REF] Fitzgerald | The interaction of bacterial pathogens with platelets[END_REF][START_REF] Sun | The interaction between pathogens and the host coagulation system[END_REF]. Rather, thrombus formation appears to be triggered by the extrinsic pathway of blood coagulation through the rapid expression of TF/CD142 by monocytes. Accordingly, in meningococcemia cases [START_REF] Osterud | Increased tissue thromboplastin activity in monocytes of patients with meningococcal infection: related to an unfavourable prognosis[END_REF]) and the humanized model used here, monocytes express TF at early stages of infection. In contrast and despite massive local bacterial accumulation, human endothelial cells did not display TF expression in vivo.

Interestingly, the contact (intrinsic) pathway of blood coagulation was also shown to be activated during meningococcal infections [START_REF] Wuillemin | Activation of the intrinsic pathway of coagulation in children with meningococcal septic shock[END_REF]. The negative charges present on the bacterial LOS or capsule could be a trigger of the contact pathway [START_REF] Oehmcke | Contact system activation in severe infectious diseases[END_REF]. This could provide an alternative explanation for fibrin deposition in proximity to bacterial colonies. In any case, vascular colonization by Neisseria meningitidis leads to the rapid formation of thrombi that restrict the efficacy of antibiotic treatment. Many pathogenic bacteria have evolved specific strategies to inhibit coagulation, a process that is thought to favor bacterial dissemination [START_REF] Popov | Biological activity of kurilostatin--an unusual alkaloid from sea sponges[END_REF]. Meningococci do not seem to be equipped to efficiently block thrombus formation, hence permitting the thrombotic occlusion of infected vessels and the sealing of the infected regions by the host. This likely reflects the main lifestyle of Neisseria meningitidis at the surface of the nasopharyngeal epithelium rather than in the vasculature.

From the clinical point of view, our observations suggest that anticoagulation therapy could potentially provide a synergistic effect in conjunction with antibiotic treatment for meningococcal infections. A number of clinical studies have tested the benefits of anticoagulants such as activated protein C in combination with antibiotics. Activated protein C acts as a negative regulator of coagulation by proteolytically inactivating coagulation cofactors Factor Va and Factor VIIIa. Unfortunately, the use of recombinant Protein C has had variable success [START_REF] Silva | Prowess-shock trial: a protocol overview and perspectives[END_REF]. Our study suggests that anticoagulation would mainly be effective in the very early stages of infection and that direct targeting of Factor Xa or thrombin may be more effective than Protein C. The efficacy of anticoagulation described here is likely to be linked to the pathogenesis of meningococcal infections and more specifically to the ability of this bacterium to adhere to the endothelium and colonize vessels.

The utility of anticoagulation is thus likely to depend on the pathogen. Although the process described here is local, in close association with infected vessels, it is a host-driven response and a likely precursor of DIC. The induction of local coagulation appears to represent a tipping point in the progression of the disease after which antibiotic therapy is no longer as efficient.

While our study suggests the potential of anticoagulant therapy for increased efficacy of antibiotic treatment, it also argues that such therapy would have to be administered early in the progression of the disease and that thrombolytic therapy is likely to be more efficacious once the bacteria have taken hold and thrombi are established. Importantly, the use of anticoagulant and fibrinolytic therapy is always associated with a potential risk of bleeding that will need to be carefully assessed. Although anticoagulation has been used in the context of sepsis in the past [START_REF] Levi | Rationale for restoration of physiological anticoagulant pathways in patients with sepsis and disseminated intravascular coagulation[END_REF][START_REF] Silva | Prowess-shock trial: a protocol overview and perspectives[END_REF], the strong implication of the cerebral vasculature render safety considerations particularly important for meningococcal infections. That said, experience from ischemic stroke shows that such risks can be managed by the use of strict inclusion criteria based on timing and individual risk/benefit assessment. Xenograft model of infection. 6-13 weeks old mice, both males and females, were grafted with human skin as previously described [START_REF] Melican | Adhesion of Neisseria meningitidis to dermal vessels leads to local vascular damage and purpura in a humanized mouse model[END_REF]. Briefly, mice were anesthetized with a mixture of ketamine hydrochloride (100 mg/kg, Boehringer Ingelheim) and xylazine hydrochloride (8.5 mg/kg, Bayer) and a graft bed of approximately 1-2 cm 2 was prepared on their flank by removing the mouse epithelium and the upper dermis layer. A human skin graft (200 µm thick) comprising the human epidermis and the papillary dermis was immediately placed over the graft bed. Grafts were fixed in place with surgical glue (Vetbond, 3M, USA) and dressings were applied for 2 weeks. Grafted mice were used for experimentation 3-6 weeks post-surgery when the human dermal microvasculature is anastomosed to the mouse circulation without evidence of local inflammation, as previously described [START_REF] Melican | Adhesion of Neisseria meningitidis to dermal vessels leads to local vascular damage and purpura in a humanized mouse model[END_REF]. All efforts were made to minimize suffering.

Materials and Methods

Mice

Neisseria meningitidis strains and mouse infection. Experiments were performed using N. meningitidis 8013 serogroup C strain (http://www.genoscope.cns.fr/agc/nemesys) [START_REF] Rusniok | NeMeSys: a biological resource for narrowing the gap between sequence and function in the human pathogen Neisseria meningitidis[END_REF]. Strains were streaked from -80 o C freezer stock onto GCB agar plates and grown overnight in a moist atmosphere containing 5% CO2 at 37 o C. For all experiments, bacteria were transferred to liquid cultures in pre-warmed RPMI-1640 medium (Gibco) supplemented with 10% FBS at adjusted OD600nm = 0.05 and incubated with gentle agitation for 2 hours at 37°C in the presence of 5% CO2. Bacteria were washed twice in PBS and resuspended to 10 8 CFU/ml in 1x PBS. Prior to infection, mice were injected intraperitoneally with 8 mg of human transferrin (Sigma Aldrich) to promote bacterial growth in vivo as previously described [START_REF] Melican | Adhesion of Neisseria meningitidis to dermal vessels leads to local vascular damage and purpura in a humanized mouse model[END_REF]. Mice were infected by retro-orbital injection of 100 µl of the bacterial inoculum (10 7 CFU total). When indicated, mice received a retro-orbital injection of 4 mg Cefotaxime (CTX, #C7912, Sigma-Aldrich) at 3h, 6h or 16h post infection and 3h prior to mouse sacrifice. No influence of mice sex on bacterial colonization have been observed.

Wholemount immunofluorescent staining. Following mouse infection with mCherry-

expressing Neisseria meningitidis for 9 hours, the human grafts were harvested and fixed in 4% PFA overnight at 4°C with gentle agitation. Skin samples were then washed and permeabilized/blocked overnight at 4°C with gentle agitation using a solution of 0.3% Triton-X100, 1% BSA, 1% normal goat serum in 1X PBS. The basal lamina of human blood vessels was stained using an AlexaFluor647-conjugated anti-human Collagen IV antibody (#51-9871-82, eBioscience, 1/100) diluted in permeabilization/blocking solution and incubated 3 days at 4°C with gentle agitation. Samples were then extensively washed and tissue transparization and mounting were performed using RapidClear 1.47 (SunJin Lab) according to manufacturer's recommendations.

Neisseria meningitidis growth curves. Bacterial liquid culture was diluted to an OD600nm of 0.05 and 150 μl of the bacterial suspension was transferred into wells of a 96-well plate.

Bacteria were then treated or not with the indicated final concentration of Cefotaxime, each performed in quintuplicates. Plates were placed in a Cytation 5 multimode reader (Biotek) at 37°C with 5% CO2 and continuous agitation and optical densities at 600 nm were recorded every 30 minutes for 20 hours.

Colony-forming units (CFU) enumeration.

To assess bacteraemia (blood-circulating bacteria) in infected animals, 10 µl of blood was sampled at the time of sacrifice. Serial dilutions of blood were plated on GCB agar plates and incubated overnight at 37 o C and in a moist atmosphere containing 5% CO2. Bacterial counts were expressed in colony-forming units (CFU) per ml of blood. To assess the extent of vascular colonization by meningococci (adherent bacteria) following mouse sacrifice at indicated times post infection, tissue biopsies were collected using a sterile dermatological biopsy puncher (approximately 4 mm 2 ), weighted and placed in 500 µl 1x PBS. Skin biopsies were dissociated and homogenized using a BeadMill4 homogenizer (Fisher Brand) and serial dilutions of skin homogenates were plated on GCB plates incubated overnight at 37 o C and in a moist atmosphere containing 5% CO2. Bacterial counts were expressed in colony-forming units (CFU) per mg of skin.

Platelet depletion. Platelets were depleted in grafted mice by retro-orbital injection of 40 µg of anti-mouse GPIba (CD42b) (#R300, Emfret Analytics) monoclonal antibodies diluted in 100 µl 1x PBS 6 hours prior to infection, as previously described [START_REF] Bergmeier | Structural and functional characterization of the mouse von Willebrand factor receptor GPIb-IX with novel monoclonal antibodies[END_REF][START_REF] Nieswandt | Identification of critical antigen-specific mechanisms in the development of immune thrombocytopenic purpura in mice[END_REF]. The efficiency of platelet depletion was assessed by counting platelets on citrated blood using an automated hemocytometer (SCIL Vet ABC plus) according to manufacturer's instructions.

Spinning disk confocal intravital imaging.

Intravital imaging of the human xenograft was adapted from [START_REF] Ho | Visualization of Plasmodium falciparum-endothelium interactions in human microvasculature: mimicry of leukocyte recruitment[END_REF] and previously described [START_REF] Manriquez | Colonization of dermal arterioles by Neisseria meningitidis provides a safe haven from neutrophils[END_REF]. Briefly, 30 min prior to surgery, mice were injected subcutaneously with buprenorphine (0.05 mg/kg, CEVA) and anesthetized by spontaneous inhalation of isoflurane in 100% oxygen (induction: 4%; maintenance: 1.5% at 0.3 L/min). A flap of skin supporting the human graft was immobilized onto a custom-made heated deck (36°C) and continuously moistened with warmed 1x PBS (36°C). Mouse body temperature was maintained at 37°C using a heating pad and oxygen saturation and heart rate were monitored using the pulse oximetry Physiosuit apparatus (Kent Scientific). Image acquisition was performed using a Leica DM6 FS upright microscope equipped with a motorized stage, a Fluotar 25x/0.95 objective (Leica), and coupled to a Yokogawa CSU-W1 confocal head modified with Borealis technology (Andor). Four laser excitation wavelengths (488, 561, 642, and 730 nm) were used in fast succession and visualized with the appropriate long-pass filters. Fluorescence signals were detected using a sCMOS 2048x2048 pixel camera (Orca Flash v2+, Hamamatsu).

Metamorph acquisition software (Molecular devices) was used to drive the confocal microscope.

The human vessels were labelled using Dylight755-conjugated UEA-1 lectin (100 µg, Vector Laboratories), platelets using DyLight488-anti-mouse GPIbb antibody (0.1 µg, #X488, Emfret Analytics) and fibrin deposits using DyLight488-anti-human/mouse fibrin antibody (8 µg, clone 59D8, Sigma-Aldrich). All antibodies and dyes were injected intravenously 15 min prior to intravital imaging. Five to ten fields of view of interest were selected per animal and timelapse z-stack series (2-2.5 µm z-step, 50-80 µm range) were captured every 30-60 min for 4-5 hours following the intravenous injection of iRFP-expressing bacteria (10 7 CFU/100 µl 1x PBS). In mice treated with anticoagulant, Fondaparinux was injected subcutaneously 5 min prior to imaging (80 µl, 1 mg).

The blood flow during vascular colonization was monitored by perfusing 1 µm diameter fluorescent microspheres (Yellow/Green Fluoresbrite carboxylate, Polysciences, 10 7 microspheres/ml 1x PBS) at a rate of 15 µl/min before each z-stack series and images acquired on a single plane at high speed (50 frames per second, 300 frames). Blood flow was calculated from the vessel diameter and the centerline microsphere mean velocity, according to [START_REF] Hidalgo | Heterotypic interactions enabled by polarized neutrophil microdomains mediate thromboinflammatory injury[END_REF], and normalized with respect to the basal blood flow.

Platelet dynamics during vascular colonization was assessed by fast (50 frames/sec) simultaneous dual-color imaging of platelets and bacteria.

Anticoagulant treatment. The inhibition of coagulation was achieved by treating grafted mice with the factor Xa inhibitor Fondaparinux (ARIXTRA Ò , ASPEN) prior to and/or during the course of the infection. For short-term infection (6h and 9h), grafted mice received a subcutaneous injection of 80 µl Fondaparinux (corresponding to 1 mg) immediately after the intravenous injection of bacteria and 3h and 6h post infection. For long-term infection (19h), grafted mice received a subcutaneous injection of 80 µl Fondaparinux (corresponding to 1 mg) immediately after the infection and the constant delivery (infusion) of Fondaparinux was then ensured by ALZET Ò osmotic pumps (200 µl, 8 µl/h during 24h, #2001D) filled with 200 µl Fondaparinux and placed subcutaneously, according to manufacturer's instructions. The efficiency of the anticoagulant treatment was assessed using a chromogenic enzymatic method (Biophen Heparin 6 kit, #221006, HYPHEN Biomed), according to manufacturer's instructions.

Flow cytometry. To monitor the surface levels of Tissue Factor and E-Selectin on human endothelial cells, biopsies of human xenograft harvested at the indicated time post infection were digested with 0.4 mg/ml Liberase TL (Sigma Aldrich) in CO2-independent medium for 60 min at 37°C with gentle agitation (100 rpm). The resulting single-cell suspension was passed through a 70-µm cell strainer (BD Bioscience), washed once with FACS buffer (1× PBS supplemented with 0.5% BSA and 2 mM EDTA) and stained in FACS buffer according to standard protocols. Briefly, human and murine Fc receptors were blocked using Human TruStain FcX (Biolegend, #422302, 1/20) and anti-mouse CD16/CD32 (FcBlock clone 2.4G2, BD Biosciences, #553141, 1/200), respectively, and cells were stained for 30 min at 4°C with the following anti-human antibodies: PerCP-Vio700-conjugated REAfinity recombinant anti-CD31 (Miltenyi Biotec, clone REA730, #130-110-673, 1/20), BV421-conjugated anti-Tissue Factor/CD142 (BD Biosciences, clone HTF-1, #744003, 1/20), and PE-conjugated anti-CD62E/E-Selectin (eBioscience, clone P2H3, #12-0627-42, 1/20).

To monitor the surface levels of Tissue Factor on circulating murine monocytes, blood was harvested at the indicated time post infection and treated with 1x RBC lysis solution (BioLegend). Murine Fc receptors were blocked using anti-mouse CD16/CD32 (FcBlock clone 2.4G2, BD Biosciences, #553141, 1/200) and cells were stained in FACS buffer for 30 min at 4°C with the following anti-mouse antibodies: PacificBlue-conjugated anti-CD45 (Biolegend, clone 30-F11, #103126, 1/100), PE-Cy7-conjugated anti-CD11b (BD Biosciences, clone M1/70, #552850, 1/200), APC-conjugated anti-CD115 (Invitrogen, clone In (E-F), data are presented as the mean ± SEM. Two-tailed Mann-Whitney test.

In (H), data are presented as violin plot showing the distribution of the dataset. Kruskal-Wallis test with Dunn's correction for multiple comparisons.

  . Bacterial sensitivity to intravenously injected antibiotics was assessed in the animal model 3h, 6h and 16h post infection by treating mice with Cefotaxime (CTX, Fig 1C), a standard of care antibiotic for meningococcal infections in patients. Mice were treated with doses reflecting the concentration used in patients (200 mg/kg) (Griffiths et al, 2018). As for most clinical meningococcal isolates, the strain used for these experiments is sensitive to low doses of Cefotaxime when cultured in medium containing serum in vitro (Fig EV1). Antibiotic administration led to a strong reduction in the numbers of blood-circulating bacteria regardless the time post infection (Fig 1D). Numbers of adherent bacteria also decreased upon antibiotic treatment 3h post infection. In contrast, 6 and 16h post infection, bacteria became strongly resistant to the antibiotic treatment. While bacterial counts were reduced by up to 11 000-fold 3h post infection (55 410 ± 12 445 versus 5 ± 3 CFU/mg of skin in non-treated versus CTX-treated animals, respectively), 16h post infection bacterial counts only decreased by 12-fold (90 731 ± 50 772 versus 7 642 ± 2 200 CFU/mg of skin in non-treated versus CTX-treated animals, respectively) (Fig 1E). Similar results were obtained at 6h post infection (Fig 1E). Taken together, these data show that between 3h and 6h post infection, bacteria adhering to the endothelium experience a change, either intrinsic or in their environment, which leads to antibiotic treatment resistance. Given the ability of meningococci to occlude infected vessels combined with their pro-coagulant activity, a straightforward hypothesis would be that a reduction in microvascular patency could alter the access of antibiotics to bacteria. Meningococcal vascular colonization leads to a rapid and complete block of blood flow Using intravital imaging, we next assessed whether meningococcal vascular colonization alters vascular perfusion and consequently the proper access of antibiotics to sites of infection. Blood flow velocities were assessed using 1 µm fluorescent microspheres injected intravenously. High temporal resolution imaging with a spinning disk confocal microscope allowed the visualization of beads freely circulating in human vessels of non-infected animals (Fig 2A and Movie EV1). Beads circulated at average speeds of 1174 ± 141 µm/s in the observed arterioles (n = 31, average diameter 35 µm) and 505 ± 219 µm/s in venules (n = 13, average diameter 54 µm). Similar values were observed for the first 2 hours of infection (Fig 2B). In contrast, a steep reduction was observed at 3 hours post infection in all infected vessels, culminating in a complete absence of flow at 4 hours post infection (Fig 2B and Movie EV1). No variation in the blood vessel diameter was observed during the infection (Fig

(

  Fig EV2D). Numbers of adherent bacteria following the antibiotic treatment were similar to the numbers observed in non-depleted control animals (6 835 ± 3 495 versus 7 642 ± 2 200 CFU/mg of skin in platelet-depleted versus non-depleted animals, respectively) (Fig EV2E to be compared to Fig 1E), suggesting that platelets do not limit the efficacy of antibiotic treatment upon vascular colonization by Neisseria meningitidis. Taken together, these results argue against a role for platelets in restricting the efficiency of antibiotic treatments upon meningococcal systemic infections.

  Fig 3A-B, Fig EV3A and Movie EV3). In absence of infection, fibrin deposition was not observed (n = 17 vessels). Thrombus formation thus coincides with the time of blood flow decrease and the emergence of the resistance to the antibiotic treatment (Fig 1E and Fig 2B). Importantly, fibrin deposition occurs in most infected vessels, reaching 64% coverage at 4h30 post infection (Fig 3B). Interestingly, in patients, circulating monocytes have been shown to express Tissue Factor (TF) during meningococcemia

  antibiotic treatment (Cefotaxime, CTX) was administrated 16h post infection (Fig 3C). As expected, Fondaparinux strongly reduced plasma Factor Xa (Fig 3D). Interestingly, Fondaparinux treatment per se did not alter, nor promote meningococci survival either in the blood or adhering to the vessel walls (Fig EV3B-C). The antibiotic treatment also efficiently decreased the numbers of circulating bacteria in conditions of Factor X inhibition (Fig 3E).Strikingly, upon inhibition of coagulation and antibiotic treatment, numbers of adherent bacteria in the human grafts were decreased by more than 500-fold when compared to mice that did not received antibiotics (29 591 ± 23 065 versus 52 ± 45 CFU/mg of skin in Fondaparinux-versus Fondaparinux+CTX-treated animals, respectively) (Fig3F).

  . SCID/Beige (CB17.Cg-Prkdc scid Lyst bg-J /Crl) mice were purchased from Charles Rivers (France) and housed under specific pathogen-free condition at Institut Pasteur. Mice were kept under standard conditions (light 07.00-19.00h; temperature 22±1 o C; humidity 50±10%) and received sterilized rodent feed and water ad libitum. All experiments were performed in agreement with guidelines established by the French and European regulations for the care and use of laboratory animals and approved by the Institut Pasteur committee on Animal Welfare (CETEA) under the protocol code DAP 180022. For all experiments, male and female mice between 9 and 16-weeks of age were used. Littermates were randomly assigned to experimental groups. Human skin. Normal human skin was obtained from adult patients (20-60 years old), both males and females, undergoing plastic surgery in the service de chirurgie reconstructrice et plastique of Groupe Hospitalier Saint Joseph (Paris, France). In accordance with the French legislation, patients were informed and did not refuse to participate in the study. All procedures were approved by the local ethical committee Comité d'Evaluation Ethique de l'INSERM IRB 00003888 FWA 00005881, Paris, France Opinion: 11-048.

Figure 1 -

 1 Figure 1 -Limited antibiotic efficiency during Neisseria meningitidis vascular colonization A Schematic representation of the human skin xenograft mouse model. B Representative whole-mount immunofluorescence pictures (maximum intensity zprojection) of the colonization of human vessels (grey, anti-human Collagen IV) by mCherryexpressing Neisseria meningitidis (green) in grafted animals 9 hours post infection (p.i.). Scale bar: 40 µm. (n = 3 mice). C Schematic representation of the experimental approach used to assess bacterial sensitivity to Cefotaxime (CTX) treatment at 3h, 6h and 16h post infection (p.i.). D, E Bacterial colony forming unit (CFU) counts from (D) blood (circulating bacteria) and (E) dissociated human xenografts (adherent bacteria) collected from mice infected for the indicated time points and treated (+) or not (-) with Cefotaxime (CTX). (n = 12, 6 and 10 mice per group at 3h, 6h and 16h p.i., respectively, pooled from N = 3 independent experiments per time point). Data information: In (D-E), data are presented as the mean ± SEM. Kruskal-Wallis test with Dunn's correction for multiple comparisons.

Figure 2 -

 2 Figure 2 -Vascular colonization by Neisseria meningitidis blocks the blood flow A Representative image of fluorescent microsphere tracking as a read-out of blood flow during the early phase of the infection (0h and 4h p.i.). Tracks were color-coded according to the mean velocity of the corresponding microsphere. Inset shows the extent of vascular colonization by iRFP-expressing Nm (green) 4h post infection. Blue arrowheads indicate immobilized microspheres. Scale bar: 50 µm. B, C Normalized blood flow (B) and vessel diameter (C) as a function of the time post infection. (n = 18 and 19 vessels for non-infected and infected animals, respectively, pooled from N = 2 mice per condition and imaged independently). D Data in (B) were classified according to the vessel type: venules (left) and arterioles (right).(n = 13 and 14 arterioles, and 5 and 5 venules in non-infected and infected mice, respectively).Data information: In (B-D), data are presented as the mean ± SEM. Two-way ANOVA test.

Figure 3 -

 3 Figure 3 -The efficiency of antibiotic treatments is restored upon inhibition of coagulation A Representative image (maximum intensity z-projection) and orthogonal view of fibrin deposition (red, anti-Fibrin) in human vessels (grey, UEA1 lectin) infected by iRFPexpressing Neisseria meningitidis (green). Scale bar: 50 µm. (n = 28 infected vessels, pooled from N = 3 mice imaged independently).

Figure EV1 -

 EV1 Figure EV1 -Neisseria meningitidis is sensitive to low doses of Cefotaxime in vitro Growth curves of N. meningitidis in the presence of the indicated final concentrations of Cefotaxime. (N = 1 experiment). Data are presented as the mean ± SEM of quintuplicates.

Figure EV2 -

 EV2 Figure EV2 -Platelets do not prevent the efficiency of the antibiotic treatment A Representative image (maximum intensity z-projection) and orthogonal view of platelets (purple, anti-GPIbb) in bacteria-free spaces in human vessels (grey, UEA1 lectin) infected by iRFP-expressing Neisseria meningitidis (green) for 4 hours. Scale bar: 25 µm. (n = 3 mice). B Schematic representation of the experimental approach used to assess the impact of platelet depletion on bacterial sensitivity to Cefotaxime (CTX) treatment. C Quantification of platelet numbers in the blood of mice treated (+) or not (-) with the platelet-depleting antibody directed against GPIba. (n = 6 and 12 mice for control and anti-GPIba-treated mice, respectively, pooled from N = 2 independent experiments). D, E Bacterial colony forming unit (CFU) counts from (D) blood (circulating bacteria) and (E) dissociated human xenografts (adherent bacteria) collected from platelet-depleted mice (anti-GPIba) infected for 19h and treated (+) or not (-) with Cefotaxime (CTX). (n = 6 mice per group, pooled from N = 2 independent experiments). Data information: In (C), data are presented as the mean ± SEM. Unpaired t test.In (D-E), data are presented as the mean ± SEM. Two-tailed Mann-Whitney test.

Figure EV3 -

 EV3 Figure EV3 -The efficiency of the antibiotic treatment is restored upon coagulation inhibition A Representative time sequence (maximum intensity z-projection) of fibrin deposition (red, anti-Fibrin) in human vessels (dashed line) infected by iRFP-expressing Neisseria meningitidis (green). Scale bar: 50 µm. (n = 28 infected vessels, pooled from N = 3 mice imaged independently). B, C Bacterial colony forming unit (CFU) counts from (B) blood (circulating bacteria) and (C) dissociated human xenografts (adherent bacteria) collected from control (-) and Fondaparinux-treated (+) mice infected for 5 minutes or 6 hours. (n = 7 mice per group, pooled from N = 2 independent experiments). Data information: In (B-C), data are presented as the mean ± SEM. (B) Kruskal-Wallis test with Dunn's correction for multiple comparisons. (C) Two-tailed Mann-Whitney test.

Figure EV4 -

 EV4 Figure EV4 -Early upregulation of Tissue Factor at the surface of circulating monocytes upon meningococcal vascular colonization A Representative flow cytometry analysis (left) and quantification (right) of the cell surface expression of murine Tissue Factor (CD142) on blood-circulating monocytes (gated as CD45 + CD11b + CD115 + Gr-1 + ) in non-infected control mice (black) and mice infected for 6h (red). (n = 3 mice). B Representative flow cytometry analysis (left) and quantification (right) of the cell surface expression of human Tissue Factor (CD142, Top) and human E-Selectin (CD62E, bottom) on CD31 + human endothelial cells isolated from human skin xenografts in non-infected control mice (black) and mice infected for 6h (red). (n = 3 mice). Data information: In (A-B), data are presented as the mean ± SEM and were normalized with respect to the mean fluorescence intensity (MFI) of the non-infected controls.

Acknowledgements

This work was supported by the Integrative Biology of Emerging Infectious Diseases (IBEID) laboratory of excellence (ANR-10-LABX-62), and the VIP European Research Councilstarting grant (310790-VIP to G.D.). D.O. was supported by a grant from the Agence Nationale de la Recherche (ANR-18-CE15-0006-MeningoChip to G.D.).

Data availability

This study includes no data deposited in external repositories.

Author contributions

Conflict of interest

The authors declare no conflict interests.


Emiliano Lorini
email: emiliano.lorini@istc.cnr.it

Cristiano Castelfranchi
email: cristiano.castelfranchi@istc.cnr.it

IJPRAIfinalversion The unexpected aspects of surprise

Keywords: belief systems, expectation and surprise, cognitive architectures

Some symbolic AI models for example BDI (belief, desire, intention) models are conceived as explicit and operational models of the intentional pursuit and belief dynamics. The main concern of these models is to provide a clear understanding of the functional roles of different kinds of epistemic and motivational states (beliefs, acceptances, expectations, intentions, goals, desires etc...), of the relational properties among them. Mental configurations of appraisal (involving different kinds of motivational and epistemic states) which correspond to particular cognitive emotions such as disappointment, fear, relief, shame etc... have been analyzed by several authors close to the BDI theoretical tradition. The main objective of this work is a formal analysis of Surprise in a BDI-like cognitive architecture. A clarification of the functional role of Surprise in a BDI-like cognitive architecture with respect to resource bounded belief revision is given.

Introduction

Some symbolic AI models and logics for example BDI models and logics [START_REF] Bratman | Intentions, plans, and practical reason[END_REF][START_REF] Cohen | Intention is choice with commitment[END_REF][START_REF] Meyer | A Logical Approach to the Dynamics of Commitments[END_REF][START_REF] Rao | Modelling rational agents within a BDI-architecture[END_REF] are conceived as explicit and operational models of the intentional pursuit and belief dynamics. The main concern of these models is to understand the functional roles of different kinds of epistemic and motivational states (beliefs, acceptances, expectations, intentions, goals, desires etc...) and the relational properties among them. Mental configurations of appraisal (involving different kinds of motivational and epistemic states) which correspond to particular cognitive emotions such as disappointment, fear, relief, shame etc... have been analyzed by several authors close to the BDI theoretical tradition. [START_REF] Castelfranchi | Affective Appraisal versus Cognitive Evaluation in Social Emotions and Interactions[END_REF][START_REF] Meyer | Reasoning about emotional agents[END_REF] In this work a formal model of Surprise based on a BDI-like cognitive architecture and in particular on a theory of expectations [START_REF] Castelfranchi | Mind as an Anticipatory Device: For a Theory of Expectations[END_REF][START_REF] Castelfranchi | Cognitive Anatomy and Functions of Expectations[END_REF][START_REF] Miceli | The Mind and the Future. The (Negative) Power of Expectations[END_REF] is developed. In order to model Surprise a propositional logic with probabilities is used. The syntax and semantics of the formal logic are presented in section 2. The process of interpretation of input data is modeled by defining an abductive procedure for selecting the best interpretation of input data. Mismatch-based Surprise will be defined in section 3 as the the reaction of alert due to a mismatch between an expectation under scrutiny (a belief about the future that the cognitive system is willing to verify) and the selected interpretation of input data; the degree of Mismatch-based Surprise will be associated to the degree of conflict between the selected interpretation of input data and the invalidated expectation under scrutiny. It will be formally derived that "the higher the probability associated to the invalidated expectation under scrutiny, the higher the intensity of the surprise due to its invalidation" and "the lower the probability associated to the selected explanation of input data (invalidating the expectation under scrutiny), the higher the intensity of the surprise due to the invalidation".

In section 4 we will analyze the surprise generated by the deeper mismatch between the selected explanation of input data and the (logical) supports of the expectation under scrutiny. A support of an expectation under scrutiny will be defined as the background belief whose content entails the content of the expectation under scrutiny. It will be formally derived that the intensity of this deeper mismatch-based surprise is always equal or lower than the intensity of first-order mismatch-based surprise (due to the invalidation of the expectation under scrutiny by the selected explanation of input data). Section 5 is devoted to discuss the importance of the present approach for the theory of belief revision. We will argue that deeper mismatch-based surprises should be conceived as later affective responses which are coupled with the revision of deeper beliefs in a belief base. We will defend the following thesis concerning the functional role of surprise in resource bounded cognitive agents: since realistic cognitive agents are non-omniscient and have not direct and instantaneous access to all their knowledge both in phase of perception and in phase of revision, some mechanism which is responsible : 1) for signaling the global inconsistency of the belief base with respect to the incoming input data, 2) for making explicit deeper layers, 3) for the revision of broader parts of the belief base, is needed. One of the functional role of surprise is exactly this. Therefore the present work provides new insights with respect to the following two scientific areas. With respect to the area of cognitive modelling a conceptual clarification of the notion of Surprise is provided. We claim that, on one side, (i) surprise is function of the probability of the invalidated fact under scrutiny; a while on the other side (ii) it is function of the probability of the perceived fact. On the one side the more a This thesis is also defended in our complementary work. [START_REF] Lorini | The cognitive structure of surprise: looking for basic principles[END_REF] certain was my scrutinized prediction, the more surprise I am. On the other side the more unpredictable, the more unexpected the incoming input, the more surprise I am. Indeed in our view current models of surprise [START_REF] Meyer | Towards a process analysis of emotions: The case of surprise[END_REF][START_REF] Macedo | Modelling Forms of Surprise in an Artificial Agent[END_REF][START_REF] Ortony | Surprisingness and expectation failure: Whats the difference?[END_REF] do not capture the first dimension of surprise. b With respect to the area of belief revision the present analysis builds the bases for developing a model of Surprise-driven progressively structured belief revision in resource bounded cognitive agents. Our work is very close to works done under the philosophically inspired [START_REF] Harman | Change in View[END_REF] local revision framework [START_REF] Hansson | Local Change[END_REF][START_REF] Wasserman | Resource-bounded Belief Revision[END_REF] which has been proposed as an advancement of standard AGM belief revision 1 for understanding belief revision in resource bounded cognitive agents.

Formal foundations

Let us give the formal specification of the model. We define in this section the formal language with the related syntax and semantics. We use a logic of probabilistic quantified beliefs with a semantics similar to the semantics given by Halpern. 10

Syntax

The primitives of the formal language are the following: -A set of agent variables AGT = {i, j,...}; -A set of propositional variables Π= {p, q,...}. The set of propositional formulas is noted by Prop and is defined by the the closure of Π under the Boolean operations ∧ and ¬. Let us use ϕ and ψ to represent propositional formulas. A primitive term is a an expression of the form P i (ϕ) where ϕ is a propositional formula. A term is an expression of the form a 1 P i (ϕ 1 ) +...+ a k P i (ϕ k ) where a 1 ,..., a k are real numbers and k ≥ 1. A basic formula is a statement of the form t ≥c where t is a term and c is a real number. A complex formula is a Boolean combination of basic formulas. Let us use f and g to represent complex formulas. Finally a conditional formula is a statement of the form ϕ ⇒ i ψ (where ϕ and ψ are propositional formulas). Finally we use the standard notation ||ϕ|| W to represent the set of worlds in W at which ϕ is true. Formally ||ϕ|| W = {v ∈ W |M, v ϕ j }.

Semantics

A model for our logic is defined as M = (W, X, π) where:

• W is a non-empty set of possible worlds;

b Other models of Surprise based on Information theory claim that Surprise is function of the distance between prior probabilities and posterior probabilities after the conditioning on the set of perceived data. [START_REF] Baldi | Surprise: A Shortcut for Attention?[END_REF] We believe that also this approach is incomplete since it only focuses on revision-based surprise and completely neglects the surprise due to the mismatch between an incoming input and the scrutinized expectation.

• π is a valuation function which assigns a truth value to propositions in possible worlds π : W × Π → B where B is the set of boolean variables B = {true, f alse} • X is a function which associates with each world w and agent i ∈ AGT a probability space (W w,i , F w,i , X w,i ) where:

-W w,i is a subset of W called sample space; -F w,i is the algebra of subsets built on W w,i , -X w,i is a probability function defined on 2

W w,i such that X w,i : 2

W w,i → [0, 1].

Truth conditions and validity

• M,w p iff π(p, w) = true.

• M,w for all w ∈ W.

• M,w ¬ϕ iff not M,w ϕ. • M,w ϕ ∧ ψ iff M,w ϕ and M,w ψ. • M,w a 1 P i (ϕ 1 ) + ... + a k P i (ϕ k ) ≥ c| ≤ c iff a 1 X w,i (||ϕ 1 || Ww,i ) + ... + a k X w,i (||ϕ k || Ww,i ) ≥ c| ≤ c • M,w ϕ ⇒ i ψ iff ∀ v ∈ W w,i if M,v ϕ then M,v ψ.
We use a standard axiom system 10 for the logic of probabilities. c

Abducible formulas, input data and expectations under scrutiny

In the present section we define some ad-hoc formal constructions. This implies an extension of our definition of model. Given the following definition 1, 2 and 3 our models are now defined by the tuple M = (W, X, π, Γ, Test , Abd ).

Definition 1: Perceived data. We define a perception function Γ as the function which assigns a set of propositional variables to agents in possible worlds. Formally Γ: AGT × W → 2 Π . The input data for an agent i at a given world w is the set Γ w,i of propositional variables. Thus the elements in Γ w,i are the data perceived by the agent i at world w and that the agent must interpret on the basis of his pre-existent belief structure.

Definition 2: Tested expectation (expectation under scrutiny). We also introduce a test function Test which assigns a single propositional formula to agents in possible worlds. d This function returns the representation that a given agent is c The following categories of axioms are given. 1) Axioms for propositional reasoning: -Modus Ponens, -Propositional tautologies. 2) Axioms for reasoning about linear inequalities. 3) Axioms for reasoning about probabilities: -

P i (ϕ) ≥ 0, -P i (true) ≥ 0, -P i (ϕ ∧ ψ) + P i (ϕ ∧ ¬ψ) = P i (ϕ) (additivity), -P i (ϕ) = P i (ψ) if P i (ϕ) ←→ P i (ψ).
d Our test function is comparable to the awareness function given in some logic of implicit beliefs. 9

scrutinizing at a certain world. Formally Test : AGT × W → P rop. The representation under scrutiny for an agent i at a given world w is denoted by T est w,i . The following is the simple semantic characterization of a Test formula. M,w T est i (ϕ) iff ϕ = T est w,i .

Definition 3: Abducible formulas. We finally introduce an abducible formula function Abd which assigns a set of propositional formulas in Prop to agents in possible worlds. Formally Abd: AGT × W → 2 P rop . Abd w,i defines the set of propositional formulas in Prop given by the function Abd for agent i and world w.

The following is the semantic characterization of Abd formulas. M,w Abd i (ϕ) iff ϕ ∈ Abd w,i

We assume that elements in Abd w,i are mutually contradictory. This is made explicit by the next Assumption.

Assumption 1. If {ϕ 1 , ..., ϕ n } = Abd w,i then P i (ϕ 1 ∨ ... ∨ ϕ n ) = P i (ϕ 1 ) +...+ P i (ϕ n ).
Elements in Abd w,i are the candidate interpretations of the data Γ w,i obtained by the agent's sensors (see definition 1). We make the assumption that an agent i at a given world w must interpret sensor data Γ w,i by means of a classical abductive procedure of explanation extraction. e

The procedure returns the candidate interpretation (abducible formula) with the highest explanatory value with respect to the input dataset Γ w,i . Before describing the procedure let us introduce the following abbreviation to define beliefs of an agent.

Definition 4: Beliefs.

Bel i ϕ = def P i (ϕ) = 1

Procedure of explanation extraction and selection.

The abductive procedure is defined according to the following two steps.

Step 1. For each abducible formula

ϕ i ∈ Abd w,i = {ϕ 1 , ..., ϕ m } with 1 ≤ i ≤ m calculate the maximal subset max -Γ ϕi w,i
of the set of input data Γ w,i whose elements are entailed by ϕ i given agent's beliefs (max -Γ ϕi w,i contains the elements of Γ w,i that ϕ i is able to explain). Formally:

• max -Γ ϕi w,i ⊆ Γ w,i ;
e With classical abductive procedure (or scheme) we mean the most straightforward conception of an inference to an explanation of the input datum d: an explanation for a datum d is some formula H logically entailing d. Classical schemes are not the only accepted for abduction. There several analysis of less standard schemes for logical abductions. [START_REF] Boutilier | Abduction as Belief Revision[END_REF][START_REF] Walliser | Abductive logics in a belief revision framework[END_REF] • for all

p ∈ Γ w,i , p ∈ max -Γ ϕi w,i if Bel i (ϕ i → p) and ¬Bel i p • for all Y, if max -Γ ϕi w,i ⊂ Y ⊆ Γ w,i then it exists q ∈ Y such that ¬Bel i (ϕ i → p) or Bel i q.
Step 2. Select the abducible formula ϕ i ∈ Abd w,i with the highest explanatory value as the best explanation of the input data. We introduce a measure of explanatory value where two different parameters are taken into account: 1. the number of perceived data that the abducible formula is able to explain and 2. the prior probability associated to the abducible formula. With respect to the second parameter we use standard probability theory. The explanatory value of each candidate explanation (abducible formula) ϕ i depends on to the posterior probability that ϕ i is true, given the disjunction ϕ 1 ∨ ... ∨ ϕ m of all candidate explanations (abducible formulas). Using Bayes theorem we obtain:

P i (ϕ i |ϕ 1 ∨ ... ∨ ϕ n ) = Pi(ϕ1∨...∨ϕm|ϕi)•P (ϕi) Pi(ϕ1∨...∨ϕm)
.

(

Given previous assumption 1 (candidate explanations are mutually contradictory) (1) is equivalent to the following formula:

P i (ϕ i |ϕ 1 ∨ ... ∨ ϕ n ) = Pi(ϕi) Pi(ϕ1)+...+Pi(ϕm) . (2)
Moreover the explanatory value of a candidate explanation should depend on the number of data that the hypothesis explains. Therefore the previous probabilistic measure is weighted for the following ratio:

γ(ϕ i , Γ w,i ) = ( |max-Γ ϕ i w,i | |Γw,i|
).

(

) 3
|Γ w,i | is the number of elements in the set of input data Γ w,i (cardinality of Γ w,i ), |max -Γ ϕi w,i | (cardinality of max -Γ ϕi w,i ) is the maximal number of input data that ϕ i explains (see in step 1 how max -Γ ϕi w,i is defined) and κ is an arbitrary constant in [0, 1]. f Therefore the explanatory value of a candidate explanation ψ with respect to the input dataset Γ w,i and given a set of contradictory candidate explanations {ϕ 1 , ..., ϕ m } {ψ} = Abd w,i is defined by the following formula which enriches our basic language given in section 2.2. Special conditions of satisfiability are given.

Definition 5: Explanatory value of an abducible formula.

M,w Explvalue i (ψ, [ϕ 1 , ..., ϕ m ])=c iff {ϕ 1 , ..., ϕ m } {ψ} = Abd w,i and f We could also assume that κ is a function which assigns for each agent i and world w a value of reliability of the perceptive source in the interval [0, 1]. Indeed when the agent perceives some input data he can have some doubt about the fact that his sensors are working well. Therefore the explanatory value of the candidate hypothesis is reduced in order to integrate this uncertainty about the reliability of the perceptual apparatus.

κ w,i • γ(ψ, Γ w,i ) • Xw,i(||ψ|| W w,i ) Xw,i(||ϕ1|| W w,i )+...+Xw,i(||ϕm|| W w,i )+Xw,i(||ψ|| W w,i ) = c
Finally we introduce the special formula selectedexpl i (ψ, [ϕ 1 , ..., ϕ m ]) in order to represent the fact that ψ is the abducible formula which has been selected by agent i as the best explanation of the input data (given a set of contradictory candidate explanations {ϕ 1 , ..., ϕ m } {ψ} = Abd w,i ). We provide the following semantic characterization. Definition 6: Selected explanation.

M,w selected -expl i (ψ, [ϕ 1 , ..., ϕ m ]) iff {ϕ 1 , ..., ϕ m } {ψ} = Abd w,i and θ w,i ≤ κ • γ(ψ, Γ w,i ) • Xw,i(||ψ|| W w,i ) Xw,i(||ϕ1|| W w,i )+...+Xw,i(||ϕm|| W w,i )+Xw,i(||ψ|| W w,i ) and ∀ ϕ i ∈ Abd w,i -{ψ} Xw,i(||ψ|| W w,i ) Xw,i(||ϕ1|| W w,i )+...+Xw,i(||ϕm|| W w,i )+Xw,i(||ψ|| W w,i ) ≥ Xw,i(||ϕi|| W w,i ) Xw,i(||ϕ1|| W w,i )+...+Xw,i(||ϕm|| W w,i )+Xw,i(||ψ|| W w,i )
Notice that in the previous semantic characterizations we introduced the constant θ w,i in [0, 1]. This constant specifies the tolerability of agent i at world w to select a best interpretation of input data with a low explanatory value (we call θ w,i tolerability threshold). Given a set Abd w,i = {ϕ 1 , ..., ϕ m }∪ {ψ} of candidate interpretations of input data the abducible formula ψ ∈ Abd w,i is selected by agent i at a given world w as the explanation of the input data if and only if ψ is the abducible formula with the highest explanatory value with respect to the set of input data (among the candidate explanations in Abd w,i ) and the explanatory value of ψ exceeds the tolerability threshold.

Invalidation of the expectation under scrutiny and surprise

The result of the previous selection process is the explanatory item ψ ∈ Abd w,i = {ϕ 1 , ..., ϕ m }∪ {ψ} with the highest explanatory value. If the selected explanation ψ conflicts with the agent's expectation under scrutiny, the agent's expectation is invalidated due to the "logical" mismatch with the incoming interpretation of the input data. This mismatch generates a certain degree of surprise. We assume here that the intensity of superficial Surprise is measured by the degree of conflict between the expectation under scrutiny and the selected explanation where the notion of degree of mismatch between arbitrary formulas is given by the following definition.

Degree -M ismatch i (ϕ, ψ) ≥ c| ≤ c = def Pi(ϕ)+Pi(¬ψ) 2 ≥ c| ≤ c
Let us characterize the notion of Mismatch-basedSurprise as the mismatch between an arbitrary expectation that ϕ under scrutiny and the invalidating selected explanation of input data ψ (given a set {ϕ 1 , ..., ϕ m } {ψ} = Abd w,i of abducible formulas).

Definition 8: Mismatch-based Surprise.

M ismatch -basedSurprise i (ϕ, ψ, [ϕ 1 , ..., ϕ m ]) ≥ c| ≤ c = def T est i (ϕ) ∧ selected -expl i (ψ, [ϕ 1 , ..., ϕ m ]) ∧ (ψ ⇒ i ¬ϕ) ∧ Degree -M ismatch i (ϕ, ψ) ≥ c| ≤ c
Given the previous two definitions we provide the following principles concerning the measure of intensity of surprise due to the mismatch between the expectation under scrutiny and the selected explanation of the input data invalidating the expectation.

Proposition 1.1. Given two agents i and j and a value of probability P 1 for agent i's expectation under scrutiny, a value of probability P 2 for agent j 's expectation under scrutiny, a value of probability P 3 for agent i's explanation of input data and a value of probability P 4 for agent j 's explanation of input data, if P 1 > P 2 and P 3 = P 4 then agent i's is more (or equally) surprised than agent j due the invalidation of the expectation under scrutiny. Formally:

|= P i (ϕ 1 ) > P j (ϕ 2 ) ∧ P i (ψ 1 ) = P j (ψ 2 ) ∧ M ismatch -basedSurprise j (ϕ 2 , ψ 2 , [ϕ 1 , ..., ϕ m ]) = c ∧ T est i (ϕ 1 ) ∧ selected -expl i (ψ 1 , [ϕ 1 , ..., ϕ m ]) ∧ (ψ 1 ⇒ i ¬ϕ 1 ) → M ismatch -basedSurprise i (ϕ 1 , ψ 1 , [ϕ 1 , ..., ϕ m ]) > c
Proposition 1.2. Given two agents i and j and a value of probability P 1 for agent i's expectation under scrutiny, a value of probability P 2 for agent j 's expectation under scrutiny, a value of probability P 3 for agent i's explanation of input data and a value of probability P 4 for agent j 's explanation of input data, if P 1 = P 2 and P 3 < P 4 then agent i's is more (or equally) surprised than agent j due the invalidation of the expectation under scrutiny. Formally:

|= P i (ϕ 1 ) = P j (ϕ 2 ) ∧ P i (ψ 1 ) < P j (ψ 2 ) ∧ M ismatch -basedSurprise j (ϕ 2 , ψ 2 , [ϕ 1 , ..., ϕ m ]) = c ∧ T est i (ϕ 1 ) ∧ selected -expl i (ψ 1 , [ϕ 1 , ..., ϕ m ]) ∧ (ψ 1 ⇒ i ¬ϕ 1 ) → M ismatch -basedSurprise i (ϕ 1 , ψ 1 , [ϕ 1 , ..., ϕ m ]) > c
Proof. The proofs of Propositions 1.1 and 1.2 are obtained straightforward from the definition 7 of degree of mismatch and definition 8

The intuitive reading of the previous propositions 1.1 and 1.2 is that "the higher the probability associated to the invalidated expectation under scrutiny, the higher P(G,R) g ¬g r 0,04 0,76 ¬r 0,2 0 the intensity of the surprise due to its invalidation" and "the lower the probability associated to the explanation invalidating the expectation under scrutiny, the higher the intensity of the surprise due to the invalidation". Propositions 1.1 and 1.2 are reasonable. Indeed intensity of surprise should depend at least on the previous two parameters: the probability of my invalidated expectation and the probability of the interpreted input data. We are very surprised when we are almost certain that a certain fact will happen and our expectation gets invalidated. We are even more surprised when our expectation is invalidated by something that we perceive and we believe to be almost impossible.

Another principle looks reasonable. Since surprise is the response to a mismatch between two representations (the anticipated and expected one and the explanation of input data), and since a mismatch not necessarily is a yes/no result but can be partial (given that the representation is a pattern of features), one might claim that: the higher is the number of mismatching features, the stronger the surprise. This principle is too simplistic and naive . Let us argue against it by the following example.

Example. Consider an agent driving with the car in a street. The agent is ready to stop at the next traffic light since he expects the next traffic light to be red and not green. The agent is actively testing his expectation that (r ∧¬g). Given the variables R = "the traffic light will be red" and G = "the traffic light will be green", he assigns joint probabilities of R and G and conditional probabilities of G given R (see Table 1 and Table 2).

Consider now two different situations. 1) Imagine that the agent perceives the traffic light to be not red and green (g ∧ ¬r). The input data invalidate the expectation that (r ∧¬g) and mismatch with two out of two propositional features of the expected representation: both propositional atoms in the (expected) conjunction r ∧ ¬g are invalidated by the input data.

2) Imagine that the agent perceives the traffic light to be red and green (r ∧ g).

Again the input data invalidate the expectation that (r ∧¬g) but only mismatch with one out of two propositional features of the expected representation: only the propositional atom ¬g in the (expected) conjunction r ∧ ¬g is invalidated by the input data.

Let us calculate the degree of mismatch (and consequently the intensity surprise) in situations 1 and 2 and compare them.

(i) Degree-Mismatch (r ∧¬g, g ∧ ¬r) = P (r∧¬g)+(P (r)+P (¬g)-P (r∧¬g)) 2 = 0.78 (ii) Degree-Mismatch (r ∧¬g, g ∧ r) = P (r∧¬g)+(P (¬r)+P (¬g)-P (¬r∧¬g)) 2 = 0.86. This example shows that not necessarily higher the number of mismatching features, more intense is the surprise. In fact, the features composing the expected and tested representation as well as the features composing the representation explaining the input data are generally dependent one from each other. Thus if the features have some degree of interdependence, if there is some internal coherence of the global representation, one might be more surprised when one feature F1 of the global representation explaining the input data (which invalidates the tested expectation) is evaluated to be very unlikely given feature F2.

Deeper surprises and crisis of doxastic supports

In this section we analyze the surprise generated by the deeper mismatch between the selected explanation of input data and the (logical) supports of the expectation under scrutiny. A support of an expectation under scrutiny is defined as the background belief whose content entails the content of the expectation under scrutiny.

Example.

Imagine that agent i has explained the input data by the fact "there is a car close to me". The explanation "there is a car close to me" has invalidated the expectation under scrutiny "there is a table close to me" (and perhaps some presupposed belief such as "there is a wooden object close to me") and generated a mismatch-based surprise with a certain intensity. The agent tries now to answer to the question: -"why am I expecting to perceive a table?" by looking for the supports of the fact "there is a table close to me", i.e. by looking for some fact which implies the fact "there is a table close to me". Imagine that the agent explains the fact "there is a table close to me" by the fact "I am in a dining room". The fact "there is a car close to me" invalidates the first-order support "I am in a dining room" of the expectation "there is a table close to me" and a second-order mismatch-based surprise is generated by the invalidation.

Let us define the notion of support of an expectation under scrutiny.

It is straightforward to prove that: 2) follows from (3) and definition 7 of Degree of Mismatch. Finally from the valid formula M ismatch 4), ( 1) and definition 10 we can conclude that:

(2) M,w |= Degree -M ismatch i (ϕ, ψ) = c ∧ (ω ⇒ i ϕ) → Degree -M ismatch i (ω, ψ) ≤ c. Indeed M,w |= (ω ⇒ i ϕ) implies that (3) M,w |= P i (ω) ≤ P i (ϕ). (
-basedSurprise i (ϕ, ψ, [ϕ 1 , ..., ϕ m ]) = c ∧ Support i (ω, ϕ) → Degree -M ismatch i (ϕ, ψ) = c ∧ (ω ⇒ i ϕ) and (2) we can establish that: (4) M,w |= M ismatch -basedSurprise i (ϕ, ψ, [ϕ 1 , ..., ϕ m ]) = c ∧ Support i (ω, ϕ) → Degree -M ismatch i (ω, ψ) ≤ c. From (
M,w |= M ismatch -basedSurprise i (ϕ, ψ, [ϕ 1 , ..., ϕ m ]) = c ∧ Support i (ω, ϕ) → DeepM ismatch -basedSurprise i (ϕ, ψ, [ϕ 1 , ..., ϕ m ] , ω) ≤ c

Surprise and cascade belief revision

The conclusive part of the work is devoted to discuss the relevance of the present approach for the theory of belief revision. We are not going to give any formal emphasis on this part. We move toward a procedural perspective and conceive deeper mismatch-based surprises as deeper affective responses which are coupled with the revision of deeper beliefs in a belief base. We want to defend here the following thesis concerning the functional role of surprise in resource bounded cognitive agents: since realistic cognitive agents are non-omniscient and have not direct and instantaneous access to all their knowledge both in phase of perception and in phase of revision, some mechanism which is responsible : 1) for signaling the global inconsistency of the belief base with respect to the incoming input data , 2) for making explicit deeper layers, 3) for the revision of broader parts of the belief base, is needed. One of the functional role of surprise is exactly this. Let us assume the existence of certain threshold Δ of acceptable surprise. This threshold determines whether a given agent decides to continue to verify the consistency of his beliefs and expectations or decides to stop with the check of consistency and revises his belief base. If an agent has felt a very intense first-order mismatchbased surprise (i.e. the degree of first-order mismatch-based surprise is above threshold Δ) due to the invalidation of the expectation under scrutiny by the selected explanation of input data then the agent checks the adequacy of deeper layers of his belief base. Otherwise the agent revises all those elements of his belief base which are entailed by the expectation under scrutiny. The check of adequacy of deeper layers in the knowledge base consists in making explicit the first-order supports of the expectation under scrutiny and is responsible for the generation of a second-order mismatch-based surprise. If again the second-order mismatch-based surprise is very intense (i.e. its degree is above threshold Δ) the agent checks the adequacy of deeper layers; otherwise he revises all those elements of his belief base which are entailed by the first-order supports of his expectation under scrutiny. Again the check of consistency consists in making explicit the direct supports of the firstorder supports of the expectation under scrutiny (which are second-order supports of the expectation under scrutiny). This generates a third-order mismatch-based surprise. Deeper and deeper beliefs (deeper and deeper supports of the expectation under scrutiny) are made explicit until the intensity of the n-order mismatch-based surprise is not intense anymore (i.e. its degree is below threshold Δ). Previous proposition 3 insures that mismatch-based surprises decreases along the top down path going toward deeper layers of the belief base (and toward deeper logical supports of the expectation under scrutiny). Given proposition 3 we can predict that degree of n-order mismatch-based surprise ≥ degree of (n+1)-order mismatch-based surprise and therefore we can predict that there will be an instant at which the agent will "come out" of the progressive descent (the existence of a stop condition is guaranteed) and will finally revise all those elements of his belief base which are entailed by the deepest retrieved supports of the expectation under scrutiny. It is easy to verify that: deeper is the descent in the belief base (deeper are the retrieved supports of the expectation under scrutiny), more elements are involved in the belief revision process. See for instance figure 1. Imagine that the the coloured circle is the invalidated expectation under scrutiny. If the first-order mismatch-based surprise is under threshold Δ, then only those elements entailed by the expectations under scrutiny are involved in the belief revision process (the elements inside the small rectangle). If the first-order mismatch-based surprise is above threshold Δ, then the agent makes explicit the first-order supports of the expectation under scrutiny and if the second-order mismatch-based surprise is below threshold Δ, then those elements entailed by the first-order supports of the expectation under scrutiny are involved in the belief revision process (the elements inside the big rectangle). Therefore the revision (eventually) associated with the second-order mismatch-based surprise involves more elements than the revision (eventually) associated with the first-order mismatch-based surprise. In our view it is implausible to assume that realistic cognitive agents revise their beliefs with ϕ every time they perceive a new fact ϕ. g Realistic and non-omniscient cognitive agents are situated in complex environments where many tasks must be solved. Since accurate belief revision requires time and considerable computational costs, realistic cognitive agents need some mechanism which is responsible for trigg Notice that standard approaches to belief revision are mainly interested on finding rationality principles and postulates driving belief change (this is the main purpose of the classical AGM theory 1 ). All those models implicitly assumes that when the agent perceives some fact ϕ the perception is always a precursor of a belief change with ϕ. Standard models of belief revision have not investigated under what conditions belief revision should be triggered after having perceived a certain fact. gering belief revision and for making it more or less deep. On one side belief change in cognitive agents is triggered by very surprising incoming input. The degree of surprise relative to the incoming input "signals" to the agent the fact that things are not going as expected and that the knowledge of the environment must be reconsidered. Indeed wrong beliefs generally lead to bad performances and to failure in the intention and goal fulfillment. On the other side realistic cognitive agents do not generally reconsider their information state when input data conform with their expectations. Indeed it is not convenient for the survival of the agent to reconsider his beliefs every time a new fact is perceived. A model of cognitive belief change should be able to account for this trade-off between extensive and deep belief change triggered by intense surprise and superficial belief change (and in some cases belief change avoidance) when perceived facts are not very unexpected. h h The prominent functional role of surprise in belief change has already been investigated in cognitive psychology. [START_REF] Meyer | Towards a process analysis of emotions: The case of surprise[END_REF] Here we are suggesting a richer analysis by arguing that the process of surprisedriven belief revision is gradual (more intense is surprise, broader is the part of the knowledge structure involved in belief revision).

Conclusion

In this conclusion we fix the guidelines for possible extensions of our analysis. Some existing psychological model of Surprise 18 establishes that intense surprise triggers a search for causal explanations, justifications and confirmations of the unexpected event. After that an intense surprise has been felt, the agent tries to cope with it by trying to justify the unexpected event, by trying to confirm it (see also Lazarus' idea of Secondary Appraisal 13 ). In the present work the problem of coping strategies is left unspecified. We have assumed that the interpretation of input data is one-step process and that once the best explanation of input data has been the selected a gradual revision of the belief structure starts. In order to deal with coping strategies we would need to develop our theory into the following direction. Similarly to Shanahan's approach [START_REF] Shanahan | A logical account of perception incorporating feedback and expectation[END_REF] we should integrate in our model a feedback mechanism for readjusting the explanatory values of the candidate explanations (abducible formulas) . We should model this feedback mechanism in terms of epistemic actions. [START_REF] Lorini | The role of epistemic actions in expectations[END_REF] After that the agent has interpreted the input data, if the selected explanation is very surprising, the agent either: 1) can try to enlarge the set of abducible formulas in order to find a less surprising explanation of input data (an explanation whose explanatory value is higher than the explanatory value of the previously selected and very surprising explanation) or; 2) given the set of abducible formulas he can try to check (by doing some epistemic action for verification) whether the interpretation selected at the first round is the right one. Imagine that a doctor is almost sure that the patient has a certain pathology x. The doctor decides to subject the patient to a rigorous test in order to confirm the diagnosis. In our model the abducible formulas would be two: "the patient has pathology x", "the patient has not pathology x". Imagine the result of the test is negative. The doctor interprets this result as "the patient has not pathology x". The doctor gets surprised since the expectation that "the patient has pathology x" under scrutiny is invalidated. How could the doctor cope with this surprise? The doctor could make some epistemic action for verification (before accepting the fact that "the patient has not pathology x" and revising his beliefs and expectations). For instance the doctor could make some further medical tests.

We have studied in the previous section the role of surprise in belief revision. In our view surprise plays also a relevant role in intention reconsideration. [START_REF] Bratman | Intentions, plans, and practical reason[END_REF] Commitments on pragmatic plans can be reconsidered when some unexpected fact is perceived and the fact is evaluated to be relevant. Imagine an agent who has planned to go to Rome by train. The agent is giving for granted that there is not train strike. He reads the newspaper and discovers that there will be a train strike the day after. This relevant fact surprises the agent and makes him to reconsider his prior intention to go to Rome by train. The triggering role of surprise with respect to intention reconsideration is not formally analyzed in this paper. We leave this issue to future work.

Definition 7 :

 7 Degree of Mismatch.

Fig. 1 .

 1 Fig. 1. Expectation under scrutiny and its supports.

Table 1 .

 1 Joint probabilities.

	P(G/R)	g	¬g
	r	0,05 0,95
	¬r	1	0

Table 2 .

 2 Conditional probabilities.

Acknowledgment

This research has been supported by the European Project MindRACES. Moreover, we would like to thank our colleagues Rino Falcone, Luca Tummolini, Maria Miceli and also Mark Bickhard, Andrew Ortony and the other participants in the panel discussion about Surprise at the AAAI Fall Symposium "From Reactive to Anticipatory Cognitive Embodied Systems".

Definition 9: Support ψ of an expectation under scrutiny that ϕ. Support i (ψ, ϕ) = def T est i (ϕ) ∧ (ψ ⇒ i ϕ)

It is easy to prove the following proposition. Proposition 2. If the selected explanation of input data invalidates the expectation under scrutiny then it invalidates all supports of this expectation. Formally:

Proof. It follows straightforward from definition 9. Indeed ψ ⇒ i ϕ in formula Support i (ψ, ϕ) can be rewritten as ¬ϕ ⇒ i ¬ψ. Therefore (for transitivity of ⇒), if M,w |= ω ⇒ i ¬ϕ then M,w |= ω ⇒ i ¬ψ Let us now introduce the notion of Deep Mismatch-based Surprise: the surprise associated with the invalidation of the support of an expectation under scrutiny.

We can now prove the following property relating mismatch-based surprise (generated by mismatch between the expectation under scrutiny and the invalidating selected explanation of the input data) with deep mismatch-based surprise (generated by the mismatch between a support of the expectation under scrutiny and the invalidating selected explanation of input data). Proposition 3. Given an expectation that ϕ under scrutiny and the belief that ω which supports ϕ, if the former is invalidated by the selected explanation of input data ψ then the surprise associated with the mismatch between the belief that ω and the invalidating ψ is equally or less intense then the surprise associated with the mismatch between the scrutinized expectation that ϕ and the invalidating ψ. Formally:

Proof. We need first of all to prove that:

This can be proved from definition 8 of Mismatch-Based Surprise and proposition 2.

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