The aim of this study was, firstly, to analyse the long-time fatigue behaviour of crowns constructed from a novel polyetherketoneketone (PEKK) polymer, using artificial prepared teeth. Secondly, to determine the effect of the material’s stiffness that used as an artificial prepared tooth on the fatigue life of the PEKK crowns in comparison to human prepared teeth.6 loading cycles were reached. The number of loading cycles until failure was recorded. Wöhler curves were created to display the fatigue limits.Veneered crowns with a PEKK framework were constructed on three different prepared teeth: artificial polymethyl methacrylate (PMMA) teeth, artificial CoCr teeth and extracted human teeth. As far as applicable, the loading protocol was based on EN ISO 14801:2007 for fatigue testing of dental implants. After initial static fracture tests on three specimens from each group, the remaining crowns were loaded with different force levels until fracture or until 2 × 10Static fracture limits as well as fatigue limits differed for all three core materials. The static fracture tests resulted in fracture limits of 1200 (± 293) N for the PMMA group, 1330 (± 219) N for the CoCr group and 899 (± 96) N for the human tooth group. Fatigue limits of 770 N, 840 N and 720 N were determined for the PMMA group, CoCr group and human tooth group, respectively.The determined fatigue limit of above 720 N  is sufficiently high and a good performance of this crown material is expected in the clinical loading life. The results showed that using artificial teeth instead of natural teeth for fatigue testing of crowns might result in an overestimation of the fatigue limits of the crown material.PEKK-made crowns offer a stable and priceworthy treatment for patients, in particular those that suffer from metal allergy. Metal and ceramic are commonly available restoration materials for single dental crown. It has proven to be a good choice to match the aesthetic and functional requirements of the patient and is considered by many dentists as a gold standard –4. NonetThe disadvantages of metal restorations can be avoided using full ceramic restorations. One advantage of the recent CAD/CAM ceramics such as zirconia is reduced treatment costs and treatment time . AnotherIn the last years, different high-performance polymers have been introduced to different medical fields. Most of these polymers belong to the polyaryletherketone (PAEK) family in general or more specific to the polyetheretherketone (PEEK) sub-family. Numerous studies documented the successful clinical performance of such polymers in orthopaedic and spine patients as well as in dental applications –16.Such high-performance polymers are also of interest for the field of dentistry. Common polymers are currently only used in very few cases as a permanent fixed restoration –19 due tRecently, a novel polymer from the polyetherketoneketone (PEKK) family has been introduced for dental applications. In a clinical study , the bioIn an in vitro study , the retIn order to estimate the mechanical performance of the crowns made from such materials during their lifetime in the oral environment, it is not sufficient to perform static loading tests but the cyclic loading of each crown has to be investigated since the cyclic loading can influence the long-term success of these crowns. Dynamic loading is essential to test dental restoration materials, as the dental restoration has to endure repeated loading during biting, swallowing and speaking.When performing such tests, a special care must be taken for choosing a suitable test environment representing—as far as possible—the clinical situation. Since it is difficult to collect enough extracted human teeth on which the crowns can be tested, artificial teeth offer an alternative approach. Nonetheless, it is not clear in how far the material properties of the prepared teeth can influence the long-term behaviour of the crowns.Hence, the aim of this study was twofold: firstly, to analyse the long-time fatigue behaviour of crowns constructed from PEKK using artificial prepared teeth and secondly, to determine the effect of the material’s stiffness that used as an artificial prepared tooth on the fatigue life of the PEKK crowns in comparison to human prepared teeth.Group 1 used 14 identical artificial teeth made from polymethyl methacrylate (PMMA).Group 2 used 11 identical artificial teeth made from a cobalt-chrome (CoCr) alloy.Group 3 used 9 extracted human wisdom teeth. The prepared human teeth varied in size and geometry. The occlusal surfaces of these teeth were individually reduced according to clinical practice to hold the crown.In the optimal case, extracted human teeth would be used for such fatigue testing. Unfortunately, in most cases, the same reasons that cause a tooth to be extracted are at the same time reasons why these extracted teeth are not suitable for fatigue testing. If a tooth defect can be repaired by creating a dental crown, it should not be extracted. Wisdom teeth are typically smaller than the first and second molars, they are sometimes not fully erupted and they are often damaged during extraction. Therefore, it is difficult to obtain a reasonable high number of human molar teeth in a sufficiently good quality with similar geometry. For this reason, three different test series were performed with different approaches to simulate the prepared teeth:The choice for the two materials for the artificial teeth—PMMA and CoCr—was done to have one material that is softer than the natural tooth as well as one material that is harder than the natural tooth. This was done to avoid that an unfortunate choice for a material distorts the fatigue testing results.For all three groups, dental crowns made from PEKK  were manufactured on the prepared teeth according to clinical practice while maintaining the manufacturer’s instructions for use. As no local dental lab was available with the required equipment for processing Pekkton, the crowns were manufactured directly at the company Cendres + Métaux SA.2O3 at a pressure of 2 bar and then cleansed with oil-free pressurised air.The crown framework was milled in wax using small milling machine  and then pressed in Pekkton® ivory . The thickness of the framework was 0.8 mm on the occlusal plane, and a wall thickness of at least 0.6 mm in all other areas Fig. . The croTo archive an identical veneering for all specimens on the artificial teeth, a two-part resin mould was used Fig. . The mouPriming of the surfaces was performed with SR Link . The primer was applied with a residence time of 2 to 3 min. Subsequently, Nexco Opaquer  was applied in 2 thin layers to the framework. The first layer was light-cured for 20 s for each segment . The final hardening was performed with Solidilite EX  for 3 min. After the final hardening, the inhibition layer was removed with a disposable sponge.The framework was positioned in the mould and covered carefully with Nexco dentine paste . Salinisation of the surfaces was performed with Monobond Plus  and primed with Luxatemp glaze&bond . The abutments of the PMMA teeth were prepared by sandblasting using 110-μm abrasive blast media at a pressure of 1 bar. After the conditioning, the crowns are cemented on top of the PMMA teeth using Multilink Automix .All specimens on artificial teeth  were prepared by a dental technician at Cendres + Métaux SA and then sent to the Materials Testing Lab of the Department of Oral Technology at the University of Bonn.For the fatigue testing, the specimens were embedded in a short piece of copper tube using resin  with a Young’s modulus of 3.0 GPa. According to the clinical situation, the specimens were embedded at the height of the cementoenamel junction Fig. . The copA total of 9 extracted wisdom teeth were collected at a dental practice in Wesseling, Germany. All teeth were controlled to have no cracks and were suitable to be prepared to receive a crown. After extraction, they were disinfected in sodium azide solution. After disinfection, any remaining soft tissue was removed from the roots, and the teeth were stored in purified water at 8 °C for further use. The teeth were returned to this storage in between all the following steps. In order to aid the preparation of a crown, the teeth were embedded in short copper tubes in the same way as performed for the artificial specimens.The preparation of the teeth was performed using coarse and fine diamonds according to common clinical practice at 40,000 rpm and with water cooling. Plaster replicas of these prepared crowns were created and send to Cendres + Métaux for the manufacture of the crowns. The creation of the crowns followed the same steps as described above. The only difference to the above specimens was that no mould was used, as each of the generated crowns followed the individual geometry of each individual tooth. The crowns were sent back to the Department of Prosthetic Dentistry, Preclinical Education and Materials Science and cemented on top of the teeth.For this purpose, the teeth were cleansed with water, dried using pressurised oil-free air, and Multilink Automix Primer A+B  was used as primer. All the following steps were performed in accordance with the crowns on artificial tooth.The fatigue tests of the PEKK crowns were performed on a commercial pneumatic setup , and the initial force level for fatigue testing was chosen as 80% of these fracture limits as required by EN ISO 14801:2007.6 loading cycles for each specimen. These parameters were chosen for the fatigue testing of the PEKK crowns as well.For tests in wet environment, the standard requires a loading frequency of 2 Hz and a total of 2 × 10For all tests, a specimen was considered to have failed fatigue testing if either a fracture occurred in the veneering and/or the framework, or a crack was visible in the veneering and/or the framework. Due to the punctual load application, indentions at the point of force application directly beneath the tip of the thrust die were to be expected. Such an indention or small cracks that did not exceed the size of the indenter were not considered as failure, as long as no crack formation could be observed outside of this location. For all tests, a specimen was excluded from testing, if a fracture of the natural or artificial tooth occurred, and the crown did not fail by the above-mentioned criteria.Based on the data gained from the fatigue tests for all three groups, Wöhler curves were generated according to EN ISO 14801:2007. The number of specimens per group and force level was not high enough to perform a statistical analysis to determine possible significant differences among the three groups.log (number of loading cycles) versus the corresponding force level [For the statistical analysis, a linear regression analysis was performed using the linearised data gained from plotting ce level . The sloThe static fracture tests resulted in fracture limits of 1200 (± 293) N for the PMMA, 1330 (± 219) N for the CoCr and 899 (± 96) N for the human tooth groups, respectively. It was decided to use the same force levels for all three groups to allow a comparison between all three groups. An initial value of 1200 N was chosen for fatigue loading with the artificial teeth, and for each following force level, the force was decreased by 150 N.All specimens of the group 1 and group 3 that were submitted to a load of 600 N survived during the fatigue test. For group 2 (CoCr), all specimens that were submitted to a load of 750 N survived. A summary of the performance of the different specimens in the groups can be seen in Table A total of 15 specimens on artificial PMMA teeth were used in the fatigue tests. Those crowns on the PMMA teeth that failed in the fatigue testing showed a mixture of cracks in the veneering and complete fractures of the crown  detached from the PMMA tooth after less than 10 cycles with a force level of 1200 N. The early failure might be an indication for a failure in the application of the cement. As the adhesion of the cement to the PMMA differs from the adhesion to natural teeth, this event was not considered as a failure of the crown.A total of 11 specimens on artificial CoCr teeth were submitted to the fatigue tests. Those crowns on the CoCr teeth that failed in the fatigue testing showed fractures of the whole crown only at a force level of 1200 N. For lower force levels, specimens failed only because of cracks visible in the veneering. No specimen on CoCr teeth had to be excluded due to a fracture of the artificial tooth.A total of 9 specimens on human teeth were used in the fatigue tests. Due to the limited number of available teeth, only two specimens were used in each but the final force level. No crowns on human teeth at all were tested at a force level of 1200 N due to the limited number of available teeth as well as due to the lower static fracture limit of these specimens.Those crowns on human prepared teeth that failed in the fatigue test showed mainly fractured crowns. Only in one case, at a force level of 900 N, a crack formation in the veneering was observed, and fatigue loading of this specimen stopped. In two cases, the whole crown including the framework fractured (at 1050 N and 900 N). Both full fractures occurred during the initial phase while the setup was adjusting the load control parameters. The remaining two fractures only involved the veneering while the framework was still partially intact (at 1050 N and 750 N). Such a separate failure of the veneering was only observed for the specimens on human teeth.One specimen had to be excluded as the tooth fractured within the first ten loading cycles with a force level of 600 N. The fractured tooth showed a noticeable enlarged pulp and in turn a rather thin-walled tooth, which might have caused the fracture. The fatigue testing was repeated with a new specimen.For the statistical analysis, the following slopes and 95% confidence intervals were determined for the three materials: − 94.6 (± 67.4) for PMMA, − 174.3 (± 60.1) for CoCr and − 48.4 (± 43.9) for the human teeth. The determined slopes for both groups with artificial teeth lay above the 95% confidence interval of the data for the crowns on human teeth. Thus, it can be deduced that the fatigue behaviour of the specimens differs depending on the chosen material for the artificial tooth . PMMA has a stiffness of 3300 MPa , while CoCr has a stiffness of 250,000 MPa . Thus, the fatigue testing on artificial teeth seems to overestimate the fatigue limit of the tested veneered polymer crowns. Nonetheless, it must be kept in mind that only a limited number of natural teeth of the necessary quality are available for fatigue testing. A fracture mode that only involved the veneering while the framework was partially intact could only be observed for the specimens on human teeth. This might be related to the fact that the cement used for fixing the crowns on the abutments is chosen for the best adhesion between tooth and framework, and its adhesive properties might differ on the artificial teeth made from PMMA or CoCr.Nicolaisen et al.  performeIn a recent study of Alsadon et al. , the fatDifferent criteria influence the magnitude of the biting force to be expected in the clinical situation. Age, sex, health, state of the dentition, the prevalence of dental restorations such as tooth- and implant-supported prostheses , stomatoDespite the high number of papers dealing with the maximum bite force, there is to our knowledge no source available for a suggested minimum fatigue limit for dental restorations. It can be assumed that not every intra-oral loading cycle reaches the maximum amount of biting force. Gibbs et al.  found inFor our knowledge, there is only limited literature available about the determination of the fatigue limit for dental crowns. There is few literature available about the cyclic loading for dental crowns; for example, a review study by Elshiyab et al. 2017  gives a A further study needs to be done to compare PEKK crowns with other crowns’ materials such as metal-ceramic and full ceramic crowns. This was a limitation of the present study.A high fatigue limit of 720 N was determined for PEKK crowns that could provide a good, expected performance of this crown material in the clinical loading life. The results showed that using artificial teeth instead of natural teeth for fatigue testing of crowns might result in an overestimation of the fatigue limits of the crown material.


In the version of the microPublication originally published online, the link to extended data file was omitted from the original article, this has been corrected. A link to the extended data file <Authier et al_GerminationData_Metadata_Micropublication 2021.csv> has been attached to the original article.A reference to the extended data file has also been added to the figure legend (1B and 1C).


This research paper designs the noninteger order SEITR dynamical model in the Caputo sense for tuberculosis. The authors of the article have classified the infection compartment into four different compartments such as newly infected unrecognized individuals, diagnosed patients, highly infected patients, and patients with delays in treatment which provide better detail of the TB infection dynamic. We estimate the model parameters using the least square curve fitting and demonstrate that the proposed model provides a good fit to tuberculosis confirmed cases of India from the year 2000 to 2020. Further, we compute the basic reproduction number as  Tuberculosis is a disease instigated by Mycobacterium tuberculosis bacteria (MTB), which commonly affects the lungs; however, it can also harm other parts of the body. It is an infectious disease; around ten percent of the latent infection develops into a highly infected disease and results in death. The transmission of TB from one person to another occurs through air droplets when an infectious person sneezes or coughs . GeneralIn the wake of the COVID-19 pandemic, India has observed a drop in detecting new TB cases by 1.3 million from 2019 to 2020 )For an integrable function f, the Caputo derivative of fractional order α with Also, the fractional integral of order , 15)15])f is given by α and For Also, the C-F fractional integral is presented by , 36)36])α is given by The L. T. of Caputo fractional differential operator of order The compartmental model for the transmission dynamics of TB is represented graphically in Fig. λ is the contact rate. At time t, The population is separated into nine compartments which are classified based on the infection status. Here, modeling , 34. TheTherefore, we can observe that the solution of system  is nonneThe overall dynamics of the population attained by adding all six equations of model  is as foSystem  is moderThe feasible region for model  is givenWe prove that the closed set Ω is the feasible region of system .The closed set Ω is a positive invariant concerning fractional system <0, see  for moreWe use the following equations to determine the equilibrium points for the fractional-order model : \documeThe algebraic solution of the equation provides equilibrium points of the system, and if n system  has a poFrom the above equations, we observe that the endemic equilibrium Figure μ. The average life span in India is 69 years mentioned in Press Information Bureau, Government of India [In order to obtain a good fit to the real data, we use the least square curve fitting algorithm given by Khan et al.  for our of India . The natFrom Fig. In this section, we use the generalized Adams–Bashforth–Moulton technique given in  to proveBy using Lemma In the ensuing theorem the kernels The kernelsatisfies the Lipschitz condition and contraction if the inequality given below holdsS and For Suppose that Thus, for Similarly, the Lipschitz condition for We take the norm of the first equation in the above system, then With Lipchitz condition , we haveAs a similar way, we obtained Thus, we can write that The existence of a solution is given in the next theorem.A system of solutions given by the fractional SEITR model  and then starts taking treatment. In Fig. In Fig. In Fig. The individual who is not diagnosed in the initial compartment may become actively infected and also spread the infection to others. People having symptoms of TB infection but not diagnosed as TB-infected and people diagnosed as positive but not starting treatment may move to treatment class after reaching more severe conditions. In Fig. Figure Figure λ of an infected individual is considered as lower than the recovery rate From Fig. λ, treatment rates Figure For Fig. In this paper, a dynamical fractional-order mathematical model in the Caputo sense is proposed for the behavioral investigation of TB infection in India. Using the nonlinear least square algorithm, we estimated the parameters used in this model using TB-infected cases reported in India from the year 2000 to 2020, which shows that the model curve provides a good fit to the real data. Moreover, the future trend of the model curve provides indication to the decision maker or policy maker for better devising disease prevention and control measures. Further, we present the endemic equilibria, locally asymptotically stable, and the TB infection-free equilibrium in terms of the basic reproduction number. The value of the basic reproduction is obtained as


A search for magnetised quark nuggets (MQN) is reported using acoustic signals from hydrophones placed in the Great Salt Lake (GSL) in the USA. No events satisfying the expected signature were seen. This observation allows limits to be set on the flux of MQNs penetrating the Earth’s atmosphere and depositing energy in the GSL. The expected signature of the events was derived from pressure pulses caused by high-explosive cords between the lake surface and bottom at various locations in the GSL. The limits obtained from this search are compared with those obtained from previous searches and are compared to models for the formation of MQNs. The extra Fermi Sphere could lower the energy state for the quarks binding the higher-mass strange quark, overcoming the tendency for the strange quark to decay, and negating the effects of surface tension seen in nuclei. Since the quarks are not constrained by the need to be held within a nucleon, the density of this state of matter could be higher than the nuclear density. Furthermore, since the lowest energy levels in the Fermi spheres would be filled in the ground state, QNs would be nearly electrically neutral. Hence the repulsive electrostatic forces which decrease the stability of higher mass nuclei would be absent for QNs, and there would not be any limit to their masses. Indeed, Fahri and Jaffe3 showed that large masses are possible.Quark Niggets (QNs), , are a novel form of matter consisting of u, d and s quarks. They were first proposed by Bodmer4 postulated that they could explain the matter–antimatter asymmetry in the Universe. Hence identification of QN DM could solve two of the major outstanding problems in cosmology.QNs could be candidates for the Dark Matter (DM) in the Universe. Furthermore, Liang and Zhitnitsky5 showed that QNs could also show ferromagnetism through the one-gluon-exchange mechanism. There should then be a large magnetic field on the surface of such QNs with a predicted strength of Bo ~ 1012±1 T , typical of that at the surface of a neutron or proton.Tatsumi6, we presented the design and supporting simulations for the experiment in the GSL. The results of the experiment are reported here.In this paper the results of a search for MQNs are described. An acoustic technique was employed using hydrophones placed in the Great Salt Lake (GSL) in the USA. In Ref.11. These experiments would not have been sensitive to MQNs which would range out in the rock overburden. The searches for non-ferromagnetic QNs by Porter et al.12 and Piotrowski, et al.13 using astronomical telescopes would have been sensitive to MQNs but their flux limits were given for non-ferromagnetic QNs. A previous study by our group14 excluded MQNs with very high surface magnetic fields. The experiment presented here is a search for MQNs with lower surface magnetic fields.Previous searches for QNs have been made using underground detectors15, were used in the published searches with astronomical telescopes for non-ferromagnetic QNs by Porter et al.12 and Piotrowski et al.13.The properties of non-ferromagnetic QNs, discussed by De Rujula and Glashow17 kg/m3). They are larger than atomic sizes for masses greater than vqn is the QN velocity, and K = 2/3 assuming the molecules are in random motion or 4/3 assuming they are stationary. Here we set K = 1, the difference reflecting the uncertainties in the assumptions. If QNs are constituents of DM, they would be expected to enter the solar system at speeds, 16, the velocity of the Solar System in its path through the Galaxy.A brief summary of these properties follows. QNs could have densities somewhat greater than nuclear density (~ 2 × 10R of a QN of mass 3 and 3).From Eq. , the ranentclass1pt{minima6. A plasma of free electrons and positive ions is created as the electrons are torn off the atoms due to friction and their interaction with the intense magnetic field. The plasma comes into equilibrium at a distance Rmp from the MQN at which the plasma pressure balances the pressure exerted by the magnetic field. Assuming that the plasma-facing surface of the magnetopause is approximately spherical, the magnetopause has been shown to have a radius, Rmp given byB0 is the magnetic field at the equator of the QN and Rmp for Rqn in Eq. , this is integrated as in Eq. .5\documeSince the size of the magnetopause is much greater than the radius of the QN, the stopping power is larger for MQNs than for non-magnetic QNs and, therefore, the range in matter is much shorter.6 and briefly updated with important new information in “As described in Supplemental Information on Methods: Sensor locations, topology and sound speed of the Great Salt Lake (GSL) Observatory for MQNs, an array of three hydrophones sensors was placed in the GSL at a depth of 2 m (GSL mean depth 4.8 m) for the purpose of detecting MQN impacts. The hydrophones and data acquisition system are described in detail in Ref.d, as Figure The shots at near distances did not have time for the ~ 250-Hz portion of the waveform to reach full amplitude; therefore, each had to be extrapolated to a fixed time . Figure cs as function of depth z. The data show a gradient Δcs/Δz ~ 0.5 s-1, which means acoustic waves in the water are refracted along a circular trajectory with radius of ~ 3.2 km. Consequently, propagation is quite complicated. Simple estimates of pressure versus distance are incorrect. The multiple layers of material in and under the GSL had to be considered to understand propagation.As explained in Supplemental Information on Methods: Sensor locations, topology and sound speed of the Great Salt Lake (GSL) Observatory for MQNs, we measured the sound speed The geology under the Great Salt Lake is complex with layers of mud, mirabilite, and rock and with a fault line separating the east and west sides of the lake. Seismic exploration and core-sampling have been focused on the east side, but our experiments had to be on the west side to avoid boat traffic.17. Three-dimensional effects, variations in the actual depth, and remaining uncertainties in the actual lake-floor geology were approximated by a simulation that assumed horizontal stratification of the water column and sediment layers.In spite of the complexity, the acoustic signals from explosive calibrations emulating an MQN impact created the relatively simple waveform shown in Fig. 3, and compressional attenuation of 2.0 dB/m to include absorption and refraction of sound in the lake 18. The simulated thickness of the layers of mud, mirabilite, and rock were consistent with extrapolations from the measurements 19 on the east side of the lake. The properties of each layer were assumed to be independent of radius and azimuth. Idealized simulations, based on a continual line source, a single receiver, and a lake of uniform depth, showed that acoustic energy was propagated in a waveguide formed by the multiple layers and produced a resonant signal at a frequency of approximately 250 Hz followed by a higher frequency signal burst, as was observed with the calibration shots. The signal properties were found to be very sensitive to the precise thicknesses of the individual layers. A combination of 5 m of brine, 0.75 m of hard mineral (corresponding to the stromatolite layer we found in this part of the lake), 3.0 m of mud, a 0.6 m transition layer into a 1.8 m mirabilite mineral layer above a 5 m rock layer, over bed-rock (to bound the simulation) reproduced the observed group velocity difference between the resonant and higher frequency components as well as the attenuation of the signals observed from the calibration shots  to 1597 m/s (at the bottom), with a density of 1200 kg/mThe experiment was deployed for a total of 146 days, The threshold for triggering the sensors depended on ambient weather conditions. The trigger threshold was reset every hour to three times the maximum amplitude of 10,000 digitized samples taken at 1,000,000 samples per second for a duration of 0.01 s. This implies the trigger level was a minimum of 13.5 standard deviations above the random ambient background. Continually resetting the threshold meant that the total observation time depended on signal amplitude. Setting the trigger level at a pressure of 2.5 Pa meant that the apparatus was sensitive for 80% of the time. The total data taking time allowing for all losses of sensitive time was Coherently produced triggered events were recorded if the amplitude exceeded this threshold for a duration of more than 3 ms. That duration was set to exclude signals that were obviously incompatible with the signatures from the explosive calibrations. Each hydrophone was triggered autonomously, recording the digitised signal as the mean of ten samples every 10 μs together with a time stamp (from the GPS).During the portion of the data taking that had all three sensors on line, sensors a1, a2, and a3 were respectively triggered 5,865,531, 1,941,741, and 142,282 times. The calibration shots showed similar behaviour for all 3 hydrophones. The rather different trigger rates associated with the 3 sensors could not be ascribed to ambient conditions but were thought to be generated electrochemically by the corrosive environment of the GSL. Sound transited the distance between sensors in ≈0.2 s. To ensure that variations in trigger levels did not exclude recording of coincident signals, a time interval of 0.7 s was used as an initial filter for the data. The number of coincident triggers in all 3 sensors within a time of 0.7 s was 1110. The signals are mostly of short duration. The 3 longest duration signals are shown in Fig. It can be seen that these signals do not resemble those from the explosive calibrations  of the simulated hit position to the hydrophone array and S is a safety factor taken to be 2. The quantities in Eq.  or give smaller signals on passage through the water of the GSL. Hence, they have smaller acceptances.Bo . The reason is that the stopping power is larger for lower MQN density . Hence, it is quite plausible that MQNs would grow quickly enough for their increasing inter-quark binding energy to prevent such decays. The growth of MQNs depends sensitively on the value of the surface magnetic field, Bo.The open dotted curve on the upper panel of Fig. Figure .12 and Piotrowski, et al.13 using astronomical telescopes. If we assume that the acceptances in these experiments for MQNs is similar to those for QNs their flux limits can also be used to limit the value of Bo. The bottom of the blue triangle in Fig. Limits have been set on the fluxes of non-ferromagnetic quark nuggets (QNs) by the experiments of Porter, et al21 but we believe that none have addressed the case of MQNs. The high-altitude search by the SLIM collaboration8 would have been sensitive to MQNs but their published flux limits are orders of magnitude higher than those published here. The deep underground searches by the MICA10, MACRO11. and OHYA9 collaborations have published limits on the QN flux which are again higher than the limits presented here due to the limited area of the detectors. However, these experiments would only have been sensitive to very high mass MQNs since the lower masses would range out in the rock overburden. The new experiments CTA22 and JEM-EUSO23 have the potential to perform searches for QNs and MQNs over a wide area. However, triggers to detect very fast meteor trails will be needed to distinguish solar system meteors from extra-galactic MQNs.There have been previous searches for non-magnetic QNs25 and measuring their radio-frequency emissions after passing through Earth’s magnetosphere26. Each method has strengths and weaknesses. Tables in Ref.14 Supplementary Information provide quantitative mass distributions for evaluating a specific detection system. The calculation in this paper assumed the mass density of Dark Matter in the solar system is the same as it is in interstellar space. That may not be true for MQNs since some of them would pass through a portion of the Sun, be slowed to less than escape velocity from the solar system, and be subsequently scattered by the combined planetary gravity to prevent their return to the Sun. These will be trapped in the solar system and enhance the density of Dark Matter. We do not yet know the enhancement factor, so the event rates described above may be pessimistic. If so, low-cost efforts to look for QNs and MQNs are recommended. A special trigger to detect fast meteors in a high-energy gamma-ray cosmic-ray detector could be used to prove or disprove, or place tighter limits on the existence of these objects.Other methods for detecting MQNs include measuring non-meteorite impacts on Earth–2 kg are excluded. In the Aggregation Model of Ref.14 magnetic fields of less than 0.12 TT are excluded by the data presented here.A search for ferromagnetic quark nuggets (MQNs) has been undertaken. No significant signal was observed and limits on the flux of these objects hitting the Earth have been set. In the fixed mass model masses of MQNs up to 10In order to detect these objects, if they exist, experiments covering larger fields of view with longer exposure times will be required. The atmosphere is a barrier for MQNs since low mass MQNs stop before reaching the Earth’s surface. The Aggregation Model predicts that most of the dark matter density will be concentrated in the high mass region whereas most of the flux will be at low masses. Hence searches in the upper atmosphere or in space will be advantageous. An experiment sensitive to fast high-altitude meteors could detect MQNs since most meteors travel with the velocity of the Earth around the solar system whilst QNs and MQNs which are of Galactic origin will have velocities about 10 times larger.It would be good to have a dedicated experimental program to search for these objects, given that either of them may carry the solution for two of the outstanding problems in cosmology, namely the nature of the dark matter in the Universe and the matter–antimatter asymmetry in the Universe.6. Each data acquisition system used a N210 USRP  manufactured by National Electronics and used the gnuradio open-source UHD  to configure each unit. The three systems were characterized and cross calibrated in the lab and during deployment in the experiment. These units were designed to operate primarily in the frequency domain for software radio. We found they operated satisfactorily in the time-domain required for our experiments if and only if the base-band frequency variable was set to 0. Otherwise, the gain is strongly dependent of this setting.The hydrophones and data acquisition system are described in detail in Ref.Supplementary Information.


Particle swarm optimization (PSO) has the disadvantages of easily getting trapped in local optima and a low search accuracy. Scores of approaches have been used to improve the diversity, search accuracy, and results of PSO, but the balance between exploration and exploitation remains sub-optimal. Many scholars have divided the population into multiple sub-populations with the aim of managing it in space. In this paper, a multi-stage search strategy that is dominated by mutual repulsion among particles and supplemented by attraction was proposed to control the traits of the population. From the angle of iteration time, the algorithm was able to adequately enhance the entropy of the population under the premise of satisfying the convergence, creating a more balanced search process. The study acquired satisfactory results from the CEC2017 test function by improving the standard PSO and improved PSO.  Scholars have applied different approaches to the increasing number of structurally complex optimization problems, which are difficult to solve using traditional means, including evolutionary algorithms such as the genetic algorithm (GA) , bee colPSO was proposed by Kennedy and Eberhart  in 1995 (1) Parameter tuning. Gunasundari et al.  proposed(2) Learning strategy. PSO is prone to getting trapped in local optima, and the search accuracy is not sufficient in late iterations because it uses only two experiences to guide particle learning. Improving this aspect of the learning strategy has attracted much attention. Liang et al.  proposed(3) Topology. Kennedy  pointed (4) Algorithm crossover. Hybrid algorithms are a key research area to improve the performance of PSO algorithms. They incorporate operators such as crossover, selection, mutation, and choice to improve the search quality of the population individuals and the general efficiency of the algorithm. For example, the use of genetic operators can improve population diversity and convergence to the global optimum. Hybrid algorithms can better escape local optima and overcome certain inherent drawbacks associated with single algorithms. Zhang et al.  proposedIn summary, the key to PSO improvement lies in balancing diversity and convergence, preventing premature convergence to a local optimum, and improving local exploitation. Previous work used entropy as an evaluation index to improve the algorithm but stopped at using entropy to constrain the population traits and did not explore a more optimal search process. Others divided up the search strategy using both new strategies or vastly different ones. This paper proposes a multi-stage search strategy to fully maintain the population diversity for the global search in the early stage and modifies the update formula in the late stage to help jump out of the local optima, improving the local search ability. While improving the algorithm exploitation ability and search accuracy, the original algorithm converges after adding the strategy. This strategy can be used as an improved operator and implemented in heterogeneous comprehensive learning particle swarm optimization (HCLPSO) and TSLPSO. Both of these algorithms divide sub-populations to balance exploration and exploitation; however, adding a new balancing strategy can have adverse effects. This paper explores these two representative algorithms, which are compatible with the new strategy, to illustrate the effectiveness of the strategy. The experimental results show that this strategy can effectively improve the probability of the algorithm converging to the global optimal solution.ith particle at moment t is 1 and r2 are random numbers between . The two factors c1 and c2, known as “acceleration coefficients”, are positive constants commonly used to determine when the cognition speed of the ith particle is accelerated towards pbest and gbest, respectively. The symbol ω denotes the inertia weight parameter which was originally developed to address the velocity explosion problem.PSO tends to a global convergence through the cooperation and competition of particles in the search space. The particle velocity and position updates are determined by the best position found by the corresponding particle (pbest) and the best position found by the whole population up to the current iteration of the algorithm (gbest). Let the total number of particles be Currently, TSLPSO is one of the algorithms that achieves superior results on the congress on evolutionary computation (CEC) test function , in whici values. The velocity of the exploration sub-population is updated asi, i, then two particles, namely ith particle. If fitness  strategy with learning probability Pc to generate random numbers for each dimension of the particles and to compare them with their respective learning probability PcSince the exploring particles are not allowed to access the information of the exploiting particles, there is no information flow from the exploiting sub-population to the exploring sub-population. To prevent loss of diversity, exploring and exploiting individuals do not interact. Inertia weights and dynamic acceleration coefficients are used for both sub-populations.The paper uses the multi-stage search strategy to improve the search results based on the original mechanisms of PSO, TSLPSO, and HCLPSO. The multi-stage search strategy is described below.Before population iteration, each dimension of each particle is given an additional attribute Ra, the value of which is 1 or −1. As shown in In the second stage, no operation is applied to the particle population. In the third stage, only the Ra of the optimal particle is kept unchanged, and all other particles have the opposite property value of the optimal particle in each dimension, i.e., they will be attracted by it.1 and c2 in the velocity update Formula (1) to achieve the expected improvement. However, from another perspective, the improvement strategy is independent of the velocity and position update Formulate (1) and (2), and this study sought to improve the algorithm in terms of exploration and exploitation and to control the shape of the population by relatively direct and simple means. From the analysis of the experimental results in In the velocity update Formula (1), particles are pulled by the gbest and pbest points. This seems similar to the attraction and repulsion of this strategy, but in fact, the PSO easily traps in local optima because particles are influenced by the current gbest point in the early stage, and the poor search accuracy in the later stage is caused by the attraction of the pbest point. Existing improvement algorithms make dynamic linear changes to learning factors cThe position and velocity of all particles are updated according to velocity and position update Formulae (1) and (2), the position of the lagging particles will change according to the fitness value of the particles, and the direction of movement is determined by the Ra value. The Ra value is an inherent property of each particle: its positive or negative value determines repulsion or attraction between particles, i.e., particles attract those with the opposite Ra value, but repel those with the same. Each particle has a Ra value for each dimension.As a result of the randomness of Ra, a lag particle action also has randomness. When the force value pow is large enough, the new interaction force between particles can dominate the movement of the lag particle swarm. The force value is calculated asTo limit premature convergence, which is the tendency to trap in local optima, the Euclidean distance between particles is incorporated in stage 1. Once the distance between any two particles reaches a threshold, the particle with the lag fitness value will be selected and a force applied to it in the opposite direction with respect to the gbest point, pushing it away from the region in which it was originally located. This strategy coexists with the strategy repulsion and attraction mentioned above, and they affect each other.As shown in In stage 3, we redefined the additional behavior of the particles, maintaining the Ra value of the current particle closest to the gbest point, and making the Ra values of all other particles the opposite, i.e., all other particles will be attracted by the particle near the gbest point, the power force decreases with the number of iterations, and it will decay to 0 at the end of the iteration.It should be noted that the optimal particle may change, so the pow of each particle will be recalculated after each iteration, and there is no Ra influence between other particles except the optimal particle.Unlike the standard PSO, in which particles are influenced by the information of the pbest and gbest points, in stage 3, particles are influenced by the information of the current optimal particle position, which does not overlap with the gbest point position. The subtle differences between the two can be seen in (1)A particle distance determination is triggered in stage 1. The Ra of the less adapted of two particles will be reassigned with the same value as the other particle;(2)At the beginning of stage 3, a particle that is not the best adapted will have an Ra value opposite to that of the best particle;(3)When the optimal particle changes in stage 3, repeat 2.Ra is randomly attached to each particle at the beginning of phase 1 and will change in the following situations:(1)In all phases, on the basis of the original algorithm velocity and position update formula, particles that are equivalent to each other will be subject to attraction from the gbest point;(2)In stage 1, the lag particle will be attracted by the pow in Formula (7) from the better particle population;(3)In stage 1, the particle with the most lag of two close particles will be subject to repulsive force;(4)In stage 3, the non-optimal particle will be attracted by the optimal particle.The interactions between particles are summarized as follows, where repulsion and attraction are reflected by the change in the direction of velocity and magnitude, which directly acts on velocity change:Input: Tmax is the maximum number of iterations, and Dim is the number of dimensions, Ra: denotes additional properties, and pow denotes force parameters.Output: GbestValue is the global optimal solution, and a and b are constants for control phase division.Algorithms 1.1 Initialize a population, initialize particle velocity, position, fitness; initialize PBest, GBest.2 Randomly assign particles Ra3 for t = 1 to Tmax do4 Update the velocity v of the particles according to the particle population update formula;5      If t < a/Tmax do6      calculate the additional offset of the particle position based on pow and Ra7      update the particle velocity v and position x again.8      update pbest, gbest.9      Distance detection, adjusting Ra.10    else If t > b/Tmax do11        adjust the Ra of the non-optimal particle opposite to the optimal particle.12        Calculate the additional offset of the particle position based on pow and Ra.13        update the particle velocity v and position x again.14    end15 endThe paper discusses the effectiveness of the strategy in terms of entropy and convergence. An increase in population confusion often means a decrease in convergence performance, but if it increases the likelihood that the population searches for the optimal value without affecting convergence, i.e., it produces small fluctuations in the convergence curve, it will help improve the algorithm. We chose population entropy and DBscan as reference tools to draw figures to establish that the multi-stage search strategy worked.Entropy is a measure of the degree of chaos in a system and population entropy is an indicator of group diversity. The general definition of entropy may not apply in high-dimensional space; therefore, we chose population entropy as the evaluation criterion of algorithm convergence. In this paper, we assume that the number of particles is M, and the best individual position of the particles is Step 1: Calculate the minimum fitness of the particle’s historical best position, Step 2: Divide the interval Step 3: Calculate the population entropy, The larger the computational results of population entropy, the more chaotic the particle distribution in space and the less convergent the algorithm.We compared PSO and the improved algorithm RaPSO using CEC2017-fun10 as the objective function and by conducting experiments in 10-dimensional space. The fitness value of DBscan can measure the degree of population disorder from the perspective of density. It defines clusters as the maximum set of densely connected points, which can divide regions with sufficient density into clusters and find clusters of arbitrary shapes in the spatial database of noise. The DBscan calculation method is described as follows:Step 1: Define the field radius e and the threshold of core point;Step 2: Start from one particle and find all particles that have a density connection. Jump out of this cluster and start to find particles again from the other particle;Step 3: Repeat step 2 until all particles have been traversed.We chose the conditions for the population entropy experiment and plotted We chose 29 test functions in CEC2017 as objective functions for optimization. Among them, Fun1 is single-peaked, Fun3–Fun10 are multi-peaked, Fun11–Fun20 are mixed, and Fun21–Fun30 are composite. The details are shown in the The 29 test functions of CEC2017 were selected as target functions for optimization tests, and experiments were conducted with a population size N = 100, repeat = 30, and D = 10, 30. Other parameter settings are shown in After adding the improvement mechanism, there were three additional controllable parameters in each algorithm: str, prop, and stage. In order to ensure the accuracy of parameter selection, the number of experimental repetitions was increased to 100.x-axis in y-axis represents the average fitness after normalization. The value range of the x-axis was obtained after a large number of experiments. The experimental range was far from that. It was the range obtained after preliminary screening. The sub-graph in the x-axis from 20 to 30. The results from The parameter str affects the magnitude of the strength of all repulsions and attractions in stages 1 and 3. The algorithm used in the experiment was the standard PSO. With the other parameters unchanged, Fun5, Fun14, and Fun23 in CEC2017 were selected as objects for the comparison experiments. The y-axis was unprocessed. Other parts are similar to those used in the experiment in The parameter prop determines the proportion of poorly adapted particles in stage 1. Because only Fun5 in CEC2017 was selected as the test function, the fitness value of the The parameter stage is the basis for deciding the division of the three stages. Similarly, we used standard PSO as the experimental algorithm in 10 dimensions. The test function was Fun25 in CEC2017. Based on the experimental results in x-axis is the fitness evaluation number . The y-axis is the average fitness value. Different colors represent different original algorithms. Solid lines represent algorithms with the strategy and dotted lines represent those without. According to the figure, algorithms adopting the strategy achieve lower (better) fitness and better search performance. Since the strategy increases population disorder during early iterations, the phenomena of slower convergence in early iterations and a poorer fitness value could be observed in certain circumstances in the figure; however, these phenomena are often improved at the end of search. Full development in the preliminary stage lays a more solid foundation for later exploration, and may in turn yield better fitness values. According to A relatively direct and simple method is proposed herein to address problems such as premature convergence and poor searching precision with PSO, i.e., improving particle swarm diversity in early iterations of the algorithm and introducing guidance for the best particle location in the population at the end of the iteration to enhance searching precision. The concept was based on PSO and its two improved algorithms and acts as a multi-stage search strategy operator. It guides the particle swarm to different goals in different stages, improves the comprehensive exploration and development ability, and upgrades population disorder without changing the convergence performance of the algorithm too much, thus effectively improving the particle’s ability to jump out of local optima. Particle swarm behavior before and after the application of the strategy was compared and described, and the effectiveness of the strategy was verified with the population entropy and DBscan tools. The contrast experiment indicated the selection process for three parameters in the proposed strategy. Moreover, according to experimental results of the algorithms on the test functions, the strategy can improve search performance.In future work, we hope to mathematically demonstrate that the selected parameter values are optimal, prove the general applicability of the strategy, and demonstrate the adequacy and necessity of the divided stages. The effectiveness of the proposed strategy was experimentally demonstrated, but a mathematical analysis and proof are required.


While the SRT model allows one to gain more information about ability than is possible based on considering only response accuracy, measurement may be confounded if persons show differences in their response speed that cannot be explained by ability, for example due to differences in response caution. In this paper we propose an adapted version of the SRT model that makes it possible to model person differences in overall speed, while maintaining the idea of the SRT model that the speed at which individual responses are given may be indicative of ability. We propose a two‐dimensional SRT model that considers dichotomized response time, which allows one to model differences between fast and slow responses. The model includes both an ability and a speed parameter, and allows one to correct the estimates of ability for possible differences in overall speed. The performance of the model is evaluated through simulation, and the relevance of including the speed parameter is studied in the context of an empirical example from formative educational assessment.With advances in computerized tests, it has become commonplace to register not just the accuracy of the responses provided to the items, but also the response time. The idea that for each response both response accuracy and response time are indicative of ability has explicitly been incorporated in the signed residual time (SRT) model (Maris & van der Maas, 2012, Psychometrika,  While standard item response theory , as it captures the speed at which a respondent provides responses on the test. Since under the hierarchical model in the population of persons response speed and ability may be correlated, the model can make use of collateral information about ability contained in the RTs to improve the precision of measurement  model . Hence, in both models the nature of the estimated ability is not altered by the inclusion of RTs in the model. Both models make use of collateral information available in the RT data to improve the precision with which this ability is estimated, but they differ in the way in which RTs are incorporated. In the case of the hierarchical model, RTs are not directly linked to ability, but rather are taken to be fully explained by a latent speed variable that in turn provides collateral information for the estimation of ability. Thus, although speed and ability may be correlated, the model still assumes there to be variability in speed conditional on the latent variable (provided the correlation is not 1 or −1) and respondents of the same ability level may differ in their response speed. In contrast, the SRT model does not include a latent speed parameter, and assumes that, conditional on ability, all respondents will operate at exactly the same speed. That is, the expected response time on a particular item is the same for all persons of a particular ability level, from which it follows that the model assumes that respondents of the same ability level all operate at the same speed.Although the SRT model is a normative model, it can be considered unlikely that respondents fully succeed in optimizing their response speed, meaning that individual differences in RTs may remain that cannot be accounted for under the model. The many applications of the hierarchical model tell us that there is often a positive manifold for the RTs, suggesting that some respondents consistently respond at a higher speed than others. Since in practice positive  item discrimination parameters might be underestimated due to the model being less able to explain person differences in observed performance. Bias in the person estimates can be expected both for respondents who display suboptimal responding behaviour in terms of their response speed (resulting in their ability being underestimated) and for respondents who behave optimally . These two issues make it important to exclude the possible presence of unmodelled person differences in response speed when using the SRT model, or, if such differences are present, to incorporate such differences in the statistical model.In this paper we propose an IRT model that – similar to the SRT model – measures ability based on both RA and RT, but unlike the SRT model also takes into account that persons may differ in their response speed. To facilitate statistical modelling, responses are categorized as being either fast or slow, resulting in four possible combinations of RA and RT. The proposed two‐dimensional discretized SRT model maintains the core idea of the SRT model, as it assigns more credit to fast correct responses than to slow correct responses and assigns less credit to fast incorrect responses than to slow incorrect responses. Like the SRT model, in the proposed model performance is explained on the person side by an ability parameter. However, the model includes an additional person parameter, which explains the relative frequency with which fast responses are given (and hence captures a person's tendency to give fast rather than slow responses). As such, it can account for person differences in performance that are not due to differences in ability but rather due to differences in a response tendency , and can correct the ability estimates for these person differences that may otherwise confound measurement. In this way, the model is designed to combine the advantages of the SRT model  and of the hierarchical model , while avoiding their limitations as discussed. Furthermore, the latent variable included may also provide test takers and test administrators with additional relevant information, since it provides insight into the extent to which a test taker deviated from the model‐implied optimal response speed. This could be useful feedback for improving future test‐taking performance, and it could also be of substantive interest in its own right, if such a latent variable relates to psychological constructs such as risk seeking or risk avoidance.The structure of the paper is the following. In Section 2The SRT model was proposed by Maris and van der Maas , and useiS is the item score , iX is the binary RA , iT is continuous RT, d is the imposed item time limit, and index i refers to a particular item. The model assumes there to be an item time limit, which in this version of the model is taken to be the same for all items. The scoring rule thus entails that the time that is left before the deadline  is either added to the total score if the response is correct, or subtracted from it if the response is incorrect. The idea of the scoring rule is to give more credit to fast correct responses than to slow correct responses, and also to discourage fast careless responding by punishing fast incorrect responses more heavily than slow incorrect responses. The SRT model is derived from the sufficiency of the person total score sp+=∑ispi for the ability of person p, the sufficiency of the item total score si+=∑pspi for the difficulty of item i, and conditional independence of the item scores given the latent variable.where i is the difficulty of item i. The SRT model has been generalized by van Rijn and Ali (i> 0In the SRT model the distribution of item RA and RT given ability (denoted by θ) is written as and Ali , who intBy including a discrimination parameter in the model, it becomes possible to account for differences between the items in terms of the strength of the relationship between the item score and ability.3To illustrate the relevance of including a secondary dimension in the SRT model capturing person differences in response speed, we present an example from educational measurement. We use data from Math Garden , the scoring rule of the discretized SRT model follows directly from the scoring rule of the continuous SRT model. This is illustrated graphically in Figure While in principle any dichotomization of Using the discretized SRT model allows one to study relevant differences between fast and slow responses. The objective of Coomans et al.  was to os is a possible realization of the random variable iS. Under this model, for any person j, is ∑isji a sufficient statistic for θ, and for any item i, ∑isji is a sufficient statistic for δi, where jis is the observed score of person j on item i.where ∑isji, one can consider a partial credit model , but rather allows for a more flexible distribution of scores. It may be noted that the parameterization that was presented above differs from the original parameterization of Masters  and and6) ani> 0). Whether the unconditional correlation between a pair of RTs is positive, negative, or absent thus depends on contingent characteristics such as the item difficulties and the ability distribution, and one would not expect to consistently observe a positive manifold if an SRT model generated the data. Thus, the positive manifold that is commonly observed in RT data and that was also found in the empirical example suggests that there may be structural differences between persons in their RTs that cannot be explained by the ability parameter in the versions of the SRT model discussed so far. Therefore, we propose to further extend the model in equation  is an indicator function that takes the value 1 if the response is fast  and 0 if it is slow. The probability of observing values of 0 or 3  increases when θ2 increases, and hence θ2 can be seen as a latent speed variable. A bivariate normal distribution is assumed for the joint distribution of the two latent variables, with the means constrained to 0 and the variances constrained to 1 for identification, and a free correlation parameter ρ.where θs is the N × K matrix of the observed item scores of N respondents on K items; α is a K‐vector of the item slope parameters in the unidimensional model and a K × 2 matrix of the item slopes in the two‐dimensional model; and β is a K × 3 matrix of the category intercept parameters of the items.The likelihood functions for the discretized SRT models without (equation ) and witIt may be noted that the proposed two‐dimensional discretized SRT model falls under the class of multidimensional nominal response models , and cm is a vector (in our case of length 4) containing the set of scores assigned for each possible value of iS in the mth dimension. For the proposed two‐dimensional discretized SRT model one would consider the item scoring functions c1 =  and c2 =  for the ability and speed dimension, respectively. While these types of multidimensional NRMs have been applied for the purpose of capturing response styles in the context of Likert scales  a null condition, where data were generated under the unidimensional discretized SRT model  for the latent variables the five sets of threshold parameters translate into the following probabilities of the item scores:  ,  ,  ,  , and  , respectively.For αi =  for all items, and for all other conditions αi = . This difference between αi1 and αi2 was used because the first dimension considers scores that have a wider range (ranging from 0 to 3) than the scores considered in the second dimension (ranging from 0 to 1). On the person side, values for the two latent variables were sampled from a bivariate normal, using θ∼N2θ;0,1ρρ1.For the null condition, mirt  and the5.2i2 = 0), where it can be observed that for both K = 20 and K = 40 the absolute bias, variance, and MSE of the ability parameters are comparable for the two SRT models. Their EAP reliability is also identical. Thus, when persons do not differ in their speed, using the two‐dimensional discretized SRT model does not worsen the quality of the estimates, and overfitting does not seem to be a problem. Both SRT models outperform the hierarchical model in terms of absolute bias, variance, MSE, and EAP reliability. This can be explained by the fact that the hierarchical model does not model the patterns in the RT data correctly. The bias in the ability estimates under the IRTree model is similar to the SRT models, but its variance is worse, resulting in a higher MSE and worse EAP reliability for the IRTree model than for the two SRT models. It may be noted that for all models the average absolute bias is notably greater than 0, which is due to the estimation procedure's shrinkage of the estimates towards the population mean. Such shrinkage can be considered desirable as it reduces prediction error , the two SRT models show differences in performance. In all these conditions, the SRT model without the extra dimension shows notably more absolute bias in the estimated ability parameter, indicating that ignoring this secondary dimension may confound measurement. The relative difference between the two SRT models in absolute bias is bigger for the longer test (K = 40) than for the shorter test (K = 20). In contrast, the SRT model without the secondary dimension shows slightly lower variance in the estimates than the model that does include this dimension. Compared to the decrease in bias, this increase in variance is relatively small, suggesting that the small loss in precision may be outweighed by the decrease in bias. This is also apparent when the MSEs are considered, which shows the two‐dimensional SRT model to perform notably better than the one‐dimensional SRT model in all conditions where persons differ in speed. The decrease in MSE obtained when including the second dimension in the model is most notable for longer tests (K = 40), where a reduction of 27%–33% was observed. It should be noted that the model‐based EAP reliability is higher for the 1D SRT model than for the 2D SRT model, which can be explained by considering that, due to being misspecified, the 1D SRT model assumes the RTs to contain more information about ability than they actually do. That is, the 1D SRT model incorrectly assumes that all patterns in the RTs have to relate to ability, inflating the estimated precision and hence resulting in a higher model‐based EAP reliability than the 2D SRT model, which does not make that assumption.When persons do differ in their speed . The models were estimated using the R package 6.2The results for each of the five models and for both discretization options are displayed in Table The results in Table Compared to the 2D SRT model, the hierarchical model shows worse fit, with AIC and BIC values being lower for both discretizations. Thus, the data suggest that the 2D SRT model should be preferred over the hierarchical model, and that the hierarchical model may not provide the best description of the structure in the data. While the model‐based EAP reliability is higher for the hierarchical model than for the 2D SRT model, this does not suggest that the hierarchical model should be preferred since the hierarchical model does not appear to describe the data well and hence it may not be safe to rely on the model‐based EAP reliability as an indicator of the quality of measurement under the hierarchical model.When comparing the 2D SRT model to the IRTree model, the differences are less pronounced. For both discretizations and according to both the AIC and the BIC, the 2D SRT model should be preferred. However, the differences appear to be smaller than when contrasting the 2D SRT model with the other models. This is also in line with the simulation results reported in the previous section, where the 2D SRT model only outperformed the IRTree model by a small margin. The model‐based EAP reliability of both models is also rather similar. However, conceptually there are relevant differences between the two models that could motivate a choice between the two models, in the sense that the latent variable that relates only to the RTs has a different meaning in the two models. In the IRTree model, its interpretation matches that of the latent speed variable in the standard hierarchical model, and hence the latent variable simply captures between‐person differences in response speed. In the context of the 2D SRT model, the RTs are also included in the measurement model for ability, and the added latent variable essentially explains between‐person differences in the extent to which respondents deviate from their model‐implied response speed; that is, the latent variable may be more relevantly interpreted as the extent to which a respondent shows risk‐seeking or risk‐averse behaviour, in terms of responding faster or more slowly than is optimal under the SRT scoring rule. Since this application focuses on a setting where respondents are informed that RTs are taken into account in their score, using the 2D SRT model rather than the IRTree model might be considered more appropriate. Choosing the 2D SRT model can also be said to provide more practically relevant information for the test taker and test administrator, in the sense that this added latent variable can provide personalized insight into whether future test performance can be improved by responding more quickly or more slowly, and might also be of substantive interest in its own right . For these reasons we will continue investigating the results obtained for the 2D SRT model, where we will focus mainly on the half‐deadline split discretization.2 was 1.33, and the estimated values for the items ranged from 0.92 to 1.68. Thus, the items discriminate persons rather well on the secondary dimension. Figure 2 against the estimated values of α1 for each item, where it can be observed that there does not appear to be a clear relationship between the two parameters. Thus, it is not the case that items for which a high estimate of α2 was obtained are generally items that discriminate relatively well or relatively poorly on the dimension of interest (θ1), and hence the degree to which an item is sensitive to a person's speed does not appear to have a notable impact on its quality as an indicator of ability.On the item side, the estimates obtained for the 2D SRT model support the conclusion that there are relevant between‐person differences in response speed that can be modelled. The average value of the estimates of αOn the person side, the correlation between the two latent variables was considered. The correlation was estimated at .16, with the 95% confidence interval ranging from .11 to .22, suggesting a weak positive association between speed and ability. Thus, for the application considered, persons who respond more speedily are on average more able than those who respond relatively slowly.The practical impact of using the 2D SRT model instead of the other models for discretized RTs can be gauged by contrasting the ability estimates obtained under the 2D SRT model with those obtained under the other models  with those obtained using the three‐way split . The correlation between the ability estimates in the two‐dimensional SRT models estimated for two‐category RTs and for three‐category RTs was .987. The estimated correlation between the speed and ability latent variables was .17 in the model for the three‐category RTs. Thus, the results obtained for the two discretizations hardly seem to differ, suggesting that the particular choice of discretization of RT need not have a notable impact on the model inferences.7In this paper a two‐dimensional discretized SRT model is considered, which maintains the idea of the traditional SRT model that RT may contain relevant information about ability, while acknowledging that persons also differ in the speed at which they take the test. By incorporating a speed latent variable, the proposed model aims both to remove the confounding effect of speed for the estimation of ability that may occur under the traditional SRT model, and to gain a more complete picture of the respondents and the way in which they take the test. That is, the model provides information not only about ability, but also about the speed at which a person takes the test. This latter latent variable captures associations between RTs that are not explained by ability. As such, one can expect it to be related to psychological attributes such as response caution and risk seeking. However, the exact interpretation of this latent variable will likely depend on the particular application that is considered, and empirical research would be needed to validate any claim about this latent variable capturing a particular response tendency such as response caution.The proposed model makes use of dichotomized RT rather than continuous RT, and as such does not have the standard SRT model as a special case. The dichotomization was considered to enable the modelling of the speed latent variable using the toolbox of multidimensional polytomous IRT. For this paper, a dichotomization at the middle of the theoretical range of RT was used, and an item scoring motivated based on the SRT model for continuous RT was implemented, in line with the scoring rule of Coomans et al. . HoweverWhile different dichotomizations of RT can be considered, it may be emphasized that if one wants to maintain a scoring rule that is based on the SRT model with continuous RT, the choice of dichotomization should be reflected in the scoring rule. An alternative could be to depart from the scoring rule suggested by the standard SRT model, for example if one feels that the difference between a fast and a slow correct response should be seen as less (or more) indicative of an ability difference than the difference between a slow correct and a slow incorrect response. Ultimately, the choice of the scoring rule should be subjected to validation research if the purpose is to obtain unconfounded measurement of a particular ability.In addition to employing different dichotomizations of RT, one can consider using different ways of discretizing RT, for example by binning it in three or more categories , as was also considered in the context of the empirical example. For such a polytomous categorization of RT one would need to carefully consider the scores that are assigned to each combination of RA and RT for both the speed and the ability dimension, but such an approach might allow for finer‐grained analysis of the response processes. Conversely, further study of the response processes might result in a more informed decision on how to discretize RT and how to score the response outcomes in the most meaningful or relevant way. However, care should also be taken to avoid issues with sparsity, which may arise if RTs are binned in more than a few categories.It may be argued that in high‐stakes testing settings, whatever scoring rule is adopted and communicated to the respondents should be maintained, and hence that it is up to the respondents to optimize their test‐taking behaviour in accordance with this scoring rule. In this sense the respondent who shows notable response caution on a test where the SRT scoring rule is adopted is simply using a suboptimal testing strategy for which perhaps no correction should be implemented. As such, it may not necessarily be appealing to employ the two‐dimensional discretized SRT model when dealing with a high‐stakes test. However, for low‐stakes testing settings the main purpose is commonly to obtain as accurate as possible a picture of the true ability of each respondent, and correcting for distorting factors such as differences in response caution can be considered desirable. Thus, the proposed model is likely to be of most use in low‐stakes testing settings, such as the application considered in the empirical example. Additionally, using the two‐dimensional model in low‐stakes practice settings could be very useful for providing test takers with feedback on the extent to which their response speed deviates from the model‐implied optimal response speed, with persons who have an extreme value on this second latent variable likely benefiting from altering their test‐taking behaviour. This kind of feedback could be helpful if the same test taker will later be faced with a high‐stakes test where performance is evaluated based on the SRT scoring rule.8All authors declare no conflict of interest.Jesper Tijmstra, Ph.D.  Maria Bolsinova .Appendix S1. This Appendix contains the R code for the simulation study and the analysis of the empirical example presented in this paper.Click here for additional data file.


The purpose of the paper is to provide practical information on the selection of heat treatment conditions and the prevention of secondary microbial contamination of liquid sugar. The guidelines included in the paper were formulated on the basis of practical experience gained. The most important aspect often refers to the unnoticed cause of secondary contamination of liquid sugar after heat treatment, during its cooling in diaphragm heat exchangers along with the indication of the possibility of elimination of this cause. The article also presents the results of microbiological analysis in the field of heat resistance of microorganisms present in raw juice, obtained from the extractor. The data indicate the difficulty of thermal inactivation of mesophilic and thermophilic spores present in the studied environment. The cited results of microbiological analysis prove the usefulness of the measures to prevent secondary microbial contamination of the finished product. Listeria monocytogenes, Salmonella, Cronobacter spp., Escherichia coli, and coagulase-positive staphylococci. The above regulation does not contain microbiological criteria for sugar in either solid or liquid form. If there are no legal requirements for the microbiological status of the finished food product, the final criteria are based on the company's experience, customer requirements, normative documents or requirements developed by the associations. The microbiological criteria for liquid sugar refer to the total number of mesophilic bacteria, yeast and mold. The sample specification for liquid sugar determines the maximum amounts of microorganisms in colony forming units per 10 g of D.S.E , which are for: mesophiles 200 CFU / 10 g of D.S.E, yeast 10 CFU / 10 g of D.S.E, mold 10 CFU / 10 g of D.S.E (Nordic Sugar Accessed 15 June 2019). The document does not specify requirements for thermophilic bacteria. However, the thermophilic spores present in sugar solutions arouse concern among manufacturers of some canned and non-alcoholic beverages. Thermophilic spores of aerobic bacteria  and anaerobic bacteria  have been associated with the spoilage of low-acid shelf-stable foods. These microorganisms are only problematic for shelf-stable foods that are distributed and stored at high ambient temperatures .Obtaining the desired microbiological status of finished food products is one of the most important goals for enterprises dealing with food processing. Enterprises in the food industry are supposed to obtain a finished product with the best possible microbiological quality. Microorganisms present in a finished food product intended for sale can multiply and change both its organoleptic and physicochemical characteristics as well as have a negative impact on its usefulness for consumption. Microorganisms that cause disease in humans may not be present in food. Food safety management is supported by various documents, including primarily legal acts. The legal documents that provide specific microbiological or chemical requirements for ready-made food products are particularly useful. The microbiological criteria for foods in Commission Regulation (EC) No 2073/2005  during its production and the effectiveness of eliminating the causes of secondary microbiological contamination of the finished product. The conditions (temperature and time of operation) of heat treatment of liquid sugar depend on its microbiological state, i.e. the type and amount of microflora that colonizes it, which is the result of conditions prevailing both at the earlier stages of the technological process and between these stages. In order to achieve the assumed level of microbiological purity of the finished product in practice, it is necessary to get acquainted with different fields of knowledge such as: food technology and microbiology, unit processes, quality management, including food safety, engineering, in particular the construction and operation of equipment used for food production. Without such interdisciplinary knowledge it is difficult to find the right solutions to effectively prevent or minimize microbial contamination to an acceptable level, and yet we know that "prevention is better than cure". The microbiological aspect should be analyzed throughout the production chain for the possible multiplication of microorganisms and the effects of their activities, the effectiveness of inactivation of microorganisms and the elimination of the possibility of secondary contamination of the finished product. In the case of liquid sugar, the huge danger of secondary microbiological contamination occurs during its cooling in the diaphragm heat exchangers. Due to the high concentration of sucrose, liquid sugar is not an environment supporting multiplication of bacteria during its storage, but it is a convenient medium for multiplication of mold and yeast Bergwall . SecondaThe literature provides no information on the elimination of the cause of secondary microbial contamination, which results from improper operation of the apparatus.The main purpose of this work is to obtain practical information on the prevention of secondary microbial contamination of liquid sugar.Current industrial practices reveal a tendency to use a higher temperature for the purpose of pasteurizing or sterilizing a medium achieving a short heat treatment time. In order to illustrate this principle, an illustrative drawing Fig.  is preseThis curve illustrates the relationship between temperature and heat treatment time. As the temperature rises, the required heat treatment time is reduced. At the same time, heat supplied to the medium during its heat treatment affects not only the microorganisms present in it, but also its chemical and physical properties. The practical experience demonstrates that in most cases heat treatment affects the properties of the medium with a specific tendency. This effect is illustrated by the dashed line specifying acceptable conditions for heat treatment of the medium, which will not cause a significant change in its properties in relation to the original state (before the process).In practice, in order to guarantee the effectiveness of sterilization or pasteurization, a longer operating time of a given temperature (i.e. above a continuous line) is used than this required for the thermal inactivation of certain forms of microorganisms present in a given medium. At the same time, in order not to allow for significant chemical changes to occur in the medium due to the effect of elevated temperature, a time shorter than that illustrated by the dashed line should be applied. When analyzing the drawing Fig. , it is cIn summary, figure Fig.  illustraDepending on its intended use, liquid sugar may be pasteurized or sterilized. Liquid sugar which is a raw material for the pharmaceutical industry must be sterilized, i.e. all forms of microorganisms must be eliminated. Pasteurized liquid sugar is used mainly in the soft drinks industry, ice cream and dairy products. Liquid sugar is used for products such as marmalade, jam, pickled vegetables, sauces, ketchup and mustard. It can also be used as substrate in industrial fermentation . Thus, prolonging the pasteurization time did not allow for inactivation of mesophilic and thermophilic spores present in the raw juice. Based on data obtained during the microbiological analysis of raw juice during its long-term storage at 80 °C, the number of the maintained thermophilic spores was at the level of 102 CFU / mL.The obtained results indicate a significant heat resistance of the population of spors of mesophilic bacteria present in the raw juice collected from the extractor, because the long-term action of the temperature of 80 °C did not allow for their elimination. From the data obtained, it result that the population of spors of mesophilic bacteria colonizing the raw juice was resistant to the tested pasteurization temperature. The number of mesophilic bacterial spors remained at 10Geobacillus stearothermophilus spores in the temperature range 115 °C–125 °C in aqueous solutions of glucose, sucrose, starch, casein, peptone, molasses and milk powder. She stated that the tested substances have a protective effect on the survival of spores, extending the time necessary to carry out effective sterilization. Another author  at pH 5.0 and 7.0 compared to distilled water.It should be emphasized that the obtained results may indicate not only a very high resistance of microorganisms present in the raw juice, but also the protective influence of environmental components on them. The author  after 3 min at 115 °C and after 20 s at 125 °C. Significant activation of Geobacillus stearothermophilus spores was also observed in real environments such as skimmed milk and asparagus juice, i.e. after 30 s at 121 °C. In the conditions cited, the number of germinable spores increased by approximately one order of magnitude.While analyzing the obtained results, one cannot omit the possibility of activating spores at this temperature, which can be confirmed by a slight increase in the number of spores of thermophilic bacteria at the end of the heat treatment process Fig.  showed tłaszczyk  found a The causes of secondary microbiological contamination of the finished product may be of different character. Most of them are well-known. The secondary microbiological contamination of the finished food product may be caused by packaging material, leakage of the packaging, surface of devices used in the final stages of production, contact with non-sterile air or errors in operating the installation. The measures to prevent secondary microbial contamination include, among others, the preparation of effective procedures for cleaning and disinfecting equipment surfaces and their handling, selection and handling of packaging materials, and effective training for employees . The relatively minute mass exchange between these heat exchanging media is not of energetic and mass significance , but it is of great importance for microbiological reasons. The minimum amount of non-sterile liquid sugar or cooling water passing into pasteurized liquid sugar causes its microbial secondary contamination. To eliminate this type of danger, the overpressure must be maintained on the side of the pasteurized liquid sugar in relation to the pressure in unpasteurized liquid sugar and cooling water.The described cause of secondary microbial contamination of liquid sugar after the pasteurization process during its cooling in diaphragm heat exchangers can be effectively eliminated by using an additional pump (marked in Fig. In order to effectively eliminate the vegetative forms of mesophiles present in liquid sugar and preserve its properties, it is recommended to pasteurize it at 110 ° C and within 5 s.Based on our own research, it was found that mesophilic and thermophilic spores present in raw juice are resistant to a pasteurization temperature of 80 °C. Two-hour pasteurization of raw juice sourced from the extractor at 80 °C did not allow for inactivation of mesophilic and thermophilic spores present in it.The use of an additional pump and appropriate valves to maintain the overpressure on the side of pasteurized liquid sugar in relation to unpasteurized liquid sugar and cooling water is an effective way to prevent the microbial contamination of pasteurized liquid sugar during its cooling in diaphragm heat exchangers using non-sterile media.


The phenomenon of ageing transitions (AT) in a Erdős–Rényi network of coupled Rulkov neurons is studied with respect to parameters modelling network connectivity, coupling strength and the fractional ratio of inactive neurons in the network. A general mean field coupling is proposed to model the neuronal interactions. A standard order parameter is defined for quantifying the network dynamics. Investigations are undertaken for both the noise free network as well as stochastic networks, where the interneuronal coupling strength is assumed to be superimposed with additive noise. The existence of both smooth and explosive AT are observed in the parameter space for both the noise free and the stochastic networks. The effects of noise on AT are investigated and are found to play a constructive role in mitigating the effects of inactive neurons and reducing the parameter regime in which explosive AT is observed. Here, each neuron is a dynamical entity. A complex dynamical system approach enables investigating the time evolution of a large number of interconnected subsystems, the dynamics of each of which are reasonably well understood in isolation but whose collective dynamical behaviour can be significantly different due to the interaction effects. The complexity arises from the interactions, on account of the coupling between the subsystems. Each subsystem is modelled as a node and the mutual interactions due to coupling are modelled as links/edges forming a complex network. One of the most widely investigated complex networked system is the Kuramoto model of phase oscillators6. The scale of such complex networked systems can range from 10 to 20 in “low” dimensional oscillator networks7 to billions of interacting neurons, for example in the human brain model8. Phenomena like synchronization11, explosive synchronization14 and explosive death16 are the hallmarks of such complex systems and have been studied across domains ranging from networks of neurons17, mechanical oscillators18 to chemical systems19.Investigating the dynamical behaviour of an animal brain, formed as an intricate web of large number of interconnected neurons1—which continue to function, maybe sub-optimally, after trauma. In this context, one of the more interesting recent developments has been to study Ageing Transitions (AT) in networks of dynamical systems. In ageing neuronal networks, some nodes (neurons) lose their functionality and become inactive dynamically. When the fraction of inactive nodes is low, the networked system might be able to compensate the loss through coupling and continue to exhibit “active” dynamical behaviour. However, as the fraction of inactive nodes increase, the system collectively transitions to a state where there is no significant “active” dynamical behaviour in the system. AT was first observed in Stuart-Landau (SL) oscillators20 and later in many systems like globally coupled networks22, locally coupled oscillators23, fractional order SL oscillators24, Hindmarsh-Rose neurons27 and time-delayed interaction systems28 among others. The nature of AT from the active state of the network to the inactive state can be gradual, monotonic or explosive and depends on several key parameters associated with the problem. A sudden or explosive transition is characterised by a sharp change in the system’s dynamical characteristics on a slight change of one of the parameters. These type of transitions are potentially detrimental as this leads to a total loss of capability of the network, without prior warning. There is therefore a need to investigate the parameter regimes at which these explosive transitions occur. An order parameter, which is a measure of the collective dynamical characteristic of a complex network in which the individual components can behave quite differently, is defined to qualitatively and quantitatively investigate the parameter regimes at which these ageing transitions are observed.Investigating the capability of a networked system to operate dynamically even when segments of the system malfunctions is of interest to estimate its robustness to external and/or internal perturbations/malfunctions. Such robustness is observed in many biological systems—such as the brainThis article focuses on investigating the phenomenon of AT in a network of Rulkov neurons, with the fraction of “inactive” neurons as one of the important parameters. As is well known, neuronal dynamics is typically characterized as a discontinuous system comprising of two distinct states—a silent or inactive state and a spiking or active state. The Rulkov neuron is a simple map based model which captures the discontinuous nature of the neuronal dynamics and unlike differential equation based models, enables numerical studies at significantly cheaper computational costs. This enables carrying out parametric studies for networked systems with a large number of Rulkov neurons. A mean field coupling model is proposed for the neuronal interconnections and the conditions for AT are investigated using measures based on standard order parameters. Subsequently, the dynamics of the networked system is investigated when the mean field coupling model is modified to be stochastic. Parametric studies on the stochastic system reveal that noise could play a constructive role in altering the dynamical characteristics of the network by delaying the onset and the nature of AT. In investigating the dynamics of ageing systems for a given set of parameters, the fraction of “inactive” nodes are taken to be constant as the time-scale of ageing is usually several orders of magnitude larger than the time-scale of the system dynamics.30 involves two state variables n” represents the n-th discrete time step. Depending on the choice of the parameters, a solitary neuron exhibits three distinct long-time dynamical states29: dead/silent (DS), tonal spiking (TS) and spiking bursting (SB). In this study, the parameters are selected such that the individual neurons exhibit either the DS or TS states. In the case of networked neurons, the map consists of additional terms 31, with each node (neuron) having a connection probability D, A is defined, which is a normalised measure for the largest amplitude of the oscillations in the steady state condition averaged across all neurons in the network; see Eqs. .The Rulkov mapmentclass2pt{minimsee Eqs. –8) that see Eqs. –6b. Thissee Eqs. –11. A caExhaustive parametric investigations have been carried out numerically. The representative interesting results that bring out the salient features for both the noise-free and the stochastic cases are discussed next.A as a function of A with respect to A is observed to decrease linearly with The noise free system is when the intensity mentclass2pt{minimmentclass2pt{minimA with respect to Next, simulations are carried out for an ER network with entclass1pt{minimaentclass1pt{minimaA at a particular value of These n, the ordinate is the neuron number i and the colour code denotes the values of the state variable A by its definition is zero. It is important to note that as the number of inactive neurons increase in the network, the effects of the active neurons on the network dynamics diminishes and beyond a critical threshold value of This sudden discontinuous transition in the state of the network is defined as explosive AT. To gain insights into the collective dynamics of the network, raster plots which show the time histories of the dynamics of each neuron are investigated; see Fig. entclass1pt{minimaA as a function of A parametric study is carried out to investigate in detail the behaviour of the network as The colour-map is same for all cases, with yellow denoting critical “death” or “inactivation” probability for A as The From the spike, Stochasticity is introduced in the system through additive noise to the mean coupling strength 0 in Eq. . The noiA as a function of Figure A for the stochastic network is greater in comparison to the noise free system for almost the entire regime indicating that the amplitude of the oscillations, on an average, increase with noise intensity D. Explosive AT is observed for higher values of the mean coupling strength for both the noise free as well as the stochastic case as can be seen in Fig. A, as The corresponding variation for the noise-free network  zero in the regime post explosive AT. This figure is similar to Fig. A does not go to zero. An inspection of Fig. D increases, the minimum values for D, exhibit smooth AT as D increases. This implies that multiplicative noise can not only increase the region in the A parametric study is next carried out to investigate the behaviour of the stochastic network as These observations are true for all network connectivities ith neuron, where An additional case that has been studied is to investigate the effects of inhomogeneity associated with each neuron. This was carried out by considering the parameter The phenomenon of explosive AT in an ER network of coupled Rulkov neurons has been studied, with network connectivity, coupling strength and the fractional ratio of inactive neurons being the parameters of interest. This has been investigated for both the noise free network as well as the stochastic network. In the latter case, the mean inter-neuronal coupling strength have been superimposed with Gaussian white noise at each iterate. An order parameter has been defined for quantifying the dynamical state of the network. It has been shown that explosive AT occurs only when at least 30, involving two state variables The Rulkov mapn” represents the n-th discrete time step. 29. For a neuron in isolation, these parameters can be treated as constants and can be simplified by putting 29: dead/silent (DS), tonal spiking (TS) and spiking bursting (SB). In DS state, the isolated neurons show no activity; see Fig. Here, In the SB state, the neuron shows bursts of spikes, which may be chaotic depending on the parameter values. The parameter regime considered in this study is limited to the DS and TS states and can be seen in Fig. i, j denote the neuron number, N denotes the total number of neurons in the network, the parameters ith node indicating the number of connections of a particular neuron, i-th and the j-the neuron at time instant n and 29 for i-th neuron and the corresponding membrane potentials For a network of coupled Rulkov neurons, Eqs. –2 are wrn and D is the noise intensity. For the noise-free case, The most general form of the time-dependent coupling strength D, D. Here, ith neuron over all time steps n. The denominator in Eq.  random graph with