Sound source localization is critical to animal survival and for identification of auditory objects. We investigated the acuity with which humans localize low frequency, pure tone sounds using timing differences between the ears. These small differences in time, known as interaural time differences or ITDs, are identified in a manner that allows localization acuity of around 1° at the midline. Acuity, a relative measure of localization ability, displays a non-linear variation as sound sources are positioned more laterally. All species studied localize sounds best at the midline and progressively worse as the sound is located out towards the side. To understand why sound localization displays this variation with azimuthal angle, we took a first-principles, systemic, analytical approach to model localization acuity. We calculated how ITDs vary with sound frequency, head size and sound source location for humans. This allowed us to model ITD variation for previously published experimental acuity data and determine the distribution of just-noticeable differences in ITD. Our results suggest that the best-fit model is one whereby just-noticeable differences in ITDs are identified with uniform or close to uniform sensitivity across the physiological range. We discuss how our results have several implications for neural ITD processing in different species as well as development of the auditory system. The ability to localize sound sources accurately is critical to the survival of many species and also contributes to the human ability to follow conversations in noisy environments, the so-called “cocktail party effect” Several factors affect the magnitude of ITDs including the distance separating the two ears , sound frequency and azimuthal position of the sound source. For all head sizes and frequencies, a sound produced at the midline reaches each ear at the same time, assuming symmetrically placed ears across the midline. ITD increases as the sound source is positioned at greater azimuth angles. For humans, a sound located at one side of the head (90° azimuth) generates a maximum ITD of around 750 µs for low frequency sounds There is some inconsistency in the literature concerning the variation of jnd ITD with angle. It is sometimes attested that poor acuity out at the sides (90°) is due to poor lateral ITD sensitivity arising from fewer neurons detecting long ITDs Acuity is dependent on both ITD sensitivity and the rate of change of angle with ITD. The rate of change of angle with ITD increases as azimuth angle increases, meaning that the same change in ITD corresponds to a larger change in angle at more lateral azimuths. Acuity displays a similar overall variation, with larger MAA values at greater azimuths. We explore the extent to which acuity variation is influenced by the dual factors of ITD sensitivity and the angular variation of ITD. Our analysis makes no assumptions about the neural procedure for identifying ITD and hence the results concerning sensitivity of ITD identification are characteristic of the entire ITD processing system rather than a single part of the auditory pathway.This study uses previously published data on azimuthal variation of auditory acuity between 0° and 90° by Mills fe−2πift) at distance r. We modelled the head as a sphere of radius a, with ears set back at 100° from the midline −1 based on an ideal gas at 15°C). The head-related transfer function, H, relates the pressure that would be present in the free field, Pfreefield, to the pressure developed at the surface of the sphere, Psurface.The acuity data from Mills mcos(A) are Legendre polynomials of order m and argument cos(A). Angle A is the angle in radians between a ray from the centre of the sphere to the sound source and a ray from the centre of the sphere to the measurement point on the surface. When considering the pressure at the two ears for a sound source at azimuth angle θ relative to the midline, A = 100°−θ for the leading ear and A = 100°+θ for the lagging ear. hm(kr) are spherical Hankel functions (first kind), order m and argument kr. h'm(ka) are the derivatives of the spherical Hankel functions, argument ka. The head-related transfer function, H, is a function of k, a, r and θ and is a complex wave with phase Φ.PThe phase for a particular k, a, r and θ can be found using the argument of H:+) and at the lagging ear (H−) we were able to determine the IPD for a given incident sound angle:By evaluating H at the leading ear  for a given frequency of sound from the relation:max value of 5 and mmax of 6 is less than 1 µs over the whole 0°<θ<180° range for a sound of 1000 Hz at a distance of 0.5 m and is less than this for lower frequencies or greater sound source distances. The interaural level difference (ILD) in decibels was calculated from the difference in magnitude of the head related transfer function at each ear:Our evaluation of the head-related transfer function used a maximum m value of 6. The difference in predicted ITDs using an mFor calculations of acuity, the rate of change of azimuth angle was determined using the following identity for total derivatives, which is valid when derivatives are continuous and non-zero and therefore can not be used at exactly 90° or 270°.2 and Corrected Akaike Information Criterion (AICc) values were calculated for best-fit solutions. AICc values are a measure of the relative quality of a statistical model for a given set of data and are founded on estimates of the information lost when a given model is used to represent the process that generates the data 2 values can not be used as a goodness of fit measure.All calculations for model predictions and best-fit comparisons to human data were performed using Wolfram Mathematica 7.0.0, Wolfram Research Inc, Champaign, IL, USA. Best-fit curves were determined using the NonLinearModelFit function for a least-squares fit of data points to constant, linear or non-linear functions. Best-fit curves were calculated using regression analysis, determining best-fit parameters to a confidence level of 95%. Variation of angle θ with ITD in the time domain was determined using numerical interpolation of data points of the angle at which a given ITD occurs. Adjusted Rth percentile values for head circumference.Comparative predictions of acuity for human newborns and adults were calculated for a distant 500 Hz sound source, ΔITD = 15 µs and head radius for newborns calculated from World Health Organization child growth standards Auditory acuity depends on both angular ITD variation and ITD sensitivity (jnd ITDs). We used the data on the variation of human acuity with azimuth angle from two studies, Mills fe−2πift where Sf is a frequency-dependent amplitude , is dependent on two factors; jnds in ITD (ΔITD) and the rate of change of angle with ITD. The latter can be calculated using our acoustic model. Our first approach used acuity data and the rate of change of angle with ITD to consider the potential variation of ΔITDs. Using the relationship shown in 2 values. The proportionality constants (kp) for best-fit linear distributions are low, ranging between 0.0045 and 0.011 for the Mills data. These kp values relate to differences in ΔITD of only 3.4 µs and 8.3 µs respectively, for sounds at 0° and 90° azimuth. We also observed that the magnitude of our predicted ΔITDs are of a similar magnitude to the just-noticeable differences in ITD found by Dominitz and Colburn The best-fits for uniform or linear model distributions are shown in max). Again, we found that the best-fit linear jnd ITD distributions have low proportionality constants and are essentially close to the uniform case. AICc values were used to evaluate how well the non-linear acuity models account for the data. The value of AICc is used to compare different models for the same data set, a lower AICc value indicates a better explanation of the data by the model. The AICc calculations take into account the the number of parameters in a model, thus including a measure of model complexity in order to prevent overfitting of data. We used the corrected version of AIC owing to the low number of data points per data set. We found that all of the data sets except one . These analyses also demonstrate that the dominant component in human pure tone localization acuity is due to the variation of ITD with angle and that the sensitivity of ITD identification by the processing system is uniform or close to uniform across the ITD range for low frequency sounds.We have taken an analytical approach to determine the most appropriate description of just-noticeable differences in ITD identification for pure tone sounds. Our analyses are based on using angular acuity data where ITD can be considered as the dominant interaural cue for the sound localization task. Our results are most consistent with uniform or near uniform ITD jnds to describe previously observed psychophysics of human pure tone source localization. Additionally, we determined that there is a limiting form of acuity variation due to the maximum sensitivity available to any binary ITD processing system. This limiting, lower bound on acuity has the same form as for uniform jnd in ITD, with good acuity at the midline which is fairly constant out to 60° but then becomes increasingly poor for more lateral pure tone sound sources.Our prediction of uniform or near uniform ITD jnds is a property of the whole ITD processing system and is consistent with the ITD jnds found by Dominitz and Colburn Our analysis required us to ascertain which acuity data sets were suitable for determining ITD jnds and led to us rejecting frequencies higher than 500 Hz owing to the likelihood of ILD as a significant localization cue and the additional potential complications of the phase ambiguity limit in source localization. An additional consideration we made in relation to the acuity data was whether the data could be used as a measure of sensitivity of angle identification. This requirement highlights the differing effects of accuracy and precision errors. Accuracy errors affect the ability of a listener to determine the absolute location of a sound source and can be manifested as bias in the identified angle of a sound source. In humans How do constant just-noticeable differences in ITD relate to processing of the ITD signal? Our conclusion that ITDs are processed with uniform jnds results from an analytical model that encompasses the entire auditory system, irrespective of how ITDs are actually coded within the nervous system. Just-noticeable differences in ITD identification are often used as a behavioural outcome to test models of binaural neural processing, which is one of the reasons for undertaking this study. Early models for binaural neural comparisons required the processing system to have fewer nerve fibres encoding for long ITDs Investigations into the most appropriate model for binaural neural processing have also demonstrated a unifying principle across species, that “the ITDs an animal encounters should be coded with maximal accuracy” Within the brainstem of birds, constant ITD jnds are suggested by physiological recordings of uniform, equally spaced ITD tuning curves in the nucleus Laminaris Sound localization acuity would be expected to show variability with head size through head size dependence of ITD. Consideration of head size raises some interesting questions concerning the plasticity of auditory acuity during development. As an individual matures and its head size increases, the maximum ITD increases. The angular variation of ITD has been experimentally determined for several different animals through their development, including cat The effect of head size on sound localization during human development can be seen by comparing maximum ITD and acuity between newborns and adults. Maximum ITD is 483 µs for a newborn (5.44 cm head radius) compared to 755 µs for an adult see Methods We have demonstrated that the most appropriate model for human sensitivity of ITD identification (just-noticeable differences in ITD) is one that is uniform or near-uniform for all physiological ITDs at a single low frequency. The non-linear angular variation of ITD creates a lower bound on acuity for a given head size and greatest available ITD sensitivity. Our results show that acuity towards 90° is always predicted to be worse than at the midline, whatever the neural basis for ITD processing.


Easy access favours the informative role that pharmacists play in Poland with regard to the proper use of medicinal products as well as preventing illness and promoting health. The aim of the present study was to define situations in which patients ask a pharmacist for advice and to identify the most important factors that affect the patients’ decisions in seeking advice from a pharmacist.n all, 101 patients  aged 19–67 years participated in the study. The study was conducted using a structured interview research method. Patients were asked to answer a set of closed-ended questions related to their habits regarding the purchase of medicines and the factors that affected their decision to seek the advice of a pharmacist.Factors determining the choice to contact a pharmacy.Patients seldom asked pharmacists for advice: 77 of the patients “rarely” or “never” went to a pharmacy to consult the pharmacist. When patients did ask the pharmacist for advice, it was mainly concerning the choice of over-the-counter medicines. The most important reason for patients visiting a pharmacy for advice was the large number of pharmacies in Poland and their ease of access; the long queues of people in busy pharmacies and the lack of confidentiality in the pharmacy were considered negative factors. The current advisory role of pharmacists in Poland seems of minimal importance to the public. To make full use of pharmacists, Polish customers should be aware of the advisory role of pharmacists. Therefore, pharmacy associations should undertake initiatives to increase patients’ awareness of the advisory role of pharmacists.The programme of pharmaceutical studies or education in Poland must place greater emphasis on the development of pharmacist counselling skills.In Poland, a patient’s privacy and confidentiality need to be better protected, i.e., through the use of separate, private counselling areas.On the first appearance of the symptoms of a disease, more than half of Poles attempt to carry out the treatment themselves, using home remedies or over-the-counter (OTC) medicines , 2. MajoIntensive advertising campaigns for OTC medicines as well as media reports  on health and self-medication with regard to minor ailments may mislead consumers or create the false impression that taking a particular type of medicine is necessary . As a coThe aim of this study was to define situations in which consumers ask pharmacists for advice and to indicate the key factors that affect the consumer’s decision about whether to seek such advice in a pharmacy.Three community pharmacies located in different socioeconomic areas of Warsaw were selected for the study. Study pharmacies constituted a convenient sample. In all, 101 patients  aged 19–67 years  took part in the study. In Table The questionnaire comprised two parts: the first part included questions on the patient’s habits with regard to purchasing medicines; the second part related to factors that may have influenced the patient’s decision to seek the advice of a pharmacist. The patients’ attitudes to various aspects of consultations with a pharmacist were determined based on a five-point Likert scale. The patients were asked to describe how often and why they visited a pharmacy, what influenced the selection of a particular pharmacy, the factors that either encouraged or discouraged the patients in seeking a pharmacist’s advice, and the situations under which the patients asked a pharmacist for advice. The questions used in the questionnaire were based on a literature review and the results of focus group discussions. The questionnaire was pretested for repeatability and validity using a convenience sample of nine people. Answers obtained in the pretest were not included to the study.p < 0.05. The statistical analysis was carried out using STATISTICA 9.1 software.Statistical analysis of the survey was carried out using nonparametric statistical tests. The choice of this approach was determined by the type of data obtained  and the lack of need to make assumptions about the distribution of the population. Interrelated variables were preliminarily identified by means of nonparametric correlation tests . After dividing the population according to grouping variables , hypotheses concerning statistically important differences were verified by means of the Kruskal–Wallis test (a nonparametric test for more than two groups). The level of significance was set at Ethical approval was not required for this study.Of the surveyed participants, 40 regularly visited a pharmacy several times a month. The remaining participants said that they went to a pharmacy once in a while, i.e., a few times a year. A quarter of the surveyed patients (26) declared that they suffered from chronic diseases.Half of the surveyed participants 52) said that their choice of pharmacy was a deliberate one. The remainder declared that they had no preference for a particular pharmacy. When choosing a pharmacy, the patients generally took into account the convenience of the location, e.g., in a shopping centre, close to their home or to a main road . The second-most important factor determining the choice of pharmacy was the price of the medicine . Table  said thaThe surveyed patients most frequently visited a pharmacy to get a prescription filled  or to buy a specific OTC medicine (60). The surveyed participants “rarely” or “never” went to a pharmacy to consult a pharmacist (77 of those surveyed gave such answers). When the patients did go to a pharmacy to consult a pharmacist, they usually asked for assistance in choosing an appropriate OTC medicine . The surveyed participants over 50 years of age asked for the pharmacist’s advice when they needed a non-prescription medicine more frequently than patients aged under 34 years .Patients with a basic vocational education stated (answering “I agree” or “I definitely agree”) that they asked the pharmacist for advice when they had a minor complaint more frequently than patients with other levels of education  and they have easy access to the pharmacist. The possibility of obtaining advice on the spot without having to make an appointment was cited by 31 of those surveyed.The most important factors that discouraged patients from consulting a pharmacist were queues in the pharmacy (42 of the participants answered “I agree” or “I definitely agree” to this point). In addition, no sense of confidentiality between the patient and pharmacist owing to the presence of other people who might overhear the conversation was cited by 32 of those surveyed. Table Polish patients prefer buying medicines in pharmacies , 8. TheyIn view of the fact that pharmacies in Poland are relatively widespread and easily accessible, Polish patients often rely on the pharmacist’s recommendation when choosing OTC medicines  and dietOur results suggest that patients in Warsaw seek the advice of pharmacists, mainly in choosing an appropriate non-prescription medicine. However, the consultative role of the pharmacist would appear to be of minor importance among the surveyed patients. This conclusion seems to be consistent with the result of another survey, which showed that only one in five respondents take expert advice into consideration when choosing where to buy products. In the same survey, only 7 % of respondents indicated that pharmacists readily provide information on medicines and treatment . CustomeIn the course of the survey, patients explained that they felt ill at ease when there was a queue in the pharmacy, and so they then opted not to consult the pharmacist. The lack of confidentiality during a consultation with the pharmacist was found to be an important factor that discourages patients from asking the pharmacist for advice. Earlier studies have also demonstrated that lack of privacy appears to be the most important factor that discourages patients from asking questions about medicines , and it The major shortcoming of the present study was that the study sample may not be representative of the population of Warsaw. The study was conducted in pharmacies, not among a representative population of Warsaw inhabitants; this may explain the relatively high prescription use among the participants. Furthermore, a high proportion of the participants had tertiary-level education. Warsaw has a significantly higher percentage of inhabitants with tertiary education and a significantly lower percentage of inhabitants with a basic vocational education than the rest of the country. According to 2002 data from the Central Statistical Office, around 30 % of people aged 19–65 years had tertiary education (including 38 % of people aged 25–34) . UnfortuIn view of the specific characteristics of the surveyed group , it is necessary to carry out this survey in rural areas to verify the results obtained.This study demonstrates that it is necessary for the public to become more aware of the advisory role of pharmacists. However, to fulfil patients’ needs, pharmacists need to be able to provide patients with comprehensive information about medicines and health-related issues. In addition, the pharmacy must be properly prepared and adequately equipped to assure that patient confidentiality is respected.


The Exact Call Chemistry for the SOLiD Next-Generation Sequencing platform augments the two-base-encoding chemistry with an additional round of ligation, using an alternative set of probes, that allows some mistakes made when reading the first set of probes to be corrected. Additionally, the Exact Call Chemistry allows reads produced by the platform to be decoded directly into nucleotide sequence rather than its two-base ‘color’ encoding.We apply the theory of linear codes to analyse the new chemistry, showing the types of sequencing mistakes it can correct and identifying those where the presence of an error can only be detected. For isolated mistakes that cannot be unambiguously corrected, we show that the type of substitution can be determined, and its location can be narrowed down to two or three positions, leading to a significant reduction in the the number of plausible alternative reads.The Exact Call Chemistry increases the accuracy of the SOLiD platform, enabling many potential miscalls to be prevented. However, single miscalls in the color sequence can produce complex but localised patterns of error in the decoded nucleotide sequence. Analysis of similar codes shows that some exist that, if implemented in alternative chemistries, should have superior performance. The collection of technologies described as Second- or Next- Generation Sequencing (NGS) platforms are characterised by the synthesis of complementary strands of DNA from clusters of homologous templates . The cheThe SOLiD sequencing chemistry consists of multiple rounds where probes, consisting of eight bases and a fluorophore, are sequentially ligated to the template sequence to build up a complementary strand. Each round consists of a priming step followed by a repeated cycle of ligating probes to the template, exciting the fluorophores and imaging the resulting emission, then cleaving the flurophore and part of the probe ready for the next cycle. The probe/fluorophore combinations are designed so that the probes interrogate the first two of the eight ligated positions in the template, with each of four fluorophore colors used to indicate four of the 16 possible nucleotide pairs at these positions. The color of the fluorophore for each template is recorded and used later to determine the sequence of the read. After imaging, those templates to which a probe failed to ligate have their previous probe decapped (i.e. dephosphorylated) so they will not be extended on future cycles. This reduces problems analogous to ‘phasing’ on the Illumina platform . The fluAfter five rounds, probes have been ligated so that every position of the template sequence has been the first position of a probe. The colors recorded and prior knowledge of one of the bases, typically the last adapter base adjacent to the template, are sufficient to determine the nucleotide sequence assuming that no errors have been made. Conventionally, one of the rounds is primed so that it starts one base before the beginning of the template sequence and so the first base interrogated is the last base of a known adapter sequence.The SOLiD Exact Call Chemistry (ECC) augments the ‘two-base-encoding’ chemistry with an extra round where a different set of probes is ligated. Each cycle of this round interrogates positions 1, 2 and 4 of each five-nucleotide block of the template. The same four fluorophores are used, each now indicating the presence of one of 16 of the possible 64 combinations of nucleotides at the positions interrogated. These color calls can be used to detect and recover from miscalls made in previous rounds.Directly decoding the colors from the first five rounds into nucleotide sequence allows catastrophic failures to occur where a single calling error creates errors in every position from then on . As wellThe mathematical basis for the error correcting properties of the SOLiD ECC is the theory of convolutional codes , a classWe start by defining the terminology we need to analyze the SOLiD code. The chemistry of the sequencing reactions and imaging mean that each fragment of DNA, or ‘source sequence’ (nucleotides), is encoded into a ‘code sequence’ (colors) of equal or greater length. The original five rounds produce as many observed colors as there are nucleotides in the fragment; the ECC round adds one-fifth as many additional color calls. The code sequence is then observed with potential errors (‘observed sequence’). Since the code sequence is longer than the source sequence, not all possible code sequences correspond to an encoded source sequence; those that do correspond are termed ‘valid’. Given an observed sequence, some procedure is used to find the closest valid code sequence (‘corrected sequence’) and the nucleotide sequence that produces this code sequence is the decoded sequence. Note that the source sequence and decoded sequence are sequences of nucleotides (‘base-space’), whereas the code, observed and corrected sequences are colors (‘color-space’).4) is defined G so c=w1c1 + w2c2 is a valid code sequence corresponding to the source sequence w1s1 + w2s2.The ECC code, and any linear code in general, is defined in terms of a F4 Table . Codes a4 is added to a call. For example, the error type ‘ + β’ transforms the color 1 into 2(≡α) since 1⊕β=α, and transforms 0(≡0) into 3(≡β) since 0⊕β=β and +0 leaves the nucleotide unchanged. The group structure ensures that all possible substitutions can be encoded in this form and, when errors are rare, the sequence of error types will consist mostly of 0s. Adding the sequence of error types to the observed sequence results in the recovery of the (true) code sequence; likewise, adding the error types to the code sequence results in the observed sequence.Rather than expressing errors in terms of a specific color miscall, we consider types of errors defined by what happens when an element of GFH that can be used to determine whether a code sequence is valid. The parity check matrix uses a subset of the elements of observed sequence to call a putative decoded sequence and then calculates the expected value of the remaining ‘parity’ elements by treating the decoded sequence as true. The expected and observed parity sequences are then summed element-wise to create the parity check sequence. H is constructed so that the parity check sequence is the sequence-matrix product xH, where x is the observed sequence. By construction, the parity check of valid code sequence is a sequence of 0s; invalid code sequences have a non-zero parity check. When errors have occurred, the observed sequence can be expressed in terms of the code sequence for the source sequence and a sequence of error types e: x=c + e. Since the parity check of a valid code sequence is a sequence of zeros, the parity check of the observed sequence only depends on the sequence of errors and not the source sequence as xH=(c + e)H=eH. The value of eH=xH is termed the ‘syndrome’ of the error, a sequence in GF4 of length n−kthat partitions the set of possible errors into equivalence classes.Each generator matrix has an associated parity check matrix σ) to the simplest possible sequence of error types that has it (eσ); this can be done by enumerating all single errors, followed by doublets, triplets, etc. until every syndrome has been observed at least once and all error types up to a given complexity have been considered. The syndrome table is a function of the code structure only and can be calculated once and distributed with the probe sets. To decode each observed sequence x, its syndrome xH is calculated and then located in the table to find the simplest sequence of error types exH that could cause it. The sequence x + exHis then a valid code sequence which can then be decoded. If there are several equally simple sequences of error types, then an error has been detected but the correction is ambiguous.Syndromes provide a simple and efficient method of decoding an observed sequence under the assumption that errors are rare and so the most plausible corrected sequence is the valid coding sequence which requires the fewest changes from the observed sequence. An error consisting of multiple changes may belong to the same equivalence class as one with fewer changes but the simpler error is the more likely and so is the best predictor of what error actually occurred. Before decoding begins, a table is constructed mapping every syndrome  can be expressed as the dot product a·ρi where a is a row vector of the bases in the window and the ρi describe the calculations to be done, with ρ1=11000 and ρ2=1β0β0. The ρiare the ‘probe generators’ for the chemistry since they define the color of fluorophore that each probe has attached to it: the probe that interrogates the sequence b has color b·ρi.The calculation for both the first and second streams (The architecture of the ECC code imposes the block structure shown in figure α0β) has the color encoding 13233(≡1βαββ) since (01α0β)GBlock=(1βαββ). Conversely, which can be used to freely convert between a sequence of five nucleotides and its encoding. For example, the nucleotide sequence ACGAT  and n the total number of color calls produced in both the two-base-encoding and ECC chemistry rounds. where HECC, corresponding to generator GECC, is The parity-check matrix, Im is the m×midentity matrix. By considering the action of this parity-check matrix on an encoded sequence, the calculation of the syndrome can be given concrete form: the first matrix of the factorised form of HECC takes the observed sequence and inverts it block by block to get a putative nucleotide decoding, with the values of the parity colors preserved. The second matrix calculates the parity colors for the putative decoding and adds them element-wise to the parity colors actually observed. The resulting sequence of length n−k is the syndrome. If the syndrome is composed entirely of 0s, i.e. the parity-check has been passed, then the code sequence and its putative nucleotide decoding are valid. If the syndrome has non-zero entries then an error has been detected. where e1 is c + e1 and the correction e2≠e1 is applied, then converting back to a nucleotide sequence using the inverse of the block generator results in When an error is known to have occurred but is ambiguous, it may be mistakenly ‘corrected’ by applying the wrong error type. The equivalence class contains the possible simple error types that could have caused the observed syndrome but only one of them is the one that actually occurred; applying any to the observed sequence will result in a valid code sequence. We can determine the pattern of nucleotide errors that mistaken correction will result in: if the observed sequence with (unknown) error The theory described examines the worst case where an error can occur anywhere and there is no additional information about which site it is likely to have affected. Real-life performance of the code depends on additional factors. Firstly, the distribution of where errors occur is not uniform, depending for example on the chemical and physical characteristics of the sequencing process, and may not even be independent between positions; the performance of a code may change depending on the error profile. Secondly, but related, the sequencing platform provides quality information in the form of Phred scores  that canA convenient way to deal with these extra complications is to simulate encoded sequence under a realistic model of how errors occur and then decode using dynamic programming, on a ‘trellis’ graph that defines all possible decodings and their relative probabilities, to find the most probable call for each position of the decoded sequence . In pracIn the absence of empirical data for how the distribution of calls and miscalls changes over probes and rounds, and how errors are correlated between positions, we have assumed that errors occur independently for every element of the code sequence and errors are picked uniformly from the three possibilities. The probability of an error for a particular round and cycle of ligation is taken to be equivalent to the quality score from a read sampled at random from a real set of data, implicitly assuming that the error characteristics for alternative probe sets (which depend on the ligation efficiencies of the different pentamers) will be the same as for the ECC probe set. The simulation scheme is as follows: 1. Sample a fragment from genome (base-space).2. Mutate bases of fragment independently with equal probability .3. Encode mutated fragment (convert to code-space).4. Sample qualities from a set of real data.5. Mutate the encoded sequence with probabilities defined by quality values.This scheme outlined has an error model similar to that implicitly assumed when analysing real data since information about both alternative calls and correlation between calls has already been lost during processing into a color-space sequence with a single quality value for each position. The scheme does not simulate the occurrence of insertions and deletions but these are relatively rare compared to substitutions and are most likely to be due to errors introduced during sample preparation rather than calling errors.If a read is error-free then it can be decoded unambiguously and the calls from the two-base-encoding chemistry and ECC codes will agree. When an error occurs in any of the two-base-encoding chemistry rounds, it translates into multiple base miscalls. The structure of the ECC code allows such errors to be detected, recovered from, and, in certain circumstances, corrected. The syndrome equivalence classes are defined by the types of error that can occur and so, by examining them, we can classify the cases where errors can be corrected unambiguously and those where additional information is needed to resolve the ambiguity.Since the elements of the syndrome consist solely of the summation of the observed and expected parity colors, there is a one-to-one correspondence between them and parity colors. An error in a block can only affect two elements of the syndrome, those corresponding to the two parity colors that overlap the block’s first and last elements. We refer to these as the upstream and downstream syndromes, respectively, so it is sufficient to concentrate on how errors occur in only one block. All statements we make about the error correcting properties of the encoding are on a per-block basis, so a code that can correct one error per block can correct multiple errors if spread between blocks. The final block only has a upstream syndrome and so has more limited error correction.βα, identical to that produced by an error of +1 in the third position. If a syndrome is unique  then, assuming only a single error has occurred in that block or parity color, that error can be determined and so corrected. Note that while errors in parity color p−, overlapping a block and the previous one, appear to have unique syndromes, this parity color is also p +  for the previous block and so the syndrome can actually be caused by multiple different errors.The value of the syndromes for all possible single-color errors that could occur in a block or its parity colors are shown in Table β, 0α and 01 can be generated by single errors at c1, c4 or p + , and so errors at these positions cannot be distinguished. The equivalence classes define the possible simple changes to the observed sequence that will produce a valid sequence that can be decoded into a string of bases, only one of which produces the correct sequence. When an error is wrongly corrected, the code sequence is changed in two places, the original error and the correction; the structure of the code ensures that the changes should be complementary .The syndrome equivalence classes show the types of single-color error that cannot be disambiguated without additional information; a error has been detected but is not correctable. For example, the syndromes 0d, then the pattern of differences p induced in base-space is given by αat the second position might be erroneously corrected by an error of type + α at the third position, so the differences are 0αα00 and the pattern induced in base-space is 00α00 (the sum of the second and third rows of α). If this occurred to the nucleotide sequence ATGCG then the sequence ATACG would result.Using the inverse of the block generator matrix, the pattern of changes that are induced in base-space by miscorrections to the observed sequence can be derived. If errors occur and are miscorrected, such that the difference between the correct and observed values of the data colors is α at the first position may be wrongly corrected at either the fourth position or the parity color, leading to the triple error 0ααα0 or the quadruple error 0αααα, respectively. Note that no pattern ever affects the first position of the block; the structure of the matrix All patterns induced by single-color errors are listed in Table 4, the pattern should be evident if it was due to simple miscall. If the pattern is not evident, the differences are either due to sequence variants or multiple miscalls.One notable feature of these more complex errors is that the error type is the same at all affected positions, a property that might help distinguish sequencing errors from genuine variants if the reads are later mapped to a reference. By adding the mapped read to the reference in GFWhile the consideration of syndromes provides useful information about a code’s properties and syndrome decoding is computationally efficient, it is not a replacement for probabilistic methods of decoding since the latter incorporates the quality information about each call and can use it to make better decisions about correction. However, syndrome decoding techniques may still be of use in conjunction with the more computationally expensive probabilistic methods since the syndrome provides a quick test of whether the observed sequence is valid (no correction needed). Syndrome equivalence classes could also be used to restrict the paths through a trellis to a plausible set, providing a heuristic to speed up the dynamic programming algorithm to find the most probable decoding.We have shown the type of errors that the ECC code can correct but have not yet addressed whether it is optimal. Before examining specific alternative codes, it is interesting to look at what can possibly be achieved and there are several mathematical results that restrict the performance of any code. Firstly the Hamming , JohnsonFor reads of 50 bases encoded into 60 letters (as with the ECC code), the lower bound guarantees that a code exists that can detect any two errors and correct any single error. The upper bounds show that no code can guarantee to correct more than three errors. For reads of length 75 bases encoded into 90 letters, then a code exists that can detect and correct any two errors but no code can guarantee to detect and correct more than four errors. There is no guarantee that a convolutional code can meet this bound and the two most common classes of codes that come close to attaining these bounds, Turbo codes  and Low-αβ, β1 and 1α do not occur. The missing syndromes are not truly unused, being generated by multiple errors, but the failure to use them to distinguish single errors suggests that there may exist alternative codes with a greater ability to correct single calling errors. The advantage would derive from using the extra syndromes to partition single-color errors more evenly into equivalence classes. Such alternative codes can be analysed using the same techniques as the ECC code.Examining the syndromes in Table Rather than consider all possible convolutional codes, we will focus our attention on a subset that satisfy some reasonable restrictions inspired by the reality of the SOLiD platform. It is desirable that any new chemistry would be backwards compatible with the two-base-encoding chemistry, meaning that the first five rounds of sequencing must use an unaltered two-base-encoding probe set. This backwards compatibility restriction is equivalent to requiring that a new code must have an unpunctured color stream.While the number of rounds and probe sets could be varied, and probes of differing lengths could be used, to remain comparable to the current ECC chemistry we will focus only on chemistries where a single additional round  will be used and the probes remain based on pentamers. Analogous to the code structure shown in figure α or β can be written as αW or βW, where W  is the probe generator of a code starting with 1, and the linearity of convolutional codes ensures that the two codes have identical sets of code words: although the mapping between nucleotide sequence and encoded sequence will differ, the error correcting characteristics will be the same.By redefining the boundaries of the block structure, the number of different alternative codes that need to be considered can be further reduced: a code whose probe generator starts with one or more 0s is identical to one starting at the first non-zero element with the tail padded with zeros. The probe generator of any code that starts with The block generator for alternative codes that satisfies all the restrictions, and its inverse, can be written as p1p2p3p4p5 is the generator for the second set of probes, L is the matrix whose upper triangle consists of 0 and whose diagonal and lower triangular elements are all 1, x=Lp and y=x5. The ECC code has the probe generator p1p2p3p4p5=1β0β0. The inverse generator only exists when y is invertible (i.e. y≠0); when y is not invertible, blocks of colors cannot be individually inverted and the syndrome analysis is not applicable. Due to the structure of L, requiring y to be invertible is the same as the sum of the elements of the probe generator not being zero.where p5=0.One further restriction will be placed on the probe generator of alternative codes: the ECC probe generator contains 0 at its fifth position and this is probably a consequence how accurately pentamers with differing final bases can be ligated to the sequence. Codes whose generators use the final position might theoretically have better error correction properties but the increased rate of calling errors, due to incorrect probes being ligated, may cancel any improvement their use may offer; consequently, we will require p1p2p3p4p5=10β00 and has similar equivalence classes to the ECC code but only uses the first three positions of the generator. While the set of syndromes for single-color errors is also incomplete  and thus the error correcting properties will be similar to the ECC code, the shorter length of the generator means that calculations on the trellis, to determine the most probable decoding, can be carried out four times quicker.There are 48 probe generators satisfying all the restrictions and the invertibility condition, of which the syndromes and equivalence class for two interesting alternatives are shown in Table β010, uses all possible syndromes and potentially has better error correcting properties than the ECC code. Comparing the equivalence classes of the new code to those for the ECC code, the new code uses the extra syndromes to split the largest class into two. Whereas the ECC code is unable to distinguish errors at the first, fourth or parity positions, the new code can unambiguously correct errors at the first position and the ambiguity of fourth and parity position errors is reduced.The second alternative code shown in Table Table β010 is more powerful than the ECC code but this does not necessarily mean that it performs better in practice since actual performance will depend on the distribution of different types of error and at which positions they occur. To help quantify the difference in performance between codes, they were compared on artificial data, simulated so that it had an error profile similar to real data but for which the original sequence of bases is known, the error profile being estimated by mapping to a SNP-corrected genome.The theory described suggests that the code with probe generator 1E. coli DH10b with qualities being sampled from a real sequencing run of the same genome. The probability of generalised error was set to be equivalent to Q34, close to observed values for prepared samples. Three sets of reads were produced, using different codes on the same set of fragments to generate the ECC information, and sequence was called into both base-space and color-space using the Maximum A Posteriori (MAP) criterion. Reads were then mapped to an appropriately encoded reference using the BWA aligner [One million fragments were sampled uniformly from the positive strand of the genome of  aligner  with an β0β0) to help call reads makes a considerable improvement in both color-space and base-space. Without any correction, 77.1% of simulated reads map to the reference in color-space and only 38.4% of these are perfect, with a further 16.7% having one error. After correction, 48.6% of reads are perfect, with 10.2% containing one error. In base-space, the number of mapped reads increases from 47.2% to 64.3%. This large increase is due to correcting single-color errors that would otherwise induce multiple base errors and prevent the read from being mapped using the ‘five differences or fewer’ criterion.Using the ECC information ; the alternative code generated by 1The alternative codes that we describe have equal or superior error correction ability to that of the ECC but the creation of a new probe set is a considerable investment and a more thorough search of alternatives codes should be undertaken before the expense is incurred. All the alternative codes considered satisfy a number of restrictions, several of which could be relaxed to improve the error correction properties. Firstly, more rounds could be used, slowing the sequencing process down but providing more redundancy with which to correct errors. Secondly, the architecture of the encoder was constrained to be backwards compatible with the two-base-encoding sequencing chemistry, completely defining one unpunctured stream. A completely new chemistry could vary both probe sets and also change the puncturing matrix, allowing much more flexibility over the design of the code architecture and potentially creating codes that are capable of unambiguously correcting multiple errors.While many sequencing errors can be corrected, their occurrence is non-uniform, being more frequent in the later cycles of each round. This leads to a tendency for the poorest reads to contain multiple errors in close proximity. Such bursts of errors cannot be successfully corrected, so the advantage of the ECC is limited to generally high quality reads containing a few sparsely distributed errors. Perhaps the major advantage of the new chemistry is that the sequencing platform can produce reliable nucleotide sequence, without the possibility of a single error causing a catastrophic decoding failure, which allows the reads produced to be analysed with the wealth of tools available that assume nucleotide sequence.http://www.ebi.ac.uk/goldman-srv/solid/, distributed under version 3 of the GNU General Public Licence, as is software to simulate two- and four-base encoded reads and some utility functions to manipulate convolutional codes in GF4. This software is provided solely for the purposes of reproducibility.Software reimplementing the decoding algorithm for the two-base- and four-base- encodings (SOLiD ECC), as well as the alternative encodings described, on a trellis to find both the maximum likelihood (Viterbi decoding) and maximum a posteriori  (forwardThis work was supported in part by Wellcome Trust Technology Development grant WT088151MA and a grant to the EBI from Life Technologies Corporation. Life Technologies Corporation had no input into the development of the study, article preparation or decision to publish. No other competing interests are declared.The study was conceived and planned by both authors. The analysis of the Exact Call Chemistry coding was done by TM, who also drafted the manuscript. Both authors read, edited and approved the final manuscript.


In this paper we describe a localization and local communication system which allows situated agents to communicate locally, obtaining at the same time both the range and the bearing of the emitter without the need of any centralized control or any external reference. The system relies on infrared communications with frequency modulation and is composed of two interconnected modules for data and power measurement. Thanks to the open hardware license under which it is released, the research community can easily replicate the system at a low cost and/or adapt it for applications in sensor networks and in robotics. A noCommunication can be separated in two categories, namely abstract or situated. Abstract communication  refers tThe majority of communication systems implemented in sensor networks make extensive use of abstract communication using radio devices. In these systems, nodes are missing useful information such as the location of the emitter of a message. This problem has been addressed in works such as , 12, butFollowing modern artificial intelligence approaches , 18, newLocalization and communication systems have been designed and studied in several previous works. Some implementations based on GPS  or on trAn ultrasonic localization system is described in , but it ° in bearing and 10.23 cm in range. However, due to the radio frequency electronics employed, the board is difficult to miniaturize. In [Finally, infrared sensors have been previously used for relative positioning systems taking advantage of their directionality. In , 29 an irize. In , 31 anotEpuck Range & Bearing board) but can be adapted to many other applications such as home automation, weather stations, security systems, and so on. Due to the many potential board usages, we focus on the hardware system abstracted from any specific application. The board is able to receive data and at the same time extract the emitter's range and bearing from the communication. All the specifications of the board are available under open hardware license which makes the board easily reproducible at a low production cost of less than 250 euros.In this paper, we pursue a relative localization system for miniaturized nodes which does not make use of any external device. We are interested in exploiting a general, robust and self-localization mechanism. The localization mechanism should allow a receiver to extract accurate distance and angle information without the need of any external device. We also focus on the design of a low consumption and accu-rate system. For these reasons, in this work we present the implementation of situated communication using an open hardware board where limited range communication and localization of emitters by receivers are achieved using infrared signals. Infrared is chosen because of the high directionality of the signals emitted, the low aperture angle of the receivers, the inexpensive transducers and the low power requirements. The board has been designed for robotics tasks  timer is initialized with a period of 1.09 Once a transmission order is sent by the master to the board, the communication module is in charge of decomposing the data for the different sensors with a preamble (2 bits), the data (8 bits) and a CRC (2 bits). If the master needs to transmit a flow of data, the communication module buffers all the messages one after the other, in a transparent manner for the transmission timer.The reception software is continuously checking if a message arrives. Once the preamble of a frame is detected by an infrared modulated receiver, the board continues receiving the data and CRC while it is charging a peak detector through an infrared photodiode. If the frame has correctly arrived (checked by the CRC), the peak detector level is read and stored in a buffer. As the aperture of the receiving sensor is wide, it is likely that several sensors receive the same data at the same time. The information given by the different peak detectors is used to calculate the orientation and distance to the emitter. These two values are then stored in a buffer to be sent to the master board. The board implements different mechanisms for signal interference and noise errors. In the first case, if two emitters are addressing the same receiver the transmission will be disrupted. Therefore, the CRC check will detect that the frame is not properly received: This will cause the frame to be discarded. Infrared noise comes mainly from light conditions in the environment. To deal with it, the board continuously measures the infrared signal in the environment. Once a frame is correctly received, the board subtracts the environment measure from the peak receptor and returns it as the frame signal intensity. This implementation allows the board to be moved from one place to another with different light conditions. Section 4. shows results for different light conditions.For the correct understanding of the localization and communication system and its replication or modification possibilities, the forthcoming subsections detail the different hardware modules implemented in the board.2.1.The board can be powered from 2.5 V to 6 V. Once the board is switched on, three isolated power lines are created: one for the digital system, one for the analog and the last one for the emission module. The three power lines are obtained from two different supplies.The first power supply is in charge of the emission module. This supply is based on a low dropout linear regulator which allows a voltage variation between 0.8 V and 3.46 V see . This poR102 and R114 are 15 KΩ resistors, and potentiometer D53 modifies its value from 0 Ω to 100 KΩ with an 8 bits SPI frame, so 256 levels of approximately 390 Ω are managed. Resistors R102 + D53 and R114 form the resistor divider network necessary to set the output voltage. With this configuration, emisV follows Equation 1:adjV has a nominal voltage of 0.4 V. emisV minimum value of 0.8 V is achieved for D53 = 0Ω, and maximum value of 3.46 V for D53 = 100KΩThe second power supply is in charge of the rest of the electronics including the microcontroller. Analog and digital lines, both of 3.3 V, are separated and short circuited just in one point to reduce noise.The power consumption of the board depends on the settings of the emission power supply. 2.2.±20°, a 100 mA forward current, a maximum power consumption of 180mW and a nominal switching time of 12ns.The emission module is composed of 12 different emitters. Each sensor set is composed of a narrow beam infrared led and logic gates to create the modulation as the one shown in Communication is based on frequency modulation with data at 10 KHz over a carrier of 455 KHz . If a higher strength signal arrives to the diode, it will continue charging the capacitor. If the signal arrived has a lower intensity than the actual value stored in the capacitor, the peak detector will keep its value (see an example in V 19) is added to the circuit for discharging the capacitor. The resets are managed trough 12 independent output pins. The complementary activities of the data and signal intensity reception submodules are sketched in The signal intensity submodule is based on a PIN diode and two operational amplifiers. 2.4.The communication module has been designed to be the slave of a main processor system. Two buses, I2C and RS232, have been incorporated for facilitating the use of the board.In the I2C communication, the range and bearing board acts as slave of the main processor system. The board takes care of the requests of transmission and is continuously checking for incoming frames. The main processor system polls continuously the board to check if any communication has been received.In the serial port communication, interruptions are enabled in both directions. The master board is able to send orders of transmission or range modifications. Once a frame is demodulated by the communication board, it interrupts the master and transmits the demodulated data, the estimated angle and the distance to the emitter.In both communication types, the master has the control of the emission range. The modification of the power supply output can be ordered at anytime and results in an immediate modification of the emission range.3.SI6 while in S8I and SI9.Due to the hardware design, one single transmission is likely to be detected by several infrared sensors. For getting the correct location information, an internal data fusion must be carried out before supplying data to the master board. max1ϕ and max2ϕ are the orientation angles of the two maximum reception value sensors and maxV̂1 and maxV̂2 are the received values on both sensors.For getting a more accurate measure on the bearing we implement a linear combination between the two sensors with highest power signal following Equation 2:max1 and ρ̃max2 from the emitter to each of the two maximum sensors from the received ADC values ν̂max1 and ν̂max2 respectively.It is difficult to rely on a single sensor to determine accurately the distance of the emitter. Therefore, we use a linear combination based on the estimated bearings and distances provided by the two sensors that detected the strongest signal intensity. To this end, we have devised an empiric relationship between the ADC values and the distance when the emitter and receiver sensors are facing each other as shown in maxiλ from the center of the board to the emitter following Equation 3 λ˜maλ of the emitter by averaging out maxiλ as shown in Equation 4:We obtain the estimated distance 4.Two different experiments have been run to characterize and validate the localization and communication system. The first experiment is developed in an obstacle-free environment with 2 nodes, while the second experiment is a multi-node network in a non-obstacle-free environment.4.1.° intervals) from the emitter to avoid bias on the emitter transmission. The emitter stays in place while the receiver is rotated at 10° intervals in each position. At each position, the receiver waits till 100 messages are received and stores 12 signal intensity values. We have repeated this test for 60 positions along each of 8 directions and 36 relative orientations for 10 different boards.One emitter and one receiver board are placed in an obstacle-free environment from 10 cm to 6 m, in 10 cm intervals as shown in ° on average and 26.87° in the worst case. The error on the range was 7.18 cm on average and 38.62 cm in the worst case  sun light, ii) incandescent light with a 75 W lamp, iii) halogen light with a 75 W lamp and iv) fluorescent light with an energy-saving lamp of 20 W. Different light conditions introduce divers infrared component signals. As mentioned in Section 2. the board implements a signal noise correction mechanism based on the ambient light conditions. The board keeps continuously measuring the infrared ambient light and subtracts it from the measure when a signal is received. This mechanism allows the board to be moved from one place to another with different light conditions. Depending on the amount of infrared light in the environment, the range of the board is slightly reduced. We have tested the board in four different light conditions environments: Notice that for distances greater than 4 meters (in a fluorescent light environment) the board receives the data but, due to the sensibility of the photodiode sensors, it is not able to estimate the distance to the emitter. Instead, it returns a tag informing that the emitter is too far, the data and a bearing estimate calculated using the position of the data receivers which have demodulated a correct frame.4.2.2 arena with 4 different obstacles as shown in The network is composed of 9 nodes in a 2×2 mNode A is offering a measure (e.g. temperature) to the network needed by node B, which also requires information about the location of node A. Each node in the network stores the information offered by node A and its relative position. A node which has received information about node A broadcasts the data received and the relative location of node A, that will be received by other nodes in the network, and so on.node A is transmitted from node i to node j. In a first step, node i transmits its estimate of the distance idy and direction iϕ of node A to node j. For the direction, the value transmitted is the angle α, obtained from (x003D5)i using the communication beam as reference axis: α = iϕ − iγ. In a second step, node j transforms the received data into its own coordinates system. First, it calculates the direction pointed by node i as ϕj = jγ + α − π. Then, nodej calculates the location jloc =  of node A related to its own reference frame using node i information and the simple trigonometric equations x = ijcosγjd + icosϕjdy and y = ijsinγjd + isinϕjdy.Since the nodes do not have an absolute reference system, they must rely on a common reference axis, the communication axis. node A they broadcast the information. In the middle of the experiment, we remove node C from the network (see Due to the physical characteristics of the infrared signals, receivers must have direct vision to the emitters to receive messages. In the example shown, nodes receive information from different paths which is averaged with data from previous communication. Once the different nodes receive information about the data and location of work see  and obsenode A stored in node B each 100 ms. Results show that data information is always received by node B with a maximum error of 5° for the bearing and of 8cm for the distance. Once a node is removed from the network, the relative position estimate of node A suffers a small change because of the removed node. This is a consequence of the increased transmission distance between the nodes. Experiments last for 60 seconds and there are 30 replications. We obtain the data and location information of 5.In this paper we have described the design of an open board for localization and local communication.° in bearing at dis-tances below one meter. For longer distances, performance degrades gracefully with a maximum error of 38.62 cm in range and 26.87° in bearing at 6 meters.The system provides a communication rate up to 5kbps with a frequency modulation which allows robustness to light conditions. The range of communication can be modified from 0 to 6 meters by software and in real time. The system gets the data and extracts range and bearing from the communication at the same time. The board operates with a maximum error of 1 cm in range and 2Although the board has been designed for robotics tasks, it can easily be used in different applications such as smart sensors, intelligent ambients, home automation, and so on. Thanks to the open hardware license under which the board is released, along with the full documentation and the low cost of production, this board provides researchers with a new and versatile communication tool for systems made of multiple interacting entities.Potentials for further improvements to the system have been explored. Increasing the number of peak reception modules will offer a better spatial information which will reduce the range and bearing error. In the near future, we plan to extend the range and bearing board to work in 3D spaces. This implementation will allow the nodes to be located in different planes and to create a 3D spatial map. It will give to the network the capability of having nodes which could move freely in the space keeping them localized. For the accomplishment of the 3D extension, a media control access will be implemented on top of the existing firmware.


Motor learning is driven by movement errors. The speed of learning can be quantified by the learning rate, which is the proportion of an error that is corrected for in the planning of the next movement. Previous studies have shown that the learning rate depends on the reliability of the error signal and on the uncertainty of the motor system’s own state. These dependences are in agreement with the predictions of the Kalman filter, which is a state estimator that can be used to determine the optimal learning rate for each movement such that the expected movement error is minimized. Here we test whether not only the average behaviour is optimal, as the previous studies showed, but if the learning rate is chosen optimally in every individual movement. Subjects made repeated movements to visual targets with their unseen hand. They received visual feedback about their endpoint error immediately after each movement. The reliability of these error-signals was varied across three conditions. The results are inconsistent with the predictions of the Kalman filter because correction for large errors in the beginning of a series of movements to a fixed target was not as fast as predicted and the learning rates for the extent and the direction of the movements did not differ in the way predicted by the Kalman filter. Instead, a simpler model that uses the same learning rate for all movements with the same error-signal reliability can explain the data. We conclude that our brain does not apply state estimation to determine the optimal planning correction for every individual movement, but it employs a simpler strategy of using a fixed learning rate for all movements with the same level of error-signal reliability. Is every individual movement that we make optimal, or is only some average behaviour optimal?Over the last few decades, many studies have examined whether optimality principles can explain human motor behaviour. Although different frameworks have been used, such as optimal (feedback) control  = 3.21; p = 0.07). The mean endpoint variance was 86 mm2; the mean variances for the Extent and Direction components were 67 mm2 and 18 mm2, respectively.The overall picture in condition M  is quiteIn all conditions, the endpoint of the first movement to a target often differed considerably from later ones. This suggests that planning of the first movement to a target was often inaccurate. The fact that only the first endpoint differed implies that the error in this movement was used to adjust planning of the next movement. In conditions H and M, the error signal was visual, and it was reliable enough to reduce the error. In condition L, the error signal arose by comparing the felt finger location to the seen target position. Idiosyncratic biases in the proprioceptive sense of finger location relative to visual targets Learning curves were constructed to quantify how quickly subjects shifted their endpoints in the beginning of a series towards the steady-state position. We calculated the Mahalanobis distance  = 15.32; p<0.0001). Another, subtle, difference between the learning curves is that the end is as good as flat in conditions H and M, whereas it rises in condition L. In all conditions, initial movements could be inaccurate in all series of a session  = 56.72; p<0.00001) but not between components  = 0.45; p>0.5), and there was no significant interaction between condition and component  = 0.91; p = 0.43). Post-hoc Tukey’s honestly significant difference tests showed that the autocorrelation differed significantly between any pair of conditions . The lag 1 cross-correlations, which express the statistical relation between the Extent component of one endpoint and the Direction component of the previous or next endpoint, are approximately zero in all conditions. This suggests that in all conditions, an error in one component did not lead to a corrective adjustment in the planning of the other component.The observed serial correlations are plotted in k (PACF(k)) is the autocorrelation between the endpoints of movements t and t – k after their linear dependence on the intervening endpoints has been removed. Partial autocorrelations of the observed endpoints are shown in In condition L, the autocorrelations at several lags greater than 1 are also positive . These cIn summary, we found that both the time constant of the learning curve and the lag 1 autocorrelation of the endpoints increased with increasing error-signal uncertainty. Since the time constant and the autocorrelation increase when smaller error corrections are made, these results confirm the earlier finding We used the Kalman filter to determine the optimal planning correction for individual movements. The task of motor planning is to generate motor commands that will bring the finger to the target. The substantial errors in the first movement to a target  indicatebe the endpoint that would result if the centrally planned motor command of movement t would directly drive the movement without being corrupted by noise. We will refer to this as the planned aim pointexecution noise. Its consequence is that the actual endpoint xt).where ct) planning correction The actual endpoint is unknown to the subject. It can be eliminated from the above equations to yield:The first equation can be viewed as a state equation with state ct)( when the new movement has been planned (with known correction) but before its endpoint is observed:We will now use the Kalman filter to determine the planning corrections Hats denote estimates and the minus symbol indicates that these are a priori values. The measurement update equations give the a posteriori values that are obtained after the endpoint is observed:Kt), we obtain:This equation shows that for this planning correction, every movement is planned such that it is expected to be accurate. The second measurement update equation (Eq. 6b) then becomes:This equation shows that after the endpoint has been observed, the estimated planned aim point of the movement just executed is corrected by an amount that is proportional to the sensed error (the difference between the sensed endpoint and the target location). When we substitute this into Eq. 7, we find that the planning correction is equal to:Kt) as:The planning correction is thus proportional to the sensed error, and the Kalman gain 0 is a covariance matrix, that, in order to produce relatively large errors, has elements that exceed those of Σpl and Σex. Since there was no systematic pattern in the direction of the initial errors , we initialized the state estimate as:To complete the specification of the Kalman filter, we have to choose the initial values of the state estimate and its error covariance. The planned aim point of the first movement in a series depends on the initial state estimate. The fact that the first movement could be quite inaccurate in each series  and 4 sur0 is a random vector that reflects the uncertainty of the initial state estimate. As a result of this large initial uncertainty, the Kalman gain, and therefore the learning rate, will initially be large and then decrease to stabilize at a lower value.The model assumes that the dynamics of the system are linear, that all noise is Gaussian, and that planning corrections are proportional to the sensed error. Previous studies sens  differed from one another by a scaling factor. This assumption was motivated by the finding of Cheng and Sabes pl and Σex directly from data, and found that both were anisotropic with a larger variance in the Extent than in the Direction component. Specifically, we parameterized these matrices as: Σpl =  swΣmot, Σex =  s(1– w)Σmot and: Σ0 =  csΣmot, where Σmot =   mm2 is a prototype covariance matrix to which Σpl, Σex and Σ0 are proportional. Parameter s scales all these matrices relative to Σsens, w determines the relative size of the planning and execution covariance matrices, and c scales the initial state-estimate uncertainty. We assumed that the  element of Σmot was 4 times as large as the  element because the ratio of Extent to Direction variance was about 4 for all subjects and in all conditions. Hence, there were three free parameters: s, w and c. For condition H, we assumed that the error-signal uncertainty was negligible (Σsens = 0). Since both the Mahalanobis distance and the autocorrelation are standardized values that are independent of the magnitude of the endpoint variability, the predictions were independent of the value of s for this condition. As a result, there were only two free parameters (w and c) for this condition.We tested whether the Kalman filter can explain the data by evaluating whether it can reproduce the observed learning curves and autocorrelations. The predictions depend on the various covariance matrices defined above. Since it is not possible to obtain accurate estimates of all of these matrices, we followed a different approach in which we essentially determined whether any set of values of the covariance matrices could reproduce both the observed learning curves and the autocorrelations. To reduce the number of free parameters, we assumed that all covariance matrices were diagonal , we used literature values or estimates obtained in a control experiment for the error-uncertainty covariance matrix Σp<0.05, two-tailed weighted t tests: For each subject and each condition, we determined the values of the free parameters that minimized the difference between the predicted and observed values of the initial value of the learning curve, the learning-curve time constant, and the lag 1 and lag 2 autocorrelations of the Extent and Direction component , Σsens, the aspect ratio of Σsens, while we also varied the aspect ratios of Σpl and Σex simultaneously such that one aspect ratio was doubled while the other was halved. We also considered the case that Σpl was isotropic; here, we assumed that Σpl =  swI, with I the identity matrix, whereas Σex =  s(1– w)Σmot, as before. The variations of Σsens were not applicable to condition H because we assumed Σsens = 0 for this condition. We therefore examined the effect of Σsens being non-zero for this condition.In summary, the Kalman filter predicts faster correction for initial errors than observed and it predicts different autocorrelations for the Extent and Direction component whereas the observed ones do not differ. We conducted a sensitivity analysis to examine whether these failures of the model can be the result of incorrect assumptions in the parameterization of the covariance matrices. In this analysis we repeated the analysis above several times, where each time the value of one or two parameters was doubled or halved. The parameters that were varied were: Σmot, Σpl and Σex have very little effect on the time constants and ACF(1)s, and the predicted time constant is always well below the observed one. The only parameter change that leads to a time constant near the observed one is assuming a non-zero Σsens (last row of sens = 0.2Tr(Σmot)I) is unrealistically large: To obtain an endpoint variance matching the data (86 mm2), we would need s ≈ 14, which corresponds to Σsens ≈  mm2. In other words, the standard deviation in the perceived size of an error of 7 mm  would be almost 4 mm. This is unrealistically large as subjects saw the target and the endpoint simultaneously , while it is possible to obtain correct ACF(1)s for the Direction and Extent components by doubling the aspect ratio of Σpl and halving that of Σex.The sensitivity analysis for condition M  shows thIn summary, the sensitivity analysis demonstrates that the failure of the Kalman filter to explain the data cannot be the result of making incorrect assumptions about the underlying covariance matrices for conditions H and M, whereas this model can explain the results of condition L only if at least three parameters are changed to extreme values. Together, this suggests that the motor system does not use a Kalman filter to determine the learning rate in every individual movement.B that is the same for all movements in a condition, but that can vary between conditions:To obtain a better understanding of the actual learning rate, we compared the observed behaviour to the predictions of a second model. This model is almost the same as the Kalman filter, but rather than using the time-varying, optimal Kalman gain as learning rate, it uses a learning rate B. There are no state-estimation equations for this model because it does not perform state estimation – there is just a fixed learning rate for each level of error-signal reliability. Since this model is an extension of the Planned Aim Point Correction (PAPC) model used in B is a diagonal matrix because all observed cross correlations were about zero.The only difference between these equations and equation 11 of the Kalman filter is that the Kalman gain has been replaced by learning rate w and s as for the Kalman filter were used to parameterize Σpl and Σex; c was not a free parameter here, but was in each simulation chosen such that it reproduced the observed initial value of the learning curve (the resulting time constant and ACF(1)s were virtually independent of this value in a wide neighbourhood around the value used). The diagonal elements of B, extB and dirB, were the other free parameters. The total number of free parameters was therefore four , but, to prevent overfitting, this number was restricted to three for each condition. For condition H, s was not a free parameter as the time constant and autocorrelations were independent of this parameter. For conditions M and L, w was not a free parameter, but its value was chosen the same as estimated for condition H (0.197).We tested the EPAPC model in the same way as the Kalman filter: by examining whether it can reproduce the observed time constants and autocorrelations. The same parameters p>0.5, two-tailed weighted t tests: t tests confirmed that none of the predicted ACF(1)s differed significantly from the observed values , and that the predicted ACF(1) was not significantly different for the Extent and Direction components (p>0.25 for each condition).t test: p = 0.29), which implies that the scalar learning rate used in p = 0.48). For condition L, the learning rates were 0.15±0.03 and 0.29±0.06, which were significantly different (p = 0.006). Learning rates decreased with increasing error-signal uncertainty , respectively. These values were not significantly different  as predicted by this model  smaller than that of the sensed endpoints). The autocorrelation of the actual endpoints is positive for this learning rate.Although we found that the learning rate was not optimal for each individual movement , it is possible that learning rates were optimal under the restriction that they were the same for all trials of the same condition, as in the EPAPC model. With optimal, we mean that they had the value that minimized the endpoint variance. To examine whether this was the case, we derived an expression for the endpoint variance Var(odel see :(16)wherThe learning rates estimated from the data (indicated in extB (0.15) was smaller than the optimal one (0.23). To check whether this was the case, we regressed for each series the Extent and Direction components of the endpoints against the trial number. We made use of the finding that proprioceptive drift has, per subject, a fixed direction β for the 24 target directions θ to the function β(θ) =  p1+ p2 sin(θ) + p3 cos(θ). Here, p1 represents drift that all series have in common, such as an increasing Extent, whereas p2 and p3 account for drift that is constant in external space, such as a rightward drift. The drift captured by this fit was subtracted from the actual endpoints, after which the whole analysis was repeated. This led to estimated learning rates of 0.16±0.03 and 0.30±0.06 for Extent and Direction, respectively. These values differ only marginally from the ones obtained without drift correction, and they differ substantially from the optimal ones. This demonstrates that proprioceptive drift cannot explain the difference between observed and optimal learning rates in condition L.It is surprising that learning rates were near-optimal in two conditions, but not in the third. This could be a result of drift in the proprioceptive sense of hand location in condition L. Although care was taken to minimize such drift by giving subjects visual feedback about their finger location at the beginning of each trial We analyzed the time-series statistics of repeated reaching movements with different levels of error-signal reliability to determine the learning rate used by the motor system for updating motor planning on the basis of observed errors. We found that the learning rate increases with increasing error-signal reliability, which agrees with the results of earlier studies One could argue that the comparison between the Kalman filter and the EPAPC model is not fair because the EPAPC model had one free parameter more. However, we parameterized the Kalman filter such that it had the highest possible number of free parameters (three): The variance of the process noise, of the measurement noise and of the initial state estimate fully determine the time constant of the learning curve and the steady-state lag 1 autocorrelation. The fact that this model is unable to reproduce both the time constant and the autocorrelation simultaneously demonstrates that subjects cannot have used a Kalman filter to determine the learning rate in every individual trial. This is confirmed by the sensitivity analysis, as this analysis showed that the Kalman filter is also unable to reproduce the data when the underlying variances are varied by large amounts. Extending the Kalman filter from a single-state to a two-state model in which the two states have different learning and retention rates We first emphasize that we cannot exclude that learning rates were not completely fixed per condition. They could have decreased at the beginning of a series, but to a smaller extent than predicted by the Kalman filter. Wei and Körding A next question is why our motor system chooses the specific learning rate that is chooses for a given error-signal reliability. It may be chosen to minimize the endpoint variance as it was close to the, for this purpose, optimal value in two of the three conditions. However, it was not optimal for the condition with the lowest error-signal reliability. We showed that this difference cannot be the result of proprioceptive drift, and the sensitivity analysis suggests that it cannot be explained by incorrect assumptions about model parameters either. Why did the learning rate differ from the optimal value in this condition? One possibility is that subjects used an incorrect estimate of the error-signal reliability, and used a learning rate that was optimal for this incorrect value. Using Eq. 29, one can show that this would mean that subjects overestimated the error-signal variance of the Extent component by more than a factor 2, and underestimated that of the Direction component by more than a factor 10. It is unlikely that subjects misestimated these variances by such large amounts as studies on visual-proprioceptive integration We found that the learning rate was the same for the Extent and Direction components for conditions H and M. This confirms that the scalar learning rate that was used in the model of van Beers Our main conclusion is that our brain does not determine the optimal learning rate for every individual movement, as a Kalman filter would do. Instead, the learning rate is approximately the same for all movements with the same level of error-signal reliability. The average behaviour is thus near optimal, but individual movements are not optimal. This conclusion applies to the particular case of determining the learning rate in motor learning, but the issue is relevant to many other cases in the sensorimotor domain as well. Examples include feedback control of on ongoing movement, integration of sensory information and bimanual coordination. Since neural algorithms that produce behaviour that is optimal for every individual movement may be different, probably more complicated, from algorithms that produce behaviour that is only optimal on average, much insight into our motor system can be gained from addressing the issue for other sensorimotor tasks. The present results suggest that our motor system may not try to optimize individual movements but cares more about the average behaviour.Eight subjects  participated in all experimental conditions. None of them reported any sensory or motor deficits, and all had normal or corrected-to-normal vision, reported being right handed, and were unaware of the purposes of the study.All subjects gave verbal informed consent (which was then documented) before participation. All experiments were conducted in agreement with the ethics and safety guidelines of the Science Faculty of Utrecht University, where the experiment was conducted, and was part of a program that received blanket approval of the Medical Ethical Test Committee of the University Medical Centre Utrecht. All data were encoded and analyzed anonymously.The same set up was used as in The task was to move the tip of the right index finger from a start position to visual targets. The start position was a pink disc (4 mm radius) at a fixed location approximately 35 cm in front of the waist. A red cursor (a 4 mm radius disc) was shown at the fingertip location when it was within 3 cm from the start position. This enabled subjects to place their finger quickly and accurately on the start position, and it also prevented drift of the perceived finger location throughout an experimental session Condition H (high error reliability): A red disc (4 mm radius) was shown at the endpoint location. It was shown alongside the target so that the error signal was highly reliable. A score was awarded based on the distance from the target see . This coCondition M (medium error reliability): A cloud of 15 red circular dots (0.8 mm radius) was shown. The dot locations were drawn independently from a circular Gaussian distribution with the actual endpoint as the mean and a standard deviation of 15 mm see . New dotCondition L (low error reliability): Subjects received no visual feedback about their movement endpoints see , but theIn all conditions, the visual feedback, if any, was shown for 1 s. After that, subjects moved their finger back to the start position to begin the next trial.A session consisted of 24 series of 30 movements each, all in the same experimental condition. The targets were located at 10 cm distance from the start position in equally spaced directions. A blocked design was used, in which the same target was used for all movements in a series. The target of the first series was randomly chosen exactly to the left of right of the start location. Each later target direction differed 105 degrees from the previous direction in the counter clockwise direction. There were breaks of at least 10 seconds between series. At the start of a session, each subject practiced the task in the condition of that session for several minutes before starting the experiment proper (with a different target than in the first series). A session lasted approximately one hour. Each subject performed one session of each condition, each on a different day. The order of conditions was randomized between subjects.We analyzed the two-dimensional movement endpoints. A small fraction of the movements  was discarded from the analysis because the recording had failed. Endpoints were transformed into an Extent component  and a Direction component . To characterize error-corrective learning, we determined two measures: Mahalanobis distance and serial correlations.Dt)) to the learning curves, where t is the movement number, a and b are constants and ct is the time constant. Since the time constant could occasionally not be determined reliably, as indicated by a large confidence interval, a weighted average over subjects was calculated by weighing each subject’s time constant by the inverse of the squared width of the confidence interval.A learning curve was constructed for each subject by calculating the Mahalanobis distance of each endpoint, and then averaging these across series as a function of the trial number in the series. Time constants of the learning curves, and their 95% confidence intervals, were estimated for individual subjects using nonlinear least-squares regression. We fit exponentials of the form k. Since the endpoints are two-dimensional, the serial correlations consist of two autocorrelation functions ACF(k), one for each component (Extent and Direction), and two cross-correlation functions CCF(k) between the components. The cross-correlation function CCFij,(k) between components i and j at lag k was calculated as:ixt). Summations start at 6 because the first 5 movements were not included. The method developed by Marshall i(k) of component i at lag k was found as: ACFi(k) =  CCFii,(k).Whereas learning curves are informative about correction for large errors in the beginning of a series, the serial correlations focus on error correction in the “steady state” when errors are small. Serial correlations were calculated from the last 25 endpoints of each series, to avoid them being influenced by the correction for the large initial errors. Serial correlations express the statistical relationship between the endpoints of movements separated by a certain lag (number of movements) Two models for trial-by-trial motor learning are described in the Results section. We ran Monte Carlo simulations to determine the predictions of each model. Each simulation consisted of 2,000 sets of 24 simulated series of 30 movements, corresponding to 2,000 subjects performing a full experiment. Random vectors were drawn from Gaussian distributions to simulate the effects of planning, execution and sensory noise, as specified in the Results section.sens was assumed to be zero because subjects received highly reliable error signals. All subjects participated in a control condition to estimate Σsens for condition M. This condition was similar to condition M, but now they saw the cloud of dots from the start of the movement and they moved their finger until they perceived it aligned with the target. Eight targets in equally spaced directions were tested. The variability in the indicated positions hardly varied between targets and subjects. Based on these results, we assumed that Σsens was  mm2 for each target. For condition L, we assumed that Σsens equalled  mm2, which is the sum of the variances of visual and proprioceptive (right hand) localization reported by van Beers et al. For condition H, the sensory-noise covariance matrix Σw, s and c . For the EPAPC model, there were four free parameters: w, s, extB and dirB, but only three of these were estimated per condition (see Results).The potentially most powerful method to estimate the parameters of a linear dynamic system from time-series data is maximum likelihood estimation using the expectation-maximization algorithm extB and dirB were varied between 0.1 and 0.9, and w between 0.0 and 0.6. Tensor product splines were then fit (function spap2 in Matlab) to the resulting ACF(1)s, ACF(2)s and time constant as a function of extB, dirB and w. Best parameter values were determined per subject by finding the values for which the sum of squares of the normalized difference between observed and predicted values of the time constant, and the ACF(1)s and ACF(2)s of Extent and Direction, was minimized. The differences were normalized by dividing each difference by the width of the confidence interval of the observed value. There is some arbitrariness in treating these five values equally. We verified that the parameter estimates hardly changed when the five values were weighted differently (such as giving the time constant a weight of 50% and the ACFs weights of 12.5%).To find the best parameter estimates, we ran simulations for a range of parameter values. For instance, to fit the EPAPC model to the data of condition H, s for conditions M and L. This probably reflects differential scaling of the assumed sensory-noise covariance matrices in these conditions. Indeed, whereas the matrix for condition M was estimated in a control condition, the matrix for condition L was estimated from literature values. Free parameter s was included exactly to account for such effects.The best estimates of the parameter values for the Kalman filter are given in t +1 as a function of the endpoint in movement t and the various types of noise:We use xT − bsens, where bsens is the bias in the sensory information . Define deviation dt)( in movement t as the difference between the endpoint and the expected endpoint:The expected endpoint is Then we have from (19):E denotes the expected value. After substitution of (21) and some algebraic manipulations, we find for k = 0:The covariance matrix function x) as the trace of the covariance matrix of the endpoints, we find i  is:Γ(0) is the covariance matrix of the deviations. Since the deviations differ a constant vector from the endpoints, it is also the covariance matrix of the endpoints. When we define the endpoint variance Var(bsens.Note that the variance and autocorrelations are independent of sensory bias 2 denotes variance. This equation can also be expressed in terms of the free parameters of the model:The fact that all matrices in the above equations are diagonal see  implies extB and dirB for which the derivatives of the variance with respect to extB and dirB are zero. These optimal learning rates are:where: