dataesr/scientific-paragraphs-categorization huggingface dataset:
Kerzman and Trummer  [17]  introduced a method to compute the Riemann mapping function using the Szegö kernel that is numerically stable while Trefethen  [30]  introduced numerical methods for solving the more speci c conformal Schwarz-Christo el transformation (conformal transformation of the upper half-plane onto the interior of a simple polygon). Furthermore, a Matlab toolkit is available in  [12]  as well as a Python translation (https://github. com/AndrewWalker/cmtoolkit) that has been used to produce the gure 5 that shows some examples of arbitrary shapes and the automatic mapping of the polar and Cartesian domains. However, even if automatic, this mapping can be perceived as not intuitive. Provided the shape are not too distorted, we'll see in the results section that ad-hoc mapping can also be used.
Discussion We've introduced a graphical, scalable and intuitive method for the placement and the connection of biological cells and we illustrated its use on three use-cases. We believe this method, even if simple and obvious, might be worth to be considered in the design of a new class of model, in between symbolic model and realistic model. Our intuition is that such topography may be an important aspect that needs to be taken into account and studied in order for the model to bene t from structural functionality. Furthermore, the proposed speci cation of the architecture as an SVG le associated with the scalability of the method could guarantee to some extent the scalability of the properties of the model. Notes: All gures were produced using the Python scienti c stack, namely, SciPy  [16] , Matplotlib  [15]  and NumPy  [32].
The second scheme uses a dedicated DDEs solver with adaptive step sizes to control error, making it theoretically more accurate. To highlight the effects of the approximation used by the first integration scheme in regard to parameter estimation and Bayesian model selection, we performed simulations of local field potentials using first, a simple model comprising 2 regions and second, a more complex model comprising 6 regions. In these simulations, the second integration scheme served as the standard to which the first one was compared. Then, the performances of the two integration schemes were directly compared by fitting a public mismatch negativity EEG dataset with different models. The simulations revealed that the use of the standard DCM integration scheme was acceptable for Bayesian model selection but underestimated the connectivity parameters and did not allow an accurate estimation of conduction delays.
We also evaluate it by using a second integration scheme of DDEs, more precise though computationally more expensive. First, we briefly introduce the temporal parameters and the system of DDEs used by the ERP neuronal model and describe the two integration schemes used to solve this system. In particular, we compare the effect of varying conduction delays on the response of the system for a simple model with 2 sources. Second, we test the two DDEs integration schemes in the context of DCM. We use simulations to generate data with various conduction delays and modulations of connection and focus on the capability to identify the generative model among a series of plausible models and to retrieve parameter estimates for connectivity as close as possible to those used to generate the data. Last, we illustrate the effects of the DDEs integration scheme on a public mismatch negativity (MMN) EEG dataset (available at www.
fil. ion. ucl. ac. uk/spm/data/eeg_mmn). The MMN paradigm has been extensively studied with DCM, with a minimal network of five cortical sources that could account for the differences observed between evoked responses to frequent and rare auditory stimuli  (Garrido et al. , 2007). This public dataset was therefore a candidate of choice to extend the LFP simulations to a real EEG case study.
Data generation To compare DDE_TA and DDE_RK methods, we generated data from the ERP neural mass model of DCM. Given this generative model and a set of parameters, each dataset was simulated in a baseline and a modulated condition. In the modulated condition, all forward and backward connection strengths were changed from their baseline respective values while other parameters stayed unchanged. Because of the theoretical reasons briefly mentioned in section 2. 2 and detailed in  (Shampine and Thompson, 2001) , the DDE_RK integration scheme was used to generate the data and reproduce accurately the temporal dynamics described by the equations of the neural mass model. The model received a brief input, described by a Gaussian function. Conduction delays between regions were systematically specified and imposed to be identical between each pair of reciprocal connection.
Observation noise was created by convolving Gaussian noise with a smoothing kernel of 4 time samples (corresponding to 4 ms duration). This observation noise was then scaled in order to get a ratio of the mean power of the simulated signal over the mean power of the simulated noise of 25 dB, which is a signal-to-noise ratio found in very good quality real ERP dataset. All simulated recordings used the LFP modality of DCM code and the pyramidal cell activity of each region was observed with unitary gain. With this configuration, the variational Bayes optimization scheme directly fits the local neuronal dynamics to infer the parameters of the model, instead of using a lead-field weighted linear combination of all neuronal activities supposed to generate EEG or MEG signals.
Fit procedure Each dataset was fitted with 3 models: two false models based on the structural architecture of the generative model but allowing respectively for modulations of forward connections only (F) and backward connections only (B), and the true generative model allowing for modulations of both forward and backward connections (FB). All models were equipped with DDE_TA integration scheme and had the same default priors. In particular, synaptic time constants and conduction delays had prior expectations as mentioned in Section 2. 1 and the same relatively low prior precision (DCM standard value of 16). This ensured that all models were offered the same freedom to move these parameters away from their prior values.
Bayesian model comparison and parameter estimates A random effect (RFX) Bayesian model comparison (BMC) was used to compute the expected posterior probability for each of the 3 models  (Penny et al. , 2010; Stephan et al. , 2009). RFX was preferred to fixed effect (FFX) modeling because it frees the inference from the assumption that datasets were generated with a common model. Having identified the model with highest expected posterior probability, a mean posterior density on modulation parameters was estimated by averaging the posterior distribution estimated for each dataset weighted by the posterior probability of the model for this dataset. Finally, these parameter estimates were compared to the true generative values. The full estimation procedure was then repeated with models equipped with DDE_RK and parameter estimates were compared to those obtained with DDE_TA.
Simulation 1: model with 2 regions In this simulation, the generative model was composed of 2 regions reciprocally connected with forward and backward connections and a stimulus input entering region 1 (Fig. 1 ). In the modulated condition of each dataset, forward and backward connections were modulated by a factor of 0. 5 and 0. 8 respectively compared to the baseline condition. These strengths of modulation 1 were chosen in order to get a notable effect of the modulation while keeping the system stable. We generated a series of 10 datasets with an SNR of 25 dB for every symmetrical 1 These are log-scale values; this means that for forward modulation, if c is the strength of 1->2 connection in the baseline condition, then the strength of 1->2 connection in the modulated condition is c*exp(0.
Simulation 2: model with 6 regions In this simulation, the generative model was composed of 6 regions reciprocally connected with forward, backward and lateral connections and a stimulus input entering regions 1 and 4 (Fig. 2 ). For each dataset, all connection strengths were uniformly sampled from the interval [0 2] 2  while conduction delays between reciprocally connected regions were uniformly sampled between 8 ms and 32 ms. In the modulated condition, all forward and backward connections were modulated with a strength of 0. 5. A series of 100 datasets was generated with an SNR of 25 dB. Typical time courses are presented in Fig. 2. Due to the number of regions interacting in this model and the full variability of connection strengths and conduction delays used to generate the data, this simulation reproduced a more realistic and non-trivial situation.
Application: DCM of mismatch negativity EEG dataset We compared the two integration schemes while fitting an ERP dataset from a subject recorded with 128 EEG channels during an auditory mismatch negativity (MMN) paradigm (dataset publicly available at www. fil. ion. ucl. ac. uk/spm/data/eeg_mmn). In this experiment, standard (500 Hz) and deviant (550 Hz) tones were randomly presented to the subject with respective frequencies of occurrence of 80% and 20%. The MMN effect refers to the difference between the evoked responses to standard and deviant tones; it corresponds to a negative component present only in the deviant condition. In this dataset, the baseline condition is the average response to standard tones. The other condition is the average response to deviant tones and is considered as the modulated condition.
Simulation 1: parameter estimation and model comparison using the model with 2 regions Using the data generated with the FB model equipped with DDE_RK presented on Fig. 1 , we proceeded to the Bayesian model comparison among F, B, and FB models across all conduction delays (Fig. 5a, left ). The FB model integrated with DDE_TA was more plausible than the F and B models, with an expected posterior probability of p ¼ 0. 58 and with a high confidence (exceedance probability, p ¼ 0. 99). This means that, for any randomly selected dataset, there was an important posterior probability (total expected posterior probability, p ¼ 0. 31 þ 0. 11 ¼ 0. 42) for this data to have been generated by either the F or the B model and therefore, that one modulation present in the data (backward or forward, respectively) was not significantly detected.
The presence of these false negatives indicates either that the absent modulation is compensated by other parameters of the model or that the potential improvement in accuracy of the FB model is not sufficient to counterbalance its additional complexity. In contrast, considering the same models integrated with DDE_RK, the FB model was clearly identified, with an expected posterior probability of p ¼ 0. 98 (Fig. 5a, right ). For each conduction delay, average posteriors were computed from the corresponding series of 10 datasets fitted with the winning FB model. Fig. 5b  shows the posterior parameter estimates (mean, 90% credibility interval, and true parameter) for forward and backward modulations. The systematic underestimation of forward and backward modulations, previously mentioned for F and B models, was also found with the FB model.
However, there was a slight but systematic overestimation of the backward conduction delays, ranging from 6 ms for shortest delays to 2 ms for largest delays. Simulation 2: parameter estimation and model comparison using the model with 6 regions Bayesian model comparison among F, B, and FB models fitted to the series of 100 datasets generated from the 6 source models is presented in Fig. 6a , left. Using DDE_TA, the posterior probability was estimated higher for the generative FB model (p ¼ 0. 53) than for the F model (p ¼ 0. 46) and the B model (p ¼ 0. 01). Despite the rather small difference between the FB and the F models, the FB model was estimated more plausible than the F model with good confidence (exceedance probability, p ¼ 0. 73). In contrast, Bayesian comparison of models fitted using DDE_RK (Fig.
6a , right) only pointed out the FB model, with expected posterior probability of p ¼ 0. 98 and exceedance probability of p ¼ 1. 00. The average posteriors of modulation (and their 90% credibility interval) are shown in Fig. 6b , separately for each forward and each backward connections. With DDE_TA, the estimation of the modulation for the forward connections 1->2, 1->3, 4->5 and 4->6 were rather close to the true value, whereas the forward connections 2->3, 5->6 and all Fig. 3. Set of dynamic causal models (DCMs) used to fit the MMN dataset. All models have the same structure composed of 5 regions (bilateral A1: primary auditory cortices; bilateral STG: superior temporal gyri; right IFG: inferior frontal gyrus) reciprocally connected with forward/backward (black lines) and lateral (gray lines) connections.
The models differ by the connections being modulated (dashed black lines) in the deviant tone condition. Stimulus input enters each model bilaterally in A1. backward modulations were very poorly estimated. In contrast, with DDE_RK, even if backward modulations were not perfectly retrieved (thus showing a similar overall tendency as with DDE_TA), forward modulations were estimated with high precision (in particular for connections 1->2, 1->3 and 4->6 where the true modulation lied in the tight 90% credibility interval). The estimates obtained by models equipped with DDE_RK were clearly the most accurate, yet the backward connections were underestimated. We interpret this result as a weak effect of the modulation of backward connections on the generated data. Indeed, because of random strengths of connection used to generate the datasets, some regions had very weak responses (in both the baseline and the modulated conditions), in which the effect of the modulation could easily be hidden by noise.
If we consider only interactions among regions 1, 2, and 3, the first effects of modulation appeared around 120 ms in the responses of regions 2 and 3 (Fig. 2 ) and corresponded to the modulation of  connections 1->2 and 1->3, which were estimated accurately. Due to synaptic and conduction delays, the following main effects of modulation started from 170 ms for the other forward (2->3) and backward connections (2->1, 3->1 and 3->2) and were estimated with less accuracy. These late components (and associated modulations), which were usually subtler than the early ones in our generated datasets, were therefore harder to detect for the optimization procedure. The associated decrease in precision also reflects a (temporo-spatial) propagation of parameter estimation errors, which is accentuated when using the DDE_TA integration scheme.
Application: Mismatch negativity EEG dataset Bayesian model comparison in terms of relative free energy is shown in Fig. 7a  (left) for the seven models and the two integration schemes tested on the publicly available MMN dataset from an example subject. The best model was FBI integrated with DDE_RK (posterior probability, p ¼ 1. 0). In addition, each model integrated with DDE_RK had higher evidence than the same model integrated with DDE_TA. Even if the BMC restricted to models using DDE_TA also selected FBI as the best model with very high confidence (posterior probability, p ¼ 0. 98), the relative ranking of other models was different (Fig. 7a , left). The ranking of the models (and their respective relative free energy) using DDE_TA was: 1: FBI (289), 2: I (285), 3: FB (253), 4: FBI1 (243), 5: F (225), 6: B (12), 7: no (0).
Although the true generative parameters for this dataset are unknown, the present results suggest the same under-estimation of modulation and the same imprecise estimation of conduction delays as observed in the simulations (Figs. 5b  and 6b ). Fig. 6. Summary of results from Simulation 2. a) RFX Bayesian model comparison between the three models: F, B and F, for the 2 integration schemes, DDE_TA (two first plots, in black) and DDE_RK (two next plots, in gray). The true FB model was identified with good confidence with DDE_TA (exceedance ¼ 0. 73) and DDE_RK (exceedance probability, p ¼ 1. 00). b) Posteriors estimation of forward (left plot) and backward (right plot) modulations using (in black) and DDE_RK (in gray), represented for each connection with their 90% credibility interval. The dotted, black, horizontal corresponds to the true generative parameters in each plot.
first simulation showed that the loss of precision in the estimation of the modulation was independent of the conduction delay. The second simulation replicated this result in a more challenging and realistic configuration using a model composed of 6 sources. The variability one can expect in DCM group studies was introduced using connections with random strengths. The effects of the modulation were non-uniformly assigned to the connections through all the datasets (a weak connection carry almost no effect of modulation) and Bayesian procedures had to gather the information available (and unequally spread) in each dataset to properly infer the generative parameters at the group level. BMS identified well the FB model as being more plausible than either the F or the B models, with high confidence (exceedance probability, p ¼ 0.
73). However, the late effects of backward modulations were not as well recovered as earlier effects of forward modulations. It could reflect that in this simulation, the late components had smaller amplitude than the early ones, relative to a same level of noise and thus exhibited more subtle effects of the modulation which were harder to detect, especially using DDE_TA. Finally, fitting seven DCM models to a MMN dataset revealed, for any model architecture taken separately, the higher goodness of fit obtained by DDE_RK compared to DDE_TA, along with a stronger deviation of the parameters (in posteriors estimates) from their prior distribution (especially for forward and backward modulations), suggesting a larger degree of freedom of the system dynamics. Assessing the complete added-value of Runge-Kutta methods for experimental data is difficult without crossed validating measures, as the ground truth remains unknown.
However, by fitting a whole series of real datasets with DDE_TA and DDE_RK and comparing their relative free energy (and its decomposition into accuracy and complexity), it would be possible to at least validate the increased goodness of fit provided by DDE_RK. The initial objective of the present study was to assess the effects of the 1st order Taylor approximation used to integrate the biophysical equations of DCM for ERPs. Briefly, with this approximation, the simulations showed that 1) the Bayesian model selection was robust enough to identify the generative model; 2) the estimation of parameters were relatively poorly estimated as compared to when a Runge-Kutta method was used. These findings were clear from simulations, and the use of a real MMN dataset gave some hints that it might also be true for experimental data.
Acknowledgments The research leading to these results has received funding from the  European Research Council  under the European Union's  Seventh Framework Programme  (FP/2007-2013)/ ERC  Grant Agreement no. 616268  F-TRACT and from the  French "Investissements d'avenir" programme  under grant numbers  ANR-11-INBS-0006  and  ANR-10-IAIHU-06.
Implementation The model was simulated with the "Brian" spiking neural network simulator  (Goodman and Brette, 2008, 2009; Stimberg et al. , 2014). Each run was repeated 4000 times, for a duration of 350ms of simulated time per run. Spikes in the first 100ms were discarded, and the CV was computed as the ratio of the standard deviation to the mean of the simulated interspike intervals for the remaining 250ms. The code is available at https://github. com/neural-reckoning/vcn_regularity.
Level dependence Classification of chopper cells is carried out at a fixel level, or at a fixed level relative to threshold. However, based on the model above, we would expect CV to vary as a function of level. For example, at a higher level, the auditory nerve fibre inputs to the chopper cell would be firing at a higher rate (increasing ρ E ). This would lead a change in both µ and σ/µ. We would therefore expect that not only would the chopper cell's firing rate be level-dependent, but also its CV. Figure  6  shows a new analysis of an existing data set where tones were presented at 20 dB re threshold and 50 dB re threshold. Almost all cells do indeed show a change in both firing rate and CV at these two different levels. In the majority of cases, a cell classified as sustained or transient at one level, would also be so classified at the other level.
Reproducibility and robustness We have made the code and an interactive version of this paper online at https://github. com/ neural-reckoning/vcn_regularity. This has several aims. Firstly, we wish to allow readers to explore the consequences of changes in the parameters. A fixed set of figures in a printed paper only allows us to show a tiny snapshot of the complete parameter space, but with the interactive version the complete space can be easily explored without having to install any software locally. Secondly, we hope that it gives confidence in the results as it can be easily verified that the results presented here do not depend on highly tuned sets of parameters. Thirdly, by making the complete code available and possible to modify and run without any burden of installing complex software locally, we hope to encourage other researchers to use and build on this model with the minimum of effort.
25/day, std = 5. 7), 115 (mean = 28. 75/day, std = 5. 3) during the second week. The photos did not provide direct geographical information, so we used the QGis software v 2. 18. 5 to point on each of the goats (not the kids) present in the pasture and then extract their spatial coordinates. Note that we did not differentiate the activity of the animals (feeding, sleeping, walking) because this was too difficult based on a picture taken from behind. All of the statistical analyses were performed using Matlab. We virtually delimited several zones in the pastures, depending on the predominant grass species. We delimited the pasture of the first flock, denoted P1, into ten different zones and the pasture of the second flock, denoted P2, into seven different zones. The dominant grass species on each of the zone is provided in Supplementary Table  S1  and a visual of the zones is provided in Supplementary Fig.
S3. The species present in the two pastures are Panicum maximum, Digitaria Decumbens, Dichanthium sp. , Paspalum dilatatu, Kyllinga sp. , Brachiaria mutica purpuresens, Achyranthes aspera, Sida acuta, Digitaria swazilandensis, and Sporobolus indicus. Note that the zones are highly heterogeneous and several species, if not all, can be found inside a same zone. Meteorological data were collected from the nearest weather station 1 km away, from the on-line platform https://intranet. inra. fr/ climatik_v2/ClimatikGwt. htmlClimatik (accessed on November 29 th , 2017. Meteorological station 97118002). A simulation framework of goats and L3 spatial distribution. We proposed a simulation framework because it allows us to describe the time that each animal spends on the different locations of the pasture during the first grazing week.
e. , the number of time a goat was detected on the quadrat divided by the number of pictures took from the drone. Let f q denote the normalised frequency for quadrat q, such that ∑ = = f 1 q nQ q 1. Time spent on a quadrat. Let define {λ g1 , …, λ gnQ } a set of normalised weights such that for all goat g: ∑ λ λ = =. = q t T 1 and for all quadrat , q nQ gq gq gq g 1 λ gq is simply the proportion of time goat g spends on quadrat q. There are unfortunately no ways to estimate this weight using our dataset or any published data and, therefore, the weights will be drawn randomly. These weights depict some aspect of the animals' behavior. For example, when they are all equal (i. e. , λ g1 = … = λ gnQ ), the goat tends to explore all the quadrats. In contrast, when only a few of the weights are non-zeros, the goat is very specific and only spends time on a few quadrats.