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 Hαkinematics of the SINGS nearby galaxies survey - I*This is the first part of an Hαkinematics follow-up survey of theSpitzer Infrared Nearby Galaxies Survey (SINGS) sample. The data for28galaxies are presented. The observations were done on three differenttelescopes with Fabry-Perot of New Technology for the Observatoire dumont Megantic (FaNTOmM), an integral field photon-counting spectrometer,installed in the respective focal reducer of each telescope. The datareduction was done through a newly built pipeline with the aim ofproducing the most homogenous data set possible. Adaptive spatialbinning was applied to the data cubes in order to get a constantsignal-to-noise ratio across the field of view. Radial velocity andmonochromatic maps were generated using a new algorithm, and thekinematical parameters were derived using tilted-ring models. Hubble Space Telescope STIS Spectra of Nuclear Star Clusters in Spiral Galaxies: Dependence of Age and Mass on Hubble TypeWe study the nuclear star clusters (NCs) in spiral galaxies of variousHubble types using spectra obtained with the STIS on board the HubbleSpace Telescope (HST). We observed the nuclear clusters in 40 galaxies,selected from two previous HST WFPC2 imaging surveys. At a spatialresolution of ~0.2" the spectra provide a better separation of clusterlight from underlying galaxy light than is possible with ground-basedspectra. Approximately half of the spectra have a sufficiently highsignal-to-noise ratio for detailed stellar population analysis. For theother half we only measure the continuum slope, as quantified by the B-Vcolor. To infer the star formation history, metallicity, and dustextinction, we fit weighted superpositions of single-age stellarpopulation templates to the high signal-to-noise ratio spectra. We usethe results to determine the luminosity-weighted age, mass-to-lightratio, and masses of the clusters. Approximately half of the sampleclusters contain a population younger than 1 Gyr. Theluminosity-weighted ages range from 10 Myr to 10 Gyr. The stellarpopulations of NCs are generally best fit as a mixture of populations ofdifferent ages. This indicates that NCs did not form in a single event,but that instead they had additional star formation long after theoldest stars formed. On average, the sample clusters in late-typespirals have a younger luminosity-weighted mean age than those inearly-type spirals (L=8.37+/-0.25 vs.9.23+/-0.21). The average mass-weighted ages are older by ~0.7 dex,indicating that there often is an underlying older population that doesnot contribute much light but does contain most of the mass. The averagecluster masses are smaller in late-type spirals than in early-typespirals (logM=6.25+/-0.21 vs. 7.63+/-0.24) and exceed the masses typicalof globular clusters. The cluster mass correlates loosely with totalgalaxy luminosity. It correlates more strongly with both the Hubble typeof the host galaxy and the luminosity of its bulge. The lattercorrelation has the same slope as the well-known correlation betweensupermassive black hole mass and bulge luminosity. The properties ofboth nuclear clusters and black holes in the centers of spiral galaxiesare therefore intimately connected to the properties of the host galaxy,and in particular its bulge component. Plausible formation scenarioshave to account for this. We discuss various possible selection biasesin our results, but conclude that none of them can explain thedifferences seen between clusters in early- and late-type spirals. Theinability to infer spectroscopically the populations of faint clustersdoes introduce a bias toward younger ages, but not necessarily towardhigher masses.Based on observations made with the NASA/ESA Hubble Space Telescope,obtained from the Data Archive at the Space Telescope Science Institute,which is operated by the Association of Universities for Research inAstronomy, Inc., under NASA contract NAS5-26555. These observations areassociated with proposals 9070 and 9783. The Star Formation Threshold in NGC 6822We investigate the star formation threshold in NGC 6822, a nearby LocalGroup dwarf galaxy, on subkiloparsec scales using high-resolution,wide-field, deep H I, Hα, and optical data. In a study of the H Ivelocity profiles we identify cool and warm neutral components in theinterstellar medium of NGC 6822. We show that the velocity dispersion ofthe cool component (~4 km s-1) when used with a ToomreQ-criterion gives an optimal description of ongoing star formation inNGC 6822 superior to that using the more conventional dispersion valueof 6 km s-1. However, a simple constant surface densitycriterion for star formation gives an equally superior description. Wealso investigate the two-dimensional distribution of Q and the starformation threshold and find that these results also hold locally. Therange in gas density in NGC 6822 is much larger than the range incritical density, and we argue that the conditions for star formation inNGC 6822 are fully driven by this density criterion. Star formation islocal, and in NGC 6822 global rotational or shear parameters areapparently not important. The structure of galactic disks. Studying late-type spiral galaxies using SDSSUsing imaging data from the SDSS survey, we present the g' and r' radialstellar light distribution of a complete sample of ~90 face-on tointermediate inclined, nearby, late-type (Sb-Sdm) spiral galaxies. Thesurface brightness profiles are reliable (1 σ uncertainty lessthan 0.2 mag) down to μ27 mag/''. Only ~10% of all galaxies havea normal/standard purely exponential disk down to our noise limit. Thesurface brightness distribution of the rest of the galaxies is betterdescribed as a broken exponential. About 60% of the galaxies have abreak in the exponential profile between  1.5-4.5 times thescalelength followed by a downbending, steeper outer region. Another~30% shows also a clear break between  4.0-6.0 times thescalelength but followed by an upbending, shallower outer region. A fewgalaxies have even a more complex surface brightness distribution. Theshape of the profiles correlates with Hubble type. Downbending breaksare more frequent in later Hubble types while the fraction of upbendingbreaks rises towards earlier types. No clear relation is found betweenthe environment, as characterised by the number of neighbours, and theshape of the profiles of the galaxies. Discovery Among the DisksNot Available Discovery of an Extended Ultraviolet Disk in the Nearby Galaxy NGC 4625Recent far-UV (FUV) and near-UV (NUV) observations of the nearby galaxyNGC 4625 made by the Galaxy Evolution Explorer (GALEX) show the presenceof an extended UV disk reaching to 4 times the optical radius of thegalaxy. The UV-to-optical colors suggest that the bulk of the stars inthe disk of NGC 4625 are currently being formed, providing a uniqueopportunity to study today the physics of star formation underconditions similar to those when the normal disks of spiral galaxieslike the Milky Way first formed. In the case of NGC 4625, the starformation in the extended disk is likely to be triggered by interactionwith NGC 4618 and possibly also with the newly discovered galaxy NGC4625A. The positions of the FUV complexes in the extended disk coincidewith peaks in the H I distribution. The masses of these complexes are inthe range 103-104 Msolar, with theirHα emission (when present) being dominated by ionization fromsingle stars. The Molecular Interstellar Medium of Dwarf Galaxies on Kiloparsec Scales: A New Survey for CO in Northern, IRAS-detected Dwarf GalaxiesWe present a new survey for CO in dwarf galaxies using the ARO Kitt Peak12 m telescope. This survey consists of observations of the centralregions of 121 northern dwarfs with IRAS detections and no known COemission. We detect CO in 28 of these galaxies and marginally detectanother 16, increasing by about 50% the number of such galaxies known tohave significant CO emission. The galaxies we detect are comparable instellar and dynamical mass to the Large Magellanic Cloud, althoughsomewhat brighter in CO and fainter in the far-IR. Within dwarfs, wefind that the CO luminosity LCO is most strongly correlatedwith the K-band and the far-infrared luminosities. There are also strongcorrelations with the radio continuum (RC) and B-band luminosities andlinear diameter. Conversely, we find that far-IR dust temperature is apoor predictor of CO emission within the dwarfs alone, although a goodpredictor of normalized CO content among a larger sample of galaxies. Wesuggest that LCO and LK correlate well because thestellar component of a galaxy dominates the midplane gravitational fieldand thus sets the pressure and density of the atomic gas, which controlthe formation of H2 from H I. We compare our sample with moremassive galaxies and find that dwarfs and large galaxies obey the samerelationship between CO and the 1.4 GHz RC surface brightness. Thisrelationship is well described by a Schmidt law withΣRC~Σ1.3CO. Therefore,dwarf galaxies and large spirals exhibit the same relationship betweenmolecular gas and star formation rate (SFR). We find that this result isrobust to moderate changes in the RC-to-SFR and CO-to-H2conversion factors. Our data appear to be inconsistent with large (orderof magnitude) variations in the CO-to-H2 conversion factor inthe star-forming molecular gas. Dwarf and Normal Spiral Galaxies: are they Self-Similar?The investigation presented here was focused on clarifying the existenceof dwarf spiral galaxies as a separate group from classical spirals.First, a list of spiral galaxies with small sizes was obtained.Information on colors, luminosities, morphologies and chemical contentwas searched for in the literature for these galaxies. Using thisinformation, it can be concluded that dwarf spirals are not likely to bethe tail of the distribution of classical galaxies. On the contrary,significant differences in some of the most important properties ofspiral galaxies, such as the metallicity gradient and the bar frecuency,were found. In any case, further and more accurate observations areneeded for a definitive answer. On the alignment between binary spiral galaxiesWe show some significance against the null hypothesis of randominteractions of binary spiral galaxies, and in favour of the alternativethat more interactions than expected occur for axes either nearlyparallel (spins being parallel or anti-parallel) or nearly orthogonal.We discuss this in the context of similar prior studies, using adifferent statistical focus in such a way that we are able toincorporate additional data. Dark Matter in Dwarf Galaxies: High Resolution ObservationsWe present observations and analysis of rotation curves and dark matterhalo density profiles in the central regions of four nearby dwarfgalaxies. This observing program has been designed to overcome some ofthe limitations of other rotation curve studies that rely mostly onlongslit spectra. We find that these objects exhibit the full range ofcentral density profiles between ρ∝ r^0 (constant density) andρ∝ r-1 (NFW halo). This result suggests that thereis a distribution of central density slopes rather than a unique halodensity profile. Neutral Hydrogen in the Interacting Magellanic Spirals NGC 4618 and NGC 4625Asymmetry is a common trait in spiral galaxies and is particularlyfrequent among Magellanic spirals. To explore how morphological andkinematic asymmetry are affected by companion galaxies, we analyzeneutral hydrogen observations of the interacting Magellanic spirals NGC4618 and NGC 4625. The analysis of the H I distribution reveals thatabout 10% of the total H I mass of NGC 4618 resides in a looping tidalstructure that appears to wrap all the way around the galaxy. Throughcalculations based on derived H I profiles, we show that NGC 4618 andNGC 4625 are no more asymmetric than the noninteracting Magellanicspirals analyzed recently by Wilcots & Prescott. We also deriverotation curves for the approaching and receding sides of each galaxy.By fitting the mean curves with an isothermal halo model, we calculatedynamical masses of 4.7×109 and9.8×109 Msolar out to 6.7 kpc for NGC 4618and NGC 4625, respectively. While the rotation curves had systematicallyhigher velocities on the receding side of each galaxy, the effect was nomore pronounced than in studies of noninteracting spirals. The degree ofinteraction-driven asymmetry in both galaxies is indistinguishable fromthe intrinsic degree of asymmetry of lopsided galaxies. A Hubble Space Telescope Census of Nuclear Star Clusters in Late-Type Spiral Galaxies. II. Cluster Sizes and Structural Parameter CorrelationsWe investigate the structural properties of nuclear star clusters inlate-type spiral galaxies. More specifically, we fit analytical modelsto Hubble Space Telescope images of 39 nuclear clusters in order todetermine their effective radii after correction for the instrumentalpoint-spread function. We use the results of this analysis to comparethe luminosities and sizes of nuclear star clusters to those of otherellipsoidal stellar systems, in particular the Milky Way globularclusters. Our nuclear clusters have a median effective radius ofre=3.5 pc, with 50% of the sample falling in the range2.4pc<=re<=5.0pc. This narrow size distribution isstatistically indistinguishable from that of Galactic globular clusters,even though the nuclear clusters are, on average, 4 mag brighter thanthe old globular clusters. We discuss some possible interpretations ofthis result. From a comparison of nuclear cluster luminosities withvarious properties of their host galaxies, we confirm that more luminousgalaxies harbor more luminous nuclear clusters. It remains unclearwhether this correlation mainly reflects the influence of galaxy size,mass, and/or star formation rate. Since the brighter galaxies in oursample typically have stellar disks with a higher central surfacebrightness, nuclear cluster luminosity also correlates with thisproperty of their hosts. On the other hand, we find no evidence for acorrelation between the presence of a nuclear star cluster and thepresence of a large-scale stellar bar. Cold dust and molecular gas towards the centers of Magellanic type galaxies and irregulars. I. The dataWe present 1300 μm continuum emission measurements and observationsof the 12CO (1-0) and (2-1) transition towards the centers of64 Magellanic type galaxies (Sdm/Sm) and irregulars (Im/I0/Irr). Thesources are selected to have IRAS flux densities S100 μm≥1000 mJy and optical diameters mainly below 180 arcsec. We wereable to detect 12CO towards 41 and the continuum emissiontowards 28 galaxies. In addition, we obtained the corresponding data fora set of 6 complementary galaxies of different morphological type.Based on observations collected at ESO, La Silla, Chile and IRAM, PicoVeleta, Spain.The full version of Figs. \ref{spec1.fig} and \ref{spec2.fig} is onlyavailable in electronic form at http://www.edpsciences.org SINGS: The SIRTF Nearby Galaxies SurveyThe SIRTF Nearby Galaxy Survey is a comprehensive infrared imaging andspectroscopic survey of 75 nearby galaxies. Its primary goal is tocharacterize the infrared emission of galaxies and their principalinfrared-emitting components, across a broad range of galaxy propertiesand star formation environments. SINGS will provide new insights intothe physical processes connecting star formation to the interstellarmedium properties of galaxies and provide a vital foundation forunderstanding infrared observations of the distant universe andultraluminous and active galaxies. The galaxy sample and observingstrategy have been designed to maximize the scientific and archivalvalue of the data set for the SIRTF user community at large. The SIRTFimages and spectra will be supplemented by a comprehensivemultiwavelength library of ancillary and complementary observations,including radio continuum, H I, CO, submillimeter, BVRIJHK, Hα,Paα, ultraviolet, and X-ray data. This paper describes the mainastrophysical issues to be addressed by SINGS, the galaxy sample and theobserving strategy, and the SIRTF and other ancillary data products. Companions of Bright Barred Shapley-Ames GalaxiesCompanion galaxy environment for a subset of 78 bright and nearby barredgalaxies from the Shapley-Ames Catalog is presented. Among the spiralbarred galaxies, there are Seyfert galaxies, galaxies with circumnuclearstructures, galaxies not associated with any large-scale galaxy cloudstructure, galaxies with peculiar disk morphology (crooked arms), andgalaxies with normal disk morphology; the list includes all Hubbletypes. The companion galaxy list includes the number of companiongalaxies within 20 diameters, their Hubble type, and projectedseparation distance. In addition, the companion environment was searchedfor four known active spiral galaxies, three of them are Seyfertgalaxies, namely, NGC 1068, NGC 1097, and NGC 5548, and one is astarburst galaxy, M82. Among the results obtained, it is noted that theonly spiral barred galaxy classified as Seyfert 1 in our list has nocompanions within a projected distance of 20 diameters; six out of 10Seyfert 2 bar galaxies have no companions within 10 diameters, six outof 10 Seyfert 2 galaxies have one or more companions at projectedseparation distances between 10 and 20 diameters; six out of 12 galaxieswith circumnuclear structures have two or more companions within 20diameters. Molecular gas in the central regions of the latest-type spiral galaxiesUsing the IRAM 30 >m telescope, we have surveyed an unbiased sampleof 47 nearby spiral galaxies of very late (Scd-Sm) Hubble-type foremission in the 12CO(1-0) and (2-1) lines. The sensitivity ofour data (a few mK) allows detection of about 60% of our sample in atleast one of the CO lines. The median detected H2 mass is1.4x 107 >msun within the central few kpc, assuming astandard conversion factor. We use the measured line intensities tocomplement existing studies of the molecular gas content of spiralgalaxies as a function of Hubble-type and to significantly improve thestatistical significance of such studies at the late end of the spiralsequence. We find that the latest-type spirals closely follow thecorrelation between molecular gas content and galaxy luminosityestablished for earlier Hubble types. The molecular gas in late-typegalaxies seems to be less centrally concentrated than in earlier types.We use Hubble Space Telescope optical images to correlate the moleculargas mass to the properties of the central galaxy disk and the compactstar cluster that occupies the nucleus of most late-type spirals. Thereis no clear correlation between the luminosity of the nuclear starcluster and the molecular gas mass, although the CO detection rate ishighest for the brightest clusters. It appears that the central surfacebrightness of the stellar disk is an important parameter for the amountof molecular gas at the galaxy center. Whether stellar bars play acritical role for the gas dynamics remains unclear, in part because ofuncertainties in the morphological classifications of our sample. An Infrared Space Observatory Atlas of Bright Spiral GalaxiesIn this first paper in a series we present an atlas of infrared imagesand photometry from 1.2 to 180 μm for a sample of bright spiralgalaxies. The atlas galaxies are an optically selected,magnitude-limited sample of 77 spiral and S0 galaxies chosen from theRevised Shapley-Ames Catalog (RSA). The sample is a representativesample of spiral galaxies and includes Seyfert galaxies, LINERs,interacting galaxies, and peculiar galaxies. Using the Infrared SpaceObservatory (ISO), we have obtained 12 μm images and photometry at60, 100, and 180 μm for the galaxies. In addition to its imagingcapabilities, ISO provides substantially better angular resolution thanis available in the IRAS survey, and this permits discrimination betweeninfrared activity in the central regions and global infrared emission inthe disks of these galaxies. These ISO data have been supplemented withJHK imaging using ground-based telescopes. The atlas includes 2 and 12μm images. Following an analysis of the properties of the galaxies,we have compared the mid-infrared and far-infrared ISO photometry withIRAS photometry. The systematic differences we find between the IRASFaint Source Catalog and ISO measurements are directly related to thespatial extent of the ISO fluxes, and we discuss the reliability of IRASFaint Source Catalog total flux densities and flux ratios for nearbygalaxies. In our analysis of the 12 μm morphological features we findthat most but not all galaxies have bright nuclear emission. We find 12μm structures such as rings, spiral arm fragments, knotted spiralarms, and bright sources in the disks that are sometimes brighter thanthe nuclei at mid-infrared wavelengths. These features, which arepresumably associated with extranuclear star formation, are common inthe disks of Sb and later galaxies but are relatively unimportant inS0-Sab galaxies. Based on observations with the Infrared SpaceObservatory (ISO), an ESA project with instruments funded by ESA MemberStates (especially the PI countries: France, Germany, Netherlands, andUnited Kingdom) and with the participation of ISAS and NASA. The UZC-SSRS2 Group CatalogWe apply a friends-of-friends algorithm to the combined Updated ZwickyCatalog and Southern Sky Redshift Survey to construct a catalog of 1168groups of galaxies; 411 of these groups have five or more members withinthe redshift survey. The group catalog covers 4.69 sr, and all groupsexceed the number density contrast threshold, δρ/ρ=80. Wedemonstrate that the groups catalog is homogeneous across the twounderlying redshift surveys; the catalog of groups and their membersthus provides a basis for other statistical studies of the large-scaledistribution of groups and their physical properties. The medianphysical properties of the groups are similar to those for groupsderived from independent surveys, including the ESO Key Programme andthe Las Campanas Redshift Survey. We include tables of groups and theirmembers. A Hubble Space Telescope Census of Nuclear Star Clusters in Late-Type Spiral Galaxies. I. Observations and Image AnalysisWe present new Hubble Space Telescope I-band images of a sample of 77nearby late-type spiral galaxies with low inclination. The main purposeof this catalog is to study the frequency and properties of nuclear starclusters. In 59 galaxies of our sample, we have identified a distinct,compact (but resolved), and dominant source at or very close to thephotocenter. In many cases, these clusters are the only prominent sourcewithin a few kiloparsecs from the galaxy nucleus. We present surfacebrightness profiles, derived from elliptical isophote fits, of allgalaxies for which the fit was successful. We use the fitted isophotesat radii larger than 2" to check whether the location of the clustercoincides with the photocenter of the galaxy and confirm that in nearlyall cases, we are truly dealing with nuclear'' star clusters. Fromanalytical fits to the surface brightness profiles, we derive thecluster luminosities after subtraction of the light contribution fromthe underlying galaxy disk and/or bulge. Based on observations made withthe NASA/ESA Hubble Space Telescope, obtained at the Space TelescopeScience Institute, which is operated by the Association of Universitiesfor Research in Astronomy, Inc., under NASA contract NAS 5-26555. Theseobservations are associated with proposal 8599. Neutral hydrogen and optical observations of edge-on galaxies: Hunting for warpsWe present 21-cm HI line and optical R-band observations for a sample of26 edge-on galaxies. The HI observations were obtained with theWesterbork Synthesis Radio Telescope, and are part of the WHISP database(Westerbork HI Survey of Spiral and Irregular Galaxies). We present HImaps, optical images, and radial HI density profiles. We have alsoderived the rotation curves and studied the warping and lopsidedness ofthe HI disks. 20 out of the 26 galaxies of our sample are warped,confirming that warping of the HI disks is a very common phenomenon indisk galaxies. Indeed, we find that all galaxies that have an extendedHI disk with respect to the optical are warped. The warping usuallystarts around the edge of the optical disk. The degree of warping variesconsiderably from galaxy to galaxy. Furthermore, many warps areasymmetric, as they show up in only one side of the disk or exhibitlarge differences in amplitude in the approaching and receding sides ofthe galaxy. These asymmetries are more pronounced in rich environments,which may indicate that tidal interactions are a source of warpasymmetry. A rich environment tends to produce larger warps as well. Thepresence of lopsidedness seems to be related to the presence of nearbycompanions. Full Fig. 13 is only available in electronic form athttp://www.edpsciences.org The Westerbork HI survey of spiral and irregular galaxies. II. R-band surface photometry of late-type dwarf galaxiesR-band surface photometry is presented for 171 late-type dwarf andirregular galaxies. For a subsample of 46 galaxies B-band photometry ispresented as well. We present surface brightness profiles as well asisophotal and photometric parameters including magnitudes, diameters andcentral surface brightnesses. Absolute photometry is accurate to 0.1 magor better for 77% of the sample. For over 85% of the galaxies the radialsurface brightness profiles are consistent with published data withinthe measured photometric uncertainty. For most of the galaxies in thesample H I data have been obtained with the Westerbork Synthesis RadioTelescope. The galaxies in our sample are part of the WHISP project(Westerbork H I Survey of Spiral and Irregular Galaxies), which aims atmapping about 500 nearby spiral and irregular galaxies in H I. Theavailability of H I data makes this data set useful for a wide range ofstudies of the structure, dark matter content and kinematics oflate-type dwarf galaxies. Based on observations made with INT operatedon the island of La Palma by the Isaac Newton Group in the SpanishObservatorio del Roque de los Muchachos of the Instituto de Astrofisicade Canarias. The tables in Appendix A are only available in electronicform at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/390/863. Thefigures in Appendix B are only available in electronic formhttp://www.edpsciences.org The Frequency of Active and Quiescent Galaxies with Companions: Implications for the Feeding of the NucleusWe analyze the idea that nuclear activity, either active galactic nuclei(AGNs) or star formation, can be triggered by interactions by studyingthe percentage of active, H II, and quiescent galaxies with companions.Our sample was selected from the Palomar survey and avoids selectionbiases faced by previous studies. This sample was split into fivedifferent groups, Seyfert galaxies, LINERs, transition galaxies, H IIgalaxies, and absorption-line galaxies. The comparison between the localgalaxy density distributions of the different groups showed that in mostcases there is no statistically significant difference among galaxies ofdifferent activity types, with the exception that absorption-linegalaxies are seen in higher density environments, since most of them arein the Virgo Cluster. The comparison of the percentage of galaxies withnearby companions showed that there is a higher percentage of LINERs,transition galaxies, and absorption-line galaxies with companions thanSeyfert and H II galaxies. However, we find that when we consider onlygalaxies of similar morphological types (elliptical or spiral), there isno difference in the percentage of galaxies with companions amongdifferent activity types, indicating that the former result was due tothe morphology-density effect. In addition, only small differences arefound when we consider galaxies with similar Hα luminosities. Thecomparison between H II galaxies of different Hα luminositiesshows that there is a significantly higher percentage of galaxies withcompanions among H II galaxies with L(Hα)>1039 ergss-1 than among those with L(Hα)<=1039ergs s-1, indicating that interactions increase the amount ofcircumnuclear star formation, in agreement with previous results. Thefact that we find that galaxies of different activity types have thesame percentage of companions suggests that interactions betweengalaxies is not a necessary condition to trigger the nuclear activity inAGNs. We compare our results with previous ones and discuss theirimplications. The So-called Bar'' in the Large Magellanic CloudWe propose that the off-centered bar'' in the Large Magellanic Cloud(LMC) is an unvirialized structure slightly misaligned with, and offsetfrom, the plane of the LMC disk. The small displacement and misalignmentare consequences of recent tidal interactions with the Small MagellanicCloud and the Galaxy. This proposal, although radical, is consistentwith the kinematics of the LMC and the near-infrared star count mapsfrom the Deep Near-Infrared Survey and the 2 Micron All-Sky Survey and,in particular, with the reported 25°-50° inclination range ofthe LMC and the east-west gradient of distance moduli of standardcandles. Contributions to LMC microlensing come mainly from the mutuallensing of stars in the disk and the bar.'' The predicted microlensingoptical depth is at levels comparable to observation, even withoutincluding contributions from MACHOs. Observational tests are suggestedto discriminate between our misaligned offset bar'' model and theconventional picture of an off-centered planar bar. Nearby Optical Galaxies: Selection of the Sample and Identification of GroupsIn this paper we describe the Nearby Optical Galaxy (NOG) sample, whichis a complete, distance-limited (cz<=6000 km s-1) andmagnitude-limited (B<=14) sample of ~7000 optical galaxies. Thesample covers 2/3 (8.27 sr) of the sky (|b|>20deg) andappears to have a good completeness in redshift (97%). We select thesample on the basis of homogenized corrected total blue magnitudes inorder to minimize systematic effects in galaxy sampling. We identify thegroups in this sample by means of both the hierarchical and thepercolation friends-of-friends'' methods. The resulting catalogs ofloose groups appear to be similar and are among the largest catalogs ofgroups currently available. Most of the NOG galaxies (~60%) are found tobe members of galaxy pairs (~580 pairs for a total of ~15% of objects)or groups with at least three members (~500 groups for a total of ~45%of objects). About 40% of galaxies are left ungrouped (field galaxies).We illustrate the main features of the NOG galaxy distribution. Comparedto previous optical and IRAS galaxy samples, the NOG provides a densersampling of the galaxy distribution in the nearby universe. Given itslarge sky coverage, the identification of groups, and its high-densitysampling, the NOG is suited to the analysis of the galaxy density fieldof the nearby universe, especially on small scales. Arcsecond Positions of UGC GalaxiesWe present accurate B1950 and J2000 positions for all confirmed galaxiesin the Uppsala General Catalog (UGC). The positions were measuredvisually from Digitized Sky Survey images with rms uncertaintiesσ<=[(1.2")2+(θ/100)2]1/2,where θ is the major-axis diameter. We compared each galaxymeasured with the original UGC description to ensure high reliability.The full position list is available in the electronic version only. Groups of galaxies. III. Some empirical characteristics.Not Available Catalogue of HI maps of galaxies. I.A catalogue is presented of galaxies having large-scale observations inthe HI line. This catalogue collects from the literature the informationthat characterizes the observations in the 21-cm line and the way thatthese data were presented by means of maps, graphics and tables, forshowing the distribution and kinematics of the gas. It containsfurthermore a measure of the HI extension that is detected at the levelof the maximum sensitivity reached in the observations. This catalogueis intended as a guide for references on the HI maps published in theliterature from 1953 to 1995 and is the basis for the analysis of thedata presented in Paper II. The catalogue is only available inelectronic form at the CDS via anonymous ftp 130.79.128.5 orhttp://cdsweb.u-strasbg.fr/Abstract.html Total magnitude, radius, colour indices, colour gradients and photometric type of galaxiesWe present a catalogue of aperture photometry of galaxies, in UBVRI,assembled from three different origins: (i) an update of the catalogueof Buta et al. (1995) (ii) published photometric profiles and (iii)aperture photometry performed on CCD images. We explored different setsof growth curves to fit these data: (i) The Sersic law, (ii) The net ofgrowth curves used for the preparation of the RC3 and (iii) A linearinterpolation between the de Vaucouleurs (r(1/4) ) and exponential laws.Finally we adopted the latter solution. Fitting these growth curves, wederive (1) the total magnitude, (2) the effective radius, (3) the colourindices and (4) gradients and (5) the photometric type of 5169 galaxies.The photometric type is defined to statistically match the revisedmorphologic type and parametrizes the shape of the growth curve. It iscoded from -9, for very concentrated galaxies, to +10, for diffusegalaxies. Based in part on observations collected at the Haute-ProvenceObservatory. A catalogue of spatially resolved kinematics of galaxies: BibliographyWe present a catalogue of galaxies for which spatially resolved data ontheir internal kinematics have been published; there is no a priorirestriction regarding their morphological type. The catalogue lists thereferences to the articles where the data are published, as well as acoded description of these data: observed emission or absorption lines,velocity or velocity dispersion, radial profile or 2D field, positionangle. Tables 1, 2, and 3 are proposed in electronic form only, and areavailable from the CDS, via anonymous ftp to cdsarc.u-strasbg.fr (to130.79.128.5) or via http://cdsweb.u-strasbg.fr/Abstract.html An image database. II. Catalogue between δ=-30deg and δ=70deg.A preliminary list of 68.040 galaxies was built from extraction of35.841 digitized images of the Palomar Sky Survey (Paper I). For eachgalaxy, the basic parameters are obtained: coordinates, diameter, axisratio, total magnitude, position angle. On this preliminary list, weapply severe selection rules to get a catalog of 28.000 galaxies, wellidentified and well documented. For each parameter, a comparison is madewith standard measurements. The accuracy of the raw photometricparameters is quite good despite of the simplicity of the method.Without any local correction, the standard error on the total magnitudeis about 0.5 magnitude up to a total magnitude of B_T_=17. Significantsecondary effects are detected concerning the magnitudes: distance toplate center effect and air-mass effect.
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# What is the physical intuition behind this energy conservation theorem?

I'm reading Quantum Theory for Mathematicians, by Brian C. Hall. Although the book is about Quantum Mechanics, it's chapter 2 is actually about Classical Mechanics, in which I encountered the following theorem:

Rephrasing in English, suppose a force acting on a particle has two components. The first component comes from a potential function and the second component is orthogonal to the velocity. Then the energy of the particle is conserved.

What is the physical intuition behind this theorem?

This is a generalization of a result commonly discussed in Electrodynamics: magnetic forces do no work.

The idea is that for the energy of the particle to change, some force must do work on the particle due to the work-energy theorem. For the force Hall chose, we can split the analysis in two parts: the potential and the "magnetic" force (I'll give it this name in analogy with Electrodynamics).

The potential can do work, but it also goes in the very definition of energy. It is what we call a conservative force. If it accelerates the particle, the potential energy drops in exchange. Notice there is some sort of equilibrium in here: this force can make the particle go faster, at the expense of the potential getting lower. In general, this means there is a maximum possible speed the particle can get by being accelerated by the force, for eventually one will "run out" of potential energy and all energy will be kinetic. Furthermore, if the particle slows down, the loss of kinetic energy is just being traded for potential energy, which can once again be converted in kinetic and so on.

For the magnetic force, things are quite different. It does not enter the definition of the energy, so in principle it could spoil this idea and either accelerate the particle indefinitely (which would be non-physical) or "steal way energy" (which is the case for friction). What the theorem is showing is that forces that are always normal to the velocity do neither: they can only change direction. The fact they are always orthogonal to the velocity means that the infinitesimal amount of work they can do will always be \begin{align} dW &= \mathbf{F \cdot} \text{d}\mathbf{l}, \\ &= \mathbf{F \cdot} \frac{\text{d}\mathbf{l}}{\text{d} t} \ \text{d} t, \\ &= \mathbf{F \cdot v} \ \text{d} t, \\ &= 0, \end{align} hence, it can't do any work. As a consequence, it can't change the energy. (I think/hope this doesn't solve Exercise 8, since this argument is usually fair game for physicists, but too sloopy for mathematicians hahaha).

The fact it can't change the energy is a different way of saying it can never lead to the particle speeding up or down. If it increased the energy, it could be possible to turn this extra energy into kinetic and get the particle to move faster. If it decreased the energy, it could be possible to slow down the particle in a similar way.

In short, one possible physical intuition is that if a force is orthogonal to the velocity, it can only change the direction, it can never lead directly or indirectly to an increase or decrease in speed. As commonly stated in Electrodynamics, "magnetic forces do no work".

Not sure if this helps, but...

If you start off with Newtonian mechanics, energy conservation has to be added in as an additional axiom. (As in "... and only forces which conserve energy are found in the wild").

If you start off with Lagrangian mechanics it is simply not possible to write a Lagrangian which does not lead to a conservative force.

At first I thought Newtonian and Lagrangian formulations were totally equivalent, but once I grasped the above it suddenly made sense why people consider Lagrangian to be more fundamental.

So the physical intuition is simply that a formulation of mechanics based around conserved energy is more fundamental than one based around force, and forces are simply derived from that.

• This is not true! Energy conservation follows from Newtons laws, for a certain class of force fields. Also you can write down Lagrangians that are not invariant under time translations and therefore describe systems which do not conserve energy. Dec 28, 2021 at 10:58
• For a single particle 2D universe F(x,y) = yi -xj is not conservative. In fact almost anything you could write down would not be in any context. So the "certain class of force fields" is just those which come from a Lagrangian. I take your point though that this only applies to Lagrangians where L is not a function of time. But I think stating that L is not a function of time is a simpler constraint than saying "only conservative force fields appear in closed systems". Dec 29, 2021 at 11:48
• I'm not sure why this has been marked down. The OP was asking for an intuitive explanation. The argument above is essentially the same as made in Susskind's Theoretical Minimum books. Of course it is possible to take examples of forces and show by calculation that they're conservative, but that doesn't really tell you why do they always turn out to be conservative, or provide any intuition. Dec 29, 2021 at 12:05
• I think the work energy theorem is only a proper theorem, ie something that is derived, in the case of a specific force. Then you can calculate work and show it to be true. In the general case, it is a principle, ie a statement about what sort of forces can exist. Why does it work? An intuitive explanation might be that energy comes first and force is derived from it. It's up to you whether you consider this to be of intuitive help. If you don't you just need to say, well it just turns out that this principle holds about this particular quantity work, derived from force, and KE. Dec 30, 2021 at 17:34
• @AlexZeffertt There is the following consideration: as we know there is no intrinsic zero point of potential energy; all calculations use difference of potential energy; choice of zero point of potential energy is arbitrary. Similarly, kinetic energy is a function of the relative velocity of two (or more) objects. In that sence attribution of kinetic energy is frame dependent. Magnitude of acceleration, on the other hand, does have an intrinsic zero point. What comes first? In physics you can often run derivations in both directions. Dec 31, 2021 at 13:12

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# Help

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37
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What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions?

Mar 27, 2021

#1
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The form of the quadratic gives us 2(x - 1)(x - 4) and 2(x - 2)(x - 2).  Since the solutions must be distinct, only 2(x - 1)(x - 4) = 2x^2 - 10x + 8 works, so k = 10.

Mar 27, 2021
#2
+228
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Err, thats wrong... I used the veita.

We use the fact that the sum and product of the roots of a quadratic equation $ax^2+bx+c=0$ are given by $-b/a$ and $c/a$, respectively. Let the two roots of the equation be $p$ and $q$. Then $p+q=k/2$. However the only other restriction on $p$ and $q$ is that $pq = 4$ and that $p$ and $q$ are distinct integers. For each such possibility $(p,q)$, we also have the possibility $(-p,-q)$ since $(-p)(-q) = pq = 4$. This gives two values of $k$: $k=2(p+q)$ and $k=2(-p-q)$. Since these occur in such pairs, the sum of all possible values of $k$ is $\boxed{0}$.Alternatively, one can note that the only way to factor 4 into 2 distinct integer factors is $4\cdot1$ and $(-4)(-1)$, so that the two possible values of $k$ are $10$ and $-10$, given a sum of $0$.

Mar 27, 2021
#3
+315
+1

wolfiechan  Mar 27, 2021
#4
+315
+1

Good try though guest!

Mar 27, 2021


text:
# Getting Started Programming Julia

To get us started with Julia we cover three basics. Arithmetic operators, name assignment.

## Arithmetic operators

The standard arithmetic operators are addition (+), subtraction (-), multiplication (*), division (/), power (^) and remainder (%). They work as expected and the only one that is different for the Python crowd is power. That one is Matlab consistent. The normal precedence of operations applies. First power. Then multiplication and division. Then remainder. Finally addition and subtraction. Parentheses can be used to change the order of operation.

1 + 3 * 2
# 7
(1 + 3) * 2
# 8
2 * 3 ^ 2
# 18

In those examples, both sides of the operator are scalars. It gets a little more interesting when at least one of them is a vector or a matrix. Not all of the above operations are defined between vectors and scalars. Only division and multiplication are defined. We create a vector using square brackets ([]) with the elements separated by commas.

[3, 1, 4] * 2
# 3-element Array{Int64,1}:
#  6
#  2
#  8

[3, 1, 4] / 2
# 3-element Array{Float64,1}:
# 1.5
# 0.5
# 2.0

The other operations are not defined and throw an error.

[3, 1, 4] ^ 2
"""
MethodError: no method matching ^(::Array{Int64,1}, ::Int64)
Closest candidates are:
^(!Matched::Float16, ::Integer) at math.jl:885
^(!Matched::Regex, ::Integer) at regex.jl:712
^(!Matched::Missing, ::Integer) at missing.jl:155
...

Stacktrace:
[1] macro expansion at .\none:0 [inlined]
[2] literal_pow(::typeof(^), ::Array{Int64,1}, ::Val{2}) at .\none:0
[3] top-level scope at In[52]:1
"""

The reasons for this design choice have to do with Julias focus on linear algebra and are not important here. If we want this operation to work in an element-wise manner, we have to force it explicitly. We can do so using the dot (.). This way we can explicitly force every operator to be applied element-wise.

[3, 1, 4] .^ 2
# 3-element Array{Int64,1}:
#  9
#  1
# 16
[3, 1, 4] .+ 2
# 3-element Array{Int64,1}:
#  5
#  3
#  6

Now that we have our arithmetic operators, let’s move on to name assignment so we can store the results of our operations.

## Name Assignment

We assign names to values with the = operator using the syntax name = value. Once a name is assigned, we can use the name instead of the value in operations.

result_one = 2 + 2
result_two = result_one + 3
# 7

Once a name is assigned to a value we can reassign that same name to a different value without problem.

result = 2 + 2
result = 10
# 10

If we want to assign a name that is not supposed to be reassigned, we can use the const keyword. If we try to reassign a constant name we get an error.

const a = 8.3144621
a = 3
"""
invalid redefinition of constant a

Stacktrace:
[1] top-level scope at In[73]:2
"""

There are a few rules about the names we can assign. Generally, Unicode characters (UTF-8) are allowed. This means we can do something like this.

δt = 0.0001

Here we are using the special character delta. If you want to quickly generate such a character, many Julia environments allow you to do this by typing \delta and hitting tab. I recommend using these sparingly, as they might confuse people transitioning from other languages that don’t allow unicode names. On the other hand they might be useful to make your code style more mathy. Not allowed as names are built-in keywords that have special meaning and trying to assign them will result in an error.

if = 3
# syntax: unexpected "="

If you are interested in more details about variables and name assignments you can take a look at the official documentation. In the next blog post we will take a look at the type system of Julia.

# Arithmetic Operations in NumPy

• NumPy arrays come with many useful methods
• All arithmetic operations that are used on arrays are performed element-wise
• NumPy code is almost always faster than native Python (.append is a notable exception)

NumPy arrays are so useful because they allow us to do math on them very efficiently. For example, NumPy arrays come with many useful methods. One such method is the sum method, which calculates the sum of all values in the array

import numpy as np
my_array = np.array([4, 3, 1])
my_array.sum()
8

There are many other methods like this and they are extremely useful. Here is a list of the most commonly used methods.

my_array = np.array([4, 3, 1])
my_array.sum()  # Calculate the sum array values
8
my_array.mean()  # Calculate the mean of array values
2.6666666666666665
my_array.std()  # Calculate the standard deviation of array values
1.247219128924647
my_array.max()  # Find the maximum value
4
my_array.min()  # Find the minimum value
1

To learn about all array methods you can call the dir() function on any array, which will list all its methods. Alternatively you can check out the documentation for the array https://docs.scipy.org/doc/numpy/reference/generated/numpy.ndarray.html

Another useful property of arrays is that they do math when they appear together with any of the arithmetic operators (+, -, *, /, **, //, %).

my_array = np.array([4, 3, 1])
my_array_plus = my_array + 2
my_array_plus
array([6, 5, 3])

Here, the array appeared together with a scalar value, the single number 2. That number was added to each value. However, we can do the same thing with two arrays, if the have the same shape.

array_one = np.array([4, 3, 1])
array_two = np.array([1, 2, 4])
array_plus_array = array_one + array_two
array_plus_array
array([5, 5, 5])

In this case, addition is again performed element-wise. Each element in array_one is added to a corresponding element in array_two. The fact that the array performs useful math in this context might seem unremarkable but remember how the native Python list behaves.

list_one = [4, 3, 1]
list_two = [1, 2, 4]
list_plus_list = list_one + list_two
list_plus_list
[3, 2, 1, 1, 2, 4]
array_plus_array = np.array(list_one) + np.array(list_two)
array_plus_array
array([5, 5, 5])

If you are in full numerical computation mode this behavior of list might seem stupid to you. But remember: Python is a general purpose programming language and list is a general purpose container to store a sequence of objects. There could be anything in those lists and addition might not be a meaningful operation for those objects. This behavior always works, a list can be concatenated to another list regardless of the objects they store. That’s why we have NumPy. Python has to implement objects in a way that suits its general purpose. NumPy implements behavior in a way that we would expect while we do numerical stuff.

### A word on performance

This is one of the rare occasions where it is worthwhile to talk about performance. When you are getting started, I strongly recommend against thinking too much about performance. Write functioning code first, then worry about readability, maintainability, reproducibility etc. etc. and worry about performance last (trust me on this one). But some of you will be working with large amounts of data and you will be delighted to hear that NumPy is much faster than native Python.

my_array = np.random.rand(100000)  # A large array with 100000 elements
my_list = list(my_array)
timeit sum(my_list)
18.1 ms ± 801 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
timeit my_array.sum()
90.3 µs ± 6.86 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

The native Python version of sum is orders of magnitude slower than the NumPy version. You might have noticed that I created a very large array to demonstrate this. Actually the performance difference will increase with increasing array size, you can verify this for yourself. The take home message here is that whenever you can replace native Python with NumPy, you gain performance. But don’t worry about optimizing your NumPy code. One exception is the .append method, but more on that later.

### Summary

We learned two essential things and one kind of interesting side-note. The first essential lesson is that arrays come with many methods that allow us to do useful math. We learned some of those methods and as you keep working with NumPy those will become second nature. The second thing we learned is that arithmetic operators are applied element-wise to arrays. This means that a scalar value is applied to each element in an array and whenever two arrays of the same shape appear together with an operator each element is applied to each corresponding element. We will learn the details of array shapes in the next blog post. Finally, we also learned that NumPy code is almost always much faster than native Python code. This is good to know. However, especially in the beginning you should focus on anything but performance.


text:
## Introduction

Going far beyond the obvious macroscopic structure of the brain, which hardly differs between human individuals, various authors postulated a coupling of structure and function1,2,3. Neurons, hubs, or in graph theory called nodes, with similar connection patterns often show similar functionality4,5. On the contrary, several studies have repeatedly shown that the brains of patients suffering from schizophrenia or other neurological diseases, have a different connectome than healthy people6,7. This is particularly evident in patients suffering from severe stroke, other lesions within the central nervous system or after a massive hemispherectomy, in which the entire right hemisphere has been removed. Despite these damages, several patients have been able to learn to speak again, and they acquired all language-related abilities8. Although their brains showed an altered structure, it included strongly preserved parts of the initial brain regions responsible for language processing8,9.

Nevertheless, and this inevitably leads to a highly discussed paradox and known as the structure dilemma2, it must be acknowledged that this structure is not exclusive. There is no ‘master plan’ of the brain, no fixed wiring diagram, nor a completely determined structure defined in each individual’s genetic code. While genetics ensures that essential, mainly macroscopic, structures develop in an appropriate spatiotemporal pattern, subsequent refining steps, as the wiring between individual neurons, are thought to be influenced to a large extent by randomness. Early anatomical studies of different brain regions support the notion of a high degree of randomness during network formation in the developing brain10,11,12,13 and the ongoing learning process in the adult brain5. Braitenberg and Schüz even describe the cortex as a ‘mixing device’, whose connections are set up in a largely random manner14. Probably the most spectacular example of a healthy human brain is that of a 44 year-old man with a massive ventricular enlargement resulting in an grossly altered structure of the entire brain15.

Indeed, also artificial neural networks (ANNs) have revealed similar representations for both sides of the dilemma16. On the one hand, calculations with ANNs, as for example random initialization methods, suggest a high degree of randomness, but it has also been found that subnets or, in other words, a pre-set structure, take over problem solving17,18. These subnetworks, in this recent publication described as winning tickets17, ‘win the initialization lottery’, as their initial weights were able to solve problems with unsurpassed accuracy. Since artificial neural networks are confronted with ever more complex problems, whereas computing power is not unlimited, techniques to reduce the general network size have been developed19,20. An elimination of unnecessary weights from a neural network is called pruning and indeed, it has been shown that the network size can be reduced by more than 90$$\%$$ without a significant loss of accuracy17,21,22. These phenomena are based on the findings that not all parts of a network are equally important. Based on these ideas, we considered these sparse (in contrast to dense networks, in which every neuron is connected to each other) artificial neural networks as graphs and separated the raw structure. That means that the connections within the network are only defined as activating, inhibiting or non-existing (illustrated in Fig. 1). Consequently, in this paper the ‘structure’ always refers to the specific connections within these sparse networks and therefore is considered as the graph theoretical connectome of the artificial neural network. The biological connectome was likewise described and analysed as a graph2,23,24,25, which allows a comparison and analysis of both network structures.

The aim of this paper is to compare information processing in biological neural networks with that in ANNs, to get a deeper insight into the structural dilemma of both, types of neural networks. For a valid comparison and an interdisciplinary approach we carried out diverse experiments with artificial neural networks, including two feed-forward and two recurrent architectures, as well as two biologically-based neural networks. As biological networks we used the wiring diagram of the male C. elegans26, as the only living organism whose neural system has been mapped in its entirety, representing the micro-connectome level, and second, the functional connectome data of healthy patients representing the macro-connectome level27. Experiments for all networks were performed with different initialization and pruning methods, on different training sets, as well as with disturbed or blocked weights and bias values during the training process. The bias value is a parameter in the calculations of artificial neural networks that provides an additive offset for each neuron to adjust its sensitivity to its activation. To avoid any misunderstandings in the context of the word bias, we will refer to it as the ‘neuron offset value’.

Through this systematic analysis we wanted to gain a deeper insight into the relevance of the structure for information processing and to understand the dilemma from a technical perspective.

## Results

### Structure contains the information

We mentioned in the introduction that in biological systems the neuronal structure apparently contributes considerably to information processing. When transferring this approach to artificial neural networks, three questions arise: (1) how much information does the structure of an ANN contain related to the weights, (2) does a functional biological network with its distinct structure can be translated in an artificial neural network, and (3) which changes arise in relation to learning performance and characteristics, when computing these technically transferred networks? To answer these questions we performed several experiments and tried to implement biological connectomes as artificial neural networks. In the following we will refer to them as biological-based neural networks (BBNN). For this purpose, we first used the micro-connectome of chemical connections of C. elegans26 and transformed it in a recurrent and a feed-forward architecture with similar distributed path lengths (see “Methods” for a detailed description). Indeed, as the learning curves in Fig. 2A shows, the raw structure of the nematode neurons were also able to learn like an ANN. The feed-forward configuration is printed in dashed lines, whereas the solid lines show the recurrent architecture. The feed-forward architecture reaches an accuracy of approximately 90$$\%$$, whereas the recurrent configuration reaches 93$$\%$$ accuracy after 30 epochs of learning.

In addition, we initialized the network with either winning ticket17 (orange) or structure implantation (purple) to investigate whether the pre-trained structure contains enough or the same information as the weights themselves. In contrast to the winning ticket initialization, in which this structure including its weights has been implanted, the structure implantation only initialized the structure with fixed values for the categories exciting, inhibiting, and no connections (explained in Fig. 1), but without any distinct information about the weights. Despite the fact that weight information was not provided, both C. elegans configurations showed approximately the same accuracy (95$$\%$$ in the recurrent architecture), whereas the faster approach was particularly pronounced in the feed-forward architecture. This result is also obtained for the commonly used LeNet 4x300 (see Fig. 2B). Here, the faster approximation of a learned state, as the gradient of the accuracy became small and there was no improvement in subsequent learning epochs, is noteworthy. Other initialization methods as dense or sparse diversity (see “Methods”) did not show significant improvements in relation to the random glorot initialization (best case of learn curve sum $$\mathrm {max}~t=1.11$$ with $$\mathrm {min}~p=0.14$$). The significance level is $$5\%$$. A minor optimisation was obtained by lightning28, which outperforms the other random initialization methods (worst case of learn curve sum $$\mathrm {min}~t=6.75$$ with $$\mathrm {max}~p=1.26 \cdot 10^{-6}$$). Figure 2C shows the learning curve of a recurrent network in a random G(np) configuration with 1194 nodes. Although all three learning methods show approximately the same behaviour and final result, both winning ticket and raw display, in contrast to the structure implantation great fluctuations in their accuracy during learning iterations. This may be caused by a possible restriction due to the specification of too concrete weights, which could be exactly the wrong ones for this particular learning process state. It seems that an initialization with only the structure and no weight information is more successful. Therefore, it can be stated that the structure already contains all needed information.

To examine this in more detail, we trained eight different networks on two training data sets with different initialization and pruning methods and compared them by their best accuracy (Fig. 3). In particular, we used recurrent networks with 1194 and 177 nodes in a random G(np) configuration or with a small world architecture, respectively (the uniform and IRNN configurations are shown in the Supplementary Material). This decision was based on recent findings in biological and artificial neural networks. Brain networks indeed show the defined high clustering and short path length topology29 as demonstrated by various neuroimaging studies30 and calculations31. Also in a technical context, it has been shown that in feed-forward networks a small world architecture positively influences learning time and error probability32,33,34,35. In addition, we calculated two different BBNNs. First,the naturally occurring micro-connectome of C. elegans26, as well as in the previously described recurrent and its feed-forward configuration; and second, technically transferred human connectomes36 (the 177 node recurrent networks were the corresponding random reference).

Leading in the comparison of the best accuracy of recurrent networks were the networks with 1194 nodes, whereby the small world architecture performed even better (worst not alpha value inflation endangered case for MNIST $$\mathrm {min}~t=5.16$$ with $$\mathrm {max}~p=3.25 \cdot 10^{-5}$$; worst not alpha value inflation endangered case for Fashion-MNIST $$\mathrm {min}~t=6.61$$ with $$\mathrm {max}~p=1.67 \cdot 10^{-6}$$). The alpha inflation endangered cases are listed in Table 1, which based on $$m = 88$$ data accesses and the Bonferroni correction $$p \le \frac{\alpha }{m} = 0.057\%$$. These findings seem to put into perspective previous results of advantages of a network with small world architecture32,33. However, it should be kept in mind that we have only considered recurrent networks, but not the modified version of a feed-forward network, as it was used in e. p.34. The size of the network must also be taken into account here. The effect of a considerable improvement only occurred in larger networks. In the smaller 177 node configuration no consistent behavior in the accuracy could be observed as shown in Table 2, whereby here pruning had strong negative effects on the accuracy. The profitable benefits of this specialized structure seem to be cancelled out if the computing capacity is not large enough. Table 3 shows additionally the statistical t-test results on the best accuracy and the learn curves sum for the different architectures. In general structure implantation and winning ticket perform similar. Structure implantation performs better related to the learn curve sum, excepted for the small 177 architectures. In some cases winning ticket is better than structure implantation related to the best accuracy, but the differences are very small. The values are only definite for the small random networks (G(np) 177 and small world 177) where winning ticket is clearly better.

Another argument supporting this hypothesis concerns the results of C. elegans. In the ranking of the best accuracy, C. elegans in its unpruned configurations can be positioned directly after the 1194 nets (worst not alpha value inflation endangered significant case for MNIST $$\mathrm {min}~t=7.88$$ with $$\mathrm {max}~p=1.52 \cdot 10^{-7}$$; worst significant case for Fashion-MNIST $$\mathrm {min}~t=3.96$$ with $$\mathrm {max}~p=4.64 \cdot 10^{-4}$$). Table 4 shows the cases that are not significant. The unrestricted network showed the best results (worst case for MNIST $$\mathrm {min}~t=7.54$$ with $$\mathrm {max}~p=2.80 \cdot 10^{-7}$$; worst case for Fashion-MNIST $$\mathrm {min}~t=11.0$$ with $$\mathrm {max}~p=1.02 \cdot 10^{-9}$$), whereas in the pruning experiments without pre-trained initialization it displayed the largest decline (worst not alpha value inflation endangered case for MNIST $$\mathrm {min}~t=4.55$$ with $$\mathrm {max}~p=1.24 \cdot 10^{-4}$$; worst significant case for Fashion-MNIST $$\mathrm {min}~t=11.0$$ with $$\mathrm {max}~p=1.09 \cdot 10^{-9}$$). Table 5 shows the not significant cases. These results are understandable since the neural structure of C. elegans was not optimised to solve the MNIST problem. Neurobiological research postulated that a specialized structure is always coupled with a certain function2,3. However, the structure needs to be changed only slightly so that it can solve this synthetic task more effectively, as evident in comparison with the C. elegans G(np) reference variant (not significant cases are shown in Table 4). The use of the pre-trained information of structure implantation improves the performance for the raw C. elegans, thereby emphasizing the argument of only minimal changes as shown in Table 6.

A final argument underlining the importance of structure is illustrated in Fig. 4 (for a detailed description of the ‘toast plots’ and its analysis see “Methods”). It turned out that for each initialization the resulting structure was always analogous to the structure with which it was initialized, independent of the experimental settings.

This phenomenon was particularly conserved in recurrent networks. Here, only a minimum of edge changes could be detected in the entirety of the connections. Also the Spearman R coefficient, which is depicted in the lower left toast and illustrates the change of weights, shows that these structures also were almost completely similar in their active weights. The lack of correlation for not pruned and implanted information (winning ticket and structure implantation) results from the analysis of the primary (part of the structure) and secondary (not part of the structure) connections, since the secondary part, which is random, is not deactivated by pruning. It thus can be concluded that the weights will follow through the implanted structure.

To give a first summary, presenting the arguments for one side of the dilemma, it can be stated that it is possible to initialize a network with a pre-learned structure, missing the information of weights, with an equal accuracy and an improved learning behavior. In addition, the edge and weight comparison of unmodified, pruned and implanted models repeatedly reveal the fact that the structure holds the information about the weights itself. As it could be seen by the transformation of a biological network into an artificial network learning successfully, biological network structure is conserved and seems to play an important role. We were thus able to show that the importance of structure is not only a phenomenon in naturally occurring, but also in artificial information processing structures.

### Structure is not exclusive

Reconsidering the figures presented thus far, it must be noted, however, that these also contain arguments for the other side of the paradox. It could be seen that the structure—although being of considerable importance—is not exclusive. For instance, it is evident from Fig. 4 that the same structure was able to solve both problems, the MNIST and the more complex Fashion-MNIST. This is indicated by the extensive yellow areas in the upper right toast, which for the most part show only minimal changes in the connections. This effect of so-called transfer-learning has been observed in recurrent networks as well as in the feed-forward networks. It is remarkable that not only the structure was left unchanged during learning of an alternative problem, but that also the weights were nearly identical (indicated through dark black areas in the lower left toast). Therefore, a more or less randomly initialized network could solve two different problems with nearly the same structure and weights. The same applies to C. elegans. As it was mentioned in the previous section, the worm was not made to solve this synthetic problems, neither its sensory neurons were made to read image information, nor its motor neurons to output numbers. Nevertheless, also in its pruned configuration, it was able to solve this task.

Also in Fig. 3, some arguments underline the non-exclusivity of structures. Thus, if the bar graph is considered in its entirety, it shows that every network can solve the two problems. Only a minimal difference of less than 5% could be observed within one calculation experiment between the networks. Especially remarkable in this context were the random G(np) graphs. Although these had by definition random connections and thus randomized structures, they did not perform worse in solving the problems. On the other hand, even the more sophisticated networks showed very small improvements or even a worse behavior in their accuracy. This lead to the hypothesis that probably any structure can be used to solve any problem, as long as the network has enough capacity. Indeed, this finding is not entirely new. Honey and colleagues claimed that a variety of functional roles can be performed by ‘computational reservoirs’ with sufficient built-in complexity1. It should be emphasized that this theory was also discussed and supported on the biological side by various studies. For example, Prinz et al.37 simulated the pyloric rhythm of the crustacean stomatogastric nervous system and found that this even tightly regulated network can result from a wide range of different underlying mechanisms and parameters. Biological networks with structural38 and functional39 variability can perform the same functions. In addition, there are findings that neurons themselves rather than their connections are tuned in the learning process. Nonsynaptic changes of membrane components, such as a modulation of voltage-dependent membrane conductances, which results in a change in excitability, is produced by learning40.

Comparing structure and their weights alone misses the point of isomorphisms. A graph can be identically connected but on a different node setting. This may appear during the analyses of edge changes and the correlation of weights as a different network. The upper right toasts in Fig. 6 show the results of a simple isomorphism test. If two graphs are isomorph, they have an identical degree sequences over all nodes. A match indicates that both graphs could be isomorph, a mismatch denotes that the graphs were not isomorph (further information on the isomorphism test can be found in “Methods”). Figure 6 shows that in most cases the variants were not isomorph. Only few pairs displayed the possibility to be identical.

### The influence of the weights and the neuron offset value

Considering the results presented thus far showing the two sides of the paradox, the question of what influence the weights have on the results remains unresolved. In fact, in all of the networks investigated, the best possible accuracy between winning ticket and structure implantation was found to be very similar. Both problems, MNIST41 and Fashion-MNIST42, were solved with identical accuracy for each data set, which disclosed the relevance of the structure even without the weights. This was additionally supported by Fig. 4, showing that during learning only minimal changes were applied to the weights.

Indeed, a quite familiar observation was drawn from Gaier and Ha25 as they figured out the importance of the architecture in contrast to the previously established interest in the weights. To put this influence of the weights to the test, we performed experiments with either disturbed or blocked weights and neuron offset values, respectively. A disturbance of the value means that a normally distributed noise was added before each learning step. In blocked runs the disturbed values were not trainable (further information can be observed in the “Methods” section). Figure 5 shows the results of this experiment. Despite adding noise to the weights, the best accuracy decreased only slightly up to a level of approximately $$10^{-2}$$ (dark blue). If the added noise was greater than $$10^{-2}$$, the accuracy dropped rapidly. As can be seen in Fig. 5E,F on this scale a range of instability is recognizable. The solver is provoked to make changes that are even disproportionately stronger in the range over $$10^{-2}$$ than the actual change caused by the noise.

For the blocked runs, the behaviour is similar, except that the best accuracy is only at around 0.5. Here the role of the bias is also indicated. Although weight changes can no longer be made, the solver is still able to change the neuron offset value and thereby reaches an accuracy of 0.5. The role of bias should thus not be underestimated.

The neuron offset value was even less sensitive to the disturbances. In both cases, disturbed and blocked bias values, the best accuracy was stable up to a strength of $$10^{-2}$$. If the results from Fig. 5E,F are also taken together, it can be seen that the neuron offset values changed by the noise are hardly changed by the solver. Changes in the bias can be easily compensated by minimal changes in the more powerful weights, as can be seen in Fig. 5B. Until the system gets unstable, the weights remain constant in the relation to the starting point. Above the stability point the weight change gets smaller, because the solver is unable to find the correct countermeasure to compensate the disturbance. The initial configuration in the weights, which follow directly from the structure, thus can not be disturbed severely without the system becoming unstable, as can derived from Fig. 5A,B. With the described conclusion that the weights follow from the structure, it must therefore be noted that this solution has only a small domain.

This may be also the reason why only very few changes in the weights were detected in the learning process. However, if it turns out that neither the structure nor the resulting weights undergo major changes in the learning process, the question arises how artificial neural networks are able to learn at all. In principle, there are two possibilities. First, it is possible that even minimal changes could have enormous effects on the entire system, i.e. it could be a chaotic system. The other, perhaps more likely explanation could be an adjustment of the neuron offset value. The bottom left toasts of Fig. 6, in which the Spearman R coefficient of the neuron offset values are displayed, providing an exemplary answer to this question for four of our analysed networks. Independent of the network, the Spearman R correlation coefficient showed mostly very low values (recognizable by an almost completely dark blue coloration), indicating the dissimilarity between individual neuron offset values. This clearly demonstrated the change in those values caused by the learning process. Thus, it could be stated that adjustments were done by the solver not as commonly assumed largely through weight or structural changes, but through changes in the neuron offset value of each artificial neuron.

This behavior could be interpreted as an increasing or decreasing sensitivity of every individual neuron to its input values. This behaviour suggested that a change in the neuron offset value was sufficient to adjust the network effectively.

In a biological context, a possible analogy for this phenomenon can also be identified. A modulation through an intracellular storage of calcium can result in a short-term signal amplification as well as an adjustment of neuronal excitability43. Although the behaviour is similar, it is doubtful whether gradient-based learning as a mechanism can be compared with learning performed by true nervous systems. Therefore, the obviously similar behaviour might have a completely different cause.

Nonetheless, the further development of artificial neural networks can make use of known functions from nature in order to extend their functionality and effectiveness in a sustainable way. From a biological point of view, results, obtained through ongoing technical simulation experiments, may also contribute to the understanding of the learning process and the structural organisation of the brain.

## Discussion

In this paper we wanted to understand the relevance of the structure for information processing. Therefore, we analysed the behavior of artificial and biological-based networks based on technical neurons and their training methods. We were able to show that the structural dilemma, which is often resumed in biology, can also be found in ANNs. We figured out that an initial structure is indeed sufficient to improve learning accuracy, with minimal importance on the weights. Nevertheless, it seems to be of less relevance how this structure is organized. Contrary to expectations, a network based on biological structural principles does not improve learning success compared to randomly initialized structures. This raises the question why biological networks show precisely these structures. Based on our findings, it is questionable that these structures are made specifically for information processing. But, this follows the question why these structures are created that way and why deviations show pathogenic patterns. To answer these questions, further experiments will be carried out based on the parallels between artificial and biological-based neural networks presented in this paper. For this it is essential to include further information levels of the biological system. Further research is therefore aimed to model the technical neurons in a more biological way based on neural coding44,45, or to include electrical synapses.

## Methods

### Biological network data

For a multi-level analysis and comparison of biological networks we used two different datasets. First, the wiring diagram of the male C. elegans (updated dataset: https://wormwiring.org/series/ published in26), representing the micro-connectome level, and second, functional connectome data of healthy patients with varying age, gender and ethnical background, representing the macro-connectome level36, (directly accessible via: http://umcd.humanconnectomeproject.org/umcd/default/browse_studies)27. Both data sets are freely accessible online.

The single data set of the male C. elegans is composed of 369 neurons, 3888 connections through chemical synapses, 134 muscle connectors and 1018 muscle connections. The analysed adjacency matrix is bidirectional and weighted. More details about this dataset can be obtained from26.

984 data sets were included in the analysis of the macro-connectome, as 17 of the provided sets were not complete. In particular, the data records of “Beijing_Zang_newid182” with the sequential number up to 198 were affected, when we downloaded the data in August 2019. We processed the provided adjacency matrix with a threshold to obtain a mean degree of around 22. This results in a number of edges of 3964. The connectome graph was undirected but weighted. We ignored the weights, because we were only interested in the structure. The undirected graph was transformed to a directed one by randomly choosing the direction and type equally for all edges.

### Artificial neural network data

In order to make well-founded statements about the behaviour of ANNs, we used both feed-forward as well as recurrent networks. As feed-forward networks, we used the established structures of the LeNet 300-100 and the LeNet 4x30046. For the class of recurrent networks, we used random G(np) graphs with either 1194 nodes corresponding to the LeNet 300-100 architecture and a comparable mean degree, or 177 nodes analogous to the connectomes and also a corresponding mean degree. These graphs were calculated additionally with a directed small-world architecture.

The adjacency matrices of the human connectome were preprocessed first, as it was described in the previous section. For C. elegans, we used the naturally occurring network, provided from the original dataset, and a randomly connected version as G(np) with an equal number of nodes. A speciality of the C. elegans model are the separated layers for neurons (including sensor neurons) and muscle attachment points. The G(np) reference holded the muscle layer and connects the neuron layer randomly. The dataset of the C. elegans holded no information about excitatory and inhibitory neurons. Based on a calculated, theoretical optimal percentage of inhibitory synapses of 30%47 and a measured value of typical 20–30%48 in mammalian brains we assumed this percentage also for C. elegans. We trained 10 different datasets with a random value between 20 and 30% of inhibitory neurons. We saved this model, but did not perform a Monte Carlo simulation to avoid over-interpreting of the distribution of the inhibiting neurons.

### Training process

#### Initialization

In order to test the effect of different conditions at the beginning of learning, the nets were initialized differently. The methods included the randomized initialization glorot49, lightning28, winning ticket17 and two new implemented methods called structure implantation and diversity. In contrast to winning ticket, in which a pretrained structure including its weights has been implanted the structure implantation only implanted the structure with the categories exciting, inhibiting and no connections but without any information about the weights. This was done by using a threshold to decide whether or not an edge in the graph existed. This threshold was adjusted dynamically. The n strongest connections (positive and negative) were used to keep the average connectivity c constant. The existing edges were divided in positive and negative weights, corresponding to an exciting or inhibiting connection. Non-existing edges were 0, existing ones were set to the same value with the sign corresponding to the connection type. Low noise of a tenth of the connection value was added on top of the resulting adjacency or bipartite matrices.

The method of the diversity initialization should represent first degrees of kinship with biology. The amount of links showed a distribution in a specific range and not a constant value. The diversity initialization generated a set of neurons with a randomly chosen number of edges to random other neurons. For this paper this type of initialization was again divided into two subgroups:

Dense diversity:

very dense connected system defined by the in-going degree $$d_\text {in}$$.

\begin{aligned} d_\text {in} = {\left\{ \begin{array}{ll} \left[ {\mathcal {N}}\left( 100, 10\right) \right] &{} \quad \text {for } 35\% \text { off all existing Neurons}\\ \left[ {\mathcal {N}}\left( 50, 5\right) \right] &{} \quad \text {otherwise} \end{array}\right. } \end{aligned}
(1)
Sparse diversity:

sparse connected system defined by the in-going degree $$d_\text {in}$$.

\begin{aligned} d_\text {in} = {\left\{ \begin{array}{ll} \left[ {\mathcal {N}}\left( 8, 2\right) \right] &{} \quad \text {for } 80\% \text { off all existing Neurons}\\ \left[ {\mathcal {N}}\left( 25, 5\right) \right] &{} \quad \text {otherwise} \end{array}\right. } \end{aligned}
(2)

#### Datasets

The artificial neural networks were trained on either the MNIST41, Fashion-MNIST42 or the CIFAR-1050 (For CIFAR-10 the results are available in the Supplementary Materials) database. Recurrent networks iterated over a specific number of calculation steps. All recurrent networks were calculated for 25 steps and the input information was constantly set. The values of the datasets were normalized to the range of $$\left[ 0, 1\right]$$ as a preprocessing step.

#### Architectures

As already described in the beginning of this chapter, we used different ANNs for our calculations.

• LeNet 300-100: 784 input neurons followed by a dense layer with 300 neurons, a dense layer with 100 neurons and 10 output layer neurons containing the output information; only the output layer had a softmax activation function, whereas the others were activated through relu.

• LeNet 4x300: 784 input neurons followed by 4 dense layers with 300 neurons each and 10 output neurons; only the output layer had a softmax activation function, whereas the others were activated through relu.

• Recurrent 1194: 1194 neurons were activated through tanh; 784 of the 1194 were defined as input neurons, to generate the output 10 neurons were calculated with softmax in an additional output layer.

• Recurrent 117: The 784 input information of the datasets were encoded with an random encoder with a distribution of $${\mathcal {U}}(-0.05, 0.05)$$, the same encoder was used for all experiments, all neurons were input neurons, to generate the output 10 neurons were calculated with softmax in an additional output layer.

• C. elegans model: The 784 input information of the datasets were encoded with an random encoder with a distribution of $${\mathcal {U}}(-0.05, 0.05)$$ to 136 sensory neurons, which were included in the recurrent layer with 369 tanh activated neurons, which were mapped to the 134 tanh activated muscle attachment points, those were decoded to 10 output values which were calculated with softmax in an additional output layer. In general, it needs to be added that the given connectivity matrix values were, due to the absence of inhibiting connections and a to wide numerical range not suitable to use, which made the definition of other values necessary.

• C. elegans LeNet model: The en- and decoding is the same as in C. elegans. The minimal distances of every node to an output was calculated and based as mapping for the feed forward layers. The encoded inputs were also added based on the minimal distance to an output mapping. Thus, this model has several inputs on all four layers. The exact configuration is shown in Table 7. 22 nodes are not added to the model, because they are not connected in the C. elegans micro-connectome.

#### Pruning

We used two different pruning techniques: iterative pruning and the newly introduced bio pruning. During the iterative pruning the graph was analysed after each epoch and the pruning mask was updated to the new graph. In opposite to that the bio pruning was aimed to take up another idea from biology, the random based creating and destroying of connections. The solver was allowed to build up maximum $$\mathrm {Pois}(20)$$ new connections, delete maximum $$\mathrm {Pois}(20)$$ weak connections and could break 0 strong connections after each epoch. Within the number of connections, resulting from the average connectivity c, strong connections can be formed. Beyond of this, weak connections can be built up and destroyed. The average connectivity may therefore variate for the bio pruning method.

#### Training

The models were trained with the solver settings given in Table 8. Categorical cross entropy was used as loss value and accuracy as metric. All models were trained for 30 epochs with shuffled batches of the size 128 for the feed forward and 256 for the recurrent networks.

All variants that we trained have learned to solve the problems. We only figured out two exceptions, which were the pruned glorot initialized feed forward networks with a mean degree less than 15. This can be observed in Table 9. The table shows the behavior of the glorot initialization based on the mean degree for the LeNet 300-100.

A weakly connected component is the amount of nodes that have paths between each other independent of the edge directions. If the fraction of the largest weakly connected component $$S = 34.3\%$$, only a third of the neurons were connected to each other in maximum.

A reason for the learning deficiency of the pruned glorot variant could be the fact that the largest weakly connected component was to small. The other feed forward initialization (lightning, dense and sparse diversity) did not show this behaviour. The values of the diversity initialization were similar to lightning.

### Disturbed and blocked training

Gradient based learning tries to minimize the loss function l by adjusting the trainable parameters. This is achieved by following the gradient $$\frac{\partial l}{\partial x_i}$$. If the learning parameters are set well, the optimizer climbs down to a local minimum by every step. Based on the start parameter the local minimums can differ. By adding noise to the trainable parameters the optimizer is forced to react on this disturbance. Related to the strength of the noise a solution is possible by compensating the distribution or climbing down an other local minimum.

In our experiments the normal distributed noise $$\epsilon \sim {\mathcal {N}}(0, \sigma )$$ with a mean of 0 and a standard deviation of $$\sigma$$ is added to the values $$\chi$$ before the optimization in every batch. Weights and neuron offset are treated separately and were not disturbed together. Thus, the optimizer has to deal with the disturbed weights or altered neuron offsets. Additionally, to show the compensating of the optimizer, in blocked runs the disturbed values were not trainable. The optimizer has to compensate the disturbance only by adjusting the unblocked values. The unblocked values were not disturbed. As architecture a pruned recurrent 1194 with an uniform initialisation were used. Two metrics were used:

This metric shows the strength of the gradient related to the added distrubtion.

\begin{aligned} \gamma = \frac{\sqrt{\frac{1}{n}\sum _{i=1} ^n \frac{\partial l}{\partial \chi _i}}}{\sigma } \end{aligned}
(3)
Relative change rms:

This metric shows the effective change of the values to the initial state by the gradient related to the effective change of the noise over N batches.

\begin{aligned} \delta = \frac{\sqrt{\frac{1}{n}\sum _{i=1} ^n \Delta \chi _i}}{\sqrt{N} \sigma } \end{aligned}
(4)

### Possible graphs and isomorphism

The difficulty in calculating an isomorphism is demonstrated in Fig. 6d. Per definition, the connectome-based recurrent networks can not have isomorph graphs, because these networks have no free nodes that are not connected to an input or output. The isomorphism test checks only the possibility of an isomorphism in general. It cannot test against the specific conditions of fixed input and output nodes. So the algorithm tends to show a higher possibility that an isomprophism exists than actually would be the case. A discussion of the absolute changes in the graphs requires a reference in order to be able to estimate how large a change actually is. For this purpose different theoretical values of the graphs were calculated. The number of possible edges

\begin{aligned} M = \sum _{l = 0}^{L} n^{l}_\text {ingoing} \cdot n^{l}_\text {outgoing} \end{aligned}
(5)

with L number of layers and $$n^{l}_\text {ingoing}$$ or $$n^{l}_\text {outgoing}$$ number of nodes in- and outgoing of layer l defines the amount of all possible connections. The number of all possinble unique graphs

\begin{aligned} P\left( M, m \right) = \frac{M!}{m!} \end{aligned}
(6)

with M as number of all possible edges and m as number of existing edges, shows how many unique forms the graph could have. For this we assume that free nodes, which are not connected to an input or output, can permuted in their position. The last paramater is the number of possible isomorph graphs

\begin{aligned} I = \prod _{l = 0}^{L} f^{l}! \end{aligned}
(7)

with $$f^{l}$$ as number of free nodes in layer l.

Table 10 (calculated with https://www.wolframalpha.com/) showed the permutation and isomorphisms values for the different architectures, which were used in this paper. The selected architectures covered a wide range and differ in their basic properties. For example, the recurrent 177 architecture had relative few possible edges and unique graphs and no free nodes. In opposite, the recurrent 1194 architecture had the most possible edges and unique graphs but not as much isomorphisms as the LeNet 4x300 architecture.

### Analytical methods

#### Toast plots

To observe the changes in the trained model with simple metrics, we focus on the change of the weights during the learning. Figure 7a shows the distribution of the weight dependencies between different states of the model. By comparing two independent initialized states the weights show no logic and distributed equally corresponding to the different layers. The comparison of the initial to the corresponding trained state shows the complete opposite. The post training weights depend directly on the initialisation. Figure 7b show the cumulative distribution function of the weight change between the different states of Fig. 7a. It demonstrates the gap between randomized and depending trained weights.

For an optimized presentation of the extensive analytical results we developed our so-called toast plots. Figure 8 shows the two types we used in this paper. They offer the advantage that several analyses as well as runs can be recorded and compared at a glance. One plot is built out of two toasts. In particularly, type 1 (see Fig. 8a) displays the Spearman R coefficient of the edge changes, meaning how similar the weights of the edges are compared to other runs. The second toast of that plot shows the number of odd edges compared to other runs. Type 2 (see Fig. 8b) shows the Spearman R coefficient of the bias values and the results of an isomorphism test (see section ‘Possible graphs and isomorphism’ for further details).

Each run is marked with a one to three letter abbreviation at the side of the graph. Each of these include four different subvariants, entailing the data set they were trained with (see more details in training section) and if it is the initial or the trained status. The abbreviations given in the legends describe how the net was initialized and pruned. In detail these are: R—reference, g—Glorot, L—lightning , D—dense diversity, S—sparse diversity initialization; c—connectome, C—C. elegans; I—iterative pruning, B—bio pruning; W—winning ticket, i—structure implantation. A detailed explanation of the abbreviations and setups are given in the training process section.