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1.
Front Physiol ; 14: 1162807, 2023.
Article in English | MEDLINE | ID: mdl-37408588

ABSTRACT

The distribution of octopuses within the Octopus vulgaris species complex remains inadequately understood. Species determination can be complex and involves characterizing a specimen's physical features and comparing its genetic makeup to other populations. In this study, we present the first genetic confirmation of Octopus insularis (Leite and Haimovici, 2008) inhabiting the coastal waters of the Florida Keys, United States. We employed visual observations to identify species-specific body patterns of three wild-caught octopuses and used de novo genome assembly to confirm their species. All three specimens exhibited a red/white reticulated pattern on their ventral arm surface. Two specimens displayed body pattern components of deimatic display (white eye encircled by a light ring, with darkening around the eye). All visual observations were consistent with distinguishing features of O. insularis. We then compared mitochondrial subunits COI, COIII, and 16S in these specimens across all available annotated octopod sequences, including Sepia apama (Hotaling et al., 2021) as a control outgroup taxon. For species exhibiting intraspecific genomic variation, we included multiple sequences from geographically distinct populations. Laboratory specimens consistently clustered into a single taxonomic node with O. insularis. These findings confirm O. insularis presence in South Florida and suggest a more extensive northern distribution than previously assumed. Whole genome Illumina sequencing of multiple specimens enabled taxonomic identification with well-established DNA barcodes while also generating the first de novo full assembly of O. insularis. Furthermore, constructing and comparing phylogenetic trees for multiple conserved genes is essential for confirming the presence and delineation of cryptic species in the Caribbean.

2.
Sci Rep ; 13(1): 11028, 2023 07 07.
Article in English | MEDLINE | ID: mdl-37419931

ABSTRACT

Geographic variation in the vocal behavior of manatees has been reported but is largely unexplored. Vocalizations of wild West Indian manatees (Trichechus manatus) were recorded with hydrophones in Florida from Florida manatees (Trichechus manatus latirostris), and in Belize and Panama from Antillean manatees (Trichechus manatus manatus) to determine if calls varied between subspecies and geographic regions. Calls were visually classified into five categories: squeaks, high squeaks, squeals, squeak-squeals, and chirps. From these five categories, only three call types (squeaks, high squeaks and squeals) were observed in all three populations. Six parameters from the temporal and frequency domains were measured from the fundamental frequency of 2878 manatee vocalizations. A repeated measures PERMANOVA found significant differences for squeaks and high squeaks between each geographic location and for squeals between Belize and Florida. Almost all measured frequency and temporal parameters of manatee vocalizations differed between and within subspecies. Variables that may have influenced the variation observed may be related to sex, body size, habitat and/or other factors. Our findings provide critical information of manatee calls for wildlife monitoring and highlight the need for further study of the vocal behavior of manatees throughout their range.


Subject(s)
Trichechus manatus , Vocalization, Animal , Trichechus manatus/physiology , Animals , Body Size , Americas , Atlantic Ocean , Ecosystem
3.
Zoo Biol ; 42(6): 723-729, 2023.
Article in English | MEDLINE | ID: mdl-37283165

ABSTRACT

Captive animals typically develop anticipatory behaviors, actions of increased frequency done in anticipation of an event such as feeding. Anticipatory behaviors can be an indicator of an animal's welfare. However, for rehabilitating animals that are expected to be reintroduced into the wild, these behaviors need to be extinguished to ensure successful release. Scheduled activities such as feeding occur daily and vocalizations could potentially be used to identify anticipatory behavior. Here, we tested the hypothesis that manatee calves modify their vocal production rate as a form of anticipatory behavior. Vocalizations of two Antillean manatee (Trichechus manatus manatus) calves were recorded for 10 min before, during, and after feeding sessions at Wildtracks, a manatee rehabilitation center in Belize. The number of calls were counted across recording sessions and three acoustic parameters were measured from calls including duration, frequency modulation, and center frequency. A repeated measures ANOVA comparing the number of calls across sessions indicated manatees produced significantly more calls before feeding sessions than during and after sessions. In addition, manatees increased the duration and lowered the frequency of calls before feeding sessions. This information can give further insight on ways to improve rehabilitation protocols and manage human interactions to increase the overall survival rate of rehabilitated manatees when released back into the wild.


Subject(s)
Trichechus manatus , Humans , Animals , Cattle , Animals, Zoo
4.
Entropy (Basel) ; 24(5)2022 Apr 23.
Article in English | MEDLINE | ID: mdl-35626476

ABSTRACT

In theoretical biology, robustness refers to the ability of a biological system to function properly even under perturbation of basic parameters (e.g., temperature or pH), which in mathematical models is reflected in not needing to fine-tune basic parameter constants; flexibility refers to the ability of a system to switch functions or behaviors easily and effortlessly. While there are extensive explorations of the concept of robustness and what it requires mathematically, understanding flexibility has proven more elusive, as well as also elucidating the apparent opposition between what is required mathematically for models to implement either. In this paper we address a number of arguments in theoretical neuroscience showing that both robustness and flexibility can be attained by systems that poise themselves at the onset of a large number of dynamical bifurcations, or dynamical criticality, and how such poising can have a profound influence on integration of information processing and function. Finally, we examine critical map lattices, which are coupled map lattices where the coupling is dynamically critical in the sense of having purely imaginary eigenvalues. We show that these map lattices provide an explicit connection between dynamical criticality in the sense we have used and "edge of chaos" criticality.

5.
Front Psychol ; 12: 689501, 2021.
Article in English | MEDLINE | ID: mdl-34621209

ABSTRACT

Humans use whistled communications, the most elaborate of which are commonly called "whistled languages" or "whistled speech" because they consist of a natural type of speech. The principle of whistled speech is straightforward: people articulate words while whistling and thereby transform spoken utterances by simplifying them, syllable by syllable, into whistled melodies. One of the most striking aspects of this whistled transformation of words is that it remains intelligible to trained speakers, despite a reduced acoustic channel to convey meaning. It constitutes a natural traditional means of telecommunication that permits spoken communication at long distances in a large diversity of languages of the world. Historically, birdsong has been used as a model for vocal learning and language. But conversely, human whistled languages can serve as a model for elucidating how information may be encoded in dolphin whistle communication. In this paper, we elucidate the reasons why human whistled speech and dolphin whistles are interesting to compare. Both are characterized by similar acoustic parameters and serve a common purpose of long distance communication in natural surroundings in two large brained social species. Moreover, their differences - e.g., how they are produced, the dynamics of the whistles, and the types of information they convey - are not barriers to such a comparison. On the contrary, by exploring the structure and attributes found across human whistle languages, we highlight that they can provide an important model as to how complex information is and can be encoded in what appears at first sight to be simple whistled modulated signals. Observing details, such as processes of segmentation and coarticulation, in whistled speech can serve to advance and inform the development of new approaches for the analysis of whistle repertoires of dolphins, and eventually other species. Human whistled languages and dolphin whistles could serve as complementary test benches for the development of new methodologies and algorithms for decoding whistled communication signals by providing new perspectives on how information may be encoded structurally and organizationally.

6.
Sci Rep ; 11(1): 5183, 2021 03 04.
Article in English | MEDLINE | ID: mdl-33664380

ABSTRACT

During the COVID-19 pandemic, the scientific community developed predictive models to evaluate potential governmental interventions. However, the analysis of the effects these interventions had is less advanced. Here, we propose a data-driven framework to assess these effects retrospectively. We use a regularized regression to find a parsimonious model that fits the data with the least changes in the [Formula: see text] parameter. Then, we postulate each jump in [Formula: see text] as the effect of an intervention. Following the do-operator prescriptions, we simulate the counterfactual case by forcing [Formula: see text] to stay at the pre-jump value. We then attribute a value to the intervention from the difference between true evolution and simulated counterfactual. We show that the recommendation to use facemasks for all activities would reduce the number of cases by 200,000 ([Formula: see text] CI 190,000-210,000) in Connecticut, Massachusetts, and New York State. The framework presented here might be used in any case where cause and effects are sparse in time.


Subject(s)
COVID-19/prevention & control , Masks , Models, Statistical , Physical Distancing , COVID-19/epidemiology , Humans , Retrospective Studies , United States/epidemiology
7.
PLoS One ; 15(6): e0235155, 2020.
Article in English | MEDLINE | ID: mdl-32584861

ABSTRACT

Tracking the origin of propagating wave signals in an environment with complex reflective surfaces is, in its full generality, a nearly intractable problem which has engendered multiple domain-specific literatures. We posit that, if the environment and sensor geometries are fixed, machine learning algorithms can "learn" the acoustical geometry of the environment and accurately track signal origin. In this paper, we propose the first machine-learning-based approach to identifying the source locations of semi-stationary, tonal, dolphin-whistle-like sounds in a highly reverberant space, specifically a half-cylindrical dolphin pool. Our algorithm works by supplying a learning network with an overabundance of location "clues", which are then selected under supervised training for their ability to discriminate source location in this particular environment. More specifically, we deliver estimated time-difference-of-arrivals (TDOA's) and normalized cross-correlation values computed from pairs of hydrophone signals to a random forest model for high-feature-volume classification and feature selection, and subsequently deliver the selected features into linear discriminant analysis, linear and quadratic Support Vector Machine (SVM), and Gaussian process models. Based on data from 14 sound source locations and 16 hydrophones, our classification models yielded perfect accuracy at predicting novel sound source locations. Our regression models yielded better accuracy than the established Steered-Response Power (SRP) method when all training data were used, and comparable accuracy along the pool surface when deprived of training data at testing sites; our methods additionally boast improved computation time and the potential for superior localization accuracy in all dimensions with more training data. Because of the generality of our method we argue it may be useful in a much wider variety of contexts.


Subject(s)
Acoustics , Dolphins/physiology , Machine Learning , Vocalization, Animal , Animals
8.
J Acoust Soc Am ; 147(2): EL80, 2020 02.
Article in English | MEDLINE | ID: mdl-32113273

ABSTRACT

Antillean manatees produce vocalizations reported to be important for communication, but their vocal behavior throughout their geographic range is poorly understood. A SoundTrap recorder (sample rates: 288/576 kHz) was deployed in Belize to record vocalizations of wild manatees in a seagrass channel and of a young rehabilitated and released manatee in a shallow lagoon. Spectral analysis revealed broadband vocalizations with frequencies up to 150 kHz and a high proportion of calls with ultrasonic components. Ultrasonic frequency components appear prevalent in their vocal repertoire and may be important to manatee communication.

9.
Sci Rep ; 9(1): 18643, 2019 Dec 04.
Article in English | MEDLINE | ID: mdl-31796884

ABSTRACT

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

10.
Sci Rep ; 9(1): 4927, 2019 03 20.
Article in English | MEDLINE | ID: mdl-30894626

ABSTRACT

In daily life, in the operating room and in the laboratory, the operational way to assess wakefulness and consciousness is through responsiveness. A number of studies suggest that the awake, conscious state is not the default behavior of an assembly of neurons, but rather a very special state of activity that has to be actively maintained and curated to support its functional properties. Thus responsiveness is a feature that requires active maintenance, such as a homeostatic mechanism to balance excitation and inhibition. In this work we developed a method for monitoring such maintenance processes, focusing on a specific signature of their behavior derived from the theory of dynamical systems: stability analysis of dynamical modes. When such mechanisms are at work, their modes of activity are at marginal stability, neither damped (stable) nor exponentially growing (unstable) but rather hovering in between. We have previously shown that, conversely, under induction of anesthesia those modes become more stable and thus less responsive, then reversed upon emergence to wakefulness. We take advantage of this effect to build a single-trial classifier which detects whether a subject is awake or unconscious achieving high performance. We show that our approach can be developed into a means for intra-operative monitoring of the depth of anesthesia, an application of fundamental importance to modern clinical practice.


Subject(s)
Awareness/drug effects , Brain/drug effects , Consciousness/drug effects , Hypnotics and Sedatives/pharmacology , Ketamine/pharmacology , Propofol/pharmacology , Anesthesia/methods , Animals , Awareness/physiology , Brain/physiology , Consciousness/physiology , Cortical Excitability/drug effects , Cortical Excitability/physiology , Electrocorticography , Electrodes, Implanted , Haplorhini , Male , Monitoring, Intraoperative/methods , Neural Inhibition/drug effects , Neural Inhibition/physiology , Neurons/drug effects , Neurons/physiology , Unconsciousness/chemically induced , Unconsciousness/diagnosis , Unconsciousness/psychology , Video Recording , Wakefulness/drug effects , Wakefulness/physiology
11.
Chaos ; 28(9): 093102, 2018 Sep.
Article in English | MEDLINE | ID: mdl-30278628

ABSTRACT

We investigate a critically-coupled chain of nonlinear oscillators, whose dynamics displays complex spatiotemporal patterns of activity, including regimes in which glider-like coherent excitations move about and interact. The units in the network are identical simple neural circuits whose dynamics is given by the Wilson-Cowan model and are arranged in space along a one-dimensional lattice with nearest neighbor interactions. The interactions follow an alternating sign rule, and hence the "synaptic matrix" M embodying them is tridiagonal antisymmetric and has purely imaginary (critical) eigenvalues. The model illustrates the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a "critical" synaptic matrix, i.e., having purely imaginary eigenvalues. In order to identify the dynamical underpinnings of these behaviors, we explored a discrete-time coupled-map lattice inspired by our system: the dynamics of the units is dictated by a chaotic map of the interval, and the interactions are given by allowing the critical coupling to act for a finite period τ , thus given by a unitary matrix U = exp ⁡ ( τ 2 M ) . It is now explicit that such critical couplings are volume-preserving in the sense of Liouville's theorem. We show that this map is also capable of producing a variety of complex spatiotemporal patterns including gliders, like our original chain of neural circuits. Our results suggest that if the units in isolation are capable of featuring multiple dynamical states, then local critical couplings lead to a wide variety of emergent spatiotemporal phenomena.

12.
Development ; 144(9): 1725-1734, 2017 05 01.
Article in English | MEDLINE | ID: mdl-28465336

ABSTRACT

Epithelial remodeling determines the structure of many organs in the body through changes in cell shape, polarity and behavior and is a major area of study in developmental biology. Accurate and high-throughput methods are necessary to systematically analyze epithelial organization and dynamics at single-cell resolution. We developed SEGGA, an easy-to-use software for automated image segmentation, cell tracking and quantitative analysis of cell shape, polarity and behavior in epithelial tissues. SEGGA is free, open source, and provides a full suite of tools that allow users with no prior computational expertise to independently perform all steps of automated image segmentation, semi-automated user-guided error correction, and data analysis. Here we use SEGGA to analyze changes in cell shape, cell interactions and planar polarity during convergent extension in the Drosophila embryo. These studies demonstrate that planar polarity is rapidly established in a spatiotemporally regulated pattern that is dynamically remodeled in response to changes in cell orientation. These findings reveal an unexpected plasticity that maintains coordinated planar polarity in actively moving populations through the continual realignment of cell polarity with the tissue axes.


Subject(s)
Cell Polarity , Cytological Techniques/methods , Epithelial Cells/cytology , Software , Animals , Automation , Cell Shape , Cell Tracking , Drosophila melanogaster/cytology , Drosophila melanogaster/embryology , Embryo, Nonmammalian/cytology , Epithelial Cells/metabolism , Genotype , Image Processing, Computer-Assisted
13.
Eur J Neurosci ; 43(6): 751-64, 2016 Mar.
Article in English | MEDLINE | ID: mdl-26663571

ABSTRACT

Natural auditory scenes possess highly structured statistical regularities, which are dictated by the physics of sound production in nature, such as scale-invariance. We recently identified that natural water sounds exhibit a particular type of scale invariance, in which the temporal modulation within spectral bands scales with the centre frequency of the band. Here, we tested how neurons in the mammalian primary auditory cortex encode sounds that exhibit this property, but differ in their statistical parameters. The stimuli varied in spectro-temporal density and cyclo-temporal statistics over several orders of magnitude, corresponding to a range of water-like percepts, from pattering of rain to a slow stream. We recorded neuronal activity in the primary auditory cortex of awake rats presented with these stimuli. The responses of the majority of individual neurons were selective for a subset of stimuli with specific statistics. However, as a neuronal population, the responses were remarkably stable over large changes in stimulus statistics, exhibiting a similar range in firing rate, response strength, variability and information rate, and only minor variation in receptive field parameters. This pattern of neuronal responses suggests a potentially general principle for cortical encoding of complex acoustic scenes: while individual cortical neurons exhibit selectivity for specific statistical features, a neuronal population preserves a constant response structure across a broad range of statistical parameters.


Subject(s)
Auditory Cortex/physiology , Auditory Perception , Models, Neurological , Acoustic Stimulation , Animals , Auditory Cortex/cytology , Evoked Potentials, Auditory , Male , Neurons/physiology , Rats , Rats, Long-Evans , Rats, Sprague-Dawley
14.
Article in English | MEDLINE | ID: mdl-26274219

ABSTRACT

Transport networks play a key role across four realms of eukaryotic life: slime molds, fungi, plants, and animals. In addition to the developmental algorithms that build them, many also employ adaptive strategies to respond to stimuli, damage, and other environmental changes. We model these adapting network architectures using a generic dynamical system on weighted graphs and find in simulation that these networks ultimately develop a hierarchical organization of the final weighted architecture accompanied by the formation of a system-spanning backbone. In addition, we find that the long term equilibration dynamics exhibit behavior reminiscent of glassy systems characterized by long periods of slow changes punctuated by bursts of reorganization events.


Subject(s)
Models, Theoretical , Computer Simulation , Datasets as Topic , Feedback , Time Factors
15.
J Neurosci ; 35(30): 10866-77, 2015 Jul 29.
Article in English | MEDLINE | ID: mdl-26224868

ABSTRACT

What aspects of neuronal activity distinguish the conscious from the unconscious brain? This has been a subject of intense interest and debate since the early days of neurophysiology. However, as any practicing anesthesiologist can attest, it is currently not possible to reliably distinguish a conscious state from an unconscious one on the basis of brain activity. Here we approach this problem from the perspective of dynamical systems theory. We argue that the brain, as a dynamical system, is self-regulated at the boundary between stable and unstable regimes, allowing it in particular to maintain high susceptibility to stimuli. To test this hypothesis, we performed stability analysis of high-density electrocorticography recordings covering an entire cerebral hemisphere in monkeys during reversible loss of consciousness. We show that, during loss of consciousness, the number of eigenmodes at the edge of instability decreases smoothly, independently of the type of anesthetic and specific features of brain activity. The eigenmodes drift back toward the unstable line during recovery of consciousness. Furthermore, we show that stability is an emergent phenomenon dependent on the correlations among activity in different cortical regions rather than signals taken in isolation. These findings support the conclusion that dynamics at the edge of instability are essential for maintaining consciousness and provide a novel and principled measure that distinguishes between the conscious and the unconscious brain. SIGNIFICANCE STATEMENT: What distinguishes brain activity during consciousness from that observed during unconsciousness? Answering this question has proven difficult because neither consciousness nor lack thereof have universal signatures in terms of most specific features of brain activity. For instance, different anesthetics induce different patterns of brain activity. We demonstrate that loss of consciousness is universally and reliably associated with stabilization of cortical dynamics regardless of the specific activity characteristics. To give an analogy, our analysis suggests that loss of consciousness is akin to depressing the damper pedal on the piano, which makes the sounds dissipate quicker regardless of the specific melody being played. This approach may prove useful in detecting consciousness on the basis of brain activity under anesthesia and other settings.


Subject(s)
Cerebral Cortex/physiology , Consciousness/physiology , Unconsciousness , Anesthetics/pharmacology , Animals , Cerebral Cortex/drug effects , Consciousness/drug effects , Electroencephalography , Haplorhini , Male , Signal Processing, Computer-Assisted
16.
Article in English | MEDLINE | ID: mdl-24723852

ABSTRACT

In this work we analyze electro-corticography (ECoG) recordings in human subjects during induction of anesthesia with propofol. We hypothesize that the decrease in responsiveness that defines the anesthetized state is concomitant with the stabilization of neuronal dynamics. To test this hypothesis, we performed a moving vector autoregressive analysis and quantified stability of neuronal dynamics using eigenmode decomposition of the autoregressive matrices, independently fitted to short sliding temporal windows. Consistent with the hypothesis we show that while the subject is awake, many modes of neuronal activity oscillations are found at the edge of instability. As the subject becomes anesthetized, we observe statistically significant increase in the stability of neuronal dynamics, most prominently observed for high frequency oscillations. Stabilization was not observed in phase randomized surrogates constructed to preserve the spectral signatures of each channel of neuronal activity. Thus, stability analysis offers a novel way of quantifying changes in neuronal activity that characterize loss of consciousness induced by general anesthetics.


Subject(s)
Anesthetics, Intravenous/pharmacology , Cerebral Cortex/drug effects , Neurons/drug effects , Propofol/pharmacology , Electroencephalography , Humans
17.
PLoS One ; 8(6): e65386, 2013.
Article in English | MEDLINE | ID: mdl-23799012

ABSTRACT

Time-reversal symmetry breaking is a key feature of many classes of natural sounds, originating in the physics of sound production. While attention has been paid to the response of the auditory system to "natural stimuli," very few psychophysical tests have been performed. We conduct psychophysical measurements of time-frequency acuity for stylized representations of "natural"-like notes (sharp attack, long decay) and the time-reversed versions of these notes (long attack, sharp decay). Our results demonstrate significantly greater precision, arising from enhanced temporal acuity, for such sounds over their time-reversed versions, without a corresponding decrease in frequency acuity. These data inveigh against models of auditory processing that include tradeoffs between temporal and frequency acuity, at least in the range of notes tested and suggest the existence of statistical priors for notes with a sharp-attack and a long-decay. We are additionally able to calculate a minimal theoretical bound on the sophistication of the nonlinearities in auditory processing. We find that among the best studied classes of nonlinear time-frequency representations, only matching pursuit, spectral derivatives, and reassigned spectrograms are able to satisfy this criterion.


Subject(s)
Acoustics , Humans , Psychophysics , Time
18.
Proc Natl Acad Sci U S A ; 110(16): E1438-43, 2013 Apr 16.
Article in English | MEDLINE | ID: mdl-23536302

ABSTRACT

Finding the first time a fluctuating quantity reaches a given boundary is a deceptively simple-looking problem of vast practical importance in physics, biology, chemistry, neuroscience, economics, and industrial engineering. Problems in which the bound to be traversed is itself a fluctuating function of time include widely studied problems in neural coding, such as neuronal integrators with irregular inputs and internal noise. We show that the probability p(t) that a Gauss-Markov process will first exceed the boundary at time t suffers a phase transition as a function of the roughness of the boundary, as measured by its Hölder exponent H. The critical value occurs when the roughness of the boundary equals the roughness of the process, so for diffusive processes the critical value is Hc = 1/2. For smoother boundaries, H > 1/2, the probability density is a continuous function of time. For rougher boundaries, H < 1/2, the probability is concentrated on a Cantor-like set of zero measure: the probability density becomes divergent, almost everywhere either zero or infinity. The critical point Hc = 1/2 corresponds to a widely studied case in the theory of neural coding, in which the external input integrated by a model neuron is a white-noise process, as in the case of uncorrelated but precisely balanced excitatory and inhibitory inputs. We argue that this transition corresponds to a sharp boundary between rate codes, in which the neural firing probability varies smoothly, and temporal codes, in which the neuron fires at sharply defined times regardless of the intensity of internal noise.


Subject(s)
Models, Biological , Neurons/metabolism , Synaptic Transmission/physiology , Computer Simulation , Diffusion , Neurons/physiology , Stochastic Processes , Time Factors
19.
Phys Rev Lett ; 110(4): 044301, 2013 Jan 25.
Article in English | MEDLINE | ID: mdl-25166166

ABSTRACT

The time-frequency uncertainty principle states that the product of the temporal and frequency extents of a signal cannot be smaller than 1/(4 π). We study human ability to simultaneously judge the frequency and the timing of a sound. Our subjects often exceeded the uncertainty limit, sometimes by more than tenfold, mostly through remarkable timing acuity. Our results establish a lower bound for the nonlinearity and complexity of the algorithms employed by our brains in parsing transient sounds, rule out simple "linear filter" models of early auditory processing, and highlight timing acuity as a central feature in auditory object processing.

20.
Phys Rev E Stat Nonlin Soft Matter Phys ; 86(2 Pt 1): 021134, 2012 Aug.
Article in English | MEDLINE | ID: mdl-23005749

ABSTRACT

Transport networks are found at the heart of myriad natural systems, yet are poorly understood, except for the case of river networks. The Scheidegger model, in which rivers are convergent random walks, has been studied only in the case of flat topography, ignoring the variety of curved geometries found in nature. Embedding this model on a cone, we find a convergent and a divergent phase, corresponding to few, long basins and many, short basins, respectively, separated by a singularity, indicating a phase transition. Quantifying basin shape using Hacks law l ~ a(h) gives distinct values for h, providing a method of testing our hypotheses. The generality of our model suggests implications for vascular morphology, in particular, differing number and shapes of arterial and venous trees.

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