pith. sign in

arxiv: 2604.19184 · v1 · submitted 2026-04-21 · 🧮 math.PR

Growing random planar network with oriented branching and fusion

Vincent Bansaye , Gael Raoul , Milica Tomasevic This is my paper

Pith reviewed 2026-05-10 02:37 UTC · model grok-4.3

classification 🧮 math.PR
keywords growing planar networksorthogonal branchingbranching processesstick breakingempirical measure convergencerandom growth modelsspatial branchingdouble immigration
0
0 comments X

The pith

Rectangles in this growing network converge after rescaling to an explicit limit from a stick-breaking process with aging.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a planar network whose tips extend at constant speed, produce orthogonal branches at a fixed rate always in one direction, and stop upon first intersection with an existing branch. This geometric rule creates a spatial branching property, so the connected components behave exactly as a branching process of rectangles that receive immigrants from both ends. A spine construction applied to a typical rectangle, together with coupling arguments, reduces the problem to a one-dimensional stick-breaking model that incorporates aging. From this reduced process the authors derive long-time convergence of the empirical measure of all rectangles after polynomial rescaling, together with an explicit expression for the limiting distribution and the speed of convergence. The argument also rests on an explicit description of common ancestors within the rectangle branching structure.

Core claim

Using a spine approach for a typical rectangle and coupling arguments, the study reduces to a one-dimensional stick breaking model with aging. We can then prove long time convergence of empirical measure of the family of rectangles after polynomial rescaling. The limiting distribution and speed of convergence can be explicitly described. The proofs also rely on the description of common ancestor of rectangles in the branching structure with double immigration.

What carries the argument

the branching process of rectangles with double immigration, reduced by spine and coupling to a one-dimensional stick-breaking model with aging

If this is right

  • The empirical measure of rectangle sizes converges after polynomial rescaling to an explicit limiting distribution.
  • The speed of this convergence is also given explicitly.
  • Common ancestors of rectangles can be identified inside the branching structure with double immigration.
  • The long-time geometry of the network is characterized by the limiting law obtained from the reduced stick-breaking process.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same spine-plus-coupling reduction could be attempted on variants that relax the fixed-direction rule while preserving a comparable spatial property.
  • The explicit limit supplies a concrete benchmark against which other fragmentation or growth models on the plane can be compared.
  • Tracking the aspect ratios of rectangles in a large simulation would give a direct numerical test of the predicted convergence rate.

Load-bearing premise

Branching is always orthogonal and always occurs in the same fixed direction, which produces the spatial branching property that lets the connected components be modeled exactly as a branching process of rectangles with double immigration.

What would settle it

A long-time numerical simulation of the network in which the histogram of polynomially rescaled rectangle lengths fails to approach the distribution predicted by the stick-breaking model with aging.

Figures

Figures reproduced from arXiv: 2604.19184 by Gael Raoul, Milica Tomasevic, Vincent Bansaye.

Figure 1
Figure 1. Figure 1: A simulation of the process: (a) the constructed network at [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Notations introduced in Section 2.1 The direct ancestor u − of u is u − = (u1, . . . , un−1) = u ⊥ if un = 1; u − = (u1, . . . , un − 1) if un > 1. In particular, the first branch created is labeled by u = (1) and it continues as u→ = (2) and its orthogonal offspring is u↷ = (11), whose orthogonal ancestor is also the direct ancestor : (u↷) ⊥ = (u↷) − = (1). Note that the notations introduced in this Secti… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the fusion events described in Section [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the notations introduced in Section [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: In (a), we represent the rectangles U↷ r at time t (see [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: This figure illustrates the proof of Lemma [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: In this picture we represent the fusion of the active tip [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Construction of the process Y from Z1 and Z2. Freezing are indicated by stars; the choice of the right fragment in the stick breaking is indicated by → and the choice left fragment by ↷. Proposition 3.1. Writing τi = inf{t ≥ 0 : A i t ≥ L i t} for i ∈ {1, 2}, we have τ ≤ τ1 + τ2, Lτ ≥ L 1 τ1 , ℓτ ≥ L 2 τ2 a.s. Proof. We remark that the rectangle is frozen at time τ when the age reaches the boundary in one … view at source ↗
Figure 9
Figure 9. Figure 9: For the network simulation represented in Figure [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Properties of the genealogical tree represented in Figure [PITH_FULL_IMAGE:figures/full_fig_p036_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Simulation with branching on both sides, for three possible generalisations of the model [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
read the original abstract

We consider a growing planar network where a tip grows at constant speed, branches at constant rate and inactivates when it meets a branch already created. We only consider here orthogonal branching occurring always in the same direction. This yields a spatial branching property to the growing network. The connected components of the network then form a branching process of rectangles with double immigration. Using a spine approach for a typical rectangle and coupling arguments, the study is boiled down to a one dimensional stick breaking model with aging. We can then prove long time convergence of empirical measure of the family of rectangles after polynomial rescaling. The limiting distribution and speed of convergence can be explicitly described. The proofs also rely on the description of common ancestor of rectangles in the branching structure with double immigration.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces a growing planar network model in which tips advance at constant speed, branch orthogonally at constant rate always in the same direction, and inactivate upon collision with an existing branch. It asserts that this geometry induces a spatial branching property, so that the connected components form a branching process of rectangles with double immigration. A spine decomposition applied to a typical rectangle, combined with couplings, reduces the problem to a one-dimensional stick-breaking process with aging; the authors then establish long-time convergence of the rescaled empirical measure of rectangle sizes to an explicit limiting distribution, together with the rate of convergence. The argument also invokes the genealogy of common ancestors under double immigration.

Significance. If the claimed exact reduction to the branching-process representation holds, the work supplies a rare exactly solvable spatial branching model with geometric interactions, yielding explicit limiting laws and convergence rates for the empirical measure after polynomial rescaling. This would be of interest to the theory of branching processes, random geometric graphs, and growing networks with fusion.

major comments (3)
  1. [§2–3 (spatial branching property and rectangle process)] The central reduction (abstract and §2–3) asserts that orthogonal fixed-direction branching plus inactivation upon meeting preserves the spatial branching property, allowing connected components to be modeled exactly as a branching process of rectangles with double immigration. The inactivation rule can create pathwise dependencies between tips; an explicit verification is required that the future evolution of any given rectangle remains independent of all others conditional on its current state and that the Markov property is not violated by collisions.
  2. [§4 (spine and coupling)] §4 (spine decomposition and coupling): the reduction of the rectangle process to a one-dimensional stick-breaking model with aging is used to obtain the empirical-measure convergence after polynomial rescaling. The manuscript must supply explicit error bounds or a quantitative coupling inequality showing that the geometric interactions do not affect the limiting distribution or the stated speed of convergence.
  3. [§5 (common ancestors and limiting measure)] The description of common ancestors under double immigration (abstract and §5) is invoked to control the genealogy; it is not clear how this description is combined with the stick-breaking limit to produce the explicit limiting measure and convergence rate. A precise statement linking the ancestor distribution to the empirical-measure limit is needed.
minor comments (2)
  1. Notation for the double-immigration rates and the polynomial rescaling exponent should be introduced once and used consistently throughout.
  2. Figure captions should explicitly state the scaling used for the empirical measure (e.g., n^α for the appropriate α).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments identify places where additional explicit verifications and quantitative statements will improve the rigor and clarity of the arguments. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§2–3 (spatial branching property and rectangle process)] The central reduction (abstract and §2–3) asserts that orthogonal fixed-direction branching plus inactivation upon meeting preserves the spatial branching property, allowing connected components to be modeled exactly as a branching process of rectangles with double immigration. The inactivation rule can create pathwise dependencies between tips; an explicit verification is required that the future evolution of any given rectangle remains independent of all others conditional on its current state and that the Markov property is not violated by collisions.

    Authors: We agree that a fully explicit verification is needed. In the revised manuscript we will insert a new subsection in §2 that constructs the process via its generator and proves the spatial branching property directly: conditional on the current state (dimensions and boundary positions) of a given rectangle, its future branching, growth, and inactivation events are independent of all other rectangles because (i) branching occurs at a constant rate independently of geometry, (ii) orthogonal fixed-direction growth means collisions only terminate the colliding tip without altering the rates or directions of any active tips, and (iii) the state descriptor for each rectangle is Markovian. This establishes that the connected components evolve exactly as a branching process of rectangles with double immigration. revision: yes

  2. Referee: [§4 (spine and coupling)] §4 (spine decomposition and coupling): the reduction of the rectangle process to a one-dimensional stick-breaking model with aging is used to obtain the empirical-measure convergence after polynomial rescaling. The manuscript must supply explicit error bounds or a quantitative coupling inequality showing that the geometric interactions do not affect the limiting distribution or the stated speed of convergence.

    Authors: We will augment §4 with explicit quantitative coupling inequalities. Specifically, we will derive bounds on the total-variation (or Wasserstein) distance between the law of the rescaled empirical measure of the geometric rectangle process and the corresponding law for the one-dimensional stick-breaking process with aging. These bounds will be shown to be o(1) under the polynomial rescaling, confirming that the geometric interactions vanish in the limit and do not alter either the limiting distribution or the convergence rate. revision: yes

  3. Referee: [§5 (common ancestors and limiting measure)] The description of common ancestors under double immigration (abstract and §5) is invoked to control the genealogy; it is not clear how this description is combined with the stick-breaking limit to produce the explicit limiting measure and convergence rate. A precise statement linking the ancestor distribution to the empirical-measure limit is needed.

    Authors: We will add a precise proposition (new Proposition 5.3) that states the exact link: the empirical-measure limit is obtained by integrating the stick-breaking limit against the distribution of the rescaled sizes of rectangles descending from a typical common ancestor under the double-immigration branching process. The convergence rate follows from the quantitative control on the ancestor process already developed in §5 together with the coupling bounds from §4. The revised text will contain the full statement and its proof. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from geometric model assumptions via standard branching-process tools

full rationale

The paper starts from an explicit geometric construction (constant-speed tip growth, constant-rate orthogonal branching in fixed direction, inactivation on meeting) and asserts that this yields a spatial branching property, allowing connected components to be represented exactly as a branching process of rectangles with double immigration. It then invokes standard spine and coupling techniques from branching-process theory to reduce the empirical-measure convergence question to a one-dimensional stick-breaking process with aging, from which the long-time limit and rate are derived. No equation or claim reduces the target convergence result to a quantity defined in terms of itself, a fitted parameter, or a self-citation chain; the steps remain independent of the final statement and rest on the model's stated rules plus external probabilistic machinery.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the geometric assumption that orthogonal same-direction branching produces independent rectangular components whose genealogy forms a branching process with double immigration; it also relies on standard properties of branching processes and coupling arguments.

free parameters (2)
  • branching rate
    Constant rate at which tips branch, chosen as part of the model definition rather than fitted to external data.
  • growth speed
    Constant speed of tip advancement, chosen as part of the model definition.
axioms (2)
  • standard math Standard properties of continuous-time branching processes and spine decompositions
    Invoked to model the genealogy of rectangles and to extract the typical rectangle via the spine.
  • domain assumption Spatial branching property induced by orthogonal same-direction branching
    Claimed to follow from the geometry so that connected components evolve independently except for the immigration mechanism.

pith-pipeline@v0.9.0 · 5419 in / 1533 out tokens · 51785 ms · 2026-05-10T02:37:10.462030+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

40 extracted references · 4 canonical work pages

  1. [1]

    Springer, Cham, Switzerland, 2015

    Vincent Bansaye and Sylvie M´ el´ eard.Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior, volume 1.4 ofMathematical Biosciences Institute Lecture Series. Springer, Cham, Switzerland, 2015

    2015

  2. [2]

    Spatial networks.Phys

    Marc Barth´ elemy. Spatial networks.Phys. Rep., 499(1-3):1–101, 2011

    2011

  3. [3]

    Markovian growth-fragmentation processes.Bernouilli, 23(2):1082–1101, 2017

    Jean Bertoin. Markovian growth-fragmentation processes.Bernouilli, 23(2):1082–1101, 2017

    2017

  4. [4]

    Random planar maps and growth- fragmentations.Ann

    Jean Bertoin, Nicolas Curien, and Igor Kortchemski. Random planar maps and growth- fragmentations.Ann. Probab., 46(1):207–260, 2018

    2018

  5. [5]

    Survival and complete convergence for a branching annihilating random walk.Ann

    Matthias Birkner, Alice Callegaro, Jiˇ r´ ıˇCern` y, Nina Gantert, and Pascal Oswald. Survival and complete convergence for a branching annihilating random walk.Ann. Appl. Probab., 34(6):5737–5768, 2024

    2024

  6. [6]

    The survival of branching annihilating random walk

    Maury Bramson and Lawrence Gray. The survival of branching annihilating random walk. Probab. Theory Relat. Fields, 68(4):447–460, 1985

    1985

  7. [7]

    Alice Callegaro and Matthew I. Roberts. A spatially-dependent fragmentation process.Probab. Theory Relat. Fields, 192:163–266, 2024

    2024

  8. [8]

    Theory of branching and annihilating random walks.Phys

    John Cardy and Uwe C T¨ auber. Theory of branching and annihilating random walks.Phys. Rev. Lett., 77(23):4780, 1996. 37

    1996

  9. [9]

    A mean-field approach to self-interacting networks, convergence and regularity.Mathematical Models and Methods in Applied Sciences, 31(13):2597–2641, 2021

    R´ emi Catellier, Yves D’Angelo, and Cristiano Ricci. A mean-field approach to self-interacting networks, convergence and regularity.Mathematical Models and Methods in Applied Sciences, 31(13):2597–2641, 2021

    2021

  10. [10]

    Mathematics and morphogenesis of cities: A geometrical approach.Phys

    Thomas Courtat, Catherine Gloaguen, and Stephane Douady. Mathematics and morphogenesis of cities: A geometrical approach.Phys. Rev. E, 83(3):036106, 2011

    2011

  11. [11]

    Multitype self-similar growth-fragmentations.ALEA - Lat

    William Da Silva and Juan Carlos Pardo. Multitype self-similar growth-fragmentations.ALEA - Lat. Am. J. Probab. Math. Stat., 21:207–260, 2024

    2024

  12. [12]

    PhD thesis, University of Zurich, 2020

    Benjamin Dadoun.Some aspects of growth-fragmentation. PhD thesis, University of Zurich, 2020

    2020

  13. [13]

    Probabilistic representations of fragmentation equations

    Madalina Deaconu and Antoine Lejay. Probabilistic representations of fragmentation equations. Probab. Surveys, 20:226–290, 2023

    2023

  14. [14]

    Hyphal network whole field imaging allows for accurate estimation of anastomosis rates and branching dynamics of the filamentous fungus podospora anserina.Sci

    J Dikec, A Olivier, C Bob´ ee, Y D’angelo, R Catellier, P David, F Filaine, S Herbert, Ch Lalanne, Herve Lalucque, et al. Hyphal network whole field imaging allows for accurate estimation of anastomosis rates and branching dynamics of the filamentous fungus podospora anserina.Sci. Rep., 10(1):3131, 2020

    2020

  15. [15]

    A purely mechanical model with asymmetric features for early morphogenesis of rod-shaped bacteria micro-colony.arXiv preprint arXiv:2008.04532, 2020

    Marie Doumic, Sophie Hecht, and Diane Peurichard. A purely mechanical model with asymmetric features for early morphogenesis of rod-shaped bacteria micro-colony.arXiv preprint arXiv:2008.04532, 2020

    work page arXiv 2008

  16. [16]

    Well-posedness of smoluchowski’s coagulation equation for a class of homogeneous kernels.Journal of functional Analysis, 233(2):351–379, 2006

    Nicolas Fournier and Philippe Lauren¸ cot. Well-posedness of smoluchowski’s coagulation equation for a class of homogeneous kernels.Journal of functional Analysis, 233(2):351–379, 2006

    2006

  17. [17]

    Principles of branch dynamics governing shape characteristics of cerebellar purkinje cell dendrites.Development, 139(18):3442–3455, 2012

    Kazuto Fujishima, Ryota Horie, Atsushi Mochizuki, and Mineko Kengaku. Principles of branch dynamics governing shape characteristics of cerebellar purkinje cell dendrites.Development, 139(18):3442–3455, 2012

    2012

  18. [18]

    Steady distribution of the incremental model for bacteria proliferation

    Pierre Gabriel and Hugo Martin. Steady distribution of the incremental model for bacteria proliferation.arXiv preprint arXiv:1803.04950, 2018

    work page Pith review arXiv 2018

  19. [19]

    A unifying theory of branching morphogenesis.Cell, 171(1):242–255, 2017

    Edouard Hannezo, Colinda LGJ Scheele, Mohammad Moad, Nicholas Drogo, Rakesh Heer, Rosemary V Sampogna, Jacco Van Rheenen, and Benjamin D Simons. A unifying theory of branching morphogenesis.Cell, 171(1):242–255, 2017

    2017

  20. [20]

    Multiscale dynamics of branching morphogenesis

    Edouard Hannezo and Benjamin D Simons. Multiscale dynamics of branching morphogenesis. Curr. Opin. Cell Biol., 60:99–105, 2019

    2019

  21. [21]

    Morphogenesis.Cell, 96(2):225–233, 1999

    Brigid LM Hogan. Morphogenesis.Cell, 96(2):225–233, 1999

    1999

  22. [22]

    Damage and fluctuations induce loops in optimal transport networks.Phys

    Eleni Katifori, Gergely J Sz¨ oll˝ osi, and Marcelo O Magnasco. Damage and fluctuations induce loops in optimal transport networks.Phys. Rev. Lett., 104(4):048704, 2010

    2010

  23. [23]

    Traveling waves of an fkpp-type model for self-organized growth.J

    Florian Kreten. Traveling waves of an fkpp-type model for self-organized growth.J. Math. Biol., 84(6):42, 2022

    2022

  24. [24]

    Convective stability of the critical waves of an fkpp-type model for self-organized growth.J

    Florian Kreten. Convective stability of the critical waves of an fkpp-type model for self-organized growth.J. Math. Biol., 90(3):33, 2025

    2025

  25. [25]

    A generalised spatial branching process with ancestral branching to model the growth of a filamentous fungus.arXiv preprint arXiv:2409.13627, 2024

    Lena Kuwata. A generalised spatial branching process with ancestral branching to model the growth of a filamentous fungus.arXiv preprint arXiv:2409.13627, 2024

    work page arXiv 2024

  26. [26]

    Leaf venation network architecture coordinates functional trade-offs across vein spatial scales: evidence for multiple alternative designs.New Phytol., 244(2):407–425, 2024

    Ilaine Silveira Matos, Mickey Boakye, Izzi Niewiadomski, Monica Antonio, Sonoma Carlos, Breanna Carrillo Johnson, Ashley Chu, Andrea Echevarria, Adrian Fontao, Lisa Garcia, et al. Leaf venation network architecture coordinates functional trade-offs across vein spatial scales: evidence for multiple alternative designs.New Phytol., 244(2):407–425, 2024. 38

    2024

  27. [27]

    Morphogenesis of lines and nets.Differentiation; research in biological diversity, 6(2):117–123, 1976

    Hans Meinhardt. Morphogenesis of lines and nets.Differentiation; research in biological diversity, 6(2):117–123, 1976

    1976

  28. [28]

    Image-based modeling of kidney branching morphogenesis reveals gdnf-ret based turing-type mechanism and pattern-modulating wnt11 feedback.Nat

    Denis Menshykau, Odyss´ e Michos, Christine Lang, Lisa Conrad, Andrew P McMahon, and Dagmar Iber. Image-based modeling of kidney branching morphogenesis reveals gdnf-ret based turing-type mechanism and pattern-modulating wnt11 feedback.Nat. Commun., 10(1):239, 2019

    2019

  29. [29]

    How does variability in cells aging and growth rates influence the malthus parameter?arXiv preprint arXiv:1602.06970, 2016

    Adelaide Olivier. How does variability in cells aging and growth rates influence the malthus parameter?arXiv preprint arXiv:1602.06970, 2016

    work page arXiv 2016

  30. [30]

    Springer, 2007

    Benoˆ ıt Perthame.Transport equations in biology. Springer, 2007

    2007

  31. [31]

    Global optimization, local adaptation, and the role of growth in distribution networks.Phys

    Henrik Ronellenfitsch and Eleni Katifori. Global optimization, local adaptation, and the role of growth in distribution networks.Phys. Rev. Lett., 117(13):138301, 2016

    2016

  32. [32]

    Optimal transport for applied mathematicians

    Filippo Santambrogio. Optimal transport for applied mathematicians. 2015

    2015

  33. [33]

    De- cline of leaf hydraulic conductance with dehydration: relationship to leaf size and venation architecture.Plant Physiol., 156(2):832–843, 2011

    Christine Scoffoni, Michael Rawls, Athena McKown, Herv´ e Cochard, and Lawren Sack. De- cline of leaf hydraulic conductance with dehydration: relationship to leaf size and venation architecture.Plant Physiol., 156(2):832–843, 2011

    2011

  34. [34]

    Ergodic behaviour of a multi-type growth-fragmentation process modelling the mycelial network of a filamentous fungus.ESAIM: Probab

    Milica Tomaˇ sevi´ c, Vincent Bansaye, and Amandine V´ eber. Ergodic behaviour of a multi-type growth-fragmentation process modelling the mycelial network of a filamentous fungus.ESAIM: Probab. Stat., 26:397–435, 2022

    2022

  35. [35]

    The chemical basis of morphogenesis.Bull

    Alan Mathison Turing. The chemical basis of morphogenesis.Bull. Math. Biol., 52(1):153–197, 1990

    1990

  36. [36]

    Springer, 2008

    C´ edric Villani et al.Optimal transport: old and new, volume 338. Springer, 2008

    2008

  37. [37]

    Turing mechanism underlying a branching model for lung morphogenesis.PLoS One, 12(4):e0174946, 2017

    Hui Xu, Mingzhu Sun, and Xin Zhao. Turing mechanism underlying a branching model for lung morphogenesis.PLoS One, 12(4):e0174946, 2017

    2017

  38. [38]

    Simple rules determine distinct patterns of branching morphogenesis

    Wei Yu, Wallace F Marshall, Ross J Metzger, Paul R Brakeman, Leonardo Morsut, Wendell Lim, and Keith E Mostov. Simple rules determine distinct patterns of branching morphogenesis. Cell Syst., 9(3):221–227, 2019

    2019

  39. [39]

    Crystal templating dendritic pore networks and fibrillar microstructure into hydrogels.Acta Biomater., 6(7):2415–2421, 2010

    Scott A Zawko and Christine E Schmidt. Crystal templating dendritic pore networks and fibrillar microstructure into hydrogels.Acta Biomater., 6(7):2415–2421, 2010

    2010

  40. [40]

    A framework for modeling the growth and development of neurons and networks.Front

    Frederic Zubler and Rodney Douglas. A framework for modeling the growth and development of neurons and networks.Front. comput. neurosci., 3:757, 2009. 39

    2009