2016, 13(3): 613-629. doi: 10.3934/mbe.2016011

Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity

1. 

Johannes Kepler University, Altenbergerstraße 69, 4040 Linz, Austria

Received  June 2015 Revised  August 2015 Published  January 2016

The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here, we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to approximate the curved boundary by means of a continuous two-piecewise linear threshold. Explicit expressions for the first passage time density towards the new boundary are provided. First, we introduce different approximating linear thresholds. Then, we describe how to choose the optimal one minimizing the distance to the curved boundary, and hence the error in the corresponding passage time density. Theoretical means, variances and coefficients of variation given by our method are compared with empirical quantities from simulated data. Moreover, a further comparison with firing statistics derived under the assumption of a small amplitude of the time-dependent change in the threshold, is also carried out. Finally, maximum likelihood and moment estimators of the parameters of the model are derived and applied on simulated data.
Citation: Massimiliano Tamborrino. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. Mathematical Biosciences & Engineering, 2016, 13 (3) : 613-629. doi: 10.3934/mbe.2016011
References:
[1]

M. Abundo, Some results about boundary crossing for Brownian motion,, Ric. Mat., 50 (2001), 283.   Google Scholar

[2]

L. Alili, P. Patie and J. Pedersen, Representation of the first hitting time density of an Ornstein-Uhlenbeck process,, Stoch. Models, 21 (2005), 967.  doi: 10.1080/15326340500294702.  Google Scholar

[3]

K. Borovkov and A. Novikov, Explicit bounds for approximation rates of boundary crossing probabilities for the Wiener process,, J. Appl. Probab., 42 (2005), 82.  doi: 10.1239/jap/1110381372.  Google Scholar

[4]

A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model,, Math. Biosci. Eng., 11 (2014), 1.   Google Scholar

[5]

A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities,, Adv. in Appl. Probab., 19 (1987), 784.  doi: 10.2307/1427102.  Google Scholar

[6]

R. M. Capocelli and L. M. Ricciardi, On the transformation of diffusion process into the Feller process,, Math. Biosci., 29 (1976), 219.  doi: 10.1016/0025-5564(76)90104-8.  Google Scholar

[7]

M. J. Chacron, A. Longtin and L. Maler, Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli,, J. Neurosci., 21 (2001), 5328.   Google Scholar

[8]

M. J. Chacron, K. Pakdaman and A. Longtin, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue,, Neural Comput., 15 (2003), 253.  doi: 10.1162/089976603762552915.  Google Scholar

[9]

M. J. Chacron, A. Longtin, M. St-Hilaire and L. Maler, Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors,, Phys. Rev. Lett., 85 (2000), 1576.  doi: 10.1103/PhysRevLett.85.1576.  Google Scholar

[10]

R. S. Chhikara and J. L. Folks, The Inverse Gaussian Distribution: Theory, Methodology, and Applications,, Marcel Dekker, (1989).   Google Scholar

[11]

D. R. Cox and H. D. Miller, The Theory of Stochastic Processes,, CRC Press, (1977).   Google Scholar

[12]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophys. J., 4 (1964), 41.  doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

[13]

M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes,, Commun. Stat. Simulat., 28 (1999), 1135.  doi: 10.1080/03610919908813596.  Google Scholar

[14]

M. T. Giraudo, L. Sacerdote and C. Zucca, A Monte Carlo method for the simulation of first passage times of diffusion processes,, Methodol. Comput. App. Probab., 3 (2001), 215.  doi: 10.1023/A:1012261328124.  Google Scholar

[15]

J. Honerkamp, Stochastic Dynamical Systems. Concepts, Numerical Methods, Data Analysis,, Wiley/VCH, (1993).   Google Scholar

[16]

R. Jolivet, A. Roth, F. Schurmann, W. Gerstner and W. Senn, Special issue on quantitative neuron modeling,, Biol. Cybern., 99 (2008), 237.  doi: 10.1007/s00422-008-0274-5.  Google Scholar

[17]

R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold,, Front. Comput. Neurosci., 3 (2009), 1.  doi: 10.3389/neuro.10.009.2009.  Google Scholar

[18]

B. Lindner, Moments of the first passage time under weak external driving,, J. Stat. Phys., 117 (2004), 703.  doi: 10.1007/s10955-004-2269-5.  Google Scholar

[19]

B. Lindner and A. Longtin, Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron,, J. Theor. Biol., 232 (2005), 505.  doi: 10.1016/j.jtbi.2004.08.030.  Google Scholar

[20]

A. Metzler, On the first passage problem for correlated Brownian motion,, Stat. Probabil. Lett., 80 (2010), 277.  doi: 10.1016/j.spl.2009.11.001.  Google Scholar

[21]

A. Molini, P. Talkner, G. G. Katul and A. Porporato, First passage time statistics of Brownian motion with purely time dependent drift and diffusion,, Physica A, 390 (2011), 1841.  doi: 10.1016/j.physa.2011.01.024.  Google Scholar

[22]

A. Novikov, V. Frishling and N. Kordzakhia, Approximations of boundary crossing probabilities for a Brownian motion,, J. Appl. Probab., 36 (1999), 1019.  doi: 10.1239/jap/1032374752.  Google Scholar

[23]

K. Pötzelberger and L. Wang, Boundary crossing probability for Brownian motion,, J. Appl. Probab., 38 (2001), 152.  doi: 10.1239/jap/996986650.  Google Scholar

[24]

R Core Team, R: A Language and Environment for Statistical Computing,, R Foundation for Statistical Computing, (2014).   Google Scholar

[25]

L. M. Ricciardi, On the transformation of diffusion processes into the Wiener process,, J. Math. Anal. Appl., 54 (1976), 185.  doi: 10.1016/0022-247X(76)90244-4.  Google Scholar

[26]

L. M. Ricciardi, Diffusion Processes and Related Topics in Biology,, Lecture notes in Biomathematics, (1977).   Google Scholar

[27]

L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling,, Math. Japonica, 50 (1999), 247.   Google Scholar

[28]

L. Sacerdote and M. T. Giraudo, Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications,, in Stochastic Biomathematical Models, (2058), 99.  doi: 10.1007/978-3-642-32157-3_5.  Google Scholar

[29]

L. Sacerdote, M. Tamborrino and C. Zucca, First passage times of two-dimensional correlated processes: Analytical results for the Wiener process and a numerical method for diffusion processes,, J. Comput. Appl. Math., 296 (2016), 275.  doi: 10.1016/j.cam.2015.09.033.  Google Scholar

[30]

L. Sacerdote, O. Telve and C. Zucca, Joint densities of first hitting times of a diffusion process through two time dependent boundaries,, Adv. Appl. Probab., 46 (2014), 186.  doi: 10.1239/aap/1396360109.  Google Scholar

[31]

T. H. Scheike, A boundary-crossing results for Brownian motion,, J. Appl. Probab., 29 (1992), 448.  doi: 10.2307/3214581.  Google Scholar

[32]

S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex,, Neural Comput., 11 (1999), 935.  doi: 10.1162/089976699300016511.  Google Scholar

[33]

T. Taillefumier and M. O. Magnasco, A fast algorithm for the first-passage times of Gauss-Markov processes with Hölder continuous boundaries,, J. Stat. Phys., 140 (2010), 1130.  doi: 10.1007/s10955-010-0033-6.  Google Scholar

[34]

M. Tamborrino, S. Ditlevsen and P. Lansky, Parameter inference from hitting times for perturbed Brownian motion,, Lifetime Data Anal., 21 (2015), 331.  doi: 10.1007/s10985-014-9307-7.  Google Scholar

[35]

H. C. Tuckwell, Recurrent inhibition and afterhyperpolarization: Effects on neuronal discharge,, Biol. Cybernet., 30 (1978), 115.  doi: 10.1007/BF00337325.  Google Scholar

[36]

H. C. Tuckwell, Introduction to Theoretical Neurobiology, Volume 2. Nonlinear and Stochastic Theories,, Cambridge University Press, (1988).   Google Scholar

[37]

H. C. Tuckwell and F. Y. M. Wan, First passage time of Markov processes to moving barriers,, J. Appl. Probab., 21 (1984), 695.  doi: 10.2307/3213688.  Google Scholar

[38]

E. Urdapilleta, Survival probability and first-passage-time statistics of a Wiener process driven by an exponential time-dependent drift,, Phys. Rev. E, 83 (2011).  doi: 10.1103/PhysRevE.83.021102.  Google Scholar

[39]

L. Wang and K. Pötzelberger, Boundary crossing probability for Brownian motion and general boundaries,, J. App. Probab., 34 (1997), 54.  doi: 10.2307/3215174.  Google Scholar

[40]

L. Wang and K. Pötzelberger, Crossing probabilities for diffusion processes with piecewise continuous boundaries,, Methodol. Comput. Appl. Probab., 9 (2007), 21.  doi: 10.1007/s11009-006-9002-6.  Google Scholar

[41]

C. Zucca and L. Sacerdote, On the inverse first-passage-time problem for a Wiener process,, Ann. Appl. Probab., 19 (2009), 1319.  doi: 10.1214/08-AAP571.  Google Scholar

show all references

References:
[1]

M. Abundo, Some results about boundary crossing for Brownian motion,, Ric. Mat., 50 (2001), 283.   Google Scholar

[2]

L. Alili, P. Patie and J. Pedersen, Representation of the first hitting time density of an Ornstein-Uhlenbeck process,, Stoch. Models, 21 (2005), 967.  doi: 10.1080/15326340500294702.  Google Scholar

[3]

K. Borovkov and A. Novikov, Explicit bounds for approximation rates of boundary crossing probabilities for the Wiener process,, J. Appl. Probab., 42 (2005), 82.  doi: 10.1239/jap/1110381372.  Google Scholar

[4]

A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model,, Math. Biosci. Eng., 11 (2014), 1.   Google Scholar

[5]

A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities,, Adv. in Appl. Probab., 19 (1987), 784.  doi: 10.2307/1427102.  Google Scholar

[6]

R. M. Capocelli and L. M. Ricciardi, On the transformation of diffusion process into the Feller process,, Math. Biosci., 29 (1976), 219.  doi: 10.1016/0025-5564(76)90104-8.  Google Scholar

[7]

M. J. Chacron, A. Longtin and L. Maler, Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli,, J. Neurosci., 21 (2001), 5328.   Google Scholar

[8]

M. J. Chacron, K. Pakdaman and A. Longtin, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue,, Neural Comput., 15 (2003), 253.  doi: 10.1162/089976603762552915.  Google Scholar

[9]

M. J. Chacron, A. Longtin, M. St-Hilaire and L. Maler, Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors,, Phys. Rev. Lett., 85 (2000), 1576.  doi: 10.1103/PhysRevLett.85.1576.  Google Scholar

[10]

R. S. Chhikara and J. L. Folks, The Inverse Gaussian Distribution: Theory, Methodology, and Applications,, Marcel Dekker, (1989).   Google Scholar

[11]

D. R. Cox and H. D. Miller, The Theory of Stochastic Processes,, CRC Press, (1977).   Google Scholar

[12]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophys. J., 4 (1964), 41.  doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

[13]

M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes,, Commun. Stat. Simulat., 28 (1999), 1135.  doi: 10.1080/03610919908813596.  Google Scholar

[14]

M. T. Giraudo, L. Sacerdote and C. Zucca, A Monte Carlo method for the simulation of first passage times of diffusion processes,, Methodol. Comput. App. Probab., 3 (2001), 215.  doi: 10.1023/A:1012261328124.  Google Scholar

[15]

J. Honerkamp, Stochastic Dynamical Systems. Concepts, Numerical Methods, Data Analysis,, Wiley/VCH, (1993).   Google Scholar

[16]

R. Jolivet, A. Roth, F. Schurmann, W. Gerstner and W. Senn, Special issue on quantitative neuron modeling,, Biol. Cybern., 99 (2008), 237.  doi: 10.1007/s00422-008-0274-5.  Google Scholar

[17]

R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold,, Front. Comput. Neurosci., 3 (2009), 1.  doi: 10.3389/neuro.10.009.2009.  Google Scholar

[18]

B. Lindner, Moments of the first passage time under weak external driving,, J. Stat. Phys., 117 (2004), 703.  doi: 10.1007/s10955-004-2269-5.  Google Scholar

[19]

B. Lindner and A. Longtin, Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron,, J. Theor. Biol., 232 (2005), 505.  doi: 10.1016/j.jtbi.2004.08.030.  Google Scholar

[20]

A. Metzler, On the first passage problem for correlated Brownian motion,, Stat. Probabil. Lett., 80 (2010), 277.  doi: 10.1016/j.spl.2009.11.001.  Google Scholar

[21]

A. Molini, P. Talkner, G. G. Katul and A. Porporato, First passage time statistics of Brownian motion with purely time dependent drift and diffusion,, Physica A, 390 (2011), 1841.  doi: 10.1016/j.physa.2011.01.024.  Google Scholar

[22]

A. Novikov, V. Frishling and N. Kordzakhia, Approximations of boundary crossing probabilities for a Brownian motion,, J. Appl. Probab., 36 (1999), 1019.  doi: 10.1239/jap/1032374752.  Google Scholar

[23]

K. Pötzelberger and L. Wang, Boundary crossing probability for Brownian motion,, J. Appl. Probab., 38 (2001), 152.  doi: 10.1239/jap/996986650.  Google Scholar

[24]

R Core Team, R: A Language and Environment for Statistical Computing,, R Foundation for Statistical Computing, (2014).   Google Scholar

[25]

L. M. Ricciardi, On the transformation of diffusion processes into the Wiener process,, J. Math. Anal. Appl., 54 (1976), 185.  doi: 10.1016/0022-247X(76)90244-4.  Google Scholar

[26]

L. M. Ricciardi, Diffusion Processes and Related Topics in Biology,, Lecture notes in Biomathematics, (1977).   Google Scholar

[27]

L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling,, Math. Japonica, 50 (1999), 247.   Google Scholar

[28]

L. Sacerdote and M. T. Giraudo, Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications,, in Stochastic Biomathematical Models, (2058), 99.  doi: 10.1007/978-3-642-32157-3_5.  Google Scholar

[29]

L. Sacerdote, M. Tamborrino and C. Zucca, First passage times of two-dimensional correlated processes: Analytical results for the Wiener process and a numerical method for diffusion processes,, J. Comput. Appl. Math., 296 (2016), 275.  doi: 10.1016/j.cam.2015.09.033.  Google Scholar

[30]

L. Sacerdote, O. Telve and C. Zucca, Joint densities of first hitting times of a diffusion process through two time dependent boundaries,, Adv. Appl. Probab., 46 (2014), 186.  doi: 10.1239/aap/1396360109.  Google Scholar

[31]

T. H. Scheike, A boundary-crossing results for Brownian motion,, J. Appl. Probab., 29 (1992), 448.  doi: 10.2307/3214581.  Google Scholar

[32]

S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex,, Neural Comput., 11 (1999), 935.  doi: 10.1162/089976699300016511.  Google Scholar

[33]

T. Taillefumier and M. O. Magnasco, A fast algorithm for the first-passage times of Gauss-Markov processes with Hölder continuous boundaries,, J. Stat. Phys., 140 (2010), 1130.  doi: 10.1007/s10955-010-0033-6.  Google Scholar

[34]

M. Tamborrino, S. Ditlevsen and P. Lansky, Parameter inference from hitting times for perturbed Brownian motion,, Lifetime Data Anal., 21 (2015), 331.  doi: 10.1007/s10985-014-9307-7.  Google Scholar

[35]

H. C. Tuckwell, Recurrent inhibition and afterhyperpolarization: Effects on neuronal discharge,, Biol. Cybernet., 30 (1978), 115.  doi: 10.1007/BF00337325.  Google Scholar

[36]

H. C. Tuckwell, Introduction to Theoretical Neurobiology, Volume 2. Nonlinear and Stochastic Theories,, Cambridge University Press, (1988).   Google Scholar

[37]

H. C. Tuckwell and F. Y. M. Wan, First passage time of Markov processes to moving barriers,, J. Appl. Probab., 21 (1984), 695.  doi: 10.2307/3213688.  Google Scholar

[38]

E. Urdapilleta, Survival probability and first-passage-time statistics of a Wiener process driven by an exponential time-dependent drift,, Phys. Rev. E, 83 (2011).  doi: 10.1103/PhysRevE.83.021102.  Google Scholar

[39]

L. Wang and K. Pötzelberger, Boundary crossing probability for Brownian motion and general boundaries,, J. App. Probab., 34 (1997), 54.  doi: 10.2307/3215174.  Google Scholar

[40]

L. Wang and K. Pötzelberger, Crossing probabilities for diffusion processes with piecewise continuous boundaries,, Methodol. Comput. Appl. Probab., 9 (2007), 21.  doi: 10.1007/s11009-006-9002-6.  Google Scholar

[41]

C. Zucca and L. Sacerdote, On the inverse first-passage-time problem for a Wiener process,, Ann. Appl. Probab., 19 (2009), 1319.  doi: 10.1214/08-AAP571.  Google Scholar

[1]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[2]

Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China. Electronic Research Archive, 2021, 29 (1) : 1661-1679. doi: 10.3934/era.2020085

[3]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

[4]

Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323

[5]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[6]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[7]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329

[8]

Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021010

[9]

Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021014

[10]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032

[11]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

[12]

Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1415-1447. doi: 10.3934/dcds.2020323

[13]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[14]

Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020360

[15]

Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264

[16]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032

[17]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

[18]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[19]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398

[20]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]