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September  2020, 25(9): 3659-3676. doi: 10.3934/dcdsb.2020085

A note on a neuron network model with diffusion

1. 

Ecole Centrale de Lyon, University Claude Bernard Lyon 1, CNRS UMR 5208, Ecully, 69130, France

2. 

School of Mathematics and Statistics, University of Hyderabad, Hyderabad, India

* Corresponding author: Suman Kumar Tumuluri

Received  January 2019 Revised  November 2019 Published  September 2020 Early access  April 2020

We study the dynamics of an inhomogeneous neuronal network parametrized by a real number $ \sigma $ and structured by the time elapsed since the last discharge. The dynamics are governed by the parabolic PDE which describes the probability density of neurons with elapsed time $ s $ after its last discharge. We prove existence and uniqueness of a solution to the model. Moreover, we show that under some conditions on the connectivity and the firing rate, the network exhibits total desynchronization.

Citation: Philippe Michel, Suman Kumar Tumuluri. A note on a neuron network model with diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3659-3676. doi: 10.3934/dcdsb.2020085
References:
[1]

L. F. Abbott, Lapicque's introduction of the integrate-and-fire model neuron (1907), Brain Res. Bull., 50 (1999), 303-304. 

[2]

P. C. Bressloff and J. M. Newby, Stochastic models of intracellular transport, Rev. Mod. Phys., 85 (2013), 135-196.  doi: 10.1103/RevModPhys.85.135.

[3]

N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates, Neural Comput., 11 (1999), 1621-1671.  doi: 10.1162/089976699300016179.

[4]

N. Brunel and M. C. W. van Rossum, Lapicque's 1907 paper: From frogs to integrate-and-fire, Biol. Cybernet., 97 (2007), 337-339.  doi: 10.1007/s00422-007-0190-0.

[5]

M. J. Cáceres, J. A. Carrillo and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states, J. Math. Neurosci., 1 (2011), Art. 7, 33 pp. doi: 10.1186/2190-8567-1-7.

[6]

G. DumontJ. Henry and C. O. Tarniceriu, Theoretical connections between mathematical neuronal models corresponding to different expressions of noise, J. Theoret. Biol., 406 (2016), 31-41.  doi: 10.1016/j.jtbi.2016.06.022.

[7]

A. A. FaisalL. P. J. Selen and D. M. Wolpert, Noise in the nervous system, Nat. Rev. Neurosci., 9 (2008), 292-303. 

[8]

W. Gerstner, Time structure of the activity in neural network models, Phys. Rev. E, 51 (1995), 738-758.  doi: 10.1103/PhysRevE.51.738.

[9] W. Gerstner and W. M. Kistler, Spiking Neuron Models, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511815706.
[10] E. M. Izhikevich, Dynamical Systems in Neuroscience, MIT Press, Cambridge, MA, 2007. 
[11]

B. K. Kakumani and S. K. Tumuluri, On a nonlinear renewal equation with diffusion, Math. Methods Appl. Sci., 39 (2016), 697-708.  doi: 10.1002/mma.3511.

[12]

M.-J. KangB. Perthame and D. Salort, Dynamics of time elapsed inhomogeneous neuron network model, C. R. Math. Acad. Sci. Paris, 353 (2015), 1111-1115.  doi: 10.1016/j.crma.2015.09.029.

[13]

B. W. Knight, Dynamics of encoding in a population of neurons, J. Gen. Physiol., 59 (1972), 734-766.  doi: 10.1085/jgp.59.6.734.

[14]

J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, (French) [Die Grundlehren der mathematischen Wissenschaften], Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.

[15]

A. Longtin, Neuronal noise, Scholarpedia 8 (2013), 1618. doi: 10.4249/scholarpedia.1618.

[16]

P. Michel, General relative entropy in a nonlinear McKendrick model, in Stochastic Analysis and Partial Differential Equations (eds. G.-Q. Chen, E. Hsu and M. Pinsky), Amer. Math. Soc., Providence, RI, 429 (2007), 205–232. doi: 10.1090/conm/429/08238.

[17]

P. MichelS. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338 (2004), 697-702.  doi: 10.1016/j.crma.2004.03.006.

[18]

P. MichelS. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001.

[19]

S. MischlerB. Perthame and L. Ryzhik, Stability in a nonlinear population maturation model, Math. Models Methods Appl. Sci., 12 (2002), 1751-1772.  doi: 10.1142/S021820250200232X.

[20]

S. Mischler and Q. Weng, Relaxation in time elapsed neuron network models in the weak connectivity regime, Acta Appl. Math., 157 (2018), 45-74.  doi: 10.1007/s10440-018-0163-4.

[21]

K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, J. Math. Neurosci., 4 (2014), Art. 14, 26 pp. doi: 10.1186/2190-8567-4-14.

[22]

K. PakdamanB. Perthame and D. Salort, Dynamics of a structured neuron population, Nonlinearity, 23 (2010), 55-75.  doi: 10.1088/0951-7715/23/1/003.

[23]

K. PakdamanB. Perthame and D. Salort, Relaxation and self-sustained oscillations in the time elapsed neuron network model, SIAM J. Appl. Math., 73 (2013), 1260-1279.  doi: 10.1137/110847962.

[24]

D. H. Perkel, A computer program for simulating a network of interacting neurons I. Organization and physiological assumptions, Comput. Biomed. Res., 9 (1976), 31-43.  doi: 10.1016/0010-4809(76)90049-5.

[25]

D. H. Perkel, A computer program for simulating a network of interacting neurons III. Applications, Comput. Biomed. Res., 9 (1976), 67-74.  doi: 10.1016/0010-4809(76)90051-3.

[26]

D. H. Perkel and M. S. Smith, A computer program for simulating a network of interacting neurons II. Programming aspects, Comput. Biomed. Res., 9 (1976), 45-66.  doi: 10.1016/0010-4809(76)90050-1.

[27]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

[28]

B. Perthame and S. K. Tumuluri, Nonlinear renewal equations, in Selected Topics in Cancer Modeling (eds. E. Angelis, M. A. J. Chaplain and N. Bellomo), {Model. Simul. Sci. Eng. Technol.}, Birkhäuser Boston, Boston, MA, (2008), 65–96.

[29]

J. PhamK. PakdamanJ. Champagnat and J.-F. Vibert, Activity in sparsely connected excitatory neural networks: Effect of connectivity, Neural Networks, 11 (1998), 415-434.  doi: 10.1016/S0893-6080(97)00153-6.

[30]

J. PhamK. Pakdaman and J.-F. Vibert, Noise-induced coherent oscillations in randomly connected neural networks, Phys. Rev. E, 58 (1998), 3610-3622.  doi: 10.1103/PhysRevE.58.3610.

[31]

C. Quiñinao, A microscopic spiking neuronal network for the age-structured model, Acta Appl. Math., 146 (2016), 29-55.  doi: 10.1007/s10440-016-0056-3.

[32]

M. N. Shadlen and W. T. Newsome, Noise, neural codes and cortical organization, Curr. Opin. Neurobiol., 4 (1994), 569-579.  doi: 10.1016/0959-4388(94)90059-0.

[33]

W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, Journal of Neuroscience, 13 (1993), 334-350.  doi: 10.1523/JNEUROSCI.13-01-00334.1993.

show all references

References:
[1]

L. F. Abbott, Lapicque's introduction of the integrate-and-fire model neuron (1907), Brain Res. Bull., 50 (1999), 303-304. 

[2]

P. C. Bressloff and J. M. Newby, Stochastic models of intracellular transport, Rev. Mod. Phys., 85 (2013), 135-196.  doi: 10.1103/RevModPhys.85.135.

[3]

N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates, Neural Comput., 11 (1999), 1621-1671.  doi: 10.1162/089976699300016179.

[4]

N. Brunel and M. C. W. van Rossum, Lapicque's 1907 paper: From frogs to integrate-and-fire, Biol. Cybernet., 97 (2007), 337-339.  doi: 10.1007/s00422-007-0190-0.

[5]

M. J. Cáceres, J. A. Carrillo and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states, J. Math. Neurosci., 1 (2011), Art. 7, 33 pp. doi: 10.1186/2190-8567-1-7.

[6]

G. DumontJ. Henry and C. O. Tarniceriu, Theoretical connections between mathematical neuronal models corresponding to different expressions of noise, J. Theoret. Biol., 406 (2016), 31-41.  doi: 10.1016/j.jtbi.2016.06.022.

[7]

A. A. FaisalL. P. J. Selen and D. M. Wolpert, Noise in the nervous system, Nat. Rev. Neurosci., 9 (2008), 292-303. 

[8]

W. Gerstner, Time structure of the activity in neural network models, Phys. Rev. E, 51 (1995), 738-758.  doi: 10.1103/PhysRevE.51.738.

[9] W. Gerstner and W. M. Kistler, Spiking Neuron Models, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511815706.
[10] E. M. Izhikevich, Dynamical Systems in Neuroscience, MIT Press, Cambridge, MA, 2007. 
[11]

B. K. Kakumani and S. K. Tumuluri, On a nonlinear renewal equation with diffusion, Math. Methods Appl. Sci., 39 (2016), 697-708.  doi: 10.1002/mma.3511.

[12]

M.-J. KangB. Perthame and D. Salort, Dynamics of time elapsed inhomogeneous neuron network model, C. R. Math. Acad. Sci. Paris, 353 (2015), 1111-1115.  doi: 10.1016/j.crma.2015.09.029.

[13]

B. W. Knight, Dynamics of encoding in a population of neurons, J. Gen. Physiol., 59 (1972), 734-766.  doi: 10.1085/jgp.59.6.734.

[14]

J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, (French) [Die Grundlehren der mathematischen Wissenschaften], Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.

[15]

A. Longtin, Neuronal noise, Scholarpedia 8 (2013), 1618. doi: 10.4249/scholarpedia.1618.

[16]

P. Michel, General relative entropy in a nonlinear McKendrick model, in Stochastic Analysis and Partial Differential Equations (eds. G.-Q. Chen, E. Hsu and M. Pinsky), Amer. Math. Soc., Providence, RI, 429 (2007), 205–232. doi: 10.1090/conm/429/08238.

[17]

P. MichelS. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338 (2004), 697-702.  doi: 10.1016/j.crma.2004.03.006.

[18]

P. MichelS. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001.

[19]

S. MischlerB. Perthame and L. Ryzhik, Stability in a nonlinear population maturation model, Math. Models Methods Appl. Sci., 12 (2002), 1751-1772.  doi: 10.1142/S021820250200232X.

[20]

S. Mischler and Q. Weng, Relaxation in time elapsed neuron network models in the weak connectivity regime, Acta Appl. Math., 157 (2018), 45-74.  doi: 10.1007/s10440-018-0163-4.

[21]

K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, J. Math. Neurosci., 4 (2014), Art. 14, 26 pp. doi: 10.1186/2190-8567-4-14.

[22]

K. PakdamanB. Perthame and D. Salort, Dynamics of a structured neuron population, Nonlinearity, 23 (2010), 55-75.  doi: 10.1088/0951-7715/23/1/003.

[23]

K. PakdamanB. Perthame and D. Salort, Relaxation and self-sustained oscillations in the time elapsed neuron network model, SIAM J. Appl. Math., 73 (2013), 1260-1279.  doi: 10.1137/110847962.

[24]

D. H. Perkel, A computer program for simulating a network of interacting neurons I. Organization and physiological assumptions, Comput. Biomed. Res., 9 (1976), 31-43.  doi: 10.1016/0010-4809(76)90049-5.

[25]

D. H. Perkel, A computer program for simulating a network of interacting neurons III. Applications, Comput. Biomed. Res., 9 (1976), 67-74.  doi: 10.1016/0010-4809(76)90051-3.

[26]

D. H. Perkel and M. S. Smith, A computer program for simulating a network of interacting neurons II. Programming aspects, Comput. Biomed. Res., 9 (1976), 45-66.  doi: 10.1016/0010-4809(76)90050-1.

[27]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

[28]

B. Perthame and S. K. Tumuluri, Nonlinear renewal equations, in Selected Topics in Cancer Modeling (eds. E. Angelis, M. A. J. Chaplain and N. Bellomo), {Model. Simul. Sci. Eng. Technol.}, Birkhäuser Boston, Boston, MA, (2008), 65–96.

[29]

J. PhamK. PakdamanJ. Champagnat and J.-F. Vibert, Activity in sparsely connected excitatory neural networks: Effect of connectivity, Neural Networks, 11 (1998), 415-434.  doi: 10.1016/S0893-6080(97)00153-6.

[30]

J. PhamK. Pakdaman and J.-F. Vibert, Noise-induced coherent oscillations in randomly connected neural networks, Phys. Rev. E, 58 (1998), 3610-3622.  doi: 10.1103/PhysRevE.58.3610.

[31]

C. Quiñinao, A microscopic spiking neuronal network for the age-structured model, Acta Appl. Math., 146 (2016), 29-55.  doi: 10.1007/s10440-016-0056-3.

[32]

M. N. Shadlen and W. T. Newsome, Noise, neural codes and cortical organization, Curr. Opin. Neurobiol., 4 (1994), 569-579.  doi: 10.1016/0959-4388(94)90059-0.

[33]

W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, Journal of Neuroscience, 13 (1993), 334-350.  doi: 10.1523/JNEUROSCI.13-01-00334.1993.

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