American Institute of Mathematical Sciences

Advanced
2014, 11(2): 385-401. doi: 10.3934/mbe.2014.11.385

Cooperative behavior in a jump diffusion model for a simple network of spiking neurons

Roberta Sirovich 1, , Laura Sacerdote 1, and  Alessandro E. P. Villa 2,

1. 

Department of Mathematics "G. Peano", University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy, Italy
2. 

Grenoble Institute of Neuroscience Inserm UMRS 836, University Joseph Fourier Grenoble, France

Received  October 2012 Revised  May 2013 Published  October 2013

The distribution of time intervals between successive spikes generated by a neuronal cell --the interspike intervals (ISI)-- may reveal interesting features of the underlying dynamics. In this study we analyze the ISI sequence --the spike train-- generated by a simple network of neurons whose output activity is modeled by a jump-diffusion process. We prove that, when specific ranges of the involved parameters are chosen, it is possible to observe multimodal ISI distributions which reveal that the modeled network fires with more than one single preferred time interval. Furthermore, the system exhibits resonance behavior, with modulation of the spike timings by the noise intensity. We also show that inhibition helps the signal transmission between the units of the simple network.
Keywords: Noisy leaky integrate and fire, multimodality., first exit times, Ornstein Uhlenbeck process, stochastic resonance, preferred time intervals.
Mathematics Subject Classification: 60G99, 60K40, 90B15, 92B2.
Citation: Roberta Sirovich, Laura Sacerdote, Alessandro E. P. Villa. Cooperative behavior in a jump diffusion model for a simple network of spiking neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 385-401. doi: 10.3934/mbe.2014.11.385
References:
[1]

L. Alili, P. Patie and J. L. Pedersen, Representations of the first hitting time density of an Ornstein-Uhlenbeck process, Stoch. Models, 21 (2005), 967-980. doi: 10.1080/15326340500294702.  Google Scholar

[2]

P. Baldi and L. Caramellino, Asymptotics of hitting probabilities for general one-dimensional pinned diffusions, Ann. Appl. Probab., 12 (2002), 1071-1095. doi: 10.1214/aoap/1031863181.  Google Scholar

[3]

A. R. Bulsara, T. C. Elston, C. R. Doering, S. B. Lowen and K. Lindenberg, Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics, Phys. Rev. E, 53 (1996), 3958-3969. doi: 10.1103/PhysRevE.53.3958.  Google Scholar

[4]

A. R. Bulsara, S. B. Lowen and C. D. Rees, Cooperative behavior in the periodically modulated Wiener process: Noise-induced complexity in a model neutron, Phys. Rev. E, 49 (1994), 4989-5000. doi: 10.1103/PhysRevE.49.4989.  Google Scholar

[5]

W. H. Calvin and C. F. Stevens, Synaptic noise and other sources of randomness in motoneuron interspike intervals, J. Neurophysiol., 31 (1968), 574-587. Google Scholar

[6]

A. Capurro, K. Pakdaman, T. Nomura and S. Sato, Aperiodic stochastic resonance with correlated noise, Phys. Rev. E, 58 (1998), 4820-4827. doi: 10.1103/PhysRevE.58.4820.  Google Scholar

[7]

G. A. Cecchi, M. Sigman, J.-M. Alonso, L. Martínez, D. R. Chialvo and M. O. Magnasco, Noise in neurons is message dependent, Proceedings of the National Academy of Sciences, 97 (2000), 5557-5561. doi: 10.1073/pnas.100113597.  Google Scholar

[8]

J. J. Collins, C. C. Chow, A. C. Capela and T. T. Imhoff, Aperiodic stochastic resonance, Phys. Rev. E, 54 (1996), 5575-5584. doi: 10.1103/PhysRevE.54.5575.  Google Scholar

[9]

J. J. Collins, C. C. Chow and T. T. Imhoff, Aperiodic stochastic resonance in excitable systems, Phys. Rev. E, 52 (1995), R3321-R3324. doi: 10.1103/PhysRevE.52.R3321.  Google Scholar

[10]

I. Duguid, T. Branco, M. London, P. Chadderton and M. Häusser, Tonic inhibition enhances fidelity of sensory information transmission in the cerebellar cortex, The Journal of Neuroscience, 32 (2012), 11132-11143. doi: 10.1523/JNEUROSCI.0460-12.2012.  Google Scholar

[11]

M. Gernert, M. Bennay, M. Fedrowitz, J. H. Rehders and A. Richter, Altered discharge pattern of basal ganglia output neurons in an animal model of idiopathic dystonia, J. Neurosci., 22 (2002), 7244-7253. Google Scholar

[12]

M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes, Comm. Statist. Simulation Comput., 28 (1999), 1135-1163. doi: 10.1080/03610919908813596.  Google Scholar

[13]

L. L. Gollo, C. R. Mirasso and A. E. P. Villa, Dynamic control for synchronization of separated cortical areas through thalamic relay, NeuroImage, 52 (2010), 947-955. doi: 10.1016/j.neuroimage.2009.11.058.  Google Scholar

[14]

M. Häusser and B. A. Clark, Tonic synaptic inhibition modulates neuronal output pattern and spatiotemporal synaptic integration, Neuron, 19 (1997), 665-678. Google Scholar

[15]

E. R. Kandel, J. H. Schwartz and T. M. Jessell, Principles of Neural Science, Vol. 4, McGraw-Hill, New York, 2000. Google Scholar

[16]

P. Lánský, On approximations of Stein's neuronal model, J. Theor. Biol., 107 (1984), 631-647. Google Scholar

[17]

M. W. Levine and J. M. Shefner, A model for the variability of interspike intervals during sustained firing of a retinal neuron, Biophysical Journal, 19 (1977), 241-252. doi: 10.1016/S0006-3495(77)85584-7.  Google Scholar

[18]

Y. Loewenstein, S. Mahon, P. Chadderton, K. Kitamura, H. Sompolinsky, Y. Yarom and M. Häusser, Bistability of cerebellar Purkinje cells modulated by sensory stimulation, Nature Neuroscience, 8 (2005), 202-211. doi: 10.1038/nn1393.  Google Scholar

[19]

A. Longtin, Stochastic resonance in neuron models, Journal of Statistical Physics, 70 (1993), 309-327. doi: 10.1007/BF01053970.  Google Scholar

[20]

A. Longtin, A. Bulsara and F. Moss, Time interval sequences in the bistable systems and the noise-induced transmission of information by sensory neurons, Phys. Rev. Lett., 67 (1991), 656-659. doi: 10.1103/PhysRevLett.67.656.  Google Scholar

[21]

M. Musila and P. Lánský, Generalized Stein's model for anatomically complex neurons, Biosystems, 25 (1991), 179-191. doi: 10.1016/0303-2647(91)90004-5.  Google Scholar

[22]

A. G. Nobile, L. M. Ricciardi and L. Sacerdote, Exponential trends of Ornstein-Uhlenbeck first-passage-time densities, J. Appl. Probab., 22 (1985), 360-369. doi: 10.2307/3213779.  Google Scholar

[23]

L. M. Ricciardi, Diffusion approximation for a multi-input model neuron, Biological Cybernetics, 24 (1976), 237-240. doi: 10.1007/BF00335984.  Google Scholar

[24]

L. Sacerdote and R. Sirovich, Multimodality of the interspike interval distribution in a simple jump-diffusion model, Sci. Math. Jpn., 58 (2003), 307-322.  Google Scholar

[25]

J. P. Segundo, J. F. Vibert, K. Pakdaman, M. Stiber and O. Diez-Martinez, Noise and the neurosciences: A long history, a recent revival and some theory, Origins: Brain and Self Organization, (1994), 299-331. Google Scholar

[26]

T. Shimokawa, K. Pakdaman and S. Sato, Time-scale matching in the response of a leaky integrate-and-fire neuron model to periodic stimulus with additive noise, Phys. Rev. E, 59 (1999), 3427-3443. doi: 10.1103/PhysRevE.59.3427.  Google Scholar

[27]

H. C. Tuckwell, Introduction to Theoretical Neurobiology: Volume 2, Nonlinear and Stochastic Theories, Cambridge University Press, 2005. Google Scholar

[28]

C. Van Vreeswijk, L. F. Abbott and G. B. Ermentrout, When inhibition not excitation synchronizes neural firing, Journal of Computational Neuroscience, 1 (1994), 313-321. Google Scholar

[29]

F. Wan and H. C. Tuckwell, Neuronal firing and input variability, J. Theor. Neurobiol., 1 (1982), 197-218. Google Scholar

[30]

K. Wiesenfeld and F. Moss, Stochastic resonance and the benefits of noise: From ice ages to crayfish and squids, Nature, 373 (1995), 33-36. doi: 10.1038/373033a0.  Google Scholar

[31]

F. Wörgötter, E. Nelle, B. Li and K. Funke, The influence of corticofugal feedback on the temporal structure of visual responses of cat thalamic relay cells, J. Physiol., 509 (1998), 797-815. doi: 10.1111/j.1469-7793.1998.797bm.x.  Google Scholar

show all references

References:
[1]

L. Alili, P. Patie and J. L. Pedersen, Representations of the first hitting time density of an Ornstein-Uhlenbeck process, Stoch. Models, 21 (2005), 967-980. doi: 10.1080/15326340500294702.  Google Scholar

[2]

P. Baldi and L. Caramellino, Asymptotics of hitting probabilities for general one-dimensional pinned diffusions, Ann. Appl. Probab., 12 (2002), 1071-1095. doi: 10.1214/aoap/1031863181.  Google Scholar

[3]

A. R. Bulsara, T. C. Elston, C. R. Doering, S. B. Lowen and K. Lindenberg, Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics, Phys. Rev. E, 53 (1996), 3958-3969. doi: 10.1103/PhysRevE.53.3958.  Google Scholar

[4]

A. R. Bulsara, S. B. Lowen and C. D. Rees, Cooperative behavior in the periodically modulated Wiener process: Noise-induced complexity in a model neutron, Phys. Rev. E, 49 (1994), 4989-5000. doi: 10.1103/PhysRevE.49.4989.  Google Scholar

[5]

W. H. Calvin and C. F. Stevens, Synaptic noise and other sources of randomness in motoneuron interspike intervals, J. Neurophysiol., 31 (1968), 574-587. Google Scholar

[6]

A. Capurro, K. Pakdaman, T. Nomura and S. Sato, Aperiodic stochastic resonance with correlated noise, Phys. Rev. E, 58 (1998), 4820-4827. doi: 10.1103/PhysRevE.58.4820.  Google Scholar

[7]

G. A. Cecchi, M. Sigman, J.-M. Alonso, L. Martínez, D. R. Chialvo and M. O. Magnasco, Noise in neurons is message dependent, Proceedings of the National Academy of Sciences, 97 (2000), 5557-5561. doi: 10.1073/pnas.100113597.  Google Scholar

[8]

J. J. Collins, C. C. Chow, A. C. Capela and T. T. Imhoff, Aperiodic stochastic resonance, Phys. Rev. E, 54 (1996), 5575-5584. doi: 10.1103/PhysRevE.54.5575.  Google Scholar

[9]

J. J. Collins, C. C. Chow and T. T. Imhoff, Aperiodic stochastic resonance in excitable systems, Phys. Rev. E, 52 (1995), R3321-R3324. doi: 10.1103/PhysRevE.52.R3321.  Google Scholar

[10]

I. Duguid, T. Branco, M. London, P. Chadderton and M. Häusser, Tonic inhibition enhances fidelity of sensory information transmission in the cerebellar cortex, The Journal of Neuroscience, 32 (2012), 11132-11143. doi: 10.1523/JNEUROSCI.0460-12.2012.  Google Scholar

[11]

M. Gernert, M. Bennay, M. Fedrowitz, J. H. Rehders and A. Richter, Altered discharge pattern of basal ganglia output neurons in an animal model of idiopathic dystonia, J. Neurosci., 22 (2002), 7244-7253. Google Scholar

[12]

M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes, Comm. Statist. Simulation Comput., 28 (1999), 1135-1163. doi: 10.1080/03610919908813596.  Google Scholar

[13]

L. L. Gollo, C. R. Mirasso and A. E. P. Villa, Dynamic control for synchronization of separated cortical areas through thalamic relay, NeuroImage, 52 (2010), 947-955. doi: 10.1016/j.neuroimage.2009.11.058.  Google Scholar

[14]

M. Häusser and B. A. Clark, Tonic synaptic inhibition modulates neuronal output pattern and spatiotemporal synaptic integration, Neuron, 19 (1997), 665-678. Google Scholar

[15]

E. R. Kandel, J. H. Schwartz and T. M. Jessell, Principles of Neural Science, Vol. 4, McGraw-Hill, New York, 2000. Google Scholar

[16]

P. Lánský, On approximations of Stein's neuronal model, J. Theor. Biol., 107 (1984), 631-647. Google Scholar

[17]

M. W. Levine and J. M. Shefner, A model for the variability of interspike intervals during sustained firing of a retinal neuron, Biophysical Journal, 19 (1977), 241-252. doi: 10.1016/S0006-3495(77)85584-7.  Google Scholar

[18]

Y. Loewenstein, S. Mahon, P. Chadderton, K. Kitamura, H. Sompolinsky, Y. Yarom and M. Häusser, Bistability of cerebellar Purkinje cells modulated by sensory stimulation, Nature Neuroscience, 8 (2005), 202-211. doi: 10.1038/nn1393.  Google Scholar

[19]

A. Longtin, Stochastic resonance in neuron models, Journal of Statistical Physics, 70 (1993), 309-327. doi: 10.1007/BF01053970.  Google Scholar

[20]

A. Longtin, A. Bulsara and F. Moss, Time interval sequences in the bistable systems and the noise-induced transmission of information by sensory neurons, Phys. Rev. Lett., 67 (1991), 656-659. doi: 10.1103/PhysRevLett.67.656.  Google Scholar

[21]

M. Musila and P. Lánský, Generalized Stein's model for anatomically complex neurons, Biosystems, 25 (1991), 179-191. doi: 10.1016/0303-2647(91)90004-5.  Google Scholar

[22]

A. G. Nobile, L. M. Ricciardi and L. Sacerdote, Exponential trends of Ornstein-Uhlenbeck first-passage-time densities, J. Appl. Probab., 22 (1985), 360-369. doi: 10.2307/3213779.  Google Scholar

[23]

L. M. Ricciardi, Diffusion approximation for a multi-input model neuron, Biological Cybernetics, 24 (1976), 237-240. doi: 10.1007/BF00335984.  Google Scholar

[24]

L. Sacerdote and R. Sirovich, Multimodality of the interspike interval distribution in a simple jump-diffusion model, Sci. Math. Jpn., 58 (2003), 307-322.  Google Scholar

[25]

J. P. Segundo, J. F. Vibert, K. Pakdaman, M. Stiber and O. Diez-Martinez, Noise and the neurosciences: A long history, a recent revival and some theory, Origins: Brain and Self Organization, (1994), 299-331. Google Scholar

[26]

T. Shimokawa, K. Pakdaman and S. Sato, Time-scale matching in the response of a leaky integrate-and-fire neuron model to periodic stimulus with additive noise, Phys. Rev. E, 59 (1999), 3427-3443. doi: 10.1103/PhysRevE.59.3427.  Google Scholar

[27]

H. C. Tuckwell, Introduction to Theoretical Neurobiology: Volume 2, Nonlinear and Stochastic Theories, Cambridge University Press, 2005. Google Scholar

[28]

C. Van Vreeswijk, L. F. Abbott and G. B. Ermentrout, When inhibition not excitation synchronizes neural firing, Journal of Computational Neuroscience, 1 (1994), 313-321. Google Scholar

[29]

F. Wan and H. C. Tuckwell, Neuronal firing and input variability, J. Theor. Neurobiol., 1 (1982), 197-218. Google Scholar

[30]

K. Wiesenfeld and F. Moss, Stochastic resonance and the benefits of noise: From ice ages to crayfish and squids, Nature, 373 (1995), 33-36. doi: 10.1038/373033a0.  Google Scholar

[31]

F. Wörgötter, E. Nelle, B. Li and K. Funke, The influence of corticofugal feedback on the temporal structure of visual responses of cat thalamic relay cells, J. Physiol., 509 (1998), 797-815. doi: 10.1111/j.1469-7793.1998.797bm.x.  Google Scholar

[1]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (1) : 1-10. doi: 10.3934/mbe.2014.11.1
[2]

Pierre Guiraud, Etienne Tanré. Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5183-5201. doi: 10.3934/dcdsb.2019056
[3]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A leaky integrate-and-fire model with adaptation for the generation of a spike train. Mathematical Biosciences & Engineering, 2016, 13 (3) : 483-493. doi: 10.3934/mbe.2016002
[4]

Samuel Herrmann, Nicolas Massin. Exit problem for Ornstein-Uhlenbeck processes: A random walk approach. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3199-3215. doi: 10.3934/dcdsb.2020058
[5]

Timothy J. Lewis. Phase-locking in electrically coupled non-leaky integrate-and-fire neurons. Conference Publications, 2003, 2003 (Special) : 554-562. doi: 10.3934/proc.2003.2003.554
[6]

Roberta Sirovich, Luisa Testa. A new firing paradigm for integrate and fire stochastic neuronal models. Mathematical Biosciences & Engineering, 2016, 13 (3) : 597-611. doi: 10.3934/mbe.2016010
[7]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013
[8]

Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285
[9]

Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020143
[10]

Michele Barbi, Angelo Di Garbo, Rita Balocchi. Improved integrate-and-fire model for RSA. Mathematical Biosciences & Engineering, 2007, 4 (4) : 609-615. doi: 10.3934/mbe.2007.4.609
[11]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049
[12]

Karl-Peter Hadeler, Frithjof Lutscher. Quiescent phases with distributed exit times. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 849-869. doi: 10.3934/dcdsb.2012.17.849
[13]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
[14]

Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451
[15]

Chang-Yuan Cheng, Shyan-Shiou Chen, Rui-Hua Chen. Delay-induced spiking dynamics in integrate-and-fire neurons. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1867-1887. doi: 10.3934/dcdsb.2020363
[16]

Benoît Perthame, Delphine Salort. On a voltage-conductance kinetic system for integrate & fire neural networks. Kinetic & Related Models, 2013, 6 (4) : 841-864. doi: 10.3934/krm.2013.6.841
[17]

Thomas Kruse, Mikhail Urusov. Approximating exit times of continuous Markov processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3631-3650. doi: 10.3934/dcdsb.2020076
[18]

Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049
[19]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142
[20]

Alessia E. Kogoj. A Zaremba-type criterion for hypoelliptic degenerate Ornstein–Uhlenbeck operators. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3491-3494. doi: 10.3934/dcdss.2020112

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy