American Institute of Mathematical Sciences

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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.  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.  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.  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.  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.   Google Scholar

[6]

A. Capurro, K. Pakdaman, T. Nomura and S. Sato, Aperiodic stochastic resonance with correlated noise,, Phys. Rev. E, 58 (1998), 4820.  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.  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.  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).  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.  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.   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.  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.  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.   Google Scholar

[15]

E. R. Kandel, J. H. Schwartz and T. M. Jessell, Principles of Neural Science,, Vol. 4, (2000).   Google Scholar

[16]

P. Lánský, On approximations of Stein's neuronal model,, J. Theor. Biol., 107 (1984), 631.   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.  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.  doi: 10.1038/nn1393.  Google Scholar

[19]

A. Longtin, Stochastic resonance in neuron models,, Journal of Statistical Physics, 70 (1993), 309.  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.  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.  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.  doi: 10.2307/3213779.  Google Scholar

[23]

L. M. Ricciardi, Diffusion approximation for a multi-input model neuron,, Biological Cybernetics, 24 (1976), 237.  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.   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.   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.  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.   Google Scholar

[29]

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

[6]

A. Capurro, K. Pakdaman, T. Nomura and S. Sato, Aperiodic stochastic resonance with correlated noise,, Phys. Rev. E, 58 (1998), 4820.  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.  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.  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).  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.  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.   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.  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.  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.   Google Scholar

[15]

E. R. Kandel, J. H. Schwartz and T. M. Jessell, Principles of Neural Science,, Vol. 4, (2000).   Google Scholar

[16]

P. Lánský, On approximations of Stein's neuronal model,, J. Theor. Biol., 107 (1984), 631.   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.  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.  doi: 10.1038/nn1393.  Google Scholar

[19]

A. Longtin, Stochastic resonance in neuron models,, Journal of Statistical Physics, 70 (1993), 309.  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.  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.  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.  doi: 10.2307/3213779.  Google Scholar

[23]

L. M. Ricciardi, Diffusion approximation for a multi-input model neuron,, Biological Cybernetics, 24 (1976), 237.  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.   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.   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.  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.   Google Scholar

[29]

F. Wan and H. C. Tuckwell, Neuronal firing and input variability,, J. Theor. Neurobiol., 1 (1982), 197.   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.  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.  doi: 10.1111/j.1469-7793.1998.797bm.x.  Google Scholar

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