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), 967-980. doi: 10.1080/15326340500294702.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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.

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