A new firing paradigm for integrate and fire stochastic neuronal models

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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

2016, 13(3): 597-611. doi: 10.3934/mbe.2016010

A new firing paradigm for integrate and fire stochastic neuronal models

Roberta Sirovich ^1, and Luisa Testa ^2,

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

Received May 2015 Revised October 2015 Published January 2016

Abstract
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A new definition of firing time is given in the framework of Integrate and Fire neuronal models. The classical absorption condition at the threshold is relaxed and the firing time is defined as the first time the membrane potential process lies above a fixed depolarisation level for a sufficiently long time. The mathematical properties of the new firing time are investigated both for the Perfect Integrator and the Leaky Integrator. In the latter case, a simulation study is presented to complete the analysis where analytical results are not yet achieved.

Keywords: Ito excursion theory., Perfect Integrator, first passage time, Leaky Integrator, firing time.

Mathematics Subject Classification: 60J99, 92Bx.

Citation: 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

References:

[1]	J. Abate and W. Whitt, The fourier-series method for inverting transforms of probability distributions,, Queueing Systems, 10 (1992), 5. doi: 10.1007/BF01158520. Google Scholar
[2]	J. Abate and W. Whitt, Numerical inversion of laplace transforms of probability distributions,, ORSA Journal on Computing, 7 (1995), 36. doi: 10.1287/ijoc.7.1.36. Google Scholar
[3]	L. Alili, P. Patie and J. L. Pedersen, Representations of the first hitting time density of an Ornstein-Uhlenbeck process,, Stochastic Models, 21 (2005), 967. doi: 10.1080/15326340500294702. Google Scholar
[4]	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
[5]	E. Bibbona and S. Ditlevsen, Estimation in discretely observed diffusions killed at a threshold,, Scandinavian Journal of Statistics, 40 (2013), 274. doi: 10.1111/j.1467-9469.2012.00810.x. Google Scholar
[6]	E. Bibbona, P. Lansky, L. Sacerdote and R. Sirovich, Errors in estimation of the input signal for integrate-and-fire euronal models,, Physical Review E, 78 (2008). Google Scholar
[7]	E. Bibbona, P. Lansky, L. Sacerdote and R. Sirovich, Estimating input parameters from intracellular recordings in the Feller neuronal model,, Physical Review E, 81 (2010). doi: 10.1103/PhysRevE.81.031916. Google Scholar
[8]	A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons,, Mathematical Biosciences and Engineering, 11 (2014), 189. Google Scholar
[9]	A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities,, Advances in Applied Probability, 19 (1987), 784. doi: 10.2307/1427102. Google Scholar
[10]	A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input,, Biological Cybernetics, 95 (2006), 1. doi: 10.1007/s00422-006-0068-6. Google Scholar
[11]	A. N. Burkitt, A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties,, Biological Cybernetics, 95 (2006), 97. doi: 10.1007/s00422-006-0082-8. Google Scholar
[12]	M. J. Caceres and B. Perthame, Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity,, Journal of Theoretical Biology, 350 (2014), 81. doi: 10.1016/j.jtbi.2014.02.005. Google Scholar
[13]	S. Cavallari, S. Panzeri and A. Mazzoni, Comparison of the dynamics of neural interactions between current-based and conductance-based integrate-and-fire recurrent networks,, Frontiers in Neural Circuits, 8 (2014). doi: 10.3389/fncir.2014.00012. Google Scholar
[14]	M. Chesney, M. Jeanblanc-Picqué and M. Yor, Brownian excursions and Parisian barrier options,, Advances in Applied Probabability, 29 (1997), 165. doi: 10.2307/1427865. Google Scholar
[15]	S. Ditlevsen and O. Ditlevsen, Parameter estimation from observations of first-passage times of the Ornstein-Uhlenbeck process and the Feller process,, Probabilistic Engineering Mechanics, 23 (2008), 170. doi: 10.1016/j.probengmech.2007.12.024. Google Scholar
[16]	S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model,, Physical Review. E (3), 71 (2005). doi: 10.1103/PhysRevE.71.011907. Google Scholar
[17]	S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Feller neuronal model,, Physical Review E, 73 (2006). doi: 10.1103/PhysRevE.73.061910. Google Scholar
[18]	G. Dumont and J. Henry, Population density models of integrate-and-fire neurons with jumps: Well-posedness,, Journal of Mathematical Biology, 67 (2013), 453. doi: 10.1007/s00285-012-0554-5. Google Scholar
[19]	G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network,, Bulletin of Mathematical Biology, 75 (2013), 629. doi: 10.1007/s11538-013-9823-8. Google Scholar
[20]	A. Elbert and M. E. Muldoon, Inequalities and monotonicity properties for zeros of hermite functions,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 129 (1999), 57. doi: 10.1017/S0308210500027463. Google Scholar
[21]	G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophysical Journal, 4 (1964), 41. doi: 10.1016/S0006-3495(64)86768-0. Google Scholar
[22]	W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511815706. Google Scholar
[23]	R. K. Getoor, Excursions of a Markov process,, Annals of Probability, 7 (1979), 244. doi: 10.1214/aop/1176995086. Google Scholar
[24]	V. Giorno, G. Nobile, L. M. Ricciardi and S. Sato, On the evaluation of first-passage-time probability densities via non-singular integral,, Advances in Applied Probability, 21 (1989), 20. doi: 10.2307/1427196. Google Scholar
[25]	M. T. Giraudo, P. Greenwood and L. Sacerdote, How sample paths of leaky integrate-and-fire models are influenced by the presence of a firing threshold,, Neural Computation, 23 (2011), 1743. doi: 10.1162/NECO_a_00143. Google Scholar
[26]	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
[27]	D. Grytskyy, T. Tetzlaff, M. Diesmann and M. Helias, A unified view on weakly correlated recurrent networks,, Frontiers in Computational Neuroscience, 7 (2013). doi: 10.3389/fncom.2013.00131. Google Scholar
[28]	J. Inoue, S. Sato and L. M. Ricciardi, On the parameter estimation for diffusion models of single neuron's activities,, Biological Cybernetics, 73 (1995), 209. doi: 10.1007/BF00201423. Google Scholar
[29]	K. Itô, Poisson point processes attached to Markov processes,, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, (1970), 225. Google Scholar
[30]	R. Jolivet, A. Rauch, H. Lüscher and W. Gerstner, Integrate-and-fire models with adaptation are good enough,, in Advances in Neural Information Processing Systems 18* (eds. Y. Weiss*, (2006), 595. Google Scholar
[31]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Vol. 113,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar
[32]	R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold,, Frontiers in Computational Neuroscience, 3 (2009). doi: 10.3389/neuro.10.009.2009. Google Scholar
[33]	A. Koutsou, J. Kanev and C. Christodoulou, Measuring input synchrony in the Ornstein-Uhlenbeck neuronal model through input parameter estimation,, Brain Research, 1536 (2013), 97. doi: 10.1016/j.brainres.2013.05.012. Google Scholar
[34]	P. Lansky, Inference for the diffusion models of neuronal activity,, Mathematical Bioscience, 67 (1983), 247. doi: 10.1016/0025-5564(83)90103-7. Google Scholar
[35]	P. Lansky and S. Ditlevsen, A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models,, Biological Cybernetics, 99 (2008), 253. doi: 10.1007/s00422-008-0237-x. Google Scholar
[36]	P. Lánskỳ, R. Rodriguez and L. Sacerdote, Mean instantaneous firing frequency is always higher than the firing rate,, Neural Computation, 16 (2004), 477. Google Scholar
[37]	P. Lansky, P. Sanda and J. He, The parameters of the stochastic leaky integrate-and-fire neuronal model,, Journal of Computational Neuroscence, 21 (2006), 211. doi: 10.1007/s10827-006-8527-6. Google Scholar
[38]	N. Lebedev, Special Functions and Their Applications,, Courier Corporation, (1972). Google Scholar
[39]	B. Lindner, M. J. Chacron and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable model of how interspike interval correlations affect neuronal signal transmission,, Physical Review E, 72 (2005). doi: 10.1103/PhysRevE.72.021911. Google Scholar
[40]	B. Øksendal, Stochastic Differential Equations,, Springer-Verlag, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar
[41]	J. Pitman and M. Yor, Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches,, Bernoulli, 9 (2003), 1. doi: 10.3150/bj/1068129008. Google Scholar
[42]	L. M. Ricciardi, Diffusion Processes and Related Topics in Biology,, Springer-Verlag, (1977). Google Scholar
[43]	L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity,, Biological Cybernetics, 35 (1979), 1. doi: 10.1007/BF01845839. Google Scholar
[44]	M. J. Richardson, Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive,, Physical Review E, 76 (2007). doi: 10.1103/PhysRevE.76.021919. Google Scholar
[45]	L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2,, Cambridge University Press, (2000). Google Scholar
[46]	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
[47]	S. Sato, On the moments of the firing interval of the diffusion approximated model neuron,, Mathematical Bioscience, 39 (1978), 53. doi: 10.1016/0025-5564(78)90027-5. Google Scholar
[48]	M. Tamborrino, S. Ditlevsen and P. Lansky, Parameter inference from hitting times for perturbed Brownian motion,, Lifetime Data Analysis, 21 (2015), 331. doi: 10.1007/s10985-014-9307-7. Google Scholar
[49]	M. Tamborrino, L. Sacerdote and M. Jacobsen, Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling,, Physica D: Nonlinear Phenomena, 288 (2014), 45. doi: 10.1016/j.physd.2014.08.003. Google Scholar
[50]	H. C. Tuckwell, Introduction to Theoretical Neurobiology. Vol. 1. Linear Cable Theory and Dendritic Structure,, Cambridge Studies in Mathematical Biology, (1988). Google Scholar
[51]	H. C. Tuckwell, Introduction to theoretical neurobiology. Vol. 2. Nonlinear and Stochastic Theories,, Cambridge Studies in Mathematical Biology, (1988). Google Scholar
[52]	Y. Yu, Y. Xiong, Y. Chan and J. He, Corticofugal gating of auditory information in the thalamus: An in vivo intracellular recording study,, The Journal of Neuroscience, 24 (2004), 3060. doi: 10.1523/JNEUROSCI.4897-03.2004. Google Scholar

show all references

References:

[1]	J. Abate and W. Whitt, The fourier-series method for inverting transforms of probability distributions,, Queueing Systems, 10 (1992), 5. doi: 10.1007/BF01158520. Google Scholar
[2]	J. Abate and W. Whitt, Numerical inversion of laplace transforms of probability distributions,, ORSA Journal on Computing, 7 (1995), 36. doi: 10.1287/ijoc.7.1.36. Google Scholar
[3]	L. Alili, P. Patie and J. L. Pedersen, Representations of the first hitting time density of an Ornstein-Uhlenbeck process,, Stochastic Models, 21 (2005), 967. doi: 10.1080/15326340500294702. Google Scholar
[4]	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
[5]	E. Bibbona and S. Ditlevsen, Estimation in discretely observed diffusions killed at a threshold,, Scandinavian Journal of Statistics, 40 (2013), 274. doi: 10.1111/j.1467-9469.2012.00810.x. Google Scholar
[6]	E. Bibbona, P. Lansky, L. Sacerdote and R. Sirovich, Errors in estimation of the input signal for integrate-and-fire euronal models,, Physical Review E, 78 (2008). Google Scholar
[7]	E. Bibbona, P. Lansky, L. Sacerdote and R. Sirovich, Estimating input parameters from intracellular recordings in the Feller neuronal model,, Physical Review E, 81 (2010). doi: 10.1103/PhysRevE.81.031916. Google Scholar
[8]	A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons,, Mathematical Biosciences and Engineering, 11 (2014), 189. Google Scholar
[9]	A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities,, Advances in Applied Probability, 19 (1987), 784. doi: 10.2307/1427102. Google Scholar
[10]	A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input,, Biological Cybernetics, 95 (2006), 1. doi: 10.1007/s00422-006-0068-6. Google Scholar
[11]	A. N. Burkitt, A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties,, Biological Cybernetics, 95 (2006), 97. doi: 10.1007/s00422-006-0082-8. Google Scholar
[12]	M. J. Caceres and B. Perthame, Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity,, Journal of Theoretical Biology, 350 (2014), 81. doi: 10.1016/j.jtbi.2014.02.005. Google Scholar
[13]	S. Cavallari, S. Panzeri and A. Mazzoni, Comparison of the dynamics of neural interactions between current-based and conductance-based integrate-and-fire recurrent networks,, Frontiers in Neural Circuits, 8 (2014). doi: 10.3389/fncir.2014.00012. Google Scholar
[14]	M. Chesney, M. Jeanblanc-Picqué and M. Yor, Brownian excursions and Parisian barrier options,, Advances in Applied Probabability, 29 (1997), 165. doi: 10.2307/1427865. Google Scholar
[15]	S. Ditlevsen and O. Ditlevsen, Parameter estimation from observations of first-passage times of the Ornstein-Uhlenbeck process and the Feller process,, Probabilistic Engineering Mechanics, 23 (2008), 170. doi: 10.1016/j.probengmech.2007.12.024. Google Scholar
[16]	S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model,, Physical Review. E (3), 71 (2005). doi: 10.1103/PhysRevE.71.011907. Google Scholar
[17]	S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Feller neuronal model,, Physical Review E, 73 (2006). doi: 10.1103/PhysRevE.73.061910. Google Scholar
[18]	G. Dumont and J. Henry, Population density models of integrate-and-fire neurons with jumps: Well-posedness,, Journal of Mathematical Biology, 67 (2013), 453. doi: 10.1007/s00285-012-0554-5. Google Scholar
[19]	G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network,, Bulletin of Mathematical Biology, 75 (2013), 629. doi: 10.1007/s11538-013-9823-8. Google Scholar
[20]	A. Elbert and M. E. Muldoon, Inequalities and monotonicity properties for zeros of hermite functions,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 129 (1999), 57. doi: 10.1017/S0308210500027463. Google Scholar
[21]	G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophysical Journal, 4 (1964), 41. doi: 10.1016/S0006-3495(64)86768-0. Google Scholar
[22]	W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511815706. Google Scholar
[23]	R. K. Getoor, Excursions of a Markov process,, Annals of Probability, 7 (1979), 244. doi: 10.1214/aop/1176995086. Google Scholar
[24]	V. Giorno, G. Nobile, L. M. Ricciardi and S. Sato, On the evaluation of first-passage-time probability densities via non-singular integral,, Advances in Applied Probability, 21 (1989), 20. doi: 10.2307/1427196. Google Scholar
[25]	M. T. Giraudo, P. Greenwood and L. Sacerdote, How sample paths of leaky integrate-and-fire models are influenced by the presence of a firing threshold,, Neural Computation, 23 (2011), 1743. doi: 10.1162/NECO_a_00143. Google Scholar
[26]	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
[27]	D. Grytskyy, T. Tetzlaff, M. Diesmann and M. Helias, A unified view on weakly correlated recurrent networks,, Frontiers in Computational Neuroscience, 7 (2013). doi: 10.3389/fncom.2013.00131. Google Scholar
[28]	J. Inoue, S. Sato and L. M. Ricciardi, On the parameter estimation for diffusion models of single neuron's activities,, Biological Cybernetics, 73 (1995), 209. doi: 10.1007/BF00201423. Google Scholar
[29]	K. Itô, Poisson point processes attached to Markov processes,, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, (1970), 225. Google Scholar
[30]	R. Jolivet, A. Rauch, H. Lüscher and W. Gerstner, Integrate-and-fire models with adaptation are good enough,, in Advances in Neural Information Processing Systems 18* (eds. Y. Weiss*, (2006), 595. Google Scholar
[31]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Vol. 113,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar
[32]	R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold,, Frontiers in Computational Neuroscience, 3 (2009). doi: 10.3389/neuro.10.009.2009. Google Scholar
[33]	A. Koutsou, J. Kanev and C. Christodoulou, Measuring input synchrony in the Ornstein-Uhlenbeck neuronal model through input parameter estimation,, Brain Research, 1536 (2013), 97. doi: 10.1016/j.brainres.2013.05.012. Google Scholar
[34]	P. Lansky, Inference for the diffusion models of neuronal activity,, Mathematical Bioscience, 67 (1983), 247. doi: 10.1016/0025-5564(83)90103-7. Google Scholar
[35]	P. Lansky and S. Ditlevsen, A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models,, Biological Cybernetics, 99 (2008), 253. doi: 10.1007/s00422-008-0237-x. Google Scholar
[36]	P. Lánskỳ, R. Rodriguez and L. Sacerdote, Mean instantaneous firing frequency is always higher than the firing rate,, Neural Computation, 16 (2004), 477. Google Scholar
[37]	P. Lansky, P. Sanda and J. He, The parameters of the stochastic leaky integrate-and-fire neuronal model,, Journal of Computational Neuroscence, 21 (2006), 211. doi: 10.1007/s10827-006-8527-6. Google Scholar
[38]	N. Lebedev, Special Functions and Their Applications,, Courier Corporation, (1972). Google Scholar
[39]	B. Lindner, M. J. Chacron and A. Longtin, Integrate-and-fire neurons with threshold noise: A tractable model of how interspike interval correlations affect neuronal signal transmission,, Physical Review E, 72 (2005). doi: 10.1103/PhysRevE.72.021911. Google Scholar
[40]	B. Øksendal, Stochastic Differential Equations,, Springer-Verlag, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar
[41]	J. Pitman and M. Yor, Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches,, Bernoulli, 9 (2003), 1. doi: 10.3150/bj/1068129008. Google Scholar
[42]	L. M. Ricciardi, Diffusion Processes and Related Topics in Biology,, Springer-Verlag, (1977). Google Scholar
[43]	L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity,, Biological Cybernetics, 35 (1979), 1. doi: 10.1007/BF01845839. Google Scholar
[44]	M. J. Richardson, Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive,, Physical Review E, 76 (2007). doi: 10.1103/PhysRevE.76.021919. Google Scholar
[45]	L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2,, Cambridge University Press, (2000). Google Scholar
[46]	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
[47]	S. Sato, On the moments of the firing interval of the diffusion approximated model neuron,, Mathematical Bioscience, 39 (1978), 53. doi: 10.1016/0025-5564(78)90027-5. Google Scholar
[48]	M. Tamborrino, S. Ditlevsen and P. Lansky, Parameter inference from hitting times for perturbed Brownian motion,, Lifetime Data Analysis, 21 (2015), 331. doi: 10.1007/s10985-014-9307-7. Google Scholar
[49]	M. Tamborrino, L. Sacerdote and M. Jacobsen, Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling,, Physica D: Nonlinear Phenomena, 288 (2014), 45. doi: 10.1016/j.physd.2014.08.003. Google Scholar
[50]	H. C. Tuckwell, Introduction to Theoretical Neurobiology. Vol. 1. Linear Cable Theory and Dendritic Structure,, Cambridge Studies in Mathematical Biology, (1988). Google Scholar
[51]	H. C. Tuckwell, Introduction to theoretical neurobiology. Vol. 2. Nonlinear and Stochastic Theories,, Cambridge Studies in Mathematical Biology, (1988). Google Scholar
[52]	Y. Yu, Y. Xiong, Y. Chan and J. He, Corticofugal gating of auditory information in the thalamus: An in vivo intracellular recording study,, The Journal of Neuroscience, 24 (2004), 3060. doi: 10.1523/JNEUROSCI.4897-03.2004. Google Scholar

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