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

## A new firing paradigm for integrate and fire stochastic neuronal models

 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

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.
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), 5-87. doi: 10.1007/BF01158520. [2] J. Abate and W. Whitt, Numerical inversion of laplace transforms of probability distributions, ORSA Journal on Computing, 7 (1995), 36-43. doi: 10.1287/ijoc.7.1.36. [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), 967-980. doi: 10.1080/15326340500294702. [4] 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. [5] E. Bibbona and S. Ditlevsen, Estimation in discretely observed diffusions killed at a threshold, Scandinavian Journal of Statistics, 40 (2013), 274-293. doi: 10.1111/j.1467-9469.2012.00810.x. [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), 011918. [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), 031916. doi: 10.1103/PhysRevE.81.031916. [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), 189-201. [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), 784-800. doi: 10.2307/1427102. [10] A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biological Cybernetics, 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6. [11] A. N. Burkitt, A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties, Biological Cybernetics, 95 (2006), 97-112. doi: 10.1007/s00422-006-0082-8. [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), 81-89. doi: 10.1016/j.jtbi.2014.02.005. [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), p11. doi: 10.3389/fncir.2014.00012. [14] M. Chesney, M. Jeanblanc-Picqué and M. Yor, Brownian excursions and Parisian barrier options, Advances in Applied Probabability, 29 (1997), 165-184. doi: 10.2307/1427865. [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), 170-179. doi: 10.1016/j.probengmech.2007.12.024. [16] S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model, Physical Review. E (3), 71 (2005), 011907, 9pp. doi: 10.1103/PhysRevE.71.011907. [17] S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Feller neuronal model, Physical Review E, 73 (2006), 061910, 9pp. doi: 10.1103/PhysRevE.73.061910. [18] G. Dumont and J. Henry, Population density models of integrate-and-fire neurons with jumps: Well-posedness, Journal of Mathematical Biology, 67 (2013), 453-481. doi: 10.1007/s00285-012-0554-5. [19] G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network, Bulletin of Mathematical Biology, 75 (2013), 629-648. doi: 10.1007/s11538-013-9823-8. [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), 57-75. doi: 10.1017/S0308210500027463. [21] G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophysical Journal, 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0. [22] W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity, Cambridge University Press, 2002. doi: 10.1017/CBO9780511815706. [23] R. K. Getoor, Excursions of a Markov process, Annals of Probability, 7 (1979), 244-266. doi: 10.1214/aop/1176995086. [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), 20-36. doi: 10.2307/1427196. [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), 1743-1767. doi: 10.1162/NECO_a_00143. [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), 1135-1163. doi: 10.1080/03610919908813596. [27] D. Grytskyy, T. Tetzlaff, M. Diesmann and M. Helias, A unified view on weakly correlated recurrent networks, Frontiers in Computational Neuroscience, 7 (2013), p131. doi: 10.3389/fncom.2013.00131. [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), 209-221. doi: 10.1007/BF00201423. [29] K. Itô, Poisson point processes attached to Markov processes, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, Univ. California Press, Berkeley, Calif., 1972, 225-239. [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, B. Sch\"olkopf and J. Platt), MIT Press, 2006, 595-602. [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. [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), p9. doi: 10.3389/neuro.10.009.2009. [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), 97-106. doi: 10.1016/j.brainres.2013.05.012. [34] P. Lansky, Inference for the diffusion models of neuronal activity, Mathematical Bioscience, 67 (1983), 247-260. doi: 10.1016/0025-5564(83)90103-7. [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), 253-262. doi: 10.1007/s00422-008-0237-x. [36] P. Lánskỳ, R. Rodriguez and L. Sacerdote, Mean instantaneous firing frequency is always higher than the firing rate, Neural Computation, 16 (2004), 477-489. [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), 211-223. doi: 10.1007/s10827-006-8527-6. [38] N. Lebedev, Special Functions and Their Applications, Courier Corporation, 1972. [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), 021911, 21pp. doi: 10.1103/PhysRevE.72.021911. [40] B. Øksendal, Stochastic Differential Equations, Springer-Verlag, 2003. doi: 10.1007/978-3-642-14394-6. [41] J. Pitman and M. Yor, Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches, Bernoulli, 9 (2003), 1-24. doi: 10.3150/bj/1068129008. [42] L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Springer-Verlag, Berlin-New York, 1977. [43] L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity, Biological Cybernetics, 35 (1979), 1-9. doi: 10.1007/BF01845839. [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), 021919. doi: 10.1103/PhysRevE.76.021919. [45] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2, Cambridge University Press, Cambridge, 2000. [46] L. Sacerdote and M. T. Giraudo, Stochastic integrate and fire models: A review on mathematical methods and their applications, in Stochastic Biomathematical Models, Lecture Notes in Math., 2058, Springer, Heidelberg, 2013, 99-148. doi: 10.1007/978-3-642-32157-3_5. [47] S. Sato, On the moments of the firing interval of the diffusion approximated model neuron, Mathematical Bioscience, 39 (1978), 53-70. doi: 10.1016/0025-5564(78)90027-5. [48] M. Tamborrino, S. Ditlevsen and P. Lansky, Parameter inference from hitting times for perturbed Brownian motion, Lifetime Data Analysis, 21 (2015), 331-352. doi: 10.1007/s10985-014-9307-7. [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), 45-52. doi: 10.1016/j.physd.2014.08.003. [50] H. C. Tuckwell, Introduction to Theoretical Neurobiology. Vol. 1. Linear Cable Theory and Dendritic Structure, Cambridge Studies in Mathematical Biology, 8, Cambridge University Press, Cambridge, 1988. [51] H. C. Tuckwell, Introduction to theoretical neurobiology. Vol. 2. Nonlinear and Stochastic Theories, Cambridge Studies in Mathematical Biology, 8, Cambridge University Press, Cambridge, 1988. [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), 3060-3069. doi: 10.1523/JNEUROSCI.4897-03.2004.

show all references

##### References:
 [1] J. Abate and W. Whitt, The fourier-series method for inverting transforms of probability distributions, Queueing Systems, 10 (1992), 5-87. doi: 10.1007/BF01158520. [2] J. Abate and W. Whitt, Numerical inversion of laplace transforms of probability distributions, ORSA Journal on Computing, 7 (1995), 36-43. doi: 10.1287/ijoc.7.1.36. [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), 967-980. doi: 10.1080/15326340500294702. [4] 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. [5] E. Bibbona and S. Ditlevsen, Estimation in discretely observed diffusions killed at a threshold, Scandinavian Journal of Statistics, 40 (2013), 274-293. doi: 10.1111/j.1467-9469.2012.00810.x. [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), 011918. [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), 031916. doi: 10.1103/PhysRevE.81.031916. [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), 189-201. [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), 784-800. doi: 10.2307/1427102. [10] A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biological Cybernetics, 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6. [11] A. N. Burkitt, A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties, Biological Cybernetics, 95 (2006), 97-112. doi: 10.1007/s00422-006-0082-8. [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), 81-89. doi: 10.1016/j.jtbi.2014.02.005. [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), p11. doi: 10.3389/fncir.2014.00012. [14] M. Chesney, M. Jeanblanc-Picqué and M. Yor, Brownian excursions and Parisian barrier options, Advances in Applied Probabability, 29 (1997), 165-184. doi: 10.2307/1427865. [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), 170-179. doi: 10.1016/j.probengmech.2007.12.024. [16] S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model, Physical Review. E (3), 71 (2005), 011907, 9pp. doi: 10.1103/PhysRevE.71.011907. [17] S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Feller neuronal model, Physical Review E, 73 (2006), 061910, 9pp. doi: 10.1103/PhysRevE.73.061910. [18] G. Dumont and J. Henry, Population density models of integrate-and-fire neurons with jumps: Well-posedness, Journal of Mathematical Biology, 67 (2013), 453-481. doi: 10.1007/s00285-012-0554-5. [19] G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network, Bulletin of Mathematical Biology, 75 (2013), 629-648. doi: 10.1007/s11538-013-9823-8. [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), 57-75. doi: 10.1017/S0308210500027463. [21] G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophysical Journal, 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0. [22] W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity, Cambridge University Press, 2002. doi: 10.1017/CBO9780511815706. [23] R. K. Getoor, Excursions of a Markov process, Annals of Probability, 7 (1979), 244-266. doi: 10.1214/aop/1176995086. [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), 20-36. doi: 10.2307/1427196. [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), 1743-1767. doi: 10.1162/NECO_a_00143. [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), 1135-1163. doi: 10.1080/03610919908813596. [27] D. Grytskyy, T. Tetzlaff, M. Diesmann and M. Helias, A unified view on weakly correlated recurrent networks, Frontiers in Computational Neuroscience, 7 (2013), p131. doi: 10.3389/fncom.2013.00131. [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), 209-221. doi: 10.1007/BF00201423. [29] K. Itô, Poisson point processes attached to Markov processes, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, Univ. California Press, Berkeley, Calif., 1972, 225-239. [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, B. Sch\"olkopf and J. Platt), MIT Press, 2006, 595-602. [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. [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), p9. doi: 10.3389/neuro.10.009.2009. [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), 97-106. doi: 10.1016/j.brainres.2013.05.012. [34] P. Lansky, Inference for the diffusion models of neuronal activity, Mathematical Bioscience, 67 (1983), 247-260. doi: 10.1016/0025-5564(83)90103-7. [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), 253-262. doi: 10.1007/s00422-008-0237-x. [36] P. Lánskỳ, R. Rodriguez and L. Sacerdote, Mean instantaneous firing frequency is always higher than the firing rate, Neural Computation, 16 (2004), 477-489. [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), 211-223. doi: 10.1007/s10827-006-8527-6. [38] N. Lebedev, Special Functions and Their Applications, Courier Corporation, 1972. [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), 021911, 21pp. doi: 10.1103/PhysRevE.72.021911. [40] B. Øksendal, Stochastic Differential Equations, Springer-Verlag, 2003. doi: 10.1007/978-3-642-14394-6. [41] J. Pitman and M. Yor, Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches, Bernoulli, 9 (2003), 1-24. doi: 10.3150/bj/1068129008. [42] L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Springer-Verlag, Berlin-New York, 1977. [43] L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity, Biological Cybernetics, 35 (1979), 1-9. doi: 10.1007/BF01845839. [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), 021919. doi: 10.1103/PhysRevE.76.021919. [45] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2, Cambridge University Press, Cambridge, 2000. [46] L. Sacerdote and M. T. Giraudo, Stochastic integrate and fire models: A review on mathematical methods and their applications, in Stochastic Biomathematical Models, Lecture Notes in Math., 2058, Springer, Heidelberg, 2013, 99-148. doi: 10.1007/978-3-642-32157-3_5. [47] S. Sato, On the moments of the firing interval of the diffusion approximated model neuron, Mathematical Bioscience, 39 (1978), 53-70. doi: 10.1016/0025-5564(78)90027-5. [48] M. Tamborrino, S. Ditlevsen and P. Lansky, Parameter inference from hitting times for perturbed Brownian motion, Lifetime Data Analysis, 21 (2015), 331-352. doi: 10.1007/s10985-014-9307-7. [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), 45-52. doi: 10.1016/j.physd.2014.08.003. [50] H. C. Tuckwell, Introduction to Theoretical Neurobiology. Vol. 1. Linear Cable Theory and Dendritic Structure, Cambridge Studies in Mathematical Biology, 8, Cambridge University Press, Cambridge, 1988. [51] H. C. Tuckwell, Introduction to theoretical neurobiology. Vol. 2. Nonlinear and Stochastic Theories, Cambridge Studies in Mathematical Biology, 8, Cambridge University Press, Cambridge, 1988. [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), 3060-3069. doi: 10.1523/JNEUROSCI.4897-03.2004.
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