# American Institute of Mathematical Sciences

September  2017, 10(3): 587-612. doi: 10.3934/krm.2017024

## Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models

 1 Departamento de Matemática Aplicada, Universidad de Granada, E-18071 Granada, Spain 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 - Bellaterra, Spain

* Corresponding author: caceresg@ugr.es

Received  February 2016 Revised  August 2016 Published  December 2016

Fund Project: The authors acknowledge support from projects MTM2011-27739-C04-02 and MTM2014-52056-P of Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (ERDF/FEDER). The second author was also sponsored by the grant BES-2012-057704.

Excitatory and inhibitory nonlinear noisy leaky integrate and fire models are often used to describe neural networks. Recently, new mathematical results have provided a better understanding of them. It has been proved that a fully excitatory network can blow-up in finite time, while a fully inhibitory network has a global in time solution for any initial data. A general description of the steady states of a purely excitatory or inhibitory network has been also given. We extend this study to the system composed of an excitatory population and an inhibitory one. We prove that this system can also blow-up in finite time and analyse its steady states and long time behaviour. Besides, we illustrate our analytical description with some numerical results. The main tools used to reach our aims are: the control of an exponential moment for the blow-up results, a more complicate strategy than that considered in [5] for studying the number of steady states, entropy methods combined with Poincaré inequalities for the long time behaviour and, finally, high order numerical schemes together with parallel computation techniques in order to obtain our numerical results.

Citation: María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic and Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024
##### References:
 [1] L. Albantakis and G. Deco, The encoding of alternatives in multiple-choice decision making, Proc Natl Acad Sci U S A, 106 (2009), 10308-10313, URL http://www.pnas.org/content/106/25/10308.full.pdf. doi: 10.1073/pnas.0901621106. [2] R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neural activity, Journal of Neurophysiology, 94 (2005), 3637-3642. [3] N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking networks, J. Comp. Neurosci., 8 (2000), 183-208. [4] N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with long firing rates, Neural Computation, 11 (1999), 1621-1671. [5] M. J. Cáceres, J. A. Carrillo and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states, Journal of Mathematical Neuroscience, 1 (2011), Art. 7, 33 pp.  doi: 10.1186/2190-8567-1-7. [6] M. J. Cáceres, J. A. Carrillo and L. Tao, A numerical solver for a nonlinear fokker-planck equation representation of neuronal network dynamics, J. Comp. Phys., 230 (2011), 1084-1099.  doi: 10.1016/j.jcp.2010.10.027. [7] M. J. Cáceres 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. [8] J. A. Carrillo, M. d. M. González, M. P. Gualdani and M. E. Schonbek, Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience, Comm. in Partial Differential Equations, 38 (2013), 385-409.  doi: 10.1080/03605302.2012.747536. [9] J. A. Carrillo, B. Perthame, D. Salort and D. Smets, Qualitative properties of solutions for the noisy integrate & fire model in computational neuroscience, Nonlinearity, 28 (2015), 3365-3388.  doi: 10.1088/0951-7715/28/9/3365. [10] J. Chevallier, Mean-field limit of generalized hawkes processes, arXiv preprint, arXiv: 1510.05620. [11] J. Chevallier, M. J. Cáceres, M. Doumic and P. Reynaud-Bouret, Microscopic approach of a time elapsed neural model, Mathematical Models and Methods in Applied Sciences, 25 (2015), 2669-2719.  doi: 10.1142/S021820251550058X. [12] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Global solvability of a networked integrate-and-fire model of mckean-vlasov type, The Annals of Applied Probability, 25 (2015), 2096-2133.  doi: 10.1214/14-AAP1044. [13] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Particle systems with a singular mean-field self-excitation. application to neuronal networks, Stochastic Processes and their Applications, 125 (2015), 2451-2492.  doi: 10.1016/j.spa.2015.01.007. [14] G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network, Bull. Math. Biol., 75 (2013), 629-648.  doi: 10.1007/s11538-013-9823-8. [15] G. Dumont, J. Henry and C. O. Tarniceriu, Noisy threshold in neuronal models: Connections with the noisy leaky integrate-and-fire model, J. Math. Biol., 73 (2016), 1413-1436, arXiv: 1512.03785. doi: 10.1007/s00285-016-1002-8. [16] W. Gerstner and W. Kistler, Spiking Neuron Models, Cambridge Univ. Press, Cambridge, 2002.  doi: 10.1017/CBO9780511815706. [17] M. d. M. González and M. P. Gualdani, Asymptotics for a symmetric equation in price formation, App. Math. Optim., 59 (2009), 233-246.  doi: 10.1007/s00245-008-9052-y. [18] C. M. Gray and W. Singer, Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex, Proc Natl Acad Sci U S A, 86 (1989), 1698-1702. [19] T. Guillamon, An introduction to the mathematics of neural activity, Butl. Soc. Catalana Mat., 19 (2004), 25-45. [20] M Mattia and P. Del Giudice, Population dynamics of interacting spiking neurons, Phys. Rev. E, 66 (2002), 051917, 19pp.  doi: 10.1103/PhysRevE.66.051917. [21] A. Omurtag, K. B. W. Sirovich and L. Sirovich, On the simulation of large populations of neurons, J. Comp. Neurosci., 8 (2000), 51-63. [22] K. Pakdaman, B. Perthame and D. Salort, Dynamics of a structured neuron population, Nonlinearity, 23 (2010), 55-75.  doi: 10.1088/0951-7715/23/1/003. [23] K. Pakdaman, B. Perthame and D. Salort, Relaxation and self-sustained oscillations in the time elapsed neuron network model, SIAM Journal on Applied Mathematics, 73 (2013), 1260-1279.  doi: 10.1137/110847962. [24] K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, The Journal of Mathematical Neuroscience (JMN), 4 (2014), 1-26.  doi: 10.1186/2190-8567-4-14. [25] B. Perthame and D. Salort, On a voltage-conductance kinetic system for integrate and fire neural networks, Kinetic and Related Models, AIMS, 6 (2013), 841-864.  doi: 10.3934/krm.2013.6.841. [26] A. V. Rangan, G. Kovačič and D. Cai, Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train, Physical Review E, 7 (2008), 041915, 13 pp.  doi: 10.1103/PhysRevE.77.041915. [27] A. Renart, N. Brunel and X. -J. Wang, Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks, in Computational Neuroscience: A comprehensive approach (ed. J. Feng), Chapman & Hall/CRC Mathematical Biology and Medicine Series, (2004), 431-490. [28] H. Risken, The Fokker-Planck Equation: Methods of Solution and Approximations, 2nd. edn. Springer Series in Synergetics, vol 18. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3. [29] C. Rossant, D. F. M. Goodman, B. Fontaine, J. Platkiewicz, A. K. Magnusson and R. Brette, Fitting neuron models to spike trains, Frontiers in Neuroscience, 5 (2011), 1-8. [30] J. Touboul, Importance of the cutoff value in the quadratic adaptive integrate-and-fire model, Neural Computation, 21 (2009), 2114-2122.  doi: 10.1162/neco.2009.09-08-853. [31] J. Touboul, Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons, SIAM J. Appl. Math., 68 (2008), 1045-1079, URL http://dx.doi.org/10.1137/070687268. doi: 10.1137/070687268. [32] H. Tuckwell, Introduction to Theoretical Neurobiology, Cambridge Univ. Press, Cambridge, 1988.

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##### References:
 [1] L. Albantakis and G. Deco, The encoding of alternatives in multiple-choice decision making, Proc Natl Acad Sci U S A, 106 (2009), 10308-10313, URL http://www.pnas.org/content/106/25/10308.full.pdf. doi: 10.1073/pnas.0901621106. [2] R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neural activity, Journal of Neurophysiology, 94 (2005), 3637-3642. [3] N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking networks, J. Comp. Neurosci., 8 (2000), 183-208. [4] N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with long firing rates, Neural Computation, 11 (1999), 1621-1671. [5] M. J. Cáceres, J. A. Carrillo and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states, Journal of Mathematical Neuroscience, 1 (2011), Art. 7, 33 pp.  doi: 10.1186/2190-8567-1-7. [6] M. J. Cáceres, J. A. Carrillo and L. Tao, A numerical solver for a nonlinear fokker-planck equation representation of neuronal network dynamics, J. Comp. Phys., 230 (2011), 1084-1099.  doi: 10.1016/j.jcp.2010.10.027. [7] M. J. Cáceres 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. [8] J. A. Carrillo, M. d. M. González, M. P. Gualdani and M. E. Schonbek, Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience, Comm. in Partial Differential Equations, 38 (2013), 385-409.  doi: 10.1080/03605302.2012.747536. [9] J. A. Carrillo, B. Perthame, D. Salort and D. Smets, Qualitative properties of solutions for the noisy integrate & fire model in computational neuroscience, Nonlinearity, 28 (2015), 3365-3388.  doi: 10.1088/0951-7715/28/9/3365. [10] J. Chevallier, Mean-field limit of generalized hawkes processes, arXiv preprint, arXiv: 1510.05620. [11] J. Chevallier, M. J. Cáceres, M. Doumic and P. Reynaud-Bouret, Microscopic approach of a time elapsed neural model, Mathematical Models and Methods in Applied Sciences, 25 (2015), 2669-2719.  doi: 10.1142/S021820251550058X. [12] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Global solvability of a networked integrate-and-fire model of mckean-vlasov type, The Annals of Applied Probability, 25 (2015), 2096-2133.  doi: 10.1214/14-AAP1044. [13] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Particle systems with a singular mean-field self-excitation. application to neuronal networks, Stochastic Processes and their Applications, 125 (2015), 2451-2492.  doi: 10.1016/j.spa.2015.01.007. [14] G. Dumont and J. Henry, Synchronization of an excitatory integrate-and-fire neural network, Bull. Math. Biol., 75 (2013), 629-648.  doi: 10.1007/s11538-013-9823-8. [15] G. Dumont, J. Henry and C. O. Tarniceriu, Noisy threshold in neuronal models: Connections with the noisy leaky integrate-and-fire model, J. Math. Biol., 73 (2016), 1413-1436, arXiv: 1512.03785. doi: 10.1007/s00285-016-1002-8. [16] W. Gerstner and W. Kistler, Spiking Neuron Models, Cambridge Univ. Press, Cambridge, 2002.  doi: 10.1017/CBO9780511815706. [17] M. d. M. González and M. P. Gualdani, Asymptotics for a symmetric equation in price formation, App. Math. Optim., 59 (2009), 233-246.  doi: 10.1007/s00245-008-9052-y. [18] C. M. Gray and W. Singer, Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex, Proc Natl Acad Sci U S A, 86 (1989), 1698-1702. [19] T. Guillamon, An introduction to the mathematics of neural activity, Butl. Soc. Catalana Mat., 19 (2004), 25-45. [20] M Mattia and P. Del Giudice, Population dynamics of interacting spiking neurons, Phys. Rev. E, 66 (2002), 051917, 19pp.  doi: 10.1103/PhysRevE.66.051917. [21] A. Omurtag, K. B. W. Sirovich and L. Sirovich, On the simulation of large populations of neurons, J. Comp. Neurosci., 8 (2000), 51-63. [22] K. Pakdaman, B. Perthame and D. Salort, Dynamics of a structured neuron population, Nonlinearity, 23 (2010), 55-75.  doi: 10.1088/0951-7715/23/1/003. [23] K. Pakdaman, B. Perthame and D. Salort, Relaxation and self-sustained oscillations in the time elapsed neuron network model, SIAM Journal on Applied Mathematics, 73 (2013), 1260-1279.  doi: 10.1137/110847962. [24] K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, The Journal of Mathematical Neuroscience (JMN), 4 (2014), 1-26.  doi: 10.1186/2190-8567-4-14. [25] B. Perthame and D. Salort, On a voltage-conductance kinetic system for integrate and fire neural networks, Kinetic and Related Models, AIMS, 6 (2013), 841-864.  doi: 10.3934/krm.2013.6.841. [26] A. V. Rangan, G. Kovačič and D. Cai, Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train, Physical Review E, 7 (2008), 041915, 13 pp.  doi: 10.1103/PhysRevE.77.041915. [27] A. Renart, N. Brunel and X. -J. Wang, Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks, in Computational Neuroscience: A comprehensive approach (ed. J. Feng), Chapman & Hall/CRC Mathematical Biology and Medicine Series, (2004), 431-490. [28] H. Risken, The Fokker-Planck Equation: Methods of Solution and Approximations, 2nd. edn. Springer Series in Synergetics, vol 18. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3. [29] C. Rossant, D. F. M. Goodman, B. Fontaine, J. Platkiewicz, A. K. Magnusson and R. Brette, Fitting neuron models to spike trains, Frontiers in Neuroscience, 5 (2011), 1-8. [30] J. Touboul, Importance of the cutoff value in the quadratic adaptive integrate-and-fire model, Neural Computation, 21 (2009), 2114-2122.  doi: 10.1162/neco.2009.09-08-853. [31] J. Touboul, Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons, SIAM J. Appl. Math., 68 (2008), 1045-1079, URL http://dx.doi.org/10.1137/070687268. doi: 10.1137/070687268. [32] H. Tuckwell, Introduction to Theoretical Neurobiology, Cambridge Univ. Press, Cambridge, 1988.
Firing rates and probability densities for $b_E^E=3$, $b_I^E=0.75$, $b_E ^I=0.5$, $b_I^I=0.25$, in case of a normalized Maxwellian initial condition with mean 0 and variance 0.5 (see (39)). $N_E$ blows-up because of the large value of $b_E ^E$
Firing rates and probability densities for $b_E^E=0.5$, $b_I^E=0.25$, $b_E ^I=0.25$, $b_I^I=1$, in case of a normalized concentrated Maxwellian initial condition with mean 1.83 and variance 0.003 (see (39)).The initial condition concentrated close to $V_F$ provokes the blow-up of $N_E$
Firing rates and probability densities for $b_E^E=3$, $b_I^E=0.75$, $b_E ^I=0.5$, $b_I^I=3$, in case of a normalized Maxwellian initial condition with mean 0 and variance 0.5 (see (39)).The blow-up of $N_E$ cannot be avoided by a large value of $b_I^I$
The function $I_2$ in terms of $N_I$, for different values of $N_E$ fixed. $I_2$ is an increasing function on $N_I$ and decreasing on $N_E$
$\mathcal{F}(N_E)$ for different parameter values corresponding to the first case of Theorem 4.1: there are no steady states (left) or there is an even number of steady states (right). Left figure: $b_E^E=3$, $b_I^E=0.75$, $b_E^I=0.5$ and $b_I^I=5$. Right figure: $b_E^E=1.8$, $b_I^E=0.75$, $b_E^I=0.5$ and $b_I^I=0.25$
$\mathcal{F}(N_E)$ for different parameter values corresponding to the second case of Theorem 4.1: there is an odd number of steady states

Left figure: $b_E^E=0.5$, $b_I^E=0.5$, $b_E^I=3$ and $b_I^I=0.5$ (one steady state). Right figure: $b_E^E=3$, $b_I^E=9$, $b_E^I=0.5$, $b_I^I=0.25$ (one steady state). Center figure: $b_E^E=3$, $b_I^E=7$, $b_E^I=0.5$ and $b_I^I=0.25$ (three steady states).

Comparison between an uncoupled excitatory-inhibitory network ($b_E^I=b_I^E=0$) and a coupled network with small $b_E^I$ and $b_I^E$. The qualitative behavior is the same in both cases

Left figure: $b_I^E=b_E^I=0$, $b_I^I=0.25$, and different values for $b_E^E$. Right figure: $b_I^E=b_E^I=0.1$, $b_I^I=0.25$ and different values for $b_E^E$.

Analysis of the number of steady states for $b_E^E=3$, $b_E^I=0.5$, $b_I^I=0.25$ and different values for $b_I^E$
Study of the limits of $\mathcal{F}(N_E)$ and $N_I^2(N_E)I(N_E)$ (see (29)) when it is finite

Left figure: $b_I^E=0.75$, $b_E^I=0.5$, $b_I^I=0.25$, $a_E=1$, $a_I=1$ for different values of $b_E^E$ and $V_F$.
Right figure: $b_E^E=1.8$, $b_I^E=0.75$, $b_E^I=0.5$, $b_I^I=0.25$ $a_E=1$ for different values of $a_I$ and $V_F$.

Firing rates for the case of two steady states (right in Fig. 6), for different initial conditions: $\rho_{\alpha}^0-1,2$ are given by the profile (40) with $(N_E,N_I)$ stationary values and $\rho_{\alpha}^0-3$ is a normalized Maxwellian with mean 0 and variance 0.25 (see (39)). For both firing rates, the lower steady state seems to be asymptotically stable whereas the higher one seems to be unstable
Stability analysis for the case of three steady states (right in Fig. 7)

Left figure: firing rates for different initial conditions: $\rho_{\alpha}^0-1,2,3$ which are given by the profile (40) with $(N_E,N_I)$ stationary values.Only the lowest steady state seems to be asymptotically stable. Center and right figure: evolution of the probability densities. (Simulations were developed considering $v\in [-6,2]$).

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