Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods

The Network of Noisy Leaky Integrate and Fire (NNLIF) model describes the behavior of a neural network at mesoscopic level. It is one of the simplest self-contained mean-field models considered for that purpose. Even so, to study the mathematical properties of the model some simplifications were necessary C\'aceres-Carrillo-Perthame(2011), C\'aceres-Perthame(2014), C\'aceres-Schneider(2017), which disregard crucial phenomena. In this work we deal with the general NNLIF model without simplifications. It involves a network with two populations (excitatory and inhibitory), with transmission delays between the neurons and where the neurons remain in a refractory state for a certain time. We have studied the number of steady states in terms of the model parameters, the long time behaviour via the entropy method and Poincar\'e's inequality, blow-up phenomena, and the importance of transmission delays between excitatory neurons to prevent blow-up and to give rise to synchronous solutions. Besides analytical results, we have presented a numerical resolutor for this model, based on high order flux-splitting WENO schemes and an explicit third order TVD Runge-Kutta method, in order to describe the wide range of phenomena exhibited by the network: blow-up, asynchronous/synchronous solutions and instability/stability of the steady states; the solver also allows us to observe the time evolution of the firing rates, refractory states and the probability distributions of the excitatory and inhibitory populations.

[1]  Xiao-Jing Wang,et al.  What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitation-inhibition balance. , 2003, Journal of neurophysiology.

[2]  Louis Tao,et al.  An embedded network approach for scale-up of fluctuation-driven systems with preservation of spike information. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[3]  M. Shelley,et al.  An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[4]  Gert Cauwenberghs,et al.  Neuromorphic Silicon Neuron Circuits , 2011, Front. Neurosci.

[5]  Duane Q. Nykamp,et al.  A Population Density Approach That Facilitates Large-Scale Modeling of Neural Networks: Analysis and an Application to Orientation Tuning , 2004, Journal of Computational Neuroscience.

[6]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[7]  Bruce W. Knight,et al.  Dynamics of Encoding in a Population of Neurons , 1972, The Journal of general physiology.

[8]  Irene M. Gamba,et al.  A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods , 2003 .

[9]  Khashayar Pakdaman,et al.  Relaxation and Self-Sustained Oscillations in the Time Elapsed Neuron Network Model , 2011, SIAM J. Appl. Math..

[10]  María J. Cáceres,et al.  Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states , 2010, Journal of mathematical neuroscience.

[11]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[12]  M. Mattia,et al.  Population dynamics of interacting spiking neurons. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Adaptation and Fatigue Model for Neuron Networks and Large Time Asymptotics in a Nonlinear Fragmentation Equation , 2014, Journal of mathematical neuroscience.

[14]  Mar'ia J. C'aceres,et al.  Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models , 2016 .

[15]  W. Singer,et al.  Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. , 1989, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Aaditya V. Rangan,et al.  Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Jonathan Touboul,et al.  Limits and Dynamics of Stochastic Neuronal Networks with Random Heterogeneous Delays , 2012, Journal of Statistical Physics.

[18]  G. Dumont,et al.  The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states , 2017, Nonlinearity.

[19]  Grégory Dumont,et al.  Population density models of integrate-and-fire neurons with jumps: well-posedness , 2012, Journal of Mathematical Biology.

[20]  Toshio Aoyagi,et al.  A model for feature linking via collective oscillations in the primary visual cortex , 2004, Biological Cybernetics.

[21]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[22]  R. Shapley,et al.  An egalitarian network model for the emergence of simple and complex cells in visual cortex , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[23]  Jonathan Touboul,et al.  Importance of the Cutoff Value in the Quadratic Adaptive Integrate-and-Fire Model , 2008, Neural Computation.

[24]  G. Edelman,et al.  Large-scale model of mammalian thalamocortical systems , 2008, Proceedings of the National Academy of Sciences.

[25]  Nicolas Brunel,et al.  Dynamics of Sparsely Connected Networks of Excitatory and Inhibitory Spiking Neurons , 2000, Journal of Computational Neuroscience.

[26]  Bertrand Fontaine,et al.  Fitting Neuron Models to Spike Trains , 2011, Front. Neurosci..

[27]  Wulfram Gerstner,et al.  Population Dynamics of Spiking Neurons: Fast Transients, Asynchronous States, and Locking , 2000, Neural Computation.

[28]  Victor A. Galaktionov.  On blow-up , 2009 .

[29]  Wulfram Gerstner,et al.  Adaptive exponential integrate-and-fire model , 2009, Scholarpedia.

[30]  Aaditya V. Rangan,et al.  DYNAMICS OF CURRENT-BASED, POISSON DRIVEN, INTEGRATE-AND-FIRE NEURONAL NETWORKS " , 2010 .

[31]  Nicolas Brunel,et al.  Dynamics of the Firing Probability of Noisy Integrate-and-Fire Neurons , 2002, Neural Computation.

[32]  A. Treves.  Mean-field analysis of neuronal spike dynamics , 1993 .

[33]  Maria E. Schonbek,et al.  Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience , 2011, 1109.1298.

[34]  Henry C. Tuckwell,et al.  Introduction to theoretical neurobiology , 1988 .

[35]  Marco De Spirito,et al.  Modifications in solvent clusters embedded along the fibers of a cellulose polymer network cause paper degradation. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  S. Nelson,et al.  An emergent model of orientation selectivity in cat visual cortical simple cells , 1995, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[37]  Lawrence Sirovich,et al.  On the Simulation of Large Populations of Neurons , 2004, Journal of Computational Neuroscience.

[38]  J Henry,et al.  Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model , 2015, Journal of mathematical biology.

[39]  J. Touboul.  Spatially Extended Networks with Singular Multi-scale Connectivity Patterns , 2013, Journal of Statistical Physics.

[40]  M. Sur,et al.  Invariant computations in local cortical networks with balanced excitation and inhibition , 2005, Nature Neuroscience.

[41]  F. Delarue,et al.  Global solvability of a networked integrate-and-fire model of McKean–Vlasov type , 2012, 1211.0299.

[42]  Nicholas J. Priebe,et al.  Contrast-Invariant Orientation Tuning in Cat Visual Cortex: Thalamocortical Input Tuning and Correlation-Based Intracortical Connectivity , 1998, The Journal of Neuroscience.

[43]  José A. Carrillo,et al.  Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience , 2015 .

[44]  B. Perthame,et al.  On a voltage-conductance kinetic system for integrate and fire neural networks , 2013, 1310.2742.

[45]  Aaditya V. Rangan,et al.  Fast numerical methods for simulating large-scale integrate-and-fire neuronal networks , 2007, Journal of Computational Neuroscience.

[46]  Khashayar Pakdaman,et al.  Dynamics of a structured neuron population , 2009 .

[47]  José A. Carrillo,et al.  Nonoscillatory Interpolation Methods Applied to Vlasov-Based Models , 2007, SIAM J. Sci. Comput..

[48]  R. Shapley,et al.  LFP power spectra in V1 cortex: the graded effect of stimulus contrast. , 2005, Journal of neurophysiology.

[49]  Julien Chevallier.  Mean-field limit of generalized Hawkes processes , 2015, 1510.05620.

[50]  Marie Doumic,et al.  Microscopic approach of a time elapsed neural model , 2015, ArXiv.

[51]  P Erdi,et al.  Statistical model of the hippocampal CA3 region II. The population framework: model of rhythmic activity in the CA3 slice. , 1998, Biological cybernetics.

[52]  María J Cáceres,et al.  Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity. , 2014, Journal of theoretical biology.

[53]  David Cai,et al.  Cascade-induced synchrony in stochastically driven neuronal networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  Jonathan Touboul,et al.  Bifurcation Analysis of a General Class of Nonlinear Integrate-and-Fire Neurons , 2008, SIAM J. Appl. Math..

[55]  David Cai,et al.  Quantifying neuronal network dynamics through coarse-grained event trees , 2008, Proceedings of the National Academy of Sciences.

[56]  Louis Tao,et al.  Efficient and Accurate Time-Stepping Schemes for Integrate-and-Fire Neuronal Networks , 2001, Journal of Computational Neuroscience.

[57]  Duane Q. Nykamp,et al.  A Population Density Approach That Facilitates Large-Scale Modeling of Neural Networks: Extension to Slow Inhibitory Synapses , 2001, Neural Computation.

[58]  Gustavo Deco,et al.  The encoding of alternatives in multiple-choice decision making , 2009, Proceedings of the National Academy of Sciences.

[59]  Wulfram Gerstner,et al.  Spiking Neuron Models , 2002 .

[60]  Nicolas Brunel,et al.  Fast Global Oscillations in Networks of Integrate-and-Fire Neurons with Low Firing Rates , 1999, Neural Computation.

[61]  Jacques Henry,et al.  A theoretical connection between the Noisy Leaky integrate-and-fire and the escape rate models: The non-autonomous case , 2017, Mathematical Modelling of Natural Phenomena.

[62]  Louis Tao,et al.  A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics , 2011, J. Comput. Phys..

[63]  F. Delarue,et al.  Particle systems with a singular mean-field self-excitation. Application to neuronal networks , 2014, 1406.1151.

[64]  Xiao-Jing Wang,et al.  Mean-Field Theory of Irregularly Spiking Neuronal Populations and Working Memory in Recurrent Cortical Networks , 2003 .

[65]  A R Bulsara,et al.  Noisy FitzHugh-Nagumo model: from single elements to globally coupled networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[66]  Grégory Dumont,et al.  Synchronization of an Excitatory Integrate-and-Fire Neural Network , 2013, Bulletin of mathematical biology.

[67]  J. Touboul.  Propagation of chaos in neural fields , 2011, 1108.2414.

[68]  Cheng Ly,et al.  Population density methods for stochastic neurons with realistic synaptic kinetics: Firing rate dynamics and fast computational methods , 2006, Network.

[69]  C. Quiñinao,et al.  On a Kinetic Fitzhugh–Nagumo Model of Neuronal Network , 2015, 1503.00492.

[70]  X. Wang,et al.  Synaptic Basis of Cortical Persistent Activity: the Importance of NMDA Receptors to Working Memory , 1999, The Journal of Neuroscience.

[71]  A. Koulakov,et al.  Model for a robust neural integrator , 2002, Nature Neuroscience.

[72]  Jonathan Touboul,et al.  Sensitivity to the cutoff value in the quadratic adaptive integrate-and-fire model , 2013 .

[73]  Chi-Wang Shu,et al.  2D semiconductor device simulations by WENO-Boltzmann schemes: Efficiency, boundary conditions and comparison to Monte Carlo methods , 2006, J. Comput. Phys..

[74]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[75]  E. Haskell,et al.  Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics: cutting the dimension down to size , 2001 .

[76]  J Rinzel,et al.  A theoretical basis for large coefficient of variation and bimodality in neuronal interspike interval distributions. , 1983, Journal of theoretical biology.

[77]  Abbott,et al.  Asynchronous states in networks of pulse-coupled oscillators. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[78]  Jonathan D. Touboul,et al.  On the Dynamics of Random Neuronal Networks , 2014, Journal of Statistical Physics.

[79]  Aaditya V. Rangan,et al.  Modeling the spatiotemporal cortical activity associated with the line-motion illusion in primary visual cortex. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[80]  Jacques Henry,et al.  Theoretical connections between mathematical neuronal models corresponding to different expressions of noise. , 2016, Journal of theoretical biology.