The Pair-Replica-Mean-Field Limit for Intensity-based Neural Networks

Replica-mean-field models have been proposed to decipher the activity of neural networks via a multiply-and-conquer approach. In this approach, one considers limit networks made of infinitely many replicas with the same basic neural structure as that of the network of interest, but exchanging spikes in a randomized manner. The key point is that these replica-mean-field networks are tractable versions that retain important features of the finite structure of interest. To date, the replica framework has been discussed for first-order models, whereby elementary replica constituents are single neurons with independent Poisson inputs. Here, we extend this replica framework to allow elementary replica constituents to be composite objects, namely, pairs of neurons. As they include pairwise interactions, these pair-replica models exhibit non-trivial dependencies in their stationary dynamics, which cannot be captured by first-order replica models. Our contributions are two-fold: $(i)$ We analytically characterize the stationary dynamics of a pair of intensity-based neurons with independent Poisson input. This analysis involves the reduction of a boundary-value problem related to a two-dimensional transport equation to a system of Fredholm integral equations---a result of independent interest. $(ii)$ We analyze the set of consistency equations determining the full network dynamics of certain replica limits. These limits are those for which replica constituents, be they single neurons or pairs of neurons, form a partition of the network of interest. Both analyses are numerically validated by computing input/output transfer functions for neuronal pairs and by computing the correlation structure of certain pair-dominated network dynamics.

[1]  G. Fayolle,et al.  Random Walks in the Quarter Plane: Algebraic Methods, Boundary Value Problems, Applications to Queueing Systems and Analytic Combinatorics , 2018 .

[2]  M. Carandini,et al.  The Nature of Shared Cortical Variability , 2015, Neuron.

[3]  A. Rybko,et al.  Poisson Hypothesis for Information Networks. II , 2005 .

[4]  J. Touboul,et al.  Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons , 2012, The Journal of Mathematical Neuroscience.

[5]  J. Jacod.  Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales , 1975 .

[6]  David E. Edmunds,et al.  Spectral Theory and Differential Operators , 1987, Oxford Scholarship Online.

[7]  M. Bramson.  PR ] 8 O ct 2 01 0 Submitted to the Annals of Applied Probability STABILITY OF JOIN THE SHORTEST QUEUE NETWORKS By , 2010 .

[8]  Richard P. Feynman.  Feynman lectures on physics. Volume 2: Mainly electromagnetism and matter , 1964 .

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

[10]  Franccois Baccelli,et al.  Replica-mean-field limits of fragmentation-interaction-aggregation processes , 2022, Journal of Applied Probability.

[11]  Paul C. Bressloff,et al.  Stochastic Neural Field Theory and the System-Size Expansion , 2009, SIAM J. Appl. Math..

[12]  Olivier D. Faugeras,et al.  A Constructive Mean-Field Analysis of Multi-Population Neural Networks with Random Synaptic Weights and Stochastic Inputs , 2008, Front. Comput. Neurosci..

[13]  Ioannis Dimitriou,et al.  Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane , 2020, Ann. Oper. Res..

[14]  Anthony J. G. Hey,et al.  Feynman Lectures on Computation , 1996 .

[15]  François Baccelli,et al.  Elements Of Queueing Theory , 1994 .

[16]  S. Amari.  Dynamics of pattern formation in lateral-inhibition type neural fields , 1977, Biological Cybernetics.

[17]  Eero P. Simoncelli,et al.  Spatio-temporal correlations and visual signalling in a complete neuronal population , 2008, Nature.

[18]  Michael A. Buice,et al.  Systematic Fluctuation Expansion for Neural Network Activity Equations , 2009, Neural Computation.

[19]  Sommers,et al.  Chaos in random neural networks. , 1988, Physical review letters.

[20]  H. McKean,et al.  A CLASS OF MARKOV PROCESSES ASSOCIATED WITH NONLINEAR PARABOLIC EQUATIONS , 1966, Proceedings of the National Academy of Sciences of the United States of America.

[21]  S. Shlosman,et al.  Spontaneous Resonances and the Coherent States of the Queuing Networks , 2007, 0708.3073.

[22]  Eero P. Simoncelli,et al.  Partitioning neuronal variability , 2014, Nature Neuroscience.

[23]  André Longtin,et al.  A stochastic-field description of finite-size spiking neural networks , 2017, PLoS Comput. Biol..

[24]  Naftali Tishby,et al.  Cortical activity flips among quasi-stationary states. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[25]  Peter Dayan,et al.  Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems , 2001 .

[26]  A. Galves,et al.  Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets , 2012, 1212.5505.

[27]  Kamiar Rahnama Rad,et al.  Mean-Field Approximations for Coupled Populations of Generalized Linear Model Spiking Neurons with Markov Refractoriness , 2009, Neural Computation.

[28]  Uri T Eden,et al.  A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. , 2005, Journal of neurophysiology.

[29]  R. L. Dobrushin,et al.  Queueing system with selection of the shortest of two queues: an assymptotic approach , 1996 .

[30]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[31]  S. Amari.  Homogeneous nets of neuron-like elements , 1975, Biological Cybernetics.

[32]  Sلأren Asmussen,et al.  Applied Probability and Queues , 1989 .

[33]  R. R. P. Jackson,et al.  Introduction to the Theory of Queues , 1963 .

[34]  E. Presutti,et al.  Hydrodynamic Limit for Interacting Neurons , 2014, 1401.4264.

[35]  M. Hp.  A class of markov processes associated with nonlinear parabolic equations. , 1966 .

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

[37]  Jean-Yves Le Boudec,et al.  A class of mean field interaction models for computer and communication systems , 2008, Perform. Evaluation.

[38]  J. Mecke,et al.  Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen , 1967 .

[39]  Franccois Baccelli,et al.  Replica-Mean-Field Limits for Intensity-Based Neural Networks , 2019, SIAM J. Appl. Dyn. Syst..

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

[41]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[42]  G. F. Newell,et al.  Introduction to the Theory of Queues. , 1963 .

[43]  W. Wildman,et al.  Theoretical Neuroscience , 2014 .

[44]  Debra Rose Wilson,et al.  A Bradford Book , 2011 .

[45]  J. Kelso,et al.  The Metastable Brain , 2014, Neuron.

[46]  J. Cowan,et al.  Field-theoretic approach to fluctuation effects in neural networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  R. Leighton,et al.  Feynman Lectures on Physics , 1971 .

[48]  Wulfram Gerstner,et al.  Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size , 2016, PLoS Comput. Biol..

[49]  F. Baccelli,et al.  Metastability of Queuing Networks with Mobile Servers , 2018 .

[50]  Olivier D. Faugeras,et al.  Noise-Induced Behaviors in Neural Mean Field Dynamics , 2011, SIAM J. Appl. Dyn. Syst..

[51]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

[52]  S. Shlosman,et al.  POISSON HYPOTHESIS FOR INFORMATION NETWORKS. I , 2006 .

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