Self-Consistent Scheme for Spike-Train Power Spectra in Heterogeneous Sparse Networks

Recurrent networks of spiking neurons can be in an asynchronous state characterized by low or absent cross-correlations and spike statistics which resemble those of cortical neurons. Although spatial correlations are negligible in this state, neurons can show pronounced temporal correlations in their spike trains that can be quantified by the autocorrelation function or the spike-train power spectrum. Depending on cellular and network parameters, correlations display diverse patterns (ranging from simple refractory-period effects and stochastic oscillations to slow fluctuations) and it is generally not well-understood how these dependencies come about. Previous work has explored how the single-cell correlations in a homogeneous network (excitatory and inhibitory integrate-and-fire neurons with nearly balanced mean recurrent input) can be determined numerically from an iterative single-neuron simulation. Such a scheme is based on the fact that every neuron is driven by the network noise (i.e., the input currents from all its presynaptic partners) but also contributes to the network noise, leading to a self-consistency condition for the input and output spectra. Here we first extend this scheme to homogeneous networks with strong recurrent inhibition and a synaptic filter, in which instabilities of the previous scheme are avoided by an averaging procedure. We then extend the scheme to heterogeneous networks in which (i) different neural subpopulations (e.g., excitatory and inhibitory neurons) have different cellular or connectivity parameters; (ii) the number and strength of the input connections are random (Erdős-Rényi topology) and thus different among neurons. In all heterogeneous cases, neurons are lumped in different classes each of which is represented by a single neuron in the iterative scheme; in addition, we make a Gaussian approximation of the input current to the neuron. These approximations seem to be justified over a broad range of parameters as indicated by comparison with simulation results of large recurrent networks. Our method can help to elucidate how network heterogeneity shapes the asynchronous state in recurrent neural networks.

[1]  Matthieu Gilson,et al.  Stability versus Neuronal Specialization for STDP: Long-Tail Weight Distributions Solve the Dilemma , 2011, PloS one.

[2]  N. Brunel,et al.  Firing frequency of leaky intergrate-and-fire neurons with synaptic current dynamics. , 1998, Journal of theoretical biology.

[3]  Moritz Helias,et al.  The Correlation Structure of Local Neuronal Networks Intrinsically Results from Recurrent Dynamics , 2013, PLoS Comput. Biol..

[4]  K. Svoboda,et al.  Neural Activity in Barrel Cortex Underlying Vibrissa-Based Object Localization in Mice , 2010, Neuron.

[5]  Wulfram Gerstner,et al.  Fluctuations and information filtering in coupled populations of spiking neurons with adaptation. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Michael A. Zaks,et al.  Mechanisms of Self-Sustained Oscillatory States in Hierarchical Modular Networks with Mixtures of Electrophysiological Cell Types , 2016, Front. Comput. Neurosci..

[7]  J J Hopfield,et al.  Rapid local synchronization of action potentials: toward computation with coupled integrate-and-fire neurons. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Francesca Mastrogiuseppe,et al.  Intrinsically-generated fluctuating activity in excitatory-inhibitory networks , 2016, PLoS Comput. Biol..

[9]  Sen Song,et al.  Highly Nonrandom Features of Synaptic Connectivity in Local Cortical Circuits , 2005, PLoS biology.

[10]  Brent Doiron,et al.  Oscillatory activity in electrosensory neurons increases with the spatial correlation of the stochastic input stimulus. , 2004, Physical review letters.

[11]  Nicolas Brunel,et al.  Dynamics of a recurrent network of spiking neurons before and following learning , 1997 .

[12]  Wyeth Bair,et al.  The Effect of a Refractory Period on the Power Spectrum of Neuronal Discharge , 1995, SIAM J. Appl. Math..

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

[14]  A. Litwin-Kumar,et al.  Slow dynamics and high variability in balanced cortical networks with clustered connections , 2012, Nature Neuroscience.

[15]  Michael A. Zaks,et al.  Sustained oscillations, irregular firing, and chaotic dynamics in hierarchical modular networks with mixtures of electrophysiological cell types , 2014, Front. Comput. Neurosci..

[16]  Benjamin Lindner,et al.  Statistical structure of neural spiking under non-Poissonian or other non-white stimulation , 2015, Journal of Computational Neuroscience.

[17]  J. Poulet,et al.  Internal brain state regulates membrane potential synchrony in barrel cortex of behaving mice , 2008, Nature.

[18]  Benjamin Lindner,et al.  Author's Accepted Manuscript , 2022 .

[19]  Carlo Fulvi Mari,et al.  Random Networks of Spiking Neurons: Instability in the Xenopus Tadpole Moto-Neural Pattern , 2000, cond-mat/0003263.

[20]  K. H. Britten,et al.  Power spectrum analysis of bursting cells in area MT in the behaving monkey , 1994, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[21]  Stefan Rotter,et al.  Solving the two-dimensional Fokker-Planck equation for strongly correlated neurons. , 2016, Physical review. E.

[22]  Marc Timme,et al.  Speed of synchronization in complex networks of neural oscillators: analytic results based on Random Matrix Theory. , 2005, Chaos.

[23]  L. Schimansky-Geier,et al.  Harmonic noise: Effect on bistable systems , 1990 .

[24]  Bard Ermentrout,et al.  When inhibition not excitation synchronizes neural firing , 1994, Journal of Computational Neuroscience.

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

[26]  Magnus J. E. Richardson,et al.  Spike-train spectra and network response functions for non-linear integrate-and-fire neurons , 2008, Biological Cybernetics.

[27]  Michael J. Berry,et al.  Weak pairwise correlations imply strongly correlated network states in a neural population , 2005, Nature.

[28]  Benjamin Lindner,et al.  Self-consistent determination of the spike-train power spectrum in a neural network with sparse connectivity , 2014, Front. Comput. Neurosci..

[29]  Boris S. Gutkin,et al.  The Effects of Spike Frequency Adaptation and Negative Feedback on the Synchronization of Neural Oscillators , 2001, Neural Computation.

[30]  D. Hansel,et al.  On the Distribution of Firing Rates in Networks of Cortical Neurons , 2011, The Journal of Neuroscience.

[31]  Partha P. Mitra,et al.  Sampling Properties of the Spectrum and Coherency of Sequences of Action Potentials , 2000, Neural Computation.

[32]  H. Sompolinsky,et al.  Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin-glasses , 1982 .

[33]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[34]  A. Aertsen,et al.  Beyond the Cortical Column: Abundance and Physiology of Horizontal Connections Imply a Strong Role for Inputs from the Surround , 2011, Front. Neurosci..

[35]  Bijan Pesaran,et al.  Temporal structure in neuronal activity during working memory in macaque parietal cortex , 2000, Nature Neuroscience.

[36]  M. J. Richardson,et al.  Dynamics of populations and networks of neurons with voltage-activated and calcium-activated currents. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  W Gerstner,et al.  Noise spectrum and signal transmission through a population of spiking neurons. , 1999, Network.

[38]  J. Fuster,et al.  Unit activity in monkey parietal cortex related to haptic perception and temporary memory , 2004, Experimental Brain Research.

[39]  Alexander B Neiman,et al.  Noise-induced transition to bursting in responses of paddlefish electroreceptor afferents. , 2007, Journal of neurophysiology.

[40]  B. Knight The Relationship between the Firing Rate of a Single Neuron and the Level of Activity in a Population of Neurons , 1972, The Journal of general physiology.

[41]  D. Hansel,et al.  How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating Inputs , 2003, The Journal of Neuroscience.

[42]  P. Hänggi,et al.  Markovian embedding of non-Markovian superdiffusion. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Benjamin Lindner,et al.  Superposition of many independent spike trains is generally not a Poisson process. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  T. Hromádka,et al.  Sparse Representation of Sounds in the Unanesthetized Auditory Cortex , 2008, PLoS biology.

[45]  Srdjan Ostojic,et al.  Two types of asynchronous activity in networks of excitatory and inhibitory spiking neurons , 2014, Nature Neuroscience.

[46]  Jan Grewe,et al.  Synchronous spikes are necessary but not sufficient for a synchrony code in populations of spiking neurons , 2017, Proceedings of the National Academy of Sciences.

[47]  Rupert Swarbrick,et al.  Firing-rate response of a neuron receiving excitatory and inhibitory synaptic shot noise. , 2010, Physical review letters.

[48]  Wulfram Gerstner,et al.  Neuronal Dynamics: From Single Neurons To Networks And Models Of Cognition , 2014 .

[49]  P. Jung,et al.  Colored Noise in Dynamical Systems , 2007 .

[50]  Klaus Obermayer,et al.  Impact of Adaptation Currents on Synchronization of Coupled Exponential Integrate-and-Fire Neurons , 2012, PLoS Comput. Biol..

[51]  Benjamin Lindner,et al.  Slow fluctuations in recurrent networks of spiking neurons. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  Brent Doiron,et al.  The mechanics of state-dependent neural correlations , 2016, Nature Neuroscience.

[53]  Benjamin Lindner,et al.  Exact analytical results for integrate-and-fire neurons driven by excitatory shot noise , 2017, Journal of Computational Neuroscience.

[54]  Opper,et al.  New method for studying the dynamics of disordered spin systems without finite-size effects. , 1992, Physical review letters.

[55]  K. Harris,et al.  Cortical state and attention , 2011, Nature Reviews Neuroscience.

[56]  M. Chacron,et al.  Firing statistics of a neuron model driven by long-range correlated noise. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  P. Dayan,et al.  Supporting Online Material Materials and Methods Som Text Figs. S1 to S9 References the Asynchronous State in Cortical Circuits , 2022 .

[58]  Jaime de la Rocha,et al.  Supplementary Information for the article ‘ Correlation between neural spike trains increases with firing rate ’ , 2007 .

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

[60]  Benjamin Lindner,et al.  Comparative study of different integrate-and-fire neurons: spontaneous activity, dynamical response, and stimulus-induced correlation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[61]  Merav Stern,et al.  Transition to chaos in random networks with cell-type-specific connectivity. , 2014, Physical review letters.

[62]  N. B,et al.  Firing Frequency of Leaky Integrate-and-fire Neurons with Synaptic Current Dynamics , 1998 .

[63]  Alexander Lerchner,et al.  Mean field theory for a balanced hypercolumn model of orientation selectivity in primary visual cortex , 2004, Network.

[64]  Eric Shea-Brown,et al.  Impact of Network Structure and Cellular Response on Spike Time Correlations , 2011, PLoS Comput. Biol..

[65]  Eric Shea-Brown,et al.  Correlation and synchrony transfer in integrate-and-fire neurons: basic properties and consequences for coding. , 2008, Physical review letters.

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

[67]  Carson C. Chow,et al.  Variability in neuronal activity in primate cortex during working memory tasks , 2007, Neuroscience.

[68]  Romain Brette,et al.  The Brian Simulator , 2009, Front. Neurosci..

[69]  Nicolas Brunel,et al.  How Connectivity, Background Activity, and Synaptic Properties Shape the Cross-Correlation between Spike Trains , 2009, The Journal of Neuroscience.

[70]  Alexander B. Neiman,et al.  Characteristic Effects of Stochastic Oscillatory Forcing on Neural Firing: Analytical Theory and Comparison to Paddlefish Electroreceptor Data , 2013, PLoS Comput. Biol..

[71]  Edward T. Bullmore,et al.  Modular and Hierarchically Modular Organization of Brain Networks , 2010, Front. Neurosci..

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

[73]  G H Wakefield,et al.  The spectral shaping of neural discharges by refractory effects. , 1993, The Journal of the Acoustical Society of America.

[74]  H. Sompolinsky,et al.  Transition to chaos in random neuronal networks , 2015, 1508.06486.

[75]  Lutz Schimansky-Geier,et al.  Maximizing spike train coherence or incoherence in the leaky integrate-and-fire model. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[78]  H. Sompolinsky,et al.  Chaos in Neuronal Networks with Balanced Excitatory and Inhibitory Activity , 1996, Science.

[79]  G Horn,et al.  An analysis of spontaneous impulse activity of units in the striate cortex of unrestrained cats , 1966, The Journal of physiology.

[80]  Brent Doiron,et al.  Theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[81]  Alexander B Neiman,et al.  Sensory coding in oscillatory electroreceptors of paddlefish. , 2011, Chaos.

[82]  Eugene M. Izhikevich,et al.  Simple model of spiking neurons , 2003, IEEE Trans. Neural Networks.

[83]  Paul M. Harrison,et al.  Experimentally Verified Parameter Sets for Modelling Heterogeneous Neocortical Pyramidal-Cell Populations , 2015, PLoS Comput. Biol..