Mesoscopic population equations for spiking neural networks with synaptic short-term plasticity

Valentin Schmutz1, Wulfram Gerstner1, Tilo Schwalger1
1Brain Mind Institute, École Polytechnique Féderale de Lausanne (EPFL), Lausanne, Switzerland

Tóm tắt

Abstract

Coarse-graining microscopic models of biological neural networks to obtain mesoscopic models of neural activities is an essential step towards multi-scale models of the brain. Here, we extend a recent theory for mesoscopic population dynamics with static synapses to the case of dynamic synapses exhibiting short-term plasticity (STP). The extended theory offers an approximate mean-field dynamics for the synaptic input currents arising from populations of spiking neurons and synapses undergoing Tsodyks–Markram STP. The approximate mean-field dynamics accounts for both finite number of synapses and correlation between the two synaptic variables of the model (utilization and available resources) and its numerical implementation is simple. Comparisons with Monte Carlo simulations of the microscopic model show that in both feedforward and recurrent networks, the mesoscopic mean-field model accurately reproduces the first- and second-order statistics of the total synaptic input into a postsynaptic neuron and accounts for stochastic switches between Up and Down states and for population spikes. The extended mesoscopic population theory of spiking neural networks with STP may be useful for a systematic reduction of detailed biophysical models of cortical microcircuits to numerically efficient and mathematically tractable mean-field models.

Từ khóa


Tài liệu tham khảo

Wilson HR, Cowan JD. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J. 1972;12(1):1.

Ben-Yishai R, Bar-Or RL, Sompolinsky H. Theory of orientation tuning in visual cortex. Proc Natl Acad Sci USA. 1995;92(9):3844–8.

Wong KF, Wang XJ. A recurrent network mechanism of time integration in perceptual decisions. J Neurosci. 2006;26(4):1314–28.

Barak O, Tsodyks M. Persistent activity in neural networks with dynamic synapses. PLoS Comput Biol. 2007;3(2):e35.

Shpiro A, Moreno-Bote R, Rubin N, Rinzel J. Balance between noise and adaptation in competition models of perceptual bistability. J Comput Neurosci. 2009;27(1):37–54.

Rubin DB, Van Hooser SD, Miller KD. The stabilized supralinear network: a unifying circuit motif underlying multi-input integration in sensory cortex. Neuron. 2015;85(2):402–17.

Markram H, Muller E, Ramaswamy S, Reimann MW, Abdellah M, Sanchez CA et al.. Reconstruction and simulation of neocortical microcircuitry. Cell. 2015;163(2):456–92.

Izhikevich EM, Edelman GM. Large-scale model of mammalian thalamocortical systems. Proc Natl Acad Sci USA. 2008;105(9):3593–8.

Potjans TC, Diesmann M. The cell-type specific cortical microcircuit: relating structure and activity in a full-scale spiking network model. Cereb Cortex. 2014;24(3):785–806.

Fiebig F, Lansner A. A spiking working memory model based on Hebbian short-term potentiation. J Neurosci. 2017;37(1):83–96.

Rössert C, Pozzorini C, Chindemi G, Davison AP, Eroe C, King J, et al. Automated point-neuron simplification of data-driven microcircuit models. arXiv e-prints. 2016.

Schwalger T, Deger M, Gerstner W. Towards a theory of cortical columns: from spiking neurons to interacting neural populations of finite size. PLoS Comput Biol. 2017;13(4):e1005507.

Harris KD, Shepherd GMG. The neocortical circuit: themes and variations. Nat Neurosci. 2015;18(2):170–81.

Lefort S, Tomm C, Sarria JCF, Petersen CCH. The excitatory neuronal network of the C2 barrel column in mouse primary somatosensory cortex. Neuron. 2009;61(2):301–16.

Gerstner W. Time structure of the activity in neural network models. Phys Rev E. 1995;51:738.

Gerstner W. Population dynamics of spiking neurons: fast transients, asynchronous states, and locking. Neural Comput. 2000;12:43.

Naud R, Gerstner W. Coding and decoding with adapting neurons: a population approach to the peri-stimulus time histogram. PLoS Comput Biol. 2012;8(10):e1002711.

Schwalger T, Chizhov AV. Mind the last spike—firing rate models for mesoscopic populations of spiking neurons. Curr Opin Neurobiol. 2019;58:155–66.

Abbott LF, Varela JA, Sen K, Nelson SB. Synaptic depression and cortical gain control. Science. 1997;275:220.

Markram H, Wang Y, Tsodyks M. Differential signaling via the same axon of neocortical pyramidal neurons. Proc Natl Acad Sci USA. 1998;95(9):5323–8.

Dittman JS, Kreitzer AC, Regehr WG. Interplay between facilitation, depression, and residual calcium at three presynaptic terminals. J Neurosci. 2000;20:1374.

Zucker RS, Regehr WG. Short-term synaptic plasticity. Annu Rev Physiol. 2002;64:355–405.

Levina A, Herrmann JM, Geisel T. Dynamical synapses causing self-organized criticality in neural networks. Nat Phys. 2007;3(12):857.

Pittorino F, Ibáñez-Berganza M, di Volo M, Vezzani A, Burioni R. Chaos and correlated avalanches in excitatory neural networks with synaptic plasticity. Phys Rev Lett. 2017;118(9):098102.

Seeholzer A, Deger M, Gerstner W. Stability of working memory in continuous attractor networks under the control of short-term plasticity. PLoS Comput Biol. 2019;15(4):e1006928.

Fortune ES, Rose GJ. Short-term synaptic plasticity as a temporal filter. Trends Neurosci. 2001;24:381.

Merkel M, Lindner B. Synaptic filtering of rate-coded information. Phys Rev E. 2010;81(4 Pt 1):041921.

Rosenbaum R, Rubin J, Doiron B. Short term synaptic depression imposes a frequency dependent filter on synaptic information transfer. PLoS Comput Biol. 2012;8(6):e1002557.

Droste F, Schwalger T, Lindner B. Interplay of two signals in a neuron with heterogeneous synaptic short-term plasticity. Front Comput Neurosci. 2013;7:86.

Tsodyks M, Pawelzik K, Markram H. Neural networks with dynamic synapses. Neural Comput. 1998;10(4):821–35.

Holcman D, Tsodyks M. The emergence of up and down states in cortical networks. PLoS Comput Biol. 2006;2(3):e23.

Galves A, Löcherbach E, Pouzat C, Presutti E. A system of interacting neurons with short term plasticity. arXiv preprint. 2019. arXiv:1903.01270.

Lindner B, Gangloff D, Longtin A, Lewis JE. Broadband coding with dynamic synapses. J Neurosci. 2009;29(7):2076–88.

Cook DL, Schwindt PC, Grande LA, Spain WJ. Synaptic depression in the localization of sound. Nature. 2003;421(6918):66.

Higley MJ, Contreras D. Balanced excitation and inhibition determine spike timing during frequency adaptation. J Neurosci. 2006;26(2):448–57.

Oswald AM, Urban NN. Interactions between behaviorally relevant rhythms and synaptic plasticity alter coding in the piriform cortex. J Neurosci. 2012;32(18):6092–104.

Gigante G, Deco G, Marom S, Del Giudice P. Network events on multiple space and time scales in cultured neural networks and in a stochastic rate model. PLoS Comput Biol. 2015;11(11):e1004547.

Kurtz TG. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Probab. 1971;8(2):344–56.

Kurtz TG et al.. Strong approximation theorems for density dependent Markov chains. Stoch Process Appl. 1978;6(3):223–40.

Pakdaman K, Thieullen M, Wainrib G. Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv Appl Probab. 2010;42(3):761–94.

Ditlevsen S, Löcherbach E. Multi-class oscillating systems of interacting neurons. Stoch Process Appl. 2017;127(6):1840–69.

Lindner B, García-Ojalvo J, Neiman A, Schimansky-Geier L. Effects of noise in excitable systems. Phys Rep. 2004;392:321.

Litwin-Kumar A, Doiron B. Slow dynamics and high variability in balanced cortical networks with clustered connections. Nat Neurosci. 2012;15(11):1498–505.

Mazzucato L, Fontanini A, La Camera G. Dynamics of multistable states during ongoing and evoked cortical activity. J Neurosci. 2015;35(21):8214–31.

Moreno-Bote R, Rinzel J, Rubin N. Noise-induced alternations in an attractor network model of perceptual bistability. J Neurophysiol. 2007;98(3):1125–39.

Jercog D, Roxin A, Barthó P, Luczak A, Compte A, de la Rocha J. UP-DOWN cortical dynamics reflect state transitions in a bistable network. eLife. 2017;6:e22425.

Tsodyks MV, Markram H. The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability. Proc Natl Acad Sci USA. 1997;94:719.

Dayan P, Abbott LF. Theoretical neuroscience: computational and mathematical modeling of neural systems. 1st ed. Cambridge: MIT Press; 2005.

Bird A, Richardson M. Long-term plasticity determines the postsynaptic response to correlated afferents with multivesicular short-term synaptic depression. Front Comput Neurosci. 2014;8:2.

Deger M, Schwalger T, Naud R, Gerstner W. Fluctuations and information filtering in coupled populations of spiking neurons with adaptation. Phys Rev E. 2014;90(6):062704.

Schwalger T, Droste F, Lindner B. Statistical structure of neural spiking under non-Poissonian or other non-white stimulation. J Comput Neurosci. 2015;39(1):29–51.

Brunel N. Sparsely connected networks of spiking neurons. J Comput Neurosci. 2000;8:183.

Romani S, Amit DJ, Mongillo G. Mean-field analysis of selective persistent activity in presence of short-term synaptic depression. J Comput Neurosci. 2006;20(2):201.

Mongillo G, Hansel D, van Vreeswijk C. Bistability and spatiotemporal irregularity in neuronal networks with nonlinear synaptic transmission. Phys Rev Lett. 2012;108:158101.

Bird AD, Richardson MJE. Transmission of temporally correlated spike trains through synapses with short-term depression. PLoS Comput Biol. 2018;14(6):1–25.