Abstract
Spike-timing-dependent plasticity is the process by which the strengths of connections between neurons are modified as a result of the precise timing of the action potentials fired by the neurons. We consider a model consisting of one integrate-and-fire neuron receiving excitatory inputs from a large number—here, 1000—of Poisson neurons whose synapses are plastic. When correlations are introduced between the firing times of these input neurons, the distribution of synaptic strengths shows interesting, and apparently low-dimensional, dynamical behaviour. This behaviour is analysed in two different parameter regimes using equation-free techniques, which bypass the explicit derivation of the relevant low-dimensional dynamical system. We demonstrate both coarse projective integration (which speeds up the time integration of a dynamical system) and the use of recently developed data mining techniques to identify the appropriate low-dimensional description of the complex dynamical systems in our model.
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References
Appleby PA, Elliott T (2006) Stable competitive dynamics emerge from multispike interactions in a stochastic model of spike-timing-dependent plasticity. Neural Comput 18(10):2414–2464
Appleby PA, Elliott T (2007) Multispike interactions in a stochastic model of spike-timing-dependent plasticity. Neural Comput 19(5):1362–1399
Avitabile D, Hoyle R, Samaey G (2014) Noise reduction in coarse bifurcation analysis of stochastic agent-based models: an example of consumer lock-in. SIAM J Appl Dyn Syst 13(4):1583–1619
Bell CC, Han VZ, Sugawara Y, Grant K (1997) Synaptic plasticity in a cerebellum-like structure depends on temporal order. Nature 387(6630):278–281
Bi GQ, Poo MM (1998) Synaptic modifications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, and postsynaptic cell type. J Neurosci 18(24):10,464–10,472
Bliss TV, Collingridge GL (1993) A synaptic model of memory: long-term potentiation in the hippocampus. Nature 361(6407):31–39
Brette R (2006) Exact simulation of integrate-and-fire models with synaptic conductances. Neural Comput 18(8):2004–2027
Brette R, Rudolph M, Carnevale T, Hines M, Beeman D, Bower JM, Diesmann M, Morrison A, Goodman PH, Harris FC Jr et al (2007) Simulation of networks of spiking neurons: a review of tools and strategies. J Comput Neurosci 23(3):349–398
Burkitt AN, Gilson M, van Hemmen JL (2007) Spike-timing-dependent plasticity for neurons with recurrent connections. Biol Cybern 96(5):533–546
Burkitt AN, Meffin H, Grayden DB (2004) Spike-timing-dependent plasticity: the relationship to rate-based learning for models with weight dynamics determined by a stable fixed point. Neural Comput 16(5):885–940
Caporale N, Dan Y (2008) Spike timing-dependent plasticity: a hebbian learning rule. Annu Rev Neurosci 31(1):25–46
Chen CC, Jasnow D (2011) Event-driven simulations of a plastic, spiking neural network. Phys Rev E 84(3):031,908
Coifman RR, Lafon S, Lee AB, Maggioni M, Nadler B, Warner F, Zucker SW (2005) Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. Proc Natl Acad Sci USA 102(21):7426–7431
DeVille RL, Peskin CS (2008) Synchrony and asynchrony in a fully stochastic neural network. Bull Math Biol 70(6):1608–1633
Erban R, Frewen TA, Wang X, Elston TC, Coifman R, Nadler B, Kevrekidis IG (2007) Variable-free exploration of stochastic models: A gene regulatory network example. J Chem Phys 126(15):155103
Erban R, Kevrekidis I, Adalsteinsson D, Elston T (2006) Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation. J Chem Phys 124(084):106
Ermentrout GB, Terman DH (2010) Math Found Neurosci, vol 64. Springer, Berlin
Ferguson AL, Panagiotopoulos AZ, Debenedetti PG, Kevrekidis IG (2010) Systematic determination of order parameters for chain dynamics using diffusion maps. Proc Natl Acad Sci 107(31):13597–13602
Gear C (2001) Projective integration methods for distributions. NEC Technical Report TR 2001-130
Gear CW, Kevrekidis IG, Theodoropoulos C (2002) ‘Coarse’ integration/bifurcation analysis via microscopic simulators: micro-galerkin methods. Comput Chem Eng 26(7):941–963
Gerstner W, Kempter R, van Hemmen JL, Wagner H et al (1996) A neuronal learning rule for sub-millisecond temporal coding. Nature 383(6595):76–78
Gilson M, Burkitt AN, Grayden DB, Thomas DA, van Hemmen JL (2009) Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. I. input selectivity-strengthening correlated input pathways. Biol Cybern 101(2):81–102
Golowasch J, Casey M, Abbott L, Marder E (1999) Network stability from activity-dependent regulation of neuronal conductances. Neural Comput 11(5):1079–1096
Gradišek J, Siegert S, Friedrich R, Grabec I (2000) Analysis of time series from stochastic processes. Phys Rev E 62(3):3146–3155
Gütig R, Aharonov R, Rotter S, Sompolinsky H (2003) Learning input correlations through nonlinear temporally asymmetric hebbian plasticity. J Neurosci 23(9):3697–3714
Izhikevich EM, Desai NS (2003) Relating STDP to BCM. Neural Comput 15(7):1511–1523
Keener J, Sneyd J (1998) Math Phys, vol 8. Springer, Berlin
Kempter R, Gerstner W, van Hemmen JL (1999) Hebbian learning and spiking neurons. Phys Rev E 59:4498–4514. doi:10.1103/PhysRevE.59.4498
Kevrekidis IG, Gear CW, Hyman JM, Kevrekidid PG, Runborg O, Theodoropoulos C et al (2003) Equation-free, coarse-grained multiscale computation: enabling mocroscopic simulators to perform system-level analysis. Commun Math Sci 1(4):715–762
Laing C, Frewen T, Kevrekidis I (2007) Coarse-grained dynamics of an activity bump in a neural field model. Nonlinearity 20:2127
Laing C, Frewen T, Kevrekidis I (2010) Reduced models for binocular rivalry. J Comput Neurosci 28(3):459–476
Laing CR, Kevrekidis IG (2008) Periodically-forced finite networks of heterogeneous globally-coupled oscillators: a low-dimensional approach. Phys D 237(2):207–215
Lee SL, Gear CW (2007) Second-order accurate projective integrators for multiscale problems. J Comput Appl Math 201(1):258–274. doi:10.1016/j.cam.2006.02.018
Lim S, Rinzel J (2010) Noise-induced transitions in slow wave neuronal dynamics. J Comput Neurosci 28(1):1–17. doi:10.1007/s10827-009-0178-y
Lubenov EV, Siapas AG (2008) Decoupling through synchrony in neuronal circuits with propagation delays. Neuron 58(1):118–131
Markram H, Lübke J, Frotscher M, Sakmann B (1997) Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs. Science 275(5297):213–215
Marschler C, Faust-Ellsässer C, Starke J, van Hemmen JL (2014) Bifurcation of learning and structure formation in neuronal maps. EPL (Europhysics Letters) 108(4):48,005
Marschler C, Sieber J, Berkemer R, Kawamoto A, Starke J (2014) Implicit methods for equation-free analysis: convergence results and analysis of emergent waves in microscopic traffic models. SIAM J Appl Dyn Syst 13(3):1202–1238
Meffin H, Besson J, Burkitt A, Grayden D (2006) Learning the structure of correlated synaptic subgroups using stable and competitive spike-timing-dependent plasticity. Phys Rev E 73(4):041,911
Mikkelsen K, Imparato A, Torcini A (2013) Emergence of slow collective oscillations in neural networks with spike-timing dependent plasticity. Phys Rev Lett 110(20):208,101
Morrison A, Aertsen A, Diesmann M (2007) Spike-timing-dependent plasticity in balanced random networks. Neural Comput 19(6):1437–1467
Ragwitz M, Kantz H (2001) Indispensable finite time corrections for Fokker-Planck equations from time series data. Phys Rev Lett 87(254):501
Roberts PD, Bell CC (2002) Spike timing dependent synaptic plasticity in biological systems. Biol Cybern 87(5–6):392–403
Rubin J, Lee D, Sompolinsky H (2001) Equilibrium properties of temporally asymmetric hebbian plasticity. Phys Rev Lett 86(2):364–367
Setayeshgar S, Gear C, Othmer H, Kevrekidis I (2005) Application of coarse integration to bacterial chemotaxis. Multiscale Model Simul 4(1):307–327
Smith JC, Abdala A, Koizumi H, Rybak IA, Paton JF (2007) Spatial and functional architecture of the mammalian brain stem respiratory network: a hierarchy of three oscillatory mechanisms. J Neurophysiol 98(6):3370–3387
Sonday BE, Haataja M, Kevrekidis IG (2009) Coarse-graining the dynamics of a driven interface in the presence of mobile impurities: effective description via diffusion maps. Phys Rev E 80(031):102
Song S, Abbott L (2001) Cortical development and remapping through spike timing-dependent plasticity. Neuron 32(2):339–350
Song S, Miller K, Abbott L (2000) Competitive hebbian learning through spike-timing-dependent synaptic plasticity. Nat Neurosci 3:919–926
Sriraman S, Kevrekidis I, Hummer G (2005) Coarse nonlinear dynamics and metastability of filling-emptying transitions: water in carbon nanotubes. Physical review letters 95(13):130,603
Turrigiano GG, Nelson SB (2004) Homeostatic plasticity in the developing nervous system. Nat Rev Neurosci 5(2):97–107
Van Rossum M, Bi G, Turrigiano G (2000) Stable hebbian learning from spike timing-dependent plasticity. J Neurosci 20(23):8812–8821
Zou Y, Fonoberov V, Fonoberova M, Mezic I, Kevrekidis I (2012) Model reduction for agent-based social simulation: coarse-graining a civil violence model. Phys Rev E 85(6):066,106
Acknowledgments
The work of CRL was supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. The work of IGK was supported by the US National Science Foundation.
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Laing, C.R., Kevrekidis, I.G. Equation-free analysis of spike-timing-dependent plasticity. Biol Cybern 109, 701–714 (2015). https://doi.org/10.1007/s00422-015-0668-0
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DOI: https://doi.org/10.1007/s00422-015-0668-0