×

Firing-rate models for neurons with a broad repertoire of spiking behaviors. (English) Zbl 1402.92019

Summary: Capturing the response behavior of spiking neuron models with rate-based models facilitates the investigation of neuronal networks using powerful methods for rate-based network dynamics. To this end, we investigate the responses of two widely used neuron model types, the Izhikevich and augmented multi-adapative threshold (AMAT) models, to a range of spiking inputs ranging from step responses to natural spike data. We find (i) that linear-nonlinear firing rate models fitted to test data can be used to describe the firing-rate responses of AMAT and Izhikevich spiking neuron models in many cases; (ii) that firing-rate responses are generally too complex to be captured by first-order low-pass filters but require bandpass filters instead; (iii) that linear-nonlinear models capture the response of AMAT models better than of Izhikevich models; (iv) that the wide range of response types evoked by current-injection experiments collapses to few response types when neurons are driven by stationary or sinusoidally modulated Poisson input; and (v) that AMAT and Izhikevich models show different responses to spike input despite identical responses to current injections. Together, these findings suggest that rate-based models of network dynamics may capture a wider range of neuronal response properties by incorporating second-order bandpass filters fitted to responses of spiking model neurons. These models may contribute to bringing rate-based network modeling closer to the reality of biological neuronal networks.

MSC:

92B20 Neural networks for/in biological studies, artificial life and related topics
92C20 Neural biology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Al-Mohy, AH; Higham, NJ, A new scaling and squaring algorithm for the matrix exponential, SIAM Journal on Matrix Analysis and Applications, 31, 970-989, (2009) · Zbl 1194.15021 · doi:10.1137/09074721X
[2] Avermann, M.; Tomm, C.; Mateo, C.; Gerstner, W.; Petersen, CCH, Microcircuits of excitatory and inhibitory neurons in layer 2/3 of mouse barrel cortex, Journal of Neurophysiology, 107, 3116-3134, (2012) · doi:10.1152/jn.00917.2011
[3] Blomquist, P.; Devor, A.; Indahl, UG; Ulbert, I.; Einevoll, GT; Dale, AM, Estimation of thalamocortical and intracortical network models from joint thalamic single-electrode and cortical laminar-electrode recordings in the rat barrel system, PLoS Computational Biology, 5, e1000,328, (2009) · doi:10.1371/journal.pcbi.1000328
[4] Brette, R.; Gerstner, W., Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, Journal of Neurophysiology, 94, 3637-3642, (2005) · doi:10.1152/jn.00686.2005
[5] Brunel, N., Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, Journal of Computational Neuroscience, 8, 183-208, (2000) · Zbl 1036.92008 · doi:10.1023/A:1008925309027
[6] Brunel, N.; Chance, FS; Fourcaud, N.; Abbott, LF, Effects of synaptic noise and filtering on the frequency response of spiking neurons, Physical Review Letters, 86, 2186-2189, (2001) · doi:10.1103/PhysRevLett.86.2186
[7] Burkitt, A. N., A Review of the Integrate-and-fire Neuron Model: I. Homogeneous Synaptic Input, Biological Cybernetics, 95, 1-19, (2006) · Zbl 1161.92315 · doi:10.1007/s00422-006-0068-6
[8] Burkitt, A. N., A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties, Biological Cybernetics, 95, 97-112, (2006) · Zbl 1161.92314 · doi:10.1007/s00422-006-0082-8
[9] Casti, A.; Hayot, F.; Xiao, Y.; Kaplan, E., A simple model of retina-LGN transmission, Journal of Computational Neuroscience, 24, 235-252, (2008) · doi:10.1007/s10827-007-0053-7
[10] Chance, FS; Abbott, LF; Reyes, AD, Gain modulation from background synaptic input, Neuron, 35, 773-782, (2002) · doi:10.1016/S0896-6273(02)00820-6
[11] Coombes, S., Waves, bumps, and patterns in neural field theories, Biological Cybernetics, 93, 91-108, (2005) · Zbl 1116.92012 · doi:10.1007/s00422-005-0574-y
[12] Ermentrout, B., Neural networks as spatio-temporal pattern-forming systems, Reports on Progress in Physis, 61, 353-430, (1998) · doi:10.1088/0034-4885/61/4/002
[13] FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1, 445-466, (1961) · doi:10.1016/S0006-3495(61)86902-6
[14] Funke, K.; Wörgötter, F., On the significance of temporally structured activity in the dorsal lateral geniculate nucleus (LGN), Progress in Neurobiology, 53, 67-119, (1997) · doi:10.1016/S0301-0082(97)00032-4
[15] Gerstein, GL; Mandelbrot, B., Random walk models for the spike activity of a single neuron, Biophysical Journal, 4, 41-68, (1964) · doi:10.1016/S0006-3495(64)86768-0
[16] Gewaltig, MO; Diesmann, M., NEST (NEural Simulation Tool), Scholarpedia, 2, 1430, (2007) · doi:10.4249/scholarpedia.1430
[17] Haider, B.; Häusser, M.; Carandini, M., Inhibition dominates sensory responses in the awake cortex, Nature, 493, 97-100, (2013) · doi:10.1038/nature11665
[18] Heiberg, T.; Kriener, B.; Tetzlaff, T.; Casti, A.; Einevoll, GT; Plesser, HE, Firing-rate models capture essential response dynamics of LGN relay cells, Journal of Computational Neuroscience, 35, 359-375, (2013) · Zbl 1382.92060 · doi:10.1007/s10827-013-0456-6
[19] Helias, M.; Kunkel, S.; Masumoto, G.; Igarashi, J.; Eppler, JM; Ishii, S.; Fukai, T.; Morrison, A.; Diesmann, M., Supercomputers ready for use as discovery machines for neuroscience, Frontiers in Neuroinformatics, 6, 26, (2012) · doi:10.3389/fninf.2012.00026
[20] Higham, NJ, The scaling and squaring method for the matrix exponential revisited, SIAM Journal on Matrix Analysis and Applications, 26, 1179-1193, (2005) · Zbl 1081.65037 · doi:10.1137/04061101X
[21] Hodgkin, AL; Huxley, AF, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, 117, 500-544, (1952) · doi:10.1113/jphysiol.1952.sp004764
[22] Ikegaya, Y.; Sasaki, T.; Ishikawa, D.; Honma, N.; Tao, K.; Takahashi, N.; Minamisawa, G.; Ujita, S.; Matsuki, N., Interpyramid spike transmission stabilizes the sparseness of recurrent network activity, Cerebral Cortex, 23, 293-304, (2013) · doi:10.1093/cercor/bhs006
[23] Izhikevich, E.M. (2003a). Figure 1.m MATLAB script. http://www.izhikevich.org/publications/figure1.m, last accessed 18 Aug 2017.
[24] Izhikevich, E. M., Simple model of spiking neurons, IEEE Transactions on Neural Networks, 14, 1569-1572, (2003) · doi:10.1109/TNN.2003.820440
[25] Izhikevich, EM, Which model to use for cortical spiking neurons?, IEEE Transactions on Neural Networks, 15, 1063-1070, (2004) · doi:10.1109/TNN.2004.832719
[26] Izhikevich, EM, Hybrid spiking models, Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences, 368, 5061-5070, (2010) · Zbl 1211.37108 · doi:10.1098/rsta.2010.0130
[27] Jansen, BH; Rit, VG, Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns, Biological Cybernetics, 73, 357-366, (1995) · Zbl 0827.92010 · doi:10.1007/BF00199471
[28] Johannesma, P. I. M., Diffusion Models for the Stochastic Activity of Neurons, 116-144, (1968), Berlin, Heidelberg · doi:10.1007/978-3-642-87596-0_11
[29] Jolivet, R.; Rauch, A.; Lüscher, HR; Gerstner, W., Predicting spike timing of neocortical pyramidal neurons by simple threshold models, Journal of Computational Neuroscience, 21, 35-49, (2006) · Zbl 1118.92013 · doi:10.1007/s10827-006-7074-5
[30] Jolivet, R.; Schürmann, F.; Berger, TK; Naud, R.; Gerstner, W.; Roth, A., The quantitative single-neuron modeling competition, Biological Cybernetics, 99, 417-426, (2008) · Zbl 1161.92009 · doi:10.1007/s00422-008-0261-x
[31] Jones, E., Oliphant, T., Peterson, P., et al. (2001). SciPy: open source scientific tools for Python. http://www.scipy.org/, [Online; Accessed 09 March 2015].
[32] Kobayashi, R.; Tsubo, Y.; Shinomoto, S., Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Frontiers in Computational Neuroscience, 3, 9, (2009) · doi:10.3389/neuro.10.009.2009
[33] Kunkel, S.; Schmidt, M.; Eppler, JM; Plesser, HE; Masumoto, G.; Igarashi, J.; Ishii, S.; Fukai, T.; Morrison, A.; Diesmann, M.; Helias, M., Spiking network simulation code for petascale computers, Frontiers in Neuroinformatics, 8, 78, (2014) · doi:10.3389/fninf.2014.00078
[34] Lapicque, L., Considérations préalables sur la nature du phénomene par lequel l’électricité excite les nerfs, Journal de Physiologie et de Pathologie Générale, 9, 565-578, (1907)
[35] Lefort, S.; Tomm, C.; Sarria, JCF; Petersen, CCH, The excitatory neuronal network of the C2 barrel column in mouse primary somatosensory cortex, Neuron, 61, 301-316, (2009) · doi:10.1016/j.neuron.2008.12.020
[36] Mainen, ZF; Sejnowski, TJ, Influence of dendritic structure on firing pattern in model neocortical neurons, Nature, 382, 363-366, (1996) · doi:10.1038/382363a0
[37] Markram, H.; Toledo-Rodriguez, M.; Wang, Y.; Gupta, A.; Silberberg, G.; Wu, C., Interneurons of the neocortical inhibitory system, Nature Reviews Neuroscience, 5, 793-807, (2004) · doi:10.1038/nrn1519
[38] Moler, C. (2012). A balancing act for the matrix exponential. http://blogs.mathworks.com/cleve/2012/07/23/a-balancing-act-for-the-matrix-exponential/.
[39] Morris, C.; Lecar, H., Voltage oscillations in the barnacle giant muscle fiber, Biophysical Journal, 35, 193-213, (1981) · doi:10.1016/S0006-3495(81)84782-0
[40] Morrison, A.; Straube, S.; Plesser, HE; Diesmann, M., Exact subthreshold integration with continuous spike times in discrete time neural network simulations, Neural Computation, 19, 47-79, (2007) · Zbl 1116.92016 · doi:10.1162/neco.2007.19.1.47
[41] Muller, E., Davison, A.P., Brizzi, T., Bruederle, D., Eppler, J.M., Kremkow, J., Pecevski, D., Perrinet, L., Schmuker, M., Yger, P. (2009). NeuralEnsemble.Org: Unifying neural simulators in Python to ease the model complexity bottleneck. In Frontiers in Neuroscience Conference Abstract: Neuroinformatics 2009. https://doi.org/10.3389/conf.neuro.11.2009.08.104.
[42] Nordbø, Ø; Wyller, J.; Einevoll, GT, Neural network firing-rate models on integral form: effects of temporal coupling kernels on equilibrium-state stability, Biological Cybernetics, 97, 195-209, (2007) · Zbl 1125.92014 · doi:10.1007/s00422-007-0167-z
[43] Nordlie, E.; Tetzlaff, T.; Einevoll, GT, Rate dynamics of leaky integrate-and-fire neurons with strong synapses, Frontiers in Computational Neuroscience, 4, 149, (2010) · doi:10.3389/fncom.2010.00149
[44] Østergaard, J.; Kramer, MA; Eden, UT, Capturing spike variability in noisy Izhikevich neurons using point process generalized linear models, Neural Computation, 30, 125-148, (2018) · doi:10.1162/neco_a_01030
[45] Ostojic, S.; Brunel, N., From spiking neuron models to linear-nonlinear models, PLoS Computational Biology, 7, e1001,056, (2011) · doi:10.1371/journal.pcbi.1001056
[46] Paninski, L.; Pillow, JW; Simoncelli, EP, Maximum likelihood estimation of a stochastic integrate-and-fire neural encoding model, Neural Computation, 16, 2533-2561, (2004) · Zbl 1180.62179 · doi:10.1162/0899766042321797
[47] Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; Vanderplas, J.; Passos, A.; Cournapeau, D.; Brucher, M.; Perrot, M.; Duchesnay, E., Scikit-learn: machine learning in python, Journal of Machine Learning Research, 12, 2825-2830, (2011) · Zbl 1280.68189
[48] Petersen, C.; Crochet, S., Synaptic computation and sensory processing in neocortical layer 2/3, Neuron, 78, 28-48, (2013) · doi:10.1016/j.neuron.2013.03.020
[49] Pillow, JW; Paninski, L.; Uzzell, VJ; Simoncelli, EP; Chichilnisky, EJ, Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model, Journal of Neuroscience, 25, 11,003-11,013, (2005) · doi:10.1523/JNEUROSCI.3305-05.2005
[50] Plesser, HE; Diesmann, M., Simplicity and efficiency of integrate-and-fire neuron models, Neural Computation, 21, 353-359, (2009) · Zbl 1178.68456 · doi:10.1162/neco.2008.03-08-731
[51] Plesser, H.E., Diesmann, M., Gewaltig, M.O., Morrison, A. (2013). NEST: the neural simulation tool. In Jaeger, D, & Jung, R (Eds.) Encyclopedia of Computational Neuroscience. Berlin: Springer, DOI https://doi.org/10.1007/SpringerReference_348323, (to appear in print).
[52] Richardson, MJE, Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive, Physical Review E, 76, 1-15, (2007)
[53] Richardson, MJE; Swarbrick, R., Firing-rate response of a neuron receiving excitatory and inhibitory synaptic shot noise, Physical Review Letters, 105, 178,102, (2010) · doi:10.1103/PhysRevLett.105.178102
[54] Rotter, S.; Diesmann, M., Exact digital simulation of time-invariant linear systems with applications to neuronal modeling, Biological Cybernetics, 81, 381-402, (1999) · Zbl 0958.92004 · doi:10.1007/s004220050570
[55] Roxin, A., The role of degree distribution in shaping the dynamics in networks of sparsely connected spiking neurons, Frontiers in Computational Neuroscience, 5, 8, (2011) · doi:10.3389/fncom.2011.00008
[56] Sakata, S.; Harris, KD, Laminar-dependent effects of cortical state on auditory cortical spontaneous activity, Frontiers in Neural Circuits, 6, 1-10, (2012)
[57] Shimazaki, H.; Shinomoto, S., Kernel bandwidth optimization in spike rate estimation, Journal of Computational Neuroscience, 29, 171-182, (2010) · doi:10.1007/s10827-009-0180-4
[58] Song, S.; Sjöström, P.; Reigl, M.; Nelson, S.; Chklovskii, D., Highly nonrandom features of synaptic connectivity in local cortical circuits, PLoS Biology, 3, e68, (2005) · doi:10.1371/journal.pbio.0030068
[59] Stein, RB, A theoretical analysis of neuronal variability, Biophysical Journal, 5, 173-194, (1965) · doi:10.1016/S0006-3495(65)86709-1
[60] Tuckwell, H.C. (1988). Introduction to theoretical neurobiology Vol. 1. Cambridge: Cambridge University Press.
[61] Weber, AI; Pillow, JW, Capturing the dynamical repertoire of single neurons with generalized linear models, Neural Computation, 29, 3260-3289, (2017) · doi:10.1162/neco∖_a∖_01021
[62] Wilson, HR; Cowan, JD, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, 12, 1-24, (1972) · doi:10.1016/S0006-3495(72)86068-5
[63] Wolfram, S. (1999). The mathematica book, 4t. Cambridge: Wolfram Media/Cambridge University Press.
[64] Yamauchi, S.; Kim, H.; Shinomoto, S., Elemental spiking neuron model for reproducing diverse firing patterns and predicting precise firing times, Frontiers in Computational Neuroscience, 5, 42, (2011) · doi:10.3389/fncom.2011.00042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.