Dynamics of the exponential integrateandfire model with slow currents and adaptation
 Victor J. Barranca,
 Daniel C. Johnson,
 Jennifer L. Moyher,
 Joshua P. Sauppe,
 Maxim S. Shkarayev,
 Gregor Kovačič,
 David Cai
 … show all 7 hide
Abstract
In order to properly capture spikefrequency adaptation with a simplified pointneuron model, we study approximations of HodgkinHuxley (HH) models including slow currents by exponential integrateandfire (EIF) models that incorporate the same types of currents. We optimize the parameters of the EIF models under the external drive consisting of AMPAtype conductance pulses using the currentvoltage curves and the van Rossum metric to best capture the subthreshold membrane potential, firing rate, and jump size of the slow current at the neuron’s spike times. Our numerical simulations demonstrate that, in addition to these quantities, the approximate EIFtype models faithfully reproduce bifurcation properties of the HH neurons with slow currents, which include spikefrequency adaptation, phaseresponse curves, critical exponents at the transition between a finite and infinite number of spikes with increasing constant external drive, and bifurcation diagrams of interspike intervals in timeperiodically forced models. Dynamics of networks of HH neurons with slow currents can also be approximated by corresponding EIFtype networks, with the approximation being at least statistically accurate over a broad range of Poisson rates of the external drive. For the form of external drive resembling realistic, AMPAlike synaptic conductance response to incoming action potentials, the EIF model affords great savings of computation time as compared with the corresponding HHtype model. Our work shows that the EIF model with additional slow currents is well suited for use in largescale, pointneuron models in which spikefrequency adaptation is important.
 Abbott, LF, Kepler, TB (1990) Statistical Mechanics of Neural Networks. Springer, Berlin, Heidelberg, New York
 Avermann, M, Tomm, C, Mateo, C, Gerstner, W, Petersen, CCH (2012) Microcircuits of excitatory and inhibitory neurons in layer 2/3 of mouse barrel cortex. Journal of Neurophysiology 107: pp. 31163134 CrossRef
 Badel, L, Lefort, S, Berger, TK, Petersen, CCH, Gerstner, W, Richardson, MJE (2008a) Extracting nonlinear integrateandfire models from experimental data using dynamic iv curves. Biological Cybernetics 99: pp. 361370 CrossRef
 Badel, L, Lefort, S, Brette, R, Petersen, CCH, Gerstner, W, Richardson, MJE (2008b) Dynamic iv curves are reliable predictors of naturalistic pyramidalneuron voltage traces. Journal of Neurophysiology 99: pp. 656666 CrossRef
 Brette, R, Gerstner, W (2005) Adaptive exponential integrateandfire model as an effective description of neuronal activity. Journal of Neurophysiology 94: pp. 36373642 CrossRef
 Brown, TH, Johnston, D (1983) Voltageclamp analysis of mossy fiber synaptic input to hippocampal neurons. Journal of Neurophysiology 50: pp. 487507
 Burkitt, AN (2006a) A review of the integrateandfire neuron model: I. homogeneous synaptic input. Biological Cybernetics 95: pp. 119 CrossRef
 Burkitt, AN (2006b) A review of the integrateandfire neuron model: Ii. inhomogeneous synaptic input and network properties. Biological Cybernetics 95: pp. 97112 CrossRef
 Cai, D, Rangan, AV, McLaughlin, DW (2005) Architectural and synaptic mechanisms underlying coherent spontaneous activity in V1. Proceedings of the National Academy of Science (USA) 102: pp. 58685873 CrossRef
 Carandini, M, Mechler, F, Leonard, CS, Movshon, JA (1996) Spike train encoding by regularspiking cells of the visual cortex. Journal of Neurophysiology 76: pp. 34253441
 Clopath, C, Jolivet, R, Rauch, A, Lüscher, H, Gerstner, W (2007) Predicting neuronal activity with simple models of the threshold type: Adaptive exponential integrateandfire model with two compartments. Neurocomputing 70: pp. 16681673 CrossRef
 Destexhe, A, Pare, D (1999) Impact of network activity on the integrative properties of neocortical pyramidal neurons in vivo. Journal of Neurophysiology 81: pp. 15311547
 Destexhe, A, Contreras, D, Steriade, M (1998) Mechanisms underlying the synchronizing action of corticothalamic feedback through inhibition of thalamic relay cells. Journal of Neurophysiology 79: pp. 9991016
 Ermentrout, B, Pascal, M, Gutkin, B (2001) The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators. Neural Computing 13: pp. 12851310 CrossRef
 FourcaudTrocme, N, Hansel, D, van Vreeswijk, C, Brunel, N (2003) How spike generation mechanisms determine the neuronal response to fluctuating inputs. Journal of Neuroscience 23: pp. 1162811640
 Geisler, C, Brunel, N, Wang, XJ (2005) Contributions of intrinsic membrane dynamics to fast network oscillations with irregular neuronal discharges. Journal of Neurophysiology 94: pp. 43444361 CrossRef
 Gerstner, W., & Kistler, W.M. (2002). Spiking neuron models — single neurons, populations, plasticity. New York: Cambridge University Press.
 Gerstner, W, Naud, R (2009) Neuroscience. How good are neuron models?. Science 326: pp. 379380 CrossRef
 Gutkin, B, Ermentrout, B, Reyes, A (2005) Phase response curves give the response of neurons to transient inputs. Journal of Neurophysiology 94: pp. 16231635 CrossRef
 Hassard, B (1978) Bifurcation of periodic solutions of the HodgkinHuxley model for the squid giant axon. Journal of Theoretical Biology 71: pp. 401420 CrossRef
 Hodgkin, AL, Huxley, AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology (London) 117: pp. 500544
 Izhikevich, EM (2003) Simple model of spiking neurons. IEEE Transactions on Neural Networks 14: pp. 15691572 CrossRef
 Jancke, D, Chavance, F, Naaman, S, Grinvald, A (2004) Imaging cortical correlates of illusion in early visual cortex. Nature 428: pp. 423426 CrossRef
 Jin, W, Xu, J, Wu, Y, Ling, H, Wei, Y (2006) Crisis of interspike intervals in HodgkinHuxley model. Chaos, Solitons, and Fractals 27: pp. 952958 CrossRef
 Jolivet, R, Kobayashi, R, Rauch, A, Naud, R, Shinomoto, S, Gerstner, W (2008) A benchmark test for a quantitative assessment of simple neuron models. Journal of Neuroscience Methods 169: pp. 417424 CrossRef
 Kistler, WM, Gerstner, W, van Hemmen, JL (1997) Reduction of the HodgkinHuxley equations to a singlevariable threshold model. Neural Computation 9: pp. 10151045 CrossRef
 Koch, C. (1999). Biophysics of Computation. Oxford: Oxford University Press.
 Lapicque, L (1907) Recherches quantitatives sur l’excitation electrique des nerfs traitee comme une polarization. Journal de Physiologie et Pathologie Général 9: pp. 620635
 McCormick, D, Wang, Z, Huguenard, J (1993) Neurotransmitter control of neocortical neuronal activity and excitability. Journal of Neurophysiology 68: pp. 387398
 McLaughlin, D, Shapley, R, Shelley, M, Wielaard, J (2000) A neuronal network model of macaque primary visual cortex (V1): Orientation selectivity and dynamics in the input layer $4{C}\alpha $4Cα. Proceedings of the National Academy of Sciences of the United States of America 97: pp. 80878092 CrossRef
 Mensi, S, Naud, R, Pozzorini, C, Avermann, M, Petersen, CC, Gerstner, W (2012) Parameter extraction and classification of three cortical neuron types reveals two distinct adaptation mechanisms. Journal of Neurophysiology 107: pp. 17561775 CrossRef
 Naud, Richard, Marcille, Nicolas, Clopath, Claudia, Gerstner, Wulfram (2008) Firing patterns in the adaptive exponential integrateandfire model. Biological Cybernetics 99: pp. 335347 CrossRef
 Nicola, W, Campbell, SA (2013) Bifurcations of large networks of twodimensional integrate and fire neurons. Journal of Computational Neuroscience.
 Pare, D, Lang, EJ, Destexhe, A (1998) Inhibitory control of somatodendritic interactions underlying action potentials in neocortical pyramidal neurons in vivo: An intracellular and computational study. Neuroscience 84: pp. 377402 CrossRef
 Pozzorini, C, Naud, R, Mensi, S, Gerstner, W (2013) Temporal whitening by powerlaw adaptation in neocortical neurons. Nature Neuroscience 16: pp. 942948 CrossRef
 Rangan, AV, Cai, D (2007) Fast numerical methods for simulating largescale integrateandfire neuronal networks. Journal of Computational Neuroscience 22: pp. 81100 CrossRef
 Rangan, AV, Cai, D, McLaughlin, DW (2005) Modeling the spatiotemporal cortical activity associated with the linemotion illusion in primary visual cortex. Proceedings of the National Academy of Sciences of the United States of America 102: pp. 1879318800 CrossRef
 Rauch, A, Camera, G, Luscher, HR, Senn, W, Fusi, S (2003) Neocortical pyramidal cells respond as integrateandfire neurons to in vivolike input currents. Journal of Neurophysiology 90(3): pp. 15981612 CrossRef
 Richardson, MJE (2009) Dynamics of populations and networks of neurons with voltageactivated and calciumactivated currents. Physical Review E 80: pp. 021928 CrossRef
 Roa, MAD, Copelli, M, Kinouchi, O, Caticha, N (2007) Scaling law for the transient behavior of typeii neuron models. Physical Review E 75: pp. 021911 CrossRef
 Somers, D, Nelson, S, Sur, M (1995) An emergent model of orientation selectivity in cat visual cortical simple cells. Journal of Neuroscience 15: pp. 54485465
 Sun, Y, Zhou, D, Rangan, AV, Cai, D (2009) Librarybased numerical reduction of the hodgkinhuxley neuron for network simulation. Journal of Computational Neuroscience 27: pp. 369390 CrossRef
 Tao, L, Cai, D, McLaughlin, D, Shelley, M, Shapley, R (2006) Orientation selectivity in visual cortex by fluctuationcontrolled criticality. Proceedings of the National Academy of Sciences of the United States of America 103: pp. 1291112916 CrossRef
 Touboul, J, Brette, R (2008) Dynamics and bifurcations of the adaptive exponential integrateandfire model. Biological Cybernetics 99: pp. 319334 CrossRef
 Troyer, T, Krukowski, A, Priebe, N, Miller, K (1998) Contrast invariant orientation tuning in cat visual cortex with feedforward tuning and correlation based intracortical connectivity. Journal of Neuroscience 18: pp. 59085927
 Tsodyks, M, Kenet, T, Grinvald, A, Arieli, A (1999) Linking spontaneous activity of single cortical neurons and the underlying functional architecture. Science 286: pp. 19431946 CrossRef
 Tuckwell, H.C. (1988a). Introduction to theoretical neurobiology: linear cable theory and dendritic structure. In Cambridge studies in mathematical biology (Vol. 1). Cambridge University Press.
 Tuckwell. H.C. (1988b). Introduction to theoretical neurobiology: nonlinear and stochastic theories. In Cambridge studies in mathematical biology (Vol. 2). Cambridge University Press.
 Van Rossum, MCW (2001) A novel spike distance. Neural Computation 13: pp. 751763 CrossRef
 Wielaard, J, Shelley, M, Shapley, R, McLaughlin, D (2001) How Simple cells are made in a nonlinear network model of the visual cortex. Journal of Neuroscience 21: pp. 52035211
 Yamada, W., Koch, C., Adams, P. (1989). Multiple channels and calcium dynamics. In Methods in neuronal modeling: from synapses to networks (pp. 97–133). MIT Press.
 Title
 Dynamics of the exponential integrateandfire model with slow currents and adaptation
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Journal of Computational Neuroscience
Volume 37, Issue 1 , pp 161180
 Cover Date
 20140801
 DOI
 10.1007/s1082701304940
 Print ISSN
 09295313
 Online ISSN
 15736873
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Adaptation current
 Integrateandfire networks
 Bifurcations
 Numerical methods
 Efficient neuronal models
 Industry Sectors
 Authors

 Victor J. Barranca ^{(1)}
 Daniel C. Johnson ^{(2)}
 Jennifer L. Moyher ^{(1)}
 Joshua P. Sauppe ^{(3)}
 Maxim S. Shkarayev ^{(4)}
 Gregor Kovačič ^{(1)}
 David Cai ^{(5)} ^{(6)}
 Author Affiliations

 1. Mathematical Sciences Department, Rensselaer Polytechnic Institute, Troy, NY, USA
 2. Division of Applied Mathematics, Brown University, Providence, RI, USA
 3. Physics Department, University of WisconsinMadison, Madison, WI, USA
 4. Department of Physics and Astronomy, Iowa State University, Ames, IA, SA
 5. Department of Mathematics, MOELSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China
 6. Courant Institute of Mathematical Sciences and Center for Neural Science, New York University, New York, NY, USA