Action Potential Onset Dynamics and the Response Speed of Neuronal Populations B. Naundorf T. Geisel F. Wolf Article

Received: 23 November 2004 Revised: 21 December 2004 Accepted: 05 January 2005 DOI :
10.1007/s10827-005-0329-8

Cite this article as: Naundorf, B., Geisel, T. & Wolf, F. J Comput Neurosci (2005) 18: 297. doi:10.1007/s10827-005-0329-8 Abstract The result of computational operations performed at the single cell level are coded into sequences of action potentials (APs). In the cerebral cortex, due to its columnar organization, large number of neurons are involved in any individual processing task. It is therefore important to understand how the properties of coding at the level of neuronal populations are determined by the dynamics of single neuron AP generation. Here, we analyze how the AP generating mechanism determines the speed with which an ensemble of neurons can represent transient stochastic input signals. We analyze a generalization of the θ-neuron, the normal form of the dynamics of Type-I excitable membranes. Using a novel sparse matrix representation of the Fokker-Planck equation, which describes the ensemble dynamics, we calculate the transmission functions for small modulations of the mean current and noise noise amplitude. In the high-frequency limit the transmission function decays as ω^{−γ} , where γ surprisingly depends on the phase θ_{s} at which APs are emitted. If at θ_{s} the dynamics is insensitive to external inputs, the transmission function decays as (i) ω^{−3} for the case of a modulation of a white noise input and as (ii) ω^{−2} for a modulation of the mean input current in the presence of a correlated and uncorrelated noise as well as (iii) in the case of a modulated amplitude of a correlated noise input. If the insensitivity condition is lifted, the transmission function always decays as ω^{−1} , as in conductance based neuron models. In a physiologically plausible regime up to 1 kHz the typical response speed is, however, independent of the high-frequency limit and is set by the rapidness of the AP onset, as revealed by the full transmission function. In this regime modulations of the noise amplitude can be transmitted faithfully up to much higher frequencies than modulations in the mean input current. We finally show that the linear response approach used is valid for a large regime of stimulus amplitudes.

Keywords population of spiking neurons spike luitiation dynamics integrate-and-five model

References Abramowitz M, Stegun IA (1972) Tables of Mathematics Functions. Dover Publications, New York.

Google Scholar Anderson JS, Lampl I, Gillespie DC, Ferster D (2000) The contribution of noise to contrast invariance of orientation tuning in cat visual cortex. Science, 290: 1968–1972.

Google Scholar Benda J, Herz AV (2003) A universal model for spike-frequency adaptation. Neural Comput. 15: 2523–2564

Google Scholar Bethge M, Silberberg G, Markram H, Tsodyks M, Pawelzik K (2001) Is population variance a signal for neocortical neurons? In: N Elsner, GW Kreutzberg, eds., Proceedings of the 4th Meeting of the German Neuroscience Society 1, Georg Thieme Verlag, 249.

Brumberg JC (2002) Firing pattern modulation by oscillatory input in supragranular pyramidal neurons. Neurosci. 114: 239–246.

Google Scholar Brunel N, Hakim V (1999) Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. 11: 1621– 1671.

Brunel N (2000) Dynamics of sparsely connected networks of excitatory and inhibitory spiking. Neurons 8: 183–208.

Google Scholar Brunel N, Chance FS, Fourcaud N, Abbott LF (2001) Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys Rev Lett. 86: 2186–2189.

Google Scholar Brunel N, Latham PE (2003) Firing rate of the noisy quadratic integrate-and-fire neuron. Neural Comput. 15: 2281–2306.

Google Scholar Coddington E, Levinson N (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York.

Google Scholar Destexhe A, Pare’ D (1999) Impact of network activity on the integrative properties of neocortical pyramidal neurons

in vivo . J Neurophysiol. 81: 1531–1547.

Google Scholar Ermentrout G, Kopell N (1984) Frequency plateaus in a chain of weakly coupled oscillators, I. SIAM J. Math. Anal. 15: 215–237.

Google Scholar Ermentrout GB, Kopell N (1986) Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM-J.-Appl.-Math. 2: 233–253.

Google Scholar Fourcaud N, Brunel N (2002) Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Comp. 14: 2057– 2110.

Google Scholar Fourcaud-Trocme’ N, Hansel D, van Vreeswijk C, Brunel N (2003) How spike generation mechanisms determine the neuronal response to fluctuating inputs. J Neurosci. 23: 11628–11620.

Google Scholar Gardiner CW (1985) Handbok of Stochastic Methods. Springer, Berlin.

Google Scholar Gerstner W (2000) Population dynamics of spiking neurons: Fast transients, asynchronous states, and locking. Neural Comput. 12: 43–89.

Google Scholar Guckenheimer J (1975) Isochrons and phaseless sets. J. Math. Biol. 1: 259–273.

Google Scholar Gutkin BS, Ermentrout GB (1998) Dynamics of membrane excitability determine interspike interval variability: A link between spike generation mechanisms and cortical spike train statistics. Neural Comput. 10: 1047–1065.

Google Scholar Izhikevich EM (2004) Which model to use for corticalspiking neurons? IEEE Transactions on Neural Networks, (in press).

van Kampen NG (1981) Itô versus Stratonovich. J Stat Phys. 24: 175–187.

Google Scholar Knight BW (1972) Dynamics of encoding in a population of neurons. J Gen Physiol. 59: 734–766.

Google Scholar Knight BW, Omurtag A, Sirovich L (2000) The approach of a neuron population firing rate to a new equilibrium: An exact theoretical result. Neural Comp. 12: 1045–1055.

Google Scholar Lapicque L (1907) Recherches quantitatives sur l’excitation electrique des nerfs traitee comme une polarization. J. Physiol. Pathol. Gen. 9: 620–635.

Google Scholar Lehoucq R, Sorensen DC, Yang C (1998) Arpack User’s Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia.

Lindner B, Schimansky-Geier L (2002) Transmission of noise coded versus additive signals through a neuronal ensemble., Phys Rev Lett. 86: 2934–2937.

Google Scholar Lindner B, Longtin A, Bulsara A (2003) Analytic expressions for rate and CV of a type I neuron driven by white gaussian noise. Neural Comput. 15: 1760–1787.

Google Scholar Mattia M, Del Giudice P (2002) Population dynamics of interacting spiking neurons. Phys. Rev. E. 66: 051917.

Google Scholar Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35: 193.

Google Scholar Naundorf B, Geisel T, Wolf F (2003) The Intrinsic Time Scale of Transient Neuronal Responses. xxx.lanl.gov, physics/0307135, Preprint.

Naundorf B, Geisel T, Wolf F (2004) Dynamical response properties of a canonicalmodel for Type-I membranes. Neurocomp (in press).

Reichl LE (1988) Transitions in the Floquet Rates of a Driven Stochastic-System. J. Stat. Phys. 53: 41.

Google Scholar Risken H (1996) The Fokker Planck Equation: Methods of Solution and Applications. Springer, Berlin.

Google Scholar Rauch A, La Camera G, Luscher HR, Senn W, Fusi S (2003) Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents. J Neurophysiol. 90: 1598–1612.

Google Scholar Silberberg G, Bethge M, Markram H, Pawelzik K, Tsodyks M (2004) Dynamics of population rate codes in ensembles of neocortical neurons. J Neurophysiol. 91: 704–709.

Google Scholar Softky W, Koch K (1993) The highly irregular ring of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci. 13: 334–355.

Google Scholar Strogatz SH (1995) Nonlinear Dynamics and Chaos. Addison Wesley.

Tateno T, Harsch A, Robinson HP (2004) Threshold firing frequency-current relationships of neurons in rat somatosensory cortex: Type 1 and type 2 dynamics. J Neurophysiol. 92: 2283–2294.

Google Scholar Trefethen NL, Bau D (1997) Numerical Linear Algebra. SIAM, Philadelphia.

Google Scholar Tuckwell H (1988) Introduction to Theoretical Neurobiology, 2 vols. Cambridge University Press, Cambridge.

Google Scholar Volgushev M, Eysel UT (2000) Noise makes sense in neuronal computing. Science 290: 1908–1909.

Google Scholar van Vreeswijk C, Sompolinsky H (1996) Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science, 274: 1724–1726.

Google Scholar Wang XJ, Buzsaki G (1996) Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. J Neurosci. 16: 6402–6413.

Google Scholar Winfree A (1967) Biological rythms and behavior of pupulations of coupled oscillators. J. Theo. Biol. 16: 15.

Google Scholar Winfree A (2001) The Geometry of Biological Time, 2nd edition. Springer-Verlag, New York.

Google Scholar © Springer Science + Business Media, Inc. 2005

Authors and Affiliations B. Naundorf T. Geisel F. Wolf 1. Max-Planck-Institut für Dynamik und Selbstorganisation and Fakultät für Physik Universität Göttingen Göttingen Germany