# American Institute of Mathematical Sciences

2016, 13(3): 569-578. doi: 10.3934/mbe.2016008

## A model based rule for selecting spiking thresholds in neuron models

 1 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, Copenhagen, 2100, Denmark

Received  February 2015 Revised  November 2015 Published  January 2016

Determining excitability thresholds in neuronal models is of high interest due to its applicability in separating spiking from non-spiking phases of neuronal membrane potential processes. However, excitability thresholds are known to depend on various auxiliary variables, including any conductance or gating variables. Such dependences pose as a double-edged sword; they are natural consequences of the complexity of the model, but proves difficult to apply in practice, since gating variables are rarely measured.
In this paper a technique for finding excitability thresholds, based on the local behaviour of the flow in dynamical systems, is presented. The technique incorporates the dynamics of the auxiliary variables, yet only produces thresholds for the membrane potential. The method is applied to several classical neuron models and the threshold's dependence upon external parameters is studied, along with a general evaluation of the technique.
Citation: Frederik Riis Mikkelsen. A model based rule for selecting spiking thresholds in neuron models. Mathematical Biosciences & Engineering, 2016, 13 (3) : 569-578. doi: 10.3934/mbe.2016008
##### References:
 [1] L. Abbott and T. Kepler, Model neurons: From Hodgkin-Huxley to Hopfield, Lect. Notes Phys., 368 (1990), p5. [2] J. A. Connor and C. F. Stevens, Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma, J. Physiol., 213 (1971), 31-53. doi: 10.1113/jphysiol.1971.sp009366. [3] P. Dayan and L. Abbott, Theoretical Neuroscience - Computational and Mathematical Modeling of Neural Systems, Computational Neuroscience, MIT Press, Cambridge, MA, 2001. [4] M. Desroches, M. Krupa and S. Rodrigues, Inflection, canards and excitability threshold in neuronal models, J. Math. Biol., 67 (2012), 989-1017. doi: 10.1007/s00285-012-0576-z. [5] S. Ditlevsen and P. Greenwood, The morris-lecar neuron model embeds a leaky integrate-and-fire model, J. Math. Biol., 67 (2013), 239-259. doi: 10.1007/s00285-012-0552-7. [6] R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. [7] J. Ginoux and B. Rossetto, Differential geometry and mechanics: Applications to chaotic dynamical systems, Int. J. Bifurcat. Chaos, 16 (2006), 887-910. doi: 10.1142/S0218127406015192. [8] A. Hodgkin and A. Huxley, A quantitative description of the membrane current and application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544. [9] E. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, The MIT Press, 2007. [10] T. Kepler, L. Abbott and E. Marder, Membranes with the same ion channel populations but different excitabilities, Biol. Cybern., 66 (1992), p381. [11] V. I. Krinsky and Yu. M. Kokoz, Analysis of equations of excitable membranes I. Reduction of the Hodgkin-Huxley equations to a second order system, Biofizika, 18 (1973), p506. [12] C. Meunier, Two and three dimensional reductions of the Hodgkin-Huxley system: Separation of time scales and bifurcations, Biol. Cybern., 67 (1992), 461-468. doi: 10.1007/BF00200990. [13] C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J., 35 (1981), 193-213. doi: 10.1016/S0006-3495(81)84782-0. [14] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. [15] M. Okuda, New method of nonlinear analysis for shaping and threshold actions, J. Phys. Soc. Jpn., 41 (1976), 1815-1816. doi: 10.1143/JPSJ.41.1815. [16] B. Peng, V. Gaspar and K. Showalter, False bifurcations in chemical systems: Canards, Phil. Trans. R Soc. Lond. A, 337 (1991), 275-289. doi: 10.1098/rsta.1991.0123. [17] L. Perko, Differential Equations and Dynamical Systems, $3^{rd}$ edition, Texts in Applied Mathematics, 7, Springer, 2000. [18] J. Platkiewicz and R. Brette, A threshold equation for action potential initiation, PLoS Comput. Biol., 6 (2010), e1000850, 16 pp. doi: 10.1371/journal.pcbi.1000850. [19] M. Sekerli, C. Del Negro, R. Lee and R. Butera, Estimating action potential thresholds from neuronal time-series: New metrics and evaluation of methodologies, IEEE T. Bio. Med. Eng., 51 (2004), 1665-1672. doi: 10.1109/TBME.2004.827531. [20] A. Tonnelier, Threshold curve for the excitability of bidimensional spiking neurons, Phys. Rev. E, 90 (2014), 022701. doi: 10.1103/PhysRevE.90.022701. [21] M. Wechselberge, J. Mitry and J. Rinzel, Canard theory and excitability, in Nonautonomous Dynamical Systems in the Life Sciences, Lecture Nones in Math., 2102, Springer, 2013, 89-132. doi: 10.1007/978-3-319-03080-7_3.

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##### References:
 [1] L. Abbott and T. Kepler, Model neurons: From Hodgkin-Huxley to Hopfield, Lect. Notes Phys., 368 (1990), p5. [2] J. A. Connor and C. F. Stevens, Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma, J. Physiol., 213 (1971), 31-53. doi: 10.1113/jphysiol.1971.sp009366. [3] P. Dayan and L. Abbott, Theoretical Neuroscience - Computational and Mathematical Modeling of Neural Systems, Computational Neuroscience, MIT Press, Cambridge, MA, 2001. [4] M. Desroches, M. Krupa and S. Rodrigues, Inflection, canards and excitability threshold in neuronal models, J. Math. Biol., 67 (2012), 989-1017. doi: 10.1007/s00285-012-0576-z. [5] S. Ditlevsen and P. Greenwood, The morris-lecar neuron model embeds a leaky integrate-and-fire model, J. Math. Biol., 67 (2013), 239-259. doi: 10.1007/s00285-012-0552-7. [6] R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. [7] J. Ginoux and B. Rossetto, Differential geometry and mechanics: Applications to chaotic dynamical systems, Int. J. Bifurcat. Chaos, 16 (2006), 887-910. doi: 10.1142/S0218127406015192. [8] A. Hodgkin and A. Huxley, A quantitative description of the membrane current and application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544. [9] E. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, The MIT Press, 2007. [10] T. Kepler, L. Abbott and E. Marder, Membranes with the same ion channel populations but different excitabilities, Biol. Cybern., 66 (1992), p381. [11] V. I. Krinsky and Yu. M. Kokoz, Analysis of equations of excitable membranes I. Reduction of the Hodgkin-Huxley equations to a second order system, Biofizika, 18 (1973), p506. [12] C. Meunier, Two and three dimensional reductions of the Hodgkin-Huxley system: Separation of time scales and bifurcations, Biol. Cybern., 67 (1992), 461-468. doi: 10.1007/BF00200990. [13] C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J., 35 (1981), 193-213. doi: 10.1016/S0006-3495(81)84782-0. [14] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. [15] M. Okuda, New method of nonlinear analysis for shaping and threshold actions, J. Phys. Soc. Jpn., 41 (1976), 1815-1816. doi: 10.1143/JPSJ.41.1815. [16] B. Peng, V. Gaspar and K. Showalter, False bifurcations in chemical systems: Canards, Phil. Trans. R Soc. Lond. A, 337 (1991), 275-289. doi: 10.1098/rsta.1991.0123. [17] L. Perko, Differential Equations and Dynamical Systems, $3^{rd}$ edition, Texts in Applied Mathematics, 7, Springer, 2000. [18] J. Platkiewicz and R. Brette, A threshold equation for action potential initiation, PLoS Comput. Biol., 6 (2010), e1000850, 16 pp. doi: 10.1371/journal.pcbi.1000850. [19] M. Sekerli, C. Del Negro, R. Lee and R. Butera, Estimating action potential thresholds from neuronal time-series: New metrics and evaluation of methodologies, IEEE T. Bio. Med. Eng., 51 (2004), 1665-1672. doi: 10.1109/TBME.2004.827531. [20] A. Tonnelier, Threshold curve for the excitability of bidimensional spiking neurons, Phys. Rev. E, 90 (2014), 022701. doi: 10.1103/PhysRevE.90.022701. [21] M. Wechselberge, J. Mitry and J. Rinzel, Canard theory and excitability, in Nonautonomous Dynamical Systems in the Life Sciences, Lecture Nones in Math., 2102, Springer, 2013, 89-132. doi: 10.1007/978-3-319-03080-7_3.
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