Abstract
Many neurons in mammalian sensory systems exhibit On and Off responses when given appropriate excitatory and inhibitory stimuli. In some instances, such neurons can also exhibit a Mixed response where diminished On and Off responses are both present. In this manuscript, we present a simple single cell model for these ubiquitous stimulus-response patterns. The model is nonautonomous consisting of two fast variables (one being the voltage), one slow recovery variable, and a time dependent stimuli current I(t). For piecewise constant I(t), On and Off responses can be reproduced and it is shown that their dependence on both the duration and the intensity of the input can be derived using singular perturbation techniques. Furthermore, we show that for certain stimuli I(t) the voltage has spike trains both during and immediately after the stimuli is presented. Such Mixed responses have also been measured experimentally, and the current model reproduces all three responses robustly for different net synaptic currents I(t).
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Bertram, R., Butte, M., Kiemel, T., Sherman, A., 1995. Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57, 413–439.
Bressloff, P., Cowan, J., 2002. Spontaneous pattern formation in primary visual cortex. In: Hogan, S.J., Champneys, A., Krauskopf, B. (Eds.), Nonlinear Dynamics: Where do We Go from Here? Institute of Physics, Bristol.
Brumberg, J.C., Nowak, L.G., McCormick, D.A., 2000. Ionic mechanisms underlying repetitive high-frequency burst firing in supragranular cortical neurons. J. Neurosci. 20, 4829–4843.
Butera, R., Clark, J., Canavier, C., Baxter, D., Byrne, J., 1995. Analysis of the effects of modulatory agents on a modeled bursting neuron: Dynamic interactions between voltage and calcium dependent systems. J. Comput. Neurosci. 2, 19–44.
Chay, T.R., Cook, D.L., 1988. Endogenous bursting patterns in excitable cells. Math. Biosci. 90, 139–153.
Connors, B.W., Gutnick, M.J., 1990. Intrinsic firing patterns of diverse neocortical neurons. Trends Neurosci. 13, 99–104.
D’Angelo, E., Nieus, T., Maffei, A., Armano, S., Rossi, P., Taglietti, V., Fontana, A., Naldi, G., 2001. Theta-frequency bursting and resonance in cerebellar granule cells: Experimental evidence and modeling of a slow K +-dependent mechanism. J. Neurosci. 21(3), 759–770.
De Vries, G., 1998. Multiple bifurcations in a polynomial model of bursting oscillations. J. Nonlinear Sci. 8, 281–316.
Ermentrout, B., 2002. Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students. SIAM, Philadelphia.
Evans, E.F., 1992. Auditory processing of complex sounds: An overview. Philos. Trans. R. Soc. Lond. B Biol. Sci. 336(1278), 295–306.
FitzHugh, R., 1961. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466.
Gibson, J.R., Belerlein, M., Connors, B.W., 1999. Two networks of electrically coupled inhibitory neurons in neocortex. Nature 402, 75–79.
Griffiths, R., Pernarowski, M., 2006. Return map characterizations of for a model of bursting with two slow variables. SIAM J. Appl. Math. 66, 1917–1948.
Hartings, J., Temereanca, S., Simons, D., 2000. High responsiveness and direction sensitivity of neurons in the rat thalamic reticular nucleus to vibrissa deflections. J. Neurophysiol. 83, 2791–2801.
He, J., 2002. Off responses in the auditory thalamus of the guinea pig. J. Neurophysiol. 88, 2377–2386.
Hubel, D.H., Wiesel, T.N., 1959. Receptive fields of single neurones in the cat’s striate cortex. J. Physiol. 148, 574–591.
Izhikevich, E., 2004. Which model to use for cortical spiking neurons. IEEE Trans. Neural Netw. 15, 1063–1070 (special issue on temporal coding).
Koch, C., 2001. Biophysics of Computation: Information Processing in Single Neurons. Oxford University Press, New York.
Kuffler, S.W., 1953. Discharge patterns and functional organization of the Mammilian Retina. J. Neurophysiol. 16(16), 37–68.
Lancaster, B., Nicoll, R.A., Perkel, D.J., 1991. Calcium acivates two types of potassium channels in Rat Hippocampal neurons in culture. J. Neurosci. 11(1), 23–30.
Latulippe, J.J., 2007. A non-autonomous bursting model for neurons. Ph.D. thesis, Department of Mathematical Sciences, Montana State University, Bozeman.
Lytton, W., Sejnowski, T.J., 1991. Simulations of cortical pyramidal neurons synchronized by inhibitory interneurons. J. Neurophysiol. 66(3), 1059–1079.
McCormick, D.A., Gray, C.M., 1996. Chattering cells: Superficial pyramidal neurons contributing to the generation of synchronous oscillations in the visual cortex. Science 274(5284), 109–113.
Morris, C., Lecar, H., 1981. Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–213.
Pernarowski, M., 1994. Fast subsystem bifurcations in a slowly varying Lienard system exhibiting bursting. SIAM J. Appl. Math. 54, 814–832.
Pernarowski, M., 1998. Fast and slow subsystems for a continuum model of bursting activity in the pancreatic islet. SIAM J. Appl. Math. 58, 1667–1687.
Pernarowski, M., 2001. Controllability of excitable systems. Bull. Math. Biol. 63, 167–184.
Pernarowski, M., Miura, R.M., Kevorkian, J., 1992. Perturbation techniques for models of bursting electrical activity in pancreatic β-cells. SIAM J. Appl. Math. 52, 1627–1650.
Rinzel, J., Lee, Y., 1987. Dissection of a model for neuronal parabolic bursting. J. Math. Biol. 25, 653–675.
Schwindt, P.C., Spain, W.J., Foehring, R.C., Chubb, M.C., Crill, W.E., 1988. Slow conductances in neurons from cat sensorimotor cortex in vitro and their role in slow excitability changes. J Neurophysiol. 59(2), 450–467.
Smolen, P., Keizer, J., 1992. Slow voltage-inactivation of Ca2+ currents and bursting mechanisms for the mouse pancreatic β-cell. J. Memb. Biol. 127, 9–19.
Steriade, M., 1997. Synchronized activities of coupled oscillators in the cerebral cortex and thalamus at different levels of vigilance. Cereb. Cortex 7, 583–604.
Terman, D., 1991. Chaotic spikes arising from a model of bursting in excitable membrane. SIAM J. Appl. Math. 51, 1418–1450.
Wang, X.-J., 1999. Fast burst firing and short-term synaptic plasticity: A model of neocortical chattering neurons. Neuroscience 89(2), 347–362.
Wang, G.-Y., Liets, L.C., Chalupa, L.M., 2003. Nitric oxide differentially modulates on and off responses of retinal ganglion cells. J. Neurophysiol. 90, 1304–1313.
Wielaard, D.J., Shelley, M., McLaughlin, D., Shapley, R., 2001. How simple cells are made in a nonlinear network model of the visual cortex. J. Neurosci. 21, 5203–5211.
Wilson, H.R., Cowan, J.D., 1972. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–24.
Wilson, H.R., Cowan, J.D., 1973. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55–80.
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Latulippe, J., Pernarowski, M. A Nonautonomous Phenomenological Model for On and Off Responses of Cells in Sensory System. Bull. Math. Biol. 71, 162–188 (2009). https://doi.org/10.1007/s11538-008-9358-6
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DOI: https://doi.org/10.1007/s11538-008-9358-6