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Travelling waves in a model of quasi-active dendrites with active spines. (English) Zbl 1190.37086
Author’s abstract: Dendrites, the major components of neurons, have many different types of branching structures and are involved in receiving and integrating thousands of synaptic inputs from other neurons. Dendritic spines with excitable channels can be present in large densities on the dendrites of many cells. The recently proposed Spike-Diffuse-Spike (SDS) model that is described by a system of point hot-spots (with an integrate-and-fire process) embedded throughout a passive tree has been shown to provide a reasonable caricature of a dendritic tree with supra-threshold dynamics. Interestingly, real dendrites equipped with voltage-gated ion channels can exhibit not only supra-threshold responses, but also sub-threshold dynamics. This sub-threshold resonant-like oscillatory behaviour has already been shown to be adequately described by a quasi-active membrane. In this paper we introduce a mathematical model of a branched dendritic tree based upon a generalisation of the SDS model where the active spines are assumed to be distributed along a quasi-active dendritic structure. We demonstrate how solitary and periodic travelling wave solutions can be constructed for both continuous and discrete spine distributions. In both cases the speed of such waves is calculated as a function of system parameters. We also illustrate that the model can be naturally generalised to an arbitrary branched dendritic geometry whilst remaining computationally simple. The spatio-temporal patterns of neuronal activity are shown to be significantly influenced by the properties of the quasi-active membrane. Active (sub- and supra-threshold) properties of dendrites are known to vary considerably among cell types and animal species, and this theoretical framework can be used in studying the combined role of complex dendritic morphologies and active conductances in rich neuronal dynamics.

37N25 Dynamical systems in biology
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
92C20 Neural biology
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[1] Cajal, R., Histology of the nervous system of man and vertebrates, (1995), Oxford University Press New York, (first published 1899)
[2] Brown, T.H.; Kairiss, E.W.; Keenan, C.L., Hebbian synapses: biophysical mechanisms and algorithms, Ann. rev. neurosci., 13, 475-511, (1990)
[3] Miller, J.P.; Rall, W.; Rinzel, J., Synaptic amplification by active membrane in dendritic spines, Brain res., 325, 325-330, (1985)
[4] Mel, B.W.; Ruderman, D.L.; Archie, K.A., Translation-invariant orientation tuning in visual ‘complex’ cells could derive from intradendritic computations, J. neurosci., 18, 4325-4334, (1998)
[5] Larkum, M.E.; Zhu, J.J.; Sakmann, B., A new cellular mechanism for coupling inputs arriving at different cortical layers, Nature, 398, 338-341, (1999)
[6] Shepherd, J., The dendritic spine: A multifunctional integrative unit, J. neurophysiol., 75, 2197-2210, (1996)
[7] Segal, M., Dendritic spines and long-term plasticity, Nature rev. neurosci., 6, 277-284, (2005)
[8] Alvarez, V.A.; Sabatini, B.L., Anatomical and physiological plasticity of dendritic spines, Ann. rev. neurosci., 30, 79-97, (2007)
[9] Segev, I.; Rall, W., Excitable dendrites and spines: earlier theoretical insights elucidate recent direct observations, Trends neurosci., 21, 11, 453-460, (1998)
[10] Baer, S.M.; Rinzel, J., Propagation of dendritic spikes mediated by excitable spines: A continuum theory, J. neurophysiol., 65, 4, 874-890, (1991)
[11] Lord, G.J.; Coombes, S., Traveling waves in the Baer and rinzel model of spine studded dendritic tissue, Physica D, 161, 1-20, (2002) · Zbl 1066.74552
[12] Coombes, S.; Bressloff, P.C., Solitary waves in a model of dendritic cable with active spines, SIAM J. appl. math., 61, 2, 432-453, (2000) · Zbl 1016.92004
[13] Coombes, S., From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves, Math. biosci., 170, 155-172, (2001) · Zbl 1005.92005
[14] Coombes, S.; Bressloff, P.C., Saltatory waves in the spike-diffuse-spike model of active dendritic spines, Phys. rev. E, 91, 028102, (2003)
[15] Timofeeva, Y.; Lord, G.J.; Coombes, S., Spatio-temporal filtering properties of a dendritic cable with active spines: A modeling study in the spike-diffuse-spike framework, J. comput. neurosci., 21, 293-306, (2006)
[16] Timofeeva, Y.; Lord, G.J.; Coombes, S., Dendritic cable with active spines: A modelling study in the spike-diffuse-spike framework, Neurocomputing, 69, 1058-1061, (2006)
[17] Johnston, D.; Narayanan, R., Active dendrites: colorful wings of the mysterious butterflies, Trends neurosci., 31, 6, 309-316, (2008)
[18] Williams, S.R.; Christensen, S.R.; Stuart, G.J.; Häusser, M., Membrane potential bistability is controlled by the hyperpolarization-activated current \(I_h\) in rat cerebellar purkinje neurons in vitro, J. physiol., 539, 469-483, (2002)
[19] Hodgkin, A.L.; Huxley, A.F., A quantitative description of membrane current and its application to conduction and excitation in nerve, J. physiol., 117, 500-544, (1952)
[20] Mauro, A.; Conti, F.; Dodge, F.; Schor, R., Subthreshold behavior and phenomenological impedance of the squid giant axon, J. gen. physiol., 55, 497-523, (1970)
[21] Ulrich, D., Dendritic resonance in rat neocortical pyramidal cells, J. neurophysiol., 87, 2753-2759, (2002)
[22] Cook, E.P.; Guest, J.A.; Liang, Y.; Masse, N.Y.; Colbert, C.M., Dendrite-to-soma input/output function of continuous time-varying signals in hippocampal CA1 pyramidal neurons, J. neurophysiol., 98, 2943-2955, (2007)
[23] Koch, C., Cable theory in neurons with active, linearized membranes, Biol. cybern., 50, 15-33, (1984)
[24] Coombes, S.; Timofeeva, Y.; Svensson, C.-M.; Lord, G.J.; Josić, K.; Cox, S.J.; Colbert, C.M., Branching dendrites with resonant membrane: A “sum-over-trips” approach, Biol. cybern., 97, 137-149, (2007) · Zbl 1122.92007
[25] Abbott, L.F.; Fahri, E.; Gutmann, S., The path integral for dendritic trees, Biol. cybern., 66, 49-60, (1991) · Zbl 0743.92010
[26] James, M.P.; Coombes, S.; Bressloff, P.C., Effects of quasi-active membrane on multiple periodic traveling waves in integrate-and-fire systems, Phys. rev. E, 67, 051905, (2003)
[27] Bressloff, P.C.; Coombes, S., Physics of the extended neuron, Internat. J. modern phys. B, 11, 2343-2392, (1997)
[28] Timofeeva, Y.; Cox, S.J.; Coombes, S.; Josić, K., Democratization in a passive dendritic tree: an analytical investigation, J. comput. neurosci., 25, 228-244, (2008)
[29] Svoboda, K.; Tank, D.W.; Denk, W., Direct measurement of coupling between dendritic spines and shafts, Science, 272, 716-719, (1996)
[30] Tsay, D.; Yuste, R., On the electrical function of dendritic spines, Trends neurosci., 24, 2, 77-83, (2004)
[31] Bloodgood, B.L.; Sabatini, B.L., Neuronal activity regulates diffusion across the neck of dendritic spines, Science, 310, 866-869, (2005)
[32] Coombes, S., The effect of ion pumps on the speed of travelling waves in the fire-diffuse-fire model of ca^2+ release, Bull. math. biol., 63, 1-20, (2001) · Zbl 1323.92068
[33] Spruston, N.; Stuart, G.; Häusser, M., Dendritic integration, ()
[34] Scott, A., Neuroscience: A mathematical primer, (2002), Springer Heidelberg · Zbl 1018.92003
[35] Narayanan, R.; Johnston, D., Long-term potentiation in rat hippocampal neurons is accompanied by spatially widespread changes in intrinsic oscillatory dynamics and excitability, Neuron, 56, 1061-1075, (2007)
[36] Cox, S.J.; Griffith, B.E., Recovering quasi-active properties of dendritic neurons from dual potential recordings, J. comput. neurosci., 11, 95-110, (2001)
[37] Bhatt, D.H.; Zhang, S.; Gan, W.-B., Dendritic spine dynamics, Ann. rev. physiol., 71, 261-282, (2009)
[38] Magee, J.; Cook, E., Somatic EPSP amplitude is independent of synapse location in hippocampal pyramidal neurons, Nature neurosci., 3, 895-903, (2000)
[39] Katz, Y.; Menon, V.; Nicholson, D.A.; Geinisman, Y.; Kath, W.L.; Spruston, N., Synapse distribution suggests a two-stage model of dendritic integration in CA1 pyramidal neurons, Neuron, 63, 171-177, (2009)
[40] Miller, R.N.; Rinzel, J., The dependence of impulse propagation speed on firing frequency, dispersion, for the hodgkin – huxley model, Biophys. J., 34, 227-259, (1981)
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