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Synapse fits neuron: joint reduction by model inversion. (English) Zbl 1379.92009

Summary: In this paper, we introduce a novel simplification method for dealing with physical systems that can be thought to consist of two subsystems connected in series, such as a neuron and a synapse. The aim of our method is to help find a simple, yet convincing model of the full cascade-connected system, assuming that a satisfactory model of one of the subsystems, e.g., the neuron, is already given. Our method allows us to validate a candidate model of the full cascade against data at a finer scale. In our main example, we apply our method to part of the squid’s giant fiber system. We first postulate a simple, hypothetical model of cell-to-cell signaling based on the squid’s escape response. Then, given a FitzHugh-type neuron model, we derive the verifiable model of the squid giant synapse that this hypothesis implies. We show that the derived synapse model accurately reproduces synaptic recordings, hence lending support to the postulated, simple model of cell-to-cell signaling, which thus, in turn, can be used as a basic building block for network models.

MSC:

92C20 Neural biology
92C05 Biophysics

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[1] Abbott LF (1994) Decoding neuronal firing and modeling neural networks. Q Rev Biophys 27:291-331 · doi:10.1017/S0033583500003024
[2] Augustine GJ, Charlton MP, Smith SJ (1985) Calcium entry and transmitter release at voltage-clamped nerve terminals of squid. J Physiol 367:163-181 · doi:10.1113/jphysiol.1985.sp015819
[3] Chapeau-Blondeau F, Chambet N (1995) Synapse models for neural networks: from ion channel kinetics to multiplicative coefficient \[w_{ij}\] wij. Neural Computation 7:713-734 · doi:10.1162/neco.1995.7.4.713
[4] Clay JR (1998) Excitability of the squid giant axon revisited. J Neurophysiol 80:903-913
[5] Dayan P, Abbott LF (2001) Theoretical neuroscience: computational and mathematical modeling of neural systems. The MIT Press, Cambridge · Zbl 1051.92010
[6] Delaleau E, Respondek W (1995) Lowering the orders of derivatives of controls in generalized state space systems. J Math Syst Estim Control 5(3):1-27 · Zbl 0852.93016
[7] Destexhe A, Mainen ZF, Sejnowski TJ (1994) Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism. J Comput Neurosci 1:195-230 · doi:10.1007/BF00961734
[8] Dorsett DA (1980) Design and function of giant fibre systems. Trends Neurosci 3:205-208 · doi:10.1016/0166-2236(80)90077-6
[9] Doya K (1999) What are the computations of the cerebellum, the basal ganglia and the cerebral cortex? Neural Netw 12(7):961-974 · doi:10.1016/S0893-6080(99)00046-5
[10] Doya K, Kimura H, Miyamura A (2001) Motor control: neural models and systems theory. Int J Appl Math Comput Sci 11(1):77-104 · Zbl 1065.93528
[11] Eliasmith C, Anderson CH (2003) Neural engineering: computation, representation, and dynamics in neurobiological systems. The MIT Press, Cambridge
[12] Ermentrout GB, Kopell N (1986) Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J Appl Math 46:233-253 · Zbl 0594.58033 · doi:10.1137/0146017
[13] FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1:445-466 · doi:10.1016/S0006-3495(61)86902-6
[14] Freedman MI, Willems JC (1978) Smooth representations of systems with differentiated inputs. IEEE Trans Autom Control 23(1):16-21 · Zbl 0376.93024 · doi:10.1109/TAC.1978.1101693
[15] van Geit W, de Schutter E, Achard P (2008) Automated neuron model optimization techniques: a review. Biol Cybern 99:241-251 · Zbl 1154.92013 · doi:10.1007/s00422-008-0257-6
[16] Glad, ST; Descusse, J. (ed.); Fliess, M. (ed.); Isidori, A. (ed.); Leborgne, D. (ed.), Nonlinear state space and input output descriptions using differential polynomials, 182-189 (1989), Berlin Heidelberg · Zbl 0682.93030 · doi:10.1007/BFb0043027
[17] Hansel D, Mato G (2001) Existence and stability of persistent states in large neuronal networks. Phys Rev Lett 86(18):4175-4178 · doi:10.1103/PhysRevLett.86.4175
[18] Henson MA, Seborg DE (eds) (1997) Nonlinear process control. Prentice Hall, Englewood Cliffs
[19] Herz AVM, Gollisch T, Machens CK, Jaeger D (2006) Modeling single-neuron dynamics and computations: a balance of detail and abstraction. Science 314:80-85 · Zbl 1226.92007 · doi:10.1126/science.1127240
[20] Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500-544 · doi:10.1113/jphysiol.1952.sp004764
[21] Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 81:3088-3092 · Zbl 1371.92015 · doi:10.1073/pnas.81.10.3088
[22] Hopfield JJ, Tank DW (1986) Computing with neural circuits: a model. Science 233(4764):625-633 · doi:10.1126/science.3755256
[23] Hunt KJ, Sbarbaro D, Zbikowski R, Gawthrop PJ (1992) Neural networks for control systems - a survey. Automatica 28:1083-1112 · Zbl 0763.93004 · doi:10.1016/0005-1098(92)90053-I
[24] Hunter IW, Korenberg MJ (1986) The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol Cybern 55:135-144 · Zbl 0611.92002
[25] Isidori A (1995) Nonlinear control systems, 3rd edn. Springer-Verlag, London · Zbl 0878.93001 · doi:10.1007/978-1-84628-615-5
[26] Izhikevich EM (2003) Simple model of spiking neurons. IEEE Trans Neural Netw 14(6):1569-1572 · doi:10.1109/TNN.2003.820440
[27] Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. The MIT Press, Cambridge
[28] Katz B, Miledi R (1967) A study of synaptic transmission in the absence of nerve impulses. J physiol 192:407-436 · doi:10.1113/jphysiol.1967.sp008307
[29] Kepler TB, Abbott LF, Marder E (1992) Reduction of conductance-based neuron models. Biol Cybern 66:381-387 · Zbl 0745.92006 · doi:10.1007/BF00197717
[30] Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice Hall, Upper Saddle River · Zbl 1003.34002
[31] Knight BW (1972) Dynamics of encoding in a population of neurons. J Gen Physiol 59:734-766 · doi:10.1085/jgp.59.6.734
[32] Kotta U, Mullari T (2005) Realization of nonlinear systems described by input/output differential equations: equivalence of different methods. Eur J. Control 11:185-193 · Zbl 1293.93175 · doi:10.3166/ejc.11.185-193
[33] Kotta U, Mullari T (2006) Equivalence of realizability conditions for nonlinear control systems. Proc Estonian Acad Sci Phys Math 55(1):24-42 · Zbl 1139.93310
[34] Kouh M, Poggio T (2008) A canonical neural circuit for cortical nonlinear operations. Neural Computation 20(6):1427-1451 · Zbl 1137.92302 · doi:10.1162/neco.2008.02-07-466
[35] Kreinovich V, Quintana C (1991) Neural networks: what non-linearity to choose? Proceedings of the 4th University of New Brunswick Artificial Intelligence Workshop, Fredericton, New Brunswick, 627-637 · Zbl 1226.92007
[36] Langley K, Grant NJ (1997) Are exocytosis mechanisms neurotransmitter specific? Neurochem Int 31:739-757 · doi:10.1016/S0197-0186(97)00040-5
[37] Llinás R, Steinberg IZ, Walton K (1976) Presynaptic calcium currents and their relation to synaptic transmission: voltage clamp study in squid giant synapse and theoretical model for the calcium gate. Proc Natl Acad Sci USA 73(8):2918-2922 · doi:10.1073/pnas.73.8.2918
[38] Llinás R, Steinberg IZ, Walton K (1981a) Presynaptic calcium currents in squid giant synapse. Biophys J 33(3):289-321 · doi:10.1016/S0006-3495(81)84898-9
[39] Llinás R, Steinberg IZ, Walton K (1981b) Relationship between presynaptic calcium current and postsynaptic potential in squid giant synapse. Biophys J 33(3):323-351 · doi:10.1016/S0006-3495(81)84899-0
[40] Messenger JB (1996) Neurotransmitters of cephalopods. Invertebr Neurosci 2:95-114 · doi:10.1007/BF02214113
[41] Morrison A, Diesmann M, Gerstner W (2008) Phenomenological models of synaptic plasticity based on spike timing. Biol Cybern 98(6):459-478 · Zbl 1145.92306 · doi:10.1007/s00422-008-0233-1
[42] Nijmeijer H, van der Schaft A (1990) Nonlinear dynamical control systems. Springer-Verlag, New York · Zbl 0701.93001 · doi:10.1007/978-1-4757-2101-0
[43] Pavlov A, Petterson KY (2008) A new perspective on stable inversion of non-minimum phase nonlinear systems. Model Ident Control 29(1):29-35 · doi:10.4173/mic.2008.1.3
[44] Pavlov A, Pogromsky A, van de Wouw N, Nijmeijer H (2004) Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Syst Control Lett 52:257-261 · Zbl 1157.34333 · doi:10.1016/j.sysconle.2004.02.003
[45] Otis TS, Gilly WF (1990) Jet-propelled escape in the squid Loligo opalescens: concerted control by giant and non-giant motor axon pathways. Proc Natl Acad Sci USA 87:2911-2915 · doi:10.1073/pnas.87.8.2911
[46] Röbenack K, Goel P (2007) Observer based measurement of the input current of a neuron. Mediterr J Meas Control 3(1):22-29
[47] Reichert H (1992) Introduction to neurobiology. George Thieme Verlag, Stuttgard, Germany
[48] Rinzel J (1985) Excitation dynamics: insights from simplified membrane models. Fed Proc 44:2944-2946
[49] Rowat PF, Selverston AI (1993) Modeling the gastric mill central pattern generator of the lobster with a relaxation-oscillator network. J Neurophysiol 70(3):1030-1053
[50] Tauc L (1997) Quantal neurotransmitter release: vesicular or not vesicular? Neurophysiol 29:219-226 · doi:10.1007/BF02461232
[51] Terrell WJ (1999a) Some fundamental control theory I: controllability, observability, and duality. Am Math Mon 106(8):705-719 · Zbl 0990.93047 · doi:10.2307/2589018
[52] Terrell WJ (1999b) Some fundamental control theory II: feedback linearization of single input nonlinear systems. Am Math Mon 106(9):812-828 · Zbl 0991.93025 · doi:10.2307/2589614
[53] Tin C, Poon CS (2005) Internal models in sensorimotor integration: perspectives from adaptive control theory. J Neural Eng 2(3):S147-S163 · doi:10.1088/1741-2560/2/3/S01
[54] Touboul J (2008) Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons. SIAM J Appl Math 68:1045-1079 · Zbl 1149.34027 · doi:10.1137/070687268
[55] Tsinias J, Kalouptsidis N (1983) Invertibility of nonlinear analytic single-input systems. IEEE Trans Autom Control 28(9):931-933 · Zbl 0529.93032 · doi:10.1109/TAC.1983.1103348
[56] Tsodyks MV, Markram H (1997) The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability. Proc Nat Acad Sci USA 94:719-723 · doi:10.1073/pnas.94.2.719
[57] Vautrin J (1994) Vesicular or quantal and subquantal transmitter release. Physiol 9:59-64
[58] Vyskocil F, Malomouzh AI, Nikolsky EE (2009) Non-quantal acetylcholine release at the neuromuscular junction. Physiol Res 58:763-784
[59] Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12(1):1-24 · doi:10.1016/S0006-3495(72)86068-5
[60] Zhao H, Chen D (1998) A finite energy property of stable inversion to nonminimum phase nonlinear systems. IEEE Trans Autom Control 43(8):1170-1174 · Zbl 0957.93044 · doi:10.1109/9.704995
[61] Zhang K, Sejnowski TJ (1999) A theory of geometric constraints on neural activity for natural three-dimensional movement. J Neurosci 19(8):3122-3145
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